Dr. Chinmoy Saha

Indian Institute of Space Science and Technology

Dr. Chinmoy Saha

Indian Institute of Space Science and Technology

Microwave Filter Design : Insertion Loss Method

Microwave Filter Design : Insertion Loss Method

IntroductionIntroduction

A filter is a network that provides perfect transmission for signal withfrequencies in certain passband region and infinite attenuation in the stopband-regions.Such ideal characteristics cannot be attained, and the goal of filter design isto approximate the ideal requirements to within an acceptable tolerance.

w

ww

1

2

V

VH =

=

w

w

1

21020

V

VLognAttenuatio

A FilterH(w)

V1(w) V2(w)

Transfer Function

Filters are used in all frequency ranges and are categorized intothree main groups:•Low-pass filter (LPF) that transmit all signals between DC andsome upper limit wc and attenuate all signals with frequenciesabove wc.

•High-pass filter (HPF) that pass all signal with frequenciesabove the cutoff value wc and reject signal with frequenciesbelow wc.

•Band-pass filter (BPF) that passes signal with frequencies inthe range of w1 to w2 and reject frequencies outside this range.The complement to band-pass filter is the band-reject or band-stop filter.

w

Attenuation/dB

0

wc

3

10

20

30

40

w

Attenuation/dB

0

w1

3

10

20

30

40 w2

Attenuation/dB

w

0

wc

3

10

20

30

40

ClassificationsClassifications

L1=g2 L2=g4

C1=g1 C2=g3RL= gN+1

1

L1=g1 L2=g3

C1=g2 C2=g4RL= gN+1g0= 1

L2L1

C1 CN

C2L2

L1C1 L3

C3

CNLN

high-pass filter

Low -pass filter

band-pass filter

ClassificationsClassifications

Filter can be further divided into active and passive type.For passive filter output power <= input powerActive filter provides power gain. Some amplifying Stage(Normally OPAMP) must be there.The characteristic of a passive filter can be described usingthe transfer function approach or the attenuation functionapproach.In low frequency circuit the transfer function (H(w))description is usedAt microwave frequency the attenuation function descriptionis preferred.At frequency below 1.0GHz, filters are usually implementedusing lumped elements such as resistors, inductors andcapacitors.

ClassificationsClassifications

Filter Characterization

Two-port Network

H(w)Input Output

Fig. 1 Two-port Network

)()()( www jeHH =

Filter Characterization

Fig. 2 Characteristics of

ideal bandpass filter

1

Freq.

lH(w)l

(w)

Characteristics of ideal bandpass filters ;

>

=

21

21

, 0

1)(

fffffor

fffforH w dand ww =)(

→ not realizable→ approximation required

Filter Characterization :Filter Characterization : Practical specificationsPractical specifications

Passband : Lower cutoff frequency f1 and upper cutoff frequency f2

Insertion loss: , must be as small as possible

Return Loss: , degree of impedance matching

Ripple: variation of insertion loss within the passband

Phase Delay and Group Delay: are measures of the amount of delay encountered by a signal as it passes through a network.

Group delay describes the delay of a packet of frequencies

Phase delay describes the delay for a single sinusoid.is the phase of H(w) in radians.

Skirt frequency characteristics :depends on the system specifications

Power handling capability

)( )(log20 dBH w

)( log20 dB

w

w

d

dd

)(=

w

w

)(=P

)(w

There are essentially two Synthesis Techniques at low-frequency : a) Image parameter Method (IPM)b) Insertion-Loss Method (ILM)

Filter Syntheses TechniquesFilter Syntheses Techniques

Image parameter Method (IPM) Provides a relatively simple filter design approachThe IPM approach divides a filter into a cascade of two-port networks, and attempt to come up with the schematic of each two-port, such that when combined, give the required frequency response. Arbitrary frequency response cannot be incorporated into the design

m=0.6 m=0.6m-derivedm<0.6

constantkT

2

1

2

1

Matchingsection

Matchingsection

High-fcutoff

Sharpcutoff

ZiTZ

iT ZiT

ZoZo

A filter response is defined by its insertion loss or power loss ratio (PLR)

Then a suitable filter schematic is synthesized.

Design procedure is based on the attenuation response or insertion loss of the filter.

