Analog Integrated Circuits (Jieh Tsorng Wu)

Ct has transient behavior its pulse width can be

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Unformatted text preview: i Vo(s) Vi (s) SC Filters =− 1 α1s + α0 sτ + 1 1 2 Vo 1 −1 CA1 ± CA2(1 − z ) CB1 + C(1 − 23-37 1 1 Vo 2 =− 1-z 1 Vo 2 2 1 Vo Vi 1 1 1 C CA2 1 C z −1 ) =− CA1 C ± CB 1 C CA2 C ∓ CA2 − 1 z C + 1 − z −1 Analog ICs; Jieh-Tsorng Wu Switch Sharing 1 1 Vi CA1 1 2 1 CB1 2 1 2 2 1 1 Vi CA2 1 1 CA1 1 Vi SC Filters 1 1 Vo 2 2 Vo 1 2 2 1 2 Vo 1 CB1 1 Vi 1 Vo 2 C C 1 CA2 2 23-38 Analog ICs; Jieh-Tsorng Wu Bilinear SC First-Order Filters CB1 1 2 1 1 Vi CA1 1 2 1 1 Vo 2 C 2 Vo 1 Vo 2 Vi 1 Vi =− =− 1 Vi CA1 1 Vi SC Filters 1 CA1 z CB1 1 1 C 1 1-z 23-39 1 CA1 + CA1z −1 CB1 + C(1 − z−1) CA1 − 1 z C CA1 C + CB 1 C + 1 − z −1 1 Vo Analog ICs; Jieh-Tsorng Wu SC Second-Order Filters 1 1/Q K Vi 1 sτ Vh 1 sτ Vb Vl 1 CB1 1 Vi CA1 1 1 C1 CA2 1 Vo Vi 1 SC Filters = 1 C2 CK2 2 2 1 1 CA1 CK 2 C1 C2 CB 1 C1 2 2 2 2 CB2 +1 + z −1 − CB 2 C1 · CK 2 C2 23-40 CA2 CK 2 C1 C2 − CB 1 C1 1 1 Vo 2 z −2 − 2 z −1 + z −2 Analog ICs; Jieh-Tsorng Wu SC Second-Order Filters CB2 CB1 1 Vi CA1 1 2 C1 CA2 1 2 1 2 2 1 1 C2 2 2 CK2 1 1 1 Vo 2 CB2 1 Vi CA1 CA2 z SC Filters CB1 1 1 C1 1 1- z 1 23-41 CK2 z 1 1 C2 1 1- z 1 1 Vo Analog ICs; Jieh-Tsorng Wu A Low-Q SC Biquad 1 K4 2 1 2 1 K6 2 1 K1 2 1 K2 2 1 2 C2 = 1 C1 = 1 1 Vi 1 2 2 K5 1 1 1 Vo 2 1 2 K3 H (z ) = SC Filters Vo(z ) Vi (z ) 2 =− (K2 + K3)z + (K1K5 − K2 − 2K3)z + K3 (1 + K6 )z2 + (K4K5 − K6 − 2)z + 1 23-42 2 =− 1 a2 z + a1 z + a0 b2z 2 + b1z + 1 Analog ICs; Jieh-Tsorng Wu A Low-Q SC Biquad We have K3 = a0 K2 = a2 − a0 K1K5 = a0 + a1 + a2 K6 = b2 − 1 K4K5 = b2 + b1 + 1 • Additional constraint can be made by K5 = 1 j ΩTs Let z = e z 1/2 K4 = K5 = or b2 + b1 + 1 = cos(ΩTs ) + j sin(ΩTs ), and Ω Ts Ω Ts + j sin = cos 2 2 z −1/2 Ω Ts Ω Ts − j sin = cos 2 2 Then 2 H ej ΩTs = − SC Filters K1K5 + j K2 sin(ΩTs ) + (4K3 + 2K2) sin (ΩTs /2) K4K5 + j K6 sin(ΩTs ) + (4 + 2K6) sin2(ΩTs /2) 23-43 Analog ICs; Jieh-Tsorng Wu A Low-Q SC Biquad Assume ΩTs 1, we have 2 H ej ΩTs ≈ − K1K5 + j K2(ΩTs ) + (K3 + K2/2)(ΩTs ) K4K5 + j K6(ΩTs ) + (1 + K6/2)(ΩTs Let K4 = K5, then K4 = K5 ≈ ωpTs • Usually, ωpTs K6 ≈ )2 2 =− α2s + α1s + α0 ω s2 + Qp · s + ω2 p p ωpTs Qp 1. • The largest capacitors are the integrating capacitors, C1 and C2. • If Qp < 1, the smallest capacitors are K4 and K5. • If Qp > 1, the smallest capacitors is K6. SC Filters 23-44 Analog ICs; Jieh-Tsorng Wu A High-Q SC Biquad K6 1 Vi 1 K1 2 1 1 2 K4 2 1 2 C1 = 1 K2 C2 = 1 2 K5 1 1 1 Vo 2 K3 H (z ) = SC Filters Vo(z ) Vi (z ) =− K3z 2 + (K1 K5 + K2 K5 − 2K3)z + (K3 − K2K5) z2 + (K4K5 + K5K6 − 2)z + (1 − K5K6) 23-45 =− a2 z 2 + a1 z 1 + a0 z 2 + b1z + b0 Analog ICs; Jieh-Tsorng Wu A High-Q SC Biquad We have K1K5 = a0 + a1 + a2 K2 K5 = a2 − a0 K3 = a2 K4K5 = 1 + b1 + b0 K5K6 = 1 − b0 • Additional constraint can be made by K4 = K5 = 1 + b1 + b0 • Less capacitance spread. In general, • For the SC biquad, it is important that the two-integrator loop have a single delay around the loop. A delay-free loop may have an excessive settling time behavior, while two delays around the loop cause difficulties in designing high-Q circuit. SC Filters 23-46 Analog ICs; Jieh-Tsorng Wu Time-Staggered SC Stages Cascaded SC Stages 1 2 1 1 2 2 2 1 1 2 1 1 2 1 2 1 2 1 1 2 2 1 Staggered Cascaded Stages 1 2 1 1 2 1 SC Filters 2 2 1 1 2 1 2 2 2 1 1 1 2 2 1 2 23-47 Analog ICs; Jieh-Tsorng Wu Capacitor Scaling V1 2 1 C1 V2 2 1 CA 2 Q3 C3 Qi Vo 1 1 2 Q4 C4 C2 For each switching cycle Qi = C1V1 + C2V2 Qi ∆Vo = − CA Q3 = C3Vo Q4 = C4Vo SC Filters 23-48 Analog ICs; Jieh-Tsorng Wu Output Capacitor Scaling If CA = kCA, C3 = kC3 , C4 = kC4, C1 and C2 unchanged, then Qi = C1V1 + C2V2 = Qi ∆Vo = − Qi CA Q3 = C3Vo = kC3 =− Vo k ∆Vo Qi = kCA k = Q3 Q4 = Q4 • If the values of all capacitors (including feedback capacitors) connected or switched to the output terminal of an opamp in an SCF are multiplied by the same constant k , then the output voltage of this opamp will be divided by k ; all other opamp output voltages remain unchanged. This follows since the described changes leave all charges flowing to and from the affected opamp unchanged. • The output capacitor scaling technique can be used to achieve optimum scaling for maximum dynamic range. SC Filters 23-49 Analog ICs; Jieh-Tsorng Wu Input Capacitor Scaling If CA = kCA, C1 = kC1 , C2 = kC2, C3 and C4 unchanged, then Qi = C1V1 + C2V2 = kC1V1 + kC2 V2 = kQi ∆Vo = − Qi CA =− kQi = ∆Vo kCA Q3 = C3Vo = C3 Vo = Q3 Q4 = Q4 • If the values of all capacitors (including feedback capacitors) connected or switched to the inverting input terminal of an opamp are multiplied by the same constant, then all voltages in the SCF remain unchanged. This is true since all voltages are affected only by the ratios of these capacitances. • The input capacitor scaling technique can be used to achieve optimum scaling for minimum capacitance. SC Filters 23-50 Analog ICs; Jieh-Tsorng Wu An All-Pole Low-Pass Ladder Filter I V R in V 0 V I V -1/(sC 3) V -1/(sC 5) R 6 4 1/(sL4) out in L 6 1/R L V in V 1 V 1/R S R V 3 V 2 out...
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