C H A P T E R 11
Filters and Tuned Amplifiers
Introduction
 The electronic filter is an important building block of communications
and instrumentation systems.
 The oldest technology for realizing filters makes use of inductors
and capacitors, and the resulting circuits are called passive LC filters.
 The passive LC filters work well at high frequencies; however, in
low-frequency application (dc to 100 KHz) the required inductors are
large and physically bulky. Additionally, it is incompatible with modern
VLSI technology.
The inductor-less filter is of interest and we shall study active-RC
filters and switched-capacitor filters.
2
Filter Transmission
 The filters we are going to study are linear circuits that can
represented by a general 2-port network.
 The transfer function T(s) is the ratio of output voltage Vo(s) to the
input voltage Vi(s).
 The filter transmission is found by evaluating T(s) and can be
expressed in terms of its magnitude and phase as
Figure 11.1
3
Filter Types
 The filters can perform a frequency-selection function:
passing signals whose frequency lies within a specified range, and
stopping signal whose frequency falls outside this range.
Figure 11.2 Ideal transmission characteristics of the four major filter types: (a) low-pass (LP), (b) high-pass (HP),
(c) bandpass (BP), and (d) bandstop (BS).
4
Filter Specification
 The transmission of a low-pass filter is specified by 4 parameters:
 The passband edge wp.  The stopband edge ws.
 The maximum allowed variation in passband transmission Amax.
 The minimum required stopband attenuation Amin.
Lower Amax, higher Amin, ws/wp closer to 1 are
desirable in an ideal filter.
Figure 11.3 Specification of the transmission characteristics of a low-pass filter. The magnitude response of a filter that just meets
specifications is also shown.
5
Filter Design Procesdure
 The following is the filter design procedure:
 Decide the filter specification.
 Find the transfer function whose magnitude meets the specification.
 The process of the above (2nd) step is known as filter approximation.
Transmission specification for a bandpass filter.
Figure 11.4 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.
6
11.2 The Filter Tranfer Function
 The filter transfer function T(s) can be written as the ratio of 2
polynomials as:
 The degree of the denominator, N, is the filter order.
 For stability consideration, M ≦ N.
 The numerator/denominator coefficients ai and bi are real numbers.
 The polynomials in the numerators/denominators can be factored
and T(s) can be written in the form:
Zero
Pole/Nature Modes
 Poles and zeros can be either a real or a complex number.
 Complex zeros and poles must occur in conjugate pair.
7
Tranfer Function of a Low-Pass Filter (1)
 Consider the filter whose transfer function is shown below.
 The filter has transmission zeros at s= ± jwl1 and s= ± jwl2.
 The transmission decreases toward -∞ as w approaches ∞. The
filter must have one or more zeros at s=∞. The number of
transmission zeros at s=∞ is N-M.
assume
the
filter is of fifth
order (N=5)
Figure 11.5 Pole–zero pattern for the lowpass filter whose transmission is sketched in Fig. 11.3. This is a fifth-order filter (N = 5).
8
Tranfer Functionof a Bandpass Filter
 Consider the filter whose transfer function is shown below.
 The filter has transmission zeros at s= ± jwl1 and s= ± jwl2.
 The transmission decreases toward -∞ as w approaches 0 and ∞.
The filter must have one or more zeros at s=∞ and s=0. Assume only
1 zero at s=∞ and s=0. The filter must be of sixth order (N=6).
Figure 11.6 Pole–zero pattern for the bandpass filter whose transmission function is shown in Fig. 11.4. This is a sixth-order filter
(N = 6).
9
Tranfer Function of a Low-Pass Filter (2)
 Consider the filter whose transfer function is shown below.
 No finite values of w at which the transmission is zero. Thus all the
transmission zeros are at s=∞.
 Such a filter is known as “all pole filter”.
assume the filter is
of fifth order (N=5)
Figure 11.7 (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)
10
Figure 11.8 The magnitude response of a Butterworth filter.
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Butterworth Filters (1)
 2 functions that are frequently used in approximating the
transmission characteristics of low-pass filters are Butterworth and
Chebyshev filters.
 Butterworth filter exhibits a monotonically decreasing transmission
with all the zeros at w =∞, making it an all-pole filter.
 The magnitude of an N-th order Butterworth filter with a passband
edge wp is given by
where ε determines the maximum variation in passband transmission.
12
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Butterworth Filters (2)
 It can be shown that the first 2N-1 derivatives of |T| relative to w are
zero at w=0. This property make the Butterworth response very flat
near w=0 and gives the response the name maximally flat response.
As the order increases, the
response approaches the ideal
brick-wall type of transmission.
