Electronic Circuits
電子電路
CHAPTER 13 (Part II )
Filters and Tuned
Amplifiers
Instructor: Hsun-Hsiang Chen (陳勛祥)
Department of Electronic Engineering
National Changhua University of Education
Email: chenhh@cc.ncue.edu.tw;
TEL: 04-7232105 ext. 8368, 8369(LAB)
Reference
 Microelectronic Circuits, 7th, 2016 by Sedra/Smith
 Internet reference.
2
Filters and Tuned Amplifiers
Filter Transmission, Types, and Specification
 The Filter Transfer Function
 Butterworth and Chebyshev Filters
 First-Order and Second-Order Filter Functions
 The second-Order LCR Resonator
 Second-Order Active Filters Based on Inductor Replacement
 Second-Order Active Filters Based on the Two-integrator-Loop
Topology
 Single-Amplifier Biquadratic Active Filters
 Sensitivity
 Switched-Capacitor Filters
 Tuned Amplifiers

3
Second-Order Active Filters Based on
Inductor Replacement
 In this section, we study a family of op-amp RC
circuits that realize the various second-order filter
functions. The circuits are based on an op amp RC
resonator obtained by replacing the inductor L in the
LCR resonator with an op amp RC circuit that has an
inductive input impedance.
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.
4
The Antoniou InductanceSimulation Circuit
Z in ( s ) ≡ V1 / I1 =
sC4 R1 R3 R5 / R2
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.
5
The Antoniou InductanceSimulation Circuit
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.
6
The Op Amp-RC Resonator
ω0 1/=
LC6 1/ C4C6 R1 R3 R5 / R2
Q ω=
R6
0 R6 C6
C6 R2
C4 R1 R3 R5
 Select C4 = C6 = C and
R 1 = R2 = R 3 = R5 = R
ω0 = 1/ CR
Q = R6 / R
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.
7
Realization of the Various Filter
Types
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,
8
Realization of the Various Filter
Types
Figure 12.22 (Continued) (d) notch at ω0, (e) LPN, ωn ≥ ω0, (f) HPN, ωn ≤ ω0, and (g) all pass. The circuits are based on the LCR circuits in Fig.
12.18. Design equations are given in Table 12.1.
9
Realization of the Various Filter Types

Complementary
 Two circuits whose transfer functions are in this fashion :
AP = 1− (BP with a center - frequency gain of 2)
are said to be complementary.
The all pass filter with unity flat gain is the complement of the
band-pass filter with a center frequency gain of 2.
 A simple procedure exists for obtaining the complement of a
given linear circuit is:

 Disconnect all the circuit nodes that are connected to ground and
connect them to vi, and disconnect all the nodes that are connected
to vi and connect them to ground. That is , interchanging input and
ground in a linear circuit generates a circuit whose transfer function
is the complement of that of the original circuit.
10
Realization of the Various Filter Types

A simple procedure exists for obtaining the complement of a
given linear circuit
 Disconnect all the circuit nodes that are connected to ground and
connect them to vi, and disconnect all the nodes that are connected
to vi and connect them to ground. That is , interchanging input and
ground in a linear circuit generates a circuit whose transfer function
is the complement of that of the original circuit.
11
12
Second-Order Active Filters Based on
The Two-Integrator-Loop Topology
 In this section, we study another family of op amp-RC
circuits that realize second-order filter function. The
circuits are based on the use of two integrators
connected in cascade in an overall feedback loop and
thus known an two-integrator-loop circuits.
 Consider the second-order high pass transfer function.
Vhp
Ks 2
= 2
s + s (ω0 / Q) + ω02
Vi
[ s 2 + s (ω0 / Q) + ω02 ]Vhp = ( Ks 2 )Vi
[1 + (ω0 / Q) / s + ω02 / s 2 ]Vhp = KVi

  ω02
1  ω0
Vhp +  Vhp  +  2 Vhp  = KVi
Q s

 s
ω02
1 ω0
Vhp = KVi −
Vhp − 2 Vhp
Q s
s
Figure 12.23 Derivation of a block diagram realization of the two-integrator-loop biquad.
13
Second-Order Active Filters Based on
The Two-Integrator-Loop Topology

