1. ECES 352 Winter 2007Ch. 8 Feedback – Part 51 Feedback Amplifier Stability *Feedback analysis l Midband gain with feedback l New low and high 3dB frequencies l Modified input and output resistances, e.g. *Amplifier’s frequency characteristics *Feedback amplifier’s gain *Define Loop Gain as β f A l Magnitude l Phase A(  ) AoAo A fo Amplifier becomes unstable (oscillates) if at some frequency  i we have

2. ECES 352 Winter 2007Ch. 8 Feedback – Part 52 *Amplifier with one pole l Phase shift of - 90 0 is the maximum l Cannot get - 180 0 phase shift. l No instability problem *Amplifier with two poles l Phase shift of - 180 0 the maximum l Can get - 180 0 phase shift ! *For stability analysis, we define two important frequencies l  1 is where magnitude of β f A goes to unity (0 dB) l  180 is where phase of β f A goes to -180 o l Instability problem if  1 =  180 Feedback Amplifier Instability β f A(dB) 0 dB 0 - 45 0 - 90 0 - 135 0 - 180 0 General form of Magnitude plot of β f A General form of Phase plot of β f A

3. ECES 352 Winter 2007Ch. 8 Feedback – Part 53 Gain and Phase Margins *Gain and phase margins measure how far amplifier is from the instability condition *Phase margin *Gain margin *What are adequate margins? l Phase margin = 50 0 (minimum) l Gain margin = 10 dB (minimum) 0 dB 0 - 45 0 - 90 0 β f A(dB) - 135 0 - 180 0 Phase margin Gain margin - 30 Φ(ω1)Φ(ω1) Φ(ω 180 )

4. ECES 352 Winter 2007Ch. 8 Feedback – Part 54 Numerical Example - Gain and Phase Margins Phase margin Gain margin β f A(dB) Phase Shift (degrees) Note that the gain and phase margins depend on the feedback factor β f and so the amount of feedback and feedback resistors. * Directly through the value of β f. * Indirectly since the gain A o and pole frequencies are influenced by the feedback resistors, e.g. the loading effects analyzed previously. Pole 1 Pole 2 23.5 dB

5. ECES 352 Winter 2007Ch. 8 Feedback – Part 55 Amplifier Design for Adequate Gain and Phase Margins *How much feedback to use? What β f ? *A change of β f means redoing β f A(  ) plots! *Is there an alternate way to find the gain and phase margins? YES l Plot magnitude and phase of gain A(  ) instead of β f A(  ) vs frequency l Plot horizontal line at 1/ β f on magnitude plot (since β f is independent of frequency) l Why? At intersection of 1/ β f with A(  ),  =  1 l So this intersection point gives the value where ω =  1 ! l Then we can find the gain and phase margins as before. *If not adequate margins, pick another β f value, draw another 1/ β f line and repeat process. *Note: β f is a measure of the amount of feedback. Larger β f means more feedback. l Recall for β f = 0, we have NO feedback l As β f increases, we have more feedback 0 - 45 0 - 90 0 Phase margin Gain margin A(dB) - 135 0 - 180 0 1/β f (dB) 0 dB General form of Magnitude plot of A General form of Phase plot of A A(ω 180 ) Φ(ω1)Φ(ω1) Φ(ω 180 )

6. ECES 352 Winter 2007Ch. 8 Feedback – Part 56 Example - Gain and Phase Margins Alternate Method Phase margin 1/β f (dB) A(dB) These are the same results as before for the gain and phase margins. 43.5 dB A(ω 180 ) Gain margin

7. ECES 352 Winter 2007Ch. 8 Feedback – Part 57 Feedback Amplifier with Multiple Poles *Previous feedback analysis l Midband gain l New low and high 3dB frequencies *Amplifiers typically have multiple high and low frequency poles. l Does feedback change the poles other than the dominant ones? l YES, feedback changes the other poles as well. l This is called pole mixing. l This can affect the gain and phase margin determinations ! l The bandwidth is still enlarged as described previously l But the magnitude and phase plots are changed, so the gain and phase margins are modified. A(  ) AoAo A fo

8. ECES 352 Winter 2007Ch. 8 Feedback – Part 58 Feedback Effect on Amplifier with Two Poles *Gain of a two pole amplifier *Gain of the feedback amplifier New poles of the feedback amplifier occur when Midband gain of feedback amplifier. Gain of feedback amplifier having two new, different high frequency poles.