Figure 16.9
14
Figure 11.10
Butterworth Filters (3)
 Its poles can be determined from the graphical construction.
N=2
N=3
N=4
15
Butterworth Filters (4)
 The transfer function of the Butterworth filter is
 The procedure to find a Butterworth transfer function that meets the
specification (Amin, Amax, ws(the edge of the stopband), wp):
 Determine ε from Amax.
 Determine the required filter order as the lowest integer value of N
that results in A(ws) ≧ Amin.
 Determine the N poles.
 Determine T(s).
The detailed procedure can be found in Ex. 12.1
16
Butterworth Filters (5)
 Poles of the ninth-order Butterworth filter.
Figure 11.11 Poles of the ninth-order Butterworth filter of Example 11.1.
17
The Butterworth Response
n
Normailized Denominator Polynomials in Factored Form
1
(1+s)
2
(1+1.414s+s2)
3
(1+s)(1+s+s2)
4
(1+0.765s+s2)(1+1.848s2)
5
(1+s)(1+0.618s+s2)(1+1.618s2)
6
(1+0.518s+s2)(1+1.414s+s2)(1+1.932s+s2)
7
(1+s)(1+0.445s+s2)(1+1.247s+s2)(1+1.802s+s2)
8
(1+0.390s+s2)(1+1.111s+s2)(1+1.663s+s2)(1+1.962s+s2)
9
(1+s)(1+0.347s+s2)(1+s+s2)(1+1.532s+s2)(1+1.879s+s2)
10
(1+0.313s+s2)(1+0.908s+s2)(1+1.414s+s2)(1+1.782s+s2)(1+1.975s+s2)
18
19
3rd-order Butterworth Low Pass Filter
20
21
Chebyshev Filters (1)
 The Chebyshev filter exhibits an equiripple response in the
passband and a monotonically decreasing transmission in the
stopband. It is an all-pole filter.
 While the odd-order filter has |T(0)| =1, the even-order filter exhibits
its maximum magnitude deviation at w=0.
N=4
N=5
Figure 11.12 Sketches of the transmission characteristics of representative (a) even-order and (b) odd-order Chebyshev filters.
22
Chebyshev Filters (2)
 The poles of a Chebyshev filter can be determined by
 The transfer function of the Chebyshev filter can be written as
 For the same order and the same Amax, the Chebyshev filter
provides greater stopband attenuation than the Butterworth filter.
 To meet identical specification, one requires a lower order for the
Chebyshev than for the Butterworth.
Ex. 12.2 is a useful practice.
23
11.4 First-order and second-order filters
Cascade Filter Design
 First and second order filters can be cascaded to realize a highorder filter. It is one of the most popular methods for the design of
active filters.
 Cascading does not change the transfer functions of individual
blocks if the output impedance level is low.
24
First Orer Filters (1)
 The general first-order transfer function is given by
Figure 11.13 First-order filters.
phase shifter
Figure 11.14 First-order all-pass filter.
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Second Orer Filters (1)
 The general second-order (bi-quadratic) transfer function is given by
 The natural modes (poles) are
 w0 is the radial distance of the
poles from the origin and is
known as the pole frequency.
 Q is called the pole quality
factor.
Figure 11.15
27
Second Orer Filters (2)
 The second-order filter function.
Figure 11.16 Second-order filtering functions.
28
Second Orer Filters (3)
 The second-order filter function.
Figure 11.16 (continued)
29
Second Orer Filters (4)
 The second-order filter function.
Figure 11.16 (continued)
30
The Second Orer LRC Resonator
 The nature mode of the parallel resonance circuit can be
determined by applying an excitation that does not change the
natural structure of the circuit.
Current
Excitation
Voltage
Excitation
Figure 16.17 (a)The second-order parallel LCR resonator.
(b, c) Two way s of exciting the resonator (a)
of without changing itsnatural structure:
Figure 11.17
resonator poles are those poles of
Vo/I and Vo/Vi.
31
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Realization of Transmission Zeros
 Having realized a given pair of complex conjugate natural mode,
find out where to inject the voltage signal Vi so that the transfer
function VO/Vi is the desired filter type.
The transmission zeros are the values of s at which the Z2(s) is zero
(provided Z1(s) is not simultaneously zero), and the values of s at
which the Z1(s) is infinite (provided Z2(s) is not simultaneously
infinite),
33
Realization of the Low -Pass and
High-Pass Function
Low-Pass Filter
High-Pass Filter
For low-pass filter, it has 2 zeros. One
is at the value of s for which the series
impedance becomes infinite and shunt
impedance becomes zero.
