CHAPTER 11 FILTERS AND TUNED AMPLIFIERS
Chapter Outline
11.1 Filter Transmission, Types and Specifications
11.2 The Filter Transfer Function
11.3 Butterworth and Chebyshev Filters
11.4 First-Order and Second-Order Filter Functions
11.5 The Second-Order LCR Resonator
11.6 Second-Order Active Filters Based on Inductor Replacement
11.7 Second-Order Active Filters Based on the Two-Integrator-Loop Topology
11.8 Single-Amplifier Biquadratic Active Filters
11.9 Sensitivity
11.10 Switched-Capacitor Filters
11.11 Tuned Amplifiers
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11-1
11.1 FILTER TRANSMISSION, TYPES AND SPECIFICATIONS
Filter Transfer Function
 A filter is a linear two-port network represented by the ratio of the output to input voltage.
 Transfer function T(s)  Vo(s) / Vi(s).
 Transmission : evaluating T(s) for physical frequency s = j → T(j) = |T(j)|e j ().
 Gain: 20 log|T(j)| (dB)
 Attenuation: - 20 log|T(j)| (dB)
Output frequency spectrum : |Vo(s)| = |T(s)| |Vi(s)|.
Types of Filters
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11-2
Filter Specification




Passband edge : p
Maximum allowed variation in passband transmission : Amax
Stopband edge : s
Minimum required stopband attenuation : Amin
Low-Pass Filter
Band-Pass Filter
 The first step of filter design is to determine the filter specifications.
 Then find a transfer function T(s) whose magnitude meets the specifications.
 The process of obtaining a transfer function that meets given specifications is called filter approximation.
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11-3
11.2 FILTER TRANSFER FUNCTION
Transfer Function
 The filter transfer function is written as the ratio of two polynomials:
T (s) 




The degree of the denominator → filter order.
To ensure the stability of the filter → N  M.
The coefficients ai and bj are real numbers.
The transfer function can be factored and expressed as:
T (s) 




aM s M  aM 1s M 1  ...  a0
s N  bN 1s N 1  ...  b0
aM ( s  z1 )( s  z 2 )...(s  z M )
( s  p1 )( s  p2 )...(s  p N )
Zeros: z1 , z2 , … , zM and ( NM ) zeros at infinity.
Poles: p1 , p2 , … , pN .
Zeros and poles can be either a real or a complex number.
Complex zeros and poles must occur in conjugate pairs.
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11-4
Transfer Function Examples
 Low-Pass Filter (with Stopband Ripple)
T (s) 
a4 ( s   )( s   )
s  b4 s 4  b3 s  b2 s 2  b1s  b0
2
2
l1
3
5
2
2
l2
 Low-Pass Filter (without Stopband Ripple)
aM
T (s)  5
4
s  b4 s  b3 s 3  b2 s 2  b1s  b0
| T | (dB)
zeros
poles
0
j
l2
l1
p


p
s
| T | (dB)
p
l1
l2

l1
l2
zeros
poles
0
j

p
5

p
p
 Band-Pass Filter
T (s) 
a5 s ( s   )(s   )
s  b5 s  b4 s 4  b3 s 3  b2 s  b1s  b0
5
s
| T | (dB)
2
6

2
l1
2
2
l2
2
zeros
poles
0
j
l2
p2

p1
l1
l1 s1
p1 p2

s2
l2
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l1
p1
p2

l2
11-5
11.3 BUTTERWORTH AND CHEBYSHEV FILTERS
The Butterworth Filter




Butterworth filters exhibit monotonically decreasing transmission with all zeros at  = .
Maximally flat response → degree of passband flatness increases as the order N is increased.
Higher order filter has a sharp cutoff in the transition band.
The magnitude function of the Butterworth filter is:
| T ( j ) |
1
1   ( /  p ) 2 N
2
 Required transfer functions can be defined based on filter specifications (Amax , Amin , p , s)
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11-6
Natural Modes of the Butterworth Filter
 The natural modes lines on a circle.
 The radius of the circle is 0   p (1 /  )1/ N
 Equal angle space  / N
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11-7
Design Procedure of the Butterworth Filters
Design Specifications
Amax , Amin , p , s
Design Procedure
1.
Determine  (from Amax)
| T ( j p ) |
2.
1
1 
2
 Amax [ dB]  20 log 1   2    10 Amax /10  1
Determine the required filter order N (from p , s , Amin)




