Yuan

COMPUTER METHODS FOR ANALYSIS OF MIXED-MODE SWITCHING CIRCUITS

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COMPUTER METHODS FOR ANALYSIS OF MIXED-MODE SWITCHING CIRCUITS

by

Fei Yuan Associate Professor Department of Electrical and Computer Engineering Ryerson University Toronto, Ontario, Canada Ajoy Opal Professor Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario, Canada

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

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1-4020-7923-0 1-4020-7922-2

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Contents

List of Figures List of Tables Preface Acknowledgments Part I

xi xxiii xxvii xxxi

The Fundamentals

1. AN OVERVIEW OF MIXED-MODE SWITCHING CIRCUITS 1 Classification Switched Capacitor Techniques 2 3 Switched Current Techniques 4 Characteristics of Mixed-Mode Switching Circuits 2. COMPUTER FORMULATION OF MIXED-MODE SWITCHING CIRCUITS 1 Modeling of Switches 1.1 Full-Transistor Models Voltage-Modulated Resistor Models 1.2 Ideal Switch Models 1.3 2

Formulation Methods for Mixed-Mode Switching Circuits A Historical Perspective 2.1 External Clocks 2.2 Conventions 2.3 Sub-Circuits 2.4 Matrix Stamps of Elements Without Memory 2.5 2.5.1 Controlled Sources

3 3 4 6 9 13 13 14 14 15 16 16 19 19 20 21 21

vi

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Ideal Switches 22 Matrix Stamps of Elements With Memory 23 24 Capacitors 28 Inductors Formulation of Circuits with Externally Clocked 28 Switches Formulation of Circuits with Internally Controlled 2.8 31 Switches Formulation of Circuits with Both Externally Clocked 2.9 and Internally Controlled Switches 32 Summary 33 2.5.2 2.6 2.6.1 2.6.2 2.7

3

3. NETWORK FUNCTIONS OF TIME-VARYING CIRCUITS Transfer Functions of Linear Time-Varying Systems 1 Linear Time-Varying Systems 1.1 Linear Periodically Time-Varying Systems 1.2 Transfer Functions of Nonlinear Time-Varying Systems 2 Volterra Functional Series 2.1 Multi-Frequency Network Functions 2.2 Multi-Frequency Transfer Functions 2.3 Frequency Response of Nonlinear Time-Varying Systems 3 Frequency Response of Nonlinear Periodically Time-Varying 4 Systems 5 Summary 4. NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS Linear Single-Step Predictor-Corrector Algorithms 1 Linear Multi-Step Predictor-Corrector Algorithms 2 Integration Using Numerical Laplace Inversion 3 Padé Polynomials 3.1 3.2 Numerical Laplace Inversion 3.3 Multi-Step Numerical Laplace Inversion 4 Summary

35 36 36 37 40 40 41 42 43 44 48 53 54 57 59 60 62 69 79

Contents

Part II

vii

Time Domain Analysis

5. INCONSISTENT INITIAL CONDITIONS Inconsistent Initial Conditions 1 Numerical Laplace Inversion Based Two-Step Algorithm 2 Backward Euler Based Algorithms 3 3.1 Two-Forward-Step Algorithm 3.2 Two-Step Algorithm 3.3 Four-Step Algorithm 3.4 Two-Step Algorithm for Linear Circuits Taylor Series Based Algorithm 4 Volterra Functional Series Based Algorithm 5 Existence of Dirac Impulses at Switching Instants 6 Dirac Impulses in Linear Circuits 6.1 Dirac Impulses in Nonlinear Circuits 6.2 Summary 7 6. SAMPLED-DATA SIMULATION OF PERIODICALLY SWITCHED LINEAR CIRCUITS 1 Sampled-Data Simulation of Periodically Switched Linear Circuits 2 Inconsistent Initial Conditions 3 Time-Domain Sensitivity 4 Inconsistent Initial Conditions of Sensitivity Networks 5 Statistical Analysis 5.1 Introduction First-Order Second-Moment Method 5.2 6 Noise Analysis Modeling of White Noise 6.1 The Algorithm 6.2 Examples 6.3 7 Clock Jitter Summary 8

83 84 85 90 92 96 97 98 99 102 103 104 106 107

109

1 2

Computation of Computation of

110 116 119 121 122 122 123 125 126 129 130 131 134 135 136

3

Computation of

136

viii COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

7. SAMPLED-DATA SIMULATION OF PERIODICALLY SWITCHED NONLINEAR CIRCUITS

139 140

1

Multi-Linear Theory

2 3

144 Volterra Circuits Sampled-Data Simulation of Periodically Switched Nonlinear Circuits 146 Inconsistent Initial Conditions 151 152 Sensitivity of Periodically Switched Nonlinear Circuits 155 Discussion 155 Stability 6.1 155 The Maximum Step Size 6.2 155 Accuracy 6.3 156 6.3.1 The Order of Taylor Series Expansion 157 6.3.2 The Order of Volterra Series Expansion 157 6.3.3 The Order of Interpolating Fourier Series 6.3.4 Simulation Window 158 6.3.5 Error Propagation 160 162 Examples 7.1 162 Time-Invariant Nonlinear Circuits Switched Capacitor Integrator with Nonlinear Op 7.2 Amp 166 General Periodically Switched Nonlinear Circuits 168 7.3 Summary 172

4 5 6

7

8

8. SAMPLED-DATA SIMULATION OF CIRCUITS WITH INTERNALLY CONTROLLED SWITCHES 1

2

Internally Controlled Switches and Switching Variables Diodes 1.1 MOSFETs 1.2 Static CMOS Inverters 1.3 Comparators 1.4 Switching Instants

3

Inconsistent Initial Conditions

4

Examples

5

Summary

177 178 178 179 180 181 181 182 183 189

Contents

ix

9. SAMPLED-DATA SIMULATION OF OVER-SAMPLED SIGMA-DELTA MODULATORS Introduction 1 Modeling of Clocked Quantizers 2 Modeling of Unclocked Quantizers 3 Modeling of Digital-to-Analog Data Converters 4 Modeling of Other Blocks 5 6 Simulation Methods 7 Examples 8 Summary Part III

Frequency Domain Analysis

10. ADJOINT NETWORK OF PERIODICALLY SWITCHED LINEAR CIRCUITS Tellegen’s Theorem 1 Inter-reciprocity 2 Adjoint Network 3 3.1 Ideal Switches 3.2 Resistors 3.3 Capacitors and Inductors 3.4 Controlled Sources 3.5 Operational Amplifiers 4 Transfer Function Theorem 5 Frequency Reversal Theorem 6 Examples 7 Summary 11. FREQUENCY DOMAIN ANALYSIS OF PERIODICALLY SWITCHED LINEAR CIRCUITS 1 2

191 192 194 194 196 196 197 198 201

Frequency Response Sensitivity Analysis 2.1 Direct Sensitivity Analysis 2.2 Sensitivity Analysis Using Adjoint Network 2.3 Sensitivity Analysis Using Sensitivity Network 2.4 Numerical Examples

207 208 211 212 212 212 213 214 215 216 220 223 225

229 230 235 237 240 250 255

x

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

3 4

5 6

Group Delay Analysis Noise Analysis Noise Characterization 4.1 Noise Sources 4.2 Noise Equivalent Circuits 4.3 The Algorithm 4.4 Numerical Examples 4.5 Statistical Analysis Summary

12. FREQUENCY DOMAIN ANALYSIS OF PERIODICALLY SWITCHED NONLINEAR CIRCUITS 1 Fundamentals Harmonic Distortion 1.1 Intermodulation Distortion 1.2 2 Distortion Analysis of Periodically Switched Nonlinear Circuits 3 Harmonic Distortion The First-Order Volterra Circuit 3.1 The Second-Order Volterra Circuit 3.2 The Third-Order Volterra Circuit 3.3 The Fold-Over Effect 3.4 4

5

6

Intermodulation Distortion The First-Order Volterra Circuit 4.1 4.2 The Second-Order Volterra Circuit 4.3 The Third-Order Volterra Circuit Examples 5.1 Modulator 5.2 Stray-Insensitive Switched Capacitor Integrator 5.3 Switched Capacitor Integrator With Nonlinear Op Amp Summary

260 264 266 275 277 284 286 293 297

303 306 306 307 309 314 314 315 316 318 318 319 320 320 321 321 322 325 332

List of Figures

Implementation of resistors using switched capacitors.

5

Implementation of resistors using stray-insensitive switched capacitors (the dotted line shows the connection of the parasitic bottom plate-substrate capacitor) .

6

1.3

Stray-insensitive switched capacitors integrators.

7

1.4

Stray-insensitive switched capacitors biquad.

7

1.5

Switched current memory cells. The first generation of switched current memory cell is sensitive to the effect of mismatches whereas the second generation switched current memory cell is mismatch-free. Switched current integrators. Switched current biquad. Ideal switch model.

1.1 1.2

1.6 1.7 2.1 2.2

2.3

8 9 10 15

Test circuit for demonstrating the difference between voltage-modulator resistor switch model and ideal switch model.

16

Time domain response of the circuit of Fig.2.2 with ideal and voltage-modulated resistor switch models. The ON-resistance of voltage-modulated resistor switch is varied from 0 to with step.

17

xii COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

2.4

Clock phases in a clock period clock phase).

represents the 19

Circuits with externally clocked switches with a total of K phases in a clock period are represented by K time-invariant sub-circuits operated in a timeinterleaved fashion.

21

2.6

Matrix stamps of nonlinear controlled sources.

23

2.7

Matrix stamps of ideal switches.

24

2.8

Matrix stamps of linear and nonlinear capacitors.

26

2.9

Matrix stamps of linear and nonlinear inductors.

29

2.10

Switching time is determined from solving numerically.

32

3.1

Foldover effect.

38

3.2

The spectrum of the output of nonlinear periodically time-varying system to input

48

2.5

3.3

The spectrum of the output of nonlinear periodically time-varying systems to input 49

4.1 4.2 4.3

4.4

4.5

Stable region of Euler formulae. The boundary of the unit circle is also included in the stable region.

57

Linear single-step and linear multi-step predictorcorrector algorithms.

59

Dependence of the relative error of the exact solution and that from numerical Laplace inversion of the unit step function with {N, M} = {2, 4} on the step size.

67

Dependence of the relative error of the exact solution and that from numerical Laplace inversion of exponentially decaying function with {N, M} = {2, 4} on the step size.

68

RC network.

68

List of Figures

xiii

Dependence of the relative error of the exact solution and that from numerical Laplace inversion of RC Network of Fig.4.5 with {N, M} = {2, 4} on the step size.

69

Relative error of the response of the RC network of Fig.4.5 using stepping and non-stepping algorithms.

73

Piece-wise linear approximation of a given input waveform. is the given input wave form and is the piecewise linear approximation of is chosen such that Nyquist sampling theorem is satisfied.

74

5.1

Test circuit.

85

5.2

Relative error of the exact response of the circuit of Fig.5.1 and the response of the circuit computed from numerical Laplace inversion.

87

Relative error between the exact value of of the network of Fig.5.1 and that computed from two-step algorithm with {N, M} = {2, 4}.

88

5.4

Example circuit for two-step algorithm

90

5.5

Test circuit for inconsistent initial conditions.

93

5.6

Comparison of the relative errors of algorithms for computing the consistent initial conditions of the circuit of Fig.5.1.

102

Piecewise-linear approximation of an arbitrary input waveform. is the input wave form and is the piecewise linear approximation of with step

114

Graphical illustration of two-step algorithm (one forward step and one backward step), is discontinuous at but continuous at

116

6.3

Periodically switched linear circuit example.

118

6.4

Response of the circuit of Fig.6.3. Sampled-data simulation, – Exact.

119

4.6

4.7 4.8

5.3

6.1

6.2

Legends:

xiv COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

6.5 6.6 6.7

White noise is represented by a train of pulses with random amplitude. Equivalent noise bandwidth. Time domain waveform of the thermal noise voltage of the resistor with

126 127

128 6.8 6.9 6.10

6.11 7.1

7.2

7.3 7.4

Spectrum of the thermal noise of the resistor with 512 samples. Switched capacitor integrator. Power spectral density of the output noise of the switched capacitor integrator of Fig.6.9. Legends measurement, – this book. Clock jitter where

129 131

132

is represented by step variation 133

Multi-linear equivalent circuits of nonlinear voltagecontrolled voltage source. (a) first-order, (b) secondorder, and (c) third-order.

142

The multi-linear equivalent circuits of nonlinear elements in phase of periodically switched nonlinear circuits.

144

Volterra circuits of periodically switched nonlinear circuits.

147

Interpolation window and approximation of the input of the second-order Volterra circuits using interpolating Fourier series.

148

7.7

Conversion of aperiodic sequence to periodic sequence using transformation. Propagation of interpolating error. Current-mirror amplifier.

7.8

Response of the current-mirror amplifier of Fig.7.7.

164

7.9

Circuit with nonlinear conductor.

165

7.10

Response of the circuit of Fig.7.9. The numerical numbers in the figure are the amplitude of the input.

166

7.5 7.6

159 161 163

List of Figures

xv

Absolute error between the response obtained from sampled-data simulation and that from LSS-PC of the circuit of Fig.7.9.

167

Dependence of the error on the order of interpolation used in sampled-data simulation of the circuit of Fig.7.9.

167

Comparison of CPU time of sampled-data simulation and LSS-PC algorithms of the circuit of Fig.7.9.

168

Switched-capacitor integrator with nonlinear operational amplifier.

169

Response and its sensitivity with respect to of the Switched-capacitor integrator with nonlinear operational amplifier of Fig.7.14.

170

Sensitivity of of the switched-capacitor integrator with nonlinear operational amplifier of Fig. 7.14 with respect to using both sampled-data simulation and brute-force methods. Legends : Bruteforce; – Sampled-data simulation.

171

7.17

General periodically switched nonlinear circuit.

171

7.18

Response of the circuit of Fig.7.17. The results from sampled-data simulation compare well with those from PSPICE simulation at time points other than switching instants. Legends : PSPICE simulation; – Sampled-data simulation.

172

Normalized difference of the response of the circuit of Fig.7.17 obtained from sampled-data simulation and that from PSPICE. The maximum normalized difference is below 0.5%.

173

Charge conservation at the switching instants of the circuit of Fig.7.17. Legends : total charge immediate before switching; + total charge immediate after switching.

174

7.11

7.12

7.13 7.14 7.15

7.16

7.19

7.20

xvi COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Sensitivity of of the circuit of Fig.7.17 with respect to Legends : Brute-force; – Sampleddata simulation.

175

Sensitivity of of the circuit of Fig. 7.17 with respect to Legends : Brute-force; – Sampleddata simulation.

175

8.1

Internally controlled switches - diodes.

179

8.2

Internally controlled switches - MOSFETs.

180

8.3

Internally controlled switches - static CMOS inverters.

180

8.4

Internally controlled switches - comparators.

182

8.5

Bulk linear voltage regulator (step-down chopper).

184

8.6

Linear voltage regulator

186

8.7

Linear voltage regulator

186

9.1

Configuration of clocked single-bit first-order sigmadelta modulator.

192

9.2

Quantizers.

193

9.3

Modeling of clocked quantizers with no dead zone. is the input and is the output.

195

Typical configurations of single-bit DACs. Q is the output of the quantizers

197

Single-bit second-order continuous-time over-sampled sigma-delta modulator.

199

Time-domain response of the single-bit second-order continuous-time over-sampled sigma-delta modulator of Fig.9.5.

199

Spectrum of the response of the single-bit secondorder continuous-time over-sampled sigma-delta modulator of Fig.9.5.

200

7.21

7.22

9.4 9.5 9.6

9.7

9.8

Single-bit second-order switched capacitor over-sampled sigma-delta modulator. This modulator is a switched capacitor implementation of the continuous modulator of Fig.9.5. 201

List of Figures

9.9

9.10

10.1 10.2

xvii

Spectrum of the response of the single-bit secondorder switched capacitor over-sampled sigma-delta modulator of Fig.9.8.

202

Signal-to-noise ratio of the single-bit second-order switched capacitor over-sampled sigma-delta modulator of Fig.9.8 (Ideal – ideal operational amplifier; Nonideal – operational amplifier with gainbandwidth 500kHz.)

203

Time reversal of a periodically switched linear circuit N and its adjoint network

211

Equivalent circuit of operational amplifier with singlepole model and its adjoint network.

215

10.3

Elements and their counterparts in the adjoint network. 216

10.4

Transfer function theorem - voltage output.

217

10.5

Transfer function theorem - current output.

217

10.6

Frequency reversal theorem.

220

10.7

Stray-insensitive switched capacitor integrator.

224

10.8

Adjoint network of the stray-insensitive switched capacitor integrator of Fig.10.7.

225

10.9

Switched capacitor band pass filter.

226

10.10

Adjoint network of the switched capacitor band pass filter of Fig. 10.9.

227

11.1

Fifth-order elliptic switched capacitor low pass filter.

236

11.2

Frequency response of the fifth-order elliptic switched capacitor low pass filter of Fig.11.1.

238

11.3

11.4

The passband of the frequency response of the fifthorder switched capacitor low pass filter of Fig.11.1. Dashed line - ideal operational amplifier with frequency characteristics given by Solid line - non-ideal operational amplifier with frequency characteristics given by

239

Error of discretization.

242

xviii COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

11.5 11.6 11.7 11.8

11.9 11.10

Sensitivity analysis of periodically switched linear circuits using adjoint network. Fold-over effect in sensitivity analysis of periodically switched linear circuits. Sensitivity networks of basic elements. Sensitivity network of periodically switched linear circuits. (a) Original circuit N, (b) Sensitivity networks (c) Adjoint network of N and Stray-insensitive switched capacitor integrator. Sensitivity of the response of stray insensitive switched capacitor integrator of Fig.11.9 to (real part).

243 251 253

254 256 257

11.11 Sensitivity of the response of stray insensitive switched capacitor integrator of Fig.11.9 to (imaginary part). 257 11.12

11.13

11.14 11.15 11.16

Normalized sensitivity of the response of magnitude of the stray insensitive switched capacitor integrator of Fig.11.9 to Response of stray insensitive switched capacitor integrator of Fig.11.9 and that of its adjoint network (20 sidebands are plotted). Switched capacitor band pass filter. Sensitivity of the response of the switched capacitor band pass filter of Fig.11.14 to (Real part).

258

259 260 261

Sensitivity the response of the switched capacitor band pass filter of Fig.11.14 to (Imaginary part).

261

11.17 Normalized sensitivity of the response of the switched capacitor band pass filter of Fig.11.14 to

262

11.18 Relative difference between the sensitivity of Fig.11.14 from adjoint network analysis and that from direct sensitivity analysis.

263

11.19 11.20

Response of the switched capacitor band pass filter of Fig.11.14 and that of its adjoint network.

264

Fifth-order switched capacitor low pass filter.

265

List of Figures

11.21 11.22

xix

Group delay of the fifth-order switched capacitor low pass filter of Fig.11.20.

267

Sampling of stationary random signal.

272

11.23 Aliasing effect of band-limited noise signals. The bandwidth of the input noise is and the bandwidth of the circuit is assumed to be infinite with unity gain.

274

Low-frequency noise equivalent circuit of resistors. (a) Norton equivalent, (b) Thevenin equivalent.

278

Noise equivalent circuit of BJTs in the forward active region. is the thermal noise of the base resistance is the shot noise at the emitterbase pn-junction, represents the shot noise and flicker noise in the emitter-base region.

279

Low-frequency noise equivalent circuit of MOSFET switches.

282

Equivalent noise circuit of MOSFET transistors in saturation.

283

Noise equivalent circuits of operational amplifiers. (a) Complete noise equivalent circuit, (b) Simplified noise equivalent circuit.

284

11.29

Switched capacitor low-pass.

288

11.30

Power spectral density of the switched capacitor low-pass of Fig.11.29. Legends: measurement; – this book.

288

11.31 Switched capacitor integrator and its noise equivalent circuit.

290

Power spectral density of switched capacitor integrator of Fig.11.31. Legends : measurement; – this book.

290

CPU time of the noise analysis of the switched capacitor integrator of Fig.11.31.

291

11.24 11.25

11.26 11.27 11.28

11.32

11.33

xx

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Ratio of CPU time of the brute-force approach for noise analysis of the switched capacitor integrator of Fig.11.31 to that of the adjoint network approach.

292

Noise equivalent circuit of the switched capacitor band pass filter of Fig.11.14. All noise current generators are modeled as

294

Power spectral density of the output noise of the band pass filter of Fig.11.35 due to thermal noise sources. Legends : data from Tóth and Suyama (1997); – this book.

295

11.37 Power spectral density of the output noise of the band pass filter of Fig.11.35 due to flicker noise sources. No foldover is considered in thermal noise analysis.

296

Mean of the response of the switched capacitor band pass filter of Fig.11.14. Legends: Monte Carlo; – FOSM.

297

of the response of the switched capacitor band pass filter of Fig.11.14. Legends: - Monte Carlo; – FOSM.

298

Fold-over effect in distortion analysis of periodically switched nonlinear circuits.

319

12.2

Modulator.

322

12.3

Convergence of the second-order harmonic of the modulator of Fig.12.2.

323

12.4

Stray-insensitive switched capacitor integrator.

324

12.5

Convergence of the second-order harmonic of the response of the stray-insensitive switched capacitor integrator of Fig.12.4.

326

Stray-insensitive switched capacitor integrator with nonlinear operational amplifier.

327

11.34

11.35

11.36

11.38

11.39

12.1

12.6

List of Figures

12.7

Spectrum of the output of the stray-insensitive switched capacitor integrator of Fig.12.6 with (Baseband).

Spectrum of the output of the stray-insensitive switched capacitor integrator of Fig12.6 with (First positive sideband). 12.9 Spectrum of the output of the stray-insensitive switched capacitor integrator of Fig.12.6 to two sinusoidal inputs at 1 kHz and 1.1 kHz (Baseband). 12.10 Spectrum of the output of the stray-insensitive switched capacitor integrator of Fig.12.6 to two sinusoidal inputs at 1 kHz and 1.1 kHz (Baseband, first and second positive sidebands).

xxi

329

12.8

330

332

333

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List of Tables

4.1 4.2 4.3 4.4 4.5 5.1

7.1 7.2 10.1 10.2 10.3 10.4

10.5 10.6

Padé approximates of The zeros of Padé polynomial The residues of Padé polynomial The zeros of Padé polynomial The residues of Padé polynomial Comparison of the relative errors of algorithms for computing consistent initial conditions of the circuit of Fig.5.1.

62 64 64 64 65

101

Parameter values of the current-mirror amplifier of 164 Fig.7.7. 165 Parameter values of the circuit of Fig.7.9. Transfer function theorem. 220 Frequency reversal theorem 223 Parameter values of the stray-insensitive switched capacitor integrator of Fig.10.7. 224 Transfer function and aliasing transfer functions of the stray-insensitive switched capacitor integrator 225 of Fig.10.7. Frequency response of the adjoint network of the stray-insensitive switched capacitor integrator of Fig.10.7. 226 Parameter values of the stray-insensitive switched capacitor integrator of Fig.10.9. 227

xxiv COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

10.7 11.1

Frequency response of the band pass filter of Fig.10.9 and its adjoint network.

228

Parameters of the fifth-order elliptic switched capacitor low pass filter of Fig.11.1.

237

11.2

Sensitivity of periodically switched linear circuits (Subscripts 1 and 2 identify the controlling and controlled branches of controlled sources, respectively). 249

11.3

Sensitivity of linear time-invariant circuits (Subscripts 1 and 2 identify the controlling and controlled branches of controlled sources, respectively).

250

Parameters of stray-insensitive Switched-capacitor integrator of Fig.11.9.

256

Parameters of switched capacitor band pass filter of Fig.11.14.

259

Parameters of the fifth-order switched capacitor low pass filter of Fig.11.20.

266

Parameter values of the switched capacitor lowpass of Fig.11.29.

287

Parameter values of the switched capacitor integrator of Fig.11.31.

289

11.A.1 Relationship between original periodically switched linear circuit and its adjoint network.

301

11.4 11.5 11.6 11.7 11.8

12.1

Frequency components of the response of nonlinear circuit to the input

308

12.2

Parameters of the modulator of Fig.12.2.

321

12.3

Harmonic distortion of the modulator of Fig.12.2

323

12.4

Parameter value of the stray-insensitive switched capacitor integrator of Fig.12.4.

324

Harmonic distortion of the stray-insensitive switched capacitor integrator of Fig.12.4.

325

Parameter value of the stray-insensitive switched capacitor integrator of Fig.12.6

326

12.5 12.6

List of Tables

12.7

xxv

Harmonic distortion of the stray-insensitive switched capacitor integrator of Fig.12.6 with 328

12.8

Harmonic distortion of the stray-insensitive switched capacitor integrator of Fig.12.6 with 328

Harmonic distortion of the stray-insensitive switched capacitor integrator of Fig.12.6 using adjoint network and brute-force methods. 12.10 Inter-modulation distortion of the stray-insensitive switched capacitor integrator of Fig.12.6. 12.11 Harmonic distortion of the stray-insensitive integrator with nonlinear capacitors. 12.9

330 331 332

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Preface

Mixed-mode switching circuits distinguish themselves from other circuits by including switches that are either clocked externally or controlled internally. These circuits have found broad applications in telecommunication networks, instrumentation, and power electronic systems, to name a few. It is the emergence of switched capacitor networks in the early 1970s and switched current circuits in late 1980s that sparked a broad interest in and the rapid development of computer methods and numerical algorithms for analysis and design of mixed-mode switching circuits. Recent advance in mixed analog-digital circuits and the systems-on-chip realization of complex electronic systems have further stimulated the enthusiasm of both the academia and industry in mixedmode switching circuits as these circuits provide a viable and yet economical means to realize both analog and digital systems on a silicon substrate using low-cost digitally-oriented CMOS technologies. As compared with time-invariant circuits, the time-varying characteristics, incomplete charge transfer, inconsistent initial conditions, and th under-sampling of broadband noise of mixed-mode switching circuits significantly complicate the analysis of these circuits, both in the time domain and frequency domain. Since the early 1970s, a significant effort has been made on the development of computer methods for the analysis and design of mixed-mode switching circuits. Many novel computer methods and numerical algorithms have emerged. A systematic presentation of these methods and an in-depth assessment of their advantages and limitations, however, are not available presently. This book

xxviii COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

is an attempt to summarize the recent advance in computer methods for mixed-mode switching circuits and to provide an in-depth and comprehensive assessment on the pros and cons of these methods. The book comprises of three parts. Part I is concerned with the issues that are fundamentally important to analysis of mixed-mode switching circuits. This part consists of four chapters. Chapter 1 provides an introduction of mixed-mode switching circuits and their applications. This chapter lays down a foundation for subsequent chapters. Chapter 2 describes the computer-oriented formulation of mixed-mode switching circuits. Starting with a brief review of the historic perspective of the computer formulation of mixed-mode switching circuits, the chapter details modified nodal analysis approach for computer formulation of mixed-mode switching circuits. Chapter 3 introduces the network functions of linear and nonlinear time-varying systems and their usefulness in characterization of periodically switched linear and nonlinear circuits. Chapter 4 examines numerical integration methods for differential equations. Specifically, it investigates the advantages and limitations of linear multi-step predictor-corrector algorithms, including linear single-step predictor-corrector algorithms, and introduces numerical Laplace inversion based numerical integration algorithms. It explores the advantages of numerical Laplace inversion based numerical integration method both analytically and numerically. Part II deals with the time domain analysis of mixed-mode switching circuits. This part consists of five chapters. Chapter 5 explores inconsistent initial conditions arising from ideal switching. In addition, it presents several numerical methods that yield consistent initial conditions when inconsistent initial conditions are encountered. Moreover, it explores numerical techniques that detect the existence of inconsistent initial conditions at switching instants. Chapter 6 addresses the time domain analysis of periodically switched linear circuits. A number of design objectives including response, parameter sensitivity, noise, the effect of clock jitter, and the statistical quantities, such as the mean and variance of the response of periodically switched linear circuits are

PREFACE

xxix

analyzed. The effectiveness of these methods is assessed using example circuits. Chapter 7 is concerned with the time domain analysis of periodically switched nonlinear circuits. The method presented in this chapter is based on time-varying Volterra functional series. Both the time-domain response and sensitivity of these circuits are analyzed. Chapter 8 deals with the analysis of circuits with internally controlled switches. The switching variable of internally controlled switches typically encountered in mixed-mode switching circuits that controls the state of these switches is defined. The methods that calculate the exact time instant at which internally controlled switches change their state are developed. The inconsistent initial conditions and impulsive network variables generated at switching instants are examined in detail. The detailed analysis of these circuits is illustrated using a linear voltage regulator. Chapter 9 is concerned with the analysis of a special class of mixed-mode switching circuits - over-sampled sigma-delta modulators. We show that sampled-data simulation method for periodically switched linear circuits, together with the behavioral modeling of quantizers, can be applied to analyze over-sampled sigma-delta modulators effectively and efficiently. Part III is devoted to the frequency domain analysis of mixed-mode switching circuits. This part consists of three chapters. Chapter 10 introduces Tellegen’s theorem for periodically switched linear circuits in the phasor domain. In addition, it derives the adjoint network of periodically switched linear circuits using the principle of inter-reciprocity. Frequency reversal theorem and transfer function theorem for periodically switched linear circuits are introduced, and their effectiveness in efficient calculation of both the transfer functions and aliasing transfer functions of these circuits is examined. Frequency domain analysis of periodically switched linear circuits including the response, sensitivity, group delay, the noise, mean and variance of the response of these circuits are analyzed in Chapter 11. Chapter 12 covers the frequency analysis of periodically switched nonlinear circuits. Both the harmonic distortion and intermodulation distortion of these circuits are derived. The book can serve as a reference book on computer methods for analysis of mixed-mode switching circuits. It can also serve as the text book

xxx COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

of a graduate course on computer-aided analysis of mixed-mode switching circuits. The book assumes that readers have a solid understanding of the basics of electrical networks, computer methods for analysis of electrical networks, and integrated devices and circuits. This book provides both graduate students and computer-aided design (CAD) tool development engineers with an in-depth understanding of computer methods and numerical algorithms for the analysis of mixed-mode switching circuits in both the time an frequency domains. FEI YUAN AND AJOY OPAL FEB. 2004

Acknowledgments

The authors wish to take this opportunity to express their sincere gratitude to the Natural Science and Engineering Research Council of Canada, University of Waterloo, Ryerson University, and Canada Foundation for Innovations for their financial support to both the authors and their graduate students for their research on computer aided analysis and design of mixed-mode switching circuits. Special thanks go to for many of his pioneering work on computer-aided analysis and design of integrated circuits that have inspired the authors profoundly, and to the members of our research teams both at University of Waterloo and Ryerson University, especially Drs. D. Bedrosian, B. Raahemifar, Y. Dong, for many of their original contributions to computer methods for mixed-mode switching circuits upon which this book is built. Our heartfelt appreciation also goes to the reviewers of the initial proposal of the book, (University of Waterloo), Dr. Timothy Trick (University of Illinois at Urbana-Champaign), Dr. John Swell (University of Glasgow) and Dr. Michael Nakhla (Carleton University). Canadian Microelectronics Corporation, Kingston, Ontario, deserves a special recognition for providing state-of-the-art computer-aided design tools for analysis and design of integrated circuits. The editorial staff of Kluwer Academic Publishers has, as always, been wonderfully supportive from the beginning of the project. The authors thankfully acknowledge the warm support of Mr. Michael S. Hackett, senior publishing editor, electrical engineering and optics, Kluwer Academic Publishers, during the course of the writing of the book.

xxxii COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

This book could not have been completed without the unconditional support of our families. Fei Yuan is grateful to his abiding wife, Jing, for her love, support, understanding, and patient during many long nights of writing and proofreading of the book, and to our little girl and boy, Michelle and Jonathan, for the joy that you have brought to our life. Daddy can finally have more time to play violins with you. Ajoy Opal is indebted to his grand parents Parkash and Ram Pershad Mehra for fostering an interest in science and engineering, to his parents Swadesh and Brij Kumar Opal for bringing him into this world and for raising him, to his children Ambika and Anuj for all the joy and wonderment that only children can bring.

I

THE FUNDAMENTALS

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Chapter 1 AN OVERVIEW OF MIXED-MODE SWITCHING CIRCUITS

This chapter presents an overview of mixed-mode switching circuits. Section 1 gives a general classification of mixed-mode switching circuits. In Section 2, switched capacitor techniques are introduced. Section 3 gives a brief introduction to switched current techniques. The characteristics of mixed-mode switching circuits and their impact on numerical algorithms for analysis of these circuits are investigated in Section 4.

1.

Classification

Mixed-mode switching circuits have found a broad range of applications in many areas of electrical engineering, from telecommunication networks, instrumentation, to power electronic systems. Mixed-mode switching circuits distinguish themselves from time-invariant circuits by including switches that are either clocked externally or controlled internally. These circuits can be broadly classified into the following categories:

Circuits with externally clocked switches. The switches in these circuits are controlled by external periodic clocks. Typical examples in this category include switched capacitor networks and switched current networks where switches are implemented using either NMOS transistors or CMOS pairs with external clocks applied to the gate of MOS transistors.

4

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Circuits with internally controlled switches. The switches circuits are controlled by internal network variables of the Rectifiers and switching voltage regulators where switches form of diodes and thyristors are representative examples in egory.

in these circuits. take the this cat-

Circuits with both internally controlled and externally clocked switches. Examples in this category include switched capacitor and switched current over-sampled sigma-delta modulators where the integrators are implemented using switched capacitor or switched current networks and the output of the quantizer of the modulators is set by the input of the quantizer.

2.

Switched Capacitor Techniques

Early applications of switched capacitor networks are the implementation of active filters where switched capacitors are used to replace monolithic diffusion resistors to overcome the following difficulties encountered in realization of these resistors : Large variation of resistance - Although diffusion resistors have a large resistance value, the absolute value of these resistors is heavily affected by the variation of the fabrication process by which the resistors are fabricated and has a large variance, usually in the range of ±30~40%. Such a large error significantly affects the performance of filters. Large parasitic capacitance - diffusion resistors have a large parasitic junction capacitance that exists between the n-diffusion that forms the resistors and the p-substrate upon which the resistors are fabricated. This junction capacitance is not only nonlinear but also varies greatly with the voltages at the terminals of the resistors [1]. Strongly nonlinear characteristics - the resistance of diffusion resistors is a strongly nonlinear function of the terminal voltages of the resistor, mainly due to the dependence of the effective area of the cross-section of the resistors on the voltages at the terminals of the resistors [2].

