Analog Integrated Circuits (Jieh Tsorng Wu)

Opamp will be divided by k all other opamp output

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: mpensated Switched-Capacitor Integrators,” ISCAS, 1990, pp. 2829–2832. SC Filters 23-22 Analog ICs; Jieh-Tsorng Wu Discrete-Time Signal Processing xc (t) Analog Prefilter x(n) yd (t) y(n) Discrete Time Processing Sampling y (t) c Analog Postfilter DAC Ts Xc(j Ω) xc (t) A Ω t 0 Ts 2Ts 3Ts 0 Xe x(n) Ωb Ωs 2Ωs jω A Ts n 0 SC Filters 1 2 3 0 23-23 2π 4π ω Analog ICs; Jieh-Tsorng Wu Continuous-Time Signals The Laplace transform and the continuous-time Fourier transform (CTFT) are Xc(s) = ∞ −∞ −st xc(t )e Xc(j Ω) = dt ∞ −∞ xc(t )e−j Ωt d t If the region of convergence of Xc(s) includes the imaginary axis, then Xc(j Ω) = Xc(s)|s=j Ω Sampling Theorem: To avoid aliasing, want Ωs > 2Ωb Ωs = 2πfs = 2π Ts • Ωb is the bandwidth of xc(t ), Ωs is the sampling frequency, and 2Ωb is called the Nyquist rate. SC Filters 23-24 Analog ICs; Jieh-Tsorng Wu Discrete-Time Signals In discrete-time domain, the z transform is ∞ X (z ) = x (n)z−n n=−∞ The discrete-time Fourier transform (DTFT) is ∞ X ej ω = x (n)e−j ωn n=−∞ If the region of convergence of X (z ) includes the unit circle, then X ej ω = X (z )|z=ej ω SC Filters 23-25 Analog ICs; Jieh-Tsorng Wu s-to-z Transformation Want to approximate Hc(s) with H (z ). z=e sTs 1 s= · ln z Ts ⇒ H (z ) = Hc(s)|s=(1/Ts ) ln z ≈ Hc(s)|s=T (z) Transformation error of an Integrator can be written as Hc(s) = 1 1 = s jΩ ⇒ H (z )|ej ΩTs = 1 · [1 − (Ω)] · ej φ(Ω) jΩ BE 1/2 BL s to z Vi Vo Vi z -1 FE LD SC Filters 23-26 Analog ICs; Jieh-Tsorng Wu s-to-z Transformation Backward Euler (BE) Transformation 1 s = · 1 − z −1 Ts ⇒ 1 1 = Ts · s 1 − z −1 =1− ΩTs /2 sin(ΩTs /2) ΩTs φ=+ 2 Forward Euler (FE) Transformation −1 1 1−z s= · Ts z −1 ⇒ −1 1 z = Ts · s 1 − z −1 =1− ΩTs /2 sin(ΩTs /2) φ=− ΩTs 2 Lossless Discrete (LD) Transformation −1 1 1−z s= · Ts z −1/2 SC Filters ⇒ −1/2 1 z = Ts · s 1 − z −1 23-27 =1− ΩTs /2 sin(ΩTs /2) φ=0 Analog ICs; Jieh-Tsorng Wu Bilinear s-to-z Transformation The transformation is −1 2 1−z s= · Ts 1 + z −1 ⇒ −1 1 Ts 1 + z =· s 2 1 − z −1 =1− ΩTs /2 tan(ΩTs /2) φ=0 jω let z = e , then jω ω 2 e −1 2 · s= = · j tan = jΩ 2 T s ej ω + 1 T s ω 2 Ω = tan 2 Ts • The unit circle in the z-plane is mapped to the j Ω axis in the s-plane. SC Filters 23-28 Analog ICs; Jieh-Tsorng Wu Hc(s) to H(z) Design Procedures for Bilinear Transformation j ΩTs |H (z = e )| 0 Ωp |H (s = j Ω )| 0 Ωc Ω Ω Ωp Ωc SC Filters Ω z Ω s /2 Ωs Ωz 23-29 Analog ICs; Jieh-Tsorng Wu Hc(s) to