Attenuation A(s )[dB]  20 log 1 / 1   2 (s /  p ) 2 N  10 log 1   2 (s /  p ) 2 N  Amin
3.
Determine the N natural modes
p1 , p2 , .... p N
4.
Determine T (s)
K0N
T (s) 
( s  p1 )(s  p2 )...(s  p N )
where 0   p (1 /  )1/ N
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11-8
The Chebyshev Filter





An equiripple response in the passband.
A monotonically decreasing transmission in the stopband.
Odd-order filter → | T(0) | = 1.
Even-order filter → maximum magnitude deviation at  = 0.
The transfer function of the Chebyshev filter is :
| T ( j ) |
1
1   2 cos 2 [ N cos 1 ( /  p )]
for    p
| T ( j ) |
1
1   2 cosh 2 [ N cosh 1 ( /  p )]
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for    p
11-9
Design Procedure of the Chebyshev Filters
Design Specifications
Amax , Amin , p , s
Design Procedure
1.
Determine  (from Amax)
| T ( j p ) |
2.
1
1 
2
 Amax (dB)  20 log 1   2    10 Amax /10  1
Determine the required filter order N (from p , s , Amin)


Attenuation A(s )  10 log 1   2 cosh 2 ( N cosh 1 (s /  p ))  Amin
3.
Determine the N natural modes
4.
Determine T (s)
1
1
 2k  1  
1
 2k  1  
1
pk   p sin
  cosh sinh 1 
  sinh sinh 1   j p cos
2

2

 N
N
 N
N
T (s) 
K pN
 2 N 1 ( s  p1 )(s  p2 )...(s  p N )
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11-10
11.4 FIRST-ORDER AND SECOND-ORDER FILTER FUNCTIONS
Cascade Filter Design
 First-order and second-order filters can be cascaded to realize high-order filters.
 Cascade design is one of the most popular methods for the design of active filters.
 Cascading does not change the transfer functions of the individual blocks if the output resistance is low.
First-Order Filters
 Bilinear transfer function T ( s) 
a1s  a0
s  b0
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11-11
First-Order Filters (Cont’d)
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11-12
Second-Order Filters
 Biquadratic transfer function
a2 s 2  a1s  a0
T ( s)  2
s  (0 / Q) s  02
 Pole frequency: 0
 Pole quality factor: Q
 Poles:
p1 , p2  
0
2Q
 j0 1 
1
4Q 2
 Bandwidth:
BW  2  1 
0
Q
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11-13
Second-Order Filters (Cont’d)
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11-14
Second-Order Filters (Cont’d)
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11-15
11.5 THE SECOND-ORDER LCR RESONATOR
The Resonator Natural Modes
Parallel Resonator
Current Excitation
Voltage Excitation
Current Excitation
Vo 1
1
s/C
 