  ω02
1  ω0



Vhp +  Vhp  +  2 Vhp  = KVi
Q s

 s
Use the inverting
op-amp Miller
integrator
ω02
1 ω0
Vhp = KVi −
Vhp − 2 Vhp
s
Q s
Figure 12.23 Derivation of a block diagram realization of the two-integrator-loop biquad.
14
Second-Order Active Filters Based on
The Two-Integrator-Loop Topology
Vhp
Ks 2
= 2
s + s (ω0 / Q) + ω02
Vi
Thp ( s ) =
Vhp
(−ω0 / s )Vhp
Vi
Vi
(ω02 / s 2 )Vhp
Vi
Kω 0 s
= 2
= Tbp ( s )
2
s + s (ω0 / Q) + ω0
Kω02
= Tlp ( s )
= 2
2
s + s (ω0 / Q) + ω0
Figure 12.23 Derivation of a block diagram realization of the two-integrator-loop biquad.
Universal active filter
15
Circuit Implementation
Replace each integrator with a Miller integrator circuit having
CR = 1/ω0.
 Replace the summer block with an op-amp summing circuit that
is capable of assigning both positive and negative weights to its
inputs.
 The resulting circuit, know as the Kerwin-Huelsman-Newcomb
2



ω
ω
1
0
0
or KHN biquad after its inventors. Vhp = KVi +  − Vhp  −  2 Vhp 

ω02
1 ω0
Vhp = KVi −
Vhp − 2 Vhp
Q s
s
Q 
s
Vbp
 s
 


Vlp
16
Circuit Implementation
Select suitably practical values for the components of the
integrators C and R so that CR = 1/ω0.
 To determine the values for the components of the resistors
associated with the summer

 Use superposition to express the output of the summer Vhp in terms of its
inputs, Vbp = -(ω0/s) Vhp = and Vlp = (ω02/s2) Vhp.
 R f  ω02

R3  R f 
R2  R f  ω0
 − Vhp  −
1 +
Vi +
1 +
 2 Vhp 
Vhp =
R2 + R3 
R1 
R2 + R3 
R1  s
 R1  s

  ω02

1  ω0



Vhp = KVi +  − Vhp  −  2 Vhp 
Q s
 s

Vbp
R f / R1 = 1
Vlp
R3 / R2 = 2Q − 1
K = 2 − (1 / Q) ← K is fixed to this value
17
Circuit Implementation

The KHN biquad can be used to realize notch and all-pass
functions by summing weighted versions of the three outputs,
LP, BP, and HP. As shown in Fig. for the summer we can write

R

R
R
R
R
R
Vo = − F Vhp + F Vbp + F Vlp  = −Vi  F Thp + F Tbp + F Tlp 
RB
RL
RB
RL 
 RH

 RH
substituting for Thp, Tbp, Tlp gives the overall transfer function
( RF / RH ) s 2 − s ( RF / RB )ω0 + ( RF / RL )ω02
Vo
= −K
Vi
s 2 + s (ω0 / Q) + ω02
18
Circuit Implementation
 RF

 RF
RF 
RF
RF
RF
Tbp +
Tlp 
Thp +
Vlp  = −Vi 
Vbp +
Vo = −
Vhp +
RL 
RB
RL
RB
 RH

 RH
Vo
( RF / RH ) s 2 − s ( RF / RB )ω0 + ( RF / RL )ω02
= −K
Vi
s 2 + s (ω0 / Q) + ω02
Figure 12.24 (a) The KHN biquad circuit, obtained as a direct implementation of the block diagram of Fig. 12.23(c). The three basic filtering
functions, HP, BP, and LP, are simultaneously realized. (b) To obtain notch and all-pass functions, the three outputs are summed with appropriate
weights using this op-amp summer.
19
An Alternative Two-IntegratorLoop Biquad Circuit
An alternative two-integrator-loop biquad circuit in which all
three op amps are used in a single-ended mode.
 Introduce an additional inverter

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.
20
An Alternative Two-IntegratorLoop Biquad Circuit
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.
21
An Alternative Two-IntegratorLoop Biquad Circuit
Perform the summation at the virtual-ground input of the first
integration.
 No high-pass function available (disadvantage)
 The circuit is known as the Tow-Thomas biquad, after its
originators.

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.
22
An Alternative Two-IntegratorLoop Biquad Circuit

The circuit is known as the Tow-Thomas biquad, after its
originators.
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.
23
An Alternative Two-IntegratorLoop Biquad Circuit

The circuit is known as the Tow-Thomas biquad, after its
originators.
  ω02

1  ω0
Vhp = KVi +  − Vhp  −  2 Vhp 
Q s
 s

Vbp
Vlp
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.
24
An Alternative Two-IntegratorLoop Biquad Circuit
An economical feed-forward scheme can be employed with the
Tow-Thomas circuit to realized the notch and all pass function
without using a fourth op amp.
 The virtual ground available at the input of each of the three op
amp permits the input signal to be fed to all three op amps. As
the Fig. shown below.

