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1 Feedback: 8.8 The Stability Problem

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Microelectronic Circuits - Fifth Edition Sedra/Smith2 Copyright 2004 by Oxford University Press, Inc. Figure 8.1 General structure of the feedback amplifier. This is a signal-flow diagram, and the quantities x represent either voltage or current signals. 8.8.1 Transfer Function of the Feedback Amplifier

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Microelectronic Circuits - Fifth Edition Sedra/Smith3 Copyright 2004 by Oxford University Press, Inc. Suppose that there is a frequency,, for which Then 8.8.1 Transfer Function of the Feedback Amplifier Loop Gain:

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Microelectronic Circuits - Fifth Edition Sedra/Smith4 Copyright 2004 by Oxford University Press, Inc. If, then (Armstrong’s “regeneration”) If, then (Armstrong’s “oscillation”) 8.8.1 Transfer Function of the Feedback Amplifier

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Microelectronic Circuits - Fifth Edition Sedra/Smith5 Copyright 2004 by Oxford University Press, Inc. 8.9.1 Stability and Pole Location A quick review

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Microelectronic Circuits - Fifth Edition Sedra/Smith6 Copyright 2004 by Oxford University Press, Inc. Figure 8.29 Relationship between pole location and transient response.

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Microelectronic Circuits - Fifth Edition Sedra/Smith7 Copyright 2004 by Oxford University Press, Inc. 8.9.2 Poles of the Feedback Amplifier Recall the closed loop transfer function (transfer function with feedback): Poles:

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Microelectronic Circuits - Fifth Edition Sedra/Smith8 Copyright 2004 by Oxford University Press, Inc. 8.9.3 Amplifier with a Single- Pole Response Simple example: suppose an amplifier without feedback has a single pole at : With negative feedback, we saw earlier that, assuming constant, the amplifier gain is:

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Microelectronic Circuits - Fifth Edition Sedra/Smith9 Copyright 2004 by Oxford University Press, Inc. The feedback has moved the pole along the axis from to. 8.9.3 Amplifier with a Single- Pole Response

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Microelectronic Circuits - Fifth Edition Sedra/Smith10 Copyright 2004 by Oxford University Press, Inc. Let. When, 8.9.3 Amplifier with a Single- Pole Response

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Microelectronic Circuits - Fifth Edition Sedra/Smith11 Copyright 2004 by Oxford University Press, Inc. Summary: when, 8.9.3 Amplifier with a Single- Pole Response

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Microelectronic Circuits - Fifth Edition Sedra/Smith12 Copyright 2004 by Oxford University Press, Inc. Figure 8.30 Effect of feedback on (a) the pole location and (b) the frequency response of an amplifier having a single-pole open-loop response. 8.9.3 Amplifier with a Single- Pole Response

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Microelectronic Circuits - Fifth Edition Sedra/Smith13 Copyright 2004 by Oxford University Press, Inc. Figure 8.36 Bode plot for the loop gain A illustrating the definitions of the gain and phase margins. Instability: AND 8.10.1 Gain and Phase Margins

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Microelectronic Circuits - Fifth Edition Sedra/Smith14 Copyright 2004 by Oxford University Press, Inc. 8.10.2 Effect of Phase Margin on Closed-Loop Response Typical phase margin design value: Value of phase margin affects shape of closed- loop gain versus frequency plot.

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Microelectronic Circuits - Fifth Edition Sedra/Smith15 Copyright 2004 by Oxford University Press, Inc. 8.10.2 Effect of Phase Margin on Closed-Loop Response Example: suppose real Closed loop gain at low frequencies:

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Microelectronic Circuits - Fifth Edition Sedra/Smith16 Copyright 2004 by Oxford University Press, Inc. 8.10.2 Effect of Phase Margin on Closed-Loop Response Supposefor some frequency. where. Thus, we can write

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Microelectronic Circuits - Fifth Edition Sedra/Smith17 Copyright 2004 by Oxford University Press, Inc. 8.10.2 Effect of Phase Margin on Closed-Loop Response

