design of rf and microwave filters

8/10/2019 Design of Rf and Microwave Filters

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RF and Microwave Filters


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Microwave & Millimeter-wave Lab. 2

1. Overview

1.1 Types of FiltersA. Lowpass Filters B. Highpass Filters

C. Bandpass Filters D. Bandstop Filters

attenuation

passband transition

bandstopband

freq

attenuation

passbandtransition

bandstopband

freq

cutoffc; cutoff

atten

pass-

band

transition

band

stop-

band

freq

atten

pass-

band

transition

band

stop-

band

freqf1

stop-

band

transition

band

f2

pass-

band

transition

band

f1

f2


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2. Filter Characterization(1)

Two-port Network ;

H()Input Output

Fig. 1 Two-port Network

)()()( jeHH


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Fig. 2 Characteristics of

ideal bandpass filter

1

Freq.

lH()l

()

Characteristics of ideal bandpass filters ;

21

21

,0

1)(

fffffor

fffforH dand )(

not realizableapproximation required


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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 matching4) Ripple

; variation of insertion loss within the passband

2f

)()(log20 dBH

1f

)(log20 dB


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5) Group delay

; time to required to pass the filter

6) Skirt frequency characteristics

; depends on the system specifications

7) Power handling capability

d

dd

)(


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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


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3.1 Filter Design(2)

Approximation methods :

1) Maximally Flat (Butterworth) response

2) Chebyshev response

3) Elliptic Function response


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3.2 Approximation Methods(2)

B. Chebyshev response: equal ripple response in the passband

: Chebyshev Polynomial of order

0

221

NLR TkP

NT N

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

21

3

3

2

21

xTxxTxTxxxTxTxxT

nnn


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10 c

PLR

Chebyshev Response, N=4-1

1+k2

s

s

p

Elliptic function response N=5

attenuation

Fig. 5 Chebyshev and Elliptic Function response

; ripple (0.01 dB, 0.1 dB, etc.); order of filter

degree of freedom=2 (ripple and order)

2

kN


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C. Elliptic Function responseequal ripple passband in both passband andstopband

: stopband minimum attenuation

: transmission zero at stopband

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

s

s

s


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4. Lowpass Prototype Filter

; normalized to 1

...

...

R gN

g0=1g1

g2

g3

g5

g4

g6a

a'

...

...

R

gN

g0=1

g1

g2

g3

g5

g4g6

a

a'

g7

Fig. 5 Lowpass prototype

sradgRL /1,1 c0


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4. Lowpass Prototype Filter(2)

Maximally Flat response ;

Equal Ripple response ;

11 02 gRP LN

LR

),(,2,1,,2

12sin2 FHNi

N

igi

even1212

odd1)(1

220

22

Nkkk

NgRTkP

LNLR

11

11ln,

2sinh,

2

12sin,

4

2

2

2

11

1

k

k

Nb

N

ia

gb

aag ii

ii

ii

i


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4. Lowpass Prototype Filter(3)

Type

Element NoButterworth

0.1 dB ripple

Chebyshev

0.5 dB ripple

Chebyshev

1 0.6180 1.1468 1.7058

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

Table1. Element values for Butterworth and chebyshev filters


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5. Impedance and freq. mapping

5.1 Impedance Scaling

Impedance level 50

; same reflection coefficient maintained

series branch(impedance) elements ;

shunt branch(admittance) elements ;

501 LL RR

iiii gggjgj 5050

50/50/ rrrr gggjgj


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5. Impedance and freq. mapping(2)

5.2 Frequency Expansion

cutoff frequency 1 lowpass cutoff frequency

mapping function ;

series and shunt branch elements ;

c

iciici gggjgj

cf )(


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PLR

'1-1

PLR

c-

c

PLR

c-c

PLR

1

-02

01 2

(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


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5.3 Lowpass to Highpass transformation

(lowpass cutoff freq. 1 highpass cutoff freq. )

mapping function ;

series branch(impedance) elements ;

shunt branch(admittance) elements ;

...

...

R

gN' RL=1

g1'g

3'g

5'

g4' g

2'

c

/)( cf

)/(1)/( iciici gggjgj

)/(1)/( rcrrcr gggjgj

Fig. 7 Highpass filter derived from lowpass prototype



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5.4 Lowpass to bandpass transformation

(low cutoff freq. , high cutoff freq. )

mapping function ;

1

2

0

012

0)(f

12210

21

0

and,1'

0'



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series branch element : impedance

shunt branch element : admittance

s

s

ii

iiCj

Ljj

ggjgjggj

1;

2

00

0

0

1

p

p

rr

rrrLj

Cjj

ggjgjggj

1

;200

0

0

...

...

R

CN

RL=1

C1

L1

L3

L5

C4

L4

C5

C2

C3

L2LN

Fig. 8 Bandpass filter derived from the lowpass prototype



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Example :Design a bandpass filter having a 0.5dBequal-ripple response, with N=3. The f0is 1GHz,bandwidth is 10%, and the input and outputimpedance 50.

step 1 : from the element values of lowpass prtotype(0.5dB ripple Chebyshev)

step 2 : apply impedance scaling

0000.1,5963.1,0967.1,5963.1 4321 gggg

HZgL

FZgCHZgL

815.79

,022.0/,815.79505963.1

031

022011



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5.5 Lowpass to bandstop transformation

(low cutoff freq. , high cutoff freq. )

mapping function ;

inverse of bandpass mapping function

1

2

1

0

00

12)(

f

12210

21

0

and

,1'

0'



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series branch element : admittance

shunt branch element : impedance

s

s

ii

iiLj

Cjj

ggjgjggj

1;

2

0

-1

0

00

1

p

p

rr

rrrCj

Ljj

ggjgjggj

1;

20

-1

0

00

Fig. 9 Bandstop network derived from the lowpass prototype

...

...

R

CNR

L=1

C1

L1L3L5 C4

L4

C5

C2

C3

L2

LN


design of rf and microwave filters

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