Harmonic Oscillators: Definition

Chapter 12 Harmonic Oscillators Definition: Electronic Oscillators = Electronic circuits that produces repetitive signals. Observations: 1. They transform DC energy into AC energy 2. According to output signal, the electronic oscillators are divided into two major classes: •harmonic oscillators - sinusoidal output; •relaxation oscillators – non-sinusoidal output.

Lucian Balut

1

Chapter 12 Harmonic Oscillators 12.1 Preliminary 12.1.1 Classification 12.1.2 Principle of Operation. Barkhausen Relationship 12.2 RC Oscillators 12.2.1 Wien Oscillators 12.2.2 Phase Shift Oscillators 12.3 LC Oscillators 12.3.1 Hartley Oscillators (inductive feedback) 12.3.2 Colpitts Oscillators (capacitive feedback) 12.4 Cristal Oscillators 12.4.1 Quartz 12.4.2 Pierce Oscillator Lucian Balut

2

Harmonic Oscillators

12.1 Preliminary 12.1.1 Classification

Principal Characteristics: •Oscillation frequency; •The amplitude of the oscillation; •Oscillation condition; •Degree of distortion; •Amplitude stability; •Frequency stability. Principal problems: •Condition of priming oscillations; •Oscillation frequency evaluation; •Oscillation amplitude evaluation; •Distortion factor evaluation.

Lucian Balut

3

Harmonic Oscillators

12.1 Preliminary 12.1.1 Classification

The main classes of oscillators are defined by: - reactive type of elements contained in the structure; - frequency range of operation. According to reactive type of elements contained in the structure the oscillators are divided into: •RC oscillators; •LC oscillators. •Cristal oscillators According to frequency range of operation •Audio frequency; •Radio frequency; •Microwave.

Lucian Balut

4

Harmonic Oscillators 12.1 Preliminary 12.1.2 Principle of Operation. Barkhausen Criterion Xi  0

a.) Block Diagram

X

b.) Notation

+

Xi Xo Xε Xf

a- 

f

- input signal - output signal - error signal - feedback signal

X0

Xf

Xo  +

a

f

- transfer function of the basic amplifier

X

Xf

- transfer function of the feedback network

Xo

Lucian Balut

5

Harmonic Oscillators 12.1 Preliminary 12.1.2 Principle of Operation. Barkhausen Criterion c.) Barkhausen Criterion Xi  0

Problem:

X +

Find the relationship between a and f if Xo ≠ 0 and Xi = 0

Xf

Xo  +

a

f

Solution: Let it be: A 

X0 Xi

- transfer factor of the amplifier (with feedback)

A

X0 Xi

A

Xi  0 Lucian Balut

6

Harmonic Oscillators 12.1 Preliminary 12.1.2 Principle of Operation. Barkhausen Criterion c.) Barkhausen Criterion Observation:

Xi  0 Xo  0

Xi  0

X

A

+

X  Xi  Xf

 +

Xf

A Calculus

Xo

a

Xi  Xf  X

A

f

X0 Xi

A

X0 X  X f



1 X Xo



Lucian Balut

Xf Xo



1 1 f a



a 1  af

7

Oscilatoare armonice 12.1 Preliminary 12.1.2 Principle of Operation. Barkhausen Criterion c.) Barkhausen Criterion (cont.) Xi  0

X +

A A

a 1  af

Xf

af  1

Xo  +

a

f

a  a exp  ja 

a f 1

f  f exp  jf 

a  f  2k

Amplitude condition kZ

Phase condition

af  1

Lucian Balut

8

Harmonic Oscillators

12.1 Preliminary 12.1.2 Principle of Operation. Barkhausen Criterion c.) Barkhausen Criterion (cont.)

af  1

Observations

a f 1 a  f  2k

Amplitude condition kZ

Phase condition

- The minimum value of the amplifier gain in order to start-up oscillation may be evaluated using the amplitude condition. - The frequency of oscillation may be evaluated using phase condition.

Lucian Balut

9

12.2 RC Oscillators

Harmonic Oscillators

They are used in low frequency range. Only two types of RC oscillators are presented: - Wien Oscillators - Phase Shift Oscillators

Lucian Balut

10

Harmonic Oscillators 12.2 RC Oscillators 12.2.1 Wien Oscillators a.) circuit b.) parts duties R1 , R2 , C 1 , C2

Wien network

c.) circuit analysis Problems: •Oscillation frequency evaluation; •Amplifier gain evaluation. Simplification - the basic amplifier is an ideal amplifier, thus: •Input impedance very high; •Output impedance very low; •Voltage gain av is real.