Allows a high degree of control over the pass band and stop band amplitude and phase characteristics, with a systematic way to synthesize a desired response.

The necessary design trade-offs can be evaluated to best meet the application requirements.

Out of lot of Choices (Butterworth, Chebyshev, Elliptic Function, Linear Phase etc.) are there to the designers. Based on the design specification and constrain an optimum design is to be chosen.

Insertion-Loss Method (ILM) Insertion-Loss Method (ILM)

if a minimum insertion loss is most important, a binomial response can be used

if a sharp cutoff is needed, a Chebyshev response is better

in the insertion loss method a filter response is defined by its insertion loss or power loss ratio

Insertion Loss MethodInsertion Loss Method

It begins with a complete specification of a physically realizable frequency characteristic

Normally the design starts with a normalized low-pass prototype (LPP). The LPP is a low-pass filter with source and load resistance of 1W and cutoff frequency of 1 Radian/s.

Impedance transformation and frequency scaling are then applied to denormalize the LPP and synthesize different type of filters with different cutoff frequencies.

There are a number of standard approaches to design a normalized LPP that approximate an ideal low-pass filter response with cutoff frequency of unity.

well known methods are:Maximally flat or Butterworth function.Equal ripple or Chebyshev approach.Elliptic function.

Power Loss Ratio (PLR)

[Sij]

Pin

Prefl

Ptrans

is an even function of w , so

inintrans

ininrefl

PSPTP

PSPP

2

21

2

2

11

2

==

==

2)(1

1

w=

tran

inLR

P

PP

Filter Design :Insertion Loss MethodFilter Design :Insertion Loss Method

IL = 10 log PLR

)()(

)()(

22

22

ww

ww

NM

M

=

2)(w

← network synthesis procedures are required

)(

)(1

1

12 w

w

D

N

P

PP

tran

inLR =

=

M (w2) and N (w2) are real polynomials

Type of responses for n-section prototype filter

A. Maximally flat response (Butterworth Response)

The pass band extends from w = 0 to w = wc

At the band edge the power loss ratio is 1 + k2.

For this point to be -3 dB point, we have k = 1.

N

c

LR kP

2

21

=

w

w

Provides the flattest possible passband response for a given filter complexity, or order. For a low-pass filter, it is specified by

k2 :passband tolerance

N:order of filter

N

c

LR kP

2

2

w

wFor w >>wc

So the insertion loss increases at the rate of 20N dB/decade

21 k

For N = 3

B. Equal ripple response (Chebyshev Response)

Type of responses for n-section prototype filter

TN (x) Chebyshev Polynomial of order NOscillates between -1 to +1 for

)()(2)(

34)( ,12 ,)(

21

33

221

xTxxTxT

xxxTxTxxT

nnn =

===

1x

For N = 3

=

0

221w

wNLR TkP

For large x,So for w>>wc insertion loss

Nn xxT )2(

2

1)( =

N

c

LR

kP

22 2

4

w

w Insertion loss increases at the

rate of 20N dB/decade

Sharper cut-off compared to Butterworth Passband response contain ripples of amplitude 1+k2

For w>>wc, Insertion loss is times greater than Butterworth filter( ) /2 42N

C. Elliptic Function

Type of responses for n-section prototype filter

It has equal-ripple responses in both passband and stopband

Maximum attenuation in the passband is Amax

Minimum attenuation in the stopband is Amin

are difficult to synthesize

linear phase response is necessary to avoid signal distortion

there is usually a tradeoff between the sharp-cutoff response

and linear phase response

a linear phase characteristic can be achieved with the phase

response

=

N

c

pA

2

1)(w

www

D. Linear Phase Response

Type of responses for n-section prototype filter

The group delay is given by

this is also a maximally flat function, therefore, signal distortion is

reduced in the passband

==

N

c

d NpAd

d2

)12(1w

w

w

We start with designing the low pass filter prototypes which are

normalized in terms of impedance and frequency

The designed prototypes are then scaled in frequency and impedance

Lumped-elements will be replaced by distributive elements for

microwave frequency operations

Design Steps

Filter Specifications

Low PassPrototype Design

Scaling and Conversion

Implementation

Maximally Flat Low Pass Filter Prototype (Butterworth)

we will derive the normalized element values, L and C, for a maximally flat response. Assume : source impedance of 1 Ω

Cut off frequency wc = 1The desired power loss ratio (k =1) with N= 2 is

Consider the two-element (N=2) low-pass filter prototype shown

41 w=LRP

Input impedance2221

)1(

1 CR

RCjRLj

RCj

RLjZin

w

ww

ww

=

=

Reflection Co-efficient is given by

1

1

=

in

in

Z

Z

The power loss ratio is given by

Maximally Flat LPF Prototype (Butterworth)… Contd

This expression is a polynomial in ω2

Comparing to the desired response R =1, since PLR=1 for ω=0.

