5 Design of RF and Microwave Filters [호환 모드]home.sogang.ac.kr/sites/eemic/lecture/note02/Lists... · PTPSP PPSP 2 21 2 2 11 2 == =G= () 1 1 1 2w w D N P P P tran in LR =+-G

Design of RF and Microwave Filters

초고주파공학

(교재*의 5장)

서강대학교 전자공학과윤상원 교수

* “RF Circuit Design: Theory and Applications”, R. Ludwig & P. Bretchko

차 례

1. Introduction ; types of Filters --------------------------------------2. Characterization of Filters ------------------------------------------3. Approximate Design Methods --------------------------------------4. Lowpass Prototype Network ---------------------------------------5.1. Impedance Scaling ------------------------------------------------5.2. frequency Expansion ----------------------------------------------5.3. Lowpass to highpass transformation -----------------------------5.4. Lowpass to bandpass transformation ----------------------------5.5. Lowpass to bandstop transformation -----------------------------5.6. Immitance Inverters ----------------------------------------------5.7. Bandpass filters using J-, K- inverters ---------------------------6.1. LC filters -----------------------------------------------------------6.2. Distribute filters ---------------------------------------------------

34815181921222527283340

Microwave & Millimeter-wave Lab. 2

1. Introduction ; types of Filters --------------------------------------2. Characterization of Filters ------------------------------------------3. Approximate Design Methods --------------------------------------4. Lowpass Prototype Network ---------------------------------------5.1. Impedance Scaling ------------------------------------------------5.2. frequency Expansion ----------------------------------------------5.3. Lowpass to highpass transformation -----------------------------5.4. Lowpass to bandpass transformation ----------------------------5.5. Lowpass to bandstop transformation -----------------------------5.6. Immitance Inverters ----------------------------------------------5.7. Bandpass filters using J-, K- inverters ---------------------------6.1. LC filters -----------------------------------------------------------6.2. Distribute filters ---------------------------------------------------

34815181921222527283340

1. Introduction

Types of Filters

A. Lowpass Filters B. Highpass Filters

C. Bandpass Filters D. Bandstop Filters

attenuation

passband transitionband

stopband

freq

attenuation

passbandtransitionband

stopband

freqcutoffwc ; cutoff


Types of Filters

A. Lowpass Filters B. Highpass Filters

C. Bandpass Filters D. Bandstop Filters

attenuation

passband transitionband

stopband

freq

attenuation

passbandtransitionband

stopband

freqcutoffwc ; cutoff

atten

pass-band

transitionband

stop-band

freq

atten

pass-band

transitionband

stop-band

freqf1

stop-band

transitionband

f2

pass-band

transitionband

f1 f2

2. Filter Characterization

n Two-port Network ;

H(w)Input Output )()()( wqww jeHH =


H(w)Input Output

Fig. 1 Two-port Network

)()()( wqww jeHH =

1

Freq.

lH(w)l

q(w)

Characteristics of ideal bandpass filters ;

îíì

><££

=21

21

, 0 1

)(fffffor

fffforH w dand wtwq -=)(


Fig. 2 Characteristics of

ideal bandpass filter

1

Freq.

lH(w)l

q(w)

→ not realizable→ approximation required

n Practical specifications ;

1) Passband

; lower cutoff frequency - upper cutoff frequency

2) Insertion loss

; must be as small as possible

3) Return Loss

; degree of impedance matching

4) Ripple

; variation of insertion loss within the passband

2f

)( )(log20 dBH w1f


1) Passband


2) Insertion loss


3) Return Loss


4) Ripple


)( 1log20 dBr

5) Group delay

; time to required to pass the filter

6) Skirt frequency characteristics

; depends on the system specifications

7) Power handling capability

wwqt

dd

d)(

-=


5) Group delay





3. Approximate Design Methods

1) based on Amplitude characteristics

A. Image parameter method B. Insertion loss method

a) J-K inverters

b) Unit element - Kuroda identity

2) based on Linear Phase characteristics


1) based on Amplitude characteristics

A. Image parameter method B. Insertion loss method

a) J-K inverters

b) Unit element - Kuroda identity

2) based on Linear Phase characteristics

3.1 Filter Design based on the insertion loss

Definition of Power Loss Ratio (PLR) ; impedance matching as well as frequency selectivity

