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Microwave Filter

Microwave EngineeringMicrowave EngineeringCHO, Yong HeuiCHO, Yong Heui

Microwave Microwave EngineeringEngineering

EM Wave EM Wave LabLab

2

Circuit Resonator



3

Applications

1. LC resonator

Filter Oscillator Frequency meter Tuned amplifier



4

LC resonator: ideal resonator

Input impedance

Input power

Resonant frequency: Wm = We

C

jLjZ

in

C

jLjIIZVIP

22

in*

in 2

1

2

1

2

1

LC

1

1. LC resonator



5

Series resonator

R, L, C Input impedance

Input power

Resonant frequency

C

jLjRZ

in

C

jLjRIIZVIP

22

in*

in 2

1

2

1

2

1

LC

1

1. LC resonator



6

Quality factor

Definition

3 dB bandwidth

Q in terms of R, L, C

dloss/seconEnergy

stored energe AverageQ

BW0fQ

RCR

L

P

WQ

l

m

0

00

12

1. LC resonator



7

Perturbation

Input impedance

LjRLjRZ 2

2

20

2

in

1. LC resonator



8

Parallel resonator

R, L, C Input admittance

Input power

Resonant frequency

CjL

j

RY

1in

Cj

L

j

RVVYVIP

1

2

1

2

1

2

1 22*in

*in

LC

1

1. LC resonator



9

Quality factor

Q in terms of R, L, CRC

L

R

P

WQ

l

m0

00

2

1. LC resonator



10

Perturbation

Input admittance

Cj

RCj

RY 2

112

20

2

in

1. LC resonator



11

Loaded Q

Unloaded Q: resonant circuit itself Loaded Q: External load resistor

QQQ eL

111

1. LC resonator



12

Short-circuited half-wave line

2. Tx line resonator

Transmission line Input impedance: lossy medium

)tanh()tan(1

)tan()tanh(

tanh

0

0in

llj

ljlZ

ljZZ



13

Approximation

Low-loss transmission line

Phase:

ll )tanh(

2/ ,0 l

00

)tan()tan(

l




14

Equivalence

Input impedance

Quality factor

jLR

jlZlj

jlZZ

2

)/(1

)/(

00

0

00in

220

lR

LQ




15

Open-circuited half-wave line

Transmission line Input impedance: lossy medium

)tan()tanh(

)tanh()tan(1

coth

0

0in

ljl

lljZ

ljZZ




16

Approximation

Low-loss transmission line

Phase:

ll )tanh(

2/ ,0 l

00

)tan()tan(

l




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Equivalence

Input impedance

Quality factor

jCR

jl

Z

jl

ljZZ

2/1

1

)/()/(

)/(1

0

0

0

00in

220 l

RCQ




18

Rectangular waveguide

3. Waveguide cavity

Metallic wall Propagation constant

Resonant condition

22

2

b

n

a

mkmn

ldmn



19

Resonant wavenumber

Resonant wavenumber

TE101 mode and TM110 mode Q of cavity

222

d

l

b

n

a

mkmnl

l

e

P

WQ

20

3. Waveguide cavity



20

Circular waveguide

Metallic wall Propagation constant Resonant condition

TE111 mode and TM110 mode

ldmn

3. Waveguide cavity



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Dielectric material

4. Dielectric cavity

High Q Fringing field High permittivity: magnetic wall Mechanical tuning TE01 mode Notation

1/2 gL



22

5. Mirror

Fabry-Perot resonator

Two mirrors High Q Laser Millimeter and optical applications



23

Microwave Filter



24

Characteristics

1. Filter

2 port network: S parameters Pass band and stop band Return loss and insertion loss Ripple and selectivity (skirt) Pole and zero Group delay



25

Characteristics

1. Filter

Phase response Signal distortion



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Classification

LPF (Low Pass Filter) HPF (High Pass Filter) BPF (Band Pass Filter) BSF (Band Stop Filter): notch filter

1. Filter



27

Filter response

Maximally flat (Butterworth) filter Chebyshev filter Elliptic function filter Bessel function filter

1. Filter



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2. Filter design

Design process

Filter specifications Design of low pass filter Scaling and conversion Design of transmission line Implementation



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Insertion loss method

Precise design method Power loss ratio: transducer gain

Reflection coefficient

Results:

2LR)(1

1

load todeliveredPower

source from availablePower

P

)()(

)()(

22

22

NM

M

)(

)(1

2

2

LR

N

MP

2. Filter design



30

Filter responses: LPF

Maximally flat response

Equal ripple response

Chebyshev polynomial

N

c

kP2

2LR 1

cNTkP

22

LR 1

)cos()(cos nTN

2. Filter design



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Example

Design 2-poles low pass filter in terms of the insertion loss method where

1,1 Sc Z

4LR 1 P

2in )(1

)1(

CZ

CZjZLjZ

L

LL

2. Filter design



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Impedance scaling

LL

s

ZZZ

ZZ

Z

CC

LZL

0

0

0

0

2in )(1

)1(

CZ

CZjZLjZ

L

LL

R

LQ 0

LC

10

Series RLC resonator

Example

2. Filter design



33

Frequency scaling for LPF

Basic equation

c

PP LRLR )(

CjCjjB

LjLjjX

c

c

c

c

CC

LL

2. Filter design



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Frequency scaling for HPF

Basic equation

cPP LRLR )(

LjCjjB

CjLjjX

c

c

1

1

LC

CL

c

c

1

1

2. Filter design



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Frequency scaling for BPF

Basic equation:

12

00

0LRLR Q ,)(

QPP

22

0

0

11

0

0

1

1

LjCjCjQjB

CjLjLjQjX

210

2. Filter design



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Frequency scaling for BSF

Basic equation:

12

0

1

0

0LRLR Q ,

1)(

QPP

1

22

1

0

0

1

11

1

0

0

1

1

LC

jCQ

jjB

CL

jLQ

jjX

210

2. Filter design



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Example

Design 5-poles low pass filter with a cutoff frequency of 2 [GHz], impedance = 50 [Ohms], insertion loss = 15 dB at 3 [GHz]

618.0

618.1

2

618.1

618.0

5

4

3

2

1

g

g

g

g

g

Maximally flat response

2. Filter design



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Richard’s transformation

Transformation

Input impedance: stub

pv

ll

tan)tan(

)tan( ljLLjjX

)tan( ljCCjjB

2. Filter design



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LC to stubs

2. Filter design

)tan( ljLLjjX

)tan( ljCCjjB



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Stub characteristics

Resonance: wavelength/8 related to the cutoff frequency

Attenuation pole: wavelength/4

Period: wavelength/2

)tan(1 l

2. Filter design



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Kuroda’s identity

Stub transformation: shunt and series stub

Series to shunt stub transform: microstrip line

Implementation

1

21Z

ZN

2. Filter design



42

Kuroda’s identity

2. Filter design

Stub transformation



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Equivalent transmission line

2. Filter design

Series to shunt stub transform: microstrip line Implementation: realization



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Materials

3. Implementation

Microstrip line Dielectric resonator Waveguide Semiconductor MEMS (Micro ElectroMechanical System) LTCC (Low Temperature Cofired Ceramic) SAW (Surface Acoustic Wave) FBAR (Film Bulk Acoustic Resonator) Superconductor