An Overview of Mixed-Mode Switching Circuits

5

To overcome these difficulties, switched-capacitor techniques that synthesize diffusion resistors using capacitors that are clocked periodically emerged in the early 1970s. The essence of this technique can be demonstrated conceptually using the circuit of Fig.1.1 where the capacitor C is connected to two constant voltages and The connection is controlled by two externally clocked switches. It is trivial to show that the average current flowing through the capacitor in a clock period is given

by

where

C is the capacitance of the capacitor,

is the clock period,

and is the equivalent resistance of the switched capacitor. It is seen that the equivalent resistance can be controlled by (i) the value of the capacitor and (ii) the clock frequency. For example, with C = 1 pF and we have Since monolithic IC techniques can ensure that the ratio of floating capacitors be controlled very accurately, usually in the range of ±0.01%, switched capacitor networks are always designed in such a way that the response of the networks is only a function of (i) the clock frequency and (ii) the ratio of the capacitors, rather than the absolute value of the capacitors, such that the behavior of filters can be controlled accurately.

6

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

This is clearly a big advantage, as compared with active filters realized using diffusion resistors.

The bottom plate of the capacitors usually has a large parasitic capacitance to the substrate, particularly if the capacitors are implemented using poly-diffusion capacitors. To minimize the effect of the parasitic bottom plate-substrate capacitance of switched capacitor networks, strayinsensitive switched capacitor techniques shown in Fig.1.2 were proposed. Consider the circuit in Fig.1.2a, in phase 1 where the two terminals of the bottom plate-substrate capacitor are shorted to ground. In phase 2 where they are connected to both the ground and the virtual ground of the following operational amplifier. As a result, this parasitic capacitor has no effect on the operation of the network. Fig.1.3 shows the basic configurations of inverting and non-inverting stray-insensitive switched capacitor integrators [3]. The basic switched capacitor integrators can be combined to form more complex switched capacitor networks, such as the biquad shown in Fig.1.4.

3.

Switched Current Techniques

The implementation of switched capacitor networks requires linear capacitors that are realized using two floating conducting layers. These floating conducting layers, however, do not exist in standard digitallyoriented CMOS technologies in which most digital CMOS circuits are

An Overview of Mixed-Mode Switching Circuits

7

realized. For most applications, the majority of the system blocks are digital. It can not be economically justified to implement these system

8

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

blocks using analog or mixed-mode IC technologies, as they are much more costly as compared with standard digitally-oriented CMOS technologies. Instead, it is highly desirable to realize analog blocks using standard digitally-oriented CMOS technologies. Switched current techniques emerged in the late 1980s [3] provide such a solution. Shown in Fig.1.5 are the first generation and second generation switched current memory cells. It is seen that switched current memory cells make use of the gate capacitance of the sampling transistor to hold the sampled input current, in the form of charge stored in and then exports it to the output in the subsequent clock phase in the form of current. Note that in these implementations, no linear capacitors are needed. Only MOS transistors are used. Switched current techniques are therefore fully compatible with and can be implemented using standard digitallyoriented CMOS technologies.

Using the basic memory cells, other building blocks, such as integrators, can be realized conveniently using switched current techniques, as shown in Fig.1.6 [4]. The basic switched current integrators can be combined to form more complex switched current networks, such as the biquad shown in Fig.1.7.

An Overview of Mixed-Mode Switching Circuits

4.

9

Characteristics of Mixed-Mode Switching Circuits

Since the early 1970s, a significant effort has been made on the development of numerical algorithms and computer methods for analysis and design of mixed-mode switching circuits. Many computer methods, such as frequency domain analysis of ideal switched capacitor networks [5, 6], frequency domain analysis of non-ideal switched capacitor networks and general periodically switched linear circuits [7, 8, 9, 10, 11, 12, 13], time domain analysis of nonlinear circuits with internally controlled switches [14, 15], adjoint network-based noise analysis of periodically switched lin-

10 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

ear circuits [16], Volterra series-based distortion analysis of periodically switched nonlinear circuits [17], sampled-data simulation algorithms for analysis of periodically switched linear and nonlinear circuits and sigmadelta modulators [18, 19, 20], to name a few, have emerged. As compared with time-invariant circuits and other time-varying circuits, mixed-mode switching circuits exhibit the following distinct characteristics : Time-varying topology - The topology of mixed-mode switching circuits varies with time. For circuits with externally clocked switches, such as switched capacitor networks and switched current networks, the time instants at which the topology of the circuits changes are known a priori. For circuits with internally controlled switches, the time instants at which the topology of the circuits varies, however, is not known at the start of simulation and is determined by the numerical value of the network variables controlling the state of switches, such as the voltage across diodes and the effective gate-source voltage of MOSFET switches, during simulation. Dual-time systems - Periodically switched circuits are dual-time systems that contain rapidly varying clocks and slowly varying signals. They have to be simulated over a large number of clock cycles with fine steps in order to obtain their time domain characteristics accurately and reliably.

An Overview of Mixed-Mode Switching Circuits

11

Instantaneous charge (flux) distribution - When the loop resistance is neglected, the re-distribution of charge in ideal switched capacitor networks, which are composed of a limited set of elements including ideal switches, independent voltage sources, voltage-controlled voltage sources, and capacitors, takes place at switching instants only. This unique characteristic of ideal switched capacitor networks enables the analysis of these networks using simple algebraic equations derived from the charge conservation of the networks at switching instants. Incomplete charge (flux) transfer - When the loop resistance becomes comparable to the duration of clock phases, the assumption that the network variables reach their steady value of a clock phase before reaching the end of the clock phase is violated. In this case, the behavior of the circuits can not be depicted using algebraic equations that only take into account the value of the network variables at discrete time instants, i.e. the clocking instants. Instead, differential equations that adequately depict the behavior of the circuits continuously in the entire clock phase are needed. The value of the network variables at the end of the clock phases must therefore be determined by solving the differential equations using numerical integration. This not only increases the cost of simulation but also complicates the analysis. Inconsistent initial conditions - Because practical switched capacitor networks and switched current networks are designed in such a way that the initial transient portion of the time domain response of these networks in each clock phase is not important and the network variables reach their steady value of the clock phase before reaching the end of the clock phase, i.e. the loop time constant is much small as compared with the duration of the clock phases. In this case, it is computationally advantageous to model switches as an ideal device, leading to ideal switching. Ideal switching, however, may cause an abrupt change in nodal voltages, as will be detailed in Chapter 5. The abrupt change in the capacitor voltage gives rise to an impulsive capacitor current. Similarly, ideal switching may give rise to an

12

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

abrupt change in loop currents, resulting in impulsive inductor voltages. The impulsive currents (voltages) re-distribute charge (flux) in the networks at switching instants instantaneously. As a result, the value of the network variables of these circuits immediately before switching differs from that immediately after switching, leading to inconsistent initial conditions, which can not be handled by conventional numerical integration methods. High output noise power- for circuits with externally clocked switches, the clocking frequency is usually much higher than the frequency of input signals, mainly due to the need to avoid spectrum overlapping of the sampled signals (Nyquist theorem). The equivalent noise bandwidth of these circuits, however, is usually several orders of magnitude higher than the sampling frequency. The under-sampling of the broad-band noise sources in these circuits, such as shot noise and thermal noise that are white in nature, gives rise to the fold-over effect where the output noise power of these circuits is not determined by the in-band noise power of the noise sources, but rather dominated by the noise power of the sideband components of these white noise sources that is folded over to the baseband. As a result, mixed-mode switching circuits exhibit a significantly high level of output noise power in the baseband.

Chapter 2 COMPUTER FORMULATION OF MIXED-MODE SWITCHING CIRCUITS

This chapter deals with computer-oriented formulation of mixed-mode switching circuits. Section 1 investigates the modeling of switches. The advantages and limitations of various switch models are examined. Section 2 presents a historic perspective of the computer-oriented formulation of mixed-mode switching circuits. We show that although many computer-oriented formulation methods were proposed, only modified nodal analysis (MNA) remains popular. This section also examines the implementation of the computer formulation of mixed-mode switching circuits using modified nodal analysis. Specifically, the matrix stamps of elements used in modified nodal analysis formulation are developed. Special attention is given to circuit elements with memory, as the energy stored in these elements is intrinsic to the operation of switching circuits. The chapter is summarized in Section 3.

1.

Modeling of Switches

Mixed-mode switching circuits distinguish themselves from time-invariant circuits by including switches. Switches appears physically in the form of NMOS transistors or as a pair of CMOS transistors in externally clocked circuits, such as switched capacitor networks and switched current networks, and diodes and thyristors in circuits with internally controlled switches. They can be characterized at different levels of circuit abstraction, among which, full-transistor model, voltage-modulated resistor

14

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

model, and ideal switch model are the most widely used circuit-level switch models.

1.1

Full-Transistor Models

A full-transistor model of MOS switches takes into account both the intrinsic and parasitic parameters of MOS devices [21]. It is capable of capturing the rapidly varying characteristics of the time-domain behavior of the switches. Simulation based on the full-transistor model, however, is costly, mainly due to the large size of the equivalent circuit of MOS switches and the small value of the parasitic parameters of MOS switches. The same holds true for other types of switches as well. The use of the full transistor model is therefore warranted only if a detailed analysis of the transient behavior of the circuits is required. For practical applications, because most mixed-mode switching circuits are designed in such a way that the transient behavior of the circuits in a clock phase dies out before reaching the end of the clock phase, the full-transistor models are rarely needed.

1.2

Voltage-Modulated Resistor Models

When the rapidly changing transient characteristics of the circuits are not of a critical concern, switches can be modeled as a voltage-modulated resistor that has a small resistance in the ON state, usually a few depending upon the technology in which MOS switches are realized and the width of the switches, and a large resistance in the OFF state, often in the range of several hundred The large difference between the ON and OFF resistances of the voltage-modulated resistor gives rise to largely distinct time constants for the ON and OFF states, respectively. As a result, stiff systems with numerically ill-conditioned circuit matrices occur. The time-domain analysis of these circuits requires fine time steps in order to capture the rapid transient behavior of the circuits in the ONstate. On the other hand, simulation needs to be executed over a long period of time in order to account for the large time constant associated with the OFF state, resulting in excessive computation and exceedingly long simulation time.

Computer Formulation of Mixed-Mode Switching Circuits

1.3

15

Ideal Switch Models

To avoid the drawbacks arising from the voltage-modulated resistor models, the ideal switch model that has zero resistance in the ON state and infinite resistance in the OFF state, as shown in Fig.2.1, is of particular interest.

The difference between the voltage-modulated resistor models and the ideal switch models manifests itself in the transient response of mixedmode switching circuits and is best illustrated using the circuit given in Fig.2.2, where and Capacitor is initially charged to 2V and is initially at rest, i.e. and The MOSFET switch closes at The response of the circuit with the ideal switch models and that with the voltagemodulated resistor switch models are shown in Fig.2.3. It is seen that the response with the ideal switch models does not have the initial rising transient portion of the response that the response of the circuit with the voltage-modulator resistor models does. Due to the rapidly timechanging characteristics, the analysis of the transient portion requires a large number of small time steps in order to capture the time-varying response. As a result, an excessive amount of computation time, usually up to 90 % of the total simulation time, is used up in the initial rising transient [22]. Also observed is that in the limiting case where the time domain response with the voltage-modulator resistor switch model approaches that with the ideal switch model. Most switched capacitor and switched current networks are designed in such a way that the initial transient portion of the time domain re-

16

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

sponse is not important and the network variables reach their steadystate values before reaching the end of clock phases. In these cases, it is computationally advantageous to model switches as an ideal device, leading to ideal switching. Ideal switching, however, may cause an abrupt change in nodal voltages, as observed in Fig.2.3. The abrupt change in the capacitor voltage gives rise to an impulsive capacitor current that redistributes the charge instantaneously. In a similar manner one can show that when inductors are encountered, ideal switching may give rise to an abrupt change in loop currents, resulting in impulsive inductor voltages.

2.

2.1

Formulation Methods for Mixed-Mode Switching Circuits A Historical Perspective

In the early stages of the development of computer methods for circuits with externally clocked switches, in particular, ideal switched capacitor networks that are composed of ideal operational amplifiers, ideal voltage sources, voltage-controlled voltage sources, capacitors, and ideal switches only, many formulation methods were proposed to deal with the time-varying topology of these circuits. In ideal switched capacitor

Computer Formulation of Mixed-Mode Switching Circuits

17

networks, due to zero loop time constants, the charge redistribution process among capacitors takes place at switching instants instantaneously. Once the ON resistance of MOS switches are considered, the redistribution of charge is governed by the loop time constant. These circuits exhibit distinct characteristics as compared with ideal switched capacitor networks and are called periodically switched circuits to distinguish them from ideal switched capacitor networks. Clearly, ideal switched capacitor networks are a subset of general periodically switched circuits. Due to incomplete charge transfer, numerical integration algorithms are needed to determine the voltage of capacitors and the current of inductors in these circuits at the end of each clock phase. Many computer-oriented formulation methods were proposed for analysis of ideal switched capacitor networks and general switched networks. Among them, equivalent-circuit [23, 24], transmission matrix [25, 26, 27], switching matrix [28, 29], signal flow diagram [30, 31, 32], state-space

18 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

[33], two-graph [34, 5], and modified nodal analysis are most cited. The equivalent-circuit approach represents switched capacitor networks with a set of building blocks that have known characteristics. This approach is effective for small switched capacitor networks only. Transmission matrix approach maps the building blocks of switched capacitor networks to a set of matrices so that the networks can be analyzed conveniently. It is effective for networks of small size. Signal-flow diagram approach makes use of Mason’s rule [35] to yield the transfer function from a given input node to an arbitrary output node. This approach provides many insights of the operation of networks, it, however, is not particularly convenient for computer analysis of switched networks. Modified nodal analysis (MNA), an extension of the nodal analysis, has been used extensively in analysis of electrical networks since its emergence in 1970s [36]. For ideal switched capacitor networks, due to the existence of impulsive currents at switching instants in these networks, nodal charge conservation law is used at switching instants [37, 28]. This approach possesses many advantages over other formulation methods including [38] : Ease in circuit formulation. MNA formulation inherits the intrinsic advantages of Tableau formulation in circuit formulation [39]. The MNA formulation for arbitrarily large circuits is straightforward. No manipulation of equations is necessary. Branch voltages and many branch currents are eliminated in MNA formulation as compared with Tableau formulation, leading to much smaller circuit matrices. Circuit matrices are sparse. The sparsity of circuit matrices enables the use of sparse matrix solvers to significantly reduce the cost of computation. Convenience in calculating parameter sensitivity [5]. Time derivatives of independent sources are never needed.

Computer Formulation of Mixed-Mode Switching Circuits

2.2

19

External Clocks

External clocks used for clocking switches are multi-phase and nonoverlapping, as shown in Fig.2.4, where denotes the period of the clock and K denotes the number of phases in a clock period. Note that and

2.3

Conventions

To avoid any ambiguity, the following conventions are adopted in this book. If a switch changes its state at the time instant we shall use to denote the time instant immediately before switching and to denote the time instant immediately after switching.

20 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

denotes the vector of the network variables of mixed-mode switching circuits in the time domain. denotes the vector of the network variables of mixed-mode switching circuits in the frequency domain. denotes the vector of the network variables of mixed-mode switching circuits in phase denotes the vector of the Fourier transform of the network variables of mixed-mode switching circuits in phase denotes the vector of the first-order time derivative of the network variables of mixed-mode switching circuits in phase denotes the vector of the network variables of mixedmode switching circuits at the time instant It can also be written as where denotes Dirac impulse function.

2.4

Sub-Circuits

The topology of an externally clocked circuit changes from one clock phase to another and remains unchanged during each clock phase. This observation suggests that an externally clocked circuit with a total of K phases in a clock period can be considered as an assembly of K timeinvariant sub-circuits that are interconnected via the initial conditions of elements with memory (capacitors and inductors), as depicted graphically in Fig.2.5. These sub-circuits are operated in a time-interleaved fashion such that

and

The time duration in which the input is specified by the window function

is connected to the sub-circuit defined as

Computer Formulation of Mixed-Mode Switching Circuits

The input to the sub-circuit

21

is therefore given by

2.5

Matrix Stamps of Elements Without Memory

2.5.1

Controlled Sources

Memoryless elements are those whose response is determined by their present inputs only. Typical memoryless elements include resistors and controlled sources. As an example, consider the nonlinear voltage-controlled voltage source characterized by

where spectively,

and and

are the controlling and controlled voltages, reare constants. The primitive schematic of non-

22

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

linear voltage-controlled voltage sources in phase is shown in Fig.2.6. The behavior of the controlled source is completely characterized by the following relations called constitutive equations

and

Eqs.(2.7) and (2.8) can be written in a matrix format, leading to the matrix stamps of the nonlinear controlled source, as shown in Fig.2.6. The missing entries of the added rows and columns are zeros. Note that nonlinear terms are purposely placed in the vector on the right hand side of the equations such that the matrices and vectors on the left hand side of the equation are all linear. The matrix stamps of other nonlinear controlled sources can also be derived in a similar manner and they are shown in Fig.2.6. The matrix stamps of linear controlled sources can be obtained from that of the corresponding nonlinear controlled sources by simply setting the nonlinear coefficients to zero. 2.5.2

Ideal Switches

As shown in Fig.2.7, there are only two distinct states associated with the operation of an ideal switch connected to nodes and : the ON state and the OFF state. Ideal switches are characterized by the following constitutive equations: The ON state is characterized by The OFF state is characterized by These characteristics can be mapped to the equations given in Fig.2.7. The matrices for the ON and OFF states can also be combined by introducing the switching variable

Computer Formulation of Mixed-Mode Switching Circuits

2.6

23

Matrix Stamps of Elements With Memory

An element is said to have memory if its response is a function of both its present inputs and its previous states. Capacitors and inductors are typical elements in this category. The energy storage capability of these elements is intrinsic to the operation of mixed-mode switching

24

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

circuits. In this section, we examine the matrix stamps of capacitors and inductors in circuits with externally clocked switches. 2.6.1

Capacitors

Consider a linear time-invariant capacitor C. Let the initial voltage of the capacitor be The capacitor is connected to nodes and with its current flowing from node to node The constitutive equation governing the capacitor in the Laplace domain is given by

where and are the Laplace transform of the capacitor current and that of the voltage respectively. Eq.(2.10) can be written equivalently in the time domain as

Computer Formulation of Mixed-Mode Switching Circuits

25

where is the Dirac impulse function. Now, consider a linear capacitor in phase of a periodically clocked linear circuit. Let the voltage of the capacitor at be Note that is the initial condition of the capacitor in phase Making use of (2.11), we arrive at

The time interleaved operation of externally clocked circuits, as detailed in Fig.2.5, requires that be zero outside phase To ensure that is zero for the input to the sub-circuit of phase must be removed. Also, the charge of the capacitor at the end of the clock phase, i.e. must be extracted completely. This is accomplished by subtracting from

In MNA, the above equation can be written in a matrix format, as shown in Fig.2.8. Most nonlinear capacitors encountered in integrated mixed-mode switching circuits are pn-junction induced. The small-signal behavior of a nonlinear capacitor is often characterized using its charge-voltage relation, usually up to the order of three [2]

where and are the AC components of the charge stored in the capacitor and the voltage across the capacitor, respectively. and are constants and are functions of the DC operation point of the capacitor. To exemplify the procedures of how (2.14) is derived, consider the junction capacitance of an abrupt pn-junction

26

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where is the junction capacitance at zero biasing voltage, is the total reverse biasing voltage, and is the built-in potential of the pnjunction. and are functions of the doping of the and that of the extrinsic silicon of the junction and the temperature of the junction [40]. Let where and are the DC and AC components of respectively. Further we assume

Computer Formulation of Mixed-Mode Switching Circuits

27

Expanding at the DC operation point in its Taylor series and truncating the series at its third-order term yield

where

The first term on the right had side of (2.16) gives the junction capacitance at the DC operating point

whereas the remaining terms quantify the AC capacitance.

It is seen that the AC capacitance is a nonlinear function of both the DC operating point and the AC amplitude of the reverse biasing voltage. If a nonlinear capacitor has initial charge the capacitor is characterized in the Laplace domain by

or equivalently in the time domain

Now consider a nonlinear capacitor in an externally clock circuit. Let denote the charge of the capacitor at

28

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

This is the initial condition of the capacitor in phase the capacitor in phase is given by

The current of

Following the same arguments as those for linear capacitors, to ensure that vanishes outside phase the charge stored in the capacitor at the end of phase denoted by must be extracted from

The MNA formulation of the nonlinear capacitor is based on (2.14) and (2.23) exclusively and is given in Fig.2.8. 2.6.2

Inductors

Inductors in mixed-mode switching circuits can be handled in a similar manner as that for capacitors. The effect of the magnetic flux stored in the inductors in phase on the behavior of circuits in the following phase must be considered. Fig. 2.9 shows the matrix stamps of linear and nonlinear inductors.

2.7

Formulation of Circuits with Externally Clocked Switches

As stated earlier that to facilitate analysis, an externally clocked circuit with input and an external clock of K clock phases can be considered as an assembly of K time-invariant sub-circuits interconnected via their initial and final conditions of the elements of the circuit with memory. Each sub-circuit must ensure that its network variables satisfy

Computer Formulation of Mixed-Mode Switching Circuits

29

To achieve this, the following criteria are followed in formulation of these sub-circuits:

30

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

The input is connected to the sub-circuit of phase only and is disconnected from the sub-circuit outside phase This is consummated by making use of the window function defined earlier. The effect of the initial voltage (charge) of the capacitors and that of the initial current (flux) of the inductors of the sub-circuit must be accounted for as they have an impact on the behavior of the circuit in phase The voltage (charge) of the capacitors and the current (flux) of the inductors at the end of phase must be extracted so that they will not affect the behavior of the sub-circuits in subsequent clock phases. The circuit matrices of each sub-circuit are obtained by applying appropriate matrix stamps introduced in the preceding sections to each circuit element. In implementation, the matrix stamps of each type of elements are coded as functions with element type, element values, and their electrical connections as input parameters of the functions so that the overall circuit matrices can be formulated in an automated manner.

where is the input of the circuit whose connection in phase is specified by the constant vector is the window function for the input, is the network vector in phase which may contain nodal voltages, branch currents, the charge of nonlinear capacitors, and the flux of nonlinear inductors. is the linear conductance matrix whose entries are made of linear elements and the linear portion of nonlinear elements, is the linear capacitance matrix in phase whose entries are from linear capacitors and inductors; is a nonlinear vector containing all nonlinear terms of the Taylor series expansion of

Computer Formulation of Mixed-Mode Switching Circuits

31

the nonlinear elements in the circuit. Clearly, for a linear circuit we have The Dirac delta function term at takes into account the contribution of the initial voltage (charge) of capacitors and the initial (current) flux of inductors at the beginning of phase whereas the Dirac delta function term at resets all capacitors and inductors so that the output of the sub-circuit of phase vanishes outside the phase.

2.8

Formulation of Circuits with Internally Controlled Switches

Unlike circuits with externally clocked switches, the time instants at which the topology of circuits with internally controlled switches changes are determined by the state of the switching variable of the switches in the circuits. The switching variable controls the state of the switch in accordance with

As an example, of a NMOS transistor is the effective gate-source voltage given by where is the threshold voltage of the transistors. Because the value of the switching variables is not known at the start of simulation, an initial state is assumed and the circuit is formulated accordingly. is then computed and monitored in every time step of the simulation of the circuit. A change in the polarity of indicates a change in the state of the corresponding switch in the current time step. Iterative algorithms, such as Newton-Raphson, are then employed to accurately compute the exact time instant by solving numerically [14], as shown in Fig.2.10. Once the exact time is detected, the topology of the circuit is updated and the initial conditions of the new circuit are computed. It should be noted that because ideal switching gives rise to impulse currents or impulse voltages, as will be shown in Chapter 5, the impulses generated at switching instants may trigger other switches in the circuit and change the topology of the circuit subsequently. In analysis of these circuits, it is important to further

32

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

examine the state of each internally controlled switches after a switch changes its state.

2.9

Formulation of Circuits with Both Externally Clocked and Internally Controlled Switches

Formulation of circuits with both externally clocked and internally controlled switches can be handled conveniently by integrating the approaches for circuits with externally clocked switches and those with internally controlled switches. Specifically, during each clock phase, the circuits are handled using the approach for circuits with internally controlled switches. At switching instants, they are handled using the approach for circuits with externally clocked switches. Since the state of internally controlled switches is not known a priori, this requires the reformulation of the circuit equations at each switching instant. It should be noted that current or voltage impulses generated at the clocking instants may trigger switches that are internally controlled. Detection of impulsive network variables at clocking instants is critical. The current or voltage impulses generated by the switching of internally controlled switches, on the other hand, have no effect on the operation of externally clocked switches.

Computer Formulation of Mixed-Mode Switching Circuits

3.

33

Summary

In this chapter, we have examined the advantages and disadvantages of various switch models. We have shown that full-transistor models are rarely used in analysis of mixed-mode switching circuits due to the insignificance of the transient portion of the response and the high computational cost associated with these models. In comparison with the fulltransistor models, the voltage-modulator resistor models are much simpler and yet are able to capture the essential characteristics of switches. The voltage-modulator resistor models, however, give rise to stiff systems that have two largely distinct time constants for the ON and OFF states of switches, leading to excessive simulation time. Ideal switch model removes this difficulty by using an open-circuit for the OFF state and a short-circuit for the ON state. Ideal switching, however, may cause an abrupt variation in nodal voltages or loop currents, resulting in inconsistent initial conditions and impulsive network variables that can not be handled by conventional numerical integration methods. To formulate the circuit equations of mixed-mode switching circuits, we have examined the reasons why only modified nodal analysis formulation method continues to remain popular. The matrix stamps of both memoryless elements and elements with memory have been developed, and the computer-oriented formulation of circuits with externally clocked switches and those with internally controlled switches have been developed. Special attention has been given to elements with memory, i.e. inductors and capacitors, as the energy storage capability of these elements is intrinsic to the operation of mixed-mode switching circuits.

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Chapter 3 NETWORK FUNCTIONS OF TIME-VARYING CIRCUITS

This chapter examines mathematical tools that are fundamentally important to the analysis of mixed-mode switching circuits. In Section 1, the time-varying network function of linear time-varying systems is introduced and its usefulness in characterization of linear time-varying systems is investigated. The aliasing transfer functions for characterizing linear periodically time-varying systems are also introduced. We show that although the input to a linear periodically time-varying systems is a single tone, the response of the system has an infinite number of tones located at the baseband and sideband frequencies. This characteristic of linear periodically time-varying systems differs fundamentally from that of linear time-invariant systems. In Section 2, Volterra functional series for characterizing nonlinear time-invariant and time-varying systems is introduced first. We then introduce multi-frequency time-varying network functions and multi-frequency network functions to characterize the behavior of nonlinear time-varying systems in both the time and frequency domains. Section 3 derives the frequency response of nonlinear time-varying systems. Section 4 presents the frequency domain response of a special class of nonlinear time-varying systems, nonlinear periodically time-varying systems, to both single-tone inputs for harmonic distortion and multi-tone inputs for intermodulation. The chapter concludes in Section 5. A proof of the periodicity of the time-varying

36

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

network function of linear periodically time-varying systems is given in Appendix 3.A.

1.

1.1

Transfer Functions of Linear Time-Varying Systems Linear Time-Varying Systems

In steady state, the behavior of a linear time-varying system in time domain is described by the impulse response of the system which is a function of both the excitation time at which the impulse is launched and the observation time at which the response of the system to the impulse is measured. The input and output of the system are related to each other in the time domain by

For linear time-invariant systems, Eq.(3.1) becomes

Note that the impulse response is evaluated at the excitation time only in (3.2). The behavior of linear time-varying systems is characterized by the time-varying network function introduced by Zadeh in [41]. is defined as

The impulse response can be derived from transform

Substituting (3.4) into (3.1) gives

using the inverse

Network Functions of Time-Varying Circuits

where

1.2

37

is the Fourier transform of

Linear Periodically Time-Varying Systems

If a system is linear periodically time-varying, we show in Appendix 3.A of this chapter that time-varying network function is also periodic in Assume that satisfies Dirichlet-Jordan criterion, i.e. is bounded, piecewise continuous, and has at most a finite number of minima, maxima, and discontinuities per period, can be represented in the Fourier series

where

and

is the period. The coefficients are determined

from

The frequency response of the output, denoted by taking Fourier transform of (3.5)

Further making use of (3.6), Eq.(3.8) becomes

Because

is obtained by

38 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

we arrive at

Eq.(3.11) reveals that : The frequency response of a linear periodically time-varying system at frequency consists of the contribution of the input signal at both and This characteristic differs fundamentally from linear time-invariant circuits. If the input to a linear periodically time-varying circuit is a broadband signal, such as thermal and shot noise, then the response of the circuit at a given baseband frequency contains the contribution of the input at both the baseband frequency and that at corresponding sideband frequencies, resulting in much higher output power. The quantity characterizes the relation between the input at the side band frequency and the output at the base band frequency It is called the aliasing transfer function. Note that aliasing transfer functions are a generalization of the transfer functions for linear time-invariant circuits where both the input and output are evaluated at the same frequency, i.e. the frequency of the input.

If the system is linear time-invariant, Eq.(3.11) is simplified to

Network Functions of Time- Varying Circuits

39

Clearly, for linear time-invariant systems, there is a one-to-one mapping between the input and output frequency components whereas for linear periodically time-varying systems, the mapping is multiple-toone. If the input of the linear time-varying system is an exponential function of time where is the frequency of the input, because the Fourier transform of is given by

using (3.5), we obtain the response of the system

If the system is further linear periodically time-varying, then can be represented in the Fourier series given in (3.6) with replaced by Substituting the results into (3.14) yields

where is the coefficient of the Fourier series given by (3.7). The following important observations are made from the preceding derivation: The response of a linear periodically time-varying system to a input at the frequency contains an infinite number of frequency components at frequency and corresponding sideband frequencies where specifies the magnitude of the frequency component at the frequency in the complex plane. It is the phasor representation of at the frequency

40

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

For linear time-invariant systems, phasors are defined at the input frequency only. For linear periodically time-varying systems, they are defined at both the input frequency and the sideband frequencies This is a distinct characteristic of these systems.

2.

2.1

Transfer Functions of Nonlinear Time-Varying Systems Volterra Functional Series

Nonlinear time-invariant systems are most often analyzed using power series [42] where the response of the system, denoted by is represented as the power series of the excitation

where are constants. It is seen that the response is related to the inputs at the present time only. Because it does not take into account the effect of the past states of elements with memory, such as capacitors and inductors, power series approach is only valid for circuits consisting of memoryless elements or circuits with memory elements but operated at sufficiently low frequencies such that the contribution of the past states of elements with memory is negligible. For high-frequency applications, the effect of past states must be considered. In this case, the coefficients of the power series become frequency-dependent. The response in this case is characterized by Volterra functional series, also known as power series with memory [43]

where

Network Functions of Time-Varying Circuits

41

and is the Volterra kernel. When the system is nonlinear time-varying, the Volterra series become [44]

2.2

Multi-Frequency Network Functions

It was shown in the preceding section that a set of time-varying Volterra kernels, are needed to characterize the behavior of nonlinear time-varying systems in the time domain. To depict the behavior of the systems in the frequency domain, we extend the definition of the time-varying network functions for linear time-varying systems given earlier to nonlinear time-varying systems by defining the following multi-frequency time-varying network functions

The

Volterra kernel is obtained from the inversion transform

Substitute (3.21) into (3.17)

where is the Fourier transform of with frequency As an example, consider a nonlinear time-varying system with the input The response is obtained from

42 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

It is evident from the above equation that multi-frequency network functions quantify the fundamental and harmonic components of the frequency response of nonlinear time-varying systems.

2.3

Multi-Frequency Transfer Functions

It is well known that the transfer function of linear time-invariant systems completely characterizes the behavior of the systems in the frequency domain. It is perhaps less well known that Zadeh’s bi-frequency transfer function defined as

where and are the output and input frequencies, respectively, depicts the behavior of linear time-varying systems in the frequency domain effectively. The output is obtained from

The corresponding time-varying network function is obtained from the inverse transform

If i.e. is a single-tone at with unit amplitude. Because we have This result reveals that represents the aliasing transfer function of the system from the input at the frequency to the output at the frequency To analyze nonlinear time-varying systems in the frequency domain, we extend the definition of Zadeh’s bi-frequency for linear time-varying systems to nonlinear time-varying systems by introducing the following multi-frequency transfer function

Network Functions of Time-Varying Circuits

43

The corresponding time-varying network function is obtained from the inverse transform

3.

Frequency Response of Nonlinear Time-Varying Systems

The frequency response of nonlinear time-varying systems is obtained from the Fourier transform of (3.17)

It was shown earlier that the Fourier transform of the 1st-order response is given by

where denotes Fourier transform operator. The Fourier transform of is obtained from

Continuing this process and substituting these results into (3.29) give

44

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

The following comments are made with respect to the preceding development: Eq.(3.32) is a frequency-domain representation of the time-varying Volterra series. Analogous to the time-domain representation of timevarying Volterra series, can be considered as the kernel of the order frequency-domain Volterra series. Clearly seen is that once is known, the spectrum of the response of nonlinear time-varying systems to the input is defined completely. Consider the special case where

because

we have

Eq.(3.34) provides a general means to characterize the behavior of nonlinear time-varying system in the frequency domain. In the next section, we will make use of these results to obtain the frequency response of a special class of nonlinear time-varying systems, namely, nonlinear periodically time-varying systems.

4.

