Filters and Tuned
Amplifiers
1
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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12.2 The Filter Transfer Function
s  z1  s  z2    s  z M 

T  s   aM
 s  p1  s  p2    s  pN 
T  s   aM
T  s   aM
s
 s
  s

 1
  1   1  
z1 z2  zM  z1
  z2
  zM

p1 p2  pN  s

 s
  s
 1 
 1  
 1

 p1
  p2
  pN

 1
M
z1 z2  zM
p1 p2  pN  1 N
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
1 


1 

s 
s  
s 
1     1 

z1 
z2  
zM 
s 
s  
s 

1 
  1 
p1 
p2  
pN 
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12.2 The Filter Transfer Function
T  s   aM
 1
M
z1 z2  zM
p1 p2  pN  1 N

1 


1 

s 
s  
s 
1     1 

z1 
z2  
zM 
s 
s  
s 

1 
  1 
p1 
p2  
pN 
If we include zeroes at infinity, then M = N:
T  s   aM
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z1 z2  z N
p1 p2  pN
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
1 


1 

s 
s 
s 

1    1 
z1 
z2  
zN 
s 
s  
s 

1 
  1 
p1 
p2  
pN 
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5
5th Order Low Pass Filter
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).
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6th Order Band Pass Filter:
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).
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All-pole filter (no finite zeroes):
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).
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12.3 Butterworth Filter:
T  j  
1
 
1  2 


 P
2N
At    P
T  j P  
T  j P 
T  j   max
Amax
1
1  2
 T  j P  
1
1
2
1
 T  j  
max
  20 log10
 20 log10 
 T  j P  



1  2
  10 log
10
1   
2
Figure 12.8 The magnitude response of a Butterworth filter.
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12.3 Butterworth Filter:
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.
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12.3.2 The Chebyshev Filter
1


2
2
1

1


cos
N
cos
  P 


T  j   
1

 1   2 cosh 2  N cosh 1    
P 


At    P
T  j P  
T  j P 
T  j   max
Amax
for    P
1
1  2
 T  j P  
1
1
2
1
 T  j  
max
  20 log10
 20 log10 
 T  j P  


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for    P
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
1  2

 10 log10 1   2 
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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
T s 
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a1 s  a0
s  0
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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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12.4.2 Second-Order Filters
a2 s 2  a1 s  a0
T s  2
s   0 Q  s   02
p1 , p2 
0
2Q
 j  0 1  1 4 Q 2 
For filters, usually
Q  0.5  complex-conjugate poles
Figure 12.15 Definition of the parameters 0 and Q of a pair of complex-conjugate poles.
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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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12.5 Second-Order LCR Filters
a2 s 2  a1 s  a0
T s  2
s   0 Q  s   02
o 
1
LC
 02 
1
LC
Q  o R C 
o
Q

1
RC
1
C
RC  R
L
LC
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.
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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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12.6.1 The Antoniou InductanceSimulation Circuit
L  C4 R1 R3 R5 R2
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.
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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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Second-Order Active Filters
Based on the Two-integrator-loop
Biquad
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.
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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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fc  1 Tc
f 
sig max
qC1  C1 vi
qC1  qC 2
C1 vi
v
T
 Req  i  c
Tc
iav C1
Tc
C2
integrator time constant = Req C2 
C2  Tc
C1
C1
iav 
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.
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12.10 Switched Capacitor Filters
C2
integrator time constant  Tc
C1
Note that the integrator time constant depends on:
• the ratio of capacitances, not their absolute value
• the clock period
MOS example: C1  0.1 pF ; C2  1 pF ; Tc  100 kHz;
C2
1
1 pF
integrator time constant  Tc
 5
 104 sec
C1 10 Hz 0.1 pF
integrator time constant  0.1ms
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12.10 Switched Capacitor Filters
Recall the integrator from the ECE 1002 Final Project:
integrator time constant  10 nF 10 k  0.1ms
Thus switched capacitor filters can work in the audio frequency
range 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.
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