Design of RF and Microwave Filters -...

Design of RF and Microwave Filters

서강대학교 전자공학과

윤상원 교수

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

J,K inverters

Microwave & Millimeter-wave Lab. 2

1. Immitance Inverters

K ; impedance inverter →

J ; admittance inverter →

ex. simplest form of inverter : λ/4 transformer series LC → J-inverter + shunt LC shunt LC → K-inverter + series LC

K(or J)

immittance

inverter

ZL(or Y

L)Z

in(or Y

in)

Fig. 10 Immitance inverter

Lin ZKZ /2

Lin YJY /2


2. Bandpass filters using J-, K-inverters

g0 R

n+1

Lossless

Lowpass

NetworkZin() or low

gn+1

R0

Lossless Bandpass Network

Zin'() or

band

Fig. 11 Equivalent Network for lowpass prototype and bandpass network

Reflection coefficient ;

lowpass :

bandpass :

If (mapping relation)

1/)'(

1/)'(

0

0

gZ

gZ

in

in

Low

1/)(

1/)(

0

0

RZ

RZ

in

in

Band

)()'(/)(/)'( 00 BandLowinin RZgZ


...

...

R0

g1 gn+1

gn

gn-

1g5

g4

g3

a

a'

g2

Z'in(')

Fig. 12 Lowpass and bandpass network


1

3

2

1

1

1

1

1

1

n

n

in

gjgj

gj

gj

gj

Y

1

43

32

21

1

1

1

1

1

1

nn

in

ggj

ggj

ggj

ggj

gY

Input admittance of LPF Prototype


j

jj

rjL

CLjB

2

21

0

in in

A

Y Y

G G

B

nn

rn

r

r

r

in

G

JjB

JjB

JjB

JjB

JY

2

1,

2

34

3

2

23

2

2

12

1

2

01

B

rnnn

rr

rr

rr

r

in

G

BJj

BBJj

BBJj

BBJj

BJY

/

/

/

/

/

2

1,

43

2

34

32

2

23

21

2

12

1

2

01

1 101

0 1 0 1

A r AG B G bW

Jg g g g

1 1

, 1 2

11

rj rj j j

j j

j jj j

B B b bJ W

g gg g

, 1

1 0 1

B rn B nn n

n n

G B G bWJ

g g g g

bj=ω0Cr

Input admittance of BPF using J-Inverter


From the partial fraction expansion including bandpass mapping relation

: fractional bandwidth, : center frequency

In the same manner, K-inverter values are derived as

1

01

1,

1

1

2

0

1,

10

100

01 , ,

nn

nn

nn

ii

ii

iigg

LWRK

gg

LLWK

gg

LWRK

1

01

1,

1

1

2

0

1,

10

100

01 , ,

nn

nn

nn

ii

ii

iigg

CWGJ

gg

CCWJ

gg

CWGJ

W 0


B=positiveY

0

/2 /2

=negative

Y0

B=negative

/2 /2

=positive

Fig. 13 Admittance(J-) inverters

CJ LJ /1

-C

C

-C -L

L

-L


Fig. 14 Impedance(K-) inverters

CK /1 LK

-L

L

-L

X=negative

=positive

Z0

-C

C

-C

X=positive

=negative

Z0


3. LC filters

A. C-coupled bandpass filters

...

...

Y0

Yb

CnL

nL

2L1 L

4

C1

C2

J01

J12

Jn,n+1

Fig. 14 Bandpass filter network using ideal J-inverters

...

...

Y0

Yb

CnL

nL

2L1

C1

C2

J01

Jn,n+1

J-inverter

-C12

C12

Fig. 15 Bandpass filter network containing practical inverters


Ya

L1

C1-C

a'

J01 Y

aL

1C

1

Yin1

Yin2

C01

Ca'

Fig. 16 Inverters of first and last stages

2

01

01

2

01

2

01

2

01

2

2

01

1

/1/1

/

/1/1

1

'

aa

a

a

in

a

a

in

YC

Cj

YC

YC

CjYY

CjY

JY

By equating the real and imaginary part of and 1inY 2inY

aa YCifCJCC 01010101 ,'


B. L-coupled bandpass filter

......

Zb

Cn+1

C1 C

3C

2

Cp1

CpnC

p2

Lp1 L

p2L

pn

Za

Fig.17 C-coupled bandpass filter

......

Zb

Ln+1

L1 L

3L

2

Cp1

CpnC

p2

Lp1 L

p2L

pn

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 1 : from the element values of lowpass prototype

step 2 : apply impedance scaling

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 pF

9.5 nH air-coil 9.5 nH air-coil 6.8 nH air-coil6.8

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-coupling LC filter

L-coupling LC filter

+ =


27 dB

Design of RF and Microwave Filters -...

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