 2
I i Y 1 / sL  sC  1 / R s  (1 / RC ) s  1 / LC
0  1 / LC
Voltage Excitation
Vo
( R || 1 / sC )
1 / LC

 2
Vi ( R || 1 / sC )  sL s  (1 / RC ) s  1 / LC
Q  0 RC
 The LCR resonator can be excited by either current or voltage source.
 The excitation should be applied without change the natural structure of the circuit.
 The natural modes of the circuits are identical (will not be changed by the excitation methods).
 The similar characteristics also applies to series LCR resonator.
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11-16
Realization of Transmission Zeros
 Values of s at which Z2(s) = 0 and Z1(s)  0
→ Z2 behaves as a short
 Values of s at which Z1(s) =  and Z2(s)  
→ Z1 behaves as an open
Realization of Filter Functions
Low-Pass Filter
T ( s) 
1 / LC
Vo
 2
Vi s  (1 / RC ) s  1 / LC
High-Pass Filter
T ( s) 
Vo
s2
 2
Vi s  (1 / RC ) s  1 / LC
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Bandpass Filter
T ( s) 
(1 / RC ) s
Vo
 2
Vi s  (1 / RC ) s  1 / LC
11-17
11.6 SECOND-ORDER ACTIVE FILTERS (INDUCTOR REPLACEMENT)
Second-Order Active Filters by Op Amp-RC Circuits
 Inductors are not suitable for IC implementation
 Use op amp-RC circuits to replace the inductors
 Second-order filter functions based on RLC resonator
The Antoniou Inductance-Simulation Circuit
 Inductors are realized by op amp-RC circuits with negative feedbacks
 The equivalent inductance is given by
Z in 
V1
 sC4 R1 R3 R5 / R2  sLeq
I1
Leq  C4 R1 R3 R5 / R2
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11-18
The Op Amp-RC Resonator
 The inductor is replaced by the Antoniou circuit
 The pole frequency and the quality factor are given by
0  1 / C4C6 R1 R3 R5 / R2 and Q  0C6 R6  R6
C6 R2
C4 R1 R3 R5
 The pole frequency and quality factor for a simplified case where R1 = R2 = R3 = R5 = R and C4 = C
0  1 / RC and
Q  R6 / R
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11-19
Filter Realization
High-Pass Filter
Low-Pass Filter
Notch Filter
Bandpass Filter
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11-20
HPN Filter
LPN Filter
All-Pass Filter
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11-21
11.7 SECOND-ORDER ACTIVE FILTERS (TWO-INTEGRATOR-LOOP)
Derivation of the Two-Integrator-Loop Biquad
Ks 2
 2
Vi
s  (0 / Q) s  02
Vhp
Vhp 
2
1 0
( Vhp )  ( 20 Vhp )  KVi
Q s
s
 High-pass implementation:
Vhp  KVi 
1 0
2
Vhp  20 Vhp
Q s
s
 Bandpass implementation:
(0 / s)Vhp
Vi

 K0 s
 Tbp ( s)
s 2  (0 / Q) s  02
 Low-pass implementation:
(02 / s 2 )Vhp
Vi

K02
 Tlp ( s )
s 2  (0 / Q) s  02
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11-22
Circuit Implementation (I)
Vhp 
Rf
Rf
R f 02
R3

R2
(1  )Vi 
(1  )( 0 Vhp ) 
( Vhp )
R2  R3
R1
R2  R3
R1
s
R1 s 2
 R f / R1  1
R3 / R2  2Q  1
K  2 1/ Q
 High-pass transfer function:
Thp ( s) 
Vhp
Vi

s 2 [2 R3 /( R2  R3 )]
s 2  s[2 R2 /( R2  R3 )]0  02
 Bandpass transfer function:
Tbp ( s) 
Vbp
Vi

s[2 R3 /( R2  R3 )]0
s  s[2 R2 /( R2  R3 )]0  02
2
 Low-pass transfer function:
Tbp ( s) 
Vbp
Vi

[2 R3 /( R2  R3 )]02
s 2  s[2 R2 /( R2  R3 )]0  02
 Notch and all-pass transfer function:
T ( s) 
Vo
2 R3 ( RF / RH ) s 2  ( RF / RB )0 s  ( RF / RL )02

Vi
R2  R3
s 2  s[2 R2 /( R2  R3 )]0  02
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11-23
Circuit Implementation (II) – Tow-Thomas Biquad





Use an additional inverter to make all the coefficients of the summer the same sign.
All op amps are in single-ended mode.
The high-pass function is no longer available.
It is known as the Tow-Thomas biquad.
An economical feedforward scheme can be employed with the Tow-Thomas circuit.
T ( s) 
Vo

Vi
s2
C1
11
r 
1
  2
 s  
C
C  R1 RR3  C RR2
1
1
 2 2
s2  s
QCR C R
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11-24
11.8 SINGLE-AMPLIFIER BIQUADRATIC ACTIVE FILTERS
Characteristics of the SAB Circuits




Only one op amp is required to implement biquad circuit.
Exhibit a greater dependence on the limited gain and bandwidth of the op amp.
More sensitive to the unavoidable tolerances in the values of resistors and capacitors.
Limited to less stringent filter specifications with pole Q factors less than 10.
Synthesis of the SAB Circuits
 Use feedback to move the poles of an RC circuit from the negative real axis to the complex conjugate
locations to provide selective filter response.
 Steps of SAB synthesis:
 Synthesis of a feedback loop that realizes a pair of complex conjugate poles characterized by 0 and Q.
 Injecting the input signal in a way that realizes the desired transmission zeros.
 Natural modes of the filter:
t ( s )  N ( s ) / D( s )
L( s)  At ( s)  AN ( s) / D( s)
The closed-loop characteristics equation:
1  L( s)  0  t ( sP )  1 / A  0
 The poles of the closed-loop system are identical to the zeros of the RC network.
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11-25
RC Networks with complex transmission zeros
1
1  1
1
s 2  s   
 C1 C2  R3 C1C2 R3 R4
t (s) 
 1
1
1 
1
 
