Microwave Filter

(1)

Microwave Filter

Microwave Engineering

CHO, Yong Heui

(2)

Circuit Resonator

(3)

EM Wave Lab

3

Applications

1. LC resonator

 Filter

 Oscillator

 Frequency meter

 Tuned amplifier



(4)

 Input impedance

 Input power

 Resonant frequency: Wm = We



C L j

j

Z

_in

   

 

 



 



 C

L j j

I I

Z VI

P

_in ^* _in ² ²

 

2 1 2

1 2

1  1



(5)

EM Wave Lab

5

Series resonator

 R, L, C

 Input power

 Resonant frequency



C L j

j R

Z

_in

    

 

 



  



 C

L j j

R I

I Z

VI

P

_in ^* _in ² ²

 

2 1 2

1 2

1 LC

 1



1. LC resonator

(6)

 Definition

 3 dB bandwidth

 Q in terms of R, L, C



d loss/secon Energy

stored energe

Average

 Q 

BW f

0

Q 

(7)

EM Wave Lab

7

Perturbation



 



     



 



 



 R j L R j L

Z

₂

2

2 0 2

in

1. LC resonator

(8)

 R, L, C

 Input admittance

 Input power

 Resonant frequency



C L j

j

Y R 

 ^



 1

in

 

 



  



 j C

L j V R

V Y

VI

P 

 1

2 1 2

1 2

1

_* ² ²

in

* in

(9)

EM Wave Lab

9

Quality factor

 Q in terms of R, L, C



L RC R

P Q W

l m

0 0

0

2 

   



1. LC resonator

(10)

 Input admittance



 



     



 



 



 j C

C R R j

Y 1 1 2

2 2 0 2

in

(11)

EM Wave Lab

11

Loaded Q

 Unloaded Q: resonant circuit itself

 Loaded Q: External load resistor



Q Q

Q

_L _e

1 1

1  

1. LC resonator

(12)

 Transmission line

 Input impedance: lossy medium



 

) tanh(

) tan(

1 ) tan(

) tanh(

tanh

0

0 in

l l

j

l j

Z l

l j

Z Z













 





(13)

EM Wave Lab

13

Approximation

 Low-loss transmission line

 Phase:



l l 

 )  tanh(

2 /

0

 , 



    l 

0 0

) tan(

)

tan( 





 

 l     ^

2. Tx line resonator

(14)

 Quality factor





 



















 

 



 



 





 

jL R

j l

l Z j

j Z l

Z

2 ) /

( 1

) /

(

0 0

0 0 0

in











2 2

0

 

 R l

Q L

(15)

EM Wave Lab

15

Open-circuited half-wave line

 Transmission line

 Input impedance: lossy medium



 

) tan(

) tanh(

) tan(

1 coth

0

0 in

l j

l

l l

Z j

l j

Z Z











 





(16)

 Low-loss transmission line

 Phase:



l l 

 )  tanh(

2 /

0

 , 



    l 

0 0

) tan(

)

tan( 





 

 l     ^

(17)

EM Wave Lab

17

Equivalence

 Quality factor





















 



 







 

jC R

j l

Z j

l

l Z j

Z

2 /

1

1 ) /

( )

/ (

) /

( 1

0 0

0 0 0

in







 

2

0

 2 

 RC l

Q

(18)

 Metallic wall

 Propagation constant

 Resonant condition



2 2

2





 



 

 

 



 

 b

n a

k m

mn



 





_mn

d  l

(19)

EM Wave Lab

19

Resonant wavenumber

 Resonant wavenumber

 TE₁₀₁ mode and TM₁₁₀ mode

 Q of cavity



2 2 2

 

 



 

 

 



 

 

 



 

d l b

n a

k

_mnl

m   

l e

P Q 2 W



0



3. Waveguide cavity

(20)

 Metallic wall

 Propagation constant

 Resonant condition

 TE₁₁₁ mode and TM₁₁₀ mode







_mn

d  l

(21)