Frequency Response of Nonlinear Periodically Time-Varying Systems

In this section, we make use of the results from the preceding section to derive the frequency response of nonlinear periodically time-varying

Network Functions of Time- Varying Circuits

45

systems to both single-tone and dual-tone inputs. In Appendix 3.A of this chapter, we have shown that of nonlinear periodically time-varying systems is periodic in with the period equal to the switching period They can be represented in Fourier series

where

If the input is given by

we obtain

Consequently

In the base band where the fundamental component of the response is given by whereas the second-, third-, ..., and harmonic components are given by and respectively. The similar pattern repeats in side bands where To obtain the harmonic components of the response of nonlinear time-varying systems to a sinusoidal input because

we have

46

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Note that we have assumed symmetrical kernels in the above derivation. The 2nd- and 3rd-order harmonic distortion, denoted by and respectively, are computed from

If the system is nonlinear time-invariant, then (3.41) and (3.42) simplify to the familiar expressions of and of nonlinear time-invariant systems [45]. If the system is further nonlinear periodically time-varying, we have

Network Functions of Time-Varying Circuits

47

The complete spectrum of the response is given in Fig.3.2. The 2nd-order and 3rd-order harmonic distortions in the base band are computed from

provided that If the input of a nonlinear time-varying system contains two different frequencies

with the spectrum of the response can also be obtained in a similar manner. It can be shown that the 3rd-order intermodulation distortion, denoted by is computed from

If the system is further nonlinear periodically time-varying, then it can be shown that the spectrum of the response is given in Fig.3.3. The 3rd-order intermodulation distortion at in the base band is computed from

where

has been assumed.

48

5.

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Summary

Mathematical tools that are fundamentally important to the analysis of mixed-mode switching circuits have been presented. Specifically, the time-varying network function of linear time-varying systems has been introduced and its usefulness in characterization of the time-domain behavior of linear time-varying systems from its frequency-domain excitation has been investigated. Aliasing transfer functions that charac-

Network Functions of Time-Varying Circuits

49

terize linear periodically time-varying systems has been introduced. We have demonstrated mathematically that the response of a linear periodically time-varying system to an input of single frequency contains an infinite number of tones located at both the input frequency and corresponding sideband frequencies. This characteristic of linear peri-

50

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

odically time-varying systems differs fundamentally from that of linear time-invariant systems. To analyze nonlinear systems, we have shown that power series approaches are valid for characterizing nonlinear systems without memory or general nonlinear systems at low frequencies at which the contribution of the past states of elements with memory is negligible. When the effect of the past states of elements with memory can not be neglected, Volterra functional series must be employed. We have introduced multi-frequency time-varying network function to characterize time-varying nonlinear systems in the time domain, and multi-frequency network functions to characterize the behavior of these systems in the frequency domain. The use of these network functions allows us to derive the spectrum of nonlinear periodically time-varying systems to inputs of both single-tone and multiple tones.

APPENDIX 3.A: Periodicity of The Network Functions of Nonlinear Periodically Time-Varying Systems In this appendix, we show that the time-varying network functions of nonlinear periodically time-varying systems are periodic in time with the same period. Consider a linear periodically time-varying system characterized by

where is the input and is the response. The periodicity of the system is characterized by where is the period of time variation. Note that although we have chosen scalar form for the purpose of simplicity the results can be readily extended to vector form. The solution of (3.A.1) is given by [46]

where

Without losing generality, let the input of the system be in Fourier series

. Representing

APPENDIX 3.A

where is the coefficient of the Fourier series and into (3.A.2) and carrying out integration give

where

and

51

Substituting (3. A.4)

are constant and

and is the order coefficient of the Fourier series expansion of comments with respect to the above results are made. of

A few

is periodic in with period Similarly, one can also show that the reciprocal denoted by is also periodic in with period

The necessary condition for the system to be asymptotically stable is This ensures is bounded as In the steady state, the first and second terms in (3.A.5) vanish, and the third term gives the steady-state response of the system

Since is periodic in with period it is evident that is also periodic in with period By comparing (3.A.7) with (3.14), the relation follows. We therefore conclude that is periodic in with period Having proved the periodicity of we now examine the periodicity of higher order time-varying network functions. Because for asymptotically stable systems the zero-input response dies down in the steady state, in the following analysis we will only focus upon the zero-state response in the steady state. Consider a nonlinear periodically time-varying system represented by

where is a nonlinear function of If the nonlinearities encountered are mild, can be approximated using only a truncated Taylor series expansion of the nonlinear equations of the nonlinearities, mathematically

52

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where and are constants. Using Volterra series, equation (3.A.8) can be represented by the following set of linear periodically time-varying systems

where and series expansion of

Since

are the 1st-, 2nd-, and third-order terms of the Volterra respectively. and If we have

is periodic in

with period

we have

where is the Fourier series coefficient. The input of the second order Volterra circuit, denoted by is given by

Using (3.A.5) we obtain the zero-state response of the second order Volterra circuit

where as

is a constant. If the system is asymptotically stable, then Thus, the steady-state response is given by

Clearly, is periodic in with period Comparing (3.A.11) with (3.A.15) we conclude that is also periodic in with period In a very like manner, one can show that is periodic in with period

Chapter 4 NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS

The behavior of mixed-mode switching circuits is depicted in the time domain using differential equations. This chapter is concerned with numerical integration algorithms for differential equations. The conventional linear single-step predictor-corrector (LSS-PC) algorithms and linear multi-step predictor-corrector (LMS-PC) algorithms are reviewed in Sections 1 and 2. We show that although these algorithms are robust in solving both linear and nonlinear circuits, the accuracy of these methods is limited by the order of polynomials used in extrapolation and the use of the first-order derivatives. In Section 3, we show that numerical Laplace inversion that derives the time domain solution from its counterpart is an elegant high-order numerical integration methods for linear circuits. The accuracy of this method is orders of magnitude higher as compared with that of LMS-PC algorithms. In addition, this method is capable of handling both impulses and discontinuities in network variables that can not be handled by LMS-PC algorithms. This unique characteristic makes numerical Laplace inversion algorithm particularly attractive for analysis of mixed-mode switching circuits where impulses and discontinuities might be encountered at switching instants. We examine the properties of numerical Laplace inversion by first introducing Padé polynomials. Our focus is then shifted to the use of Padé approximation in numerical Laplace inversion. The dependence of the accuracy of numerical Laplace inversion on the time displacement from the time

54

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

origin is studied in detail and a stepping algorithm that can provide superior numerical accuracy in computing the time domain response of linear circuits over a time interval of arbitrary length is introduced. The chapter is summarized in Section 4.

1.

Linear Single-Step Predictor-Corrector Algorithms Consider the first-order differential equation

where denotes time and Assume that is continuously differentiable with respect to time. Let the solution of (4.1) at time instants and be given by and respectively. Expand in Taylor series at with the time displacement

If is sufficiently small, the above series can be truncated at the first order without introducing a large truncation error

Because (4.3) derives from the known point it is known as the forward Euler formula. The forward Euler formula is explicit in the sense that it computes directly from the present point whose value and the first-order derivative are known. Forward Euler formula is also known as the first-order predictor. Because the truncation error given by where is directly proportional to (i) the square of the step size and (ii) the second-order derivative of if varies rapidly with time, the step size must be kept sufficiently small such that a reasonably good prediction of can be obtained.

Numerical Integration of Differential Equations

55

We can also expand at with the time displacement and truncate the series at the first order

Eq.(4.4) is known as the backward Euler formula. Note that the backward Euler formula computes using which is also unknown. To find an initial guess of is used. The correct value of is obtained by solving (4.4) iteratively using Newton-Raphson. Eq.(4.4) is therefore also known as the corrector. It is well understood that Newton-Raphson provides quadratic convergence provided that the initial estimate is sufficiently close to the solution. In order to achieve fast convergence, the predictor is often used to provide a starting point for the corrector, leading to the predictorcorrector numerical integration algorithms. If the forward Euler formula is used as the predictor and the backward Euler is used as the corrector, the algorithm is known as the linear single-step predictor-corrector (LSS-PC) algorithm. The essential steps of LSS-PC algorithm are given as follows: For a given time point with known and Euler formula is used to obtain an estimate of The first-order time derivative of from

at

the forward

is then obtained

These quantities are then substituted into (4.4), which is then solved iteratively using Newton-Raphson. It was shown in [5] that the numerical stability of the forward and backward Euler formulae can be best studied using the following benchmark equation

where given by

and

denotes the complex domain. The solution of (4.5) is

56

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Eq.(4.6) shows that the system is stable if where denotes the real part of a complex variable. In what follows we examine the stability of the LSS-PC algorithm. Let us consider the forward Euler formula first. Applying the forward Euler formula to (4.5) gives

where when

Let

To have a bounded response for a given initial condition the following condition must be met

where

and

denotes the real domain, we have

In order to have a stable numerical solution, the step size must be such that is confined within the unit circle centered at (-1,0), as shown in Fig.4.1. It is seen that for large the step must be small enough in order to ensure numerical stability of the forward Euler formula. The following important conclusions are drawn with respect to the stability of forward Euler formula: If

the system itself is unstable, from forward Euler formula yields unstable response.

we have

If (4.9) is violated, even though the system itself is stable, i.e. the computed response from the forward Euler formula will be diverging. In a very like manner, one can shown that the stable region of backward Euler formula is given by

Numerical Integration of Differential Equations

57

The above equation reveals that in order to have a stable numerical solution, the step size must be such that is outside the unit circle centered at (1,0), as shown in Fig.4.1. It should be noted that if i.e. the system itself is unstable, if the step size is chosen such that the response computed from backward Euler formula will, however, be stable. We conclude from the preceding analysis that: Forward Euler formula could yield an erroneous diverging response for a stable system. Backward Euler formula could provide a false converging response for an unstable system.

2.

Linear Multi-Step Predictor-Corrector Algorithms

The linear single-step predictor-corrector algorithms suffer from both the poor accuracy and the lack of computational efficiency due to the following reasons:

58 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Only the first-order derivatives are used. With only the first-order derivatives, for rapidly changing the step size must be kept sufficiently small in order to meet the stability and accuracy requirements. Only the information of the present and that of the most recent past data point are used in estimation of Both the accuracy and speed can be improved if more past data are used to predict It is evident that in order to increase the step size and in the mean time to improve the accuracy, more past data points and higher-order derivatives should be used in prediction of Calculation of highorder derivatives numerically is generally not only difficult and but also costly. For this reason, in practice, multiple past data points are used, but only the first-order derivatives at these data points are usually used as a compromise to estimate leading to the linear multi-step predictor-corrector (LMS-PC) algorithms. Let a total of past data points, denoted by and are known. Also assume the first-order time derivatives at these past data points, denoted by and are known as well. The predictor of LMS-PC that predicts the next data point is given by

Note that the predictor uses only the known information of the past points. The corrector of LMS-PC algorithm is given by

The corrector contains and that are both unknown. Eq. (4.12) needs to be solved iteratively using Newton-Raphson iterations. Substituting (4.12) into (4.1)

Numerical Integration of Differential Equations

Because and of (4.13) with respect to

are constant for gives

59

the derivative

Newton-Raphson iterations proceed as follows

and

Integration stops when

3.

is sufficiently small.

Integration Using Numerical Laplace Inversion

LMS-PC algorithms improve the accuracy by employing more past data. Its accuracy, however, is still limited because only the first-order derivatives of the past data points are used. For linear circuits, the accuracy of numerical integration can be significantly improved by considering high-order derivatives. This section introduces a numerical Laplace inversion based high-order numerical integration method for linear circuits.

60

3.1

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Padé Polynomials

Padé approximates are a special type of rational fraction approximation to a given function. For a given power series

The Padé fraction

where and are polynomials in of orders N and M, respectively, is used to approximate The coefficients of is determined from

Eq.(4.19) indicates that the first M + N + 1 terms of matches those of and the remaining terms are negligibly small. Assume

where M > N. Substituting these results into (4.19) yields

Making use of the identity that a polynomial is zero if and only all the coefficients of the polynomial are identically zero, we arrive at

Numerical Integration of Differential Equations

61

from which the coefficients of the Padé approximates can be determined. They are given by

A special Padé polynomial is that for

Because

Using (4.24), one can show that the Padé polynomial of

is given by

62

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where (M + N), match the corresponding coefficients of the Taylor series expansion of The remaining terms differ. The low-order Padé polynomials of are tabulated in Table 4.1.

3.2

Numerical Laplace Inversion

Laplace inversion derives the time domain response of a circuit from its counterpart

where

and are Laplace transform and inverse Laplace transform operators, respectively. The analytical solution of Laplace inverse transform is available for a set of known functions only. For arbitrary functions, numerical approximation is needed. Since varies from cir-

Numerical Integration of Differential Equations

63

cuits to circuits, it is a natural choice to apply Padé approximation to rather than Let where we arrive at

Next, we approximate

using Padé polynomial

given earlier

It was shown in [5] that under the condition all the poles of are simple and are located in the right half of the complex plane. As a result,

where and are the residues and poles of respectively. The integral in Laplace inverse transform can thus be evaluated using the residue theorem by closing the path of integration along an infinite arc around the poles in the right half plane.

where is the displacement from the time origin. If then

is a real function,

where and Let us examine the properties of the preceding numerical Laplace inversion in detail: Because and where and are Dirac impulse function and unit step function, respectively, the Laplace

64

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

transform of both Dirac impulse function and that of unit step function are well defined in Laplace domain, numerical Laplace inversion thus provides an effective way to handle Dirac impulses and discontinuities that exist in mixed-mode switching circuits and can not be handled by conventional LMS-PC integration methods in the time domain. The computation of the time-domain response involves the frequencydomain evaluation of the network function at frequencies Since is independent of the network and time, the value of and that of do not vary with circuits and can therefore be pre-computed to a very high degree of accuracy and stored. Tables 4.2-4.5 tabulate the value of and that of for { N , M } = {2,4} and {N, M} = {8,10} with 15 digits, the maximum accuracy of most numerical packages at the present time.

The accuracy of numerical Laplace inversion depends upon

Numerical Integration of Differential Equations

65

The order of Padé polynomial. When {N, M} = {8, 10} is used, the corresponding order of integration is M + N +1 = 19, i.e. the first 19 terms of the Taylor series expansion of are considered. The order of integration is hence 18. Numerical Laplace inversion is thus a high-order numerical integration method. Because approximates the Taylor series expansion of at the time origin to minimize the error, the time displacement from the time origin should be kept small. The method does not apply to nonlinear circuits simply because it is generally difficult to obtain the response of nonlinear circuits. In what follows we use several examples to demonstrate the effectiveness and accuracy of numerical Laplace inversion. A. Dirac Impulse Function

Consider Dirac impulse function at and zero at Because is obtained from

The function takes infinite value the time-domain value at

When {N, M} = {2, 4} is employed, because when 15 digits are used, we have This agrees with the theoretical result. We conclude that numerical Laplace inversion yields the correct

66

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

results of Dirac impulse function. B. Unit Step Function

The unit step function is defined as

Its Laplace transform is given by inversion, its time-domain value at tained from

Using numerical Laplace with {N, M} = {2, 4} is ob-

The relative error defined as

with is plotted in Fig.4.3. It is seen that numerical Laplace inversion yields the accurate results of even for large step sizes. The error is in the range of C. Exponentially Decaying Function

The exponentially decaying function, teristics of RC networks. Because

its time-domain response at puted from

represents the charac-

with {N, M} = {2, 4} can be com-

Numerical Integration of Differential Equations

67

The response is computed using Matlab with the maximum accuracy (15 digits) [71]. Fig.4.4 plots the relative error. It is seen that numerical Laplace inversion yields very accurate results of when the step is small. The error increases drastically when the step size is large. This agrees with our early statements that the error of numerical Laplace inversion grows when the displacement from the time origin is large. D. RC Network

Consider the RC network shown in Fig.4.5. Let C = 1F, and the input is a unit step function applied at Assuming zero initial voltage of the capacitor. The exact output voltage is given by

68

Its

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

counterpart is given by

The dependence of the normalized error between the exact solution and that computed from numerical Laplace inversion with {N, M} = {2, 4} on the step size is plotted in Fig.4.6. It is seen that the error decreases monotonically when the size step is increased from to approxi-

Numerical Integration of Differential Equations

69

mately The error, however, increases drastically when the step size is large. Also observed is that an optimal step size exists. The large relative error when is small is mainly due to The large error when is large, however, is due to the large time displacement from the time origin.

3.3

Multi-Step Numerical Laplace Inversion

Because numerical Laplace inversion is based on Padé approximation of in the vicinity of the time origin, its accuracy deteriorates if the time point has a large displacement from the time origin, as observed in the preceding section. Practical applications usually require the time domain behavior of circuits over a long period of time, Eq.(4.31) therefore can not be applied directly. An effective way to reduce the error due to a large time displacement from the time origin is to divide the long time interval into multiple small sub-intervals of equal width such that the results obtained from numer-

70 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

ical Laplace inversion is sufficiently accurate over these small intervals. As long as the time origin is reset to the start of each sub-interval and the effect of the initial conditions of the circuit at the start of the subinterval is accounted for, numerical Laplace inversion will yield accurate results. To illustrate this, let us consider the linear time-invariant circuit depicted by

where G and C are conductance and capacitance matrices formulated using the modified nodal analysis, is the network variable vector, and g is a constant vector specifying the nodes to which the input is connected. Laplace transform of is obtained from

where

The time interval in which we are interested in the behavior of the circuit is first divided into multiple small sub-intervals of equal width The first sub-interval is given by (0, The circuit in this sub-interval is depicted by (4.41). The response of the circuit at is given by

where

Numerical Integration of Differential Equations

71

and

In the second step, we reset the time origin to The input becomes and the initial condition becomes subsequently. The circuit in the second sub-interval is depicted

by

Following the similar steps as those for the first sub-interval, one can show that the response of the circuit at the end of the second sub-interval is given by

Continuing this process, we obtain the response of the circuit at the end of the sub-interval

The preceding algorithm computes the response of linear circuits over a time interval of arbitrary length in a stepping manner, and is hence termed the stepping algorithm. It is seen from (4.49) that is the transition matrix of the circuit as it links the present state to the next state in the absence of the input. on the other hand, quantifies the response of the circuit when the initial state is zero, and is hence termed the zero-state vector. The response of the circuit is completely defined by the transition matrix and the zero-state vector.

72 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

As an example, consider the RC network of Fig.4.5 studied earlier. This time, the stepping algorithm is employed to compute the output voltage of the network. When the initial condition of the capacitor is considered, the output voltage in is given by

As seen in Fig.4.6 that when the step size is a small error in the range of can be achieved. The response of the circuit from to is computed with step size (100 steps). Assuming that in the first step The response of the circuit at is given by

In the second step, the voltage across the capacitor is given by the response is computed from

In the

and

step

The relative error of the response computed using the non-stepping algorithm and that using the stepping algorithm are shown in Fig. 4.7. It is evident that the relative error is significantly reduced at large step sizes when the stepping algorithm is employed. In what follows we examine the properties of the stepping algorithm. Input Waveform Although in the preceding development, an exponential signal was used as the input in the derivation of the stepping algorithm, the input waveform is not restricted to be either sinusoidal or exponential.

Numerical Integration of Differential Equations

73

A given input waveform can be represented by the linear combination of a set of basic functions, possibly infinite, that are linearly independent. The response of the circuits to each of these basis functions is first computed separately and the complete response of the circuit is then obtained by summing up its response to the basis functions. As an illustration, consider a linear circuit. The input waveform is shown in Fig.4.8. The input waveform is approximated with a piecewise-liner input waveform shown in Fig.4.8. The input in the interval

is represented by

where is the unit step function. Note that the time origin of (4.54) is at It can be shown that under the condition the response is obtained from

74

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where

and

are constant vectors for fixed step size The two basis functions in this case are the unit step function defined earlier and the unit ramping function defined as

This approach can be generalized to use an infinite number of basis functions, such as Fourier series and wavelet, to represent arbitrary input waveforms.

Numerical Integration of Differential Equations

75

Efficiency Because and are constant for fixed they only need to be computed once and can be computed to high precision prior to the start of simulation. Only one matrix multiplication and one matrix addition are required in each step of simulation. No costly Newton-Raphson iterations are required. The stepping algorithm is computationally efficient. Accuracy Eq.(4.49) is accurate provided and are computed to high precision. To ensure a high degree of accuracy, and should be computed using numerical Laplace inversion of order {N, M} = {8, 10} and fine step size. The followings give an computationally efficient and yet numerically accurate algorithm that computes these two quantities over a large number of fine steps. To compute to high precision, we first divide into multiple fine steps with equal step size From the definition of given in (4.43), we see that the solution of the circuit

at

gives the first column of by we have

Denote the first column of

76 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Similarly, the second column of from the response of the circuit

at

Other columns of

denoted by

is obtained

can be obtained in a similar manner.

To compute the time origin is reset from to The initial condition becomes and the circuit is depicted by

Note that there is no input in this step because the original input dies out for The solution of the circuit at gives

Similarly we have

Combining these results, we arrive at

Numerical Integration of Differential Equations

77

where

Continuing this process, one can show that

The cost for evaluating (4.66) can be greatly reduced if to be where is an integer. For instance, if be obtain in only ten matrix multiplications. Similarly,

is chosen can

is obtained from

where

The preceding analysis shows that both and can be computed efficiently over a large number of fine steps to achieve high accuracy. Stability To investigate the stability of numerical Laplace inversion, we follow the approach for Euler formulae and the same benchmark equation (4.5) is used. The response of (4.5) at is obtained by using numerical Laplace inversion

78 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

By letting

we have

In the second step, the circuit is depicted by

It can be shown that the response at the end of the second step is given by

Continuing this process, we arrive at

To ensure that (4.73) is bounded for must be met

With

this is equivalent to

the following condition

Numerical Integration of Differential Equations

79

It was shown in [5] that for M – N = 2, if numerical Laplace inversion will yield a stable response. Numerical Laplace inversion is therefore an absolutely stable (A-stable) numerical integration algorithm.

4.

Summary

In this chapter, we have reviewed the linear single-step predictorcorrector algorithms and linear multi-step predictor-corrector algorithms, their imitations in analysis of mixed-mode switching circuits. Numerical Laplace inversion that derives the time domain solution from its response has been introduced. We have shown that Padé approximation based numerical Laplace inversion is a high-order numerical integration method for linear circuits with accuracy orders of magnitude higher as compared with LMS-PC algorithms. In addition, we have shown that this method is capable of handling both impulses and discontinuities in network variables that can not be handled by LMS-PC algorithms. The dependence of the accuracy of numerical Laplace inversion on the time displacement from the time origin has been studied in detail. For a given function, an optimal step size at which the error is the minimal, exists. To improve accuracy, we have shown that the stepping algorithm with the step size set to the optimal step size can provide superior numerical accuracy in computing the time domain response of linear circuits over a time interval of arbitrary length.

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II

TIME DOMAIN ANALYSIS

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Chapter 5 INCONSISTENT INITIAL CONDITIONS

Inconsistent initial conditions arise from ideal switching and cause the value of network variables immediately before switching to differ from that immediately after switching. Because the topology of switching circuits may change at switching instants, the value of network variables immediately before switching, denoted by can not be used to compute the response of circuits after switching. Instead, the value of the network variables immediately after switching, denoted by must be used to continue integration. This chapter investigates computer methods that compute from Section 1 investigates the cause of the inconsistent initial conditions of mixed-mode switching circuits. Section 2 examines the two-step algorithm derived from numerical Laplace inversion and its applications. We show that the two-step algorithm is an accurate and computationally efficient algorithm that yields from of linear circuits. Section 3 shows that the conventional backward Euler formula is capable of yielding from This approach is applicable to mixedmode switching circuits with both linear and nonlinear elements. In addition, it is capable of computing the area of impulses at switching instants. Section 4 presents a Taylor series based approach for deriving from In Section 5, we show that the inconsistent initial conditions encountered in nonlinear circuits can also be handled using Volterra functional series based methods. Section 6 is concerned with

84

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

the detection of the existence of impulses at switching instants. The determination of the existence of impulses at switching instants is critical to the analysis of circuits with internally controlled switches as impulses generated at switching instants are usually high enough to change the states of other switches in the circuits. The chapter is summarized in Section 7.

1.

Inconsistent Initial Conditions

It was shown in Chapter 2 that the full-transistor model is capable of capturing the rapidly time-varying transient characteristics of circuits. Simulation based on the full-transistor model, however, is costly. The use of this model is warranted only if the transient behavior of the circuit in the vicinity of switching instants is critically needed. Analysis based on the voltage-modulated resistor model, on the other hand, is much less expensive computationally. The large ratio of the ON and OFF resistances of the voltage-modulated resistor model, however, gives rise to stiff systems that have ill-conditioned circuit matrices and require excessive simulation time. This drawback can be eliminated if the ideal switch model that has zero resistance in the ON state and infinite resistance in the OFF state is employed. Ideal switching, however, may cause an abrupt change in nodal voltages due to impulsive capacitor currents, as shown in Fig.2.3. Similarly when inductors are encountered, ideal switching may give rise to an abrupt change in loop currents, resulting in impulsive inductor voltages. As a result, not only the value of branch currents and nodal voltages immediately before switching may differ from that immediately after switching, impulses are also generated at switching instants. Difficulties arising from ideal switching include : i) Impulses and discontinuities encountered at switching instants can not be handled by conventional linear multi-step predictor-corrector algorithms as these algorithms usually require the continuity of network variables. ii) Because switched circuits change their topology at switching instants, and (immediately before switching) can not be used to

Inconsistent Initial Conditions

continue integration after switching. Instead, diately after switching) must be used.

85

or

(imme-

iii) Impulses generated at switching instants may trigger the switching of other switches that are controlled by internal variables of the circuits. Numerical algorithms that compute from and the detection of the existence of impulses at switching instants are the focuses of this chapter.

2.

Numerical Laplace Inversion Based Two-Step Algorithm

It was shown in Chapter 4 that numerical Laplace inversion is a highorder numerical integration method that yields the accurate time domain response of linear circuits. An important advantage of this approach is that it is capable of handling both the impulses and discontinuities that can not be handled by conventional numerical integration methods in the time domain.

To illustrate this, consider the circuit shown in Fig.5.1. The output voltage of the circuit is given by

Its time-domain response is given by

86 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

It is seen that the response contains both a Dirac impulse function and an exponentially decaying function. Also Its time-domain response at using {N, M} = {2, 4} is obtained from

If we evaluate (5.3) by opening the brackets

then with 15 digits, the first term of (5.4) vanishes and the second term is identical to (4.38) except the negative sign. The relative error between the analytical results and those from numerical Laplace inversion is the curve-b in Fig.5.2, which is identical to Fig.4.4. However, if (5.3) is evaluated without separating the terms associated with the Dirac impulse function and those associated with exponentially decaying function, the relative error is the curve-a in Fig.5.2. It is observed that the relative error is much higher in this case, as compared with curve-b. Because for arbitrary networks, it is general not trivial to separate the terms associated with the Dirac impulse function and those without, the above results reveal that the existence of a Dirac impulse will significantly increase the error of numerical Laplace inversion when the time step is small. Because a Dirac impulse may exist at switching instants in mixed-mode switching circuits, special algorithms are needed to minimize the error in computing from Also observed is that the relative error is large when the step size is small and decreases monotonically with the increase in the step size until a minimum point is reached. The error then increases rapidly if the step size is further increased. This observation suggests that to achieve high accuracy in the presence of impulses, the step size should not be too large, nor too

Inconsistent Initial Conditions

87

small. The use of large step size although avoiding the effect of Dirac impulses at only yields a poor approximation of simply because the time point is distance away from the switching instant It was shown in [47] that this difficulty can be overcome by first taking a large forward step from to where As long as is sufficiently large, the effect of Dirac impulse at dies out and the result obtained at has good accuracy (in this example, ). Note that the function must be continuous at This ensures Second, a backward step of the same size is taken from to the circuit in this case has no input nor its response has a Dirac impulse. The capacitor, however, carries the initial voltage as shown in Fig.5.1. The output voltage of the circuit in this step is given by

88

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

and the response at the end of the backward step is obtained from

The normalized error between the analytical result and that computed from numerical Laplace inversion using the two-step method is plotted in Fig. 5.3. It is seen that a minimum error exists. For example, when the normalized error is in the range of This result demonstrates that for this particular circuit, if is used, the relative error between and is in the range of

To demonstrate the effectiveness of the preceding two-step algorithm, consider the circuit shown in Fig.5.4. Let the input be

It can be shown that the output voltage is given by

Inconsistent Initial Conditions

89

and

The response contains a Dirac impulse at We have and The step size and {N, M} = {2, 4} are used to compute the consistent initial condition. In the forward step, the transfer function is given by

In the backward step, the time origin is reset to The Dirac impulse in the input has died out in the forward step and the input of the circuit in the backward step becomes Also, the capacitor at the onset of the backward step carries the initial voltage calculated from

The is given by

expression of the output voltage in the backward step

It is evident that we arrive

With the step size

90 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

It is seen that the two-step algorithm yields very accuracy result with the relative difference

The preceding example reveals an important characteristic of mixedmode switching circuits at switching instants, that is the expression of network variables used in the forward step may differ from that used in the backward step. The reasons for this are as follows : i) The input of circuits immediately before switching may differ from that immediately after switching. This is particularly true if the input contains impulses at switching instants and step functions with the step transition at switching instants. ii) Impulsive network variables may exist at switching instants. Dirac impulses make their appearance in the forward step only and die out at the end of the forward step.

iii) The initial condition of circuit elements with memory at the onset of the backward step may differ from that of these element at the onset of the forward step.

3.

Backward Euler Based Algorithms

The preceding numerical Laplace inversion based two-step algorithm is most effective for linear circuits, and can not be applied to nonlinear

Inconsistent Initial Conditions

91

circuits simply because it is generally difficult to obtain the response of these circuits. Nonlinear circuits are most often analyzed using LMS-PC algorithms in the time domain, among which LSS-PC algorithm, i.e. forward Euler as the predictor and backward Euler as the corrector, is widely used. Although LMS-PC algorithms require the continuity of network variables, it was demonstrated in [48] that the backward Euler

is capable of handling Dirac impulses, subsequently inconsistent initial conditions encountered at switching instants. To illustrate this, consider step function It represents the discontinuity and its first-order derivative gives the Dirac impulse function

Using the backward Euler, the derivatives of unit step function is obtained from

Note that the first-order derivative was obtained from

Eq.(5.14) reveals that the derivatives of can be computed correctly using backward Euler despite of the discontinuity at Applying backward Euler to the linear circuit depicted by

where G and C are the conductance and capacitance matrices respectively and is the input vector, we obtain

92

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

In what follows we use the circuit shown in Fig.5.5 to demonstrate the usefulness of (5.17) in handling impulses encountered at switching instants. The switch was initially at position A and changes its position from node A to node B at As a result, inductor carries an initial current of 2A whereas inductor is initially at rest, i.e. and also, The circuit in is depicted by

from which we obtain the time-domain response

It is seen that both at

and

and

contain a Dirac impulse of strength

on the other hand, are impulse-free. Also

observed is that and are discontinuous at

3.1

and

Two-Forward-Step Algorithm

Because no input to the circuit for we have step immediately after Eq.(5.17) becomes

In the

Inconsistent Initial Conditions

93

Its solution is given by

In the second step, and carry the initial currents respectively. Eq.(5.17) becomes

The solution is given by

and

94 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Continuing this process, one can show that

Let us examine these results in detail prior to further development : The impulses of and encountered at only affect and as when They die out after the first step and have no effect on the subsequent steps. For small and can be quite large. If the circuit contained switches that are internally controlled, such as diodes, they could be activated and alter the topology of the circuits. Detection of the existence of impulses at switching instants is therefore critical. and provide an approximation of the area of the impulses. The error of such an approximation can be determined from

These products can be used to detect the existence of impulses. If when no impulse exists at Otherwise, an

Inconsistent Initial Conditions

95

impulse exists. This observation also suggests that the response of circuits at the end of the step immediately after switching can be written in the following general form

where

is a constant, quantifying the strength of the impulse, and is an impulse-free function, i.e. when The subscripts “ici” specifies “inconsistent initial conditions” whereas “ci” represents “consistent initial conditions”. From the analytical results, we have Clearly, and and as well.

and

and

can not be used to obtain and on the other hand, provide the correct value of when The same holds for and

If and are used to approximate respectively, the error can be determined as follows :

Since

we have

and

96

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where we have neglected higher order terms because that the relative error defined as

is given by and and respectively, we have

It is seen

are used to approximate

The relative error is quantified by The preceding analysis shows that the use of more forward steps does not warrant an improvement in the accuracy. In fact the accuracy is reduced if more forward steps are employed. Also observed is that the relative error is linearly proportional to the step size

3.2

Two-Step Algorithm

To improve the accuracy, a two-step algorithm similar to that using numerical Laplace inversion for switched linear circuits was proposed in [48]. In this approach, a forward step from to is taken first. It is then followed by a backward step of the same step size. Re-writing (5.22) with step size gives

Inconsistent Initial Conditions

97

The solutions of (5.32) are given by

Because

It is seen that the relative error of the two-step algorithm is given by which is proportional to Since usually the two-step algorithm with one step forward and one equal step backward provides better accuracy as compared with the preceding algorithm with two equal forward steps.

3.3

Four-Step Algorithm

To investigate whether the use of more steps can lead to an improvement in accuracy, a four-step approach in which two forward steps of equal step size are taken first, followed by two backward steps of the equal step size, is examined in this section. Using (5.23) as the initial conditions of the first backward step, we have

The responses of (5.35) are given by

98

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Further taking another equal step backward with (5.36) as the initial conditions yields

The error is obtained from

The relative error is

As compared with the error of the two-step

algorithm presented earlier, which is given by accuracy is achieved.

3.4

no improvement in

Two-Step Algorithm for Linear Circuits

The above two-step algorithm is applicable to both linear and nonlinear circuits. For linear circuits, the consistent initial conditions can be obtained as follows. Assume that switching occurs at and the initial condition of the circuit is given by In the forward step, Eq.(5.17) becomes

Inconsistent Initial Conditions

99

In the backward step, the initial condition vector is given by

Substituting (5.39) into (5.40) gives

where

For a fixed and are constant and can be computed in a pre-processing step before the start of simulation. can therefore be computed from directly.

4.

Taylor Series Based Algorithm

It was shown in the preceding section that when inconsistent initial conditions are encountered at switching instant Dirac impulses make their appearance only in the step immediately after switching and do not affect the response in steps thereafter. This observation suggests that the response of the circuit in the step immediately after switching can be written in the following form [48]

The first term in (5.43) represents the inconsistent initial conditions at whereas the second term represents the consistent initial condi-

100 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

tions. In the second step, the effect of the Dirac impulse vanishes and only the impulse-free component remains.

