Filters and Tuned Amplifiers

Chapter 11

Filters and Tuned Amplifiers

Passive LC Filters

Inductorless Filters

Active-RC Filters

Switched Capacitors

Filter Transmission, Types and Specification

Linear Filters

Transfer Function

T s( )Vo s( )

Vi s( )

The Filter Transmisson found by evaluating T(s) for physical frequencies

s j T j T j e j ( )

Gain Function

G 20 log T j dB

Attenuation Function

A 20 log T j dB

Specification of the transmission characteristics of a low-pass filter. The magnitude response of a filter that just meets specifications is also shown.

Filter Specification

Frequency-Selection functionPassingStoppingPass-BandLow-PassHigh-PassBand-PassBand-StopBand-Reject

Summary – Low-pass specs-the passband edge-the maximum allowed variation in passband, Amax-the stopband edge-the minimum required stopband attenuation, Amin

Passband rippleRipple bandwidth

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.

Filter Specification

Exercises 11.1 and 11.2

Pole-zero pattern for the low-pass filter whose transmission is shown.

This filter is of the fifth order (N = 5.)

The Filter Transfer Function

transfer function zeros or transmission zeros

T s( )aM s z1 s z2 s z3 s zM

s p1 s p2 s p3 s pN

transfer function poles or the natural poles

Pole-zero pattern for the bandpass filter whose transmission is shown. This filter is of the sixth order (N = 6.)

The Filter Transfer Function

The magnitude response of a Butterworth filter.

Butterworth Filters

Magnitude response for Butterworth filters of various order with = 1. Note that as the order increases, the response approaches the ideal brickwall type transmission.

Butterworth Filters

Graphical construction for determining the poles of a Butterworth filter of order N. All the poles lie in the left half of the s-plane on a circle of radius 0 = p(1/)1/N, where is the passband deviation parameter :

(a) the general case, (b) N = 2, (c) N = 3, (d) N = 4. 10 110Amax /e j

Butterworth Filters

Sketches of the transmission characteristics of a representative even- and odd-order Chebyshev filters.

Chebyshev Filters

First-Order Filter Functions


Fig. 11.14 First-order all-pass filter.


Second-Order Filter Functions



Realization of various second-order filter functions using the LCR resonator of Fig. 11.17(b): (a) general structure, (b) LP, (c) HP, (d) BP, (e) notch at 0, (f) general notch, (g) LPN (n 0), (h) LPN as s , (i) HPN (n < 0).

The Second-order LCR Resonator

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.

The Second-Order Active Filter – Inductor Replacement

Realizations for the various second-order filter functions using the op amp-RC resonator of Fig. 11.21 (b). (a) LP; (b) HP; (c) BP, (d) notch at 0;


(e) LPN, n 0; (f) HPN, n 0; (g) all-pass. The circuits are based on the LCR circuits in Fig. 11.18. Design equations are given in Table 11.1.


The Second-Order Active Filter – Two-Integrator-Loop

VhpVi

K s2

s2 soQ

o2

Two integrations of signal

with time constant

os

VhpVhp

1Q

os

Vhp

o

2

s2Vhp

K Vi1

o

Vhp K Vi1Q

os

Vhpo

2

s2Vhp Summing Point


Circuit Implementation


Circuit Design and Performance

T

a 1 2 40 b 1 2 20 j 1

w0 2 103 K 3

wa 100 700a Qb 0.1 0.2 b

Ta b

K j wa 2

j wa 2 j wa w0Qb

w02


Exercise 11.21

Derivation of an alternative two-integrator-loop biquad in which all op amps are used in a single-ended fashion. The resulting circuit in (b) is known as the Tow-Thomas biquad.


Fig. 11.26 The Tow-Thomas biquad with feedforward. The transfer function of Eq. (11.68) is realized by feeding the input signal through appropriate components to the inputs of the three op amps. This circuit can realize all special second-order functions. The design equations are given in Table 11.2.

Fig. 11.37 A two-integrator-loop active-RC biquad and its switched-capacitor counterpart.

Fig. 11.47 Obtaining a second-order narrow-band bandpass filter by transforming a first-order low-pass filter. (a) Pole of the first-order filter in the p-plane. (b) Applying the transformation s = p + j0 and adding a complex conjugate pole results in the poles of the second-order bandpass filter. (c) Magnitude response of the firs-order low-pass filter. (d) Magnitude response of the second-order bandpass filter.

Fig. 11.48 Obtaining the poles and the frequency response of a fourth-order stagger-tuned narrow-band bandpass amplifier by transforming a second-order low-pass maximally flat response.

Engineering

Filters and Tuned Amplifiers