Feedback (and control) systems

Behavior of Under-damped System Y() s s b y 0 M s 2n y0 2 2 2 b k s 2nsn s M s M Damping ratio Natural frequency If < 1 s s 2 1, 2 n n 1 s, s j 1 1 2 n n 2 cos 1 2n n n j jn jn 1 2 1 2 Copyright 2007-2008 Stevens Institute of Technology - All rights reserved 22-2/23

Nyquist Plot Example 8.10 Consider an amplifier with frequency independent feedback and an openloop transfer function given by As () 10 4 s 1 10 4 3 Copyright 2007-2008 Stevens 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 10 As () 4 s 1 10 The Nyquist plot of this system for =.01 is 4 3 s 0 s j Copyright 2007-2008 Stevens Institute of Technology - All rights reserved 22-4/23

Nyquist Plot Example 8.10 Consider an amplifier with frequency independent feedback and an openloop transfer function given by As () 10 4 s 1 10 4 3 The Nyquist plot of this system for =.01 is s j 180 Copyright 2007-2008 Stevens Institute of Technology - All rights reserved 22-5/23

Nyquist Plot Example 8.10 Consider an amplifier with frequency independent feedback and an openloop transfer function given by As () 10 4 s 1 10 4 3 Gain < 1 when phase shift = -180 o The Nyquist plot of this system for =.005 is Copyright 2007-2008 Stevens Institute of Technology - All rights reserved 22-6/23

Stability and Pole location Consider a system with poles at s = o + j n Transient response is vt e e e o n n () t j t j t j o < 0 Copyright 2007-2008 Stevens Institute of Technology - All rights reserved 22-7/23

Stability and Pole location Consider a system with poles at s = o + j n Transient response is vt e e e o n n () t j t j t j o = 0 Copyright 2007-2008 Stevens Institute of Technology - All rights reserved 22-8/23

Stability and Pole location Consider a system with poles at s = o + j n Transient response is vt e e e o n n () t j t j t j o > 0 Copyright 2007-2008 Stevens Institute of Technology - All rights reserved 22-9/23

Feedback Amplifier Poles Poles of closed loop feedback system are determined by: 1 As ( ) ( s) 0 in + + A(s) out - (s) Copyright 2007-2008 Stevens Institute of Technology - All rights reserved 22-10/23

Feedback Amplifier Poles Consider a system with a single pole in the open loop response As () A0 s 1 p in + + A(s) out - (s) Copyright 2007-2008 Stevens Institute of Technology - All rights reserved 22-11/23

Feedback Amplifier Poles Consider a system with a single pole in the open loop response As () A0 s 1 p in + + A(s) out - (s) A f A0 1 A0 () s s 1 (1 A ) p 0 Copyright 2007-2008 Stevens Institute of Technology - All rights reserved 22-12/23

Feedback Amplifier Poles Consider a system with a single pole in the open loop response A0 As () s 1 p Gain of feedback system is reduced in + + A(s) out - (s) A f A0 1 A0 () s s 1 (1 A ) p 0 Frequency characteristic of feedback system is changed Copyright 2007-2008 Stevens Institute of Technology - All rights reserved 22-13/23

Feedback Amplifier Poles As () A0 s 1 p in + + A(s) out - j (s) A f A0 1 A0 () s s 1 (1 A ) p 0 (1 ) pf p A0 p pf db A o A f0 A(s) log() Copyright 2007-2008 Stevens Institute of Technology - All rights reserved p pf 22-14/23

Approximating system response A f () s A 1 0 f s pf db A f0 For s A ( s) A pf f 0 f For s A ( s) pf log() f A 0 f s pf pf Copyright 2007-2008 Stevens Institute of Technology - All rights reserved 22-17/23

System with two pole response As () A 0 s s 1 1 p1 p2 Closed loop response poles: 2 s s p 1 p2 A0 p 1p2 1 0 Solving for poles: s 2 1 1 2 p1 p2 2 p 1 p2 41 A0 p 1p2 Copyright 2007-2008 Stevens Institute of Technology - All rights reserved 22-19/23

Electronic Circuits EE359A Bruce McNair B206 bmcnair@stevens.edu 201-216-5549 Lecture 16 0

Signal Generators and Waveform-shaping Circuits Ch 17 1

Input summing, output sampling voltage amplifier Series voltage summing Shunt voltage sensing 2

Using negative feedback system to create a signal generator A A ( ) 1 A ( ) 3

Basic oscillator structure 4

Basic oscillator structure With positive feedback A f () s As () 1 A( s) ( s) 5

Basic oscillator structure With positive feedback Loop gain A f () s As () 1 A( s) ( s) A() s () s 6

Basic oscillator structure With positive feedback Loop gain A f () s As () 1 A( s) ( s) A() s () s Define loop gain L(s) Ls () As () () s 7