[Sij]Pin

Prefl

Ptrans

Fig. 3 General filter network


[Sij]Pin

Prefl

Ptrans

Fig. 3 General filter network

← network synthesis procedures are required

inintrans

ininrefl

PSPTP

PSPP2

212

211

2

==

=G=

)()(1

11

2 ww

DN

PP

Ptran

inLR +=

G-=º

Approximation methods :

1) Maximally Flat (Butterworth) response

2) Chebyshev response

3) Elliptic Function response


Approximation methods :

1) Maximally Flat (Butterworth) response

2) Chebyshev response

3) Elliptic Function response

A. Maximally flat response

11.50.5 10 w/w c

PLR

Chebyshev

Maximally flat

Where, ;passband tolerance

; order of filter

Usually

→ degree of freedom=1 (order N)

N

cLR kP

221 ÷÷

ø

öççè

æ+=

ww

3.2 Approximation Methods


11.50.5 10 w/w c

PLR

Chebyshev

Maximally flat

Where, ;passband tolerance

; order of filter

Usually

→ degree of freedom=1 (order N)

Fig. 4 Comparison Between Maximally Flat and Chebyshev response

2kN

12 =k

B. Chebyshev response : equal ripple response in the passband

: Chebyshev Polynomial of order÷÷ø

öççè

æ+=

0

221ww

NLR TkPNT N


)()(2)(34)( ,12 ,)(

21

33

221

xTxxTxTxxxTxTxxT

nnn -- -=-=-==

; ripple (0.01 dB, 0.1 dB, etc.)

; order of filter

→ degree of freedom=2 (ripple and order)

2kN

10 w/wc

PLR

Chebyshev Response, N=4-1

1+k2

ws w

as

ar

wp

Elliptic function response N=5

attenuation


10 w/wc

PLR

Chebyshev Response, N=4-1

1+k2

ws w

as

ar

wp

Elliptic function response N=5

attenuation

Fig. 5 Chebyshev and Elliptic Function response

C. Elliptic Function responseequal ripple passband in both passband and stopband

: stopband minimum attenuation : transmission zero at stopband

degree of freedom=3 (order N, ripple, transmission zero at stopband )

sa

sw


C. Elliptic Function responseequal ripple passband in both passband and stopband

: stopband minimum attenuation : transmission zero at stopband

degree of freedom=3 (order N, ripple, transmission zero at stopband )sw

4. Lowpass Prototype Filter

; normalized to 1

...

...

R gN

g0=1g1

g2

g3g5

g4g6a

a'

sradgRL / 1 , 1 c0 =W== w


...

...

R gN

g0=1g1

g2

g3g5

g4g6a

a'

...

...

R

gNg0=1

g1

g2

g3g5

g4g6

a

a'

g7

Fig. 5 Lowpass prototype

n Maximally Flat response ;

n Equal Ripple response ;

11 02 ==®+= gRP LN

LR w

),( , ... 2, 1, ,2

12sin2 FHNiNigi =-

= p


n Maximally Flat response ;

n Equal Ripple response ;

ïî

ïíì

+-+==®+=

even 1212

odd 1,1)(1

22022

Nkkk

NRgTkP LNLR w

÷÷ø

öççè

æ

++

-+==

-==

--

-

1111ln ,

2sinh ,

212sin ,

42

22

11

1

kk

Nb

Nia

gbaa

g iiii

iii bbp

TypeElement No

Butterworth0.1 dB rippleChebyshev

0.5 dB rippleChebyshev

1 0.6180 1.1468 1.7058

Table1. Element values for Butterworth and chebyshev filters (n=5)


2 1.6180 1.3712 1.2296

3 2.0000 1.9750 2.5408

4 1.6180 1.3712 1.2296

5 0.6180 1.1468 1.7058

5. Impedance Scaling and Frequency Mapping

5.1 Impedance Scaling

Impedance level × 50; same reflection coefficient maintained

series branch (impedance) elements ;

shunt branch (admittance) elements ;

W=®W= 50 1 LL RR


5.1 Impedance Scaling

Impedance level × 50; same reflection coefficient maintained

series branch (impedance) elements ;


