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1 Lecture 290 Feedback Analysis using Return Ratio (3/20/02) Page 2901 LECTURE 290 FEEDBACK CIRCUIT ANALYSIS USING RETURN RATIO (READING: GHLM ) Objective The objective of this presentation is: 1.) Illustrate the method of using return ratio to analyze feedback circuits 2.) Demonstrate using examples Outline Concept of return ratio Closedloop gain using return ratio Closedloop impedance using return ratio Summary ECE 6412 Analog Integrated Circuit Design II P.E. Allen 2002 Lecture 290 Feedback Analysis using Return Ratio (3/20/02) Page 2902 Concept of Return Ratio Instead of using twoport analysis, return ratio takes advantage of signal flow graphs (theory). The return ratio for a dependent source in a feedback loop is found as follows: 1.) Set all independent sourceo zero. 2.) Change the dependent source to an independent source and define the controlling variable as,, and the source variable as. 3.) Calculate the return ratio designated as RR = /. a a Fig ECE 6412 Analog Integrated Circuit Design II P.E. Allen 2002

2 Lecture 290 Feedback Analysis using Return Ratio (3/20/02) Page 2903 Example 1 Calculation of Return Ratio Find the return ratio of the op amp with feedback shown if the input resistance of the op amp i i, the output resistance i o, and the voltage gain is a v. v r = v s R S Fig v s R S r o v x r i a v v x (a v )R S r i r o R S r i RR = v r (a v )R S r i = r o R S r i vo R S r o v r r i a v ECE 6412 Analog Integrated Circuit Design II P.E. Allen 2002 Lecture 290 Feedback Analysis using Return Ratio (3/20/02) Page 2904 ClosedLoop Gain Using Return Ratio Consider the following general feedback amplifier: s ic ks ic Fig Note that = ks ic. Assume the amplifier is linear and express s ic and as linear functions of the two sources, and. s ic = B 1 H = d B 2 where B 1, B 2, and H are defined as B 1 = s ic =0 = s ic k=0, B 2 = =0, and H = s ic =0 ECE 6412 Analog Integrated Circuit Design II P.E. Allen 2002

3 Lecture 290 Feedback Analysis using Return Ratio (3/20/02) Page 2905 ClosedLoop Gain Using Return Ratio Continued Interpretation: B 1 ihe transfer function from the input to the controlling signal with k = 0. B 2 ihe transfer function from the controlling signal to the output with = 0. H ihe transfer function from the output of the dependent source to the controlling signal with = 0 and multiplied times a 1. d is defined as, d = =0 = k=0 d = ihe direct signal feedthrough when the controlled source in A is set to zero (k=0) Closedloop gain ( / ) can be found as, s ic = B 1 H = B 1 kh s ic s ic B 1 = 1 kh = d B 2 = d kb 2 s ic = d B 1kB 2 1 kh 2.) A = = B 1kB 2 1 kh d = B 1kB 2 1 RR d = g 1 RR d where RR = kh and g = B 1 kb 2 (gain from to if H = 0 and k = 0) ECE 6412 Analog Integrated Circuit Design II P.E. Allen 2002 Lecture 290 Feedback Analysis using Return Ratio (3/20/02) Page 2906 ClosedLoop Gain Using Return Ratio Continued Further simplification: g g A = 1 RR d = g d(1rr) 1 RR = g d RR 1 RR d 1 RR = RR drr 1 RR d 1 RR Define A = g RR d RR 3.) A = A 1 RR d 1 RR Note that as RR, that A = A. A ihe closedloop gain when the feedback circuit is ideal (i.e., RR or k ). Block diagram of the new formulation: d Σ b = RR A Σ Note that b = RR A is called the effective gain of the feedback amplifier. 1 A Fig ECE 6412 Analog Integrated Circuit Design II P.E. Allen 2002

4 Lecture 290 Feedback Analysis using Return Ratio (3/20/02) Page 2907 Example 2 Use of Return Ratio Approach to Calculate the ClosedLoop Gain Find the closedloop gain and the effective gain of the transistor V CC feedback amplifier shown using the previous formulas. Assume that the BJT g m = 40mS, r π = 5kΩ, and r o = 1MΩ. The smallsignal model suitable for calculating A and d is shown. i in= = ks ic i F r π v be =s ic r o R C = g m v be Fig i in = 20kΩ R C = 10kΩ Fig A = k= = i in g m = =? f = i F v in =0 = 1 d = k=0 = i in g m =0 = = Remember that A = a 1af 1 f Therefore, A = = 20kΩ r π r π (r o R C ) (r or C ) 5kΩ 5kΩ20kΩ1MΩ10kΩ (1MΩ10kΩ) = 1.42kΩ as a. ECE 6412 Analog Integrated Circuit Design II P.E. Allen 2002 Lecture 290 Feedback Analysis using Return Ratio (3/20/02) Page 2908 Example 2 Continued What is left io calculate the RR. A smallsignal model for this is shown below. = g m r π v be =v r r o R C g m v be Fig r o R C v r = (g m ) r π r o R C r π v r r o R C = (g m r π ) r π r o R C RR = v r r o R C 1MΩ10kΩ = (g m r π ) r π r o R C = (200) 5kΩ20kΩ1MΩ10kΩ = Now, the closed loop gain is found to be, RR A = A 1 RR d 1 RR = (20kΩ) kΩ = 19.63kΩ The effective gain is given as, b = RR A = 56.74(20kΩ) = 1135kΩ ECE 6412 Analog Integrated Circuit Design II P.E. Allen 2002