In addition, the coefficient of ω2 must vanish

coefficient of ω4 to be unity

Specification

Maximally Flat LPF Prototype (Butterworth)… Maximally Flat LPF Prototype (Butterworth)… ContdContd

41 w=LRP

This procedure can be extended to find the element values for higher N (?)Not practical for large value of N. The element values for the ladder-type circuits is tabulated . (for normalized low-pass design, source impedance is 1 and ωc =1 rad/sec)

N= 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , . . .N= 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , . . .

((bb))

((aa))

N= 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , . . . …N= 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , . . . …ContdContd

Element Values for Maximally Flat LPF PrototypesElement Values for Maximally Flat LPF Prototypes

(g0 =1, ωc =1, N =1 to 10)

G. L. Matthaei, L. Young, and E. M. T. Jones,Microwave Filters, Impedance-Matching Networks, and Coupling Structures, Artech House, Dedham, Mass., 1980

),( , 2, 1, ,2

12sin2 FHNi

N

igi =

=

Performance with higher NPerformance with higher N

Attenuation versus normalized frequency for maximally flat filter prototypes

Equal Ripple Low Pass Filter Prototype (Chebyshev)

For an equal-ripple low-pass filter with a cutoff frequency ωc =1 rad/sec, the power loss ratio

where 1+k2 is the ripple level in the passband.

Since the Chebyshev polynomials have the property that

At ω = 0

=

21

1

kPLR

=1

0)0(NT For N odd

For N even

For N oddFor N even

)(1 22 wNLR TkP =

There are two cases to consider, depending on N odd or even

Consider the two-element (N=2) low-pass filter prototype shown

Low pass prototype for Low pass prototype for N = N = 22

22222

2 )12(1)(1 == ww kTkPLR

)144(1 242 = wwk ()(

,12 22

TxxT

xT

=

Can be solved for R,L, and C if the ripple level (determined by k2) is specified

= )144(1 242 wwk

Equating

Equating at ω =0 R

Rk

4

12

2 =

Equating the coefficients ofω2 and ω4

2222

4

14 RCL

Rk =

)2(4

14 22222 LCRLCR

Rk =

can be used to find L and C

R is not unity leading to impedance mismatch if the load is unity.

Solutions : quarter wave transformerAdd an additional filter element to make N odd

22 1221 kkkR = (For N even)

= )144(1 242 wwk

Low pass prototype for Low pass prototype for N = N = 2 ….2 ….ContdContd

Element Values for EqualElement Values for Equal--Ripple LPF Prototypes (Ripple LPF Prototypes (gg0 0 = 1,= 1,ωωc c = 1, = 1, NN=1 to 10)=1 to 10)

0.5 dB ripple 1+k2 = 10^(0.5/10)=1.122

For odd N , gN+1=1, Impedance Matched

For even N, gN+1 = 1.9841

Element Values for EqualElement Values for Equal--Ripple LPF Prototypes (Ripple LPF Prototypes (gg0 0 = 1,= 1,ωωc c = 1, = 1, NN=1 to 10)=1 to 10)

3 dB ripple 1+k2 = 10^(3/10)=2

For odd N , gN+1=1, Impedance Matched

For even N, gN+1 = 5.8095

11

14

=ii

iii

gb

aag

N

i

Nbi

b 22 sin2

sinh =

=

11

11ln

2

2

k

kb

N

ag

2sinh

2 11 b

=

N

iai

2

12sin

=

Parameter ExtractionParameter Extraction

11010 =IL

k

0.5 dB Ripple Level

Attenuation Attenuation vsvs Normalized FrequencyNormalized Frequency

3 dB Ripple Level

Attenuation Attenuation vsvs Normalized FrequencyNormalized Frequency

Filters having a maximally flat time delay, or a linear phase response, can be designed inthe same way, but things are somewhat more complicated because the phase of the voltage transfer function is not as simply expressed as is its amplitude. Design values have

