1. Comparing two ways to find loop gain in feedback circuits Rob Fox University of Florida

2. R3R3 R1R1 vovo vsvs R2R2 Analyzing FB circuits. The old way.

3. R3R3 R1R1 vovo vsvs R2R2 Analyzing FB circuits. The old way. (I think it’s the wrong way.)

4. R3R3 R1R1 vovo vsvs R2R2 Analyzing FB circuits. The old way. First, recognize this connection is series shunt. This determines which of four methods we’ll use.

5. R1R1 vsvs RiRi RoRo GmviGmvi vivi vovo R2R2 R3R3 Replace the amp with its small-signal equivalent circuit

6. R1R1 R2R2 vsvs RiRi RoRo GmviGmvi vivi vovo R2R2 vfvf R3R3 R1R1 Because it’s shunt at the output, we short the right side of R 2 to find the circuit for loading at the input side. Series feedback at the input means we cut R i out but leave R 1 at the right (output) side.

7. R1R1 R2R2 vsvs RiRi RoRo GmviGmvi vivi vovo R2R2 vfvf R3R3 R1R1 Because it’s shunt at the output, we short the right side of R 2 to find the circuit for loading at the input side. Series feedback at the input means we cut R i out but leave R 1 at the right (output) side. Analyze to find A = v o /v s and β = v f /v o. The loop gain T should be Aβ = v f /v s.

8. R1R1 ioio vsvs R2R2 R3R3 Now it’s series series. Note also that there are other places we could define series outputs, giving different results.

9. R1R1 vsvs RiRi RoRo GmviGmvi vivi ioio R2R2 R3R3 Replace with equivalent circuit. Series series.

10. R1R1 R2R2 vsvs RiRi RoRo GmviGmvi vivi R2R2 vfvf R3R3 ioio R3R3 R1R1 Changing output to series means we now leave R 3 in series with R 2 at the input side. This gives a new value for the loop gain. Series series.

11. R1R1 ioio isis R2R2 R3R3 Now it’s shunt at the input, series at output.

12. R1R1 isis RiRi RoRo GmviGmvi vivi ioio R2R2 R3R3

13. R1R1 R2R2 RiRi RoRo GmviGmvi vivi R2R2 R3R3 ioio isis ifif R3R3 Because it’s shunt at the input, we short the right side of R 2 at the output. Yet another value for Aβ. Now it’s shunt at the input, series at output.

14. R1R1 isis vovo R2R2 R3R3 Last permutation is shunt shunt.

15. R1R1 isis RiRi RoRo GmviGmvi vivi vovo R2R2 R3R3

16. R1R1 RiRi RoRo GmviGmvi vivi R2R2 isis R2R2 R3R3 This gives a fourth circuit and a fourth equation for Aβ. How does the circuit know it’s supposed to have different loop gains depending on where we put the inputs and define what we want to use as outputs?? vovo R2R2 ifif Last permutation is shunt shunt.

17. R1R1 RiRi RoRo GmviGmvi vivi R2R2 isis R2R2 R3R3 This gives a fourth circuit and a fourth equation for Aβ. Are the results the same for the four methods? Let’s try it with numbers for the four cases. vovo R2R2 ifif Last permutation is shunt shunt.

18. R1R1 R2R2 RiRi RoRo GmviGmvi vivi R2R2 R3R3 ioio isis ifif R3R3 Shunt series. Let R 1 =10k R 2 =9k R 3 =5k R i =20k R o =2k G m =25 mA/V

19. R1R1 RiRi RoRo GmviGmvi vivi R2R2 isis R2R2 R3R3 vovo R2R2 ifif Shunt shunt. Let R 1 =10k R 2 =9k R 3 =5k R i =20k R o =2k G m =25 mA/V

20. R1R1 R2R2 vsvs RiRi RoRo GmviGmvi vivi vovo R2R2 vfvf R3R3 R1R1 Series shunt. Let R 1 =10k R 2 =9k R 3 =5k R i =20k R o =2k G m =25 mA/V

21. R1R1 R2R2 vsvs RiRi RoRo GmviGmvi vivi R2R2 vfvf R3R3 ioio R3R3 R1R1 Series series. Let R 1 =10k R 2 =9k R 3 =5k R i =20k R o =2k G m =25 mA/V

22. R1R1 R2R2 vsvs RiRi RoRo GmviGmvi vivi R2R2 vfvf R3R3 ioio R3R3 R1R1 Four analyses; four different results for loop gain. Shunt series: T = 15.466 Shunt shunt: T = 13.116 Series shunt: T = 14.135 Series series: T = 13.535 Let R 1 =10k R 2 =9k R 3 =5k R i =20k R o =2k G m =25 mA/V

23. R1R1 R2R2 vsvs RiRi RoRo GmviGmvi vivi R2R2 vfvf R3R3 ioio R3R3 R1R1 Four analyses; four different results for loop gain. Are any of these actually results correct? No. The actual loop gain is