Because is continuous and impulse-free, we expand at in its Taylor series with the time displacement and truncate the series to the first order

Similarly,

can then be obtained from by solving (5.43), (5.45), and (5.46) simultaneously. Once they are available, can be obtained by taking a backward step from where

Apply this method to the circuit in the preceding section, we have

The solutions of (5.48) are given by

Inconsistent Initial Conditions

101

Substituting these results into (5.47) yields

The error is obtained by expending (5.50) with

The relative error given by is higher as compared with that of the two-step algorithm presented earlier. Also, in comparison with the two-step method, this approach has the drawbacks of the high cost of computation due to the need for and The relative errors of the two forward-step, two-step, four-step, and Taylor series based methods in computing are tabulated in Table 4 and plotted in Fig.5.6. It is seen that the relative error of these algorithms increases monotonically with the increase of the step size.

102 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

The two-step algorithm with one forward step and one backward step provides the best accuracy.

5.

Volterra Functional Series Based Algorithm

It is well known that Volterra functional series is an effective means for the analysis of weakly nonlinear circuits in both the time and frequency domain. It was shown in [20] that the response of a weakly nonlinear circuit can be obtained by summing up the response of its Volterra circuits

where is the response of the Volterra circuit. Writing (5.52) at the time instant immediately before switching and after switching give

Inconsistent Initial Conditions

103

and

Eqs.(5.53) and (5.54) reveal that the consistent initial condition of a periodically switched nonlinear circuit can be obtained from that of its Volterra circuits. An important advantage of this approach is that because Volterra circuits are linear, the numerical Laplace inversion based two-step algorithm can be employed to yield accurate from We will come back to this in Chapter 7 in detail when we deal with the analysis of periodically switched nonlinear circuits.

6.

Existence of Dirac Impulses at Switching Instants

The strength of impulsive voltage or currents generated by the switching of one switch may be quite large. Such a large current or voltage may trigger other switches in the circuits that are controlled internally. It is therefore critical to detect whether a Dirac impulse exists at every switching instant in analysis of circuits with internally controlled switches. It was shown earlier that the response of circuits in the step immediately after switching can be decomposed into a consistent initial condition component, denoted by which is impulse-free, and an inconsistent initial condition component, denoted by mathematically

Because

is continuous, we have

104 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Also because

we arrive at

These results indicate that the existence of the impulsive component at switching instants can be sensed by monitoring the integral

The integral is nonzero if contains a Dirac impulse at and zero, otherwise. In what follows we show how (5.59) is evaluated numerically.

6.1

Dirac Impulses in Linear Circuits

For linear circuits, the integration of (5.59) can be evaluated conveniently using numerical Laplace inversion. Notice that

we have

is thus obtained by first integrating then from to

from

to

and

Inconsistent Initial Conditions

105

It should, however, be noted that in the time interval usually differ from that in the time interval due to the following reasons : Dirac impulses encountered at exist in the time interval only and die out at the end of this time interval. In other words, there will be no Dirac impulses in the backward step. Circuit elements with memory at the onset of the backward step carry initial conditions (capacitors have initial voltages and inductors have initial currents). These initial conditions make their appearance in in the backward step. In the forward step, we have the step size

where the subscript specifies that the expression of is for the network in the forward step, i.e. only. In the backward step, we have the step size and the expression of is given by

As an example, consider the circuit in Fig.5.4. We have shown in Chapter 5 that the response of the circuit in the forward step is given by

Choose

we have

106 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

In the backward step, the Dirac impulse in the input has died out. Also, the capacitor carried a initial voltage of The expression of the output voltage becomes

The integration in the backward step is thus computed from

The area of the impulse is then obtained from

The relative difference between the exact value of the area of the impulse, which is -5, and that computed from the preceding steps is only

6.2

Dirac Impulses in Nonlinear Circuits

For nonlinear circuits, the simplest approach to integrate over the switching instant is the backward Euler formula. As pointed out in the preceding section that Dirac impulses encountered at are represented by a rectangular pulse of width and height Eq. (5.55) can be written as

The integration over the step immediately after switching is obtained from

If the result is zero when no impulse exists. Otherwise, an impulse exists and the area of the impulse is quantified by

Inconsistent Initial Conditions

7.

107

Summary

We have shown in this chapter that due to ideal switching, and may differ, leading to inconsistent initial conditions and the creation of impulses at switching instants. Because the topology of circuits changes before and after switching, instead of must be used to continue integration after switching. Several computer methods that compute from have been examined in detail in this chapter. We have shown that for linear circuits, can be computed efficiently using the numerical Laplace inversion based two-step algorithm to achieve a high degree of accuracy. The accuracy of this method depends upon the step size used in the integration. Both too small and too large steps should be avoided. To handle the inconsistent initial conditions of nonlinear circuits, numerical Laplace inversion based approach is ineffective because it is difficult to obtain the response of these circuits. Backward Euler formula, on the other hand, yields the correct consistent initial conditions for both linear and nonlinear circuits immediately after switching. The initial conditions can be obtained in two consecutive forward steps from switching instants, one forward step and one backward step of equal step size. The latter provides better accuracy. We have also shown that two forward step, followed by two backward steps of identical step size, through yield the correct consistent initial conditions, does not lead to an improvement in accuracy. The Taylor series based approach also fails to offer better accuracy. The consistent initial conditions can also be obtained using Volterra functional series based approach where the consistent initial conditions are obtained from those of corresponding Volterra circuits that are linear. An advantage of this approach is that numerical Laplace inversion based two-step algorithm can be used for better accuracy. In analysis of circuits with internally controlled switches, because the state of these switches is determined by the network variables associated with the switches and because the strength of the impulses generated at switching instants can be quite large, the switching of one switch may activate the switching of other switches in the circuits. This process continues until no switching occurs. The detection of impulses at

108 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

switching becomes critical to the analysis of circuits with internally controlled switches. The numerical methods for the detection of impulses in both linear and nonlinear circuits have been presented.

Chapter 6 SAMPLED-DATA SIMULATION OF PERIODICALLY SWITCHED LINEAR CIRCUITS

This chapter introduces an efficient and accurate numerical integration algorithm called sampled-data simulation for the time domain analysis of periodically switched linear circuits. Section 1 develops this algorithm using Laplace transform and its inverse. Section 2 is concerned with inconsistent initial conditions encountered in periodically switched linear circuits and the algorithms that yield consistent initial conditions of these circuits. Section 3 analyzes the sensitivity of the response of periodically switched linear circuits. The inconsistent initial conditions of sensitivity networks are addressed in Section 4. In section 5, a computationally efficient statistical analysis method for computing the mean and variance of the response of periodically switched linear circuits is presented. Section 6 addresses the noise analysis of periodically switched linear circuits in the time domain. We show that by modeling white noise using random pulses with the pulse width set by the noise bandwidth of the circuit to be analyzed and the amplitude set by the output noise power of the circuits due to the noise sources, the time domain response of periodically switched linear circuits to both signals and noise sources can be computed efficiently using sampled-data simulation. Section 7 analyzes the effect of clock jitter on the response of periodically switched linear circuits. By assuming that clock jitter is much smaller than clock period, the response of periodically switched linear circuits in

110 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

the presence of clock jitter is computed efficiently. The chapter concludes in Section 8.

1.

Sampled-Data Simulation of Periodically Switched Linear Circuits

Periodically switched linear circuits can be analyzed using LMS-PC algorithms introduced in Chapter 4. The inconsistent initial conditions encountered at switching instants can also be handled using BackwardEuler based methods presented in Chapter 4. A major drawback of these algorithms is their limited accuracy. High accuracy can be obtained at the cost of small step size, subsequently long simulation time. In this section, we introduce an accurate and computationally efficient method for time domain analysis of periodically switched linear circuits. The essential part of this section was originally reported in [18]. Consider a periodically switched linear circuit with the input The circuit in phase is formulated using modified nodal analysis

where is the network variable vector in phase and are the conductance and capacitance matrices in phase respectively, and is a constant vector specifying the nodes to which the input is connected in phase Laplace transform of (6.1) gives

where and is the Laplace transform of The time-domain response is obtained from the inverse Laplace transform of (6.2)

where

Sampled-Data Simulation of Periodically Switched Linear Circuits

111

and

Observed from (6.3) that : is the state transition matrix quantifying the response of the circuit from the initial state to the present state when the input is removed. The first term on the right hand side of (6.3) is hence the zero-input response of the circuit

is the zero-state vector specifying the response of the circuit to the input with zero initial conditions. The second term on the right hand side of (6.3) is thus the zero-state response of the circuit

is independent of the input. For a given circuit topology, it needs to be computed once only. is both topology and input-dependent. It must be re-computed each time the input varies. Both and are functions of the time displacement from the time origin. If the time displacement changes, both and must be re-computed. Without the loss of generality, let the input be (6.3) for where T is the time step, yields

Writing

112 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

In the where the time origin is shifted from to Subsequently, the input is changed to The initial condition of the circuit becomes The circuit in this step is depicted by

Following the same procedures as those for

we arrive at

Let us examine the properties of (6.11): If the frequency of the input and the step size T are kept unchanged, and are constant. They only need to be computed once and can be computed in a pre-processing step prior to the start of simulation. The computation required in each step is only one matrix-vector multiplication and one vector addition. The response of the circuit is obtained at time points of the fixed time interval T efficiently. The method is henceforth referred to as sampleddata simulation. The method is exact. No approximation is made in derivation of the method. In order to obtain the accurate time domain response of circuits numerically, both and must be computed to high precision. In Chapter 4 we showed that a high degree of accuracy can be achieved in computing and if the multi-step numerical Laplace inversion algorithm is employed. If the circuit has multiple exponential inputs with different frequencies

Sampled-Data Simulation of Periodically Switched Linear Circuits

113

the response of the circuit can be obtained using the principle of superposition. Note that the zero-input response remains unchanged as it is independent of inputs. The zero-state response, however, becomes

where is the zero-state vector to the input is obtained from

and

where is a constant vector specifying the connection of the input source to the circuit and

The complete response is given by

A special case of interest is when corresponding to a unit step input. The response of the circuit to the unit step input is computed from

Sinusoidal inputs can also be handled conveniently. For example, if the response is given by

114 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Similarly, if

we have

The input waveform is not restricted to be either sinusoidal or exponential. A given input waveform can be represented by a set of elementary basis functions. The response of circuits to each of these basis functions is first computed separately. The complete response of the circuit is then obtained by summing up its response to each of these basis functions using the principle of superposition.

As an example, consider a linear circuit with its input approximated by the piecewise-linear waveform with step as shown in Fig.6.1. The input in the time interval is given by

where

Sampled-Data Simulation of Periodically Switched Linear Circuits

115

and is the unit step function. Note that should be chosen in such a way that Nyquist theorem is not violated in representation of using The time origin is reset to and the circuit in the time interval is depicted by

Following the same procedures as those for circuits with an exponential input, one can show that the response of the circuit is given by [49]

where

and

For fixed step size T, both and are constant and need to be computed only once. In Appendix 6.A of this chapter, the algorithms for computing and are given. Also note that T should be chosen no larger than

116 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

2.

Inconsistent Initial Conditions

It was shown in Chapter 2 that if the initial transient portion of the response of mixed-mode switching circuits is not of a critical concern, it is advantageous computationally to model switches as an ideal device. Ideal switching, however, may cause an abrupt change in nodal voltages or branch currents, giving rise to inconsistent initial conditions. In Chapter 5, several numerical algorithms for computing from of both linear and nonlinear switching circuits were examined in detail. For linear circuits, the two-step method that is based on numerical Laplace inversion not only yields the correct consistent initial conditions at switching instants, it also provides better accuracy as compared with those that are based on backward Euler formula. In this section we show that by incorporating the numerical Laplace inversion based twostep algorithm into the preceding sampled-data simulation algorithm for periodically switched linear circuits, the inconsistent initial conditions of these circuits can be handled effectively [50].

Sampled-Data Simulation of Periodically Switched Linear Circuits

117

Assume a switching occurs at and the network variables of the circuit immediately before switching is given by To yield from a forward step from to is taken first, as shown graphically in Fig.6.2. The response at is obtained from

The step size T should be chosen such that no switching at i.e. Step size different from that of other steps can be used in this step to ensure the continuity of at Following the forward step, the time origin is reset to The initial condition is subsequently given by and the input becomes A backward step of the same step size from to is then taken to compute Because Laplace transform is defined from to only, to accommodate the backward step, the transform

is employed. In the the time origin is the input becomes The circuit in the is depicted by

and during

Using Laplace transform in the backward step, we obtain the response at the time instant immediately after switching

where

118 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

is the transition matrix for the backward step and

is the zero-state vector for the backward step. Similar to and if the step size T is fixed, ˆ and are constant and need to be computed only once for a given network topology. Once is available, integration continues from in a sampleddata manner. As an example, consider the circuit shown in Fig.6.3 with and The input is a unit step current source. The clock frequency is 5 Hz. The circuit is constructed in such a way that no floating node exists in either clock phases. Both and are modeled as ideal switches. The voltage at node 3 is computed using the sampled-data simulation and is plotted in Fig.6.4, together with the response from analytical analysis.

Sampled-Data Simulation of Periodically Switched Linear Circuits

3.

119

Time-Domain Sensitivity

The need for quantifying the effect of the variation of the value of circuit parameters on the performance of the circuits is signified by the trend that the minimum feature size of MOS devices in modern CMOS technologies is being scaled down much more aggressively as compared with the improvement in process tolerance [51]. For example, the resistance of poly resistors in a typical CMOS process has an error of ±20% approximately and that of n-well resistors has an error of ±30% approximately. Analysis of these effects is vital to both the performance of circuits and the yield of production. As compared with costly statistical analysis, such as Monte Carlo analysis, sensitivity analysis provides a deterministic and effective measure of the effect of the variation of one circuit parameter on the performance of the circuits.

120 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

The time domain sensitivity of the response of a periodically switched linear circuit with respect to a circuit parameter is defined as

The normalized sensitivity defined as

removes the impact of the value of the parameter to which sensitivity is calculated on the sensitivity. The sensitivity to parasitic parameters whose value is usually extremely small can also be computed from

To obtain the time-domain sensitivity of the response of periodically switched linear circuits to a circuit parameter we differentiate (6.1) with respect to

Comparing the circuit depicted by (6.1) and that delineated by the above expression one can see that both circuits are linear and have the same topology but distinct inputs. The unknown of the circuit depicted by the above equation is sensitivity and the circuit is hence called sensitivity network of For each circuit parameter, there is a corresponding sensitivity network. The sensitivity networks have the identical topology but distinct inputs. Also, these sensitivity networks are periodically switched linear circuits with the same clock frequency as that of the original circuit. The time domain response of the sensitivity network is obtained by differentiating (6.11) with respect to

Sampled-Data Simulation of Periodically Switched Linear Circuits

121

The algorithm of (6.34) has the following properties: For fixed T, and are constant. They can be computed in a pre-processing step prior to the start of simulation, the sensitivity of the circuit at time points of a fixed time interval can be computed efficiently in two matrix-vector multiplications and three vector additions. The algorithms that compute are given in Appendix 6.B of this chapter. To compute

and

must be available a priori.

No approximation is made in the derivation of the method. The method is thus exact. To yield accurate sensitivity numerically, and their derivatives to circuit parameters must be computed to high precision. This is accomplished by using the multistep numerical Laplace inversion, as detailed in Appendix 6.B of this chapter. The algorithm computes the sensitivity to a single element in one analysis of sensitivity network. For different circuit elements, and must be re-computed. The method becomes costly if sensitivities to a large number of circuit parameters are needed.

4.

Inconsistent Initial Conditions of Sensitivity Networks

Similar to the inconsistent initial conditions encountered in analysis of the time domain response of periodically switched linear networks, discontinuity may occur at switching instants of sensitivity networks. We need to compute from Since sensitivity networks are periodically switched linear circuits, the two-step algorithm effective for computing the consistent initial conditions of periodically switched linear circuits is applicable. In what follows, we detail this algorithm. To compute from a forward step of step size T is taken from is computed using (6.34) in the forward step.

122 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

The step size T should be such that is continuous at i.e. In the backward step, the sensitivity at the time instant immediately after switching can be solved by differentiating (6.29) with respect to

where

and

Similar to

and

for fixed T,

and are constant and need to be computed only once. They can be computed along with in a pre-processing step prior to the start of simulation. Once these matrices and vectors are known, the time domain sensitivity at time points of an equal interval can be obtained efficiently.

5.

5.1

Statistical Analysis Introduction

Sensitivity analysis is not capable of analyzing the joint effect of the variation of multiple circuit parameters, especially when correlation exists. In addition, the results obtained are only valid at the nominal value of circuit parameters and may not be valid at other values of

Sampled-Data Simulation of Periodically Switched Linear Circuits

123

these random parameters. Worst-case analysis determines the boundary of the design objective using the so-called corner technique. Due to the over-estimation nature of the method, the results obtained usually lead to expensive over-designs [52]. Monte Carlo analysis, available in most commercial CAD tools for circuit design, yields a converging results only if the number of samples, each obtained from one circuit analysis, is large. The effectiveness of these methods is often quickly offset by the excessive computation, especially for periodically switched circuits where the cost of computation of each circuit analysis is high. Most recently, interval analysis has been applied to statistical analysis of linear analog circuits [53]. Its application to statistical analysis of time-varying circuits, specifically periodically switched linear circuits, however, is yet to be exploited. In practice, to quantify the effect of random variation of circuit parameters, designers often rely on costly multiple runs of SPICE-type simulators to determine the upper and lower bounds of circuit parameters, resulting in exceedingly long design cycles. In this section, we introduce an explicit and computationally efficient non-Monte Carlo method for statistical analysis of periodically switched linear circuits in the time domain. The method computes the first and second moments of the time domain response of periodically switched linear circuits with normally distributed circuit parameters at time points of an equal interval.

5.2

First-Order Second-Moment Method

In analysis of the mean and variance of the response of circuits, it is often assumed that the value of circuit elements is normally distributed with the mean to be the nominal value of the elements. When the coefficient of variance of the circuit parameters, defined as the ratio of the standard deviation of the circuit parameters to the mean of the parameters, is small, the response of the circuit at denoted by where x is the vector containing all random variables of the circuits, can be approximated by only considering the first and secondorder terms of its Taylor series expansion at the mean of x

124 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Where x, is normally distributed with and are the gradient and Hessian of at respectively, and denote the real and complex domains, respectively, and M is the number of random variables in the circuit. The mean of is obtained by taking expectation of (6.39) and neglecting the moments whose order is higher than two

where is the covariance of response, denoted by is obtained form

The variance of the

Substituting (6.40) into (6.41) and neglecting the moments whose order exceeds 2 give

Eqs.(6.40) and (6.42) demonstrate that when up to the second-order moments are considered, and are linear functions of the covariance of the circuit parameters. The method is hence referred to as the first-order second-moment (FOSM) method. It should be noted that in derivation of FOSM method, the objective function is expended at the mean of x. Recent study shows that approximation can also be

Sampled-Data Simulation of Periodically Switched Linear Circuits

125

made at the boundary of the design objective and yields the advanced first-order second-moment (AFOSM) method [54]. The accuracy of the method depends upon (i) the accuracy in computing the derivatives of the response with respect to circuit parameters and (ii) the accuracy of FOSM method. It was shown earlier that computation of the derivatives of the response to circuit parameters can be made very accurate if the multi-step numerical Laplace inversion algorithms are employed. The accuracy of the method is thereby mainly determined by that of FOSM method, which is determined by both the coefficient of variance of circuit parameters, the degree of their correlation, and the characteristics of the circuits. Because only up to the second-order moments were considered in the derivation of FOSM, the error of FOSM method will rise if or equivalently the coefficient of variance of the circuit parameters are large.

6.

Noise Analysis

Noise considered here is the small fluctuation of currents or voltages that are generated within devices themselves. Noise coupled externally, such as the switching noise generated by digital circuits and electromagnetic interference, is excluded. Noise encountered in integrated circuits typically includes thermal noise, shot noise, and flicker noise among which thermal noise and shot noise are white in nature. They have constant power spectral density over a wide frequency range. The power spectral density of flicker noise, however, is inversely proportional to frequency. The corner frequency of the flicker noise of MOS devices is in the range of several MHz [2, 1]. Due to the under-sampling of white noise sources by and the large bandwidth of periodically switched linear circuits, the total output noise of these circuits is usually dominated by the noise power folded over from the sideband components of the white noise sources in the circuits [55]. For this reason, in what follows only white noise is considered. The two key issues encountered in time domain noise analysis are (i) the generation of random noise signals in the time domain and (ii) the accuracy of numerical integration methods used for solving circuits with white noise sources.

126 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

6.1

Modeling of White Noise

There are two widely used methods for representing white noise in the time domain. The first approach models a white noise source using a set of sinusoids. It was shown in [56] that a stationary random process in can be represented by

where

and

Note that is a random variable. The usefulness of this approach in practice, however, is rather limited, mainly due to the high cost of computation because the circuit needs to be solved at frequencies in order to yield the output noise power of the circuit.

The second approach represents a white noise source with a train of pulses with the pulse width set by the noise bandwidth of the circuit where

is the equivalent noise bandwidth of the circuits

[19], and the pulse amplitude A set by where N(0,1) is a normally distributed random number generator with zero mean and unit standard deviation, and is the total output noise power of the

Sampled-Data Simulation of Periodically Switched Linear Circuits

circuit due to the noise source, as shown graphically in Fig.6.5. obtained from

127

is

where is the power spectral density of the noise source and is the equivalent noise bandwidth of the circuit. For white noise whose power spectral sensitivity is independent of frequency, the equivalent noise bandwidth, as shown in Fig.6.6, is determined from [2],

where is the transfer function from the white noise source to the output of the circuit. For circuits whose frequency behavior is characterized using the single-pole system with the pole at

we have

128 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

As an illustration, consider a resistor R with one-sided power spectral density where is Boltzmann constant and T is the absolute temperature in degrees Kelvin. Let the equivalent noise bandwidth of the circuit in which R resides be The step size used in representation of the thermal noise of the resistor in the timedomain is given by Nyquist theorem

seconds and the

total noise power of the resistor is calculated from The time-domain noise waveform of the resistor is shown in Fig.6.7.

The power spectral density of the thermal noise of the resistor is obtained by performing FFT on its time domain wave form. The spectrum of the thermal noise of the resistor with 512 samples is shown in Fig.6.8, the interval between two adjacent frequency samples can be calculated from the noise waveform has a mean of The averaged power is

The average of ten sets of data of and standard deviation of For reliable simulation, the

Sampled-Data Simulation of Periodically Switched Linear Circuits

129

period of the pseudo-random number generator should be much larger than simulation time, usually 10 times the simulation time.

6.2

The Algorithm

Numerical integration methods whose accuracy is sufficiently higher than the amplitude of noise signals are needed for time domain noise analysis. Conventional numerical integration methods that are based on LMS-PC fail to provide the needed accuracy for noise analysis of periodically switched linear circuits. It was shown earlier that sampleddata simulation algorithm is an efficient and exact algorithm for time domain analysis of linear circuits. In this section, we make use of this algorithm to analyze the noise of periodically switched linear circuits. Consider a periodically switched linear circuit with an input and a set of white noise sources In the time interval in phase the time origin is set to The circuit in the time interval is depicted by

130 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where and are constant vectors specifying the nodes of the input and those of the noise source in phase respectively. The time domain response is obtained from

where

and

An efficient algorithm for computing is given in Appendix 6.A of this chapter. At the boundary of adjacent clock phases, the twostep algorithm is employed to compute from The spectrum of the output noise of the circuit is obtained in a post FFT analysis on a set of time-domain response data of the circuit in the steady state.

6.3

Examples

Consider the switched capacitor integrator shown in Fig.6.9. The clock frequency is 10 kHz. The clock has four equal phases per clock period. The MOS switches are modeled as a resistor in series with an ideal switch. The operational amplifier is modeled using the macromodel [5] with 700 kHz unit-gain frequency. The noise of the operational

Sampled-Data Simulation of Periodically Switched Linear Circuits

131

amplifiers is represented by the thermal noise of an equivalent resistor at the non-inverting input terminal of the operational amplifier with the resistance The output noise power spectrum is obtained using the method presented in this section and the results are shown in Fig. 6.10, together with the measurement results extracted from [57]. An excellent agreement is obtained.

7.

Clock Jitter

Clock jitter causes clock period to deviate from its nominal value to in a random manner, where is a random variable with For practical circuits, usually holds. To analyze the effect of clock jitter on the response of periodically switched linear circuits, a change in the clock period is represented by a random variation in the step size T of the sampled-data simulation, as shown in Fig.6.11. Note that

where M is the number of steps that sampled-data simulation algorithm takes in one clock period. The step size is changed from T to for the step, where and Subsequently, both

132

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

and are changed to and respectively. Because and can be approximated from their Taylor series expansions of a finite terms

and

The algorithms for computing the time derivatives of and are given in Appendix 6.C of this chapter. For small clock jitter, only the first-order derivatives are usually needed. Note that for fixed T, the

Sampled-Data Simulation of Periodically Switched Linear Circuits

133

derivatives need to be computed only once. The truncation errors can be easily estimated from

and

where Once these derivatives are available, the response of the circuit in the presence of clock jitter is obtained from

In simulation,

is generated from

134 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where is the standard deviation of the clock jitter and N(0,1) is a normal random number generator with zero mean and unity standard deviation.

8.

Summary

Sampled-data simulation, an efficient, accurate, and absolutely stable time domain algorithm for periodically switched linear circuits has been developed in this chapter. We have shown that this algorithm computes response, sensitivity, mean and variance of the response, output noise power in the presence of white noise sources, and the effect of clock jitter of periodically switched linear circuits at time points of a fixed time interval. In computation of the time domain response, only one matrix multiplication and one vector addition are needed in each time step. Once the transition matrix and zero-state response vector are computed to high precision, the response can be obtained with a very high degree of accuracy. The inconsistent initial conditions arising from ideal switching are handled using numerical Laplace inversion based two-step algorithm effectively. In sensitivity analysis, the sensitivity of the response of periodically switched linear circuits to a circuit parameter is computed at time points of a fixed interval. No approximation is made. A drawback of this method in sensitivity analysis is that it yields the sensitivity of the response to one circuit parameter, rather than to all circuit parameters, in one network analysis. It is therefore computationally costly if sensitivities to a large number of circuit parameters are needed. In statistical analysis, the first-order second-moment method that yields the mean and variance of the response of periodically switched linear circuits without multiple analyses of the circuits has been introduced. As compared with Monte Carlo based methods, the method is computationally efficient. Because only up to the second-order moments were considered in the derivation of the first-order second-moment method, the accuracy of the method deteriorates if the coefficient of variance of circuit parameters is large. In noise analysis, white noise signals of a given circuit are represented by a train of pulses with the pulse width set by the noise bandwidth of the circuit and the pulse amplitude set by the output noise power of the circuits

APPENDIX 6.A: Computation of

and

135

due to the noise signals. Using sampled-data simulation, the response of periodically switched linear circuits to both input signals and noise sources can be computed to high precision. The power spectrum of the output noise of the circuit is obtained in a post FFT analysis.

APPENDIX 6.A: Computation of

and

In this appendix, we develop efficient and accurate algorithms for computing the vectors and introduced in this chapter.

1.

Computation of Consider the circuit

Laplace transform of (6.A.1) is given by

It is seen that the response of the circuit is To compute for an arbitrary T with a high degree of accuracy, a multi-step numerical Laplace inversion approach similar to that for computing and given in the preceding chapters is taken here. The time interval in which we are interested in the behavior of the circuit is first divided into multiple small sub-intervals of equal width such that the error of numerical Laplace inversion over each sub-interval is sufficiently small. The first sub-interval is given by (0, Using numerical Laplace inversion, we obtain the response of the circuit at

In the second step, the time origin is shift from step is depicted by

to

The circuit in this

136 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS Following the same steps as those for

we obtain

where

Continuing this process, we arrive at

2.

Computation of To compute

we consider the circuit

Laplace transform of (6.A.7) is given by

It is seen that the response of the circuit is that for one can show that

3.

Computation of

To compute

we consider the circuit

Laplace transform of (6.A.10) is given by

Following the same approach as

137

APPENDIX 6.B

It is seen that the response of the circuit is In step 1, the circuit is depicted by (6.A.10) and its time domain response is given by

In step 2, the time origin is reset to

and the circuit is represented by

It can be shown that at

Continuing this process one obtain

Notice that in computing and numerical Laplace inversion is performed in the first step only. Once and are available, and can be computed efficiently and accurately in a recursive manner.

APPENDIX 6.B: Computation of Parameter Derivatives of and It was shown in Chapter 4 that and can be computed using multistep numerical Laplace inversion to high precision as follows

and

relates to

by

138 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS and

The derivatives with respect to a circuit element (6.B.1) and (6.B.2) with respect to directly

are obtained by differentiating

and

APPENDIX 6.C: Computation of Time Derivative of and The derivative terms can be computed using numerical Laplace inversion for high precision. Making use of the property

we arrive at

and

These quantities can be computed simultaneously with additional computation.

and

with little

Chapter 7 SAMPLED-DATA SIMULATION OF PERIODICALLY SWITCHED NONLINEAR CIRCUITS

Due to the aggressive reduction in both the feature size of devices and the supply voltage of modern CMOS technologies, mixed-mode switching circuits exhibit increasingly nonlinear characteristics. These nonlinear characteristics include junction capacitances, nonlinear channel current of MOSFET devices due to velocity saturation and mobility degradation [58], the finite slew rate, clock feed-through and charge injection [59], to name a few. To analyze the nonlinear behavior of these circuits, general-purpose analog simulators, such as PSPICE and Spectre [60] that use linear multi-step predictor-corrector algorithms [60, 61] as their simulation engines, can be used. These algorithms, however, exhibit the following deficiencies when analyzing periodically switched nonlinear circuits, particularly when switches are modeled as an ideal device : Inability to handle inconsistent initial conditions arising from ideal switching [60]. Inconsistent initial conditions that may occur at switching instants not only generate current or voltage impulses but also cause network variables to exhibit discontinuous characteristics. Both can not be handled by LMS-PC algorithms. Reliance on Newton-Raphson iterations in every time step of integration to find the correct solution. Not only the cost of NewtonRaphson is normally high, the step size used in these algorithms has

140 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

also to be kept sufficiently small so that a better starting point, and subsequently quadratic convergence, can be achieved. The accuracy of general-purpose analog simulators is limited by the local truncation error (LTE) of the numerical methods used. Due to stability constraints, the order of numerical integration algorithms used by general-purpose analog simulators is low. Most SPICE-like simulators employ only the first and second-order algorithms (backward Euler and Trapezoidal). This chapter investigates non LMS-PC algorithms for time-domain analysis of periodically switched nonlinear circuits, specifically, it extends the sampled-data simulation algorithm for periodically switched linear circuits given in the preceding chapter to periodically switched nonlinear circuits with weakly nonlinearities. In addition, it extends the two-step algorithm for periodically switched linear circuits to periodically switched nonlinear circuits to handle inconsistent initial conditions encountered at switching instants. Both the response and sensitivity of periodically switched nonlinear circuits at equally spaced time points are investigated in this chapter. The chapter is organized as follows: Section 1 presents the multi-linear model of nonlinear elements typically encountered in periodically switched nonlinear circuits. Section 2 derives the Volterra circuits of periodically switched nonlinear circuits. In Section 3, a sampled-data simulation algorithm for periodically switched nonlinear circuits is developed. A two-step algorithm for handling the inconsistent initial conditions of these circuits is derived in Section 4. Section 5 is concerned with the sensitivity analysis of periodically switched nonlinear circuits. In Section 6, factors that affect the accuracy and efficiency of the method are examined. The simulation results of example circuits are presented in Section 7. The chapter concludes in Section 8.

1.

Multi-Linear Theory

A large number of circuits encountered in telecommunication systems operate at a fixed DC operating point and the signals to be processed by these circuits are usually of small amplitude. The behavior of nonlinear elements in these circuits can be characterized adequately using the

Sampled-Data Simulation of Periodically Switched Nonlinear Circuits

141

truncated Taylor series expansion of these nonlinear elements at their DC operating points [45]. Elements of such a characteristic are said to be weakly nonlinear. To illustrate this, consider a nonlinear voltagecontrolled voltage source in a periodically switched nonlinear circuit with the input

where and are the controlling and controlled voltages of the voltage-controlled voltage source in phase respectively, as shown in Fig.7.1, and are constants. In phase the circuit is nonlinear time-invariant. The network variable of the circuit can be represented in their Volterra series to the order of 3

where is the Volterra series expansion of Representing the voltages of both the controlling branch and that of the controlled branch of the voltage-controlled voltage source in Volterra series of the input using (3.17) of the order of three, where is a nonzero constant, and substituting the results into (7.1) give

Note that the second subscript in (7.3) specifies the order of Volterra series expansion. Eq.(7.3) is a power series in Since a power series equals to zero if and only if all the coefficients of the power series are identically zero, we obtain

142 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Note that in derivation of (7.4), we discarded Volterra series expansion terms whose order is higher than 3. Eq.(7.4) reveals that the behavior of the nonlinear voltage-controlled voltage source can be characterized equivalently by three linear voltage-controlled voltage sources, as shown in Fig.7.1. The effect of the nonlinear characteristics is accounted for by the embedded nonlinear voltage sources in the second and third-order circuits.

Sampled-Data Simulation of Periodically Switched Nonlinear Circuits

143

To derive the multi-linear model of nonlinear elements with memory, let us consider a nonlinear capacitor modeled by

where and are the charge and voltage of the capacitor in the phase, respectively, and are constants. To ensure that the current of the capacitor vanishes outside phase the effect of the charge of the capacitor at the onset of phase denoted by and that at the end of phase denoted by must be accounted for. This leads to

Representing and using Volterra series to the order of 3 with the input substituting the results into (7.6), and equating the terms of the same power in we arrive at

where are the

and terms of the Volterra series expansions of and respectively. In a similar manner, one can show that (7.5) can be written as

144 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

So the behavior of nonlinear capacitors in phase of a periodically switched nonlinear circuit is completely characterized by the linear relationships given by (7.7) and (7.8). Other nonlinear elements can be handled in a similar manner and their multi-linear equivalent circuits are shown in Fig.7.2.

