Analog Integrated Circuits (Jieh Tsorng Wu)

Signal level scaling is to maximize dynamic range

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Unformatted text preview: ing and reconstruction filters. Active-RC Filters 21-10 Analog ICs; Jieh-Tsorng Wu Single-Amplifier 2nd-Order Filters —Sallen-Key LP Biquad Let a = 1, R1 = R2 = R , C1 = C2 = C, A = ωu/s, H (s) = K · ω2 1+1 p K/A 2 s2 + sωp 3 − K 1+1 + ω2 p K/A ≈K · ωp(1 − K/A) s2 + sωp 3 − K (1 − K/A) + ω2 p 2 2 ⇒ H (s) ≈ K · ωp ωp = √ 1+ ≈ ωp 1 − ωp(1 − sK/ωu ) s2 (1 2 + ) + sωp(3 − K ) + = ωp − ∆ωp ω2 p =K · Q =Q 1+ ωp (1 − K/ωu) s2 + sωp/Q + ωp2 ≈Q 1+ 2 = Q + ∆Q 2 ωp K K2 = = ωu |A(j ωp )| • H (s) has an additional positive zero at ωu/K . • The Sallen-Key biquad is a good low-Q LP filter with small ωu-caused deviations. Active-RC Filters 21-11 Analog ICs; Jieh-Tsorng Wu State-Variable Second-Order Filters 1 1/Q Vi 1 sτ K Vh Vb 1 sτ Vl 2 2 Vh s s = +K · =K · 2 + s/(Qτ ) + 1/τ 2 Vi s s2 + sωp /Q + ω2 p Vb Vi = −K · s/τ s2 + s/(Qτ ) + 1/τ 2 = −K · sωp s2 + sωp /Q + ω2 p 1 1 ωp = √ = τ1 τ2 τ 2 ωp Vl 1/τ 2 = −K · = −K · 2 + s/(Qτ ) + 1/τ 2 Vi s s2 + sωp /Q + ω2 p Active-RC Filters 21-12 Analog ICs; Jieh-Tsorng Wu State-Variable Second-Order Filters For integrators with finite quality factors, let − 1 1 →− sτ τ (sα1 + σ1) + 1 1 →+ sτ τ (sα2 + σ2) The new ωp and Q are 2 ωp2 = ωp 1+ α1α2 Q= ωp 1 σ2 σ1 σ2 · + Q ωp ω2 p Q · ωp α2 + Q · α2 σ1+α1σ2 ω p Active-RC Filters 21-13 Analog ICs; Jieh-Tsorng Wu Tow-Thomas (TT) Biquad R RQ R/K Rx C C R Vi A1 Rx A2 Vl A3 Vb Vb Vi = −K · 2 ωps s2 + sωp/Q + ω2 p ωp Vl = −K · Vi s2 + sωp /Q + ω2 p 1 ω= RC The sensitivities for any passive component x are ωp Sx = −1/2 Active-RC Filters 21-14 Q Sx ≤ 1 Analog ICs; Jieh-Tsorng Wu Tow-Thomas (TT) Biquad Let A1 = ωu1/s, A2 = ωu2/s, and A3 = ωu3/s, then 1 1 − →− sτ τ (sα1 + σ1) α1 = 1 + 1 1 + →+ sτ τ (sα2 + σ2) ωp = ∆ωp ωp ωu1 α2 = 1 + Assuming matched opamps and ωp ωp − ωp ωp 2 ω σ1 = − ωu1 1 1+K + Q ωp ωu2 2 2 ω ω σ2 = − −2 ωu2 ωu3 ωu, we have 1 2 + K ωp 2+K · · ≈− =− ωu 2 2 |A(j ωp )| 1 Q ≈ ω Q 1 − 4Q · ωp ← Q Enhancement u Active-RC Filters 21-15 Analog ICs; Jieh-Tsorng Wu Ackerberg-Mossberg (AM) Biquad R Rx C RQ R/K Rx A3 C R Vi Vl A2 A1 Vb Let A1 = ωu1/s, A2 = ωu2/s, and A3 = ωu3/s, then − Active-RC Filters 1 1 →− sτ τ (sα1 + σ1) + 1 1 →+ sτ τ (sα2 + σ2) 21-16 Analog ICs; Jieh-Tsorng Wu Ackerberg-Mossberg (AM) Biquad where α1 = 1 + If Q ωp ωu1 2 ω σ1 = − ωu1 1 1+K + Q α2 = 1 + ωp ωu2 2 2 2ω ω − ωu3 ωu2 σ2 = + 1, we have ωp − ωp ωp = ∆ωp ωp 1 + ∆ωp/ωp Q ≈ ω Q 