Upload
buituyen
View
220
Download
4
Embed Size (px)
Citation preview
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-1/23
Feedback (and control) systemsStability and performance
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-2/23
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
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-3/23
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
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-4/23
Nyquist Plot
• Example 8.10
– Consider an amplifier with frequency independent feedback
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
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-5/23
Nyquist Plot
3
4
4
10( )1
10
A ss
180s j
• Example 8.10
– Consider an amplifier with frequency independent feedback
and an open- loop transfer function given by
– The Nyquist plot of this system for
= .01 is
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-6/23
Nyquist Plot
3
4
4
10( )1
10
A ss
• Example 8.10
– Consider an amplifier with frequency independent feedback
and an open- loop transfer function given by
– The Nyquist plot of this system for
= .005 is
Gain < 1 when phase shift = -180o
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-7/23
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
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-8/23
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
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-9/23
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
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-10/23
Feedback Amplifier Poles
1 ( ) ( ) 0A s s
• Poles of closed loop feedback system are determined by:
A(s)
(s)
+in out
-
+
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-11/23
Feedback Amplifier Poles
0( )1
p
AA s s
• Consider a system with a single pole in the open loop response
A(s)
(s)
+in out
-
+
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-12/23
Feedback Amplifier Poles
• Consider a system with a single pole in the open loop response
0( )1
p
AA s s
A(s)
(s)
+in out
-
+
0
0
0
1( )1
(1 )
f
p
AAA s s
A
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-13/23
Feedback Amplifier Poles
• Consider a system with a single pole in the open loop response
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
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-14/23
Feedback Amplifier Poles
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
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-15/23
Approximating system response
dB
log()
|Af0 |
pf
0For ( )pf f fs A s A
0( )1
ff
pf
AA s s
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-16/23
Approximating system response
dB
log()
|Af0 |
pf
0For ( ) f pfpf f
As A s
s
0( )1
ff
pf
AA s s
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-17/23
Approximating system response
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
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-18/23
System with two pole response
0
1 2
( )1 1
p p
AA ss s
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-19/23
System with two pole response
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:
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-20/23
System with two pole response
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
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-21/23
Pole quality
21 2 0 1 21 0p p p ps s A
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-22/23
Pole quality
21 2 0 1 21 0p p p ps s A
2 200 0s s
Q
Copyright ©2007-2008Stevens Institute of Technology - All rights reserved 22-23/23
Pole quality
0 1 2
1 2
1 p p
p p
AQ
2 200 0s s
Q
j
0
2Q
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
Basic oscillator structure
( )( )1 ( ) ( )f
A sA sA s s
With positive feedback
6
Basic oscillator structure
( )( )1 ( ) ( )f
A sA sA s s
With positive feedback
( ) ( )A s s
Loop gain
7
Basic oscillator structure
( )( )1 ( ) ( )f
A sA sA s s
With positive feedback
( ) ( )A s s
Loop gain
Define loop gain L(s)
( ) ( ) ( )L s A s s
8
Basic oscillator structure
( )( )1 ( ) ( )f
A sA sA s s
With positive feedback
( ) ( )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
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)
1
f o
f o
o o
x x
Ax x
A x xA
11
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)
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
Nonlinear oscillator amplitude control
17
Nonlinear oscillator amplitude control
18
Nonlinear oscillator amplitude control
19
Wein-Bridge oscillator (without amplitude stabilization)
20
Wein-Bridge oscillator (without amplitude stabilization)
A
(s)
21
Wein-Bridge oscillator (without amplitude stabilization)
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
Wein-Bridge oscillator (without amplitude stabilization)
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
Wein-Bridge oscillator (without amplitude stabilization)
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
Wein-Bridge oscillator (without amplitude stabilization)
A
(s)
2 11( )13
R RL jj CR
CR
1
1
oo
o
CRCR
CR
Oscillation at o if
25
Wein-Bridge oscillator (without amplitude stabilization)
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
Quadrature oscillator
Limiting circuit
Integrator 2
Integrator 1
33
Quadrature oscillator
Limiting circuit
Integrator 2
Integrator 1
2 2 2
0
1( )
1
L ss C R
CR
34
Quadrature oscillator
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
Piezoelectric oscillator
Quartz crystalschematic symbol
Equivalentcircuit
44
Piezoelectric oscillator
Quartz crystalschematic symbol
Equivalentcircuit
Reactance
45
Piezoelectric oscillator
Seriesresonance
Parallelresonance1
ssLC
1p
s p
s p
C CL
C C
46
Piezoelectric oscillator
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)