2.

Volterra Circuits

Having derived the multi-linear equivalent circuits of weakly nonlinear elements, in this section we introduce the Volterra circuits of periodically switched nonlinear circuits. Consider a periodically switched linear

Sampled-Data Simulation of Periodically Switched Nonlinear Circuits

circuit with the input using modified nodal analysis

The circuit in phase

145

is formulated

where is the network variable vector consisting of nodal voltages, some branch currents, the charge of nonlinear capacitors, and the flux of nonlinear inductors; and are the conductance and capacitance matrices, respectively, containing linear elements and the first-order term of the Taylor series expansion of the nonlinear elements. The higher order terms are embedded in the nonlinear function and is a constant vector specifying the nodes to which the input is connected in phase Representing the network variable vector of the circuit using Volterra series expansion

where

is the term of the Volterra series expansion of Evaluating the above expression at yields Volterra series expansion of the initial condition of the circuit

Substituting these results into (7.9) yields,

with the initial condition

where

146 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

and is a nonlinear function derived from the characteristics of the nonlinear elements. The preceding development reveals that : The periodically switched nonlinear circuit can be characterized equivalently by a set of periodically switched linear circuits of the same topology but distinct inputs. These periodically switched linear circuits are called Volterra circuits. The effect of the nonlinearities is accounted for by the embedded nonlinear excitations of highorder Volterra circuits. The input of the Volterra circuit is a nonlinear function of lower-order Volterra circuits only. To solve the Volterra circuits, the response of Volterra circuits of orders must be available. The time domain solution of the nonlinear circuit is obtained by summing up that of the corresponding Volterra circuits, as shown graphically in Fig.7.3.

3.

Sampled-Data Simulation of Periodically Switched Nonlinear Circuits

The response of the first-order Volterra circuits in phase using sampled-data simulation directly

is obtained

To solve the second-order Volterra circuits, we use an interpolating function that interpolates to approximate

Sampled-Data Simulation of Periodically Switched Nonlinear Circuits

147

There are many interpolation techniques available. Polynomial-based interpolation, such as, Lagrange interpolation and Newton finite difference interpolation [62], are effective for low-order interpolation and become unstable once the order is high [63]. Exponential interpolation [64] is effective only if the exponents are known. Fourier series interpolation is an efficient interpolation method [63, 65]. The order of the interpolation can be well above 1000 while still preserving the numerical stability. An important advantage of high-order Fourier series interpolation is the high accuracy of approximation. To minimize the cost of computation, a simulation window of width as shown in Fig.7.4, is employed. in the window is approximated using an interpolating Fourier series

where

is given by [64, 66]

148 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

and

is the interpolation error. The coefficients

are determined from

where

and

Sampled-Data Simulation of Periodically Switched Nonlinear Circuits

149

Neglecting the interpolation error, the behavior of the second-order Volterra circuits in the simulation window is depicted by

where is a constant vector specifying the nodes to which the input of the second-order Volterra circuit is connected. Eq.(7.24) is solved using the sampled-data simulation and the principle of superposition

150 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where

For a different order of Volterra circuit, only a new set of coefficients and need to be computed. They are obtained from

The response of higher-order Volterra circuits in the simulation window can be computed in a similar manner

The response of the nonlinear circuit at time points in the window is obtained by summing up that of the corresponding Volterra circuits

Sampled-Data Simulation of Periodically Switched Nonlinear Circuits

151

Because and are constant, the only computation required for each simulation window is to obtain and

4.

Inconsistent Initial Conditions

Similar to periodically switched linear circuits, inconsistent initial conditions may occur at switching instants in periodically switched nonlinear circuits. In this section, we extend the two-step algorithm for computing from of periodically switched linear circuits to periodically switched nonlinear circuits. Suppose a switching occurs at and the initial conditions of the Volterra circuit immediately before the switching instant are given by The forward step from to yields

and

The step size must be properly chosen such that A backward step from to taken to yield

is then

152 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

and

where

Finally the consistent initial conditions are obtained from

5.

Sensitivity of Periodically Switched Nonlinear Circuits

The time domain sensitivity of periodically switched nonlinear circuits to a circuit parameter is obtained by differentiating with respect to

Sampled-Data Simulation of Periodically Switched Nonlinear Circuits

153

The above result reveals that the sensitivity of a periodically switched nonlinear circuits can be obtained by summing up that of the corresponding Volterra circuits. The sensitivity of the first-order Volterra circuit is obtained from differentiating (7.15) with respect to

To get the sensitivity of the (7.28) with respect to

To calculate

and

order Volterra circuit, we differentiate

we differentiate (7.27) with respect to

Since is a function of is a function of and and can be computed once the response and sensitivity of the lower-order Volterra circuits are available.

154 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

At the switching instants, the two-step algorithm is applied to obtain the consistent initial conditions of sensitivity networks. Assume a switching occurs at In the forward step, the sensitivity of the Volterra circuits at is calculated using (7.36) and (7.37). In the backward step, we calculate the sensitivity of the Volterra circuits at is obtained by differentiating (7.32) and (7.33) with respect to

and

Finally, the sensitivity immediately after switching is obtained by summing up that of all Volterra circuits

Sampled-Data Simulation of Periodically Switched Nonlinear Circuits

155

Discussion

6.

In this section we examine factors that affect the efficiency and accuracy of sampled-data simulation of periodically switched nonlinear circuits.

6.1

Stability

The stability of the method can be examined from that of the interpolating Fourier series and that of the sampled-data simulation of linear circuits. Interpolating Fourier series has superior numerical stability over polynomial-based interpolation schemes, as demonstrated in [63]. It was shown in [18] that sampled-data analysis for linear circuits is an A-stable numerical integration algorithm. For a stable linear circuit it guarantees a stable numerical solution.

6.2

The Maximum Step Size

The upper bound of the step size is subject to the constraint set by Nyquist theorem. Specifically, because the highest frequency of the input signal of the Volterra circuits is the frequency of the highest-order term of the interpolating Fourier series given by

the lower bound of the sampling frequency is therefore given by

from which the maximum step size is obtained

The actual step size is determined from both the simulation accuracy and CPU time and is usually much smaller than

6.3

Accuracy

The accuracy of the method depends upon the following factors : The order of Taylor series expansion in representation of the characteristics of nonlinear elements.

156 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

The order of Volterra series expansion in depicting nonlinear circuits. The order of interpolating Fourier series in approximation of the input of high-order Volterra circuits. Simulation window. Error propagation. We examine each of them in detail. 6.3.1

The Order of Taylor Series Expansion

The order of the Taylor series expansion of nonlinear elements depends upon the characteristics of the nonlinearities, the amplitude of the input signal, and the tolerance of the error of approximation. When the Taylor series expansion is employed to characterize the nonlinear element, the truncation error is given by

where is the displacement from the operating point and As an example, consider a forward biased diode characterized by

where and are the forward biasing voltage and current of the diode, respectively, is the saturation current, and is the thermal voltage. Let where and are the DC and AC components of respectively. Expanding in Taylor series at the DC operating point gives

where is the AC component of If the 4th-order Taylor series expansion is used, the truncation error is estimated from

Sampled-Data Simulation of Periodically Switched Nonlinear Circuits 6.3.2

157

The Order of Volterra Series Expansion

The order of Volterra series expansion in depicting the nonlinear circuits depends upon the nonlinear characteristics of the circuits and the error of approximation. Consider a nonlinear resistor characterized by

where and are the voltage and current of the resistor in phase respectively, and are constants. Representing and in their Volterra series expansions to the order of 3 and substituting the results into (7.50) yields

Eq.(7.51) reveals that the nonlinear resistor can be represented by three linear resistors, together with added voltage sources quantifying the nonlinear effect. If the 4th-order Volterra series expansion is considered, we will have

The difference is the last equation in (7.52) that accounts for the effect of the 4th-order nonlinear characteristic. Eq.(7.51) will be considered adequate if the difference between the response of the circuit with the 3rd-order Volterra series expansions and that with the 4th-order Volterra series expansions considered is negligible. 6.3.3

The Order of Interpolating Fourier Series

The solution of the first-order Volterra series is accurate provided that and are computed to high precision. The error in solving

158 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

the second-order Volterra circuit is dominated by the interpolation error To estimate this error, the input of the second-order Volterra circuit is derived with the step sizes T and respectively. The order of interpolation is determined from the normalized mean square error (NMSE) [67]

where and are the response of the circuit with the step sizes of T and respectively. In order to compute and and are needed. The relationship

and

can be employed to reduce computational cost. 6.3.4

Simulation Window

When a simulation window is employed it is assumed that the data series sampled by the window is periodic. The rate of convergence of the interpolating Fourier series depends upon the boundary condition of the data series. It was shown in [67, 68] that a large error will exist if the data series is discontinuous at the boundary, known as Gibb effect. The error will be further increased if the derivatives of the function are also discontinuous [69]. To minimize the error due to the discontinuity of the function value at the boundary, the window should be chosen such that the input of Volterra circuits is periodic with respect to the simulation window. For circuits with only one sinusoidal input, the window size that is the same as the period of the sinusoidal input meets this requirement

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159

because the frequency components in the circuits consist of the frequency of the input and its harmonics only. For circuits with non-sinusoidal inputs or multiple sinusoidal inputs, this does not hold. Consider the input of the second-order Volterra circuit as shown in Fig.7.5. To reduce Gibb effect, we introduce a new function

where is a constant, and impose is therefore given by

The value of

Because is periodic, interpolating Fourier series can be employed to derive without introducing a large error. Once is available, can be obtained from the inverse of the transform

160 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Using this technique the second-order Volterra circuit is depicted by

where and are the coefficients of the interpolating Fourier series that interpolates The response of the circuit in the window is obtained from

This approach can be further developed to minimize the error due to the discontinuity of the derivatives of the function [69]. 6.3.5

Error Propagation

Because the error due to the truncation of Taylor series and that of Volterra series are usually insignificant for mildly nonlinear circuits. In what follows only the interpolation error is considered. The interpolation error in deriving the input of the second-order Volterra circuit, denoted by as shown in Fig.7.6, gives rise to the error of the response of the second-order Volterra circuit, is given by

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161

where The sampled-data value of the input of the third-order Volterra circuit is computed from

from which the input is approximated using interpolating Fourier series

The response of the third-order Volterra circuit is given by where

162 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

and and are Laplace transform of and respectively. The total error is obtained by summing up that of the second and third-order Volterra circuits

7.

Examples

In this section, the response and sensitivity of several periodically switched nonlinear circuits are analyzed using the algorithms presented in this chapter .

7.1

Time-Invariant Nonlinear Circuits

The first example considered is the current-mirror amplifier shown in Fig.7.7 with the value of the circuit parameters given in Table 7.7. The amplifier is a building block for continuous-time current-mode circuits [70]. Neglecting channel-length modulation and other second-order effects, we obtain the AC component of the channel current

where

is the linear transconductance and

is the second-order nonlinear transconductance, is the surface mobility of free electrons, is the gate capacitance per unit area, W and L

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163

are the width and length of the transistors, respectively, and are the biasing voltage and the threshold voltage, respectively. The AC circuit of the amplifier is shown in Fig.7.7 where only intrinsic capacitances are considered. The output current is computed using both sampleddata simulation and LSS-PC. The results are shown in Fig.7.8 with and without the gate-to-source capacitance and gate-to-drain capacitance considered. It is seen that when the capacitance is not considered, the amplifier realizes When the capacitance is considered, the output current is reduced. The results from sampled-data simulation are in a good agreement with those from LSS-PC analysis.

The second example of time-invariant nonlinear circuit is shown in Fig.7.9 with the value of the circuit parameters given in Table 7.9. The nonlinear conductor is characterized by

164 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

The voltage across is computed using both sampled-data simulation and LSS-PC algorithms. The number of steps per simulation window for sampled-data simulation and LSS-PC is 20 and 100, respectively. It was observed that if the number of steps per simulation window in LSS-PC is lower than 100, a significant error exists. The results are shown in Fig.7.10 for various input amplitudes. The difference between sampled-

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165

data simulation and LSS-PC is plotted in Fig.7.11. It is seen that the maximum normalized difference is nearly 1%.

The error due to the order of the interpolating Fourier series is shown in Fig.7.12 with the input amplitude 0.5A, It is observed that an increase in the order of the interpolating Fourier series lowers the error. The computational efficiency is demonstrated in Fig.7.13 where the CPU time is plotted as a function of the number of simulation windows. The CPU time is measured for both sampled-data simulation and LSSPC programs coded in Matlab, an interactive mathematical language that runs in an interpretive mode [71]. The program was executed on Sun Ultra 1 workstation with 450 MHz CPU and 256MB RAM. It is seen the initial cost of computation of sampled-data simulation is higher than that of LSS-PC, mainly due to the cost of the pre-processing step. The CPU time of LSS-PC analysis arises rapidly with the number of simulation windows, whereas that of sampled-data simulation arises slowly,

166 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

indicating that the computational cost of sampled-data simulation is nearly independent of the number of simulation windows.

7.2

Switched Capacitor Integrator with Nonlinear Op Amp

Consider the switched capacitor integrator shown in Fig. 7.14. The nonlinear gain of the operational amplifier is modeled as The clock frequency is 100 kHz and the input is a cosine wave. The output response and its sensitivity to are obtained using sampled-data simulation and the results are shown in Fig.7.15. The sensitivity results are computed using sampled-data simulation and brute-force approach with 1% parameter variation. The results are as plotted in Fig.7.16. A good agreement is observed.

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167

168 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

7.3

General Periodically Switched Nonlinear Circuits

Consider the circuit shown in Fig.7.17 that contains two externally clocked switches. The clock phases are non-overlapping. The voltages of the two capacitors are well defined inside either clock phases but discontinue at switching instants. The clock frequency of the circuit is 5 Hz. Each clock period has two phases of equal width. Zero initial conditions were assumed for all capacitors. The nonlinear conductor was modeled as

where

are constants and their values are given by The input was a unit step current source. To simplify the simulation, the width of the simulation window was set to be the same as that of the clock phase. The number of samples per window

Sampled-Data Simulation of Periodically Switched Nonlinear Circuits

169

was 20. The voltages at all nodes were solved using sampled-data simulation and the voltage at node 3 are plotted in Fig.7.18. As shown in the figure, the voltage at node 3 is discontinuous at where inconsistent initial conditions are encountered and continuous at where only consistent initial conditions are encountered. This observation demonstrates that sampled-data simulation can handle both the inconsistent and consistent initial conditions at the switching instants. The results from PSPICE analysis are also plotted in the figure for comparison. The normalized difference between the results from sampled-data simulation and those from PSPICE are measured. Two different situations are considered. Inside each clock phase where no discontinuity in the response, the normalized difference for time points except switching instants is defined as

170 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where and are the response of sampled-data simulation and PSPICB, respectively. It can be obtained directly from the responses, and is shown in Fig.7.19. It is seen that the maximum normalized difference is below 0.5%. At the switching instants, a direct comparison of the results from sampled-data simulation and PSPICE is difficult. The reason is as follows: If a switching action causes response discontinuity at although the inconsistent initial conditions immediately before and after switching and can be obtained from sampled-data simulation, they can not be obtained from PSPICE. For traditional integration methods, such as those used in PSPICE, the continuity of network variables is required. In order to handle a rapid change in the response, very small time steps must be taken to ensure the accuracy of integration. No time point can be clearly defined as a switching instant. At the switching instants, the initial conditions from sampled-data simulation are verified by checking the charge conservation before and after switch-

Sampled-Data Simulation of Periodically Switched Nonlinear Circuits

171

ing. The total charge on capacitors and immediately before and after switching is computed and plotted in Fig.7.20. The sensitivities of the output voltage with respect to conductor and capacitor were calculated using sampled-data simulation with the same settings as those used in response simulation. They were also calculated using the brute-force method as follows

172 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

1% variation in the nominal value of conductor and capacitor was used to calculate the sensitivity in the brute-force. The results are shown in Fig. 7.21 for the sensitivity to and Fig. 7.22 for the sensitivity to It is seen that the results from the sampled-data simulation and the

brute-force match well.

8.

Summary

Sampled-data simulation of the response and sensitivity of periodically switched circuits with weak nonlinearities has been presented. The method characterizes the behavior of these circuits using a set of periodically switched linear circuits called Volterra circuits that have identical topology but distinct inputs. The input of high-order Volterra circuits is a nonlinear function of the response of lower order Volterra circuits only. The analytical expression of the input of high-order Volterra cir-

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173

cuits is approximated using interpolating Fourier series that interpolates the value of the input of the Volterra circuits at time points of a fixed interval. The sampled-data simulation algorithm for linear circuits is used to solve the Volterra circuits. In addition, the two-step algorithm for periodically switched linear circuits is utilized to compute the consistent initial condition of the Volterra circuits, subsequently, that of periodically switched nonlinear circuits. Both the response and sensitivity of periodically switched nonlinear circuits are obtained at equally spaced intervals of time. The method achieves superior computational efficiency by avoiding costly Newton-Raphson iterations. Also, high numerical accuracy is achieved by employing high-order interpolating Fourier series. The method is most effective and computationally efficient for periodically switched nonlinear circuits with mild nonlinearities whose behavior can be characterized sufficiently using the low-order Volterra series expansion. For circuits with harsh nonlinearities, such as comparators and

174 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Schmitt triggers, other methods such as those that are based on behavior modeling [18], can be used instead. The method is a general computeroriented formulation method that can be applied to mildly nonlinear circuits with externally clocked switches including switched capacitor networks and switched current networks.

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Chapter 8 SAMPLED-DATA SIMULATION OF CIRCUITS WITH INTERNALLY CONTROLLED SWITCHES

Internally controlled switches are switches whose state (ON or OFF) is controlled by the internal network variables associated with the switches. The switching time at which internally controlled switches change their state is not known at the start of simulation. This differs fundamentally from circuits with externally clocked switches where the state of switches in these circuits is solely determined by the state of external clocks and is known a priori. More importantly, the topology of circuits with internally controlled switches may change at any time, depending upon the value of the switching variable of internally controlled switches that controls the state of these switches. Moreover, the switching of one internally controlled switch may trigger other internally controlled switches in the circuits, subsequently further altering the topology of the circuits. As compared with the analysis of circuits with externally clocked switches, the evaluation of the switching variable of internally controlled switches must be carried out in each time step of simulation and at each time instant at which an internally controlled switch changes its state in order to detect any change in the topology of the circuits. This significantly complicates the analysis. This chapter is concerned with the analysis of circuits with internally controlled switches. In Section 1, the switching variable of internally controlled switches typically encountered in mixed-mode switching circuits is defined. We show that the switching variable of each internally

178 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

controlled switch must be evaluated in each step of simulation in order to determine the state of the switch. In Section 2, methods that compute the time instant at which internally controlled switches change their state are developed. Section 3 examines the generation of impulsive network variables at switching instants and its complication in analysis of circuits with internally controlled switches. We show that impulses generated at switching instants not only give rise to inconsistent initial conditions, they may also initiate a sequence of switching activities of other internally controlled switches in the circuits. In Section 4, a linear voltage regulator is used as an example to illustrate the analysis of circuits with internally controlled switches in detail. The chapter is summarized in Section 5 with concluding remarks.

1.

Internally Controlled Switches and Switching Variables

Unlike circuits with externally clocked switches, the time instants at which the topology of circuits with internally controlled switches changes are determined by the state of the switching variable of the switches in the circuits only. The switching variable of an internally controlled switch controls the state (ON or OFF) of the switch in accordance with the following criterion

In what follows we examine internally controlled switches typically encountered in mixed-mode switching circuits and the switching variable characterizing the state of these switches.

1.1

Diodes

Ideal diodes are the simplest internally controlled switches. The operation of an ideal diode is controlled by the voltage across the diode, as shown in Fig.8.1. The switching variable of an ideal diode is the forward biasing voltage of the diode.

Sampled-Data Simulation of Circuits with Internally Controlled Switches

179

Diodes belong to the category of voltage-controlled switches. constitutive equation of the diode is given by

The

1.2

MOSFETs

MOSFETs switches are controlled by the effective gate-source voltage of the devices, as shown in Fig.8.2. The switching variable of a NMOS switch is defined as

where is the threshold voltage of the NMOS switch. The constitutive equation of the NMOS switch is given by

MOSFET switches also fall into the category of voltage-controlled switches.

180 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

1.3

Static CMOS Inverters

The switching variable of the complementary static CMOS inverter shown in Fig.8.3 is defined as

where

is the threshold voltage of the inverter. For most applications,

and is achieved by the appropriate sizing of the NMOS and PMOS transistors. The output of the inverter is solely controlled by its input in accordance with

Sampled-Data Simulation of Circuits with Internally Controlled Switches

1.4

181

Comparators

Comparators are used extensively in sigma-delta modulators as quantizers. Both voltage-mode and current-mode comparators are available, as shown in Fig.8.4. The output of clocked comparators updates itself at the clocking instants only and remains unchanged during clocking phases, whereas the output of unclocked comparator is transparent to the inputs. The switching variable of an ideal voltage-mode comparators is defined

as

and the constitutive equation of the comparator is given by

where is a user-defined voltage. It is evident that a voltage-mode comparator is a voltage-controlled switch. The switching variable of a current-mode comparator is given by

where is the input current to the comparator. The constitutive equation of the current-mode comparator is given by

Current-mode comparators are current-controlled switches. The switching variable of other types of internally controlled switches, such as thyristors, can also be defined in a similar way.

2.

Switching Instants

It was seen in the preceding section that a change in the polarity of the switching variable of an internally controlled switch indicates a

182 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

change in the state of the switch. At the switching instant, the following equation holds [14, 15]

The switching time is obtained by solving (8.12) using iterative algorithms, such as Newton-Raphson [14].

3.

Inconsistent Initial Conditions

It was shown in Chapter 5 that when a switching occurs at the value of network variables immediately after the switching can be written as

where denotes the consistent initial condition and denotes the inconsistent initial condition. Note both and are finite and can be stored. Also, only makes its appearance at the switching instant. If no impulsive network variables will exist in the circuit. Otherwise, inconsistent initial conditions will be encountered. together with are used to test each internally controlled switch at switching instants to determine whether a violation of the switching condition occurs. Because the switching of one internally controlled switch may trigger the switching of other internally controlled switches, either due to or the evaluation of the switching condition of all internally controlled switches must continue until there

Sampled-Data Simulation of Circuits with Internally Controlled Switches

183

is no switching condition violation. It is only after no switching condition violations exist, the final topology of the circuit at the switching instant can be established, and the two-step algorithms introduced in Chapter 5 can be employed to compute from Once is available, simulation proceeds forward in time. It should be emphasized that at the switching instant of each internally controlled switch, not only the detection of whether impulsive network variables may be generated at the switching instant must be conducted, the value of the network variables at the time instant immediately after the switching, i.e must also be carried out to update the initial condition of the circuit so that the detection of the switching of other internally controlled switches at the same time instant can be carried out.

4.

Examples

In this section we use the linear voltage regulator shown in Fig.8.5 as an example to demonstrate the analysis of circuits with internally controlled switches. The basic operation of the voltage regulator is as follows : The MOSFET switch is controlled by an external clock with variable duty cycle. When the supply voltage E supplies a current to the series inductor L and the load resistor R. Because the output is current, the value of the load resistor R is small. The functionality of the shunt capacitor C is to sustain the output voltage, whereas that of the series inductor L is to sustain the output current in the absence of E. The value of L and that of C must, therefore, be sufficiently large so that the ripple of the output current and that of the output voltage during the absence of E are small. When E is disconnected from the load. The output current is sustained by the magnetic energy stored in the series inductor and the electric energy stored in the shunt capacitor. By adjusting the duty cycle of the switching MOSFET given by the average output current can be controlled and is given by

184 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Since chopper.

The voltage regulator is also known as step-down

For a detailed analysis of the operation of the regulator, we examine the operation of the regulator in the following four different time regions: i) ii)

iii) iv)

The MOSFET switch is ON. The MOSFET switch is turning OFF. The MOSFET switch is OFF. The MOSFET switch is turning ON.

The regulator exhibits distinct characteristics in these four different regions. Assume during the clock phase The schematic of the voltage regulator when is shown in Fig.8.6. Reset the time origin from to The circuit at is depicted by

Sampled-Data Simulation of Circuits with Internally Controlled Switches

185

or symbolically

where

and

Note that because for the diode is OFF during the clock phase. The time domain response of the circuit can be computed conveniently using the sampled-data simulation of linear circuits given in Chapter 6.

where M(T) and U(T) were defined earlier. When the MOSFET switch turns OFF at the circuit schematic is shown in Fig.8.7. Reset the time origin from to Note that the inductor carries an initial current and the capacitor has an initial voltage The behavior of the circuit at the switching instant is depicted by

186 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

The state of the diode can be determined by evaluating If the diode will turn ON. Otherwise, it remains OFF. Due to the effect of Dirac impulse,

holds for any value of and The diode will turn ON immediately after the MOSFET switch turns OFF. The voltage regulator will have a new topology after the diode is ON. Numerically, the determination of whether the circuit change its topology at the switching instant is achieved by first examining whether the switching variable of the diode contains a Dirac

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187

impulse function. This is achieved by using the method given in Chapter 5, specifically, by evaluating the following integral

Because

and

If the switching variable of the diode contains a Dirac impulse. Otherwise, only consistent initial conditions will be encountered at the switching instant. The consistent initial condition is obtained by employing the two-step algorithms given in Chapter 5. Once and are available, the switching variable of the diode given by is evaluated at In this case, since and is non-zero, an impulsive forward biasing voltage exists across the diode and the diode turns ON at When the MOSFET switch is open for Again, we reset the time origin from at is depicted by

or symbolically

to

The circuit

188 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

The response of the regulator with

can be obtained from

When the MOSFET switch turns ON at we reset the time origin from to It can be shown that

No impulsive network variables will be encountered at the switching instant. Only the consistent initial conditions exist. The initial conditions can then be updated easily and simulation proceeds forward to the next clock cycle. The preceding example exemplifies the procedures of the analysis of circuits with internally controlled switches, which can be summarized as follows : Step 1 At the start of simulation, an initial state is assigned to each internally controlled switch in the circuit. Simulation starts with a given initial condition Step 2

is computed using the sampled-data simulation, along with the evaluation of the switching variable vector

Step 3 When a sign change of is detected, revealing the switching of an internally controlled switch in the time interval iteration algorithms are employed to allocate the exact switching instant The solution of the circuit at is saved. Step 4 Determine whether impulsive network variables are generated at the switching instant by evaluating the integral Also, the two-step algorithms given in Chapter 5 are employed to compute

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189

Step 5 Re-evaluate the switching variable vector with the newly obtained initial condition. If experiences a sign change from the time step immediately before switching and go to Step 4. Otherwise, go to Step 6. Step 6 Sampled-data simulation proceeds.

5.

Summary

The analysis of circuits with internally controlled switches has been presented. The definition of the switching variable of internally controlled switches typically encountered in mixed-mode switching circuits has been presented. We have shown that the switching variable of each internally controlled switch must be evaluated in each step of simulation in order to determine the state of the switch. We have also shown that impulses generated at switching instants may lead to the violation of switching conditions of other internally controlled switches in the circuits and trigger the switching of these switches. This process continues until there are no more violations of the switching condition of internally controlled switches. The final topology of the circuit at the switching instant can then be determined and the consistent initial can be determined. The method presented in this chapter does not involve any approximation, nor does it require the storage of the numerical value of Dirac impulse function. The sampled-data simulation algorithms, together with the methods of this chapter can be integrated to analyze linear circuits with both internally controlled and externally clocked switches. For nonlinear circuits with both internally controlled and externally clocked switches, the method allows us to determine the topology of the circuit at switching instants and the consistent initial conditions of the new circuit so that algorithms for nonlinear circuits, such as Newton-Raphson and the method presented in Chapter 7, can be employed to analyze these circuits effectively.

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Chapter 9 SAMPLED-DATA SIMULATION OF OVER-SAMPLED SIGMA-DELTA MODULATORS

This chapter applies the sampled-data simulation algorithm for periodically switched linear circuits presented in Chapter 6 to a special class of nonlinear circuits - over-sampled sigma-delta modulators. Section 1 presents the characteristics of over-sampled sigma-delta modulators and the difficulties encountered in analysis of these circuits. In Section 2, the behavior modeling of clocked quantizers of over-sampled sigmadelta modulators that convert analog signals into a sequence of digital bits is investigated. Unclocked quantizers are addressed in Section 3. We show that by utilizing the methods for circuits with internally controlled switches, sigma-delta modulators with unclocked quantizers can be analyzed. Section 4 investigates the modeling of single-bit digitalto-analog data converters. The modeling of other blocks of sigma-delta modulators are dealt with in Section 5. In Section 6, the simulation methods that utilizing both the behavioral modeling of the quantizers and the sampled-data simulation of periodically switched linear circuits are addressed in detail. In Section 7, both continuous-time and switched capacitor over-sampled sigma-delta modulators are analyzed using the method developed in this chapter and the results are presented. Concluding remarks are given in Section 8.

192 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

1.

Introduction

Over-sampled sigma-delta modulators are widely encountered in analogto-digital data conversion[72]. These modulators trade off speed for accuracy, and can achieve a high precision resolution without requiring precisely matched analog components. In addition, they can be fabricated on the same substrate with other digital circuitry using standard digital fabrication technologies. A typical configuration of a clocked firstorder over-sampled sigma-delta modulator is shown in Fig.9.1. The key element of over-sampled sigma-delta modulators is the quantizers, implemented using comparators with one input from the output of the integrator and the other the ground, that quantizes the incoming analog signal and outputs a stream of digital data. These serial digital data stream is then converted to parallel data using a decimator. The main function of the integrator in the forward path is to sense the difference between the incoming analog signal and the output of digital-to-analog converter and drive the difference to zero in an integrating manner. Single-bit oversampled sigma-delta modulators are the most widely used, owing to their intrinsic advantages of the ease in implementation and insensitivity to non-idealities of integrators.

Sampled-Data Simulation of Over-Sampled Sigma-Delta Modulators

193

Due to the over-sampling nature of over-sampled sigma-delta modulators, switched capacitor networks or switched current networks with the clock frequency that is much higher than that of the frequency of input signals are the circuit techniques for the realization of these modulators. The over-sampling ratio, defined as the ratio of the clock frequency to the frequency of the input signals, is usually in the range of 16~64. Although over-sampled sigma-delta modulators are relatively simple circuits, simulation of these circuits is rather difficult, mainly due to the following reasons : Comparators are harsh nonlinear elements, as shown in Fig.9.2. The output-input relation of the quantizers can not be characterized using Taylor series expansion. As a result, methods for linear circuits and those introduced in Chapter 7 for circuits with mildly nonlinear elements can not be used to analyze these circuits.

Sigma-delta modulators are dual-time circuits. They contain a highfrequency clock and a slowly varying input signal. The circuit has to be simulated over a large number of clock cycles with fine time steps in order to obtain the performance characteristics, such as signal-tonoise ratio and dynamic range, of these circuits reliably, resulting in excessive simulation time. High simulation accuracy requirement. For example, the resolution of a 16-bit analog-to-digital data converter with a 1V constant voltage

194 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

input is only The accuracy of numerical algorithms must be sufficiently higher than this limit in order to analyze these circuits.

2.

Modeling of Clocked Quantizers

Quantizers are the key element of over-sampled sigma-delta modulators. For a reliable operation, switched capacitor and switched current networks are designed in such a way that the network variables of these circuits reach their steady value of the clock phase before reaching the end of the clock phase. The quantizer of over-sampled sigma-delta modulators implemented using switched capacitor or switched current techniques thereby also operates in a clocked manner, i.e. the output of the quantizer changes only at the clocking instants, and remains unchanged during the clock phase, as shown in Fig.9.3. This observation suggests that a clocked quantizer can be modeled effectively using the following behavioral model:

where

and are the input and output of the quantizer at respectively, and are two constant reference voltages. One of the main advantages of modeling quantizers using behavioral models is the elimination of the difficulties encountered in circuit-level modeling of quantizers.

3.

Modeling of Unclocked Quantizers

The analysis of preceding sigma-delta modulators with clocked quantizers is significantly simplified as we only need to worry the output of the quantizers at clocking instants. For sigma-delta modulators with unclocked quantizers, the exact time instants at which the quantizers flip must be determined with high precision. As pointed out in Chapter 1 that sigma-delta modulators with unclocked quantizers fall into the category of circuits with both externally clocked and internally controlled

Sampled-Data Simulation of Over-Sampled Sigma-Delta Modulators

195

switches. Unclocked quantizers behave as an internally controlled switch with the switching variable given by

where and are the voltages at the inverting and non-inverting input terminals of voltage comparators in phase The switching variable of current-mode quantizers can also be defined in a similar manner. The time instant at which the quantizer changes its output can be determined as follows : If and have distinct signs, a change in the polarity of the output of the quantizer occurs in the time interval The exact time instant at which the output of the quantizer varies is obtained by solving numerically using iteration approaches, such as Newton Raphson. Once is allocated, the step size is determined and is determined. Note that this process requires the computation of and using numerical Laplace inversion. Once is available, the two-step algorithm introduced in Chapter 5 is employed to compute from

196 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Once is available, simulation proceeds in a sampled-data manner. As compared with the analysis of sigma-delta modulators with clocked quantizes, the following additional computation in the analysis of sigma-delta modulators with unclocked quantizers is required : i) Evaluation of the switching variable of the quantizer in each integration step. ii) If a change in the sign of the switching variable is detected, the exact switching time is allocated by solving

and subsequently using the iii) Computation of sampled-data simulation for periodically switched linear circuits. iv) Computation of the consistent initial condition at time instants immediately after switching using the two-step algorithms presented in Chapter 5.

4.

Modeling of Digital-to-Analog Data Converters

The function of a digital-to-analog converter (DAC) in a single-bit sigma-delta modulator is to select two distinct reference voltages (or currents), based on the output of the quantizers. Fig.9.4 shows the typical configurations of DACs in single-bit sigma-delta modulators. The single-bit DAC is a switched linear circuit with the controlling signal be the output of the quantizer of the sigma-delta modulator and two voltage inputs with constant voltages for voltage-mode DACs or two current inputs with constant currents for current-mode DACs.

