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ECE 6412 - Analog Integrated Circuit Design - II © P.E. Allen - 2002
LECTURE 290 – FEEDBACK CIRCUIT ANALYSIS USING RETURN
RATIO
(READING: GHLM – 599-613)
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
- Closed-loop gain using return ratio
- Closed-loop impedance using return ratio
- Summary
Lecture 290 – Feedback Analysis using Return Ratio (3/22/04) Page 290-
Concept of Return Ratio
Instead of using two-port analysis, return ratio takes advantage of signal flow graph theory.
The return ratio for a dependent source in a feedback loop is found as follows:
1.) Set all independent sources to zero.
2.) Change the dependent source to an independent source and define the controlling variable as, sr, and the source variable as st.
3.) Calculate the return ratio designated as RR = - s (^) r/st.
sr asr st
sr
st
asr
Fig. 290-
sin sout
sout
Rest of feedback amplifier
sr asr st
sr
st
sin sout
sout
Rest of feedback amplifier
ECE 6412 - Analog Integrated Circuit Design - II © P.E. Allen - 2002
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 is ri, the output resistance is ro, and the voltage gain is a (^) v.
R (^) S
R (^) F
v (^) s
v (^) o
R (^) S R (^) F
v (^) s v (^) o
vx r^ i
r (^) o
-a (^) v v (^) x
R (^) S R (^) F
v (^) o
vr r^ i
r (^) o
-a (^) v v (^) t
Fig. 290-02 -
Solution
vr =
(-avvt)RS||ri ro + RF + RS||ri →^ RR^ = -^
vr vt =^
(av)RS||ri ro + RF + RS||ri
Lecture 290 – Feedback Analysis using Return Ratio (3/22/04) Page 290-
Closed-Loop Gain Using Return Ratio
Consider the following general feedback amplifier:
sic ksic soc
sr
soc
sin sout
sout
Rest of feedback amplifier Fig. 290-
Note that soc = ks (^) ic.
Assume the amplifier is linear and express sic and sout as linear functions of the two
sources, sin and soc.
sic = B 1 sin - H soc sout = d sin + B 2 soc
where B 1 , B 2 , and H are defined as
B 1 =
sic sin
soc=0 =^
sic sin
k=0 ,^ B^2 =^
sout soc
sin=0 ,^ and^ H^ = -^
sic soc
sin=
ECE 6412 - Analog Integrated Circuit Design - II © P.E. Allen - 2002
Example 2 – Use of Return Ratio Approach to Calculate the Closed-Loop Gain
Find the closed-loop gain and the effective gain of the transistor feedback amplifier shown using the previous formulas. Assume that the BJT gm = 40mS, r π = 5kΩ, and ro = 1MΩ.
Solution
The small-signal model suitable for calculating A∞ and d is
shown.
A∞ =
sout sin
k=∞ =^
vo iin
gm=∞ =?^ Remember that^ A^ =^
a 1+af →^
f as^ a^ →^ ∞.
f =
v (^) o i (^) F
vin= 0 =
RF Therefore,^ A∞^ = -^ RF^ = -20kΩ
d =
sout sin
k= 0 =^
vo iin
gm= 0 =^
r π r π+RF+(ro||RC) (ro||RC)
5kΩ 5kΩ+20kΩ+1MΩ||10kΩ (1MΩ||10kΩ) = 1.42kΩ
i (^) in
v (^) o
VCC
R (^) C = 10kΩ R (^) F = 20kΩ
-^ Fig. 290-
v (^) be=s (^) ic v (^) o =s (^) out
i (^) in= s (^) in
R (^) F
rπ r (^) o R (^) C g (^) mv (^) be = ksic Fig. 290-
i (^) F
Lecture 290 – Feedback Analysis using Return Ratio (3/22/04) Page 290-
Example 2 – Continued
What is left is to calculate the RR. A small-signal model for this is shown below.
v (^) be=v (^) r v (^) o
R (^) F
rπ ro R (^) C g (^) m v (^) be = g (^) m v (^) t Fig. 290-
vr = (-g (^) mvt)
r (^) o||R (^) C r π+RF+ro||RC r^ π^ →^
vr vt = (-g^ m^ r^ π)
r (^) o||R (^) C r π+RF+ro||RC
RR = -
vr vt = (g^ m^ r^ π)
r (^) o||R (^) C r π+RF+ro||RC = (200)^
1MΩ||10kΩ 5kΩ+20kΩ+1MΩ||10kΩ = 56.
Now, the closed loop gain is found to be,
A = A∞
RR
1 + RR +^
d 1 + RR = (-20kΩ)
1 + 56.74 +^
1.4kΩ 1 + 56.74 = -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
Closed-Loop Impedance Formula using the Return Ratio (Blackman’s Formula)
Consider the following linear feedback circuit where the impedance at port X is to be calculated.
ksic
Fig. 290-
sic = s (^) r
st
sic =sr
st
vx
Rest of feedback amplifier
ix
Port X
Port Y
sy
Expressing the signals, vx and sic as linear functions of the signals ix and sy gives,
vx = a 1 ix + a 2 sy sic = a 3 ix + a 4 sy
The impedance looking into port X when k = 0 is,
Zport(k=0) =
vx i (^) x
k= 0 =^
v (^) x i (^) x
sy= 0
Lecture 290 – Feedback Analysis using Return Ratio (3/22/04) Page 290-
Closed-Loop 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 (i (^) x = 0).
sic = a 4 sy = a 4 st
Also,
sr = ksic → sr = ka 4 st → RR(port open) = -
sr st = -^ ka^4
2.) The second condition is when port X is shorted (vx = 0).
i (^) x = -
a 2 a 1 sy^ = -^
a 2 a 1 st
∴ sic = a 3 ix + a 4 sy =
a 4 -
a 2 a 3 a 1 st
The return signal is
sr = ksic = k
a 4 -
a 2 a 3 a 1 st^ →^ RR(port shorted) = -^
s (^) r st = -^ k
a 4 -
a 2 a 3 a 1
3.) The port impedance can be found as (Blackman’s formula),
4.) Zport =
vx i (^) x =^ a^1
1 - k
a 4 -
a 2 a 3 a 1 1 - a 4 ⇒^ Zport^ =^ Zport(k=0)
1 + RR(port shorted) 1 + RR(port open)
SUMMARY
- Return ratio is associated with a dependent source. If the dependent source is converted to an independent source, then the return ratio is the gain from the dependent source variable to the previously controlling variable.
- The closed-loop gain of a linear, negative feedback system can be expressed as
A = A∞
RR
1 + RR +^
d 1 + RR where A∞ = the closed-loop gain when the loop gain is infinite RR = the return ratio d = the closed-loop gain when the amplifier gain is zero
- The resistance at a port can be found from Blackman’s formula which is
Zport = Zport(k=0)
1 + RR(port shorted) 1 + RR(port open) where k is the 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.
Small-signal analysis is generally quicker and easier than the two-port approach or the return ratio approach.
- Why study feedback? Because it is a great tool for understanding a circuit and for knowing how to modify the performance in design.
