@10_Frequency Response Analysis and Filters
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Transcript of @10_Frequency Response Analysis and Filters
11-1
CHAPTER 11 FILTERS AND TUNED AMPLIFIERS
Chapter Outline11.1 Filter Transmission, Types and Specifications11.2 The Filter Transfer Function11.3 Butterworth and Chebyshev Filters11.4 First-Order and Second-Order Filter Functions11.5 The Second-Order LCR Resonator11.6 Second-Order Active Filters Based on Inductor Replacement11.7 Second-Order Active Filters Based on the Two-Integrator-Loop Topology11.8 Single-Amplifier Biquadratic Active Filters11.9 Sensitivity11.10 Switched-Capacitor Filters11.11 Tuned Amplifiers
11-2
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
11-3
Filter Specification Passband edge : p
Maximum allowed variation in passband transmission : Amax
Stopband edge : s
Minimum required stopband attenuation : Amin
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.
Low-Pass Filter Band-Pass Filter
11-4
11.2 FILTER TRANSFER FUNCTION
Transfer Function The filter transfer function is written as the ratio of two polynomials:
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:
Zeros: z1 , z2 , … , zM and ( NM ) 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.
01
1
01
1
......)(
bsbsasasasT N
NN
MM
MM
))...()(())...()(()(
21
21
N
MM
pspspszszszsasT
11-5
Transfer Function Examples Low-Pass Filter (with Stopband Ripple)
Low-Pass Filter (without Stopband Ripple)
Band-Pass Filter
012
23
34
45
22
221
24 ))(()(
bsbsbsbsbsssasT ll
0
p s
l1 l2
| T | (dB)
0
p s
| T | (dB)
0
p1
s1
| T | (dB)
p2s2l1 l2
j
zerospoles
p1
p2
l1
l2
l1p1
p2
l2
j
zerospoles
5
p
p
j
zerospoles
p
l1
l2
p
l1
l2
012
23
34
45)(
bsbsbsbsbsasT M
012
23
34
45
56
22
221
25 ))(()(
bsbsbsbsbsbssssasT ll
11-6
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:
Required transfer functions can be defined based on filter specifications (Amax , Amin , p , s)
Np
jT22 )/(1
1|)(|
11-7
Natural Modes of the Butterworth Filter The natural modes lines on a circle. The radius of the circle is Equal angle space
Np
/10 )/1(
N/
11-8
Design Procedure of the Butterworth Filters
Amax , Amin , p , s
Design Specifications
Design Procedure1. Determine (from Amax)
2. Determine the required filter order N (from p , s , Amin)
3. Determine the N natural modes
4. Determine T (s)
1101log20][1
1|)(| 10/2max2
max
Ap dBAjT
min2222 )/(1log10)/(1/1log20])[(n Attenuatio AdBA N
psN
pss
Np
N
N
pspspsKsT /1
021
0 )/1( e wher))...()((
)(
Nppp .... ,, 2 1
11-9
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 :
p
p
forN
jT
)]/(cos[cos1
1|)(|122 p
p
forN
jT
)]/(cosh[cosh1
1|)(|122
11-10
Design Procedure of the Chebyshev Filters
Amax , Amin , p , s
Design Specifications
Design Procedure1. Determine (from Amax)
2. Determine the required filter order N (from p , s , Amin)
3. Determine the N natural modes
4. Determine T (s)
1101log20)(1
1|)(| 10/2max2
max
Ap dBAjT
min122 ))/(cosh(cosh1log10)(n Attenuatio ANA pss
))...()((2)(
211
NN
Np
pspspsK
sT
1sinh1cosh
212cos1sinh1sinh
212sin 11
NNkj
NNkp ppk
11-11
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
0
01)(bsasasT
11-13
Second-Order Filters Biquadratic transfer function
Pole frequency: 0
Pole quality factor: Q Poles:
Bandwidth:
200
201
22
)/()(
sQs
asasasT
200
21 411
2,
Qj
Qpp
QBW 0
12
11-16
11.5 THE SECOND-ORDER LCR RESONATOR
The Resonator Natural Modes
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.