( )iiii gggjgj 5050 ®® ww

( )50/50/ rrrr gggjgj ®® ww

5.2 Frequency Expansion

cutoff frequency 1 → lowpass cutoff frequency

mapping function ;series and shunt branch elements ;

cw

( )iciici gggjgj wwww ®®

www cf =)(


( )iciici gggjgj wwww ®®

PLR

w'1-1

PLR

wwc-wc

PLR

wc-wc

w

PLR

w-w1

-w0-w2w0w1 w2

(a) Lowpass Prototype response

(d) Lowpass to Bandpass Transformation

(b) Frequency expansion

(c) Lowpass to Highpass transformation


PLR

w'1-1

PLR

wwc-wc

PLR

wc-wc

w

PLR

w-w1

-w0-w2w0w1 w2

(a) Lowpass Prototype response

(d) Lowpass to Bandpass Transformation

(b) Frequency expansion

(c) Lowpass to Highpass transformation

Fig. 6 Various mapping relations derived from lowpass prototype network

5.3 Lowpass to Highpass transformation

(lowpass cutoff freq. 1 → highpass cutoff freq. ) mapping function ; series branch (impedance) elements ;


cwwww /)( cf -=

( ))/(1 )/( iciici gggjgj wwww ®-®


...

...

R

gN' RL=1

g1'g3'g5'

g4' g2'

( ))/(1 )/( rcrrcr gggjgj wwww ®-®

Fig. 7 Highpass filter derived from lowpass prototype

5.4 Lowpass to bandpass transformation

(low cutoff freq. , high cutoff freq. )

mapping function ;

1w 2w

÷÷ø

öççè

æ-

-=

ww

ww

www

w 0

012

0)(f

12210

21

0

and

,1'0'

wwwwww

wwwwwww

-=D=®

±±=®±=±=®=


12210

21

0

and

,1'0'

wwwwww

wwwwwww

-=D=®

±±=®±=±=®=

n series branch element : impedance

n shunt branch element : admittance

ss

iiii Cj

Ljjggjgjggj

ww

www

ww

ww

ww

www 1 ;

200

0

01 +=

D+

D®÷÷

ø

öççè

æ-

D=

pp

rrrrr Lj

Cjjggjgjggj

ww

www

ww

ww

ww

ww

w 1 ; 200

0

0 +=D

+D

®÷÷ø

öççè

æ-

D=


n series branch element : impedance

n shunt branch element : admittance

pp

rrrrr Lj

Cjjggjgjggj

ww

www

ww

ww

ww

ww

w 1 ; 200

0

0 +=D

+D

®÷÷ø

öççè

æ-

D=

...

...

R

CN RL=1

C1L1L3L5

C4L4

C5

C2

C3

L2LN

Fig. 8 Bandpass filter derived from the lowpass prototype

Example : Design a bandpass filter having a 0.5dB equal-ripple response, with N=3. The f0 is 1GHz, bandwidth is 10%, and the input and output impedance 50Ω.

step 1 : from the element values of lowpass prototype

(0.5dB ripple Chebyshev)

step 2 : apply impedance scaling

step 3 : apply bandpass transformation

0000.1 ,5963.1 ,0967.1 ,5963.1 4321 ==== gggg


Example : Design a bandpass filter having a 0.5dB equal-ripple response, with N=3. The f0 is 1GHz, bandwidth is 10%, and the input and output impedance 50Ω.


(0.5dB ripple Chebyshev)


step 3 : apply bandpass transformation

HZgLFZgCHZgL 815.79 , 022.0/ , 815.79505963.1 031022011 =====´==

012

022

202

3101

3011

/)( 91.34/' 726.0/'

' 199.0/'' 127/'

wwww

www

-=D=D==D=

==D===D=

pFCCnHCL

CpFLCLnHLL R=50 W

RL=50 W

L3'=127nH

C3'=0.199pF

L2'=0.726nH C2'=34.91pF

L1'=127nH

C1'=0.199pF

5.5 Lowpass to bandstop transformation

(low cutoff freq. , high cutoff freq. )

mapping function ;

inverse of bandpass mapping function

1w 2w1

0

00

12)(-

÷÷ø

öççè

æ-

-=

ww

ww

www

wf


12210

21

0

and

,1'0'

wwwwww

wwwwwww

-=D=®

±±=®±=±=®=

n series branch element : admittance

n shunt branch element : impedance

ss

iiii Lj

Cjjgg

jgjggjw

www

ww

www

ww

www 1 ;

20

-1

0

001 +=

D+

D®÷÷

ø

öççè

æ-

D=

pp

rrrrr Cj

Ljjggjgjggj

ww

www

ww

ww

ww

www 1 ;

20

-1

0

00

+=D

+D

®÷÷ø

öççè

æ-

D=


pp

rrrrr Cj

Ljjggjgjggj

ww

www

ww

ww

ww

www 1 ;

20

-1

0

00

+=D

+D

®÷÷ø

öççè

æ-

D=

Fig. 9 Bandstop network derived from the lowpass prototype

...