s 2  s


 C1 R3 C2 R3 C1 R4  C1C2 R3 R4
1
1  1
1
s 2  s  

 R1 R2  C4 R1 R2C3C4
t ( s) 
 1
1
1 
1
 
s 2  s


 R2C4 R2C3 R1C4  R1 R2C3C4
Characteristics Equation of the Filter
0
1
1  1
1
 02  s 2  s   
Q
 C1 C2  R3 C1C2 R3 R4
1
 0 
C1C2 R3 R4
s2  s
 CC R R
Q   1 2 3 4
R3

1
1 
  
 C1 C2 
1
2
Let C1 = C2 = C , R3 = R , R4= R/m  m  4Q
 CR  2Q / 0
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11-26
Injection the Input Signal
 The method of injection the input signal into the feedback loop through the grounded nodes.
 A component with a ground node can be disconnected from the ground and connected to the input source.
 The filter transmission zeros depends on the components through which the input signal is injected.
Vo

Vi
 s( / C1 R4 )
 1
1  1
1
s 2  s   
 C1 C2  R3 C1C2 R3 R4
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11-27
Generation of Equivalent Feedback Loops
Va
t
Vb
Va
 1 t
Vc
Equivalent Loop
Characteristics Equation:
1  L( s)  0  1  At ( s)  0
Characteristics Equation:
1
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A
(1  t )  0  1  At ( s)  0
A 1
11-28
Generation of Equivalent Feedback Loops (Cont’d)
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11-29
11.9 SENSITIVITY
Filter Sensitivity
 Deviation in filter response due to the tolerances in component values
 Especially for RC component values and amplifier gain
Classical Sensitivity Function
 Definition:
y / y
x / x
y x
S xy 
x y
S xy  lim
x 0
 For small changes:
S xy 
y / y
x / x
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11-30
11.10 SWITCHED-CAPACITOR FILTERS
Basic Principle
 A capacitor switched between two nodes at a sufficiently high rate is equivalent to a resistor
 The resistor in the active-RC integrator can be replaced by the capacitor and the switches
 Equivalent resistor:
iav 
v
T
C1vi
 Req  i  c
iav C1
Tc
 Equivalent time constant for the integrator = Tc(C2/C1)
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11-31
Practical Circuits
 Can realize both inverting and non-inverting integrator
 Insensitive to stray capacitances
 Noninverting switched-capacitor (SC) integrator
 Inverting switched-capacitor (SC) integrator
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11-32
Filter Implementation
 Circuit parameters for the two integrators with the same time constant
Tc
C
C2
 Tc 1  C1  C2  C  C3  C4  KC
C4
C3
0 
1
1

C1C2 R3 R4 TC
C3 C4 K

C2 C1 Tc
Q
C
R5 C4

 C5  0Tc
Q
R4 C5
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11.11 TUNED AMPLIFIERS
Characteristics of a Tuned Amplifier
 Center frequency: 0
 3-dB bandwidth: B
 Skirt selectivity: S
(the ratio of the 30-dB bandwidth to the 3-dB bandwidth).
 The 3-dB bandwidth is less than 5% of 0 in many applications
 narrow-band characteristics.
 The circuits studied are small-signal voltage amplifiers in which
transistors operate in class A mode.
The Basic Principle for Single-Tuned Amplifiers
 Parallel RLC circuit is used as the load for an amplifier.
 The amplifier exhibits a second-order bandpass characteristic.
 g mVi
 g mVi