Figure 12.26 The Tow–Thomas biquad with feedforward. The transfer function of Eq. (12.68) 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 12.2.
25
The Tow–Thomas biquad with
feedforward
  Vo
Vi  rVi   −1 
1
  V   −1 
+
−
−
−
Vo =−  + sC1   i  + 
V

−
 o  

R
sC
sQCR
sCR
sCR
R
sCR




 

2 
3 
 
 1 
Vo
1
  Vi   Vi  rVi   −1 
+
=
−
+
+ 
Vo + 
V
sC

−
 o
1 



2
(
)
sQCR
sCR
R
sC
SCR
R
sCR








2 
3 

 s
  V   V  srVi   −1 

s
1 
+ 2 2 =
−  + s 2C1   i  +  i  −
Vo  s 2 +


QCR C R 

 R1
  C   CR2  R3   CR 
 2 C1 s  1
r 
1 
=
−Vi  s
+
−
+
 C C  R RR  C 2 R R 
3 
2
 1


1 1
1
r 
C 
s2  1  + s  −
+

C  R1 RR3  C 2 RR2
Vo
C
= −
1
1
Vi
+ 2 2
s2 + s
QCR C R
Figure 12.26 The Tow–Thomas biquad with feedforward. The transfer function of Eq. (12.68) 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 12.2.
26
Final Remark
Two-integrator loop biquads are extremely versatile and easy to
design.
 Their performance is adversely affected by the finite bandwidth
of the op amps. Special techniques exist for compensating the
circuit for such effects. (see the SPICE simulation in section
12.12)

27
An Alternative Two-IntegratorLoop Biquad Circuit

R3  R f 
R2  R f  ω0
+
+
+
−
1
1
V
V

 i


hp 
R2 + R3 
R1 
R2 + R3 
R1   s

V=
hp
R f  ω02

−
V

hp 
R1  s 2

 ω0

R3  R f   ω0 
Vbp =
1 +
  −  Vi
 − Vhp  =
+
s
R
R
R
s 


2
3 
1 
+
R f  ω0   ω0

R2  R f  ω0 
+
−
−
−
−
1
V
V
(
)


 bp


bp 
R2 + R3 
R1   s 
R1  s   s

R3  R f  ω0 
1 +
 −  Vi
R2 + R3 
R1   s 
R3  R1 + R f  
1 

 −
 Vi
R2 + R3  R1   sRC 
= −
2Q − 1
1 
( 2 ) 
 Vi
1 + 2Q − 1  sRC 
2Q − 1  1 
 1 
=
−
−K 

 Vi =
 Vi
R f / R1 = 1
Q  sRC 
sRC




1
= −
V
s
R
K
C
(
/
)


R3 / R2 = 2Q − 1 K = 2 − (1 / Q)
R3 / R2 = KQ
28
Single Amplifier Biquadratic Active Filters

The op amp RC biquadratic circuits
 Good performance, Versatile, Easy design
 No economic, requiring three or four amplifiers per second-order stage

In this section, a class of second-order filter circuits that
required only on op amp per biquad.
 Suffer a greater dependence on the limited gain and bandwidth of the op-
amp
 More sensitive to the values of resistor and capacitor

The single amplifier biquads
 Limited to the less stringent filter specifications – for example, pole Q
factors less than about 10

The synthesis process
 Synthesis of a feedback loop that realizes a pair of complex-conjugate
poles characterized by a frequency ω0 and a Q factor Q.
 Injecting the input signal in a way that realizes the desired transmission
zeros.
29
Synthesis of the Feedback Loop
Assume : except for having a finite gain A, the op is ideal.
 The t(s) is the open-circuit voltage transfer function of the RC
network n.
N (s)

t (s) =
D( s)
 The roots of N(s) are the transmission zeros of the RC network, and the
roots of D(s) are its poles.