9. ECES 352 Winter 2007Ch. 8 Feedback – Part 59 *Feedback amplifier poles *For no feedback (β f = 0), *For some feedback (β f A o > 0), Q increases and 1- 4Q 2  0 so poles move towards each other as β f increases. *For β f such that Q = 0.5, then 1- 4Q 2 = 0 so two poles coincide at *For larger β f such that Q > 0.5, 1- 4Q 2 < 0 so poles become complex frequencies. Feedback Effect on Amplifier with Two Poles Root-Locus Diagram s =  + j  jj  jj 

10. ECES 352 Winter 2007Ch. 8 Feedback – Part 510 *Amplifier with some feedback ( β f A o > 0 ), as β f increases 1 - 4Q 2  0 so poles move towards each other. For sufficient feedback, Q = 0.5 and 1 - 4Q 2 = 0 so the poles meet at the midpoint. Frequency Response - Feedback Amplifier with Two Poles A f (dB) Original amplifier * For Q = 0.707, we get the maximally flat characteristic * For Q > 0.7, get a peak in characteristic at corresponding to oscillation in the amplifier output for a pulsed input (undesirable). Original amplifier Increasing feedback, poles meet for Q = 0.5 Q>0.7 Original poles before feedback

11. ECES 352 Winter 2007Ch. 8 Feedback – Part 511 Frequency Response of Feedback on Amplifier with Two Poles *For no feedback (β f = 0), Q is a minimum and we have the original two poles. *For some feedback (β f A o > 0), Q > 0 and 1- 4Q 2  0 so poles move towards each other. *For β f such that Q = 0.5, then 1 - 4Q 2 = 0 so two poles coincide at -  o /2Q. *For larger β f such that Q > 0.5, 1- 4Q 2 < 0 and the poles become complex l For β f such that Q = 0.7, we get the maximally flat response (largest bandwidth). l For larger β f such that Q > 0.7, we get a peak in the frequency response and oscillation in the amplifier’s output transient response (undesirable). 3dB frequency increases as feedback (Q and β f ) increases! - 3dB

12. ECES 352 Winter 2007Ch. 8 Feedback – Part 512 *DC bias analysis *Configuration: Shunt-Shunt (A Ro = V o /I s ) *Loading: Input R 1 = R f, Output R 2 = R f *Midband gain analysis Example - Feedback Amplifier with Two Poles R s =10 K R B1 =90 K R B2 = 10 K R C =1.5 K R f =8 K 10 V  =75 r x =0 C  =7 pF C  =0.5 pF R B = 9 K IfIf

13. ECES 352 Winter 2007Ch. 8 Feedback – Part 513 Example - Feedback Amplifier with Two Poles *Determine the feedback factor β f *Calculate gain with feedback A Rfo *High frequency ac equivalent circuit IsIs RsRs RBRB R1R1 R2R2

14. ECES 352 Winter 2007Ch. 8 Feedback – Part 514 Original High Frequency Poles IsIs RsRs RBRB R1R1 R2R2 C  Pole C  Pole RsRs RBRB R1R1 R2R2 See how we did the analysis for the C  2 high frequency pole for the Series-Shunt feedback amplifier example.