...

R

CN

RL=1C1

L1L3L5 C4

L4

C5

C2

C3

L2LN

5.6 Immitance Inverters

n K ; impedance inverter →

n J ; admittance inverter →

K(or J)

immittanceinverter

ZL(or YL)Zin(or Yin)

Fig. 10 Immitance inverter

Lin ZKZ /2=


n K ; impedance inverter →

n J ; admittance inverter →

ex. simplest form of inverter : λ/4 transformer

series LC → J-inverter + shunt LC shunt LC → K-inverter + series LC

Lin ZKZ /2=

Lin YJY /2=

5.7 Bandpass filters using J-, K-inverters

g0 Rn+1

LosslessLowpassNetwork

Zin(w) or Glow

gn+1 R0

Lossless Bandpass Network

Zin'(w) orGband

Fig. 11 Equivalent Network for lowpass prototype and bandpass network

Reflection coefficient ;

lowpass :

bandpass :

If (mapping relation)

G


Reflection coefficient ;

lowpass :

bandpass :

If (mapping relation)

G

1/)'(1/)'(

0

0

+-

=GgZgZ

in

inLow w

w

1/)(1/)(

0

0

+-

=GRZRZ

in

inBand w

w

)()'(/)(/)'( 00 wwww BandLowinin RZgZ G=G®=

...

...

R0

g1 gn+1

gn

gn-

1

g5

g4

g3

a

a'

g2

Zin(w')


...

...

R0

Rn+1

CnLnL2L1

L4

C1 C2

K01 K12 Kn,n+1

Zin(w)

Fig. 12 Lowpass network and bandpass network

n From the partial fraction expansion including bandpass mapping relation

: fractional bandwidth, : center frequency

n In the same manner, J-inverter values are derived as

1

011,

1

12

01,

10

10001 , ,

+

++

+

++ ===

nn

nnnn

ii

iiii gg

LWRK

ggLL

WKggLWR

Kwww

W 0w


n From the partial fraction expansion including bandpass mapping relation

: fractional bandwidth, : center frequency

n In the same manner, J-inverter values are derived as

1

011,

1

12

01,

10

10001 , ,

+

++

+

++ ===

nn

nnnn

ii

iiii gg

CWGJ

ggCC

WJggCWG

Jwww

n Typical immittance inverters ;

CK w/1= LK w=

-L

L

-L-C

C

-C


Fig. 13 Impedance(K-) inverters

CK w/1= LK w=

X=negative

F

F=positive

Z0 X=positive

F

F=negative

Z0

CJ w= LJ w/1=

-C

C

-C -L

L

-L


B=positiveY0

F/2 F/2

F=negative

Y0

B=negative

F/2 F/2

F=positive

Fig. 14 Admittance(J-) inverters

6.1. LC filters

A. C-coupled bandpass filters

...

...

Y0

Yb

CnLnL2L1 L4

C1C2J01 J12 Jn,n+1

Fig. 14 Bandpass filter network using ideal J-inverters


Fig. 14 Bandpass filter network using ideal J-inverters

...

...

Y0

Yb

CnLnL2L1C1

C2J01 Jn,n+1

J-inverter

-C12

C12

Fig. 15 Bandpass filter network containing practical inverters

YaL1

C1-Ca'J01 Ya L1C1

Yin1Yin2

C01

Ca'

Fig. 16 Inverter of first and last stages

( ) ( )201

012

01

201

2

012

201

1

/1/1/

/1/11

'

aa

a

ain

aa

in

YCCj

YCYC

CjYY

CjYJ

Y

ww

ww

w

w

++

+=

+=

+=


( ) ( )201

012

01

201

2

012

201

1

/1/1/

/1/11

'

aa

a

ain

aa

in

YCCj

YCYC

CjYY

CjYJ

Y

ww

ww

w

w

++

+=

+=

+=

By equating the real and imaginary part of and1inY 2inY

aa YCifCJCC <<== 01010101 ,' ww

B. L-coupled bandpass filter

......

Zb

Cn+1C1 C3C2

Cp1CpnCp2

Lp1 Lp2 Lpn

Za

Fig.17 C-coupled Bandpass filter


B. L-coupled bandpass filter

......