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Microelectronic Circuits - Fifth Edition Sedra/Smith18 Copyright 2004 by Oxford University Press, Inc. 8.10.2 Effect of Phase Margin on Closed-Loop Response Closed loop gain at frequency :

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Microelectronic Circuits - Fifth Edition Sedra/Smith19 Copyright 2004 by Oxford University Press, Inc. 8.10.2 Effect of Phase Margin on Closed-Loop Response Magnitude of the closed loop gain at frequency : For a phase margin of

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Microelectronic Circuits - Fifth Edition Sedra/Smith20 Copyright 2004 by Oxford University Press, Inc. 8.10.2 Effect of Phase Margin on Closed-Loop Response Magnitude of the closed loop gain at frequency : Thus the gain peaks at 131% of its value at low frequencies with a phase margin of 45 .

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Microelectronic Circuits - Fifth Edition Sedra/Smith21 Copyright 2004 by Oxford University Press, Inc. 8.10.2 Effect of Phase Margin on Closed-Loop Response

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Microelectronic Circuits - Fifth Edition Sedra/Smith22 Copyright 2004 by Oxford University Press, Inc. Construct a Bode plot for rather than for. 8.10.3 An Alternative Approach for Investigating Stability (Bode)

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Microelectronic Circuits - Fifth Edition Sedra/Smith23 Copyright 2004 by Oxford University Press, Inc. Example: 8.10.3 An Alternative Approach for Investigating Stability (Bode)

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Microelectronic Circuits - Fifth Edition Sedra/Smith24 Copyright 2004 by Oxford University Press, Inc. Figure 8.37 Stability analysis using Bode plot of |A|. 8.10.3 An Alternative Approach for Investigating Stability (Bode) 50 60 85

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Microelectronic Circuits - Fifth Edition Sedra/Smith25 Copyright 2004 by Oxford University Press, Inc. 8.10.3 An Alternative Approach for Investigating Stability (Bode) Note that instability occurs for smaller values of. Result that we will not prove: –If the horizontal line intersects the curve on a segment for which the slope is –20 dB/decade, then the phase margin will be a minimum of 45º.

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Microelectronic Circuits - Fifth Edition Sedra/Smith26 Copyright 2004 by Oxford University Press, Inc. 8.11 Frequency Compensation Frequency compensation modifies the open-loop transfer function of an amplifier (with three or more poles) so that the closed-loop transfer function is stable for any value chosen for the closed- loop gain.

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Microelectronic Circuits - Fifth Edition Sedra/Smith27 Copyright 2004 by Oxford University Press, Inc. 8.11 Frequency Compensation: Theory The simplest method of frequency compensation modifies the original open- loop transfer function,, by introducing a new low frequency pole at to form a new open-loop transfer function,, which has a slope of –20 dB/decade at the intersection of the curve and the curve.

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Microelectronic Circuits - Fifth Edition Sedra/Smith28 Copyright 2004 by Oxford University Press, Inc. Figure 8.38 Frequency compensation for = 10 2. The response labeled A is obtained by introducing an additional pole at f D. 8.11 Frequency Compensation: Example: Stability for closed-loop gains of 40 dB or higher by adding a new pole at.

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Microelectronic Circuits - Fifth Edition Sedra/Smith29 Copyright 2004 by Oxford University Press, Inc. Figure 8.38 Frequency compensation for = 10 2. The response labeled A is obtained by introducing an additional pole at f D. 8.11 Frequency Compensation: Difficulty: Stability achieved, but high open-loop gain, and hence the benefits of negative feedback, only for:

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Microelectronic Circuits - Fifth Edition Sedra/Smith30 Copyright 2004 by Oxford University Press, Inc. 8.11.3 Miller Compensation and Pole Splitting The Miller effect allows shifting to and shifting to a higher frequency. Stability and high open-loop gain for Figure 8.38 Frequency compensation for = 10 2. The response labeled A is obtained by introducing an additional pole at f D. The A response is obtained by moving the original low-frequency pole to f D.