5.

Modeling of Other Blocks

When the input to the integrators is small, the integrators can be treated as a linear element, and can be modeled at the circuit level using linear elements such as resistors, inductors, capacitors, dependent and independent sources, and ideal switches. The non-idealities of integrators, such as finite bandwidth, finite input impedance, and non-zero output impedance of the operational amplifier of the integrators, can also be modeled conveniently using circuit-level models.

Sampled-Data Simulation of Over-Sampled Sigma-Delta Modulators

6.

197

Simulation Methods

Sigma-delta modulators are partitioned into (i) a linear block consisting of the integrators, digital-to-analog converters, and other linear components, and (ii) a nonlinear block composed of the quantizer. The linear block is modeled at the circuit level and formulated using modified nodal analysis while the nonlinear part is modeled using the behavioral model given earlier and incorporated in simulation without using conventional matrix approaches. Such a partition allows us to analyze the behavior of these circuits in the time domain effectively in the following way : The linear part can be formulated and analyzed effectively using the sampled-data simulation method for periodically switched linear circuits presented in Chapter 6. There are two inputs to the linear blocks : the input signal of the sigma-delta modulator and the output of the quantizer. The input signal of the sigma-delta modulators varies with time continuously. The output of the clocked quantizer changes with time in a piecewise constant manner and is treated as a step input for each clock phase. Because the output of the clocked quantizer is transparent to its input only at switching instants, the output of the linear block is needed only at the clocking instants when the quantizer updates its output. For sigma-delta modulators

198 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

with an unclocked quantizer, the output of the linear block must be evaluated in every time step. The output of the linear block is fed to the quantizer. For sigma-delta modulators with a clocked quantizer, the output of the quantizer changes only at the edge of the clock phases and remains unchanged until the arrival of the edge of the next clock phase. Simulation proceeds by calculating the output of the linear block after each clocking instant, updating the state of the quantizer based on the output of the linear block, and repeating the process for the next clock phase. For sigma-delta modulators with an unclocked quantizer, the exact time instant at which the output of the quantizer changes must be determined and the consistent initial conditions of the modulators after the quantizer changes its state must be calculated. Simulation then proceeds from the consistent initial conditions. Because sampled-data simulation yields the exact time-domain response of linear circuits regardless of the step size, also because clocked quantizers update their output only at the clocking instants, over-sampled sigma-delta modulators with clocked quantizers can therefore be analyzed effectively using the sampled-data simulation of periodically switched linear circuits presented in Chapter 6. The step size of sampled-data simulation can be set to the width of clock phases to speed up simulation.

7.

Examples

The first example is the single-bit second-order continuous-time oversampled sigma-delta modulator shown in Fig.9.5. The comparator is clocked at 1.024kHz. The operational amplifiers are considered to be ideal. The input signal is a 1kHz sinusoid of amplitude 2V. The timedomain response of the modulator is shown in Fig.9.6 and the output spectrum of the modulator obtained from transient-FFT analysis is shown in Fig.9.7. The noise shaping characteristics of the second-order sigma-delta modulator are evident. The second example is a single-bit second-order switched capacitor over-sampled sigma-delta modulator shown in Fig.9.8 [73, 74]. It is

Sampled-Data Simulation of Over-Sampled Sigma-Delta Modulators

199

actually a switched capacitor implementation of the preceding singlebit second-order continuous-time over-sampled sigma-delta modulator. Note that the resistors in the preceding continuous-time over-sampled

200 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

sigma-delta modulator are implemented using stray-insensitive switched capacitor technique with the resistance value given by [5]

where

is the equivalent resistance of the switched capacitor network,

is the clock frequency and C is the value of the switched capacitor. Again, the operational amplifiers are considered to be ideal. The spectrum of the modulator is shown in Fig.9.9. These results were obtained by simulating the circuit for 74k clock cycles, discarding the first 10k data points to remove any transients, and performing a post FFT analysis on the remaining 64k data points. The signal-to-noise ratio of the modulator is obtained by performing the preceding simulation with variable input amplitude. For each input amplitude, the output of the modulator is obtained by performing transient-FFT analyses and the signal-to-noise ratio (SNR) is calculated.

Sampled-Data Simulation of Over-Sampled Sigma-Delta Modulators

201

The result is shown in Fig.9.10. The frequency of the input is 1kHz and a 4kHz bandwidth was assumed in SNR calculation.

8.

Summary

In this chapter, we have first reviewed the characteristics of oversampled sigma-delta modulators and the difficulties encountered in analysis of these special circuits. We have shown that the difficulties encountered in modeling quantizers at the circuit level can be overcome using behavior modeling. Other blocks of sigma-delta modulators, however, can be dealt with conveniently using conventional circuit-level formulation techniques. The simulation methods that incorporate both the behavioral modeling of the quantizers and the sampled-data simulation for linear circuits have been detailed. A continuous-time over-sampled sigma-delta modulators and its switched capacitor counterpart have been analyzed and comparable results have been obtained.

202 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Sampled-Data Simulation of Over-Sampled Sigma-Delta Modulators

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III

FREQUENCY DOMAIN ANALYSIS

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Chapter 10 ADJOINT NETWORK OF PERIODICALLY SWITCHED LINEAR CIRCUITS

This chapter extends Tellegen’s theorem for linear time-invariant circuits and that for ideal switched capacitor networks to general periodically switched linear circuits and derives the adjoint network of these circuits using a phasor-domain approach. We show that by using this approach the derivation of the adjoint network of periodically switched linear circuits becomes straightforward and bears a strong resemblance to that of linear time-invariant circuits [75]. We further show that the transfer functions from multiple inputs to the output of a periodically switched linear circuit can be obtained efficiently by solving the adjoint network. More importantly, we reveal that the aliasing transfer functions from inputs at the side band frequencies of wideband inputs, such as white noise sources, to the output in the base band of a given periodically switched linear circuit can be obtained efficiently from the corresponding aliasing transfer function of the adjoint network with the input at the base band frequency and the output at the corresponding sideband frequencies. The chapter is organized as follows: Section 1 introduces Tellegen’s theorem for periodically switched linear circuits in the phasor domain. The inter-reciprocity and the adjoint network of periodically switched linear circuits are developed in Sections 2 and 3, respectively. In Section 4, the transfer function theorem of periodically switched linear circuits is derived and its usefulness is exploited. Section 5 introduces the fre-

208 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

quency reversal theorem of periodically switched linear circuits with a rigorous proof. Both theorems are assessed using example circuits in Section 6. The chapter concludes in Section 7.

1.

Tellegen’s Theorem

Tellegen’s theorem, introduced by B. Tellegen half a century ago, is a fundamental law for lumped electrical networks [76, 77, 78]. In the time domain, it is given by

where and denote the branch voltages and currents of the circuit, respectively, and B is the total number of branches in the network. Eq.(10.1) also holds for any two circuits N and having the same incidence matrix

and

where and denote respectively the branch currents and voltages of and are the time variables of N and respectively. Eqs.(10.2) and (10.3) are called the strong form of Tellegen’s theorem. It is intuitive to show that following weak form of Tellegen’s theorem also holds

The weak form of Tellegen’s theorem is of particular usefulness because it incorporates the branch voltages and currents of two separate networks N and in a single and closed expression.

Adjoint Network of Periodically Switched Linear Circuits

209

Periodically switched circuits differ from time-invariant networks by including externally clocked switches. For a given periodically switched circuit N, we can construct another periodically switched network of the same topology as that of N irrespective of switching. To find out the relationship between the phasor of the network variables of N and that of let the clock frequency of be identical to that of N. Representing each variable in (10.4) using its phasors

where and are respectively the phasors of and at the frequency and are the phasors of and at the frequency respectively. To extract the relationship between the phasor of the network variables of N and that of we first eliminate by imposing the constraint

on

and then integrate (10.5) with respect to

Because

where

is Kronecker delta function defined as

from 0 to

210 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

we arrive at

Following the similar procedures, one can show that

and

THEOREM 10.1 In the steady state, for a given lumped periodically switched linear circuit, there exists another lumped periodically switched linear circuit having the same topology, switching frequency, and reversed time. The weak and strong forms of Tellegen’s theorem for periodically switched linear circuits in the phasor domain are given by (10.10), (10.11) and (10.12), respectively. Note that : The phasor of the branch voltages and currents of both N and that of are evaluated at the same frequency. The constraint imposed on the time variable of in (10.6) reveals that the switching clock sequence of is reversed as compared with that of N, as shown in Fig. 10.1. This time reversal property of the adjoint network of periodically switched linear circuits is similar to that of linear time-invariant circuits [75] and ideal switched capacitor networks [11, 79]. The difference between Tellegen’s theorem for periodically switched linear circuits and that of linear time-invariant circuits is the summation operator representing the fundamental characteristics of periodically switched linear circuits.

Adjoint Network of Periodically Switched Linear Circuits

2.

211

Inter-reciprocity

Consider a periodically switched linear circuit N. The independent sources and output branches are separated from the remaining branches, which are called internal branches. The weak form of Tellegen’s theorem for periodically switched linear circuits becomes

where and denote the number of input/output ports and that of internal branches, respectively. To find out the relation between the inputs and outputs of N, it is desirable to construct in such a way that the second summation in the above equation, i.e. the term associated with the internal branches, vanishes. To achieve this, for each internal element in N, we construct its counterpart in such that the voltages and currents of the elements satisfy

212 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

or equivalently in the time domain

For elements with more than one branch, such as, controlled sources, the network variables of all the branches of the element should be included in (10.14) and (10.15). Elements that satisfy (10.14) or (10.15) are said to be inter-reciprocal. The circuit constructed in this way is called the adjoint network of N.

3.

Adjoint Network

Adjoint network is a powerful tool for efficient noise and sensitivity analysis of linear time-invariant circuits [75] and ideal switched capacitor networks[79]. In this section we make use of Tellegen’s theorem for periodically switched linear circuits in the phasor domain to derive the adjoint network of periodically switched linear circuits.

3.1

Ideal Switches

An ideal switch has at most two distinct states : OPEN and CLOSED. An OPEN switch is characterized by Substituting these conditions into (10.15), we obtain So an OPEN switch in .N is also an OPEN switch in In the CLOSED state, the switch is characterized by Substituting these conditions into (10.15) yields A CLOSED switch in N maps to a CLOSED switch in

3.2

Resistors

Representing the voltage and current of a resistor in a periodically switched linear circuit in the phasor form and submitting them into Ohm’s law give

Adjoint Network of Periodically Switched Linear Circuits

213

Using the property of the orthogonality of exponential series, it can be shown that the phasors of the voltage and current of the resistor are related by Substituting this relation into (10.14) we obtain

where and are the phasors of the voltage and current of the resistor in the adjoint network, respectively. In order to have (10.17) hold for arbitrary we set

Eq.(10.18) reveals that the resistor R in N maps to a resistor in the same resistance.

3.3

with

Capacitors and Inductors

Following the similar procedures as those for resistors one can show that the phasors of the voltage and current of a capacitor in periodically switched linear circuits is related to each other by

where C is the capacitance. Substituting (10.19) into (10.14)

To validate (10.20) for arbitrary

we impose

214 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Eq.(10.21) indicates that the capacitor in N maps to a capacitor of the same capacitance in the adjoint network. In a very like manner as that for capacitors, one can show that an inductor in periodically switched linear circuits is also an inductor of the same inductance in the adjoint network and is characterized by

3.4

Controlled Sources

The derivation of the adjoint network of controlled sources is illustrated using a voltage-controlled voltage source with the voltage of the controlling branch be denoted by and that of the controlled branch be denoted by where A is the voltage gain. Note the current of the controlling branch is zero whereas that of the controlled branch is determined by the circuit in which the controlled branch resides. Writing (10.14) for the controlled source

To have the above expression hold for arbitrary we impose

and

Clearly the element governed by (10.24) is a current-controlled current source. Note the magnitude of the current gain is the same as the voltage gain. The controlling and controlled branches are interchanged. The adjoint network of other controlled sources can be derived in a similar

Adjoint Network of Periodically Switched Linear Circuits

215

manner. The results are the same as those for linear time-invariant circuits.

3.5

Operational Amplifiers

Operational amplifiers are usually modeled using macro models [5, 80] to minimize the simulation cost and yet to preserve the essential characteristics of these devices. For linear operational amplifiers, these macro models are essentially linear time-invariant circuits consisting of resistors, capacitors, and controlled sources that quantify the essential characteristics of operational amplifiers, such as bandwidth and output impedance. The adjoint network of operational amplifiers can therefore be constructed on an element-by-element basis. Fig.10.2 shows a singlepole macro model linear operational amplifiers and its adjoint network. The pole is determined by R and C.

216 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

4.

Transfer Function Theorem

Having derived the adjoint network of periodically switched linear circuits, in this section we make use of this theory to develop an efficient method to compute transfer functions from multiple inputs to one output of periodically switched linear circuits. Consider a periodically switched linear circuit N having current sources and voltage sources. Both current and voltage sources are

Adjoint Network of Periodically Switched Linear Circuits

217

single-tones at frequency as shown in Figs.10.4 and 10.5. Let the input of the adjoint network be a current source applied to the port corresponding to the output port of N. Further, let the outputs of be the voltage across and current through the ports corresponding to the input ports of N. Due to inter-reciprocity, the second summation in (10.13) vanishes and the expression becomes

Because

218 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

and

we obtain

where becomes

and

Consequently, (10.25)

Because the inputs of N are single tones at frequency

we have

and

Also, since the input of we obtain

is a single-tone of unity strength at frequency

Incorporating these conditions in (10.29) and noting that

Adjoint Network of Periodically Switched Linear Circuits

219

and

where and are the transfer functions from the current input to the current output and the voltage output of respectively, we arrive at

The output of N can also be obtained using superposition

where and are the transfer functions from the voltage input and the current input to the voltage output of N, respectively. Comparing (10.35) and (10.36) we obtain

and

So the amplitude of the transfer functions of N at frequency is the same as that of the corresponding transfer functions of at the same frequency. Transfer functions of other input and output configurations have also been derived and the results are summarized in Table 10.1.

220 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

THEOREM 10.2 (TRANSFER FUNCTION THEOREM) The magnitude of the transfer functions from the multiple input sources to the single output of a periodically switched linear circuit N at frequency is equal to that of the transfer functions of the adjoint network from the single input at the output port of N to the multiple outputs located at the input ports of N at frequency

With the transfer function theorem, we need to solve the adjoint network at the frequency at which the transfer functions from multiple inputs to the single output of the original circuit are to be evaluated only once to yield all the transfer functions of the original circuit.

5.

Frequency Reversal Theorem

Adjoint Network of Periodically Switched Linear Circuits

221

Consider a single-input single-output periodically switched linear circuit N shown in Fig. 10.6. Its adjoint network is also shown in the figure. Write (10.25) in the time domain

where the subscripts and identify the network variables of the output and input branches, respectively. Because and Eq.(10.39) becomes

Representing the responses of N and using (3.5), substituting the inputs of N and into (10.40), and making use of the time reversal characteristic of we obtain

Because

the output of N at frequency

is given by

where is the aliasing transfer function from the voltage input at frequency to the voltage output at frequency Further utilizing the relationship between and gives

222 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Similarly, since

we have

where denotes the aliasing transfer function from the current input at frequency to the current output at frequency Using the time reversal characteristic of we can further simplify (10.45) as

Summing up (10.47) and (10.44) and making use of (10.41) yield

Consequently

Eq.(10.49) reveals that the magnitude of the aliasing transfer function of N from the input at frequency to the output at frequency is equal to that of from the input at frequency to the output at frequency Similar results were also obtained for circuits with other input/output configurations and are tabulated in Table 10.2. THEOREM 10.3 (FREQUENCY REVERSAL THEOREM) The magnitude of the aliasing transfer functions from multiple inputs at frequency

Adjoint Network of Periodically Switched Linear Circuits

223

to the single output at frequency of a periodically switched linear circuit is equal to that of the aliasing transfer functions of the adjoint network from the single input at frequency to the multiple output at frequency The significance of the frequency reversal theorem is that the computation of the aliasing transfer functions from inputs at frequency to the output at of N requires solving the circuit at multiple frequencies With the frequency reversal theorem, these aliasing transfer functions can be obtained by solving the adjoint network at only.

6.

Examples

Two switched capacitor networks are used as examples in this section to assess both the transfer function theorem and frequency reversal theorem. Consider the bi-phase stray-insensitive switched capacitor integrator in Fig.10.7 and its adjoint network shown in Fig.10.8 [81]. The value of the circuit parameters is given in Table 10.3. All MOSFET switches are modeled as an ideal switch in series with a noisy resistor. The operational amplifiers are considered to be ideal. The transfer and aliasing transfer functions of the circuits are solved using Watsnap, an interactive computer program for general switched linear circuits [8] and the results are presented in Tables 10.4 and 10.5. As can be seen that (i) the magnitude of the transfer function from

224 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

to of N is the same as that from to of (ii) The magnitude of the aliasing transfer functions of N match the corresponding frequency components of the output of

The second example is the second-order band pass filter shown in Fig.10.9 [81] with the value of the circuit parameters given in Table 10.6. Its adjoint network is shown in Fig.10.10. The base band frequency considered is 1000 Hz. The transfer functions and aliasing transfer functions of the band pass filter and its adjoint network are solved using Watsnap and the results are presented in Table 10.7. The numerical resolution is set to 10 digits to capture the difference between the responses of the two circuits. It is observed that the magnitude of both the transfer and aliasing transfer functions of the band pass filter match those of its

Adjoint Network of Periodically Switched Linear Circuits

225

adjoint network to seven digits. The difference after the seven digits is mainly attributed to numerical noise.

7.

Summary

Tellegen’s theorem for periodically switched linear circuits in the phasor domain has been introduced and a general theory of the adjoint network of multi-phase periodically switched linear circuits has been de-

226 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

veloped. Two new theorems, namely, frequency reversal theorem and transfer function theorem, have been introduced with rigorous proof. The application of these theorems allow us to derive transfer functions and aliasing transfer functions from the multiple input sources to the sin-

Adjoint Network of Periodically Switched Linear Circuits

227

gle output at the base band frequency of a given periodically switched linear circuit by solving its adjoint network at the base band only once. As to be seen in Chapter 11, the cost of computation in both noise and sensitivity analyses of periodically switched linear circuits can be reduced significantly by employing these theorems. Several switched capacitor networks have been solved to validate these theorems.

228 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Chapter 11 FREQUENCY DOMAIN ANALYSIS OF PERIODICALLY SWITCHED LINEAR CIRCUITS

Frequency domain analysis is generally more efficient as compared with time domain analysis, particularly for linear circuits. In this chapter, we investigate the frequency analysis of periodically switched linear circuits. In Section 1, an exact frequency analysis method for multiphase periodically switched linear circuits is presented. Sensitivity analysis of these circuits using direct sensitivity analysis, adjoint network, and sensitivity network is investigated in detail in Section 2. Group delay of periodically switched linear circuits is studied in Section 3. In Section 4, noise sources encountered in periodically switched linear circuits and their characterization in the frequency domain are investigated. The noise equivalent circuits of semiconductor devices typically encountered in periodically switched linear circuits are presented. The behavior of linear periodically time-varying systems in the presence of noise inputs is studied in detail and the average power of the response of linear periodically time-varying systems to stationary noise inputs is derived. An adjoint network-based noise analysis algorithm is developed and its effectiveness and efficiency are assessed using practical examples. Statistical analysis of periodically switched linear circuits using the first-order second-moment is presented in Section 5. The chapter is summarized in Section 6.

230 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

1.

Frequency Response

The frequency response of ideal linear switched capacitor networks is obtained from of algebraic equations obtained from charge conservation at switching instants [34]. Because switched capacitor networks are a subset of general periodically switched circuits, they can be handled by methods for general periodically switched circuits. For this reason, frequency analysis of ideal switched capacitor networks will not be presented here. Interested readers are referred to references at the end of the book, such as [5], for the details on the analysis of ideal switched capacitor networks. Unlike ideal switched capacitor networks, the incomplete charge transfer characteristics of periodically switched linear circuits, due to the inclusion of resistors, inductors, and non-ideal operational amplifiers in the configuration of these circuits, requires that the circuits be depicted by differential equations. The state of the network variables at the end of each clock phase be determined by solving these differential equations using numerical integration. It was shown in Chapter 2 that the behavior of a periodically switched linear circuit in the time domain with input and a total of K clock phases can be depicted by

for The two Dirac delta functions represent the injection of the initial conditions of elements with memory at the beginning of the clock phase accounting for the effect of the initial charge of capacitors and the initial flux of inductors, and the extraction of the final conditions of these elements at so that vanishes outside the clock phase Eq.(11.1) is thus valid for The frequency domain response of the circuit is obtained by applying Fourier transform

Frequency Domain Analysis of Periodically Switched Linear Circuits

where

231

denotes Fourier transform operator,

for continuous inputs and It is seen that to obtain the frequency domain response of the circuit, the Fourier transform of the network variables at the end of each clock phase is needed. To obtain the Fourier transform of we notice that the sub-circuit associated with the clock phase is essentially a linear time-invariant circuit with the initial conditions given by To analyze this circuit, we reset the time origin to The input of the circuit becomes and the circuit is depicted by

and for

Its response at the end of the clock phase from the sampled-data simulation with the time step

is obtained

232 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where

Eq.(11.6) quantifies the time domain behavior of the circuit at the time instants at which switching occurs. The numerical algorithms for accurately computing and were given in Appendix 6.B of Chapter 6. Applying Fourier transform to (11.6) and defining

we arrive at

where

Frequency Domain Analysis of Periodically Switched Linear Circuits

233

and I denotes an identity matrix of appropriate dimensions. Note that M is frequency because (11.9) holds for dependent. P, on the other hand, is only related to the input frequency. Once is available, can be obtained from (11.2) and the complete response is obtained from

Let us now examine the properties of the preceding frequency analysis algorithm in detail : The infinite summation of Dirac impulse function terms in (11.9) reveals that the frequency response of periodically switched linear circuits to a single-tone input at frequency contains an infinite number of frequency components at frequencies The frequency component at frequency is called the baseband component and that at frequency is called the sideband component. The method is exact. No approximation is made in the derivation of the method. To obtain the accurate frequency response numerically, both and must be computed to high precision. They can be computed using the multi-step numerical Laplace inversion algorithms given in Chapter 4. Compared with the analysis of ideal switched capacitor networks, the following additional computation is required in the frequency analysis of periodically switched liner circuits Computation of

and

Computation of LU-decomposition of backward substitutions in solving

and subsequent forward and

Frequency analysis of periodically switched linear circuits is thereby more costly.

234 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

To minimize the cost of computation, it is observed that all entries of M matrix on the right hand side of (11.9) except the one at the bottom right corner is independent of frequency. This observation suggests that it is possible to transfer most of the computation into a preprocessing stage of the algorithm so that computation per frequency point is minimized. To achieve this, we wish to reduce the system matrix to the upper block diagonal form by making all blocks below the main diagonal identically zero. This is accomplished by the following block elimination technique. Pre-multiply the first row of M by and add the result to the second row. A zero is created in the position (2,1), position (2,K) becomes and the product is added to the second row of the right hand side vector. Next, Pre-multiply the second row by and add to the third row. A zero block is created in position (3,2), the position (3,K) is filled with and the third row of the right hand side vector becomes Continuing this process, we arrive at

where

Note that both E and are independent of frequency. They can be pre-computed and stored in a pre-processing step prior to the start of simulation. Once is available, can be obtained from

In applications where clock frequency is much higher than signal frequency, the variation of network variables within each clock phase

Frequency Domain Analysis of Periodically Switched Linear Circuits

235

is usually small and can be considered constant approximately. As a result, for and The following advantages are gained from this approximation: LU-decomposition of and subsequent forward and backward substitutions are no longer needed. This significantly lowers simulation cost. The conventional adjoint network approach, similar to that of ideal switched capacitor networks [79], can be employed conveniently to efficiently compute parameter sensitivity and noise, as will be seen in the following section of the chapter. A similar approach was proposed in [82, 83] where circuit equation is discretized in the time domain using backward difference formulae, usually backward Euler, with the step size equal to the width of clock phases or smaller. The resultant discrete network equations are analyzed in Note that both approaches will give a large error if the clock frequency is comparable to that of the signal frequency. As an example, consider the fifth-order elliptic switched capacitor low pass filter shown in Fig.11.1. The clock frequency is 32 kHz with two clock phases of equal width. The circuit parameters are tabulated in Table 11.1. The frequency response of the low pass is shown in Fig.11.2 with its passband shown in Fig.11.3. 0.28 dB ripple in the passband is observed.

2.

Sensitivity Analysis

The importance of small-signal sensitivity analysis was stated earlier in Chapter 6 when the time domain sensitivity analysis of periodically switched linear circuits was investigated. This section deals with sensitivity analysis of periodically switched linear circuits in the frequency domain. The normalized small-change sensitivity is defined as [5]

236 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where is the frequency response, or other design objectives, such as the zeros or poles of the transfer function of a given circuit, is usually a circuit element to which the sensitivity is evaluated. Sensitivity to parasitic elements can also be defined in a similar manner

In practice, particularly in filter design, the normalized sensitivity of the magnitude of the response to circuit elements is often needed. This sensitivity is computed from

Frequency Domain Analysis of Periodically Switched Linear Circuits

237

where

To determine the quality of a designed circuit, the sensitivities of one variable with respect to a large number of the elements are usually required. An issue essential to sensitivity analysis is how to compute these sensitivities both accurately and efficiently.

2.1

Direct Sensitivity Analysis

It was shown in Chapter 6 that sensitivity to the circuit parameter that is independent of frequency is obtained from differentiating (11.1) with respect to

238 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

One can see that the network depicted by (11.20) has the same topology as that of the original circuit. The only difference is the input given by the terms in the brackets on the right hand side of (11.20). Fourier transform of the above equation gives the frequency domain sensitivity

Frequency Domain Analysis of Periodically Switched Linear Circuits

239

To obtain the Fourier transform of the last two terms on the right hand side of (11.21), we differentiate (11.9) with respect to

where

240 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

and

Once is available, can be solved from (11.21). An notable advantage of this approach is that no approximation is made in the derivation of the method. The method is therefore exact. It yields accurate results provided that M, P and their derivatives are computed to high precision. The main drawback is that one analysis of the network only yields the sensitivity to one element. If the sensitivities of the response with respect to a large number of elements are needed, the cost of computation becomes excessive.

2.2

Sensitivity Analysis Using Adjoint Network

Sensitivity analysis of ideal switched capacitor networks using adjoint network offers superior computational efficiency as it yields the sensitivities to all circuit parameters in one network analysis of the original network and one network analysis of the corresponding adjoint network [7, 79, 84, 85]. The adjoint network approach for ideal switched capacitor networks can be applied to sensitivity analysis of periodically switched linear circuits directly when the clock frequency of these circuits is much higher than the signal frequency. We detail this in the followings. It was shown in the preceding section that when the clock frequency is much higher than the signal frequency, the frequency response

Frequency Domain Analysis of Periodically Switched Linear Circuits

of the circuit can be approximated by response of the circuit N obtained from

241

The sensitivity of the

where d is a contact vector specifying the nodes at which the output is taken and the subscript T denotes matrix transpose, is obtained from

By defining the adjoint network

of the circuit N as

where is the Fourier transform of the network variable vector of the adjoint network at switching instants, we have

It is seen that because for different and can be obtained conveniently, the sensitivity of the response to all circuit elements can be computed in one analysis of the original circuit for and one analysis of the adjoint network for It should be noted that when the clock frequency is comparable to that of the signal frequency, a large error exists if is approximated from as shown graphically in Fig.11.4. This limits the applications of this approach. To investigate Tellegen’s theorem in the presence of perturbations, let the original circuit N be made of the following types of elements : resistors, capacitors, inductors, four types of controlled sources, ideal

242 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

switches, and independent voltage sources, as shown in Fig.11.5. Further let the change in the response of N due to the changes in the value of the elements of the circuit be denoted by Approximating using Taylor series expansion to the first order yields

where and denote the voltage gain, transconductance, current gain, and transimpedance of voltage-controlled voltage sources, voltage-controlled current sources, current-controlled current sources, and current-controlled voltage sources, respectively. It was shown in Chapter 3 that the weak form of the Tellegen’s theorem for periodically switched linear circuits is given by

Also, we showed that the network variable of a periodically timevarying linear system can be written as

Frequency Domain Analysis of Periodically Switched Linear Circuits

where variation of given by

243

is the phasor of at frequency The of N due to a perturbation in element values is therefore

244 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where is the variation of the phasor of Writing (11.31) using (11.33) gives

at

Eq.(11.34) characterizes the relationship between the variation of the branch voltages and currents of the original circuit N and those of its adjoint network It is termed the incremental weak form of Tellegen’s theorem for periodically switched linear circuits in the phasor domain. It is particularly useful in deriving the sensitivity of periodically switched linear circuits, as will be seen shortly. Also note that the perturbation of circuit element value only occurs in N. In what follows we apply (11.34) to the elements of the periodically switched linear circuit of Fig.11.5 and derive the sensitivity of the response of the circuit. Resistors : A linear resistor in N and its counterpart in acterized by

are char-

Let there be a perturbation in the resistor value. Representing the voltage and current of the resistor using Taylor series expansion to the first-order and substitute the results into (11.31)

Capacitors : The behavior of a linear capacitor of periodically switched linear circuits in the phasor domain is characterized by

Frequency Domain Analysis of Periodically Switched Linear Circuits

245

The first-order approximation of the variation of capacitor current due to a perturbation gives

Inductors : In a like manner as that for capacitors, one can show that for linear inductors

Controlled Sources : Consider a voltage-controlled voltage source with voltage gain in N. Its counterpart in is a current-controlled current source with controlling and controlled branches interchanged and gain as shown in Chapter 3. Let the controlling and controlled branches be identified with the subscripts 1 and 2, respectively. Because

we have in the phasor domain

246 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

As a result,

Expressions for other types of dependent sources can be derived in a similar manner and are left as an exercise to readers. Ideal Switches : Let there be a perturbation in the element values of N such that the voltage and current of an ideal switch vary from and respectively. If the switch is CLOSED at then, and Moreover, since its counterpart in is also a CLOSED switch at then As a result

where and are the voltage and current of the switch in respectively. Similarly, one can show that (11.43) also holds if the switch is OPEN. Writing (11.43) in the phasor domain gives

Frequency Domain Analysis of Periodically Switched Linear Circuits

247

Inputs : Because the input of N is an ideal voltage source and the corresponding branch in is a short-circuit, we have

or equivalently in the phasor domain

As a result

Output : The output branch of N is an open-circuit, This results in Because the input of is a single tone of unity strength at frequency we have

Thus

Substituting the preceding results into (11.34) yields

248 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

By comparing (11.50) with (11.30), we obtain the sensitivities of the response of the circuit with respect to the elements. The results are given in Table 11.2. Several comments are made with respect to the preceding development: Once the frequency responses of N and are available, the sensitivity of the output of N with respect to any element of N can be computed conveniently by substituting appropriate network variables. Only two network analyses, one for N and the other for are needed in computing all sensitivities. Both the baseband and sideband components of the response of the original circuit and its adjoint network contribute to sensitivities of the original circuit in the baseband, as shown in Fig.11.6. This is a unique characteristic of periodically switched linear circuits. If there is no switching, i.e. is always zero, Table 11.2 will simplify to the well-known sensitivity of linear time-invariant circuits [75]. For practical periodically switched linear circuits, the convergence of sensitivity is ensured by the low-pass characteristics of RC networks formed by the channel resistance of MOSFET switches and their parasitic shunt capacitances at high frequencies because the amplitude of network variables decreases asymptotically with frequency and eventually dies off. The rate of convergence of sensitivity, however, depends upon the frequency characteristics of the network variables with which sensitivities are associated.

Frequency Domain Analysis of Periodically Switched Linear Circuits

249

The cost of sensitivity analysis of periodically switched linear circuits is mainly the computation needed for solving N and at the input frequency. In Appendix 11.A of this chapter we show that the adjoint network can be solved efficiently by utilizing the intrinsic relationship between N and

250 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

2.3

Sensitivity Analysis Using Sensitivity Network

In the preceding section, the adjoint network based sensitivity analysis method for periodically switched linear circuits was developed rigorously. In this section, we show that the same results can also be obtained in a more illustrative manner using a technique known as sensitivity network initially developed by T. Trick for linear time-invariant circuits [75] and R. Davis for ideal switched-capacitor networks [86]. The essence of the sensitivity network approach is as follows : The fundamental laws governing a lumped circuit N are KCL and KVL [5]

Frequency Domain Analysis of Periodically Switched Linear Circuits

251

where A is the incidence matrix, and are the branch current and voltage vectors, respectively, and is the nodal voltage vector. Differentiating (11.51) with respect to a circuit element gives

Eqs.(11.52) are the governing equations of a derived network whose network variables are the derivatives of those of N with respect to is called the sensitivity network of Clearly, the solution of gives the sensitivity of N with respect to Note that by changing the element to we obtain the sensitivity network of Continuing this process, one can derive all sensitivity networks. Important to note that these sensitivity networks have the same topology as that of N. As an illustration, consider a capacitor C. Since to derive the sensitivity network of the capacitor, we differentiate the above expression with respect to C

252 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

The variables associated with the capacitor in the corresponding sensitivity network are the sensitivity current

and sensitivity voltage

The capacitance remains the same. The constitutive equation of the capacitor in the sensitivity network is the same as that in the original circuit. An ideal current source is added in parallel with the capacitor. It is the input of the sensitivity network. The substitution for the capacitor in the corresponding sensitivity network is given in Fig.11.7. Similarly, one can derive the substitutions for other basic elements in corresponding sensitivity networks. The results are shown in Fig.11.7. To compute the sensitivity of the response of a periodically switched linear circuit N with respect to capacitor C, the corresponding sensitivity network is needed. The input of is a current source connected in parallel with C of current

Because

therefore

This is equivalent to have an infinite number of current sources of value connected in parallel with the capacitor, as depicted in Fig.11.8. Each of these current sources generates an output that also contains an infinite number of frequency components. The complete output of at frequency is obtained by summing up the contributions of all the added current sources at frequency

where source at frequency

is the aliasing transfer functions from the current to the output of at frequency A care-

Frequency Domain Analysis of Periodically Switched Linear Circuits

253

ful inspection shows that there exist two difficulties in solving (11.56). First, to obtain has to be solved at frequency Secondly, if the sensitivities of the response

254 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

with respect to M elements are required, then a total of M sensitivity networks have to be constructed and solved. This amounts to excessive computation. To avert these difficulties, we notice that these sensitivity networks are topologically identical. This observation suggests that only one common adjoint network needs to be constructed for all sensitivity networks. In the above capacitor case, the magnitude of is equal to that of the frequency component of the capacitor voltage in the adjoint network provided that the input of is of unity strength. Because the adjoint network of is the same as that of N, we have

Frequency Domain Analysis of Periodically Switched Linear Circuits

255

where is the frequency component of the voltage of the capacitor in Substituting (11.57) into (11.56) gives

Note (11.58) is identical to the result given in Table 11.2. The above analysis reveals that the sensitivities of periodically switched linear circuits developed using the adjoint network approach can also be derived using the sensitivity network technique. The differences between the sensitivity networks of linear time-invariant and periodically switched linear circuits also become apparent. For each element of a linear timeinvariant circuit, there is only one corresponding sensitivity network. However, for each element of a periodically switched linear circuit, there are an infinite number of sensitivity networks.

