Feedback (and control) systems

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2 Behavior of Under-damped System Y() s s b y 0 M s 2n y 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 n n 2 cos 1 2n n n j jn jn Copyright Stevens Institute of Technology - All rights reserved 22-2/23

3 Nyquist Plot Example 8.10 Consider an amplifier with frequency independent feedback and an openloop transfer function given by As () 10 4 s Copyright Stevens Institute of Technology - All rights reserved 22-3/23

4 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 Stevens Institute of Technology - All rights reserved 22-4/23

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

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

7 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 Stevens Institute of Technology - All rights reserved 22-7/23

8 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 Stevens Institute of Technology - All rights reserved 22-8/23

9 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 Stevens Institute of Technology - All rights reserved 22-9/23

12 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 Stevens Institute of Technology - All rights reserved 22-12/23

13 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 Stevens Institute of Technology - All rights reserved 22-13/23

14 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 Stevens Institute of Technology - All rights reserved p pf 22-14/23

19 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 p1 p2 2 p 1 p2 41 A0 p 1p2 Copyright Stevens Institute of Technology - All rights reserved 22-19/23

24 Electronic Circuits EE359A Bruce McNair B Lecture 16 0

25 Signal Generators and Waveform-shaping Circuits Ch 17 1

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

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

28 Basic oscillator structure 4

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

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

31 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

32 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

33 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

34 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

35 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 If gain is sufficient, frequency of oscillation is determined only by phase response 11

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

37 Oscillator amplitude j L(j o ) < 1 f( t) j t a L(j o ) > 1 f( t) t a

38 Oscillator amplitude j L(j o ) = 1 f() t 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

39 Nonlinear oscillator amplitude control 15

40 Nonlinear oscillator amplitude control 16

41 Nonlinear oscillator amplitude control 17

42 Nonlinear oscillator amplitude control 18

43 Wein-Bridge oscillator (without amplitude stabilization) 19

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

45 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

46 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 p s s p 2 1 s p 1 R2 R1 Ls () 1 1 1R 22 sc sc R

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

48 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

49 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 R

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

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

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

53 Phase shift oscillator -A (s) 29

54 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

55 Quadrature oscillator 31

56 Quadrature oscillator Limiting circuit Integrator 2 Integrator 1 32

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

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

59 LC oscillator Colpitts oscillator 35

60 LC oscillator Hartley oscillator 36

61 LC oscillator Colpitts oscillator Frequency determining element Hartley oscillator 37

62 LC oscillator Colpitts oscillator Gain stage Hartley oscillator 38

63 LC oscillator Colpitts oscillator Feedback voltage divider Hartley oscillator 39

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

65 Practical LC (Colpitts) oscillator 41

66 Piezoelectric oscillator Quartz crystal schematic symbol 42

67 Piezoelectric oscillator Quartz crystal schematic symbol Equivalent circuit 43

68 Piezoelectric oscillator Quartz crystal schematic symbol Equivalent circuit Reactance 44

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

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

71 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

Electronic Circuits EE359A

Electronic Circuits EE359A Bruce McNair B206 bmcnair@stevens.edu 201-216-5549 Lecture 16 404 Signal Generators and Waveform-shaping Circuits Ch 17 405 Input summing, output sampling voltage amplifier Series