Filters and Tuned Amplifiers 1

sedr42021_1201.jpg Figure 12.1 The filters studied in this chapter are linear circuits represented by the general two-port network shown. The filter transfer function T(s) º Vo(s)/Vi(s).

sedr42021_1202a.jpg Figure 12.2 Ideal transmission characteristics of the four major filter types: (a) low-pass (LP), (b) high-pass (HP), (c) bandpass (BP), and (d) bandstop (BS).

12.2 The Filter Transfer Function

12.2 The Filter Transfer Function If we include zeroes at infinity, then M = N:

5th Order Low Pass Filter sedr42021_1203.jpg Figure 12.3 Specification of the transmission characteristics of a low-pass filter. The magnitude response of a filter that just meets specifications is also shown. Figure 12.5 Pole–zero pattern for the low-pass filter whose transmission is sketched in Fig. 12.3. This is a fifth-order filter (N = 5).

6th Order Band Pass Filter: sedr42021_1204.jpg Figure 12.4 Transmission specifications for a bandpass filter. The magnitude response of a filter that just meets specifications is also shown. Note that this particular filter has a monotonically decreasing transmission in the passband on both sides of the peak frequency. Figure 12.6 Pole–zero pattern for the band-pass filter whose transmission function is shown in Fig. 12.4. This is a sixth-order filter (N = 6).

All-pole filter (no finite zeroes): sedr42021_1207a.jpg Figure 12.7 (a) Transmission characteristics of a fifth-order low-pass filter having all transmission zeros at infinity. (b) Pole–zero pattern for the filter in (a).

12.3 Butterworth Filter: At sedr42021_1208.jpg Figure 12.8 The magnitude response of a Butterworth filter.

12.3 Butterworth Filter: sedr42021_1209.jpg Figure 12.9 Magnitude response for Butterworth filters of various order with e = 1. Note that as the order increases, the response approaches the ideal brick-wall type of transmission.

12.3.2 The Chebyshev Filter At

sedr42021_1212a.jpg Figure 12.12 Sketches of the transmission characteristics of representative (a) even-order and (b) odd-order Chebyshev filters.

12.4.1 First-order Filters

sedr42021_1212b.jpg Figure 12.13 First-order filters.

sedr42021_1214a.jpg Figure 12.14 First-order all-pass filter.

12.4.2 Second-Order Filters For filters, usually sedr42021_1215.jpg Figure 12.15 Definition of the parameters w0 and Q of a pair of complex-conjugate poles.

sedr42021_1216a.jpg Figure 12.16 Second-order filtering functions.

sedr42021_1216b.jpg Figure 12.16 (Continued)

sedr42021_1216c.jpg Figure 12.16 (Continued)

12.5 Second-Order LCR Filters sedr42021_1217a.jpg Figure 12.17 (a) The second-order parallel LCR resonator. (b, c) Two ways of exciting the resonator of (a) without changing its natural structure: resonator poles are those poles of Vo/I and Vo/Vi.

sedr42021_1218a.jpg Figure 12.18 Realization of various second-order filter functions using the LCR resonator of Fig. 12.17(b): (a) general structure, (b) LP, (c) HP, (d) BP, (e) notch at w0, (f) general notch, (g) LPN (wn ³ w0), (h) LPN as s ® ¥, (i) HPN (wn < w0).

sedr42021_1219.jpg Figure 12.19 Realization of the second-order all-pass transfer function using a voltage divider and an LCR resonator.

12.6.1 The Antoniou Inductance-Simulation Circuit sedr42021_1220a.jpg Figure 12.20 (a) The Antoniou inductance-simulation circuit. (b) Analysis of the circuit assuming ideal op amps. The order of the analysis steps is indicated by the circled numbers.

sedr42021_1221a.jpg Figure 12.21 (a) An LCR resonator. (b) An op amp–RC resonator obtained by replacing the inductor L in the LCR resonator of (a) with a simulated inductance realized by the Antoniou circuit of Fig. 12.20(a). (c) Implementation of the buffer amplifier K.

sedr42021_1222a.jpg Figure 12.22 Realizations for the various second-order filter functions using the op amp–RC resonator of Fig. 12.21(b): (a) LP, (b) HP, (c) BP,

sedr42021_1222d.jpg Figure 12.22 (Continued) (d) notch at w0, (e) LPN, wn ³ w0, (f) HPN, wn £ w0, and (g) all pass. The circuits are based on the LCR circuits in Fig. 12.18. Design equations are given in Table 12.1.

Second-Order Active Filters Based on the Two-integrator-loop Biquad

Second-Order Active Filters Based on the Two-integrator-loop Biquad Kerwin-Huelsman-Newcomb biquad

Second-Order Active Filters Based on the Two-integrator-loop Biquad sedr42021_1225a.jpg Figure 12.25 (a) Derivation of an alternative two-integrator-loop biquad in which all op amps are used in a single-ended fashion. (b) The resulting circuit, known as the Tow–Thomas biquad.

12.10 Switched Capacitor Filters 12.10.1 The Basic Principle: A capacitor switched between two circuit nodes at a sufficiently high rate is equivalent to a resistor connecting these two nodes.

sedr42021_1235a.jpg Figure 12.35 Basic principle of the switched-capacitor filter technique. (a) Active-RC integrator. (b) Switched-capacitor integrator. (c) Two-phase clock (nonoverlapping). (d) During f1, C1 charges up to the current value of vi and then, during f2, discharges into C2.

12.10 Switched Capacitor Filters Note that the integrator time constant depends on: the ratio of capacitances, not their absolute value the clock period MOS example:

12.10 Switched Capacitor Filters Recall the integrator from the ECE 1002 Final Project: Thus switched capacitor filters can work in the audio frequency range with pF capacitors.

sedr42021_1236a.jpg Figure 12.36 A pair of complementary stray-insensitive switched-capacitor integrators. (a) Noninverting switched-capacitor integrator. (b) Inverting switched-capacitor integrator.

sedr42021_1237a.jpg Figure 12.37 (a) A two-integrator-loop active-RC biquad and (b) its switched-capacitor counterpart.