5 Lecture 290 Feedback Analysis using Return Ratio (3/20/02) Page 2909 ClosedLoop Impedance Formula using the Return Ratio (Blackman s Formula) Consider the following linear feedback circuit where the impedance at port X io be calculated. v x Port X s ic = s ic = ks ic Port Y s y Fig Expressing the signals, v x and s ic as linear functions of the signals and s y gives, v x = a 1 a 2 s y s ic = a 3 a 4 s y The impedance looking into port X when k = 0 is, Z port (k=0) = v x k=0 = v x s y =0 ECE 6412 Analog Integrated Circuit Design II P.E. Allen 2002 Lecture 290 Feedback Analysis using Return Ratio (3/20/02) Page ClosedLoop Impedance Formula using the Return Ratio Continued Next, compute the RR for the controlled source, k, under two different conditions. 1.) The first condition is when port X is open ( = 0). s ic = a 4 s y = a 4 Also, = ks ic = ka 4 RR(port open) = = ka 4 2.) The second condition is when port X is shorted (v x = 0). = a 2 a s 1 y = a 2 a s 1 t a 2 a 3 s ic = a 3 a 4 s y = a 4 a 1 The return signal is a 2 a 3 = ks ic = ka 4 a 1 s t RR(port shorted) = a 2 a 3 = ka 4 a 1 3.) The port impedance can be found as (Blackman s formula), 4.) Z port = x x a 2 a 3 1 k a 4 a 1 = a 1 1 a 4 Z port = Z port (k=0) 1 RR(port shorted) 1 RR(port open) ECE 6412 Analog Integrated Circuit Design II P.E. Allen 2002

6 Lecture 290 Feedback Analysis using Return Ratio (3/20/02) Page Example 3 Application of Blackman s Formula Use Blackman s formula to calculate the output resistance of Example 2. We must calculate three quantities. They are R out (g m =0), RR(output port shorted), and RR(output port open). Use the following r π v be =v r r o R C model for calculations: g m v be = g m R out (g m =0) = r o R C (r π ) = 7.09kΩ Fig RR(output port shorted) = 0 because = 0. RR(output port open) = RR of Example 2 = RR(port shorted) 1 R out = R out (g m =0) 1 RR(port open) = 7.09kΩ = 129Ω ECE 6412 Analog Integrated Circuit Design II P.E. Allen 2002 Lecture 290 Feedback Analysis using Return Ratio (3/20/02) Page Example 4 Output Resistance of a SuperSource Follower Find an expression for the smallsignal output resistance of the circuit shown. The appropriate smallsignal model is shown where g m2 = k. v 2 g m1 v 2 v r in ds1 g m2 v 1 v 1 rds2 ut v 2 g m1 v 2 v r in ds1 g m2v 1= v 1 = v r g m2 Fig R out (g m2 =0) = r ds2 and RR(output port shorted) = 0 because = 0. RR(output port open) = = v r v r = ut (g m1 v 2 )r ds1 = ut g m1 r ds1 (ut ) = ut (1 g m1 r ds1 ) ut = g m2 r ds2 v r = (1 g m1 r ds1 )g m2 r ds2 RR(output port open) = v r = (1 g m1 r ds1 )g m2 r ds2 r ds2 ut v in I 2 I 1 Fig V DD M1 ut I 2 I 1 M2 ECE 6412 Analog Integrated Circuit Design II P.E. Allen 2002

7 Lecture 290 Feedback Analysis using Return Ratio (3/20/02) Page RR(port shorted) 10 R out = R out (g m2 =0) 1 RR(port open) = r ds2 1(1 g m1 r ds1 )g m2 r ds2 1 g m1 r ds1 g m2 ECE 6412 Analog Integrated Circuit Design II P.E. Allen 2002 Lecture 290 Feedback Analysis using Return Ratio (3/20/02) Page SUMMARY Return ratio is associated with a dependent source. If the dependent source is converted to an independent source, then the return ratio ihe gain from the dependent source variable to the previously controlling variable. The closedloop gain of a linear, negative feedback system can be expressed as RR A = A 1 RR d 1 RR where A = the closedloop gain when the loop gain ifinite RR = the return ratio d = the closedloop gain when the amplifier gain is zero The resistance at a port can be found from Blackman s formula which is 1 RR(port shorted) Z port = Z port (k=0) 1 RR(port open) where k ihe gain of the dependent source chosen for the return ratio calculation This stuff is all great but of little use as far as calculations are concerned. Smallsignal analysis is generally quicker and easier than the twoport approach or the return ratio approach. Why study feedback? Because it is a great tool for understanding a circuit and for design. ECE 6412 Analog Integrated Circuit Design II P.E. Allen 2002

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