EM Wave Lab

21

Dielectric material

4. Dielectric cavity

 High Q

 Fringing field

 High permittivity: magnetic wall

 Mechanical tuning

 TE_01dmode

 Notation



1 /

2 

 L 

_g

d

(22)

 Two mirrors

 High Q

 Laser

 Millimeter and optical applications



(23)

EM Wave Lab

23

Microwave Filter

(24)

 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)

EM Wave Lab

25

Characteristics

1. Filter

 Phase response

 Signal distortion



(26)

 LPF (Low Pass Filter)

 HPF (High Pass Filter)

 BPF (Band Pass Filter)

 BSF (Band Stop Filter): notch filter



(27)

EM Wave Lab

27

Filter response

 Maximally flat (Butterworth) filter

 Chebyshev filter

 Elliptic function filter

 Bessel function filter



1. Filter

(28)

 Filter specifications

 Design of low pass filter

 Scaling and conversion

 Design of transmission line

 Implementation



(29)

EM Wave Lab

29

Insertion loss method

 Precise design method

 Power loss ratio: transducer gain

 Reflection coefficient

 Results:



LR 2

) ( 1

1 load

to delivered

Power

source from

available Power





 

 P

) (

) ) (

(

₂ ₂

2 2



 

N M

M

 



) (

)

1 (

₂

2

LR



 N

P   M

2. Filter design

(30)

 Maximally flat response

 Equal ripple response

 Chebyshev polynomial

N

c

k P

2 2

LR

1 



 



 

 



 

 



 



c

T

N

k

P 

2



2

LR

1 

(31)

EM Wave Lab

31

Example

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

 

_c

 1 , Z

_S

 1

4 LR

 1   P

in 2

) (

1 ) 1

(

C Z

C Z j

L Z j

Z

L L

L



 



 



(32)



L L

s

Z Z Z

Z Z

Z C C

L Z L

0 0 0 0

 

_in ₂

) (

1 ) 1

(

C Z

C Z j

L Z j

Z

L L

L



 



 



R Q _ 

⁰

L

 1



Example

(33)

EM Wave Lab

33

Frequency scaling for LPF

 Basic equation



 

 



 



c

P

P 



_LR



LR

( )

C j

jB

L j

jX

c c

 



 



 



 



c c

C C L L



 



(34)

 Basic equation



 

 

 

 



 P 

^c

P

_LR

( )

_LR

C L j

j jX

c c







 







1 1

L C



c

1 1

 



(35)

EM Wave Lab

35

Frequency scaling for BPF

 Basic equation:



1 2

0 0

0 LR

LR

( ) , Q



 

 

 





 



 



 



 

  P Q

P

2 2

0 0

1 1

0 0

1 1

L C j

j C

jQ jB

C L j

j L

jQ jX

 

 

 

 



 



 

 

 

 



 



 



 



 

2 1

0

 

 

(36)

 Basic equation:



1 2

0 1

0 0

LR

1 , Q

)

(  



 

 

 





 





 

 



 

 



P Q P

1 1

0

1











 

 



  

 

 

 



 

 C

j L Q L

jX j



 



 

2 1

0

 

 

(37)

EM Wave Lab

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)

 Transformation

 Input impedance: stub



 





 



 





v

p

l  l

 ) tan tan(

) tan( l jL

L j

jX    

) tan( l jC

C j

jB    

(39)

EM Wave Lab

39

LC to stubs

) tan( l jL

L j

jX    

) tan( l jC

C j

jB    



(40)

 Resonance: wavelength/8 related to the cutoff frequency

 Attenuation pole: wavelength/4

 Period: wavelength/2



) tan(

1   l





(41)

EM Wave Lab

41

Kuroda’s identity

 Stub transformation: shunt and series stub

 Series to shunt stub transform: microstrip line

 Implementation



1

2

Z N   Z

(42)



(43)

EM Wave Lab

43

Equivalent transmission line

 Series to shunt stub transform: microstrip line

 Implementation: realization



(44)

 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

