R5310206-linear And Discrete Systems Analysis

1

Code No: R5310206

III B.Tech I Semester(R05) Supplementary Examinations, May 2009 LINEAR AND DISCRETE SYSTEMS ANALYSIS (Electrical & Electronics Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Obtain the state equation for a system described by a pair of simultaneous differential equations. y¨1 + a1 (y˙ 1 − y˙ 2 ) + a2 (y1 − y2 ) = 0 y¨2 + b1 (y˙ 2 − y˙ 1 ) + b2 (y2 − y1 ) = k [f (t) − y2 ] is described (b) A system   by the matrices.  0 1 0 0 £ ¤ 1 , b =  0 , c = 1 2 0 A= 0 0 0 −2 −3 1 Determine its transfer function.

[8+8]

2. (a) Calculate the Impedance consisting of R and L and the P.F of a circuit whose expression for voltage and current are e(t) = 254 sin (314t) + 50 sin (942t + 30) i(t) = 17.7 sin (314t − 45◦ ) + 1.583 sin (942t − 41.6◦ ) (b) Explain clearly the significance of the term “Mirror image symmetry” used in determining Fourier series of a given waveform. [12+4] 3. (a) Discuss the convergence fourier transform. (b) Show that area under the curve of sine function is unity.

[8+8]

4. (a) State and prove the convolution theorem. (b) Determine the LT for the following functions. i. Unit step ii. Impulse function. iii. Ramp functions.

[8+8]

5. (a) Explain Sturm’s theorem. (b) Test whether the following function is a positive real function and the polynomials are Hurwitz or not using Sturm’s test. F (s) =

2s4 +7s3 +11s2 +12s+4 . s4 +5s3 +9s2 +11s+6

[8+8]

6. Find the networks for the following functions in one Foster and one Cauer form (a) Y (s) =

(s+1)(s+3) (s+4)(s+2)

(b) Z(s) =

2(s+0.5)(s+4) . s(s+2)

[2 × 8]

7. If m(t) is band limited that is M(ω)=0 for |ω| > ω m then show that

R∞ −∞

|m(t)|2 dt = T s

∞ P n=−∞

where Ts = π/ωM .

[m(nTs )]2 [16]

8. Determine the convolution of the following pairs of signals by means of the Z-transform. ³ ´n (a) x1 (n) = 1/2 u(n), x2 (n) = Cos(πn)u(n) (b) x1 (n) = nu(n), x2 (n) = 2n u(n − 1).

[8+8] ?????

2

Code No: R5310206

III B.Tech I Semester(R05) Supplementary Examinations, May 2009 LINEAR AND DISCRETE SYSTEMS ANALYSIS (Electrical & Electronics Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. Consider the differential equation. ... y + 7¨ y + 14y˙ + 8y = 6u where y is the output and u the input. (a) Obtain state space representation. (b) Determine the characteristic equation and the eigen values. (c) Obtain the transfer function and state transition matrix. (d) Determine the unit step response at t = 0.1 second for given initial conditions. [4+4+4+4] 2. A rectangular waveform of magnitude 10V,duty ratio 75% and frequency 50Hz is applied across a resistance of 1ohm in series with an inductance of 100mH. Determine the steady state current in the circuit. Also find the power and P.F of the load current. [8+4+4] ¡t¢ 3. (a) Use the differentiation property of F.T, find the F.T. of g(t) = 2AtΠ τ (b) Compute the FT for the following: ¡ ¢ h 2Π(t−1) i i. sinΠtΠt sinΠ(t−1) ii. t2 e−2t u(t).

[8+8]

4. (a) Find the Inverse LT of x(s) =

S 2 e−2S +e−3S S(S 2 +3S+2)

(b) Find LT of e−t u(t).

[8+8]

5. (a) State and explain the properties of positive real function. (b) Check whether given polynomial H(s) = 2s4 + 5s3 + 6s2 + 2s + 1 is Hurwitz or not.[8+8] 6. Realize Z(s) =

s(s2 +2)(s2 +4) (s2 +1)(s2 +3)(s2 +5)

in all four forms.

[16]

7. Determine and plot the autocorrelation sequence of each of the following sequences. From the completed auto-correlation sequences, find the sequences that are periodic and also their periods. (a) x1 [n] = sin(2Πn/M), where M is a positive integer (b) x2 [n] = nmodulo7.

[16]

8. Evaluate the following integral: H 1 1+2z −1 −z −2 z 3 dz Where the contour of integration ‘c’ is the unit circle (a) 2πj 1− 12 z −1 )(1− 23 z −1 ) ( c (b) Find the inverse Z-transform of X(z) =

1 , 1−α10 z −10

?????

|z| > |α|.

