1

Filters and TunedAmplifiers

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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) Vo(s)/Vi(s).

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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).

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1 21 2

1 2

1 2

1 1 1

1 1 1

MMM

N

N

s s sz z zz z z

T s ap p p s s s

p p p

12.2 The Filter Transfer Function

1 2

1 2

MM

N

s z s z s zT s a

s p s p s p

1 21 2

1 2

1 2

1 1 11

11 1 1

M

MMM N

N

N

s s sz z zz z z

T s ap p p s s s

p p p

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12.2 The Filter Transfer Function

1 21 2

1 2

1 2

1 1 11

11 1 1

M

MMM N

N

N

s s sz z zz z z

T s ap p p s s s

p p p

If we include zeroes at infinity, then M = N:

1 21 2

1 2

1 2

1 1 1

1 1 1

NNM

N

N

s s sz z zz z z

T s ap p p s s s

p p p

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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).

5th Order Low Pass Filter

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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).

6th Order Band Pass Filter:

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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).

All-pole filter (no finite zeroes):

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Figure 12.8 The magnitude response of a Butterworth filter.

2

2

1

1N

P

T j

12.3 Butterworth Filter:

At P

2

1

1PT j

2max

11

1

PP

T jT j

T j

2 2max

max 10 10 1020 log 20 log 1 10 log 1P

T jA

T j

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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 Butterworth Filter:

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12.3.2 The Chebyshev Filter

2 2 1

2 2 1

1for

1 cos cos

1for

1 cosh cosh

P

P

P

P

NT j

N

2

1

1PT j

At P

2max

11

1

PP

T jT j

T j

2 2max

max 10 10 1020 log 20 log 1 10 log 1P

T jA

T j

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

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12.4.1 First-order Filters

1 0

0

a s aT s

s

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Figure 12.13 First-order filters.

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Figure 12.14 First-order all-pass filter.

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Figure 12.15 Definition of the parameters 0 and Q of a pair of complex-conjugate poles.

12.4.2 Second-Order Filters

2

2 1 02 2

0 0

a s a s aT s

s Q s

201 2 0, 1 1 4

2p p j Q

Q

For filters, usually

0.5 complex-conjugate polesQ

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Figure 12.16 Second-order filtering functions.

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Figure 12.16 (Continued)

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Figure 12.16 (Continued)

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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.

12.5 Second-Order LCR Filters

2

2 1 02 2

0 0

a s a s aT s

s Q s

20

1

LC

1o

Q RC

1o

LC

1o

CQ RC RC R

LLC

Microelectronic Circuits - Fifth Edition Sedra/Smith 21Copyright 2004 by Oxford University Press, Inc.

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 0, (f) general notch, (g) LPN (n 0), (h) LPN as s , (i) HPN (n 0).

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

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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.

12.6.1 The Antoniou Inductance-Simulation Circuit

4 1 3 5 2L C R R R R

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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.

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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,

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Figure 12.22 (Continued) (d) notch at 0, (e) LPN, n 0, (f) HPN, n 0, and (g) all pass. The circuits are based on the LCR circuits in Fig. 12.18. Design equations are given in Table 12.1.

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

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

Kerwin-Huelsman-Newcomb biquad

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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.

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

loop Biquad

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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.

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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 1, C1 charges up to the current value of vi and then, during 2, discharges into C2.

1 1C iq C v 1 2C Cq q

1

1

i i cav eq

c av

C v v Ti R

T i C

22 2

1 1

integrator time constant = ceq c

T CR C C T

C C

max

1c c sigf T f

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12.10 Switched Capacitor Filters

2

1

integrator time constant c

CT

C

Note that the integrator time constant depends on:• the ratio of capacitances, not their absolute value• the clock period

MOS example: 1 20.1 ; 1 ; 100 ;cC pF C pF T kHz

425

1

1 1integrator time constant 10 sec

10 0.1c

C pFT

C Hz pF

integrator time constant 0.1ms

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12.10 Switched Capacitor FiltersRecall the integrator from the ECE 1002 Final Project:

integrator time constant 10 10 0.1nF k ms

Thus switched capacitor filters can work in the audio frequencyrange with pF capacitors.

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

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