Linear Phase LowLinear Phase Low--Pass Filter PrototypesPass Filter Prototypes

been derived for such filters [1], however, again for the ladder circuits of Figure 8.25, and they are given in Table 8.5 for a normalized source impedance and cutoff frequency (ωc =1rad/sec). The resulting normalized group delay in the passband will be τd = 1/ωc =1 sec.

Filter TransformationFilter Transformation

For the LPF prototypes discussed so far we had Rs =1Ω and cutoff frequency ωc =1 rad/sec. So we need impedance and frequency scaling .Also, conversion mechanism to get high-pass, bandpass, or bandstop characteristics.

LL

S

RRR

RR

R

CC

LRL

0

0

0

0

=

=

=

=

Impedance Scaling

Illustration (Impedance level × 50)

same reflection coefficient maintained

Series branch(impedance) elements

Shunt branch(admittance) elements ;

W=W= 50 1 LL RR

iiii gggjgj 5050 ww

50/50/ rrrr gggjgj ww

Series Reactance

Filter Transformation: Filter Transformation: Frequency Scaling (LPF to LPF)

To change the cutoff frequency of a low-pass prototype from unity to ωc we replace ω by ω/ωc

kK

c

k LjLjjX == ww

w

cw

ww

Shunt Susceptance

kK

c

k CjCjjB == ww

w

So the new element values are

c

kk

CC

w=

c

kK

LL

w=

If both impedance and frequency scaling is required

c

kk

R

CC

w0

=

c

kK

LRL

w0=

Frequency scaling for low-pass filters

Series Reactance

Filter Transformation: Filter Transformation: (LPF to HPF)

k

Kc

kCj

LjjX

==ww

w 1

w

ww c

Shunt Susceptance

So the new element values are

kc

kC

Lw

1=

kc

kL

Cw

1=

If both impedance and frequency scaling is required

Transformation of a LPF into HPF

k

kc

kLj

CjjB

==ww

w 1

kc

kLR

Cw0

1=

kc

kC

RL

w0=

Low-pass prototypes can also be transformed to have the bandpass or bandstop responses.

BandpassBandpass and and BandstopBandstop TransformationsTransformations

LPFBPF:

Where ω1 and ω2 denote the edges of the passband,

LPF BPF BSF

Series reactance

Shunt Susceptance

LPF to BPFLPF to BPF

LPF to BSFLPF to BSF

LPFBSF:

=

=

w

w

w

w

w

w

w

w k

k

kk

L

L

jLjjX

0

0

0

0

So the series inductors of the low-pass prototype are converted to parallel LC circuits with

Similarly the shunt capacitors is changed to

Summary of Prototype Filter TransformationSummary of Prototype Filter Transformation

Design Problem 1:Design Problem 1: Calculate the inductance and capacitance values for a maximally-flat low-pass filter that has a 3dB bandwidth of 400MHz. The filter is to be connected to 50 ohm source and load impedance. The filter must has a high attenuation of 20 dB at 1 GHz.

=

n

kgk

2

12sin2

Step 1: Determine the order of the filter

c

A

Nww /log2

110log

110

10/10

=

=

N

c

A

2

11log10w

w

2

400/1000log2

110log

10

10/2010 >

=

Thus choose an integer value , i.e N=3

Step 2: Determine the element values (If not supplied)

1

32

12sin21 =

=

gg0 = g 3+1 = 1

2

32

122sin22 =

=

g

1

32

132sin23 =

=

g

Step 2: Determine the element values (If not supplied) ….

nHgR

LLc

o 9.19104002

1506

113 =

===

w

pFR

gC

co

9.1510400250

26

22 =

==

w

Step 3: Transformation for Scaling

15.9pF

19.9nH

50 ohm

50 ohm 19.9nH

Step 3: Circuit

nHgR

Lc

o 8.39104002

2506

22 =

==

w

pFR

gCC

co

95.710400250

16

113 =

===

w

7.95pF

39.8nH

50 ohm

50 ohm

7.95pF

Design Problem 2 :Design Problem 2 : Design a maximally flat low-pass filter with a cutoff frequency of 2 GHz, impedance of 50 Ω, and at least 15 dB insertion loss at 3 GHz.