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Microelectronic Circuits - Fifth Edition Sedra/Smith31 Copyright 2004 by Oxford University Press, Inc. Figure 8.40 (a) A gain stage in a multistage amplifier with a compensating capacitor connected in the feedback path and (b) an equivalent circuit. Note that although a BJT is shown, the analysis applies equally well to the MOSFET case. 8.11.3 Miller Compensation and Pole Splitting

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Microelectronic Circuits - Fifth Edition Sedra/Smith32 Copyright 2004 by Oxford University Press, Inc. 8.11.3 Miller Compensation and Pole Splitting Node equations: B: C: Transresistance amplifier:

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Microelectronic Circuits - Fifth Edition Sedra/Smith33 Copyright 2004 by Oxford University Press, Inc. 8.11.3 Miller Compensation and Pole Splitting Collect terms: B: C:

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Microelectronic Circuits - Fifth Edition Sedra/Smith34 Copyright 2004 by Oxford University Press, Inc. 8.11.3 Miller Compensation and Pole Splitting MATLAB: %Declare symbolic variables syms s R1 R2 C1 C2 Cf gm Ii Vb Vc Vo A b x %Define equations: Ax = b A=[1/R1+s*(C1+Cf) -s*Cf; gm-s*Cf 1/R2+s*(Cf+C2)]; b=[Ii; 0]; %Solve the equations: x=A\b; Vo=x(2); Vo=simplify(Vo)

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Microelectronic Circuits - Fifth Edition Sedra/Smith35 Copyright 2004 by Oxford University Press, Inc. 8.11.3 Miller Compensation and Pole Splitting MATLAB: Vo = R1*Ii*R2*(-gm+s*Cf)/ (s*R1*Cf*R2*gm+1+s*R1*C1+s*R1*Cf+s*R2*Cf+s^2*R2*Cf*R1*C1 +s*R2*C2+s^2*R2*C2*R1*C1+s^2*R2*C2*R1*Cf)

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Microelectronic Circuits - Fifth Edition Sedra/Smith36 Copyright 2004 by Oxford University Press, Inc. 8.11.3 Miller Compensation and Pole Splitting (3 capacitors, 2 poles? Miller’s Theorem.) Suppose one pole is dominant:

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Microelectronic Circuits - Fifth Edition Sedra/Smith37 Copyright 2004 by Oxford University Press, Inc. 8.11.3 Miller Compensation and Pole Splitting By comparison: Prepare for logarithmic differentiation:

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Microelectronic Circuits - Fifth Edition Sedra/Smith38 Copyright 2004 by Oxford University Press, Inc. 8.11.3 Miller Compensation and Pole Splitting Logarithmic differentiation: Thus,, so the Miller capacitor lowers the frequency of the lower frequency pole.

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Microelectronic Circuits - Fifth Edition Sedra/Smith39 Copyright 2004 by Oxford University Press, Inc. 8.11.3 Miller Compensation and Pole Splitting What about ? By comparison:

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Microelectronic Circuits - Fifth Edition Sedra/Smith40 Copyright 2004 by Oxford University Press, Inc. 8.11.3 Miller Compensation and Pole Splitting Prepare for logarithmic differentiation:

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Microelectronic Circuits - Fifth Edition Sedra/Smith41 Copyright 2004 by Oxford University Press, Inc. 8.11.3 Miller Compensation and Pole Splitting Note:

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Microelectronic Circuits - Fifth Edition Sedra/Smith42 Copyright 2004 by Oxford University Press, Inc. 8.11.3 Miller Compensation and Pole Splitting For the Miller effect to be large, :

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Microelectronic Circuits - Fifth Edition Sedra/Smith43 Copyright 2004 by Oxford University Press, Inc. 8.11.3 Miller Compensation and Pole Splitting The Miller effect can split the two poles, pushing the frequency of the lower one lower and pushing the higher frequency higher. Shifting the lower frequency pole gives stability without the addition of an extra pole.

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Microelectronic Circuits - Fifth Edition Sedra/Smith44 Copyright 2004 by Oxford University Press, Inc. 8.11.3 Miller Compensation and Pole Splitting Shifting the higher frequency pole shifts the – 20 dB/decade segment to the right and thus gives broader bandwidth.

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