Transmission Line Based Microwave Resonator

Dr. Chinmoy Saha

Indian Institute of Space Science and Technology

Transmission Line Based Microwave Resonator

Dr. Chinmoy Saha

Indian Institute of Space Science and Technology

Series and Parallel Resonator Circuits

L

R C

Zin V

22 2*

2 2 2

2 2 2

2

1

1 1 1 1 1

2 2 2 2

1 1 1 1

2 2 21 1 1 1

2 2 2

2 ( )

in

in in inin

loss m e

Z R j L jC

VP VI Z I Z I R j L j

Z C

I R I j L I jC

I R j I L IC

P j W W

2 2 2 2

2

1 1 1 1 1, ,

2 4 4 4loss e m cP I R W I L W V C IC

2 2

02

2 ( )

2 2 ( )

/ 2

Resonance occurs when .

1 ,

/ 2

in loss m e

in loss m ein

m e

lossin

P P j W W

P P j W WZ

I I

W W

PZ R

LCI

00 0

0

2 20

in 2

2 1 : Quality factor

1Z 1

m e m

loss loss

W W W LQ

P P R RC

R j L R j LLC

0

0

0 0

22

A resonator with loss can be treated as a lossless resonator

whose resonant frequency has been replaced with a complex

effective resonant frequency:

12

The half-po

in

RQZ R j L R j

j

Q

0

wer fractional bandwidth: 2 .

1

BW

BWQ

of valuesmallfor 2

)2())(( 0020

2

Parallel Resonant Circuit

)(2

1

4

1

4

1 W,

4

1 W,

2

1

1

2

1

1

2

1

2

1

2

1

11

2

22

m

2

e

2

2

*

22*

1

emlossin

Lloss

ininin

in

WWjPPL

VLICVR

VP

CjL

j

RV

ZVIZVIP

CjLjR

Z

ENEE 482 Spring 2001 6Q

BW

Q

j

jQ

R

RCj

R

CjR

CjL

j

R

CjCjLjR

Z

RCL

R

P

WQ

I

WWjP

I

PZ

in

loss

memlossinin

1 ,

2

R Zsuch that

frequnciesat occur bandwidth power -half The ,2

1

/2121

211

/11

small is where, Let

2 ,

2/

)(22

22

in

00

0

11

02

1

00

0

0

00

022

Transmission Line : Equivalent Circuit

Transmission Line : Input Impedance

20

0 0 20

0 00

0 0

00

0

00

0

( ) 1 (2.43)

( ) 1

( ) ( )

( ) ( )

cos sin

cos sin

tan

tan

j l j l j l

in j l j l j l

j l j lL L

j l j lL L

L

L

L

L

VV l e e eZ Z Z

I l V e e e

Z Z e Z Z eZ

Z Z e Z Z e

Z l jZ lZ

Z l jZ l

Z jZ lZ

Z jZ l

(2.44)

Lossless Transmission Line

Special Cases of Terminated Transmission LinesMatched Line

Matched Line

Special Cases of Terminated Transmission Lines

Short-Circuited Line ZL = 0 Γ= -1

0 0

0 0

0 0

( ) 2 sin ,

( ) 2 cos

j z j z

j z j z

V z V e e jV z

V VI z e e z

Z Z

0 tan inZ jZ l

13

Impedance

Short-circuited transmission Line

Voltage

Current

Open-Circuited Line

0 0

0 0

0 0

( ) 2 cos ,

2( ) sin (2.46)

j z j z

j z j z

V z V e e V z

V jVI z e e z

Z Z

0 cot inZ jZ l

ZL = ∞ Γ= 1

Impedance

Voltage

Current

Open-Circuited transmission Line

l = λ/2, Zin = ZL

If the line is a quarter-wavelength long, or,

l = λ/4+ nλ/2 (n = 0,1,2,3…),

Zin = Z02/ZL

Terminated transmission lines with special lengths

Quarter-Wave Transformer

Terminated transmission lines with special lengths

Problem

Equivalent circuit Short-Circuited transmission Line

Consider Losses!!

‘R’ will be there

Equivalent circuitOpen-Circuited transmission Line

Consider Losses!!

‘R’ will be there

Transmission Line Resonators

Short-circuited Transmission Line

For Series Resonance Zin

Comparing

Quality Factor

Short-circuited Transmission Line … Contd

Short-circuited Transmission Line

For Parallel Resonance

Comparing

Quality Factor

Short-circuited Transmission Line … Contd

Open-Circuited Transmission Line

Comparing

Open-Circuited Transmission Line

Quality Factor

A Half-Wave Microstrip Resonator

Dielectric Resonators

Coupling

Modes

a

d

Ground plane

DRA

a

d

Ground plane

DRA

Dielectric Resonator Antenna

?