1

ENE 428

Microwave

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.

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.

2b

1 2

34

z

y y

xaa/20

b

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 z

y

j z

x

xE A e

a

A xH e

Z 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 neP nE x x y y z ze

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 z

y

j z

x

j z

y

j z

x

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 for

the forward waves as

and for the backward waves as

where Note that the electric dipole excites the same

fields in both directions but the magnetic dipole excites

oppositely polarized fields in forward and backward directions.

00 2

10 10 10

1( ) ( ),m

Vn y y x x e

j AA E J H M dv

P P Z

e

00 2

10 10 10

1( ) ( ),m

Vn y y x x e

j AA E J H M dv

P P Z

e

10

10

.ab

PZ

))2cos(1(2

1sin 2

11

(note: details of integration calculation from the previous slide….

dxdya

x

Za

xdszheP )sin(

1)sin(2ˆ2

10

101010

aa b

dxa

x

Z

bdydx

a

x

Z 010

0 0

2

10

)]2

cos(1[2

12)(sin

2

10Z

ab

Vxxyyn dvMHJE

PA )(

1

10

dvzbya

xAjea

xAdvJE eo

zj

yy )()()2

()sin( e

dzzedybydxa

xa

xAAj zj

eo )()()2

()sin(

e

Ajea

aAj eo

j

eo e

e 01)2

sin(

12

dvzbya

xAZ

je

a

x

Z

AdvMH mozj

xx )()()2

()sin(1010

dzzedybydxa

xa

xj

Z

A zj

mo )()()2

()sin(2

10

mo

j

mo jZ

Ae

a

aj

Z

A

2

10

0

2

10

1)2

sin(

Vxxyyn dvMHJE

PA )(

1

10

][2

10

10moeo j

Z

AAj

ab

Ze

)(1

2

1010 ZAj

P

moeo

e

13

Microwave Resonator

A resonator is a device or system that exhibits resonance

or resonant behavior, that is, it naturally oscillates at some

frequencies, called its resonant frequency, with greater amplitude than at others.

Resonators are used to either generate waves of specific frequencies or to select specific frequencies from a signal.

The operation of microwave resonators is very similar to

that of the lumped-element resonators of circuit theory.

14

Basic characteristics of series RLC resonant circuits (1)

The input impedance is

The complex power delivered to the resonator is

1.inZ R j L j

C

21 1 1( ).

2 2inP VI I R j L j

C

15

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

Because

21.

2lossP I R

21.

4mW I L

2 2

2

1 1 1.

4 4e cW V C I

C

VjCdt

dVCI )(

)( jC

IV

C

I

jC

IV

)(

16

2.

1

2

lossin

PZ R

I

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

)(2 emlossin WWjPP

2

)(2222

I

WWjP

I

PZ emlossin

in

LCo

1At resonance:

em WW and so

17

Q is “quality factor” which is a measure of the loss of a resonance circuit.

High Q means low loss.

Therefore, at resonance,

so that Q decreases as R increases.

At “near” resonance: where is small,

RCR

L

P

WQ

o

o

loss

mo

12

o

)()1()1

1(2

22

2

2

2

oo

in LjRLjRLC

LjRZ

2)2())((22

ooo

o

in

RQjRLjRZ

22

Substitute with , so :

18

A resonator with loss can be treated as a “lossless” resonator with

replaced by . For example, Zin with no loss (R = 0) is

o

)2

1(Q

jo

))(()()

1(

1 2

oooin jLjL

LCjL

CjLjZ

)(22

ojLjL

)2

1(Q

jo

o )2

(2Q

jLjZ ooin

)(2 oo LjQ

L

LjR 2

19

Half-power fractional bandwidth of resonator

When frequency is such that , the average (real) power

delivered to the circuit is that delivered at resonance.

222RZ in

2

1

o

BW

2

222)( RBWjRQR and

QBW

1so

20

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

0

1

LC

o

in

RQjRLjRZ

22

21

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

22

Basic characteristics of parallel RLC resonant circuits (1)

The input impedance is

The complex power delivered to the resonator is

11 1

.inZ j CR j L

21 1 1( ).

2 2in

jP VI V j C

R L

Anti-resonance

23

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

Because , so

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

21

.2

loss

VP

R

2 2

2

1 1 1.

4 4m LW I L V

L

21.

4eW V C

IjLdt

dILV )(

L

VI

24

2.