H(z) Design Procedures for Bilinear Transformation • Prewarp the filter specifications from Ω to Ω . Ωp T s 2 Ωp = tan 2 Ts Ωc T s 2 Ωc = tan 2 Ts Ωz T s 2 Ωz = tan 2 Ts • Find Hc(s ). • The H (z ) is obtained by H (z ) = Hc SC Filters −1 2 1−z s= · Ts 1 + z −1 23-30 Analog ICs; Jieh-Tsorng Wu Switched-Capacitor Filter Systems Xo(t) Xi(t) Anti-Aliasing Filter Sampled Data Filter (Limits BW) Reconstruction Filter (Smooths output) • Discrete-time (or sampled-data) analog filters. • Filters consist of analog switches, capacitors and opamps. • Filter response is determined by ratios of capacitance. SC Filters 23-31 Analog ICs; Jieh-Tsorng Wu Design Constraints • Switched-C “resistor” cannot be the only feedback around an opamp. Since the path is not continuous, it won’t stabilize the opamp. • No floating node. Otherwise charge can accumulate. • Capacitor bottom plate must always be driven from a low impedance (voltage sources or ground). • Connect non-inverting opamp input to a dc bias. Otherwise response is sensitive to parasitic capacitances. SC Filters 23-32 Analog ICs; Jieh-Tsorng Wu Periodic Time-Variance in Biphase SC Filters n C 1 Vi1 C1 1 2 1 Vi2 C2 1 2 φ2 0 Vo 1 1 Vo 2 1 n+1 φ1 2 1 n+1/2 2 Vo 1 Vi1 1 Vi2 2 0 Vo The circuit is periodic time-variant if Vo1[n · Ts ] = Vo2 SC Filters n+ 23-33 1 · Ts 2 Analog ICs; Jieh-Tsorng Wu Periodic Time-Invariance in Biphase SC Filters n C 1 Vi1 C1 2 1 1 Vi2 C2 1 2 φ2 0 Vo 1 1 Vo 2 1 n+1 φ1 1 2 n+1/2 2 Vo 1 Vi1 1 Vi2 2 0 Vo The circuit is periodic time-invariant if Vo1[n · Ts ] = Vo2 1 · Ts n+ 2 • SC filters are more robust when designed to be time-invariant. SC Filters 23-34 Analog ICs; Jieh-Tsorng Wu Active Switched-Capacitor Integrators C 1 Vi1 C1 1 2 1 Vi2 1 1 2 C2 2 1 1 Vi1 C1 1 Vo 2 1 Vi2 2 Vo 1 1 Vi3 C2 z 1 C 1 C3 ( 1- z 1 1- z 1 1 Vo 1 ) 2 φ1 1 Vi3 1 C3 φ2 1 n Vo1 = 1 C 1 − z −1 n+1/2 n+1 · −C1Vi 1 + C2z −1Vi 1 − C3 1 − z −1 Vi 1 1 2 3 Vo2 = Vo1 · z −1/2 SC Filters 23-35 Analog ICs; Jieh-Tsorng Wu Active Switched-Capacitor Integrators • Vi 1 to Vo1 is a Backward Euler (−BE) integrator. 1 1 2 1 1 2 1 • Vi 1 to Vo is a Lossless Discrete (−LD) integrator. • Vi 2 to Vo is a Forward Euler (+FE) integrator. • Vi 2 to Vo is a Lossless Discrete (+LD) integrator. SC Filters 23-36 Analog ICs; Jieh-Tsorng Wu SC First-Order Filters 1 Vi 1 Vi α1s + α0 1 sτ Vo CA1 1 Vi CA2 ( 1- z 1 1 Vi 1 CA1 2 1 CB1 2 CB1 1 ) 1 V...
View Full Document

This note was uploaded on 03/26/2013 for the course EE 260 taught by Professor Choma during the Winter '09 term at USC.

Ask a homework question - tutors are online