Microwave Filter

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




17

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



21

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



26

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



28

2. Filter design

Design process

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



29

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



31

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



32

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



34

Frequency scaling for HPF

Basic equation

cPP LRLR )(

LjCjjB

CjLjjX

c

c

1

1

LC

CL

c

c

1

1

2. Filter design



35

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



36

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



37

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



38

Richard’s transformation

Transformation

Input impedance: stub

pv

ll

tan)tan(

)tan( ljLLjjX

)tan( ljCCjjB

2. Filter design



39

LC to stubs

2. Filter design

)tan( ljLLjjX

)tan( ljCCjjB



40

Stub characteristics

Resonance: wavelength/8 related to the cutoff frequency

Attenuation pole: wavelength/4

Period: wavelength/2

)tan(1 l

2. Filter design



41

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



43

Equivalent transmission line

2. Filter design

Series to shunt stub transform: microstrip line Implementation: realization



44

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

Microwave Filter

Technology

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