=

n

kgk

2

12sin2

Step 1: Determine the order of the filter

c

A

Nww /log2

110log

110

10/10

=

=

N

c

A

2

11log10w

w

42/3log2

110log

10

10/1510 >

=

Thus choose an integer value , i.e N=5

Step 2: Determine the element values (If not supplied)

618.0

52

12sin21 =

=

gg0 = g 5+1 = 1

618.1

52

122sin22 =

=

g

2

52

132sin23 =

=

g

Step 2: Determine the element values (If not supplied) ….

618.1

52

142sin24 =

=

g

618.0

52

152sin25 =

=

g

pFR

gCC

co

984.0102250

618.09

115 =

===

w

nHgR

LLc

o 438.61022

618.1509

224 =

===

w

pFR

gC

co

183.3102250

29

33 =

==

w

Step 3: Transformation for Scaling

Step 3: Circuit

Alternatively you can go for other type .

Design Problem 3Design Problem 3Design a bandpass filter having a 0.5 dB equal-ripple response, with N=3. The center frequency is 1 GHz, the bandwidth is 10%, and the impedance is 50.

From TableImpedance and Frequency Transformation

BPF Circuit

Design Problem 3Design Problem 3

Design a 3 section Chebyshev low-pass filter that has a ripple of 0.05dB and cutoff frequency of 1 GHz.

11

14

=ii

iii

gb

aag

N

i

Nbi

b 22 sin2

sinh =

=

11

11ln

2

2

k

kb

N

ag

2sinh

2 11 b

=

N

iai

2

12sin

=

11010 =IL

k

Extraction EquationsExtraction Equations

k = 0.1076078715 b= -5.850582044 137134.12

sinh =N

b

29307.12

sinh 2 =N

b

879.0137134.1

2

12

1 =

=g 1132.14

11

212 ==

gb

aag

879.04

22

323 ==

gb

aag nHLL 7

102

8794.050931 =

==

pFC 543.3

10250

1132.192 =

=

48

Design Problem 4Design Problem 4Design a band-pass filter having a 0.5 dB ripple response, with N=3. The center frequency is 1GHz, the bandwidth is 10%, and the impedance is 50W.

Solution

Extract the Parameters

go=1 , g1=1.5963, g2=1.0967, g3= 1.5963, g4= 1.000

Let’s first and third elements are equivalent to series inductance and g1=g3, thus

nHgZ

LLo

oss 127

1021.0

5963.1509

131 =

=

W==

w

pFgZ

CCoo

ss 199.05963.150102

1.09

131 =

=

W==

w

kok gZL =

Continue

49

Second element is equivalent to parallel capacitance, thus

nHg

ZL

o

op 726.0

0967.1102

501.09

22 =

=

W=

w

pFZ

gC

oop 91.34

1021.050

0967.19

22 =

=

W=

w

o

kk

Z

gC =

50W 127nH 0.199pF

0.726nH 34.91pF

127nH 0.199pF

50W

MicrostripMicrostrip Implementation : Why ?Implementation : Why ?

The lumped-element filter sections generally work well at low frequencies only.At higher RF and microwave frequencies lumped-element inductors and capacitors are not available (Available only for a limited range of values), and can be difficult to implement at microwave frequencies. Distributed elements such as open-circuited or short-circuited transmission line stubs, are often used to approximate ideal lumped elements.At microwave frequencies the distances between filter components is not negligible. (Additional Challenge)

Lumped elements can be converted to transmission linesections with Richards’ transformation.

Kuroda’s identities is used to physically separate filterelements by using transmission line sections.

Such additional transmission line sections do not affect the filterresponse, this type of design is called redundant filter synthesis.

It is possible to design microwave filters that take advantage ofthese sections to improve the filter response.

