Analog Integrated Circuits (Jieh Tsorng Wu)

Transmission zeros by voltage feedforward r rx c rq

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Unformatted text preview: nsitivity We have |H (j ω)| Sx θ(ω) θ (ω)Sx =− 2 1− 2 2 ωn 2 1 − ω2 n 2 + ωn/Qp + ωn/Qp 2 ωn 1+ ∂θ (ω) = =x ∂x 1 − ω2 n 2 ωn/Qp ωp · Sx + 2 1 − ω2 n 2 ωn/Qp + ωn/Qp 2 · 2 + |H | SQ p Qp + ωn/Qp 1− ωp Sx 2 2 ωn 2 · Sx 2 ωn /Qp Qp Sx 1 − ω2 n 2 + ωn/Qp Qp 2 · Sx And |H (j ω)| Sx ⇒ Filters |H | Sωp = − 2 1− 2 2 ωn 1 − ω2 n 2 = |H | Sωp · ωp Sx · 2 + ωn/Qp + ωn/Qp + |H | 2 20-40 SQ = p ωn/Qp 1 − ω2 n 2 2 + ωn/Qp 2 Analog ICs; Jieh-Tsorng Wu Second-Order Filter Sensitivity |H | |H | Sωp |H | max{Sωp } |H | min{Sωp } Filters ≈ Qp 1 + 1/Qp ≈− Qp 1 − 1/Qp SQ at ωn ≈ 1 + 1 2Qp at ωn ≈ 1 − p 1 2Qp 20-41 |H | max{SQ } = 1 p at ωn = 1 Analog ICs; Jieh-Tsorng Wu Second-Order Filter Sensitivity • Small variations of ωp are far more important than small change in Qp. • Since the errors increase with Q, low-Q filters are easier to design with less accurate components than high-Q filters. • Sensitivities are strong functions of frequency, and the passband edges are very critical. Filters 20-42 Analog ICs; Jieh-Tsorng Wu High-Order Filter Sensitivity A 6th-order Butterworth bandpass filter • For cascade design, H (s) = H1(s)H2 (s) · · · Hn(s) H (s) SH (s) = 1 and j H (s) Sx Hj (s) = Sx The sensitivity of H (s) to x is as large as sensitivity of sub-block Hj (s) to x . • Feedback paths around low-order sections in a multiple-feedback (MF) filter topology can reduce sensitivities in the passband. In the stopbands, where feedback paths lose their effectiveness, MF and cascade sensitivities are approximately the same. Filters 20-43 Analog ICs; Jieh-Tsorng Wu Active-RC Filters Jieh-Tsorng Wu ES A October 17, 2002 1896 National Chiao-Tung University Department of Electronics Engineering Capacitor Integrators Vo I R C Vo 1 1 1 = = = I j ωC + G j ωC 1 − j G j ωC 1 − j Q 1ω) ωC ( I QI (ω) = ωC G The transfer function of an integrator can be expressed as 1 1 1 1 H (j ω) = = = = F (j ω) j Im[F (j ω)] + Re[F (j ω)] j ωτ + q j ωτ 1 − j 1 Q (ω) I Im[F (j ω)] ωτ QI (ω) = = q Re[F (j ω)] • QI is the quality factor of the integrator. • For an ideal integrator, QI → ∞ and q → 0. Active-RC Filters 21-2 Analog ICs; Jieh-Tsorng Wu Active-RC Inverting Integrators R Vi C A(s) Vo Vo Vi (s) = − 1 1 · sRC 1 + 1 1 + 1 sRC A(s) Let A(s) = ωu/s, then Vo 1 1 1 1 · · (s) = − ≈− Vi sRC 1 + s/ωu + 1/(ωu RC) sRC 1 + s/ωu