Analog Integrated Circuits (Jieh Tsorng Wu)

2 input power p1 i1 j 2rezi nj 2 maximum input

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Unformatted text preview: jω M K σ ωM H (s) = ωM = K s2 s2 + (ωp/Qp)s + ω2 p ωp M= 1 − 1/(2Q2) Filters ω 20-11 KQ 1 − 1/(4Q2) Analog ICs; Jieh-Tsorng Wu Second-Order Band-Pass (BP) Filter |H (j ω)| jω K √ K/ 2 σ ωp H (s) = ω K (ωp /Qp)s s2 + (ωp/Qp)s + ω2 p ωp 3 dB Bandwidth = Qp Filters 20-12 Analog ICs; Jieh-Tsorng Wu Second-Order Band-Reject (BR) Filter — Low-Pass Notch (LPN) |H (j ω)| jω M σ K ωM ωz 2 H (s ) = Filters ω 2 K (s + ωz ) s2 + (ωp /Qp)s + 20-13 ω2 p ωz > ωp Analog ICs; Jieh-Tsorng Wu Second-Order Band-Reject (BR) Filter — High-Pass Notch (HPN) |H (j ω)| jω M K σ ωz ωM H (s ) = Filters K (s2 + ω2) z s2 + (ωp /Qp)s + 20-14 ω2 p ω ωz < ωp Analog ICs; Jieh-Tsorng Wu Second-Order Band-Reject (BR) Filter — Symmetrical Notch |H (j ω)| jω K √ K/ 2 σ ωz = ωp H (s) = K (s2 + ω2) z s2 + (ωp /Qp)s + Filters 20-15 ωz = ωp ω2 p 3 dB Notch Width = ω ωp Qp Analog ICs; Jieh-Tsorng Wu Second-Order All-Pass (AP) Filter jω σ ∠H (j ω) |H (j ω)| 0 180 ωp Filters ω 360 20-16 Analog ICs; Jieh-Tsorng Wu Second-Order All-Pass (AP) Filter 2 H (s ) = K · 2 s − (ωp/Qp)s + ωp s2 + (ωp/Qp)s + −1 φ(ωn) = −2 tan d φ(ω) Group Delay = τ = − dω ω2 p ωn/Qp 1 − ω2 n |H (j ω)| = K ω ωn = ωp 2 1 + ωn 2 τn(ωn) = ωpτ (ωn) = · Qp (1 − ω2 )2 + (ωn /Qp)2 n √ • For Qp = 1/ 3, the delay curve is maximally flat. √ • For Qp > 1/ 3, τ has a peaking, τn,max ≈ 4Qp/ωp at ωn ≈ • For 2nd-order filters, Filters 2 1 − 1/(4Qp). 1 τn,(LP,HP,BP,BR )(ωn) = τn,AP (ωn) 2 20-17 Analog ICs; Jieh-Tsorng Wu Maximally Flat (Butterworth) Filters 2 |H (j ω)| jω 1 1/(1 + 2 ) 1 1 σ 2 1/(1 + δ ) 1 ω ωs |H (j ω)|2 = 1 1 + 2ω2N 2k + N − 1 π Poles = sk = −1/N · exp j 2N Filters 20-18 k = 1, 2, · · · , N Analog ICs; Jieh-Tsorng Wu Maximally Flat (Butterworth) Filters The relationship between the filter order, N , and the steepness of the magnitude response is log δ − log N≥ log ωs • Good flatness in passband. • Poor phase linearity. • Moderate attenuation slope steepness. Filters 20-19 Analog ICs; Jieh-Tsorng Wu Equi-Ripple (Chebyshev) Filters |H1(j ω)|2 2 |H2(j ω)| Chebyshev 1 1 1/(1 + 2 Inverse Chebyshev ) N=3 N=4 2 1/(1 + δ ) 1 ωs ω Chebyshev = |H1(j ω)|2 = 1 20-20 ω 1 1+ Inverse Chebyshev = |H2(j ω)|2 = Filters ωs 1 2C 2 (ω) N 22 CN (1/ω) 2 + 2CN (1/ω) Analog ICs; Jieh-Tsorng Wu Equi-Ripple (Chebyshev) Filters The function CN is = cos[N cos−1(ω)] for ω ≤ 1 = cosh[N cosh−1(ω)] for ω > 1 = CN (ω) 2ωCN −1(ω) − CN −2(ω) The relationship between the filter order, N , and the steepness of the magnitude response is −1 cosh (δ/ ) ln(2δ/ ) N≥ ≈ cosh−1 ωs ln ωs + ω2 − 1 s • Good steepness of the attenuation slope. • Poorer phase linearity and passband flatness than the Butterworth filters. • Inverse Chebyshev filters have better phase and delay performance. Filters 20-21 Analog ICs; Jieh-Tsorng Wu Elliptic (Cauer) Filters 2 |H (j ω)| 1 1/(1 + 2 ) 2 1/(1 + δ ) 1 ω ωs |H (j ω)|2 = Filters 1 1+ 20-22 2 R 2 (ω) N Analog ICs; Jieh-Tsorng Wu Elliptic (Cauer) Filters The function RN is N/2 RN (ω) = k 2 2 ω − (ωs /ωzi ) ω2 i =1 (N −1)/2 = kω 2 2 RN (ω) for N even 2 2 ω − (ωs /ωzi ) ω2 i =1 In the stopband, if − ω2 zi − ω2 zi for N odd 1, 20 log δ ≈ 20 log |RN (ωs )| • Best steepness of the attenuation slope. • Poor phase linearity. Filters 20-23 Analog ICs; Jieh-Tsorng Wu Comparison of the Classical Filter Responses Comparing filters that satisfy the same δ and requirements: • The Cauer filter has the lowest order, while the Butterworth filter has the highest order. • The Butterworth filter has the best passband performance, and the inverse Chebyshev filter is a close second. • The Cauer filter has the largest pole quality factor; next is the Chebyshev filter, followed by the inverse Chebyshev and the Butterworth filters. • The Chebyshev filter has the worst group delay variation; next is the inverse Chebyshev filter, followed by the Butterworth and the Cauer filters. • The Butterworth and the Chebyshev are all-pole filter, while the inverse Chebyshev and Cauer filters have finite transmission zeros. • The inverse Chebyshev filters have low order, modest Q values, good delay performance, and minimal passband attenuation, making them most attractive. Filters 20-24 Analog ICs; Jieh-Tsorng Wu Linear-Phase (Bessel-Thomson) Filters 2 |H (j ω)| 1 1/(1 + 2 ) 2 1/(1 + δ ) 1 H (s) = bo D (s) N D (s ) = bi s i ω ωs bi = i =0 (2N − i )! 2N −i i !(N − i )! i = 0, 1, · · · , N − 1 D (s) is related to Bessel polynomials. D (s) = (2N − 1)DN −1 + s2DN −2 Filters 20-25 Analog ICs; Jieh-Tsorng Wu Linear-Phase (Bessel or Thomson) Filters • Approximate the linear-phase response. • Poor steepness of the attenuation slope. • It is usually more efficient to use a Butterworth, Chebyshev or a Cauer filter cascaded with an all-pass filter to achieve required gain and linear-phase response. Filters 20-26 Analog ICs; Jieh-Tsorng Wu All-Pass Filter (Delay Equalizer) Specifications |H (j ω)| (dB) jω PB 1 σ ωcL ωcH ω H (j ω) = |H (j ω)|ej φ(ω) d φ(ω) Group Delay = ...
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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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