34
Realization of the All -Pass
 The all-pass transfer function
can be written as
The disadvantage of the circuit is the
lack of a common ground terminal
between the input and the output.
Figure 11.19 Realization of the second-order all-pass transfer function using a voltage divider and an LCR resonator.
35
The Antoniou Inductance-Simulation
Circuit
 It is a amp-RC circuits that could simulate the operation of an
inductor. Prof. Antoniou has proved that this circuit is the best (tolerant
of the finite gain and bandwidth of a real OP) among other approaches.
Figure 11.20 (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.
36
The Op Amp-RC Resonator
 An op amp-RC resonator obtained by replacing the inductor L with a
simulated inductance.
 Vr is not a convenient node to use as the filter output terminal
because connecting a load there would change the filter
characteristics.
 This problem can be solved by utilizing a buffer amplifier.
Figure 11.21 (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. 11.20(a). (c) Implementation of the buffer amplifier K.
37
Realization of the Various Filter Types (1)
38
Realization of the Various Filter Types (2)
39
Realization of the Various Filter Types (3)
 The all-pass circuit with a flat gain of unity can be written as
AP = 1- (BP with a center-frequency gain of 2)
A simple procedure to obtain the
complement of a given linear circuit.
 Disconnect
all
the
nodes
connected to ground and then
connect them to Vi.
 Disconnect all the nodes
connected to Vi and then connect
them to ground.
 That is, interchange the input and
ground.
2 circuits whose transfer functions
are related in this fashion are said to
be complementary.
40
11.7 Second-Order Active Filters Based on
the two-Integrator-Loop
 It is another family of op amp-RC circuits that realize second-order
filter function. First consider a high-pass transfer function.
The negative sign in the integrator
blocks means the use of inverting Miller
integrator to implement each integrator.
Figure 11.23 Derivation of a block diagram realization of the two-integrator-loop biquad.
41
Derivation of the Two-Integrator-Loop
Biquad
 Rearrange the equation, the high-pass function can be written as
High-Pass Function
It is known as universal active
filter
Band-Pass Function
Low-Pass Function
Figure 11.23 Derivation of a block diagram realization of the two-integrator-loop biquad.
42
Figure 11.24 (a) The KHN biquad circuit, obtained as a direct implementation of the block diagram of Fig. 11.23(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.
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Circuit Implementation-KHN Biquad
 The circuit known as KHN biquad can implement such function.
44
Circuit ImplementationTwo-Thomas Biquad
 An alternative two-integrator-loop biquad circuit in which all the three
op amps are used in a single-ended mode can be developed as follows.
Coefficients of the summer
have the same sign.
High-pass function is no
longer available.
Figure 11.25 (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.
45
Figure 11.26 The Tow–Thomas biquad with feed forward. 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.
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11.8 Single-Amplifier Biquadratic
Active Filters
 The previous op amp-RC biquadratic circuits are versatile and are
easy to design. However, they are not economic in their use of op amps.
It is a problem in applications where the power-supply current is to be
conserved.
 Therefore the circuit with only one op amp is desirable. These minimal
realizations suffer 2 problems than multiple op-amp biquads.
 A dependence of limited gain and bandwidth of the op amp.
 Sensitive to the unavoidable tolerances in the values of R and C.
 The single-amplifier-biquads (SABs) are therefore limited to the less
stringent filter specifications.
 The synthesis of SABs follows a 2-step process.
 Synthesis of a feedback loop that realizes a pair of complex-conjugate
poles characterized by w0 and Q.
 Injecting the input signal in a way that realizes the desired zeros.
47
Synthesis of Feedback Loop
 The synthesis of the feedback loop.
The
closed-loop
equation is
characterization
The poles of the closed-loop system are
identical to the zeros of the RC network.
Figure 11.27 (a) Feedback loop obtained by placing a two-port RC network n in the feedback path of an op amp. (b) Definition of the
open-circuit transfer function t(s) of the RC network.
48
Networks with Complex Zeros
 2 RC networks (bridged-T networks) that can have complex
transmission zeros are shown below.
Figure 11.28 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.
49
Feedback Loop Using Bridged-T Network
The pole polynomial of the active-filter circuit will be equal to the
numerator polynomial of the bridged-T network;
Figure 11.29 An active-filter feedback loop generated using the bridged-T network of Fig. 11.28(a).
50
Injecting the Input Signal
 Any circuit node connected to ground can instead be connected to the
input voltage source without causing the poles to change.
 Depending on the components through which the input signal is
injected, different transmission zeros are obtained.
Figure 11.30 (a) The feedback loop of Fig. 11.29 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.