YL
sC  1 / R  1 / sL
Vo
gm
s

2
Vi
C s  s (1 / RC )  1 / LC
Vo 
0  1 / LC
B  1 / RC
Q  0 / B  0 RC
Vo ( j0 )
  gm R
Vi ( j0 )
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Inductor Losses
Q
L
rs
 Q0 
Y ( j 0 ) 
0 L
rs
1
1
1
1 1  j (1 / Q0 )


rs  j0 L jL 1  j (1 / Q0 ) jL 1  (1 / Q0 ) 2
1 
1 
1
1
1  j  
For Q0  1  Y ( j0 ) 

j0 L 
Q0  j0 L 0 LQ0
Q0  0 L / rs  1
RP  0 L  / rs
2
RP  0 LQ0  0 L  / rs
2





Power loss in a practical inductor is represented by a series resistance rs.
Rather than specifying the value of rs, a inductor is specified by the Q factor at the frequency at interest.
It is desirable to replace the series connection of L and rs with an equivalent parallel connection of L and RP.
RP is expressed in terms of rs, L and the frequency of interest 0 under the assumption that Q0 >>1.
In fact, the value of resulting RP varies with frequency, however, it is approximated as a constant resistance
at frequencies around 0 to simplify the analysis.
 High inductor Q factor means low inductor loss
 rs as small as possible for a given L.
 RP as large as possible for a given L.
NTUEE Electronics – L. H. Lu
11-35
The Use of Transformers
 In many cases, the required values of inductance and capacitance are not practical
 use a transformer to effect an impedance change.
 use a tapped coil (autotransformer).
n2
n=n2 /n1
C’=C /n2
R
L
C
n1
R
L
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L’=n2L
11-36
Amplifiers with Multiple Tuned Circuits
 Multiple tuned circuits are used if high selectivity is required  tuned circuit at both input and output.
 Radio-frequency choke (RFC) is frequently used for bias circuit
 RFC behaves as a short-circuit at dc  provide dc biasing.
 RFC exhibit high impedance at frequency of interest  eliminate the loading effect of dc bias circuit.
 The Miller capacitance C may results in many problems in the multiple tuned circuit
 The Miller impedance at the input is complex since the load is not simply resistive.
 The reflected impedance will cause detuning of the input circuit.
 Skewing of the response of the input circuit.
 May results in circuit oscillation.
 Using additional feedback circuits to provide
current cancellation can neutralize the effect of C .
 An alternative approach is to change the circuit
configuration to avoid Miller effect.
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11-37
The Cascode and CC-CB Cascade
 Two amplifier configuration do not suffer the Miller Effect and are suitable for multiple tuned circuits:
 Cascode configuration
 CC-CB cascade configuration
 The CC-CB cascade is usually preferred in IC implementations because of its differential structure.
NTUEE Electronics – L. H. Lu
11-38
Synchronous Tuning
 Assume the stages do not interact  the overall response is the product of the individual responses.
 Synchronous tuning  cascading N identical resonant circuits.
 The 3-dB bandwidth B of the overall amplifier is related to that of the individual tuned circuit (0 /Q):
B
0
Q
21/ N  1
 Bandwidth-shrinkage factor: 21/ N  1
 Bandwidth decreased by the factor of 21/ N  1
 Q factor increased by the factor of 21/ N  1
 Given B and N, we can determine the bandwidth required of the individual stages.
NTUEE Electronics – L. H. Lu
11-39
Stagger-Tuning
 Stagger-tuned amplifiers exhibit overall response with maximal
flatness around the center frequency f0.
 Such a response can be obtained by transforming the response of
a Butterworth low-pass filter to 0.
 The transfer function of a second-order bandpass filter can be
expressed in terms of its poles as:
T (s) 
a1s



1
1 
 s  0  j0 1 
 s  0  j0 1 
2

4Q 2
4Q 
2Q
2Q

j




+ j0
 For a narrow-band filter Q >> 1, and for values of s in the neighborhood of +j0
the transfer function is approximated as (narrow-band approximation):
T ( s) 
a1
a1 / 2
a1 / 2


s  0 / 2Q  j0 2 j0  s  0 / 2Q  j0 s  j0   0 / 2Q
(s  p2) ≈ 2j0

 j0
 The response of the second-order bandpass filter in the neighborhood of its center frequency s = j0 is
identical to a first-order low-pass filter with a pole at (0/2Q) in the neighborhood of p = 0.
 The transformation p = s  j0 can be applied to low-pass filters of order greater than one.
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11-40
 The second-order narrow-band bandpass filter:
s = p + j0
s = p + j0
 The fourth-order stagger-tuned narrow-band bandpass amplifier:
s = p + j0
s = p + j0
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11-41