1

ENE 428Microwave Engineering

Lecture 11 Excitation of Waveguides and Microwave Resonator

2

Excitation of WGs-Aperture coupling

WGs can be coupled through small apertures such as for directional couplers and power dividers

(a)

wg1

wg2

coupling aperture

feed wg cavity

(b)

coupling aperture microstrip1

microstrip2

Ground planeer

er er

wg stripline

(c) (d)

3

A small aperture can be represented as an infinitesimal electric and/or magnetic dipole.

Both fields can be represented by their respective polarization currents.

The term ‘small’ implies small relative to an electrical wavelength.

Fig 4.30

4

Electric and magnetic polarization

Aperture shape

e m

Round hole

Rectangular slot

(H across slot)

0 0 0 0ˆ ( ) ( ) ( ),e e nP nE x x y y z ze ))))))))))))))

0 0 0( ) ( ) ( ).m tmP H x x y y z z ))))))))))))))))))))))))))))

e is the electric polarizability of the aperture.m is the magnetic polarizability of the aperture.(x0, y0, z0) are the coordinates of the center of the aperture.

302

3

r 304

3

r

2

16

ld 2

16

ld

5

From Maxwell’s equations, we have

Thus since and has the same role as and , we can define equivalent currents as

and

Electric and magnetic polarization can be related to electric and magnetic current sources, respectively

0 0

0

m

e

E j B M j H j P M

H j D J j E j P J

))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

e

M))))))))))))))

J))))))))))))))

0 mj P))))))))))))))

ej P))))))))))))))

eJ j P))))))))))))))))))))))))))))

0 mM j P))))))))))))))))))))))))))))

6

Coupling through an aperture in the broad wall of a wg (1)

Assume that the TE10 mode is incident from z < 0 in the lower guide and the fields coupled to the upper guide will be computed.

y

xaa/20

b

2b

1 2

34

z

y

7

Coupling through an aperture in the broad wall of a wg (2) The incident fields can be written as

The excitation field a the center of the aperture at x = a/2, y = b, z = 0 can be calculated.

10

sin ,

sin .

j zy

j zx

xE A e

aA x

H eZ a

10

,

.

y

x

E A

AH

Z

8

Coupling through an aperture in the broad wall of a wg (3) The equivalent electric and magnetic dipoles for coupling to the fields in the upper guide are

Note that we have excited both an electric and a magnetic dipole.

0

0

10

( ) ( ) ( ),2

( ) ( ) ( ).2

y e

mx

aJ j A x y b z

j A aM x y b z

Z

e

0 0 0( ) ( ) ( ).m tmP H x x y y z z ))))))))))))))))))))))))))))

0 0 0 0( ) ( ) ( ),e ne

P nE x x y y z z ))))))))))))))

e

9

Coupling through an aperture in the broad wall of a wg (4) Let the fields in the upper guide be expressed as

where A+, A- are the unknown amplitudes of the forward and backward traveling waves in the upper guide, respectively.

10

10

sin , 0,

sin , 0,

sin , 0,

sin , 0,

j zy

j zx

j zy

j zx

xE A e for z

a

A xH e for z

Z a

xE A e for z

a

A xH e for z

Z a

10

Coupling through an aperture in the broad wall of a wg (5) By superposition, the total fields in the upper guide due to the electric and magnetic currents can be found forthe forward waves as

and for the backward waves as

where

00 2

10 10 10

1( ) ( ),m

Vn y y x x ej A

A E J H M dvP P Z

e

00 2

10 10 10

1( ) ( ),m

Vn y y x x ej A

A E J H M dvP P Z

e

1010

.ab

PZ

11

Microwave Resonator A resonator is a device or system that exhibitsresonance or resonant behavior, that is, it naturallyนoscillates at some frequencies , called its resonant frequency , with greater amplitude than at others.

Resonators are used to either generate waves of sp ecific frequencies or to select specific frequencies fro

m a signal.

The operation of microwave resonators is very similar to that of the lumped-element resonators of circuit theory.

12

Basic characteristics of series RLC resonant circuits (1)

The input impedance is

The complex power delivered to the resonator is

AC

R L

CZin

I

1.inZ R j L j

C

21 1 1( ).