Parallel Resonator Current Excitation Voltage Excitation
LCsRCsCs
RsCsLYIV
i
o
/1)/1(/
/1/111
2
LCsRCsLC
sLsCRsCR
VV
i
o
/1)/1(/1
)/1||()/1||(
2
RCQ 0
LC/10 Current Excitation
Voltage Excitation
11-17
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
LCsRCsLC
VVsT
i
o
/1)/1(/1)( 2
LCsRCs
sVVsT
i
o
/1)/1()( 2
2
LCsRCs
sRCVVsT
i
o
/1)/1()/1()( 2
Low-Pass Filter High-Pass Filter Bandpass Filter
11-18
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
25314
253141
1
/
/
RRRRCL
sLRRRRsCIVZ
eq
eqin
11-19
The Op Amp-RC Resonator The inductor is replaced by the Antoniou circuit The pole frequency and the quality factor are given by
The pole frequency and quality factor for a simplified case where R1 = R2 = R3 = R5 = R and C4 = C
2531640 //1 RRRRCC531
2
4
66660 RRR
RCCRRCQ
RC/10 RRQ /6and
and
11-22
11.7 SECOND-ORDER ACTIVE FILTERS (TWO-INTEGRATOR-LOOP)
Derivation of the Two-Integrator-Loop Biquad
High-pass implementation:
Bandpass implementation:
Low-pass implementation:
ihphphp KVVs
VsQ
V )()(12
200
200
2
2
)/(
sQsKs
VV
i
hp
hphpihp Vs
VsQ
KVV 2
2001
)()/(
)/(200
200 sT
sQssK
VVs
bpi
hp
)()/(
)/(200
2
20
220 sT
sQsK
VVs
lpi
hp
11-23
Circuit Implementation (I)
High-pass transfer function:
Bandpass transfer function:
Low-pass transfer function:
Notch and all-pass transfer function:
QKQRRRRf /12 12/ 1/ 231
)())(1()1( 2
20
1
0
132
2
132
3hp
fhp
fi
fhp V
sRR
VsR
RRR
RVRR
RRRV
200322
2323
2
)]/(2[)]/(2[)(
RRRss
RRRsVV
sTi
hphp
200322
20323
)]/(2[)]/(2[)(
RRRssRRRs
VV
sTi
bpbp
200322
2
20323
)]/(2[)]/(2[)(
RRRssRRR
VV
sTi
bpbp
200322
2
200
2
32
3
)]/(2[)/()/()/(2)(
RRRssRRsRRsRR
RRR
VVsT LFBFHF
i
o
11-24
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.
222
22
31
12
11
111
)(
RCQCRss
RRCRRr
RCs
CCs
VVsT
i
o
11-25
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:
)(/)()()( sDsANsAtsL
)(/)()( sDsNst
0/1)(0)(1 AstsL P
The closed-loop characteristics equation:
The poles of the closed-loop system are identical to the zeros of the RC network.
11-26
RC Networks with complex transmission zeros
Characteristics Equation of the Filter
43210
1RRCC
4321413231
2
4321321
2
1111
1111
)(
RRCCRCRCRCss
RRCCRCCss
st
1
213
4321 11
CCRRRCC
Q
4321413242
2
4321421
2
1111
1111
)(
CCRRCRCRCRss
CCRRCRRss
st
4321321
220
02 1111RRCCRCC
ssQ
ss
Let C1 = C2 = C , R3 = R , R4= R/m 24Qm
0/2 QCR
11-27
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.
4321321
2
41
1111)/(
RRCCRCCss
RCsVV
i
o
11-28
Generation of Equivalent Feedback Loops
tVV
b
a tVV
c
a 1
Equivalent Loop
0)(10)(1 sAtsL 0)(10)1(1
1
sAttA
ACharacteristics Equation: Characteristics Equation:
11-30
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:
For small changes:yx
xyS
xxyyS
yx
x
yx
/
/lim0
xxyyS y
x //
11-31
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:
Equivalent time constant for the integrator = Tc(C2/C1)1C
TivR c
av
ieq
c
iav T
vCi 1
11-32
Practical Circuits Can realize both inverting and non-inverting integrator Insensitive to stray capacitances Noninverting switched-capacitor (SC) integrator
Inverting switched-capacitor (SC) integrator
11-33
Filter Implementation Circuit parameters for the two integrators with the same time constant
cC TK
CC
CC
TRRCC
1
4
2
3
43210
11
KCCCCCCCCT
CCT cc 4321
4
1
3
2
QCTC
CC
RRQ c05
5
4
4
5
11-34
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.
LCRCsss
Cg
VV m
i
o
/1)/1(2
sLRsCVg
YVgV im
L
imo /1/1
RCB /1
LC/10
RgjVjV
mi
o )()(
0
0
RCBQ 00 /
11-35
Inductor Losses
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.
20
0
000 )/1(1
)/1(11)/1(1
111)(QQj
LjQjLjLjrjY
s
ss rLQ
rLQ 0
0
sP rLLQR /2000
0000000
11111)(1For LQLjQ
jLj
jYQ
sP
s
rLR
rLQ
/
1/2
0
00
11-36
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).
R L CR
L
C’=C /n2
L’=n2L
n1
n2n=n2 /n1
11-37
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.
11-38
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.
11-39
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):
Bandwidth-shrinkage factor: Bandwidth decreased by the factor of Q factor increased by the factor of
Given B and N, we can determine the bandwidth required of the individual stages.
12 /10 N
QB
12 /1 N
12 /1 N
12 /1 N
11-40
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:
For a narrow-band filter Q >> 1, and for values of s in the neighborhood of +j0
the transfer function is approximated as (narrow-band approximation):
The response of the second-order bandpass filter in the neighborhood of its center frequency s = j0 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 j0 can be applied to low-pass filters of order greater than one.
200
200
1
411
2411
2
)(
Qj
Qs
Qj
Qs
sasT
Qjsa
jQsa
jjQsasT
2/2/
2/2/
22/)(
00
1
00
1
000
1
+ j0
j0
j
(s p2) ≈ 2j0