15. ECES 352 Winter 2007Ch. 8 Feedback – Part 515 Gain and Phase Margins Phase margin Gain margin 1/β f (dB) A(dB) 38.5 dB A(ω 180 ) Φ(ω1)Φ(ω1) Φ(ω 180 )

16. ECES 352 Winter 2007Ch. 8 Feedback – Part 516 High Frequency Poles for Feedback Amplifier * These new poles are complex numbers ! * Original poles were at 2.1x10 7 and 2.3x10 8 rad/s. * This means the amplifier output will have a tendency to oscillate (undesirable!) * We have too much feedback ! * Q is too large, because β f is too large, because R f is too small (β f = - 1/R f )! * How to change R f so that β f and Q are not too large? Need to increase R f so that β f and Q are smaller. How much to increase R f ? Note that this Q value is bigger than 0.707 so the amplifier will tend to oscillate (undesirable).

17. ECES 352 Winter 2007Ch. 8 Feedback – Part 517 Frequency Response with and without Feedback Original amplifier (without feedback) Amplifier with feedback

18. ECES 352 Winter 2007Ch. 8 Feedback – Part 518 What is the Optimum Feedback and R f ? * Select Q = 0.707 and work backward to find β f and then R f * Q = 0.707 gives maximally flat response ! New poles are still complex numbers, but Q = 0.707 so okay.

19. ECES 352 Winter 2007Ch. 8 Feedback – Part 519 Gain and Phase Margins for Optimal Bandwidth Phase margin Gain margin 1/β f (dB) A(dB) 38.5 dB A(ω 180 )

20. ECES 352 Winter 2007Ch. 8 Feedback – Part 520 Frequency Response for Optimum Feedback Note: * Better midband gain than before * Improved bandwidth and * No oscillation tendency ! Original amplifier (without feedback) Amplifier with feedback

21. ECES 352 Winter 2007Ch. 8 Feedback – Part 521 Design for Minimum Phase Shift (50 o ) *Construct plots of A(  ) in dB and phase shift versus frequency as before. *Determine frequency  180 for -180 o phase shift *For minimum 50 o phase shift, l Measure up from -180 o to -130 o on phase shift plot, draw horizontal line and determine frequency where this is reached. This gives  1. l Find corresponding magnitude of A(  ) in dB at this frequency (  1 ) on the magnitude plot. Draw a horizontal line. This gives the magnitude of 1/β f in dB. l Convert 1/β f in dB to a decimal l Calculate R f from β f A(dB) Phase margin Gain margin 1/β f (dB) *Check gain margin to see if it is okay. YES, 40 dB ! *But Q = 0.94 > 0.707 so some tendency to oscillate. A(ω 180 )

22. ECES 352 Winter 2007Ch. 8 Feedback – Part 522 Design for Minimum Gain Margin (10 dB) *Construct plots of A(  ) in dB and phase shift versus frequency *Determine frequency  180 for -180 o phase shift *For minimum 10dB gain margin, l Draw vertical line on magnitude plot at  180 frequency. Where it intersects the magnitude plot, draw a horizontal line. l Measure upward from this horizontal line by 10 dB and draw a second horizontal line. This gives the magnitude of 1/β f in dB (-12dB here). l Convert 1/β f in dB to a decimal l Calculate R f from β f Phase margin Gain margin 1/β f (dB) A(dB) *Check phase margin to see if it is okay. NO, only 12 o ! *Q is also very large 5.1 >> 0.707 so strong oscillation tendency !

23. ECES 352 Winter 2007Ch. 8 Feedback – Part 523 Summary of Feedback Amplifier Stability *Feedback amplifiers are potentially unstable l Can break into oscillation at a particular frequency  i where *Need to design feedback amplifier with adequate safety margin l Minimum gain margin of 10 dB l Minimum phase margin of 50 o. l Adjust β f by varying size of R f. «Smaller R f is, the larger is the amount of feedback. A(  ) AoAo A fo Oscillation instability is a design problem for any amplifier with two or more high frequency poles. This is the case for ALL amplifiers since each bipolar or MOSFET transistor has two capacitors and each capacitor gives rise to a high frequency pole!