2 2inP VI I R j L jC

13

Basic characteristics of series RLC resonant circuits (2) The power dissipated by the resistor, R, is

The average magnetic energy stored in the inductor, L, is

The average electric energy stored in the capacitor, C, is

Resonance occurs when the average stored magnetic and electric energies are equal, thus

21.

2lossP I R

21.

4mW I L

2 2

2

1 1 1.

4 4e cW V C IC

2.

12

lossin

PZ R

I

14

The quality factor, Q, is a measure of the loss of a resonant circuit. At resonance,

Lower loss implies a higher Q

the behavior of the input impedance near its resonant frequency can be shown as

01

LC

0

2.in

RQZ R j

15

A series resonator with loss can be modeled as a lossless resonator 0 is replaced with a complex effective resonant frequency.

Then Zin can be shown as

This useful procedure is applied for low loss resonators by adding the loss effect to the lossless input impedance.

0 0 1 .2

j

Q

02 ( ).inZ j L

16

Basic characteristics of parallel RLC resonant circuits (1)

The input impedance is

The complex power delivered to the resonator is

AC RL CZin

I

11 1

.inZ j CR j L

21 1 1( ).

2 2inj

P VI V j CR L

17

Basic characteristics of parallel RLC resonant circuits (2) The power dissipated by the resistor, R, is

The average magnetic energy stored in the inductor, L, is

The average electric energy stored in the capacitor, C, is

Resonance occurs when the average stored magnetic and electric energies are equal, thus

21

.2loss

VP

R

2 2

2

1 1 1.

4 4m LW I L VL

21.

4eW V C

2.

12

lossin

PZ R

I

18

The quality factor, Q, of the parallel resonant circuit At resonance,

Q increases as R increases

the behavior of the input impedance near its resonant frequency can be shown as

01

LC

0

.1 2 1 2 /in

R RZ

j RC jQ

19

A parallel resonator with loss can be modeled as a lossless resonator. 0 is replaced with a complex effective resonant frequency.

Then Zin can be shown as

0 0 1 .2

j

Q

0

1.

2 ( )inZj C

20

Loaded and unloaded Q

An unloaded Q is a characteristic of the resonant circuit itself.

A loaded quality factor QL is a characteristic of the resonant circuit coupled with other circuitry.

The effective resistance is the combination of R and the load resistor RL.

RL

Resonant circuit Q

21

The external quality factor, Qe, is defined.

Then the loaded Q can be expressed as

0

0

eL

Lfor series circuits

RQ

Rfor parallel circuits

L

1 1 1.

L eQ Q Q

22

Transmission line resonators: Short-circuited /2 line (1)

The input impedance is

0 0

tanh tantanh( ) .

1 tan tanhin

l j lZ Z j l Z

j l l

ZinZ0,,

l

23

Transmission line resonators: Short-circuited /2 line (2) For a small loss TL, we can assume l << 1 so tanl l. Now let = 0+ , where is small. Then, assume a TEM line,

For = 0, we have

or

0 .l

p p p

l ll

v v v

00 0

0 0

( / )( )

1 ( / )in

l jZ Z Z l j

j l

2 .inZ R j L

24

Transmission line resonators: Short-circuited /2 line (3) This resonator resonates for = 0 (l = /2) and its input impedance is

Resonance occurs for l = n/2, n = 1, 2, 3, …

The Q of this resonator can be found as

0 .inZ R Z l

0 .2 2

LQ

R l

25

Transmission line resonators: Short-circuited /4 line (1) The input impedance is

Assume tanhl l for small loss, it gives

This result is of the same form as the impedance of a parallel RLC circuit

0

1 tanh cot.

tanh cotin

j l lZ Z

l j l

0 00

0

0

/ 2 ).

/ 2 )( )

2

in

l j l ZZ Z

l jl j

1.

12

inZj C

R

26

Transmission line resonators: Short-circuited /4 line (2) This resonator resonates for = 0 (l = /4) and its input impedance is

The Q of this resonator can be found as

0 .inZ

Z Rl

0 .4 2

Q RCl