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

11-12

First-Order Filters (Cont’d)

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

11-14

Second-Order Filters (Cont’d)

11-15

Second-Order Filters (Cont’d)

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

11-20

Filter Realization

Low-Pass Filter High-Pass Filter

Bandpass Filter Notch Filter

11-21

LPN Filter HPN Filter

All-Pass Filter

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:

11-29

Generation of Equivalent Feedback Loops (Cont’d)

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

11-41

The second-order narrow-band bandpass filter:

The fourth-order stagger-tuned narrow-band bandpass amplifier:

s = p + j0 s = p + j0

s = p + j0 s = p + j0