1 + ωp + Q · D ωp ωp 1 ≈ − (1 + K ) + ωu1 ωu2 2 u2 D= 2ωp ωu3 − ωp ωu1 − ωp ωu2 − ω2 p ωu1ωu2 + ω2 (1 + K ) p ωu1 1 2 − ωu3 ωu2 For matched opamps, we have Q ≈ Q Active-RC Filters 1− 1+ K 2 1+ ωp ωu + QK 21-17 ωp ωu ωp 2 ωu Analog ICs; Jieh-Tsorng Wu Arbitrary Transmission Zeros by Summing 1 1/Q 1 sτ K Vi Vh Vb Vi = a0 + −a1 · K sωp − a2 · K ω2 p Active-RC Filters s2 + sωp/Q + ω2 p Vl a2 a0 Vo 1 sτ = a1 Vo a0s2 + s(ωp /Q)[a0 − a1(K Q)] + ω2 [a0 − a2K ] p s2 + sωp /Q + ω2 p 21-18 Analog ICs; Jieh-Tsorng Wu Arbitrary Transmission Zeros by Voltage Feedforward R Rx C RQ R/K A3 C R Vi Vo2 A2 A1 aC Rx Vo1 R/c R/b 2 2 as + sωp (K − b) + cωp Vo1 =− Vi s2 + sωp/Q + ω2 p Active-RC Filters 21-19 Analog ICs; Jieh-Tsorng Wu High-Order Filter Using Cascade Topology Vi T1(s) Vo,1 T2(s) Vo,2 Tn(s) Vo |t (j ω)| jω Passband M σ m ωmi n Active-RC Filters 21-20 ωL ωU ωmax ω Analog ICs; Jieh-Tsorng Wu High-Order Filter Using Cascade Topology • Each stage is a biquad, i.e, 2 Ti (s) = ki · a2,i s + a1,i s + a0,i s2 + sωp,i /Qp,i + ω2 p,i = ki · ti (s) |ti (j ωp,i )| = 1 ki is defined as gain constant, such that |ti (j ωp,i )| = 1. • No interaction between stages, therefore H (s ) = Vo(s) Vi (s) n = T1(s) · T2(s) · T3(s) · · · = n Ti (s) = i =1 ki ti (s) i =1 • Easy to tune. • Sensitive to component variation in the passband for high-order filter, e.g., order > 8. Active-RC Filters 21-21 Analog ICs; Jieh-Tsorng Wu High-Order Filter Using Cascade Topology To maximize dynamic range want max|Vo,i | < Vo,max 0≤ω<∞ and min|Vo,i | → max ωL ≤ ω ≤ ωU i Vo,i (s) = Vi (s) · i Tj (s) = Vi (s) · Hi (s) Hi (s) = j =1 Tj (s) i = 1, · · · , n j =1 • Vo,max is the maximum undistorted signal level, which is limited by power supply or by the slew rate of the opamps. • Large signal even outside the passband must not overload the opamps. • Signal-to-noise ratio is of no interest in the stopband. Active-RC Filters 21-22 Analog ICs; Jieh-Tsorng Wu Cascaded Filter Design Procedures 1. Pole-Zero Pairing. Every |ti (j ω)| should be as flat as possible in the ω of interest, i.e., max log M (ti ) m(ti ) ← Minimize i = 1, · · · , n • A good suboptimal solution is assigning each zero or zero pair to the closest pole. 2. Section Ordering. Every |Vo,i (j ω)| or |Hi (j ω)| should be as flat as possible in the ω of interest, i.e., M (Hi ) ← Minimize i = 1, · · · , n max log m(Hi ) • The...
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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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