Basic oscillator structure With positive feedback Loop gain A f () s As () 1 A( s) ( s) A() s () s Characteristic equation 1 Ls ( ) 0 Define loop gain L(s) Ls () As () () s 8

Criteria for oscillation For oscillation to occur at o L( j ) A( j ) ( j ) 1 o o o The Barkhausen criteria: At o, the loop gain has a magnitude 1 and the phase shift is 0 (for positive feedback) 9

Criteria for oscillation For oscillation to occur at o L( j ) A( j ) ( j ) 1 o o o The Barkhausen criteria: At o, the loop gain has a magnitude 1 and the phase shift is 0 (for positive feedback) x f Ax f A x x o A 1 x o o x o 10

Oscillation frequency dependence on phase response A steep phase response ( () ) produces a stable oscillator 12

Oscillator amplitude j L(j o ) < 1 f( t) 2 0 2 j 0 1 2 3 4 5 t a 0.2 2 L(j o ) > 1 f( t) 0 2 0 1 2 3 4 5 t a 0.2 13

Oscillator amplitude j L(j o ) = 1 f() t 2 0 2 0 1 2 3 4 5 a 0 t 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? 14

Nonlinear oscillator amplitude control 15

Nonlinear oscillator amplitude control 16

Nonlinear oscillator amplitude control 17

Nonlinear oscillator amplitude control 18

Wein-Bridge oscillator (without amplitude stabilization) 19

Wein-Bridge oscillator (without amplitude stabilization) A (s) 20

Wein-Bridge oscillator (without amplitude stabilization) A (s) Ls () A () s R A 1 R () s Z 2 1 p Z p Z R Z 2 p Ls () 1 R1 Zp Z s s 21

Wein-Bridge oscillator (without amplitude stabilization) A (s) R Z 2 p Ls () 1 R1 Zp Zs 1R R 1R R Ls () 1Z Z 1Z Z 1 R R Ls () 1 ZY 2 1 2 1 p s s p 2 1 s p 1 R2 R1 Ls () 1 1 1R 22 sc sc R

Wein-Bridge oscillator (without amplitude stabilization) A (s) Ls () Ls () L( j) 1 R2 R1 1 1 1R sc sc R 1 R2 R1 R 1 sc 1 scr R scr sc 1 R2 R1 1 3 jcr CR23

Wein-Bridge oscillator (without amplitude stabilization) A L( j) 3 1 R2 R1 1 jcr CR (s) Oscillation at o if CR o 1 o CR 1 CR o 24

Wein-Bridge oscillator (without amplitude stabilization) A L( j) 3 1 R2 R1 1 jcr CR (s) Oscillation if 1 R L( j) 3 R R 2 2 1 R 2 1 25

Wein-Bridge oscillator (with amplitude stabilization) A (s) stabilization 26

Wein-Bridge oscillator (with amplitude stabilization) f 0 o o o 1 CR 1 9 3 (16 10 F)(10 10 ) 6250 rad/sec 1000 Hz R R R 2 1 R 2 1 2 20.3 10 2.03 27

Wein-Bridge oscillator (with alternative stabilization) D 1 and D 2 reduce R f at high amplitudes 28

Phase shift oscillator -A (s) 29

Phase shift oscillator -A (s) Phase shift of each RC section must be 60 o to generate a total phase shift of 180 o K must be large enough to compensate for the amplitude attenuation of the 3 RC sections at o 30

Quadrature oscillator 31

Quadrature oscillator Limiting circuit Integrator 2 Integrator 1 32

Quadrature oscillator Limiting circuit 1 Ls () scr 1 0 CR 2 2 2 Integrator 2 Integrator 1 33

Quadrature oscillator sin( 0t) cos( t) 0 34

LC oscillator Colpitts oscillator 35

LC oscillator Hartley oscillator 36

LC oscillator Colpitts oscillator Frequency determining element Hartley oscillator 37

LC oscillator Colpitts oscillator Gain stage Hartley oscillator 38

LC oscillator Colpitts oscillator Feedback voltage divider Hartley oscillator 39

LC oscillator Colpitts oscillator 0 1 CC 1 2 L C C 1 2 Hartley oscillator 0 1 1 2 L L C 40

Practical LC (Colpitts) oscillator 41

Piezoelectric oscillator Quartz crystal schematic symbol 42

Piezoelectric oscillator Quartz crystal schematic symbol Equivalent circuit 43

Piezoelectric oscillator Quartz crystal schematic symbol Equivalent circuit Reactance 44

Piezoelectric oscillator s 1 LC s Series resonance Parallel resonance p 1 CC s p L C s C p 45

Piezoelectric oscillator s 1 LC s Series resonance Parallel resonance p 1 CC s p L C s C p r << Z L 46

Pierce crystal oscillator CMOS inverter (high gain amplifier) DC bias circuit (near V DD /2) LPF to discourage harmonic/overtone oscillation Frequency determining elements (but C S dominates) 47