Analog Circuits and Systems

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1 Analog Circuits and Systems Prof. K Radhakrishna Rao Lecture 4: Dynamic Behavior of Feedback Systems

2 Current Follower using MOSFET G Common-gate amplifier = g andg = 0 m g g Loop gain = =? g ds G G m and m 0 g g ds ds is valid within the active region of the n-channel enhancement MOSFET Active region: 0 to I Dmax

3 Current Follower using MOSFET (Biasing) If the dynamic range is to be maximized the MOSFET has to be biased at quiescent current of I Dmax 3

4 MOSFET Current Follower as First-Order System If the gate-to-source capacitance is taken into consideration G = g and G = 0 m Loop gain G f G G 0 ( ) g + s C + C = ds gs ds g ( ) g + s C + C Io G0G = = = I + + ( + ) i G0G g ds s Cgs Cds + g m m ds gs ds 4

5 MOSFET Current Follower as First-Order System (contd.,) Gf = s( C ) s gs + C ds + + gm T gm where T =, gain - bandwidth product of the system C + C gs ds 5

6 Automatic Gain Controller (Dynamic Behavior) 6

7 Automatic Gain Controller (Dynamic Behavior) (contd.,) It is a dc voltage follower from V i to the output of the squarer G G AV V = ; G = sca R p po f G G = 0 = + G G 00 sca R AVV 0 p po = 00CR s + s + VV GB p po 7

8 Frequency Follower 8

9 Frequency Follower (contd.,) G L = V V p 0s p V C Φ o Φ t o = = o = = Φ Φ 0s s i i i + + t GB V V p p V =V =0V Vc p f V C = p 0000Hz / volt 5 GB = π0 radians/sec 9

10 FSK Dynamics 0

11 Simulation

12 Simulation

13 Simulation 3 3

14 Second Order System Consider a second order system G G= with < s s Transfer function of the system with feedback G = constant 4

15 Second Order System (contd.,) G G f = = +G G s s + G + + G G 0 Gf0 G = = f0 s s s s ζ + G 0G G0G n n where n is the natural frequency of the system and is equal to G G 0, and ζis the damping factor 5

16 Second Order System (contd.,) Damping factor ζ is given by ζ= + G 0G ζ is also defined as quality factor Q Q= G G 0 + 6

17 Second Order System (contd.,) If? then can be neglected in comparison to, ζ and Q then become ζ= ; Q= G G G G 0 0 Q = wheng 0G = 7

18 Second Order System: Time Response Input output relationship in time domain of this second order feedback system is governed by dx i dt n n dx X= + i +X o G Q dt When X =u(t), a unit step function i n - t X ( t ) = -e Q sin - t+ψ o G n 4Q i 8

19 Second Order System: Time Response (contd.,) For Q>, i.e. ζ< which represents under damping For a step input at t = 0, o ( ) ψ= 0 X t =0 at t=0 and therefore. The response of a second order feedback system to a step input with Q > 9

20 Second Order System: Time Response (contd.,) 0

21 Simulation (Q=)

22 Simulation (Q=0)

23 Second Order System: Time Response (contd.,) For Q < the response of the system to a step input is given by X X ( t ) = i0 - αe +αe o G n n - + n - t - - n - t Q 4Q Q 4Q For Q = the output X ( t) in response to a step input is o given by X ( ) i0 ( - t -t) X t = - αe n +αte G o n n 3

24 Step response of the second order system Q < andq= 4

25 Second Order System: Time Response (contd.,) If Q< the rate of rise is lower. When Q> the rate of rise is higher but there will be more peaks and results in higher settling time. There are ten visible peaks (count up to 0. of the first peak) when Q = 0. This can be generalized to say if Q = n, there will be n visible peaks in the transient response. The most desirable step response of a feedback is obtained for a value of Q=. The response is characterized by good rate of rise with one small peak 5

26 Second Order System: Frequency Response The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Xo G = Gf = X i s s + + nq n Gf0 nq Gf = φ= Gf = tan + n n nq 6