Lucian Balut

11

Harmonic Oscillators 12.2 RC Oscillators 12.2.1 Wien Oscillators c.) circuit analysis - Barkhausen criterion is used: R1 fv evaluation

C2

Vt

f v ( j ) 

Vo Vt

C1

R2

Vo

1

 1

R1 C2  1     j R 1C 2  R 2 C1  R 2 C1 

Lucian Balut

12

Harmonic Oscillators 12.2 RC Oscillators 12.2.1 Wien Oscillators c.) circuit analysis Calculul frecventei de oscilatie

f v ( j ) 

Vo Vt

1

 1

a v f v ( josc )  1

R1 C2  1     j R 1C 2  R 2 C1  R 2 C1 

fv (jωosc) is real

f osc 

oscR 1C 2 

1 2 R 1R 2C1C 2

R1  R 2  R

1 0 oscR 2 C1

f osc 

osc 

1 R1R 2C1C2

1 2RC

C1  C2  C

Lucian Balut

13

Harmonic Oscillators 12.2 RC Oscillators 12.2.1 Wien Oscillators c.) circuit analysis av Evaluation f v ( josc ) 

1 1

R1  R 2  R

R1 C2  R 2 C1

f v ( josc ) 

1 3

C1  C2  C

a v f v ( josc )  1 1 f v ( josc )  3

av  3

Lucian Balut

Harmonic Oscillators 12.2 RC Oscillators 12.2.1 Wien Oscillators d.) example

C

d1). circuit

+EC R1

RC1

r

RC2

C1 T1 C2

R2

T2

T 3 CO

RE1 RE

CE RE2

RL

Vo

d2). parts duties; R1 , C1 , R2 , C2

(Wien network)

Lucian Balut

15

Harmonic Oscillators

12.2 RC Oscillators 12.2.2 Phase Shift Oscillators Observation: Two types of phase shift network +

Vt

+

C R

-

Vo

High pass network

Vt

R C

-

Vo

Low pass network

Observation Only phase shift oscillator with high-pass network will be presented

Lucian Balut

16

Harmonic Oscillators

12.2 RC Oscillators 12.2.2 Phase Shift Oscillators a.) circuit

b.) parts duties R, C feedback network

Lucian Balut

17

Harmonic Oscillators

12.2 RC Oscillators 12.2.2 Phase Shift Oscillators c.) circuit analysis Problems: •Oscillation frequency evaluation; •Amplifier gain evaluation.

Simplification - the basic amplifier is an ideal amplifier, thus: •Input impedance very high; •Output impedance very low; •Voltage gain av is real.

Lucian Balut

18

Harmonic Oscillators

12.2 RC Oscillators 12.2.2 Phase Shift Oscillators

I1 C

c.) circuit analysis fv Evaluation

+

Vt

I2 C

I3 C

I4

I5

R

R

-

R

Vo

I1  I 4  I 2

I 2  I3  I5 Vt  I1

1  I4R jC

1  I5 R  I 4 R jC 1 0  I3  I3R  I5 R jC Vo  I3R 0  I2

f v ( j ) 

Vo Vt



1 180   1 6   1  2 2 2   j 3 3 3     R C    R C RC 

Lucian Balut

19

Harmonic Oscillators

12.2 RC Oscillators 12.2.2 Phase Shift Oscillators

I1 C

c.) circuit analysis

+

Vo

1 f v ( j )   180   1 6  Vt  1  2 2 2   j 3 3 3     R C    R C RC 

a v f v ( josc )  1

fv (jωosc) is real

0

Vt

I2 C

I3 C

I4

I5

R

R

R

-

1 6  osc3R 3C3 oscRC

osc 

Vo

1 6RC

av evaluation f v ( josc ) 

1  180  1     2 R 2C2  osc   1 osc  6RC

f v (osc )  

1 29

a v  29

Lucian Balut

20

Harmonic Oscillators

12.2 RC Oscillators 12.2.2 Phase Shift Oscillators d.) example +EC R1

RC T2

C1

C

C

C

T1

R2

RE

RO

R

R

R

Lucian Balut

vO

21

12.3 LC Oscillators

Harmonic Oscillators

They are used in radio frequency range. presented:

Only two types of LC oscillators are

- Hartley Oscillators (inductive feedback) - Colpitts Oscillators (capacitive feedback)

Lucian Balut

22

12.3 LC Oscillators

Harmonic Oscillators

12.3.1 Hartley Oscillators (inductive feedback) a.) circuit b.) parts duties Rp resistance loss of the oscillating circuit C, L1, L2 feedback network. c.) circuit analysis Barkhausen criterion will be used. The feedback network will be cut. A voltage test Vt source will be introduced at the amplifier input. The output signal Vf will be picked from the feedback network output. Barkhausen criterion leads to:

Vt  V f V  V t

f

Lucian Balut

23

12.3 LC Oscillators

Harmonic Oscillators

12.3.1 Hartley Oscillators (inductive feedback) c.) circuit analysis (cont.)

Hartley Oscillator

Hartley Oscillator – mesh cut

24

12.3 LC Oscillators

Harmonic Oscillators

12.3.1 Hartley Oscillators (inductive feedback) c.) circuit analysis (cont.)

V f  aY Vt Z e

L2 L1  L2

1 1 1   jC  Ze Rp j ( L1  L2 ) V f  aY Vt

jR p ( L1  L2 )

