feedback (and control) systems -...

Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-1/23

Feedback (and control) systemsStability and performance


Behavior of Under-damped System

00

2 22

2( )

2n

n n

bs y s yMY sb k s ss sM M

Damping ratio Natural frequency

21 2, 1n ns s

If

< 1

21 2, 1n ns s j

j

21nj

21nj

2 n

n

n

1cos


Nyquist Plot

• Example 8.10

– Consider an amplifier with frequency independent feedback

and an open- loop transfer function given by

3

4

4

10( )1

10

A ss


Nyquist Plot

• Example 8.10


and an open-loop transfer function given by

– The Nyquist plot of this system for

= .01 is

3

4

4

10( )1

10

A ss

s j

0s


Nyquist Plot

3

4

4

10( )1

10

A ss

180s j

• Example 8.10




= .01 is


Nyquist Plot

3

4

4

10( )1

10

A ss

• Example 8.10




= .005 is

Gain < 1 when phase shift = -180o


Stability and Pole location

( ) o n nt j t j tv t e e e

• Consider a system with poles at s = o + jn

• Transient response is

j

o < 0






j

o = 0






j

o > 0


Feedback Amplifier Poles

1 ( ) ( ) 0A s s

• Poles of closed loop feedback system are determined by:

A(s)

(s)

+in out

-

+



0( )1

p

AA s s

• Consider a system with a single pole in the open loop response

A(s)

(s)

+in out

-

+




0( )1

p

AA s s

A(s)

(s)

+in out

-

+

0

0

0

1( )1

(1 )

f

p

AAA s s

A




0( )1

p

AA s s

A(s)

(s)

+in out

-

+

0

0

0

1( )1

(1 )

f

p

AAA s s

A

Gain of feedback systemis reduced

Frequency characteristic of feedback systemis changed



0

0

0

1( )1

(1 )

f

p

AAA s s

A

0( )1

p

AA s s

A(s)

(s)

+in out

-

+

dB

log()

A(s)

p

|Ao |

|Af0 |

pf

j

ppf

0(1 )pf p A


Approximating system response

dB

log()

|Af0 |

pf

0For ( )pf f fs A s A

0( )1

ff

pf

AA s s



dB

log()

|Af0 |

pf

0For ( ) f pfpf f

As A s

s

0( )1

ff

pf

AA s s



0( )1

ff

pf

AA s s

0For ( )pf f fs A s A dB

log()

|Af0 |

pf

0For ( ) f pfpf f

As A s

s


System with two pole response

0

1 2

( )1 1

p p

AA ss s



21 2 0 1 21 0p p p ps s A

Closed loop response poles:

21 11 2 1 2 0 1 22 2 4 1p p p p p ps A

0

1 2

( )1 1

p p

AA ss s

Solving for poles:



1 2

2p p

j

p1p2

21 11 2 1 2 0 1 22 2 4 1p p p p p ps A

Root-locus diagram


Pole quality

21 2 0 1 21 0p p p ps s A


Pole quality

21 2 0 1 21 0p p p ps s A

2 200 0s s

Q


Pole quality

0 1 2

1 2

1 p p

p p

AQ

2 200 0s s

Q

j

0

2Q

0

Electronic Circuits – EE359A

Bruce McNair

[email protected]

201-216-5549

Lecture 16

1

Signal Generators and Signal Generators and WaveformWaveform--shaping Circuitsshaping Circuits

Ch 17Ch 17

2

Input summing, output sampling voltage amplifier

Series voltage summing Shunt voltage sensing

3

Using negative feedback system to create a signal generator

A

| ( ) | 1( )

AA

4

Basic oscillator structure

5


( )( )1 ( ) ( )f

A sA sA s s

With positive feedback

6


( )( )1 ( ) ( )f

A sA sA s s


( ) ( )A s s

Loop gain

7


( )( )1 ( ) ( )f

A sA sA s s


( ) ( )A s s

Loop gain

Define loop gain L(s)

( ) ( ) ( )L s A s s

8


( )( )1 ( ) ( )f

A sA sA s s


( ) ( )A s s

Loop gain

Define loop gain L(s)