if ωu 1 RC Vo 1 1 (j ω) = − =− Vi j ωτ + q j ωRC − ω2RC/ωu τ = RC Active-RC Filters ωRC ω2RC =− q=− ωu |A(j ω)| 21-3 ωu ωτ QI = =− = −|A(j ω)| q ω Analog ICs; Jieh-Tsorng Wu Actively Compensated Inverting Integrator C Vo (s) = − Vi A2 R Vi Vo A1 1 sRC 1+1/A2(s) + + 1A sRC (s) 1 1 ≈− sRC 1 − A 1s) + ( 2 1 A2 (s) 2 − + ··· 1 A3 (s) 2 + + 1A sRC (s) 1 Let A1(s) = ωu1/s, and A2(s) = ωu2/s, Vo Vi 1 (j ω) ≈ − j j ωRC 1 − ωω − u2 ω2 ω22 u ωu2 2 RC −j ω −ω ωu1 1 ≈− j ωRC 1 + ω 1 u1 RC Active-RC Filters 3 + jω + · · · 3 − ω2 ω22 u 21-4 2 + ωωRC 1 − u2 ω2 ω22 u ω − ωu2 u1 Analog ICs; Jieh-Tsorng Wu Actively Compensated Inverting Integrator Thus 2 τ ≈ RC ω RC q= ωu2 ⇒ 2 1− ω ω22 u QI = ωu2 − ωu1 ωu2 ωRC 1− ≈ ωu1 |A2 (j ω)| ωτ |A2 (j ω)| = ω q 1 − ωu2 u1 If A1(s) = A2(s) = A(s) = ωu/s, then τ ≈ RC ⇒ Active-RC Filters q=− ω4 RC ω3 u ω QI = − ωu 21-5 ωRC =− |A(j ω)|3 3 = −|A(j ω)|3 Analog ICs; Jieh-Tsorng Wu Noninverting Integrator R1 C R Vi C R R1 A1 A2 Vo Let A1 = A2 = A R1 Vi A1 A2 Vo Let A1 = A2 = A Vo 1 1 · = Vi sRC 1 + 3 + 1 + 22 + 2 2 A sRCA A sRCA Vo Vi 1 QI = − |A(j ω)| 3 Active-RC Filters R1 = 1 1 · sRC 1 + 1 + 1 A sRCA QI = −|A(j ω)| 21-6 Analog ICs; Jieh-Tsorng Wu Phase-Lead Noninverting Integrator R1 C A2 R1 R Vi Vo (s) = Vi sRC Vo A1 1 1 1+2/A2(s) + A 1s) + sR ( 1 1 1 CA1 (s) If A1(s) = A2(s) = A(s) = ωu/s, then QI ≈ + Active-RC Filters ωu = +|A(j ω)| ω 21-7 Analog ICs; Jieh-Tsorng Wu First-Order Filters State-Variable Topology 1 Vi α1s + α0 Fully-Differential Active-RC Filter R2 C1 1 sτ Vo R1 Active-RC Filter C1 Vi R2 Vi Vi Vo C R1 Vi R1 C C C1 Vo R2 α1s + α0 Vo ±sC1 + G1 R2 ±sR1 C1 + 1 =− =− =− · sτ + 1 Vi sC + G2 R1 sR2 C + 1 Active-RC Filters 21-8 Analog ICs; Jieh-Tsorng Wu Single-Amplifier 2nd-Order Filters —Sallen-Key LP Biquad C1 G1 C1 a G1 G2 Vi Vo A G2 Vi A C2 C2 RA H (s ) = RB (1-a) G 1 RA RB K G1G2 1+1 K/A Vo = Vi s2 C1C2 + s C2(G1 + G2) + C1C2 1 − K 1+1 K/A K =a· 1+ Active-RC Filters Vo RB RA 21-9 + G1 G2 a≤1 Analog ICs; Jieh-Tsorng Wu Single-Amplifier 2nd-Order Filters —Sallen-Key LP Biquad Let A = ∞ and C1 = C2 = C, then H (s ) = K G1G2/C s2 ω2 p 2 2 + s[G1 + G2(2 − k )]/C + G1G2 = G1 G2 C2 Q= /C2 G1 G2 G1 + G2(2 − K ) =K · ωp s2 + sωp /Q + ω2 p K =a· 1+ RB RA If a = 1, R1 = R2 = R , we have 1 ωp = RC 1 Q= 3−K Q SK = 3Q − 1 • Minimal use of opamp, at the expense of more passive components. • Sensitive to parasitic capacitors. • Widely used to realize the on-chip anti-alias...
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