HO16_214W09_FB_FreqResp_2pp

HO16_214W09_FB_FreqResp_2pp - Handout #16 EE 214 Winter...

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R. Dutton, B. Murmann EE215 Winter 2007-08 1 Frequency Response of Feedback Amplifiers B. Murmann and B. A. Wooley Stanford University Handout #16 EE 214 Winter 2009 B. Murmann, B. Wooley EE214 Winter 2008-09 2 Gain and Bandwidth Consider a feedback amplifier with a single pole in the response of the forward-path amplifier and feedback that is frequency independent. a(s) f v i v o v fb + a(s) = a 0 1 ! 1 A(s) ! v o (s) v i (s) = a(s) 1 + a(s) ! f = a(s) 1 + T(s) T(s) ! a(s) ! f = "Loop Gain" where
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B. Murmann, B. Wooley EE214 Winter 2008-09 3 A(s) = a 0 1 ! 1 1 + a 0 f 1 ! 1 = a 0 1 ! s p 1 + a 0 f = a 0 1 + a 0 f " 1 1 ! s p 1 1 1 + a 0 f # $ % & ( = A o " 1 1 ! s p 1 (1 + T 0 ) ) * + , - . A 0 = A(0) = a 0 1 + a 0 f T 0 = T(0) = a(0) ! f = a 0 f = "Low Frequency Loop Gain" where B. Murmann, B. Wooley EE214 Winter 2008-09 4 Thus, feedback reduces the gain by (1+T 0 ) and increases the –3dB bandwidth by (1+T 0 ) for a “one-pole” forward-path amplifier. The Gain x Bandwidth (GBW) product remains constant. 20 log 10 |A(j ! )| |p 1 | (1+T 0 )|p 1 | 20 log 10 a 0 20 log 10 |a(j ! )| 20 log 10 (1+T 0 ) 20 log 10 " 0
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B. Murmann, B. Wooley EE214 Winter 2008-09 5 Locus of the pole of A(s) in the s-plane: Pole “moves” from p 1 to (1+T 0 )p 1 Note that 20 ! log 10 a 0 " 20 ! log 10 A 0 = 20 ! log 10 a 0 A 0 # $ % & ( = 20 ! log 10 (1 + T 0 ) " 20 ! log 10 T 0 when T 0 >> 1 j ! " x x p 1 (1+T 0 )p 1 s-plane B. Murmann, B. Wooley EE214 Winter 2008-09 6 At the frequency ! 0 = (1+T 0 )|p 1 | a(j ! 0 ) = A 0 and therefore T(j ! 0 ) = a(j ! 0 ) " f = A 0 f = a 0 f 1 + a 0 f # 1 Thus, ! 0 is the unity-gain bandwidth of the loop gain, T(s). To ensure the stability of the feedback loop, the phase shift in T(s) must be less than 180 o at, and below, ! 0 .
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B. Murmann, B. Wooley EE214 Winter 2008-09 7 Instability At a frequency where the phase shift around the loop of a feedback amplifier reaches ±180 o the feedback becomes positive. In that case, if the loop gain is greater than unity, the circuit is unstable. For a “single-pole” forward path amplifier stability is assured because the maximum phase shift is 90 o . However, if a(s), or in general T(s), has multiple poles, the the amount of loop gain that can be used is constrained. The stability of a feedback amplifier can be assessed from: Nyquist diagram Bode plots (plots of gain and phase as functions of frequency) Locus of poles (root locus) of A(s) in the s-plane B. Murmann, B. Wooley EE214 Winter 2008-09 8 Nyquist Diagram For a feedback amplifier with loop gain T(j ! ) = a(j ! ) " f(j ! ) plot the magnitude vs. phase of T(j ! ) on a polar plot, with ! as a parameter. The resulting plot is symmetric for positive and negative ! . There follow examples for “single-pole” and “3-pole” amplifiers.
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B. Murmann, B. Wooley EE214 Winter 2008-09 9 Nyquist Diagram for a Single-Pole Amplifier a(s) = a 0 1 ! 1 , f = CONSTANT ! = 0 ! = |p 1 | ! positive ! negative |T(j ! )| # [T(j ! )] Re[T(j ! )] Im[T(j ! )] T 0 ! = ! B. Murmann, B. Wooley EE214 Winter 2008-09 10 Nyquist Diagram for a 3-Pole Amplifier Next consider a 3-pole amplifier where a(s) = a 0 (1 ! 1 )(1 ! s p 2 ! s p 3 ) f = CONSTANT T(s) = a(s) ! f T 0 ! T(0) = a 0 f
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B. Murmann, B. Wooley EE214 Winter 2008-09 11 Nyquist Diagram for 3-Pole Amplifier, cont’d B. Murmann, B. Wooley EE214 Winter 2008-09 12 Nyquist Criterion “If the Nyquist plot encircles the point (–1,0), then the amplifier is unstable.” The Nyquist Criterion amounts to a test for whether there are poles of A(s) in the right half of the s-plane.
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This note was uploaded on 08/13/2009 for the course EE EE214 taught by Professor Borismurmann during the Winter '08 term at Stratford.

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HO16_214W09_FB_FreqResp_2pp - Handout #16 EE 214 Winter...

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