24. R1R1 R2R2 vsvs RiRi RoRo GmviGmvi vivi R2R2 vfvf R3R3 ioio R3R3 R1R1 Four analyses; four different results for loop gain. Shunt series: T = 15.466 Shunt shunt: T = 13.116 Series shunt: T = 14.135 Series series: T = 13.535 Actual value: T = 13.928 This is the value for T that correctly gets the closed- loop gain and all series and shunt resistances. Let R 1 =10k R 2 =9k R 3 =5k R i =20k R o =2k G m =25 mA/V

25. R3R3 R1R1 vovo vsvs R2R2 Here’s the better way to find loop gain.

26. R3R3 R1R1 R2R2 First kill the input source and remove the definition of the output variable. Note that you can no longer tell which of the four topologies this is.

27. R3R3 R1R1 R2R2 The circuit has a closed signal path so there is feedback. The feedback is negative. (Odd # of inversions.) As the schematic is drawn, the feedback signal flows clockwise. The feedback signal flow is unidirectional.

28. R3R3 R1R1 R2R2 Break the signal path in any branch, so long as it breaks all signal flow.

29. R3R3 R1R1 R2R2 Break the signal path in any branch, so long as it breaks all signal flow.

30. R3R3 R1R1 R2R2 Break the signal path in any branch, so long as it breaks all signal flow.

31. R3R3 R1R1 R2R2 Break the signal path in any branch, so long as it breaks all signal flow.

32. R3R3 R1R1 R2R2 Let’s use this breakpoint.

33. R1R1 RiRi RoRo GmviGmvi vivi R2R2 R3R3 Substitute the small-signal equivalent circuit.

34. R1R1 RiRi RoRo GmviGmvi vivi R2R2 R3R3 x y Break the branch. Call the input side the x port and the output side the y port.

35. R1R1 RiRi RoRo GmviGmvi vivi R2R2 R3R3 vxvx y Z in Temporarily short the y port and find the input impedance Z in looking into the x port.

36. R1R1 RiRi RoRo GmviGmvi vivi R2R2 R3R3 vxvx y Z in Z in = R 1 ||R i Temporarily short the y port and find the input impedance Z in looking into the x port. Here, Z in = R 1 ||R i.

37. R1R1 RiRi RoRo GmviGmvi vivi R2R2 R3R3 vxvx vyvy Z in = R 1 ||R i Z in Now place a copy of Z in across the y port, and find the loop gain as |T| = |v y /v x |.

38. R1R1 RiRi RoRo GmviGmvi vivi R2R2 R3R3 vxvx vyvy Z in = R 1 ||R i Z in Now place a copy of Z in across the y port, and find the loop gain as |T| = |v y /v x |. Don’t worry about the sign – we already know the feedback is negative.

39. R1R1 RiRi RoRo GmviGmvi vivi R2R2 R3R3 x y What about the special case that R i = 0? (Maybe the amp is current-controlled.) Alternate algorithm using current gain:

40. R1R1 RiRi RoRo GmviGmvi vivi R2R2 R3R3 x y Z out First short the x port and find Z out looking into the y port. Alternate algorithm using current gain:

41. R1R1 RiRi RoRo GmviGmvi vivi R2R2 R3R3 x y Z out Z out = R 2 + R 3 ||R o Here, Z out = R 2 + R 3 ||R o.

42. R1R1 RiRi RoRo GmviGmvi vivi R2R2 R3R3 iyiy Z out = R 2 + R 3 ||R o Z out ixix Put a copy of Z out at the x port. Now short the y port and find |T |= |i y /i x |. Same result we got before using v y and v x.

43. We started with a voltage-in voltage-out amplifier, but the first thing we did was ignore that information. Same T if we’d started with current-in current-out or any other combination. R1R1 ioio isis R2R2 R3R3

44. R3R3 R1R1 R2R2 The Aβ approach requires a new analysis if you use a different input type or redefine the output variable. The Aβ approach is ambiguous unless the circuit is an amplifier. (What if it’s just a bias circuit?) The Aβ approach gives different results for different assumptions about the feedback topology. Those results can differ significantly from the actual value of the loop gain.

45. R3R3 R1R1 R2R2 The value of T found as shown here is the one that gives correct results for the impedances looking into various ports using R CL = R OL *(1 + T) for series and R CL = R OL /(1 + T) for shunt. The value of T found as shown here does not change depending on where the inputs and ouputs are. You get the same value no matter where you break the loop or whether you use current or voltage test sources.

46. R3R3 R1R1 R2R2 This loop-gain-based approach is much easier to teach and learn. It doesn’t require a table listing the tricks to use for each of four amplifier configurations. One rule for all. Compatible with the A ∞ approach for closed-loop gain – no need for β’s with different dimensionality for each configuration.