2.4

Numerical Examples

In this section, two periodically switched linear circuits are used to demonstrate the effectiveness of adjoint network based sensitivity analysis method. The first example is the stray-insensitive switched capacitor integrator shown in Fig.11.9 with circuit parameters given in Table 11.4 The operational amplifier is modeled as a single-pole device with unity gain frequency 700 kHz. Its equivalent circuit and the adjoint network are shown in Fig.10.2. The sensitivity of the output with respect to at 1 kHz was computed using the adjoint network approach and the results are plotted in Fig.11.10 for the real part and Fig.11.11 for the imaginary part. The normalized sensitivity of the magnitude of the response to is shown in Fig.11.12. For the purpose of comparison, the same sensitivity was computed using Watsnap, a commercial CAD tool for periodically switched linear circuits [8] that calculates sensitivity using the direct sensitivity analysis approach presented earlier in this chapter. The result is :

256 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

To assist in analysis, the relative difference

defined as

where is the number of sidebands considered in adjoint network based sensitivity analysis is employed to quantify the rate of convergence of

Frequency Domain Analysis of Periodically Switched Linear Circuits

257

258 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

sensitivity. In this example, and The rapid convergence of the sensitivity is due to the fast decaying profile of the voltages of in both the integrator and those of its adjoint network, as shown in Fig.11.13. The second example is the switched capacitor band-pass filter shown in Fig.11.14 with the value of circuit parameters given in Table 11.5. The sensitivity of the response with respect to at 1 kHz was computed using the proposed method and the results are plotted in Fig.11.15 for the real part and Fig.11.16 for the imaginary part. It is seen that both the real and imaginary parts of the sensitivity converge with the increase in the number of sidebands. The rate of convergence is clearly slower as compared with the previous example. The normalized sensitivity of the magnitude of the response to is shown in Fig.11.17. The relative difference versus the number of sidebands in sensitivity analysis is plotted in Fig. 11.18. It is seen that the relative difference decreases with an increase in the number of sidebands monotonically. The voltage of of the band-pass filter and that of its adjoint network are shown in

Frequency Domain Analysis of Periodically Switched Linear Circuits

259

260 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Fig.11.19 (100 sidebands). The voltage across in the adjoint network is shifted down by 100 dB in order to fit the two voltages in the same figure. As observed, both voltages decrease slowly with frequency. As a result, more sidebands are needed to yield a converged value.

3.

Group Delay Analysis

Group delay an essential design parameter for precision switched capacitor and switched current filters, characterizes the dependence of the phase of the response of a given circuit on the frequency of the input of the filters

Making use of

we obtain

Frequency Domain Analysis of Periodically Switched Linear Circuits

261

262 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Because the response of periodically switched linear circuits to the input contains an infinite number of frequency components at frequencies the derivative of the response to frequency can be obtained by differentiating (11.2) with respect to rather than leading to the group delay network depicted by

The final conditions of the group delay network are obtained from

Frequency Domain Analysis of Periodically Switched Linear Circuits

with

263

obtained from

Because the state transition matrix have

is independent of the input, we

264

COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

can be computed in a similar manner as

given earlier. The

group delay network can be solved using the same approach as that for sensitivity networks. As an example, consider the fifth-order switched capacitor low pass filter shown in Fig.11.20. The parameters of the circuit are tabulated in Table 3. The group delay is obtained using the preceding method and is shown in Fig.11.21. It is seen that the group delay has a peak of 660 ms at 3.55 kHz approximately.

4.

Noise Analysis

The most commonly encountered types of noise in silicon integrated circuits are thermal noise, shot noise, and flicker noise [2]. These noise sources are inherent to silicon devices. In this section, the power spectral density of the output of linear periodically time-varying systems to random inputs is derived using the theorems of linear periodically timevarying systems presented in Chapter 3. Such an analysis provides much

Frequency Domain Analysis of Periodically Switched Linear Circuits

265

266 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

needed theoretical foundation for noise analysis of periodically switched linear circuits. Noise sources encountered in integrated circuits and their power spectral density are investigated in this section. Noise analysis of periodically switched liner circuits using adjoint network is presented and the output noise power of several no-ideal switched capacitor networks is analyzed using the method and the results are compared with both measurement results and the results from other computer-aided design tools.

4.1

Noise Characterization

Noise is a random signal with zero mean. The behavior of a noise signal in the time domain is characterized by its autocorrelation function and in the frequency domain by its power spectral density. The

Frequency Domain Analysis of Periodically Switched Linear Circuits

autocorrelation function of a noise signal the joint moment of and [87]

denoted by

267

is

where the asterisk denotes complex conjugation, and denote two distinct time instants. quantifies the connection between and statistically. If the

is the mean-square value or the average power of The autocovariance of denoted by is defined as the joint moment of and and it relates to the autocorrelation function by

268 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

and are said to be uncorrelated if A stochastic process is said to be stationary in the strict sense if all of its statistical properties are time-invariant. It is stationary in the wide sense if its mean is constant and its autocorrelation function satisfies

where Note that the wide-sense stationarity involves only the first and second-order moments. Because it is generally difficult to evaluate the high-order moments of a given stochastic process, the wide-sense stationarity is widely used in practice. If both the mean and autocorrelation function of a wide-sense stationary noise signal are periodic in time, the noise signal is said to be cyclo-stationary in the wide sense. In the frequency domain, the power spectral density of a noise signal denoted by depicts the spectrum of the power of the noise signal. It is defined as the Fourier transform of the autocorrelation function of

Eq.(11.71) is also known as Wiener-Khintchine theorem [56]. If where A is a constant, then Processes of such characteristics are said to be white. For a wide-sense process if i.e. then

Eq.(11.72) reveals that the area under specifies the meansquare value of The power spectral density of noise sources encountered in integrated devices has been investigated extensively and is readily computable.

Frequency Domain Analysis of Periodically Switched Linear Circuits

269

Noise analysis is thereby a task of how to compute the output noise power of a given circuit containing a large number of noise sources efficiently and accurately. In was shown in Chapter 3 that the frequency response of a linear periodically time-varying system is given by

If the input of a linear periodically time-varying system is a stochastic process, then the response of the system is also a stochastic process. The autocorrelation function of the response is given by [87]

The two-dimensional power spectral density of denoted by is defined as the two-dimensional Fourier transform of the autocorrelation function of [87]

where and are the Fourier transform of respectively. Substituting (11.75) into (11.74) gives

Eq.(11.76) is further simplified to

and

270 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Thus

Because

and

therefore

Eq.(11.81) is valid for general linear periodically time-varying systems. If is wide-sense stationary, then Consequently

Let

then

Frequency Domain Analysis of Periodically Switched Linear Circuits

271

Consequently

Substituting (11.84) into (11.81) yields

The time-varying power spectral density of the response is obtained by taking the inverse transform of the two-dimensional power spectral density with respect to the frequency variable corresponding to the excitation time

Therefore

Define

272 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

we arrive at

As can be seen that is periodic in with period Fig.11.22 illustrates the sampling of a stationary random input using a sampleand-hold mechanism.

The average power spectral density of the response over a period, denoted by is computed from

Because

we obtain

Frequency Domain Analysis of Periodically Switched Linear Circuits

273

Eq.(11.92) characterizes the average power spectral density of the response of liner periodically time-varying systems with stationary inputs. It can be used to compute the average output noise power of periodically switched linear circuits. A few comments are made : If the power spectrum of the input noise is broad-band, Nyquist theorem is violated. As a result, the sideband components of the input noise are folded back to the base band. A pictorial illustration is given in Fig.11.23 where the band width of the input noise is assumed to be i.e.

The band width of the circuit is assumed to be infinite and its gain is unity. Due to aliasing effect, the output noise power is 5 times that of the input noise, i.e.

The contribution of the sideband components of the input noise source clearly dominates the total output noise power. For practical circuits with white noise sources, the total output noise is computed from

where the maximum number of sidebands considered, N, is determined by the equivalent noise bandwidth of the circuits. If there is no switching in the circuits, i.e. Eq.(11.92) simplifies to the familiar expression for linear time-invariant circuits.

274 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

It is evident that even though in this case the input noise is broadband, the side band components of the input noise, however, do not affect the output noise power in the base band. If there are a total of M uncorrelated noise sources in a periodically switched linear circuit, the output noise power is obtained from

where N is the maximum number of sidebands considered. It is seen that the computational cost in noise analysis of periodically switched

Frequency Domain Analysis of Periodically Switched Linear Circuits

275

linear circuits arises from (i) a large number of noise sources and (ii) the aliasing effect. Methods that compute the output noise power directly from (11.97) is often referred to as the brute-force method because in this approach not only the contribution of each noise source is computed separately, the contribution of every sideband component of the same noise source is also calculated individually.

4.2

Noise Sources

Thermal noise : Thermal noise, also known as Johnson noise, in recognition of the first observation of the phenomenon by J. B. Johnson [88], is generated by the random thermal agitation of mobile carriers. It is due to the random departure and return of mobile charges in thermal equilibrium. The power of thermal noise is directly proportional to temperature. The band width of thermal noise at room temperature is around 6000 GHz and time samples separated by 0.17 ps are considered to be uncorrelated [89]. In almost all cases, thermal noise is treated as a stationary process. The distribution of thermal noise is Gaussian. The power spectral density of the thermal noise generated by a resistor is given by

where R is the resistance of the resistor, is Boltzmann constant, and T is the absolute temperature in degrees Kelvin. Eq.(11.98) was first derived by Nyquist [90] from thermodynamics and the exchange of energy between resistive elements in thermal equilibrium and is known as Nyquist law. Shot noise : Shot noise of semiconductor devices is caused by the random combination of electron-hole pairs and the random diffusion of minority carriers across depletion region [91]. This phenomenon is depicted by a stochastic process representing the sum of a large number of independent events occurring at random time instants with an average rate

276 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where is a pulse shape function and is the time at which the pulse occurs. Note is causal. The distribution of is Poisson. The power spectral density of is given by Curson’s theorem [92, 93]

where is the Fourier transform of The band width of shot noise is inversely proportional to the transit time required by the carriers to cross the depletion regions, and is in high gigahertz ranges [89]. Shot noise is often treated as a stationary white process. In the extreme case where is approximated by Dirac impulse function because we have the following Schottky’s theorem for the power spectral density of the shot noise of pn-junctions [56]

where and

is the average forward biasing current of the pn-junction is the charge of an electron. It should be noted

that the above result is only valid for where time for mobile carriers to cross the potential barriers.

is the

Flicker noise : Flicker noise, also known as noise, exists in both active devices and passive resistors. The mechanism of noise in semiconductor devices has been studied extensively and tends to converge to two basic noise theories : Carrier mobility fluctuation model of flicker noise [94] - Flicker noise is generated by the scattering of mobile carriers due to the collision with the crystal lattice of silicon and impurities. Carrier density fluctuation model of flicker noise [95] - Flicker noise is caused by the trapping and de-trapping of the mobile carriers in the traps located at the surface of gate oxide.

Frequency Domain Analysis of Periodically Switched Linear Circuits

277

For MOSFET transistors, it was found that the mobility fluctuation model only holds when the devices are operated in the triode region where the inverse layer can be approximated by a homogeneous resistor. The density fluctuation theory, on the other hand, predicts noise accurately in all regions of MOSFET transistors. Flicker noise is always associated with a direct current and is often modeled as a stationary process with power spectral density given by [2]

where I is the average current, is a process and temperature dependent constant, and are constants whose values are in the ranges of and and is frequency. For most electronic devices, flicker noise surpasses thermal and shot noise at low frequencies. Extensive experiments show that there is no change in the shape of the power spectral density of flicker noise even at extremely low frequencies [96]. The upper frequency limit of flicker noise is difficult to detect as it is usually masked by the floor of thermal noise. The corner frequency, defined as the frequency at which the power spectral density of thermal/shot noise and that of flicker noise intercept, is often used as the upper bound of the frequency of flicker noise for MOSFETs and is in the range of several MHz.

4.3

Noise Equivalent Circuits

The equivalent circuits of integrated devices with noise sources included are used in noise analysis of electronic circuits. Resistors : Physical resistors such as diffusion resistors and polysilicon resistors, generate thermal noise. At low frequencies, a resistor can be modeled as a hypothetical noise-free resistor in series with a random voltage generator or a noise-free conductor in parallel with a random current generator, as shown in Fig.11.24. At high frequencies, the parasitic capacitances associated with the resistors must also be taken into account.

278 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Bipolar Junction Transistors : Bipolar junction transistors (BJTs) are widely used for high-frequency and high-speed applications due to their large intrinsic There are four main noise sources associated with a BJT : (1) thermal noise generated by the base resistance, (2) shot noise in the base current, (3) flicker noise in the base current, and (4) shot noise in the collector current. Flicker noise of BJTs arises mainly due to the generation/recombination process in the emitterbase depletion region and the trap/de-trap of carriers by the oxide located above the emitter-base junction. The former usually predominates over the later [97]. As compared with MOSFET transistors, noise of BJTs is much smaller and manifests itself in frequency regions several decades lower [98]. The base resistance consists of the intrinsic base resistance and extrinsic base resistance. The intrinsic base resistance is usually larger than the extrinsic [97]. However, due to the effect of current crowding [2], the intrinsic resistance and the associated thermal noise can be reduced by a large collector current. The extrinsic base resistance is made up of the bulk and contact resistances. It can be reduced by increasing the number of contacts in

Frequency Domain Analysis of Periodically Switched Linear Circuits

279

the base and reducing the lateral distance between the emitter and base contacts. The thermal noise generated by other parasitic resistances of bipolar junction transistors, such as emitter and collector bulk resistances, also constitute the overall noise of the device. However, because the emitter is heavily doped, the associated thermal noise is small [99]. The thermal noise originating from the collector resistance is often surpassed by the noise of collector loads. The noise equivalent circuit of BJTs in the forward active region is given in Fig. 11.25. When a BJT is operated in an ON/OFF mode, its equivalent circuit can also be obtained by including the above identified noise sources in the large-signal equivalent circuit of BJTs.

MOSFET Transistors : MOSFET transistors are building elements of switched capacitor and switched current circuits. Recent advance in CMOS technology has also made CMOS a viable technology choice for high-speed and RF applications. Noise generated by the intrinsic part of a MOSFET transistor consists of (1) shot noise in the gate leakage current, (2) thermal noise due to the random thermal motion of mobile carriers in the inversion layer and (3) flicker noise due to channel charge density fluctuation caused by the traps at the oxidesilicon interface [100]. The shot noise of the gate leakage current is usually negligible. To ensure a stable operation, MOSFET transistors

280 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

are usually biased in strong inversion where is the surface potential and is the Fermi potential [40]. The power of the flicker noise originated in the channel is given by [98]

where is the drain-source quiescent current. W and L are the width and length of the channel, respectively, is the gate transconductance, is the gate capacitance per unit area, and is the surface mobility of charge carriers in the channel. MOSFET transistors exhibit the highest noise among all active integrated devices [98]. As compared with BJTs, the corner frequency of the flicker noise of MOSFET transistors could extend to mega Hertz ranges [2]. The thermal noise generated by a MOSFET transistor in saturation is mainly due to the fluctuation in the drift current in the channel [40]. The fluctuation in the diffusion current is negligible as the drift current predominates over the diffusion current when the device is in the strong inversion. The thermal noise of the drift current is given by [101]

where and are the gate transconductance and substrate transconductance, respectively. Eq.(l 1.104) is valid only in the saturation region. It gives erroneous results if the device is in the triode region because in this region and zero thermal noise is predicted. In [102], the channel conductance, is added to (11.104) to represent the thermal noise generated by the device in ohmic region.

Frequency Domain Analysis of Periodically Switched Linear Circuits

281

where

is the pinch-off voltage. Note varies with linearly from 1 at (where to at (where is small). The validity of (11.105) was questioned in [103] because it differs from the theoretical results given in [104, 40]. Moreover, the noise power predicated by (11.105) deviates notably from SPICE simulation when different levels of MOSFET models are used. It was shown in [40] that under the quasi-static condition, the power of the thermal noise originating from the drift current in the channel of a strongly inverted MOSFET transistor is given by [40]

where

Here

and is the channel conductance when When the first-order approximation [105] is used, it is obtained from

It should be noted that (11.107) is only valid for long channel devices. In [106], the modulation of the channel length and the degradation

282 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

of surface mobility due to the high lateral field were also included in modeling the noise of MOSFET transistors and more complex results were obtained. Switches in periodically switched linear circuits are realized using NMOS transistors operated in the triode region to minimize and the sub-threshold regions. CMOS pass-transistor gates with complementary clocks are also used extensively to maximize signal dynamic range and to minimize the ON-resistance [107, 81], however, at the expense of complex layout. In the triode region, because is small, the channel can be approximated by a homogeneous conductor with conductance given by

The thermal noise generated by the channel is computed in the same way as that of resistors. When the transistor is in the sub-threshold region, we follow the same treatment as that in CSIM [105] and BSIM [108], and set Consequently, the noise is zero. A lowfrequency noise equivalent circuit of MOSFET switches is given in Fig.11.26. This model has been used widely due to its simplicity [109, 57, 110, 111].

Noise generated by the extrinsic part of the device includes the thermal noise originating from the source and drain bulk resistances, and polysilicon gate series resistance. Among them, the thermal noise of the polysilicon gate series resistance predominates. To analyze

Frequency Domain Analysis of Periodically Switched Linear Circuits

283

the noise behavior of MOSFET transistors at high frequencies, both the intrinsic and parasitic capacitances must be included. A noise equivalent circuit of MOSFET transistors in saturation is given in Fig.11.27. At very high frequencies, the thermal noise generated by the substrate resistance should also be taken into consideration as it contributes nearly 20% of the total noise of MOS transistors [112]. It is also worth noting that the so-called gate-current fluctuation [56, 100, 113] is due to the thermal noise originating in the channel and coupled via the gate-channel capacitance at high frequencies. It should not be considered as an independent nor a correlated noise source.

Operational Amplifiers : Noise generated by an operational amplifier is mainly due to the noise generated in the differential input stage. The contribution of the noise generated in the following stages is usually negligible. To model the noise of the differential input stage in either open- or short-circuited cases, two pairs of noise-current generator and noise-voltage generator are needed at each input terminal of the operational amplifier and the operational amplifier is thereby treated as a hypothetical noise-free device, as shown in Fig. 11.28(a) in which and are current-noise generators, and are voltage-noise generators [114]. Because and represent common-mode signals, they produce virtually no differential output if the common-mode rejection ratio of the operational amplifier is high. Also, for MOSFET input stages, is negligible. They can be removed from the equivalent circuit without introducing large errors [2]. Also, because the two voltage-noise generators are in series with

284 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

the input, they can be combined into a single voltage-noise generator, i.e. provided that the correlation is small. This leads to a simplified noise equivalent circuit of the operational amplifier shown in Fig. 11.28(b) [115, 116, 117]. If the operational amplifier is realized using BJTs, then consists of the thermal noise of the base resistances, shot noise in base current and the input-referred noise originating in the collector currents of appropriate transistors in the differential stage. If the input stage is realized using MOSFET transistors, then is made of the input-referred thermal and flicker noise originating in the inversion layer of the appropriate MOS transistors in the stage. It should be noted that when the operational amplifier is operated at low frequencies, is independent of frequency. However, at high frequencies, since the gain of the differential input stage varies with frequency, the input-referred noise source also changes with frequency. Consequently, is frequency-dependent.

4.4

The Algorithm

The equivalent noise band width of periodically switched linear circuits usually exceeds the clock frequency by orders of magnitude. The under-sampling of the wide band noise results in a strong aliasing effect. The noise power folded over from the side band components of the noise sources dominates the output noise power. In the preceding section, we have shown that the response of periodically switched linear circuits with

Frequency Domain Analysis of Periodically Switched Linear Circuits

285

stationary inputs is cyclo-stationary. Its average power spectral density is time-independent and is computed from

where is the power spectral density of the noise at the frequency is the aliasing transfer function from the noise source at the frequency to the output at the frequency M is the number of noise sources, and N is the maximum number of sidebands considered. Both the transfer and aliasing transfer functions from the noise sources to the output are needed in computing the output noise power. It was shown in Chapter 10 that the transfer functions and aliasing transfer functions of periodically switched linear circuits can be computed efficiently by using the adjoint network. For a given periodically switched linear circuit N,

i) Replace all noisy elements in the circuit with their corresponding equivalent circuits. ii) Define a set of constant vectors nodes to which these noise sources are connected.

to specify the

iii) Further, the output of N is specified by a constant vector d, i.e.

where

is the response of N.

of N is constructed and the system matrices iv) The adjoint network of are obtained as per details given in Appendix 11.A.

v) Using transfer function theorem of this chapter, the transfer function from the noise source to the output of N at denoted by is obtained from

286 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where is the response of at the frequency with an input of unity amplitude at the frequency The location of the input of is specified by d. vi) Further from frequency reversal theorem, the aliasing transfer function from the noise source at the frequency to the output at the frequency denoted by is obtained from

where sponse of

is the

order frequency component of the re-

vii) The transfer and aliasing transfer functions from other noise sources to the same output of N can be obtained by substituting appropriate vectors into (11.114) and (11.115), respectively. Since all noise sources are assumed to be uncorrelated, the total output noise power of N is obtained by summing up the contributions of all noise sources.

Eq.(11.116) reveals that for a given periodically switched linear circuit, once the location and type of the noise sources are known, the power spectral density of the output of the circuit at a given base band frequency can be computed efficiently by solving the adjoint network.

4.5

Numerical Examples

The first example is the the switched capacitor low pass filter shown in Fig.11.29 with the value of the circuit elements given in Table 11.7. Only the thermal noise of MOSFET switches is considered. Five different input noise band widths, 250 kHz, 500 kHz, 1 MHz, 5 MHz, and 10 MHz were considered in analysis. This corresponds to the foldover of 50, 100, 200, 1000, and 2000 sidebands. The output noise power was

Frequency Domain Analysis of Periodically Switched Linear Circuits

287

calculated using the method presented in this section and the results are plotted in Fig.11.30, together with the measurement data extracted from [109]. It is seen that the output noise power increases with the increase in the number of sidebands folded over. It eventually converges to a finite power irrespective of any further increase in the number of sidebands. This observation reveals the existence of a finite equivalent noise bandwidth of this circuit. The finite noise bandwidth of the circuit is due to the low-pass mechanism formed by the channel resistance of the MOSFET switches and the shunt capacitances. It is the finite noise bandwidth of the circuit that results in a finite output noise power. It is also seen that simulation results agree very well with the measurements. The second example investigated is a switched capacitor integrator with four non-overlapping phases of equal width [57]. The schematic of the integrator and its noise equivalent circuit are shown in Fig.11.31 with the value of its elements given in Table 11.8. The thermal and shot noise of the operational amplifier are represented by an equivalent noisevoltage generator The flicker noise of the operational amplifier and that of the MOSFET switches were not considered in the analysis. The model of the noise-free operational amplifier is the same as that given in Chapter 10. The output noise power was calculated with input noise band width of 250 kHz, 500 kHz, 1 MHz, 5 MHz,

288 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

and 10 MHz. This corresponds to the foldover of 25, 50, 100, 500, and 1000 sidebands. The results are plotted in Fig.11.32, together with

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289

the measurement data extracted from [57]. It is seen that with the increase in the number of sidebands folded over, the total output noise power increases monotonically. It eventually converges to a finite power irrespective of any further increase in the number of sidebands. The simulation results are in a very good agreement with the measurement data. The efficiency of the adjoint network algorithm is demonstrated by comparing the CPU time of the algorithm with that of the brute-force method. Fig.11.33 shows the CPU time of the proposed algorithm on computing the output noise power of the switched capacitor integrator with (a) only the noise of M1 considered and (b) all noise sources, i.e. the noise of and operational amplifier, considered. As can be seen that the amount of time spent in both cases is nearly the same. This observation validates our earlier statements on the advantages of using the transfer function theorem in noise analysis. Also observed that the cost of computation is linearly proportional to the number of sidebands folded over.

290 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

To investigate the efficiency gained from using the frequency reversal theorem over the brute-force method, the noise source of and that

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291

of the operational amplifier are turned off, and only the noise source of is activated. The output noise power of the integrator was computed using both methods. Fig.11.34 gives the ratio of the CPU time of the adjoint network method to that of the brute-force with various step sizes used in computing and It is seen that the speedup obtained using the frequency reversal theorem is significant. Also, the speedup is step size dependent. It increases with the decrease in the step size. These results are expected since the lack of efficiency of the bruteforce method is due to the repetitive calculation of at both the base band and sideband frequencies. Also, the accuracy of the computation of is inversely proportional to the step size whereas the computational cost is directly proportional to the step size. The deficiencies of the brute-force method are eliminated once the frequency reversal theorem is employed. For every base band frequency, the adjoint network is solved only once at the frequency. In other words, one calculation of of the adjoint network per base band frequency is required. The corresponding high-order frequency components are obtained using LU-

292 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

factorization and forward/backward substitutions. The computation required for LU-factorization is much less than that of a significant speedup is therefore achieved. For instance, at step size and with 100 sidebands considered, the adjoint network-based method is nearly 15 times faster than the brute-force method when only one noise source is considered. Also observed is that with the increase in the number of sidebands to be folded over to the baseband, the speed up plot in Fig.11.34 becomes flat. This is because the cost of LU-decomposition of start to dominate the total cost of computation.

The third example is switched capacitor band pass filter shown in Fig.11.14 [118] with the value of the circuit parameters given in Table 11.5 except the unit gain frequency of the operational amplifier. The unit-gain frequency of the operational amplifier is set to be 10 GHz such that the noise band width is mainly determined by the channel resistance of the MOSFET switches and shunt capacitances. The inputreferred white noise of the operational amplifiers is The

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293

flicker noise of the operational amplifiers is at 10 Hz. Note that we have used the same parameters as those in [118] for the purpose of direct comparison. There are sixteen white noise sources associated with MOSFET switches, two white noise sources and two flicker noise sources related to the operational amplifiers. The output noise power of the band-pass filter is computed and the results are plotted in Fig. 11.36, together with the simulation results extracted from [118]. The number of sidebands folded over are 3000, 10000, 30000, 50000, 60000, and 70000, which correspond to noise band width of 0.384 GHz, 1.28 GHz, 3.84 GHz, 6.4 GHz, 7.68 GHz, and 8.96 GHz, respectively. The large equivalent noise band width is mainly due to the small channel resistance of MOSFET switches, small shunt capacitances, and large unit-gain frequency of the operational amplifiers used in the simulation. It is seen that our results agree well with those from [118]. The effect of flicker noise is investigated by computing the output noise power due to the flicker noise sources of the operational amplifiers only. The output noise power in the base band with 0, 2 and 5 sidebands folded over, are computed and the results are shown in Fig.11.37. It is seen that the effect of the flicker noise dominates at frequencies less than 500 Hz. The output noise power due to the flicker noise sources saturates when 5 sidebands are considered, indicating that only a few sidebands are needed. For the purpose of comparison, the output noise power due to the thermal noise sources with no fold-over is also plotted in Fig.11.37. It is seen that the contribution of the flicker noise sources is negligible as compared with that of the thermal noise sources at high frequencies. This is the reason why flicker noise is generally neglected in noise analysis of switched analog circuits.

5.

Statistical Analysis

Similar to the first-order second-moment method for statistical analysis of periodically switched linear circuits in the time domain, the mean and variance of these circuits in frequency domain can also be analyzed using this approach and the results are given by (11.117) for the mean and (11.118) for the variance.

294 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

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295

Incorporating these results into frequency analysis of periodically switched linear circuits with the derivatives computed using the method given in the sensitivity analysis, we are able to compute the mean and variance of these circuit in the frequency domain [119]. Consider the switched capacitor band pass filter shown in Fig.11.14. No correlation is assumed among the circuit elements. Further, a uniform coefficient of variance, denoted by is used for all circuit parameters. The dependence of the response of the circuit at on the coefficient of variance of the circuit parameters is plotted in Fig.11.38. Monte Carlo analysis is also carried out. Specifically, the value of circuit parameter is generated from

296 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where N(0,1) is a normally distributed random variable with zero mean and unity standard deviation. The value of each circuit parameter is generated in this way and the circuit is solved for each set of random circuit parameters. The number of samples used in Monte Carlo analysis is 500. The results of Monte Carlo analysis are also plotted in Fig.11.38. It is seen that the results from the first-order second-moment compare well with those of Monte Carlo analysis when the coefficient of variance of the circuit parameters is small. The variance of the response for various coefficient of variance of the circuit elements is plotted in Fig.11.39, together with those of Monte Carlo analysis with 500 samples. Again, they agree well when the coefficient of variance of the elements is low. Deviation, however, becomes noticeable once is large. This observation agrees well with our earlier statements that the error of FOSM

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297

method grows if the coefficient of variance of circuit parameters is large. Also observed is that of the response exhibits a nonlinear characteristic when of the circuit parameters is large, revealing the limitations of FOSM method.

6.

Summary

Frequency analysis of periodically switched linear circuits has been presented in this chapter. We have shown that the exact response of multi-phase periodically switched linear circuits can be obtained by using a time domain analysis for the value of network variables at the switching instants and a frequency domain analysis of the circuit equation depicting the circuits. The cost of computation is much higher as compared with that of ideal switched capacitor networks. The method handles both ideal switched capacitor networks and general periodically switched linear circuits consisting of all linear elements and switches. In sensitivity analysis of these circuits, direct sensitivity analysis, adjoint network approach, and sensitivity network approach have been

298 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

presented and their effectiveness has been compared using example circuits. We have shown that direct sensitivity analysis approach yields the exact sensitivity of periodically switched linear circuits but at the cost of computation because each network analysis only yields sensitivity to one circuit parameter. To compute the sensitivity of the response to multiple circuit elements efficiently, adjoint network approach is preferred. We have shown that the sensitivity of the response of periodically switched linear circuits at a frequency in the base band consists of the contribution of the network variables both at the frequency in the base band and those at corresponding sidebands. Similar approaches have been extended to group delay analysis of periodically switched linear circuits. To analyze the output noise power of periodically switched linear circuits, noise sources encountered in periodically switched linear circuits and their characterization in the frequency domain have been investigated in detail. Noise equivalent circuits of semiconductor devices typically encountered in periodically switched linear circuits have also been

APPENDIX 11.A

299

derived. The behavior of linear periodically time-varying systems in the presence of noise inputs has been studied using autocorrelation function and the network functions derived in Chapter 3. We have shown that although the power spectral density of the output of periodically switched linear circuits with stationary noise sources is periodically time-varying, its average value is time-invariant and is determined from the aliasing transfer functions from the noise sources to the output of the circuits. An adjoint network-based noise analysis algorithm has been developed. The effectiveness and efficiency of the method have been assessed using practical examples with measurement results. Finally, statistical analysis of periodically switched linear circuits in the frequency domain using the first-order second-moment has been developed. We have shown that the method is computationally efficient and gives accurate results when the coefficient of variance of circuit elements is low.

