Analog Integrated Circuits (Jieh Tsorng Wu)

For an ideal integrator qi and q 0 active rc filters

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Unformatted text preview: (ω) = − dω Filters 20-27 Analog ICs; Jieh-Tsorng Wu Frequency Transformations Low-Pass to High-Pass Transformation HHP(s) = HLP 1 s • For RC active filters, it is an RC-CR transformation. Low-Pass to Band-Pass Transformation 2 HBP(s) = HLP s +1 Q· s • Q = ωo/B is the quality factor, where ωo is the center frequency, B = ωcH − ωcL is the passband bandwidth. • Transformation always results in symmetrical band-pass filters. Filters 20-28 Analog ICs; Jieh-Tsorng Wu Frequency Transformations Low-Pass to Band-Reject Transformation HBR(s) = HLP s 1 · Q s2 + 1 • Q = ωo/B is the quality factor, where ωo is the center frequency, B = ωsH − ωsL is the passband bandwidth. • Transformation always results in symmetrical band-reject filters. Frequency Scaling s H (s) = H( ) a • So that ωc = a · ωc, ωs = a · ωs , ωo = a · ωo Filters 20-29 Analog ICs; Jieh-Tsorng Wu High-Order Filters Cascade Topology In H1 H2 H3 Out H4 Follow-the-Leader Feedback (FLF) Topology F1 In H1 F2 H2 F3 H3 F4 Out H4 Leapfrog (LF) Topology F2 In H1 F4 H2 H3 F3 Filters 20-30 H5 H4 Out F5 Analog ICs; Jieh-Tsorng Wu High-Order Filters Cascade Topology: H (s) = H1 · H2 · H3 · H4 Follow-the-Leader Feedback (FLF) Topology: H (s) = H1 H2 H3 H4 1 + F1H1 + F2H1H2 + F3H1H2H3 + F4H1H2H3H4 Leapfrog Topology: H (s ) = D (s ) = H1 H2 H3 H4 H5 D (s) 1 + F2H1H2 + F3H2H3 + F4H4H4 + F5H4H5 +F2 F4H1H2H3H4 + F2F5 H1H2H4H5 + F3F5H2H3H4H5 Filters 20-31 Analog ICs; Jieh-Tsorng Wu LC Ladder Filters R V S V I S Y2 1 1 Z1 Y4 Z3 Y(n-1) Z(n-2) V 2 Z(n) R L Lossless LC Network A Fifth-Order Elliptic Low-Pass Filter R V Filters S V V 1 2 R S 20-32 L Analog ICs; Jieh-Tsorng Wu LC Ladder Filters When designed for maximum power transfer, the LC ladder filters are inherently insensitive to component variations, particularly in their passband. 2 Input Power = P1 = |I1 (j ω)|2Re{Zi n(j ω)} = 2 Maximum Input Power = P1,max H (s ) = |H (j ω)|2 = 1 − 1 |VS | = 4 RS 4RS V2 N (s) · = RL VS D (s) RS − Zi n(j ω) RS + Zi n(j ω) |VS | |RS + Zi n (j ω)|2 Re{Zi n(j ω)} |V2 |2 Output Power = P2 = RL |H (j ω)| = 2 4RS RL V2 · VS 2 = 1 − |ρ(j ω)|2 ρ(s) = ± 2 ≤1 RS − Zi n(s) (RS + Zi n(s)) • ρ(s) is the reflection coefficient. Filters 20-33 Analog ICs; Jieh-Tsorng Wu Sensitivity Let P is a function of x . The sensitivity of P with respect to x is defined as: P Sx ∂P/P ∂ (ln P ) x ∂P = = =· P ∂x ∂ (ln x ) dx/x The semirelative sensitivity is defined as P Qx = ∂P ∂x/x =x· ∂P ∂x • Some useful relationships: P P2 Sx 1 Filters P P = Sx 1 + Sx 2 P /P2 Sx 1 P P = Sx 1 − Sx 2 20-34 y P P Sx = Sy · Sx Analog ICs; Jieh-Tsorng Wu Sensitivity • Let Y is a function of x1, x2, · · · , xn. dY = ∂Y ∂Y ∂Y · dx1 + · dx2 + · · · + · dxn ∂x1 ∂x2 ∂xn dxn dY Y dx1 Y dx2 Y = Sx1 · + Sx2 · + · · · + Sxn · x1 x2 xn Y • Let the forward gain T = T1 · T2, we have T ST 2 T2 ∂T · = =1 T ∂T2 With negative feedback factor H , we have T1 T2 T= 1 + HT1T2 ⇒ T ST 2 T2 ∂T 1 · = = T ∂T2 1 + HT1T2 The T sensitivity is reduced by the loop gain HT1T2 Filters 20-35 Analog ICs; Jieh-Tsorng Wu Transfer Function Sensitivity Let the transfer function be m (s − z1)(s − zi ) · · · (s − zm) N (s) ams + · · · + a1s + a0 H (s ) = =K · = bnsn + · · · + b1s + b0 D (s) (s − p1)(s − p2) · · · (s − zn) The sensitivity is H N D Sx = Sx − Sx = ∂ ln N ∂ ln D − ∂ ln x ∂ ln x ∂ +x = {[ln(s − z1) + · · · + ln(s − zm)] − [ln(s − p1) + · · · + ln(s − pn)]} ∂x ∂z1 ∂zm ∂p1 ∂pn x x ∂x x ∂x x K ∂x + · · · + + ∂x + · · · + = Sx − s − z1 s − zm s − p1 s − pn K Sx z z p p p1Sx 1 z1Sx 1 zmSx m pn Sx n K = Sx − + + ··· + + ··· + s − z1 s − zm s − p1 s − pn Filters 20-36 Analog ICs; Jieh-Tsorng Wu Transfer Function Sensitivity • Any pole or zero shift influences H (s) most strongly in the neighborhood of that pole or zero. H • Sx → ∞ at a j ω-axis transmission zero zi = j ωzi . • For frequencies s = j ω in the neighborhood of pole with large quality factor, high sensitivities are expected. • Sensitivities are normally largest at the passband corner. Filters 20-37 Analog ICs; Jieh-Tsorng Wu Second-Order Filter Sensitivity The Biquadratic function is 2 a2 s + a1 s + a0 N (s) a2(s − z1)(s − z2) H (s) = = = D (s) (s − p1)(s − p2) s2 + (ωp /Qp)s + ω2 p 1 −j 2Qp p1 = −ωp 1− 1 2 4Qp p2 = p∗ = −ωp 1 1 +j 2Qp 1− 1 2 4Qp The sensitivity of the poles are p Sx 1 = ωp Sx −j Qp Sx p Sx 2 = 2 4Qp − 1 • The pole is in Qp. Filters p Sx 1 ∗ = ωp Sx +j Qp Sx 2 4Qp − 1 2 4Qp − 1 ≈ 2Qp times more sensitive to variations in ωp than to variations 20-38 Analog ICs; Jieh-Tsorng Wu Second-Order Filter Sensitivity j θ(ω) The transfer function can be expressed as H (j ω) = |H (j ω)|e H (j ω) Sx , then ∂ ln H (j ω) ∂ ln |H (j ω)| ∂θ (ω) |H (j ω)| θ(ω) = Sx = + j θ (ω)Sx = + jx ∂x ∂ ln x ∂ ln x Consider only the effects of poles on the passband of H (s) S H (s ) H (j ω) Sx Filters x ∂D (s) =− x=− D (s) ∂x =− ωn Qp 2 +2 1− 1 − ω2 n 2 ωn 2 sωp Qp ωp 2 + 2ωp Sx − sωp Qp S Qp x s2 + (ωp/Qp)s + ω2 p ωp Sx ωn Qp − 2 + ωn/Qp s sn = ωp 2 Qp Sx =− sn Qp ωp p 2 sn + sn/Qp + 1 2 ωn ωp Sx ωn 1 + +j Qp 1 − ω2 n ω ωn = ωp 20-39 Qp s + 2 Sx − Qn Sx 2 + 1− 2 ωn + ωn/Qp Qp Sx 2 Analog ICs; Jieh-Tsorng Wu Second-Order Filter Se...
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This note was uploaded on 03/26/2013 for the course EE 260 taught by Professor Choma during the Winter '09 term at USC.

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