Network theory shows
 The poles of an RC network are restricted to lie in the negative real axis,
the zeros can in general lie anywhere in the s plane.
Figure 12.27 (a) Feedback loop obtained by placing a two-port RC network n in the feedback path of an op amp. (b) Definition of the opencircuit transfer function t(s) of the RC network.
30
Synthesis of the Feedback Loop

The loop gain L(s) of the feedback circuit in Fig. below is
L( s ) = At ( s ) =

Substituting for L(s) into the characteristic equation
1 + L( s ) = 0

AN ( s )
D( s)
t (s p ) = −
1
A
In the ideal case, A=∞ and the poles are obtained from
N (s p ) = 0
That is, the filter poles are identical to
the zeros of the RC network.
 Out objective is to realize a pair of complex
-conjugate poles, we should select an RC
network that can have complex-conjugate
transmission zeros. The simplest such
networks are the bridge T networks.

31
The bridge T networks
Figure 12.28 Two RC networks (called bridged-T networks) that can have complex transmission zeros. The transfer functions given are from b to
a, with a open-circuited.
32
Active Filter

The pole polynomial of the active filter circuit will be equal to the
numerator of the bridge-T network
s2 + s

 1
1  1
1
+ ω02 = s 2 + s +  +
Q
 C1 C2  R3 C1C2 R3 R4
ω0
Which enables us to obtain ω0 and Q as
1
ω0 =
C1C2 R3 R4

 1
1  C1C2 R3 R4 

Q =  + 

C
C
R
 1

2 
3
Design sequence: ω0 and Q can be used
to determine C1, C2, R3, and R4. (2 degree
of freedom). Selecting C1 = C2 = C. R3 = R
and R4 = R/m. we obtain
2Q
m = 4Q 2

−1
CR =
ω0
Q to determine m, ω0 to select the resulting component values
are practical.
Figure 12.29 An active-filter feedback loop generated using the bridged-T network of Fig. 12.28(a).
33
Injecting the Input Signal
We now consider connecting the input signal source to the
circuit. We wish to do this, of course, without altering the poles.
 An ideal voltage source is equivalent to a short circuit, it follows
that any circuit node that is connected to ground can instead be
connected to the input voltage source without causing the poles
to change.

Vo
=
Vi
− s (α / C1 R4 )
 1
1  1
1
s 2 + s +  +
 C1 C2  R3 C1C2 R3 R4
Figure 12.30 (a) The feedback loop of Fig. 12.29 with the input signal injected through part of resistance R4. This circuit realizes the bandpass
function. (b) Analysis of the circuit in (a) to determine its voltage transfer function T(s) with the order of the analysis steps indicated by the circled
numbers.
34
Generation of Equivalent Feedback Loops

Application of the complementary transform to a feedback loop
to generate an equivalent feedback loop is a two step process
 Nodes of the feedback network and any of the op-amp that are connected
to ground should be disconnected from ground and connected to the opamp output. Conversely, output change to ground.
 The two input terminals of the op amp should be interchanged.
Figure 12.31 Interchanging input and ground results in the complement of the transfer function.
35
Figure 12.32 Application of the complementary transformation to the feedback loop in (a) results in the equivalent loop (same poles) shown in
(b).
36
Figure 12.33 (a) Feedback loop obtained by applying the complementary transformation to the loop in Fig. 12.29. (b) Injecting the input signal
through C1 realizes the high-pass function. This is one of the Sallen-and-Key family of circuits.
37
Figure 12.34 (a) Feedback loop obtained by placing the bridged-T network of Fig. 12.28(b) in the negative-feedback path of an op amp. (b)
Equivalent feedback loop generated by applying the complementary transformation to the loop in (a). (c) A low-pass filter obtained by injecting Vi
through R1 into the loop in (b).
38
Second-Order Gm–C Filter
39
Switched-Capacitor Filters
qC1 = C1vi
iav =
C1vi
Tc
Req ≡ vi / iav
Req = Tc / C1
Time constant = C2 Req = Tc (C2 / C1 )
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 φ1, C1 charges up to the current value of vi and then, during φ2, discharges into C2.
40
Practical Circuits
Figure 12.36 A pair of complementary stray-insensitive switched-capacitor integrators. (a) Noninverting switched-capacitor integrator. (b)
Inverting switched-capacitor integrator.
41
Practical Circuits
Figure 12.37 (a) A two-integrator-loop active-RC biquad and (b) its switched-capacitor counterpart.
42
Practical Circuits
1
ω0 =
C1C2 R3 R4
R3 = Tc / C3
Tc
T
C2 = c C1
C3
C4
for C1 = C2 = C
Q=
R5 Tc / C5
=
R4 Tc / C4
C5 =
R4 = Tc / C4
C4 KC
C
=
= ω0Tc
Q
Q
Q
ω0 =
C3 = C4 = KC
1
Tc
C 3C 4
C1C2
K = ω0Tc
Center - frequency gain =
Figure 12.37 (a) A two-integrator-loop active-RC biquad and (b) its switched-capacitor counterpart.
C6
C
=Q 6
ω0TC
C5
43
Tuned Amplifier
Figure 12.38 Frequency response of a tuned amplifier.
44
Vo =