MicrostripMicrostrip Implementation : HowImplementation : How

Implementation using stub

Richard’s transformation

==W

pv

lwb tantan

btanjCCjjBc =W=

At cutoff unity frequency we have Ω =1. Then

btan1 ==W8

l=

So inductor and capacitors of the lumped –element circuit are replaced by shorted and open stub of length l/8 at wc.

btanjLLjjX L =W=

Transformation from Ω plane to w plane

Reactance of the inductor is

Susceptance of the capacitor is

An inductor can be replaced with a short-circuited stub of length bl and characteristic impedance LA capacitor can be replaced with an open-circuited stub of length βl and characteristic impedance 1/C

Richard’s transformation : Summary

The four Kuroda identities use redundant transmission line sections to achievea more practical microwave filter implementation. Physically separate transmission line stubs.

For that additional transmission lines are addedTransform series stubs into shunt stubs, or vice versaChange impractical characteristic impedances into more realizable values

Kuroda’s IdentitiesKuroda’s Identities

The Four Kuroda Identities(n2=1+Z2/Z1)

It is difficult toimplement aseries stub inmicrostripline.Using Kurodaidentity, wewould be ableto transformS.C series stubto O.C shuntstub. So the 2nd

identity is mostcrucial.

Kuroda identity :Proof1st identity

ABCD of the shunt stub

W=

=

1

01

1

01

2. Z

jYDC

BA

StubSt

ABCD of the Tr. Line of length l stub

W

W

W

W=

1

1

1

11

01

1

1

2

2 Z

jZj

Z

jDC

BA

lbtan=Wwhere

So the combined ABCD (LHS)

W

W

W

W=

2

12

21

1

2 111

1

1

1

Z

Z

ZZj

Zj

W

W

W=

=

1

1

1

1cossin

sincos

1

1

2

1

1

Z

jZj

llZ

jljZl

DC

BA

TLbb

bb

22 cot

1

Z

j

ljZY

W==

b

(1)

Kuroda identity :Proof

ABCD of the series stub

W=

=

10

1

10

12

1

.

n

ZjZ

DC

BA

StubSer

ABCD of the Tr. Line of length l

W

W

W

W=

10

1

1

1

1

12

1

2

2

22

2 n

Zj

Z

njn

Zj

DC

BA

lbtan=W

where

So the combined ABCD (RHS)

WW

W

W=

2

12

2

2

212

21

)(1

1

1

Z

Z

Z

nj

ZZn

j

W

W

W=

1

1

1

1

2

2

22

2

Z

njn

Zj

DC

BA

TL

(2)

From (1) and (2), LHS and RHS are identical if

1

22 1Z

Zn =

Kuroda identity :Proof

Other Kuroda identities can be proved in a similar fashion

Second Kuroda’s Identity

Consider the second identity in which a series stub is converted into a shunt

l/8

l/8

Z1

Zo

Z1+Zo

Zo+Zo

2/Z1

SC

OC

From Table g1= 0.7654 g2 = 1.8478 g3=1.8478

g4 = 0.7654 g5 = 1

Design Problem 5 :Design a low-pass third-order maximally flat filter using only shunt stubs. The cutoff frequency is 8GHz and the impedance is 50 W.

LPF prototype

0.765 1.848

1/1.848

=0.541 1/0.765=1.307

Applying Richard’s transformation Adding Elements

Use second Kuroda identity on left,

Use first Kuroda identity on right433.0

307.2

1

307.1

1

)1307.1

11(

122 ==

=n

Z

567.0307.2

307.11

)1307.1

11(

121 ==

=n

Z

307.21)765.0

11(

765.1765.0)765.0

11(

22

12

==

==

Zn

Zn

1.765

1.848

0.541

0.433

11

2.307

0.567Use second Kuroda identity on left,

741.0567.0)848.1

567.01(

415.2848.1)848.1

567.01(

22

12

==

==

Zn

Zn

Use second Kuroda identity on left,

309.31)433.0

11(

433.1433.0)433.0

11(

22

12

==

==

Zn

Zn

1.765

0.541

11

2.307

2.415 1.433

0.741 3.309

Apply Kuroda’s identity…..

all lines are l/8 long at 8 GHz

88.3

27.1

5050

115.4

120.8 71.7

37.1 165.5

Impedance ScalingImpedance Scaling

All elements are multiplied by 50

Microstrip implementation

5088.3 120.8 71.7

50

115.4 27.1 37.1 165.5

ii/p/p o/po/p

er