L2 R p 1   2C ( L1  L2 )  j ( L1  L2 ) L1  L2





Hartley Oscillator – mesh cut

Tacking into account that the shift frequency of the amplifier is 1800, one can say:

aY  aY

Vt  V f

V f  aY Vt

jRp ( L1  L2 )

L2 Rp 1   2C ( L1  L2 )  j ( L1  L2 ) L1  L2



1   2C(L1  L2 )  0



osc 

1 ( L1  L2 )C

Vf  a Y Vt R p

L2 L1  L 2 25

12.3 LC Oscillators

Harmonic Oscillators

12.3.1 Hartley Oscillators (inductive feedback) d.) example Where: •RB1. RB2. RE

+EC

bias network

RB1

C

RL

L1

•L1, L2

inductive divider

•RL

load

• CE

coupling capacitor

CE L2

RE

T RB2

CB

Hartley oscillator

26

12.3 LC Oscillators

Harmonic Oscillators

12.3.2 Colpitts Oscillators (capacitive feedback) a.) circuit

b.) parts duties Rp resistance loss of the oscillating circuit. L, C1, C2 feedback network.

27

12.3 LC Oscillators

Harmonic Oscillators

12.3.2 Colpitts Oscillators (capacitive feedback) c.) circuit analysis

Colpitts Oscillator

Colpitts Oscillator – mesh cut

28

12.3 LC Oscillators

Harmonic Oscillators

12.3.2 Colpitts Oscillators (capacitive feedback) c.) circuit analysis (cont.) V f  aY Vt Z e

C1 C1  C2

1 1 CC 1   j 1 2  Ze Rp C1  C2 jL

jRp L

V f  aY Vt

C1 C1  C2

  CC Colpitts Oscillator – mesh cut Rp 1   2 1 2 L   jL C1  C2   Tacking into account that the shift frequency of the amplifier is 1800, one can say:

aY  aY

Vt  V f

V f  aY Vt

jRp L  CC Rp 1   2 1 2 C1  C2 

C1C 2 1  L0 C1  C 2 2

 L   jL 

osc 

C1 C1  C2

1 CC L 1 2 C1  C2

Vf  a Y Vt R p

C1 C1  C 2 29

12.3 LC Oscillators

Harmonic Oscillators

12.3.2 Colpitts Oscillators (capacitive feedback) d.) example C1 L C2

RL

Colpitts oscillator - common base

Harmonic Oscillators

12.4 Cristal Oscillators

Previous subsections showed that RC-type cells or LC-type cells were used as resonant circuits. But it must be said that the quality factor (Q) of such circuits are relatively low, leading to low stability of the frequency oscillation. One of the methods used to improve the frequency stability of the oscillating circuit is to use quartz crystal.

+

Figura 12.16

Figura 12.17

Figura 12.18

Common quartz oscillator configurations

Harmonic Oscillators

12.4 Cristal Oscillators

.Observation: According to rules controlling the oscillation frequency and its stability, the literature reveals four subclasses oscillators also:

•Crystal Oscillator XO; also called timer or clock; The simplest oscillator (); stability ± 10 ppm. •Voltage Controlled Crystal Oscillator VCXO; stability ± 10 ppm •Temperature Compensated Crystal Oscillator TCXO; stability ± 1÷5 ppm •Oven Controlled Cystal Oscillator OCXO; stability ± 1 ppm .

There are also two other types: •Digitally Compensated Crystal Oscillators DCXO). •Microcomputer Compensated Crystal Oscillator MCXO).

Harmonic Oscillators

12.4 Cristal Oscillators 12.4.1 Quartz Crystal

R Cp

Cs L

Equivalent circuit

Equivalent impedance

s 

1 LC s

1

p  L

C sC p Cs  Cp

Harmonic Oscillators

12.4 Cristal Oscillators

12.4.2 Oscilator Pierce EC

a.) circuit R1

LC

CC

X

R2

b.) parts duties R1 , R2 RE CE LC, CC X, C1

bias divider; thermal stability; decoupling capacitor; oscillating circuit; feedback.

C1

RE

CE

Harmonic Oscillators

12.4 Cristal Oscillators

12.4.2 Oscilator Pierce  1  1  Z X   R  jL  jC s  jC p 

c.) circuit analysis EC

R1

LC

CC

X

R2

C1

RE

CE

Z B  R1 R2

Pierce Oscillator

1 r jC1

Z LC  jL C

1 jC C

Pierce Oscillator – equivalent circuit

Harmonic Oscillators

12.4 Cristal Oscillators

12.4.2 Oscilator Pierce c.) circuit analysis

Pierce Oscillator – equivalent circuit

 

VB   g mU t Z LC Z x  Z B

Z

ZB B

 ZX

Pierce Oscillator - mesh cut





 g m Z LC Z x  Z B

 Z

ZB B

 ZX

Developing of these relationships determine the oscillation frequency and priming condition

1