( ) ( ) ( )L s A s s

Characteristic equation

1 ( ) 0L s

9

Criteria for oscillation

( ) ( ) ( ) 1o o oL j A j j The Barkhausen criteria:

For oscillation to occur at o

At o , the loop gain has a magnitude 1 and the phase shift is 0 (for positive feedback)

10





1

f o

f o

o o

x x

Ax x

A x xA

11





1

f o

f o

o o

x x

Ax x

A x xA

If gain is sufficient, frequency of oscillation is determined only by phase response

12

Oscillation frequency dependence on phase response

A steep phase response ( () ) produces a stable oscillator

13

Oscillator amplitude

j

j

|L(jo ) < 1

|L(jo ) > 1

0 1 2 3 4 5

2

0

2

f t( )

ta 0.2

0 1 2 3 4 5

2

0

2

f t( )

ta 0.2

14

Oscillator amplitude

j

|L(jo ) = 1

0 1 2 3 4 5

2

0

2

f t( )

ta 0

How do you stabilize the oscillator so the output level remains constant

If the oscillator is adjustable, how is this possible across the full range?

15

Nonlinear oscillator amplitude control

16


17


18


19

Wein-Bridge oscillator (without amplitude stabilization)

20


A

(s)

21


A

(s)

2

1

2

1

( ) ( )

1

( )

( ) 1

p

p s

p

p s

L s A sRAR

Zs

Z Z

ZRL sR Z Z

22


A

(s)

2

1

2 1 2 1

2 1

2 1

( ) 1

1 1( )1 1

1( )1

1( )1 11

p

p s

p s s p

s p

ZRL sR Z Z

R R R RL sZ Z Z Z

R RL sZ Y

R RL sR sC

sC R

23


A

(s)

2 1

2 1

2 1

1( )1 11

1( ) 11

1( )13

R RL sR sC

sC RR RL s R sCsCR

R sCR sCR RL j

j CRCR

24


A

(s)

2 11( )13

R RL jj CR

CR

1

1

oo

o

CRCR

CR

Oscillation at o if

25


A

(s)

2 11( )13

R RL jj CR

CR

Oscillation if

2 1

2 1

1( )3

2

R RL j

R R

26

Wein-Bridge oscillator (with amplitude stabilization)

A

(s)

stabilization

27

Wein-Bridge oscillator (with amplitude stabilization)

9 3

0

1

1(16 10 F)(10 10 )6250 rad/sec

f 1000 Hz

o

o

o

CR

2 1

2 1

220.3 10 2.03

R RR R

28

Wein-Bridge oscillator (with alternative stabilization)

D1 and D2 reduce Rf at high amplitudes

29

Phase shift oscillator-A

(s)

30

Phase shift oscillator-A

(s)

Phase shift of each RC section must be 60o to generate a total phase shift of 180o

K must be large enough to compensate for the amplitude attenuation of the 3 RC sections at o

31

Quadrature oscillator

32


Limiting circuit

Integrator 2

Integrator 1

33


Limiting circuit

Integrator 2

Integrator 1

2 2 2

0

1( )

1

L ss C R

CR

34


0sin( )t

0cos( )t

35

LC oscillator

Colpitts oscillator

36

LC oscillator

Hartley oscillator

37

LC oscillator

Colpitts oscillator Hartley oscillatorFrequencydetermining

element

38

LC oscillator

Colpitts oscillator Hartley oscillatorGainstage

39

LC oscillator

Colpitts oscillator Hartley oscillatorFeedback

voltage divider

40

LC oscillator

Colpitts oscillator Hartley oscillator

0

1 2

1 2

1

C CLC C

01 2

1L L C

41

Practical LC (Colpitts) oscillator

42

Piezoelectric oscillator

Quartz crystalschematic symbol

43



Equivalentcircuit

44



Equivalentcircuit

Reactance

45


Seriesresonance

Parallelresonance1

ssLC

1p

s p

s p

C CL

C C

46


Seriesresonance

Parallelresonance1

ssLC

1p

s p

s p

C CL

C C

r << |ZL |

47

Pierce crystal oscillator

LPF to discourageharmonic/overtone

oscillation

Frequencydetermining

elements(but CS dominates)

DC bias circuit(near VDD /2)

CMOS inverter(high gain amplifier)

feedback (and control) systems -...

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