1

2

lossin

PZ R

I

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

)(2 emlossin WWjPP

2

)(2222

I

WWjP

I

PZ emlossin

in

LCo

1At resonance:

em WW and so

Parallel resonance circuit:

RCL

R

P

WQ o

oloss

mo

2The quality factor:

so Q increases as R increases!

25

At “near” resonance, letting , so

1))()(

11(

Cj

LjRZ o

o

in

Zin can be simplified using , where ...11

1

x

x

1))(

)1(

11(

Cj

LjR

o

o

o

o

x

1))1(1

(

CjCjLjR

Z o

o

oin

o

1

2)

11(

CjCj

LjLjRo

oo

But CjLj

o

o

1

and at resonance,LC

o

1

26

so 1

2)

11(

CjCj

LjLjRZ o

oo

in

RCj

RCj

R

21)2

1( 1

1

2)

11(

CjCj

L

j

LjRo

oo

1

2)

)1(

1(

Cj

LLC

j

R

27

The effect of the loss can be accounted for by replacing with .o )2

1(Q

jo

Since , so for lossless Zin (at R = 0), LC

o

1

12

1 )1

()1

()(

Lj

LCCj

LjlosslessZ in

))(()(1

1

2

22

2

2

22

oo

o

o

o

o

LjLjLj

LC

Lj

oo

o

LCj

LLj

22

2

)(2

1

2

1

ojCjC

28

Let’s replace with , soo )

21(

Q

jo

)2

1(2

1

)(2

1

Q

jjC

jCZ

oo

in

)2

(2

1

Q

jjC o

o

o

o

oo Qj

CQ

jCC

/21

)/(

)(2

1

)caselossy (/21

in

o

ZjQ

R

Half-power bandwidth edges occur at frequency such that

Therefore,

2

22 R

Z in

QBW

1

29

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

RCj

RCj

RZ in

21)2

1( 1

LCo

1

30

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 ( )inZ

j C

31

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

32

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

33

Transmission line resonators: Short-circuited

/2 line (1)

Assuming the line is “lossy” so the input impedance is

0 0

tanh tantanh( ) .

1 tan tanhin

l j lZ Z j l Z

j l l

Note:)tanh()tanh(1

)tanh()tanh()tanh(

BA

BABA

, and )tan( ljZZ oin if = 0 (lossless)

34

Transmission line resonators: Short-circuited /2 line

(1b)

At , the transmission line length (l): and 2

l

2where and vp is the phase velocity of transmission line

o

At near resonance, , so o

pv

p

o

p v

l

v

ll

)(

opp

o

pp

o

vvv

l

v

l

2

2

22

ooo

l

)tan()tan()tan(and so

35

Transmission line resonators: Short-

circuited /2 line (2)

For a small loss TL, we can assume l << 1 so tan(l) l.

Now let = 0+ , where is small. Then, assume a

TEM line,

For = 0, we have

(because )

which can be written in the form

which is the Zin of series RLC

2 .inZ R j L

pp

o

p

o

p v

l

v

l

v

l

v

ll

)(

)()/(1

)/(

o

o

o

ooin jlZ

lj

jlZZ

1)/( ol

36

Transmission line resonators: Short-

circuited /2 line (3)

This resonator resonates for = 0 (l = /2) and its input

impedance is

where and

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

The Q of this resonator can be found as

since at 1st resonance

0 .inZ R Z l

0 .2 2

LQ

R l

o

oZL

2

LC

o

2

1

l

37

Transmission line resonators: Short-circuited /4 line

(1)

The input impedance is

0

1 tanh cot.

tanh cotin

j l lZ Z

l j l

)cot(

)cot(

)tan()tanh(1

)tan()tanh(

lj

lj

ll

ljlZZ oin

At , the transmission line length (l): and 4

l

At near resonance, , so o

o pv

p

o

p v

l

v

ll

)(

opp

o

pp

o

vvv

l

v

l

24

2

44

ooo

l

2)

2tan()

22cot()cot(

and so

38

Assume tanh(l) l for small loss, it gives

(because )

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

RLC circuit

CjR

Z in

21

1

)2

()2/(

)2/1(

o

o

o

ooin j

l

Z

jl

ljZZ

1)2/( ol

)cot()cot(

1)cot()cot()cot(

BA

BABA

(Note: )

39

Transmission line resonators: Short-

circuited /4 line (2)This resonator resonates for = 0 (l = /4) and its input

impedance is

where and

The Q of this resonator can be found as

because at resonance.

0 .in

ZZ R

l

0 .4 2

Q RCl

ooZC

4

CL

o

2

1

2l

40