APPENDIX 11.A: Solution of Adjoint Network of Periodically Switched Linear Circuits In this appendix, we show that the adjoint network of a given periodically switched linear circuit can be solved efficiently by utilizing the intrinsic relationship of these two circuits. When taking into account the time reversal characteristic of the adjoint network, it can be shown that and of N relate to those of by

where

and are the conductance and capacitance matrices of The superscript T denotes matrix transpose. Since where A is a non-singular square matrix, also because have

we

Therefore, the LU-factorization of can be obtained directly from that of without additional computation. Also, because

300 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

The state transition matrix of can also be obtained from that of N without numerical integration. The zero-state response is input-dependent. It is the solution of the circuit at when the input is applied at and the initial condition is zero. Using numerical Laplace inversion with {N, M} = {8,10} [120, 121], it can be shown that is computed from

where

Because

we obtain

Consequently

where is a constant vector specifying the nodes to which the input of is connected. It is obtained from where is a constant vector specifying the output location of N. The above analysis shows that the zero-state response of can be obtained efficiently from that of N without numerical integration. In summary, the adjoint network of N can be solved with little extra computation given that the solution of N is available. The intrinsic relationship between N and is summarized in Table 11.A.1

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301

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Chapter 12 FREQUENCY DOMAIN ANALYSIS OF PERIODICALLY SWITCHED NONLINEAR CIRCUITS

The steady-state response of a nonlinear time-invariant circuit to a sinusoidal input contains both the frequency of the sinusoidal input and its harmonics. When two sinusoids of different frequencies are applied to the circuit simultaneously, intermodulation frequency components are generated. It was shown in Chapter 11 that due to the periodic switching, the response of periodically switched linear circuits to a single-tone input at contains frequency components at both the baseband frequency and the sideband frequencies When the nonlinear characteristics of the devices of these circuits are considered, these circuits are subject to the effects of both the nonlinearities, which give rise to harmonic and intermodulation components, and periodic switching, which generates sideband frequency components. This substantially increases both the number of the frequency components in the response of these circuits and the complexity of the analysis of these circuits. Distortion can be analyzed in either the time domain or frequency domain, depending upon the characteristics of the nonlinearities of circuits. The time domain approach first computes the steady state response of the circuits to sinusoidal inputs. The frequency components of the response of these circuits are then computed by a post fast Fourier transform analysis. Although this approach is universal, a major drawback of the approach is that the time domain response of the circuits

304 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

must be computed until a steady state is reached. The results of the computation in the transient portion, however, must be discarded. Only those in the steady-state is used in FFT analysis to yields distortion. This process could be very time consuming, especially for periodically switched nonlinear circuits because the time-domain analysis of these circuits is costly, as detailed in Chapter 7. To speed up time-domain analysis, trigonometric polynomial [122] and envelope-following integration algorithms [123] have been proposed. Finite difference method [124] and shooting methods [125, 126, 127, 128], which compute the steadystate response directly, have also been investigated extensively. As compared with the time domain approaches, frequency domain methods compute the distortion directly. These methods are more efficient computationally for circuits containing mild nonlinearities, i.e. the characteristics of the nonlinear elements can be depicted adequately using a finite number of the terms of their Taylor series expansion at the operating point. Analytical approaches [129, 130, 131] for distortion analysis are pen-and-paper approaches. They are effective for circuits of small size only. Harmonic balance approach [132, 133, 134] obtains the distortion components by solving the determining equations derived from balancing the harmonics of the response of circuits to a sinusoidal input numerically. This approach is computationally expensive because the determining equations are nonlinear algebraic equations and the number of unknowns is directly proportional to the characteristics of the nonlinearities in the circuits, i.e. the order of harmonics K considered in approximation of the response of the circuit

where is the frequency of the sinusoidal input. is then plugged into the nonlinear equations depicting the circuits and a power series in is obtained. Making use of the identity of power series that a power series is zero if and only if all the coefficients of the power series are identically zero, a set of nonlinear algebraic equations with the unknowns are derived. These nonlinear algebraic equations are called determining equations. They are solved using

Frequency Domain Analysis of Periodically Switched Nonlinear Circuits

305

Newton-Raphson iterations for To reduce the cost of computation, the piecewise harmonic balance method [133, 135] and substitution methods [136] were proposed to accelerate the computation by partitioning a nonlinear circuit into linear and nonlinear sub-circuits. The linear sub-circuit can be solved efficiently in the frequency domain while the nonlinear sub-circuit is solved using conventional harmonic balance approach. The solutions of both circuits are matched at the boundary of the two sub-circuits. Because the size of the nonlinear subcircuit is usually much smaller, a significant reduction in computation is achieved. As compared with harmonic balance methods that derive distortion from solving determining equations numerically using Newton-Raphson iterations, Volterra series based approaches [137, 138, 43] compute distortion in the frequency domain directly and are computationally efficient. They have been applied for distortion analysis of nonlinear timeinvariant circuits and ideal switched capacitor networks [86, 139]. To investigate the distortion of nonlinear switched current networks, nonlinear switched capacitor networks with parasitics and non-idealities, and general switched nonlinear circuits, it becomes indispensable to include other types of elements, such as resistors, inductors, controlled sources, and current sources in both the modeling of the nonlinear elements of these circuits and the analysis of these circuits. This chapter is concerned with distortion analysis of multi-phase periodically switched nonlinear circuits in the frequency domain. The approach presented in the chapter is based upon time-varying Volterra functional series, Schetzen’s multi-linear theory [140], and time-varying network functions and multi-frequency transfer functions of nonlinear periodically time-varying systems introduced in Chapter 3. The chapter is organized as follows: Section 1 reviews the basics of the frequency domain characteristics of nonlinear circuits. Section 2 investigates the representation of the network variables of periodically switched nonlinear circuits using Volterra functional series. It also develops an algorithm for frequency analysis of periodically switched nonlinear circuits. In Section 3, the harmonic distortion of periodically switched nonlinear circuits is derived whereas the intermodulation distortion of these circuits is ob-

306 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

tained in Section 4. In assessment of the effectiveness of the method, the distortion of several periodically switched nonlinear circuits is computed using the method presented in the chapter and the results are compared with those obtained from transient-FFT analysis using SPICE in Section 5. The chapter is summarized in Section 6.

Fundamentals

1.

1.1

Harmonic Distortion

The harmonic distortion of nonlinear circuits is obtained by applying a sinusoid to the input of the circuits and computing the harmonics of the response of the circuits. Consider a nonlinear time-invariant circuit with the input-output relationship given by

where and are constants, and are the input and output of the circuit, respectively. To obtain the harmonic distortion, let we have

The second-order harmonic distortion is obtained from

For circuits with mildly nonlinearities and when the amplitude of the input is small, we have

Eq.(12.4) is simplified to

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307

In a very like manner, one can show that

The total harmonic distortion, a measure of how closely the output waveform resembles a pure sinusoid, is obtained from

1.2

Intermodulation Distortion

Intermodulation occurs when two sinusoids of different frequencies are applied to nonlinear circuits. Consider the same nonlinear circuit. Let the input be

It is trivial to show that the response of the circuit contains the frequency components tabulated in Table 1.2 The second-order intermodulation is obtained from

Note that modulation is computed from

was assumed. The third-order inter-

308 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

It is seen that :

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309

Harmonic and intermodulation distortion is intrinsically related to each other by

Both harmonic distortion and intermodulation distortion are proportional to (i) the characteristics of the nonlinearities, which are quantified by and and (ii) the amplitude of the input signal. Third-order intermodulation components at and are of critical concern, particularly for RF applications. This is because for these applications, and are usually very close (adjacent channels), the beats of the third-order intermodulation components are very close to that of the wanted channel. The preceding power series based distortion analysis is valid for circuits consisting of elements without memory or circuits at low frequencies where the effect of the past information is negligible. For circuits at high frequencies, the effect of the past information must be taken into account when analyzing the behavior of the circuits. In this case, and become frequency-dependent. Volterra functional series should be used to characterize the behavior of nonlinear circuits.

2.

Distortion Analysis of Periodically Switched Nonlinear Circuits

Periodically switched nonlinear circuits are nonlinear periodically timevarying systems. It was shown in Chapter 3 that the network variable of a periodically switched nonlinear circuit can be represented by a time-varying Volterra series

where is the term of the Volterra series expansion of A finite number of terms are usually sufficiently if the characteristics of

310 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

the nonlinear elements are mild. Multiplying (12.12) by the window function defined in (2.4), we obtain

where

and

Further multiplying (12.12) by the Dirac delta function gives

where is the network variable at is the term of the Volterra series expansion of Fourier transform of (12.13) gives

Writing (12.17) for all clock phases and summing up the results give the complete response of the periodically switched nonlinear circuit in the frequency domain

Frequency Domain Analysis of Periodically Switched Nonlinear Circuits

311

Eq.(12.18) reveals that the frequency response of periodically switched nonlinear circuits can be obtained by summing up that of corresponding Volterra circuits. It was shown in Chapter 11 that in the analysis of periodically switched linear circuits, the difficulties associated with periodically time-varying topology are handled by decomposing the circuits into a set of linear time-invariant sub-circuits connected via the initial conditions of the network variables [8]. Analogously, in distortion analysis of periodically switched nonlinear circuits, a periodically switched nonlinear circuit can be considered as a set of nonlinear time-invariant sub-circuits connected via the initial conditions of the network variables of the circuits. To simplify presentation, we assume that the characteristics of the nonlinear elements are mild and their behavior can be depicted adequately using the third-order Taylor series expansion of the nonlinear characteristics. Further assume that the network variables of periodically switched nonlinear circuits can also be represented by their thirdorder Volterra series expansion. Using modified nodal analysis, the nonlinear time-invariant circuit in phase is characterized by

where and are constant matrices depicting respectively the second and third-order nonlinear characteristics of the circuits. The elements of the vectors and are the square and cube of the corresponding elements of respectively. Note that we have used scalar-like notations for the purpose of their self-explanation. If the input is changed from to where is a nonzero constant, Eq.(12.19) becomes

312 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Eq.(12.20) is essentially a power series in Making use of the identity of power series that a power series equals to zero if and only if all the coefficients of the power series are identically zero, and

we arrive

Frequency Domain Analysis of Periodically Switched Nonlinear Circuits

313

where

and are the response of the periodically switched linear circuits characterized by (12.23) and (12.24), respectively. Note and are vectors whose dimensions are the same as The elements of are the products of the corresponding elements of and whereas those of are the cube of the corresponding elements of Fourier transform of (12.22)-(12.24) gives

Eqs.(12.26)–(12.28) reveal that : The behavior of a periodically switched nonlinear circuit can be characterized by a set of intrinsically related periodically switched linear circuits. The circuits characterized by (12.26)–(12.28) are termed respectively as the first-, second-, and third-order Volterra circuits of the periodically switched nonlinear circuit. They are denoted by

314 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

and respectively. These Volterra circuits have the same incidence matrix but different inputs. This observation reveals that in solving these circuits, the transit matrix for all Volterra circuits are the same and need to be computed only once. The zero-state vectors, however, differ and must be computed separately. The input of the circuit characterized by (12.27) is obtained from the solution of (12.26) whereas that of the circuit depicted by (12.28) is from the solution of (12.26) and that of (12.27). Volterra circuits must be solved in a sequential order with the lower-order Volterra circuits solved first. Although the topology of and are identical, the frequency components of and differ from each other due to their distinct inputs. As a result, LU-decomposition of must be performed separately.

Harmonic Distortion

3.

In this section, the frequency response of a periodically switched nonlinear circuits to a sinusoidal input is obtained using the method presented in the preceding sections.

3.1

The First-Order Volterra Circuit

Let the input of a periodically switched nonlinear circuit be

The sinusoidal input contains two distinct frequency components at and Using the principle of superposition, it can be shown that the complete response of denoted by contains the frequency components and can be written as

Frequency Domain Analysis of Periodically Switched Nonlinear Circuits

315

where and denote the phasors of the frequency component of at frequencies and respectively. The contribution of to the fundamental component at is given by

3.2

The Second-Order Volterra Circuit

The input of written as

denoted by

is given by (12.25) and can be

where and are the phasors of and respectively. Among the frequency components of only those at contribute to the second-order harmonic. In saying so, we have assumed that and will not be the multiples of for any This is often the case in reality as is usually much higher than The secondorder harmonic of the output is obtained by summing up the response of at frequency with an input at frequencies

where is the aliasing transfer function of with the input at the frequency and the output at the frequency The complete solution of in the time domain, denoted by with the input given by (12.31), is given by

316 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where and of at frequencies and These phasor quantities are needed in solving following section.

3.3

are the phasors respectively. as to be shown in the

The Third-Order Volterra Circuit

The input of contains two components. The first component contains frequency components and Among them, only those at contributes to the third-order harmonic at The input at however, affects the fundamental. Note that it was assumed that and will not be multiples of either or for any Based on these observations, we conclude that needs to be solved with the input at frequencies and the output at the frequency for its contribution to the third-order harmonic at needs to be solved with the input at frequencies and the output at the frequency for its contribution to the fundamental at In computing the input of the solutions of at and at are needed. and therefore need to be solved at all relevant side band frequencies with the output at these frequencies. however, needs only to be solved with the input at specific frequencies because only the response at the fundamental frequency and that at the third-order harmonic frequency are of interest. The output of

at

due to

is obtained from

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317

where is the aliasing transfer function of with the input at the frequency and the output at the frequency is computed from

The third-order harmonic component generated by from

is obtained

where

and is the aliasing transfer function of with the input at the frequency and output at the frequency The other input of also contributes to both the fundamental and third-order harmonics. Its contribution can be computed in a similar manner as that of Let its contribution to the base band and the third-order harmonic be denoted by and respectively. In conclusion : The fundamental component of the response of the periodically switched nonlinear circuit is obtained by summing up the contribution of the first-order Volterra circuit with the input at frequencies and the output at the frequency and that of the third-order Volterra circuit with the input at frequencies and the output at the frequency

318 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

The second-order harmonic component of the response of the periodically switched nonlinear circuit is determined from the response of the second-order Volterra circuit with the input at frequencies and the output at the frequency The third-order harmonic component of the response of the periodically switched nonlinear circuit is obtained by summing up the contribution of with inputs and at frequencies and the output at the frequency The second-order harmonic distortion of the circuit is computed from

and the third-order harmonic distortion in obtained from

3.4

The Fold-Over Effect

Eqs.(12.32), (12.34) and (12.36) demonstrate that the high-order side band components of the inputs of and contribute to both the fundamental and harmonic components of the response in the base band. This is analogous to the fold-over effect encountered in the noise analysis of periodically switched linear circuits presented in Chapter 11. The folding effect in computing is illustrated graphically in Fig. 12.1. The frequency reversal theorem introduced in Chapter 3 can be employed to reduce the cost of computation in calculating these aliasing transfer functions.

4.

Intermodulation Distortion

Intermodulation distortion arises when the input of a periodically switched nonlinear circuit contains two or more sinusoidal signals of different frequencies. For example, in a RF receiver, the frequency difference between two adjacent channels at frequencies and denoted

Frequency Domain Analysis of Periodically Switched Nonlinear Circuits

319

by is usually orders of magnitude smaller as compared with or Due to the nonlinearities of the receiver, frequency components other than those of the carriers are generated. The harmonic components in the output of the receiver are not of concern because they will be filtered out by a downstream low-pass filter. The third-order intermodulation at frequencies and however, are of a critical concern because their spectra fall so close to those of the carriers that the downstream filter can not eliminate them effectively. The analysis of the third-order intermodulation distortion is therefore of practical importance. In this section, we compute the third-order intermodulation of periodically switched nonlinear circuits.

4.1

The First-Order Volterra Circuit

Let the input of a periodically switched nonlinear circuit be

From the principle of superposition, the time-domain response of of the periodically switched nonlinear network, denoted by contains the frequency components and and can be represented by

320 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

where are the phasors of at frequencies and respectively.

4.2

and

The Second-Order Volterra Circuit

The input of

contains frequency components and Consequently, the output of consists of frequency components and Further analysis shows that the input of at and do not contribute to the third-order inter-modulation. As a result, the output of at these frequencies is not required. The output of at other frequencies, however, must be computed. They are computed in a similar manner as that of harmonic distortion.

4.3

The Third-Order Volterra Circuit

The input of consists of the components and It can be shown that among the frequency components of only those at the frequencies and contribute to the third-order inter-modulation whereas those at the frequencies and contribute to the fundamentals at and respectively. therefore needs to be solved with the input at the frequencies and outputs at the frequencies the input at the frequencies and frequencies and respectively.

and and

respectively. and output at the

Note that the amplitude of the inputs at these frequencies must be calculated prior to solving Similarly, among the frequency components of only the input at the frequencies and are of concern as they contribute to the third-order inter-modulation.

Frequency Domain Analysis of Periodically Switched Nonlinear Circuits

the input at the frequencies to the fundamentals.

and

321

as they contribute

The frequency reversal theorem can be used in both cases to lower the cost of computation. The third-order inter-modulation distortion at is computed from

where

and

and at are the contributions of

5.

are the contributions of respectively, and and at respectively.

Examples

In this section, both the harmonic distortion and intermodulation distortion of several periodically switched nonlinear circuits are analyzed using the algorithms presented in this chapter. The results are compared with those from transient-FFT analysis of SPICE.

5.1

Modulator

322 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Consider the modulator shown in Fig.12.2 with the parameter values given in Table 12.2 [141]. The nonlinear resistor is modeled as a nonlinear current-controlled voltage source

The circuit was solved using the method presented in the preceding sections and results are shown in Table 12.3, together with those from transient-FFT SPICE simulation of SPICE. Care was taken in choosing the step size in the transient analysis, collecting steady-state response data, and selecting the number of data points and windows in transientFFT analyses of the modulator using SPICE [142, 143, 68]. As observed that the results compare well with those from SPICE. To demonstrate the fold-over effect in distortion analysis of periodically switched nonlinear circuits, the dependence of the second-order harmonic components at 1 kHz on the number of side bands considered is plotted in Fig.12.3. It is seen that the second-order harmonic component of the response of the modulator converges rapidly.

5.2

Stray-Insensitive Switched Capacitor Integrator

Consider the stray-insensitive switched capacitor integrator shown in Fig.12.4 with parameters given in Table 12.4. MOSFET switches and are modeled as a linear resistor of resistance in series with an ideal switch. is modeled as a nonlinear resistor in

Frequency Domain Analysis of Periodically Switched Nonlinear Circuits

323

series with an ideal switch. The nonlinear resistor is characterized by The model of the operational amplifier is the same as those used in Chapter 11.

324 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

The circuit was solved and the results are shown in Table 12.5, together with SPICE simulation results. To ensure that the second-order harmonic components are free of numerical errors, several SPICE simulations with different step sizes used in transient analysis were conducted. The absolute errors between the harmonic components obtained from using different step sizes in transient analysis are less than 0.2 decibels. It is seen that the method gives good prediction of both the fundamental and the second-order harmonic components. The second-order

Frequency Domain Analysis of Periodically Switched Nonlinear Circuits

325

harmonic component at 1 kHz is plotted in Fig.12.5 versus the number of side bands considered. It is seen that the second-order harmonic converges monotonically with the increase in the number of side bands. Also observed that the rate of convergence is slower as compared with that of the modulator.

5.3

Switched Capacitor Integrator With Nonlinear Op Amp

The third example is the stray-insensitive switched capacitor integrator with a nonlinear operational amplifier shown in Fig.12.6 with its parameters given in Table 12.6. The only nonlinearity is the operational amplifier characterized by

The harmonic components of the output were computed and the results are tabulated in Table 12.7 for and Table 12.8 for SPICE simulation results are also shown in the tables. It is seen that the results are in good agreement with those from SPICE simulation. Also, in Table 12.7, since zero second-order harmonic is predicted by the method as both the firstand third-order periodically switched linear circuits do not contribute to

326 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

the second-order harmonic. This agrees with SPICE simulation. SPICE results for are plotted in Figs. 12.8 and 12.7. 1 By comparing these plots with Fig.3.2, it is evident that the theory 1

The estimated time constant due to the sampling capacitor and channel resistance of MOSFET switches is about To ensure the establishment of the steady state, the first 1000

Frequency Domain Analysis of Periodically Switched Nonlinear Circuits

327

presented in Chapter 3 gives an accurate prediction to the fundamental frequency component, second-order, and third-order harmonic components of the response. Since only up to the third-order Volterra series expansions were considered, the method is not capable of predicting high-order harmonics. The CPU time for computing the distortion at a single frequency is 4.1 minutes on 450MHz, 256MB Sun Sparc stations (Matlab implementation) while SPICE consumed 2.78 hours. The efficiency gain from using the proposed method is evident. To demonstrate the efficiency gain obtained from using the frequency reversal theorem in the distortion analysis of periodically switched nonlinear circuits, the same circuit was solved with frequency reversal theorem implemented and without (brute-force). The results are tabulated in Table 12.9. It is seen that the results from both methods agree well. The efficiencies of the two algorithm are also compared in terms of their

samples were discarded from FFT analysis. Also the number of samples used in FFT analysis was 64k. Rectangular window was employed.

328 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

CPU time. It was observed that the method with the frequency reversal theorem implemented is nearly three times that of the brute-force approach. The speedup is less than those in noise analysis, as shown in

Frequency Domain Analysis of Periodically Switched Nonlinear Circuits

329

Chapter 11 separately. This is because the frequency reversal theorem is only employed in solving because needs to be solved at all relevant side band frequencies, which amounts to a significant portion of the CPU time. The inter-modulation distortion of the circuit with was also investigated. The input consists of two sinusoids of frequencies 1 kHz and 1.1 kHz, respectively. The third-order intermodulation components were computed and the results are tabulated in Table 12.10, together with SPICE simulation results. A good agreement is observed. SPICE simulation results for and are plotted in Figs. 12.9 and 12.10. A careful comparison of these plots with Fig.3.3 reveals that all major beats of the response predicted by the method given in Chapter 3 match those from SPICE simulation.

330 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

Frequency Domain Analysis of Periodically Switched Nonlinear Circuits

331

To demonstrate the analysis of harmonic distortion due to the existence of nonlinear capacitors, the circuit in Fig.12.4 is considered. The only nonlinear element is capacitor modeled by

while remains unchanged. To avoid numerical difficulties, the circuit is impedance-scaled by and frequency scaled by 100. As a result, the ON-resistance of MOSFET switches is changed from to The capacitance of is changed from to becomes

The clock frequency is changed from 100 kHz to 1000 Hz. The operational amplifier is modeled as an ideal voltage-controlled voltage source with gain 1000. The distortion of the output at frequency 100 Hz was analyzed. For the purpose of comparison, it was also computed using SPICE. Both results are tabulated in Table 12.11. As can be seen that the results are in good agreement.

332 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

6.

Summary

Volterra series based frequency-domain analysis of the distortion of general periodically switched nonlinear circuits has been presented in this chapter. We have shown that a periodically switched nonlinear circuit can be characterized by a set of periodically switched linear circuits

Frequency Domain Analysis of Periodically Switched Nonlinear Circuits

333

called Volterra circuits. The input of high-order Volterra circuits is a nonlinear function of the response of lower-order Volterra circuits only. Distortion of the periodically switched nonlinear circuit is obtained by solving corresponding Volterra circuits. This result is a generalization of the multi-linear theory known for nonlinear time-invariant circuits. We have also shown that the aliasing effect encountered in noise analysis of periodically switched linear circuits also exists in distortion analysis of periodically switched nonlinear circuits. Computation associated with the folding effect can be minimized by utilizing the adjoint network theory of periodically switched linear circuits presented in Chapter 10, in particular, the frequency reversal theorem. Distortion of several periodically switched nonlinear circuits has been analyzed and the results compare well with those from transient-FFT analysis of SPICE. As compared with conventional harmonic balance approaches, the method is more efficient computationally in computing both the harmonic and

334 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS

intermodulation distortion of periodically switched circuits with mildly nonlinearities.

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Index

adjoint network, 212 capacitors, 214 controlled sources, 214 ideal switches, 212 inductors, 214 internal branches, 211 resistors, 213 time reversal, 210 aliasing effect, 284 baseband, 35 baseband frequency, 301 basis functions, 73 behavioral modeling, 197 Boltzmann constant, 275 charge conservation, 170, 230 charge injection, 139 charge of an electron, 276 Circuits with internally controlled switches buck linear voltage regulators, 183 current-mode comparators, 181 diodes, 178 inconsistent initial conditions, 182 internally controlled switches, 177 MOSFETS, 179 switching instants, 182 switching variables, 177 voltage-mode comparators, 181 clock feed-through, 139 clock jitter, 131 CMOS pass-transistor gates, 281 coefficient of variance, 125

comparators, 173 computer oriented formulation methods, 17 equivalent-circuit, 17 modified nodal analysis, 18 signal flow diagram, 17 state-space, 17 switching matrix, 17 Tableau formulation, 18 transmission matrix, 17 two-graph, 18 constitutive equations of nonlinear voltagecontrolled voltage source, 22 corrector, 55 current crowding, 278 current-mirror amplifier, 162 Dirac impulse function, 25, 66, 233, 276 dual-time circuits, 193 dual-time systems, 10 envelope-following, 302 equivalent noise bandwidth, 127 equivalent resistance of the switched capacitor, 5 exponentially decaying function, 66, 86 external clocks, 19 extraction of final conditions, 230 fast Fourier transform (FFT), 301 finite difference method, 302 Fourier series, 75 frequency reversal theorem, 325

348 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS group delay, 258 fifth-order switched capacitor low pass filter, 263 harmonic balance approach, 302 determining equations, 302 piecewise harmonic balance method, 303 substitution methods, 303 harmonic balance approach determining equations, 302 harsh nonlinearities, 173 ideal switched capacitor networks, 16 ideal switching, 11, 16, 139 incomplete charge transfer, 11, 17, 230 inconsistent initial conditions, 11, 83, 99 backward Euler based algorithms, 91 consistent initial condition component, 103 consistent initial conditions, 100 derivatives of unit step function 91 Dirac impulses in linear circuits, 104 Dirac impulses in nonlinear circuits, 106 error of numerical Laplace inversion, 86 existence of Dirac impulses, 103 four-step algorithm, 97 impulse-free component, 100 inconsistent initial condition component, 103 Taylor series based algorithm, 99 two-forward-step algorithm, 92 two-step algorithm, 83, 88, 96 two-step algorithm for linear circuits, 98 Volterra functional series based algorithm, 102 initial conditions, 20 instantaneous charge (flux) distribution, 11 inter-reciprocal, 212 interpolation boundary conditions, 158 exponential interpolation, 147

Fourier series interpolation, 147 Gibb effect, 158 interpolating Fourier series, 147 interpolating function, 147 interpolation error, 148 Lagrange interpolation, 147 Newton finite difference interpolation, 147 polynomial-based interpolation, 147 rate of convergence, 158 simulation window, 147 interval analysis, 123 Kronecker delta function, 209 linear periodically time-varying system, 35 linear periodically time-varying systems, 264 aliasing transfer function, 38, 42 baseband frequency, 39 Dirichlet-Jordan criterion, 37 phasor representation, 39 sideband frequencies, 38, 39 linear time-invariant systems, 35 linear time-varying systems, 35 time-varying network function, 36 bi-frequency transfer function, 42 excitation time, 36 impulse response, 36 observation time, 36 time-varying network functions, 41 linear transconductance, 162 linearly independent, 73 local truncation error (LTE), 140 lumped electrical networks, 208 Matlab, 165 Matrix stamps ideal switches, 22 inductors, 28 linear time-invariant capacitors, 24 memoryless elements, 21 nonlinear capacitors, 25 switching variable, 22 matrix stamps, 22 memoryless elements, 40 mild nonlinearities, 173 modeling of switches

INDEX full-transistor model, 13, 84 ideal switch model, 14, 84 voltage-modulated resistor model, 14, 84 modeling of white noise using a set of sinusoids, 126 using random pulses, 126 modified nodal analysis, 197 modulator, 320 Monte Carlo analysis, 119, 123 MOSFET diffusion current, 280 drift current, 280 Fermi potential, 279 gate capacitance per unit area, 280 gate leakage current, 279 gate transconductance, 280 gate-current fluctuation, 282 saturation region, 280 strong inversion, 279 sub-threshold region, 282 substrate transconductance, 280 surface mobility of charge carriers, 280 surface potential, 279 triode region, 280 multi-frequency network functions, 35 multi-frequency transfer functions, 303 multi-linear model of nonlinear elements, 140 multi-linear model of nonlinear elements with memory, 143 multi-linear theory, 303 multi-step numerical Laplace inversion algorithms, 233 multi-step predictor-corrector algorithms, 139 Newton-Raphson iterations, 59, 139 nodal charge conservation law, 18 noise noise, 276 aliasing effect, 273 autocorrelation function, 266 autocovariance, 267 average output noise power of periodically switched linear circuits, 272

349 average power, 267 average power spectral density, 272 bipolar junction transistors, 277 base resistance, 278 extrinsic base resistance, 278 intrinsic base resistance, 278 brute-force method, 288 corner frequency of flicker noise, 125 Curson’s theorem, 275 cyclo-stationary in the wide sense, 268 diffusion resistors, 277 flicker noise, 125, 264, 276 carrier density fluctuation model of flicker noise, 276 carrier mobility fluctuation model of flicker noise, 276 corner frequency, 277 fold-over effect, 12 Johnson noise, 275 mean-square value, 267 noise-current generator, 283 noise-voltage generator, 283 Nyquist law, 275 Nyquist theorem, 128 polysilicon resistors, 277 power spectral density, 125, 264, 266, 268 power spectral density of the thermal noise, 275 Schottky’s theorem for pn-junctions, 276 shot noise, 125, 264, 275 stationary, 267 stationary in the wide sense, 267 thermal equilibrium, 275 thermal noise, 125, 264, 275 thermal noise of the drift current, 280 Wiener-Khintchine theorem, 268 nonlinear capacitor, 143 nonlinear conductor, 163 nonlinear periodically time-varying, 46, 47 nonlinear periodically time-varying systems, 303 nonlinear resistor, 157 nonlinear time-varying systems, 35

350 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS fundamental component, 45 harmonic components, 45 harmonic frequency components, 42 intermodulation distortion, 47 multi-frequency transfer function, 43 nonlinear voltage-controlled voltage source, 141 Numerical Laplace inversion accuracy of numerical Laplace inversion, 64 Laplace inversion, 62 residue theorem, 63 residues, 63 residues of Pade polynomial 64 residues of Pade polynomial 65 zeros of Pade polynomial 64 zeros of Pade polynomial 65 numerical Laplace inversion, 53 stepping algorithm, 72 Nyquist theorem, 155, 273 operational amplifier finite bandwidth, 197 finite input impedance, 197 macro models, 215 non-zero output impedance, 197 slew rate of operational amplifiers, 139 order of interpolating Fourier series, 156 order of Taylor series expansion, 156 order of Volterra series expansion, 156 orthogonality of exponential series, 213 over-sampled sigma-delta modulators, 192 noise shaping, 198 over-sampling ratio, 193 single-bit second-order continuoustime over-sampled sigma-delta modulator, 198 single-bit second-order switched capacitor over-sampled sigmadelta modulator, 198 Padé approximation, 53 Padé approximates, 60

Padé fraction, 60 Padé polynomials of 62 parasitic resistances of BJTs, 278 periodically switched circuits, 17 periodically switched linear circuits baseband component, 233 frequency reversal theorem, 290 initial conditions, 231 sideband components, 233 single-tone input, 233 sub-circuits, 231 periodically switched nonlinear circuits, 140 fold-over effect, 316 frequency reversal theorem, 316 fundamental component, 313 intermodulation distortion, 316 second-order harmonic, 313 third-order harmonic, 314, 315 third-order intermodulation components, 317 Volterra circuits, 311 periodically time-varying linear systems, 242 phasors, 209 piecewise-linear input waveform, 74 power series, 40 predictor-corrector algorithm linear single-step predictor-corrector (LSS-PC), 53 predictor-corrector algorithms first-order predictor, 54 forward Euler formula, 54, 55 linear multi-step predictor-corrector (LMS-PC), 53 linear multi-step predictor-corrector (LMS-PC) algorithms, 58 linear single-step predictor-corrector (LSS-PC) algorithm, 55 truncation error, 54 predictor-corrector numerical integration algorithms, 55 principle of superposition, 149, 312 PSPICE, 139 sampled-data simulation, 109, 197 sampled-data simulation of linear circuits arbitrary inputs, 114

INDEX inconsistent initial conditions, 116 inconsistent initial conditions of sensitivity networks, 121 normalized sensitivity, 120 pre-processing step, 121 sensitivity analysis, 119 sinusoidal inputs, 113 step size, 198 two-step algorithm, 116, 140 unit step input, 113 zero-input response, 111 zero-state response, 111 zero-state vector, 111 sampled-data simulation of periodically switched nonlinear circuits accuracy, 155 error propagation, 156 inconsistent initial conditions, 140, 151 lower bound of sampling frequency, 155 maximum step size, 155 normalized mean square error (NMSE), 158 sensitivity, 152 sensitivity of periodically switched nonlinear circuits, 140 simulation window, 158 stability, 155 step size, 155 two-step algorithm, 151 two-step algorithm for sensitivity analysis, 154 Schmitt triggers, 173 second-order nonlinear transconductance, 162 sensitivity analysis adjoint network, 240, 254 aliasing transfer function, 254 brute-force, 166 capacitors, 244 controlled sources, 245 current-controlled current source, 245 direct sensitivity analysis, 237 ideal switches, 246 inductors, 245 inputs, 247

351 normalized sensitivity, 258 normalized sensitivity of the magnitude of the response, 236 normalized small-change sensitivity, 235 output, 247 perturbations, 241 resistors, 244 sensitivity network, 250 stray-insensitive switched capacitor integrator, 255 voltage-controlled voltage source, 245 sensitivity network, 120 sensitivity current, 251 sensitivity voltage, 251 shooting method, 302 sideband, 35 sideband frequencies, 301 sideband frequency components, 301 signal-to-noise ratio (SNR), 200 simulation window, 156 sparse matrix, 18 Spectre, 139 statistical analysis, 119 advanced first-order second-moment (AFOSM) method, 125 first-order second-moment (FOSM) method, 124 first-order second-moment method, 293 Monte Carlo analysis, 295 stepping algorithm, 72 stiff systems, 14, 84 stray-insensitive switched capacitor integrator with a nonlinear operational amplifier, 324 sub-circuits, 20 switched capacitor band pass filter, 257, 292 switched capacitor integrator, 166, 287 switched capacitor low pass filter, 286 switched capacitor network fifth-order elliptic switched capacitor filter, 235 stray-insensitive switched capacitor integrator, 320 switching instants, 92

352 COMPUTER METHODS FOR MIXED-MODE SWITCHING CIRCUITS Tellegen’s theorem incidence matrix, 208 incremental weak form of Tellegen’s theorem for periodically switched linear circuits in the phasor domain, 244 sensitivity analysis, 241 strong form, 208 Tellegen’s theorem for periodically switched linear circuits in the phasor domain, 210 time domain - same circuit, 208 weak form, 208 Tellegen’s theorem for periodically switched linear circuits, 242 Tellegen’s theorems Tellegen’s theorem for periodically switched linear circuits, 210 time-varying network functions, 303 time-varying topology, 10 time-varying Volterra functional series, 303 time-varying Volterra series, 307 transfer functions, 38 transient-FFT analysis, 198 transit matrix, 312 transition matrix, 72 trigonometric polynomial, 302 truncation error, 156 two-dimensional Fourier transform, 269

unit ramping function, 75 unit step function 66 voltage-controlled voltage source behavioral model of quantizers, 194 clocked quantizer, 194 comparators, 192 decimator, 192 quantizer, 192, 194 Volterra circuits, 102, 146 Volterra circuit, 151 first-order Volterra circuit, 146 second-order Volterra circuit, 147, 149 Volterra circuits of periodically switched nonlinear circuits, 140, 144 Volterra functional series, 35, 40 harmonic distortion, 46 symmetrical kernels, 46 Volterra kernel, 41 Volterra series expansion, 141 Watsnap, 224, 225, 255 wavelet, 75 weakly nonlinear, 141 weakly nonlinearities, 140 window function, 308 worst-case analysis, 123 zero-state vector, 72 zero-state vectors, 312