Zb

Ln+1L1 L3L2

Cp1CpnCp2

Lp1 Lp2 Lpn

Za

Fig.18 L-coupled Bandpass filter

Example : Design a LC bandpass filter. The f0 is 2.8 GHz, bandwidth is 500 MHz, and the input and output impedance 50Ω.



step 3 : apply bandpass transformation using J-inverters

Step 4 : simulation


Step 5 : Realization

0.5 pF 0.5 pF

6.8

nH

air-coil

2.7

nH

chip 1 p

F1.5

pF

1 pF 0.5 pF

6.8

nH

air-coil

2.7

nH

chip 1 p

F1.5

pF

1 pF

6.8

nH

air-coil

2.7

nH

chip

0.5

pF

5 p

F

0.5 pF 1 pF

6.8

nH

air-coil

2.7

nH

chip

0.5

pF

5 p

F

1 pF


Insertion loss < 3.1 dB

Return loss > 15.5 dB

Attenuation @ 3.3 GHz : 15 dB

Step 6. improvement

20 pF9.5 nH air-coil 9.5 nH air-coil 6.8 nH air-coil

6.8

nH

air-coil

6.8

nH

air-coil

6.8

nH

air-coil

0.5

pF

1 p

F

1 p

F

1 p

F

1 pF 1.5 pF 0.5 pF 0.5 pF 0.5 pF 0.5 pF

6.8

nH

air-coil

6.8

nH

air-coil

6.8

nH

air-coil

2.7

nH

chip

2.7

nH

chip

2.7

nH

chip

0.5

pF 1

pF

1.5

pF


20 pF9.5 nH air-coil 9.5 nH air-coil 6.8 nH air-coil

6.8

nH

air-coil

6.8

nH

air-coil

6.8

nH

air-coil

0.5

pF

1 p

F

1 p

F

1 p

F

1 pF 1.5 pF 0.5 pF 0.5 pF 0.5 pF 0.5 pF

6.8

nH

air-coil

6.8

nH

air-coil

6.8

nH

air-coil

2.7

nH

chip

2.7

nH

chip

2.7

nH

chip

0.5

pF 1

pF

1.5

pF

C-couplingLC filter

L-couplingLC filter

+ =


27 dB

6.2 Distributed filters

At microwave frequencies :

Resonators made of Lumped elements are lossy(low Q) or bulky → Distributed Resonators

Distributed resonators ; quarter-wavelength or half-wavelength transmission lines such as microstrip lines, coaxial lines and waveguides


At microwave frequencies :

Resonators made of Lumped elements are lossy(low Q) or bulky → Distributed Resonators

Distributed resonators ; quarter-wavelength or half-wavelength transmission lines such as microstrip lines, coaxial lines and waveguides

A. Combline filters: cellular base stations as well as handy phone

Fig. 17(a) Top View of Combline Filter

conductor

air or ceramic

a


Fig. 17(a) Top View of Combline Filter

conductor

air or ceramic

a

Fig. 17(b) Side View of Combline Filter

conductor tuning screw

L

Fig. 19 (a) Top view of Combline Filter

Fig. 19 (b) Side view of Combline Filter

n Instead of lumped element inductors distributed inductors (L < λ/4) are used.

n Overall equivalent circuit :

In

out

Fig. 18 Coupled line

Yoe

Yoe

YooIn OutYoe Yoe

(Yoe-Yoo)/2

Fig. 20 Coupled line


n Instead of lumped element inductors distributed inductors (L < λ/4) are used.

n Overall equivalent circuit :

In OutYoe Yoe

(Yoe-Yoo)/2

L1 L2 L4L3

Lc1 Lc5Lc4Lc3Lc2

C1 C2 C3 C4

Cc1 Cc2 Cc3

Fig. 20 Coupled lineFig. 21 Equivalent circuit of Fig.20

Fig. 22 Equivalent circuit of Fig. 19

B. Microstrip filters: Compact, light weight and low cost

Fig. 21 Side-couple microstrip filter


Fig. 23 Side-coupled Microstrip filterFig. 21 Side-couple microstrip filter

n Practical specifications ;

1) Passband


2) Insertion loss


3) Return Loss


4) Ripple


2f

)( )(log20 dBH w1f


1) Passband


2) Insertion loss


3) Return Loss


4) Ripple


)( 1log20 dBr

5) Group delay





wwqt

dd

d)(

-=


5) Group delay





Documents

5 Design of RF and Microwave Filters [호환 모드]home.sogang.ac.kr/sites/eemic/lecture/note02/Lists... · PTPSP PPSP 2 21 2 2 11 2 == =G= () 1 1 1 2w w D N P P P tran in LR =+-G