51
Generation of Equivalent
Feedback Loops (1)
 The complementary transformation of feedback loop.
Figure 11.31 Interchanging input and ground results in the complement of the transfer function.
52
Generation of Equivalent
Feedback Loops (2)
 A 2-step process to generate an equivalent feedback loop.
 Nodes of feedback network and op-amp input that are connected to
ground should be disconnected and connected to op-amp output.
 The 2 input terminals of the op amp should be interchanged.
Characteristic Equation
Characteristic Equation
Figure 11.32 Application of the complementary transformation to the feedback loop in (a) results in the equivalent loop (same poles)
shown in (b).
53
Generation of Equivalent
Feedback Loops (3)
 An example of applying the complementary transformation to get the
equivalent feedback loop.
A second-order high-pass function is realized.
Figure 11.33 (a) Feedback loop obtained by applying the complementary transformation to the loop in Fig. 11.29. (b) Injecting the input
signal through C1 realizes the high-pass function. This is one of the Sallenand-Key family of circuits.
54
Generation of Equivalent
Feedback Loops (4)
 Another example of applying the complementary transformation to get
the equivalent feedback loop.
Injecting signal to C4 results
in a bandpass function.
Injecting signal to R1 results
in a low-pass function.
Figure 11.34 (a) Feedback loop obtained by placing the bridged-T network of Fig. 11.28(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).
55
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11.10
Switched-Capacitor Filters (1)
 Previous active-RC filters need large-valued capacitors and the
requirement of accurate RC time constants that makes production in
monolithic IC form difficult.
 The switched-capacitor filter design would lend itself more naturally to
IC implementation.
57
Switched-Capacitor Filters (2)
 Basic principle of switched-capacitor filter techniques.
Inverting
integrator
Accuracy is
obtainable
with 0.1 pF.
The integrator is stray-sensitive.
Figure 11.35
58
Figure 11.36 A pair of complementary stray-insensitive, switched-capacitor integrators. (a) Noninverting switched-capacitor integrator.
(b) Inverting switched-capacitor integrator.
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Switched-Capacitor Filters (3)
 Using switched-capacitor technique, implement the circuit shown
below so that the product of RC is 1 ms.
 The product of RC may be written as RC2 = Tc(C2/C1). RC is
determined by T, which is an accurate period derived from a crystal
oscillator and the ratio of C2 and C1 which may be within 1% on a
chip.
 Set C1 is larger than the associated parasitics, say 1pF and C2=
100 pF with T=10-5 s. The value of switched-capacitor resistor is 10
MΩ. Implementing this resistor in a CMOS process requires large
area.
60
Switched-Capacitor Filters (3)
 A 2-integrator-loop active filter is composed of one inverting and one
noninverting integrator. Therefore we need a pair of complementary
switched-capacitor integrators.
Non-inverting
integrator
Inverting
integrator
These integrators are
stray-insensitive.
61
Switched-Capacitor Filters (4)
 A 2-integrator-loop active filter biquad and its switched-capacitor
counterpart.
Figure 11.37 (a) A two-integrator-loop, active-RC biquad (b) its switched-capacitor counterpart.
62
11.11 Tuned Amplifiers
Figure 11.38 Frequency response of a tuned amplifier.
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 g mVi
 g mVi
Vo 
YL SC  1 / R  1 / SL
Vo  g m
S

Vi
C S 2  S (1 / RC)  1 / LC
Figure 11.39 The basic principle of tuned amplifiers is illustrated using a MOSFET with a tuned-circuit load. Bias details are not shown.
64
Inductor Losses
W0 L R p
Q0 

rs
W0 L
1
Y ( jw0 ) 
rs  jW0 L
Figure 11.40 Inductor equivalent circuits.
65
可得到大L, 小C, Wo 不變
Figure 11.41 A tapped inductor is used as an impedance transformer to allow using a higher inductance, L’, and a smaller capacitance, C’.
66
Figure 11.42 (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.
雙tuner, selective更好, BW ↓
Figure 11.43 A BJT amplifier with tuned circuits at the input and the output.
68
Cascode, CE+CB
CC+CB
do not suffer from the Miller effect: (a) cascode and (b) common-
Figure 11.44 Two tuned-amplifier configurations that
collector common-base cascade. (Note that bias details of the cascode circuit are not shown.)
69
N tuner cascade, BW ↓
Figure 11.45 Frequency response of a synchronously tuned amplifier.
70
max flat in pass band, BW ↑
Stagger-tuning the individual resonant circuits can result in an overall response with a passband flatter than that obtained
Figure 11.46
with synchronous tuning (Fig. 11.45).
71
Figure 11.47
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Figure 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.
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