27 Second Order System: Frequency Response The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Peak of magnitude G fmax occurs at max max =n and G f = max Q G f0 Q 4Q Magnitude G at is G =G Q f n f at f0 Maximum rate of rise in the phase shift n occurs at φ = n and = max Q n 7

28 Frequency response of the second order system Frequency response of the second order system with feedback Q= 8

29 Frequency response of the second order system (contd.,) Frequency response of the second order system with feedback Q = 0 9

30 Frequency response of the second order system (contd.,) When G fmax =G f0 then the frequency response of the amplifier is said to be G = f max G Q f0-4q maximally flat. This property can be whichoccursat Q= exploited in the design of wide band amplifiers 30

31 FLL as second order system 3

32 G Φ Frequency Follower L Φ VpV = 0s o i p V C Φo t o = = = = Φi i s s s 0s t LP GB LP + VpVp Vc GB = = ; n = GB LP; Q = s s s s LP GB GB Q ( ) LP n n V =V =0V p f V C p = 0000Hz / volt 5 GB = π0 radians/sec 3

33 Example ( ) Consider a system with G =0, G =and > =3 0 rad/sec. Determine to obtain ideal transient response to a step input? will have to be chosen to make G G X Q= 0 = o = X i = = 30 G G 0 5 = rad/sec.the value 0 of GB in this example is rad/sec 33

34 Example Consider a system with G =0 5, G = and ( > ) =3 0 6 rad/sec Determine System gain is G G to obtain ideal transient response to a step input? X o = =00 X G i If the step response of the feedback system is to remain ideal Q= 3 0 = = 6 The gain bandwidth product GB in this case is GB= G =

35 Approximation of a Second Order System Second order feedback amplifiers can be approximated to a first order system when Q is much less than / Consider G G G = f0 = f0 for s=j f s s j Q Q n n n n For = n G G = f0 for Q< f j + Q n 35

36 Approximation of a Second Order System (contd.,) G f is now a first order system with bandwidth Q n G = f G G = tan - ; = f Q Q GBG n n If Q> ; G remains a second order system. f 36

37 Approximation of a Second Order System (contd.,) Consider an Op Amp with = 6 = 6 0 f 4 0 Hz and G 0 f = 6 0 Hz,. In order to design a feedback amplifier with unity gain and Q=, the first corner frequency must be shifted from 0 6 Hz to f = 4Hz as Q G = = G f Qf 4 0 ;f = = = 4Hz f G

38 Approximation of a Second Order System (contd.,) The gain bandwidth product of the unity gain feedback amplifier is G f = 4 0 f = 4Hz, f 6. If the same compensated Op Amp 0 = Hz and G = is used to design a feedback amplifier with a gain of 00, the amplifier with gain of 00 will now have a bandwidth of ( 6 ) feedback amplifier is 4 0 = Hz. The Q of this 00 G G G G 0 Q= 0 ; 0 = =0. 3 f f f 0 + f f f 38

39 Approximation of a Second Order System (contd.,) As Q is much less than the transient response of the amplifier is very sluggish and not satisfactory. In order to increase Q to, f need to be shifted from 4 Hz to 400 Hz. This will increase GB to G G f = =

40 Second order system with a zero G + s 0 z G = with s s z > + + > Transfer function of the feedback system s Gf0 + Gf0 z G f = = s s s Gf0 s s Gf0s Gf0 z G0 G0 + G 0 s + z with = G G n 0 40

41 Second order system with a zero (contd.,) = n + n + n Q G G G G z 0 0 G G G G G G = + + G G G G z 0 0 = G G G G z 0 4

42 Second order system with a zero (contd.,) Q= G G + + G G z If = G G Q = z 0 ; z GG If w z is made equal to natural frequency of the second order system then Q = Making Q = is known as frequency compensation 4

43 Conclusion Higher (third or higher) order systems are likely to become unstable when feedback is used. Design of feedback systems should attempt to reduce the order of the system to second or first order. This is what we mean by frequency compensation of a feedback system If of the subsystems of the feedback loop gets in saturation the feedback loop gets broken. Special arrangements may have to be made to bring the loop into active region. 43

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