− g mVi
− g mVi
=
YL
sC + 1 / R + 1 / sL
The voltage gain
Vo
g
s
=− m 2
Vi
C s + s (1 / RC ) + 1 / LC

The center frequency
ω0 = 1 / LC

A 3-dB bandwidth
B = 1 / CR

A Q factor
Q ≡ ω0 / B = ω0CR

A center frequency gain
Vo ( jω0 )
= −gm R
Vi ( jω0 )
Figure 12.39 The basic principle of tuned amplifiers is illustrated using a MOSFET with a tuned-circuit load. Bias details are not shown.
45
Example
Figure 12.40 Inductor equivalent circuits.
46
Inductor Losses
Figure 12.40 Inductor equivalent circuits.
Finally, it should be noted that the
coil Q factor poses an upper limit
on the value of Q achieved by the
tuned circuit.
47
Use of Transformers
Figure 12.41 A tapped inductor is used as an impedance transformer to allow using a higher inductance, L′, and a smaller capacitance, C′.
48
Use of Transformers
Figure 12.42 (a) The output of a tuned amplifier is coupled to the input of another amplifier via a tapped coil. (b) An equivalent circuit. Note that the
use of a tapped coil increases the effective input impedance of the second amplifier stage.
49
Amplifiers with Multiple Tuned Circuits
Figure 12.43 A BJT amplifier with tuned circuits at the input and the output.
50
Amplifiers with Multiple Tuned Circuits
Figure 12.44 Two tuned-amplifier configurations that do not suffer from the Miller effect: (a) cascode and (b) common-collector common-base
cascade. (Note that bias details of the cascode circuit are not shown.)
51
The bandwidthshrinkage factor.
Figure 12.45 Frequency response of a synchronously tuned amplifier.
52
Stagger-Tuning
Stagger-tuned amplifiers are usually designed so that the
overall response exhibits maximal flatness around the
center frequency f0.
Figure 12.46 Stagger-tuning the individual resonant circuits can result in an overall response with a passband flatter than that obtained with
synchronous tuning (Fig. 12.45).
53
Figure 12.47 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 + jω0 and adding a complex-conjugate pole results in the poles of the second-order bandpass filter.
54
Figure 12.47 (Continued) (c) Magnitude response of the first-order low-pass filter. (d) Magnitude response of the second-order bandpass filter.
55
Figure 12.48 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.
56
Figure 12.48 (Continued)
57
Figure 12.49 Circuits for Example 12.5. (a) Fifth-order Chebyshev filter circuit implemented as a cascade of two second-order simulated LCR
resonator circuits and a single first-order op amp–RC circuit.
58
Figure 12.49 (Continued) (b) VCVS representation of an ideal op amp with gain A.
59
Figure 12.50 Magnitude response of the fifth-order lowpass filter circuit shown in Fig. 12.49: (a) an expanded view of the passband region; (b) a view
of both the passband and stopband regions.
60
Figure 12.51 One-pole equivalent circuit macromodel of an op amp operated within its linear region.
61
Figure 12.52 Circuit for Example 11.6. Second-order bandpass filter implemented with a Tow–Thomas biquad circuit having f0 = 10 kHz, Q = 20, and
unity center-frequency gain.
62
Figure 12.53 Comparing the magnitude response of the Tow–Thomas biquad circuit (shown in Fig. 12.52) constructed with 741-type op amps, with
the ideal magnitude response. These results illustrate the effect of the finite dc gain and bandwidth of the 741 op amp on the frequency response of the
Tow–Thomas biquad circuit.
63
Figure 12.54 (a) Magnitude response of the Tow–Thomas biquad circuit with different values of compensation capacitance. For comparison, the ideal
response is also shown.
64
Figure 12.54 (Continued) (b) Comparing the magnitude response of the Tow–Thomas biquad circuit using a 64-pF compensation capacitor and the
ideal response.
65
Figure P12.11
66
Figure P12.27
67