[8+8]

3

Code No: R5310206

III B.Tech I Semester(R05) Supplementary Examinations, May 2009 LINEAR AND DISCRETE SYSTEMS ANALYSIS (Electrical & Electronics Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. Consider the differential equation. ... y + 7¨ y + 14y˙ + 8y = 6u where y is the output and u the input. (a) Obtain state space representation. (b) Determine the characteristic equation and the eigen values. (c) Obtain the transfer function and state transition matrix. (d) Determine the unit step response at t = 0.1 second for given initial conditions. [4+4+4+4] 2. A 2 ohm resistive load is supplied from a full wave rectifier connected to 230V, 50Hz single phase supply. Determine the average and rms values of load current. Also find out the proportion of DC power and AC power to the total power in the load. Investigate the effect of adding an inductance in series with the load. [8+4+4] 3. (a) If x(t) and y(t) are two arbitrary signals with fourier transforms x(ω) and y(ω) respectively Rα Rα 1 Show that x(t)y ∗ (t)dt = 2Π x(ω)y ∗ (ω)dω −α

−α

(b) Laplace transform exists for signal that do not have fourier transform. Justify this statement. [8+8] 4. (a) State and prove the convolution theorem. (b) Determine the LT for the following functions. i. Unit step ii. Impulse function. iii. Ramp functions.

[8+8]

5. (a) What are the necessary conditions for the function to be LC-Immitance function? (b) Test whether the given polynomial is positive real function: F (s) =

s3 +10s2 +27s+18 . s2 +7s+11.25

[8+8]

6. (a) Synthesize the first and second foster forms of networks for the impedance . Z(s) = 3(s+2)(s+4) (s+1)(s+3) (b) Realize the second foster form of the driving point impedance function is given by F (s) = 2(s+1)(s+3) . [8+8] (s+4)(s+2) 7. (a) Derive the power density spectrum of a periodic signal. (b) Find the power of a signal (A + f(t)), where A is a constant and the signal f(t)= sint. [8+8] 8. (a) How is the region of convergence defined for a finite duration signal? (b) Derive the differentiation property in Z-domain. (c) Explain the relationship between S-plane & Z-plane. ?????

[4+4+8]

4

Code No: R5310206

III B.Tech I Semester(R05) Supplementary Examinations, May 2009 LINEAR AND DISCRETE SYSTEMS ANALYSIS (Electrical & Electronics Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Obtain the state variable model for a system described by the following differential equation. 3 2 10 ddt3y + 5 ddt2y + 2 dy + 3y = 20 dx + 10x dt dt (b) Determine the state transition matrix for the state matrix. " # −1 1 0 0 −1 1 A= 0 0 −2

[8+8]

2. (a) Discuss in brief about the Fourier series and its applications to network analysis. (b) The applied voltage and the resulting current in a two element series circuit is given by v(t) = 50 + 50 sin 5x103 t + 30 sin 104 t i(t) = 11.2 sin (5x103 t + 63.4◦ ) + 10.6 sin (104 t + 45◦ ) Find the average power. [12+4] 3. (a) Find the fourier transform of the following functions. i. Impulse function δ(t) ii. DC signal (b) State and prove time differentiation property of the fourier transform. 4. (a) Write short notes on “convolution Integral”. (b) What is ROC for L.T. Explain the properties of ROC. 5. (a) Show that the function N (s) =

(s+2)(s+4) (s+1)(s+3)

[2 × 4 + 8] [8+8]

is positive real.

(b) Check if the polynomial H(s) = s4 + s3 + 2s2 + 2s + 24 is Hurwitz or not. [8+8] 6. An LC network is to be designed to meet the following specifications: (a) It is to have infinite impedance at 1200 cycles/sec and at 5000 cycles/sec. (b) It is to have zero impedance at 2400 cycles/sec. (c) It is to have a reactance of 100 ohms at 3000 cycles/sec. i. Determine the impedance function. ii. Find the four canonical network realizations, including element values.

[16]

7. (a) If the Nyquist rate for x(t) is ωs, what is the Nyquist rate for each of the following signals that are derived from x(t)? i. x2 (2t) ii. x(t)cos(ω0 t). (b) Given m(t) = 2+cos2 π103 t is sampled at a rate of 5000 samples/second i. Sketch the spectrum of m(t). ii. Determine the Nyquist rate and check whether aliasing occurs or not? [8+8] 8. Consider a system described by the Differential Equation y(n) = y(n − 1) − y(n − 2) + 0.5x(n) + 0.5x(n − 1). Find the response of this system to the input x(n) = (0.5)n u(n) with initial conditions y (-1) = 0.75 & y(-2) = 0.25. [16] ?????