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Notes 13

ECE 5317-6351 Microwave Engineering

Fall 2011

Transverse Resonance Method

Prof. David R. JacksonDept. of ECE

Fall 2011

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Transverse Resonance Method

This is a general method that can be used to help us calculate various important quantities:

Wavenumbers for complicated waveguiding structures (dielectric-loaded waveguides, surface waves, etc.)

Resonance frequencies of resonant cavities

We do this by deriving a “Transverse Resonance Equation (TRE).”

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Transverse Resonance Equation (TRE)

R = reference plane at arbitrary x = x0

To illustrate the method, consider a lossless resonator formed by a transmission line with reactive loads at the ends.

We wish to find the resonance frequency of this transmission-line resonator.

R

0Z0Z

x x = x0

2 2L LZ jX1 1L LZ jX

x = L

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Examine the voltages and currents at the reference plane:

R

+V r

-+V l

-

I r I l

x = x0

TRE (cont.)

R

0Z0Z

x x = x0

2 2L LZ jX1 1L LZ jX

x = L

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TRE (cont.)

Hence:Define impedances:

r

in r

l

in l

VZ

I

VZ

I

in inZ Z

r l

r l

V V

I I

Boundary conditions:

R

+V r

-+V l

-

I r I l

x = x0 inZ

inZ� x

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in in

in in

Z Z

Y Y

TRE (cont.)

R

inZ

inZ�

or

TRE

Note about the reference plane: Although the location of the reference plane is arbitrary, a “good” choice will keep the algebra to a minimum.

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Example

0 0, rZ k k

x

2 2L LZ jX1 1L LZ jX

L

Derive a transcendental equation for the resonance frequency of this transmission-line resonator.

We choose a reference plane at x = 0+.

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Example (cont.)

R

x

2 2L LZ jX1 1L LZ jX

L

0 0, rZ k k

Apply TRE:

in inZ Z

2 01 0

0 2

tan

tanL

LL

Z jZ LZ Z

Z jZ L

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2 01 0

0 2

tan

tanL

LL

Z jZ LZ Z

Z jZ L

2 01 0

0 2

tan

tanL

LL

jX jZ LjX Z

Z j jX L

2 0 0

1 0

0 2 0

tan

tan

L r

L

L r

jX jZ k LjX Z

Z j jX k L

1 0 2 0 0 2 0 0tan tanL L r L rjX Z j jX k L Z jX jZ k L

Example (cont.)

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0 2 10 2

1 2 0

tan L Lr

L L

Z X Xk L

X X Z

After simplifying, we have

Special cases:

1 2 00L L rX X k L n

1 2 00, 2 1 / 2L L rX X k L n

Example (cont.)

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Rectangular ResonatorDerive a transcendental equation for the resonance frequency of a rectangular resonator.

The structure is thought of as supporting RWG modes bouncing back and forth in the z direction.

y

z

x

,r r

PEC boundary

a

b

h

We have TMmnp and TEmnp modes.The index p describes the variation in the z direction.

b < a < h

Orient so that

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We use a Transverse Equivalent Network (TEN):

y

z

x

,r r

PEC boundary

a

b

h

0 , zZ k

z

h

mnz zk k

,0 ,

m nTE TMZ Z

We choose a reference plane at z = 0+.

in inZ Z

0 ( )inZ

PEC bottom

0inZ

Hence

Rectangular Resonator (cont.)

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y

z

x

,r r

PEC boundary

a

b

h

0 tan 0inZ jZ h

Hence

, ,, tan 0m n m n

TE TM zjZ k h

,tan 0m nzk h

, , 1, 2m nzk h p p

2 22 m n

h k pa b

0 , zZ k

z

h

Rectangular Resonator (cont.)

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y

z

x

,r r

PEC boundary

a

b

h

Solving for the wavenumber we have

2 2 2m n p

ka b h

Hence

2 2 2

0 02 mnp r r

m n pf

a b h

2 2 21

2mnp

r r

c m n pf

a b h

or

82.99792458 10 [m/s]c

Note: The TMz and TEz modes have the same resonance frequency.

0,1,2,

0,1,2,

1,2,

, 0,0

m

n

p

m n

The lowest mode is the TE101 mode.

Rectangular Resonator (cont.)

TEmnp mode:

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y

z

x

,r r

PEC boundary

a

b

h2 2

101

1 1 1

2r r

cf

a h

TE101 mode:

0, , cos sinz

x zH x y z H

a h

The other field components, Ey and Hx, can be found from Hz.

Note: The sin is used to ensure the boundary condition on the PEC top and bottom plates:

0n zH H

Rectangular Resonator (cont.)

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Rectangular Resonator (cont.)

Lp (Probe inductance)

Tank (RLC) circuit

R L C

y

z

x

,r r

PEC boundary

a

b

h

Practical excitation by a coaxial probe

Circuit model

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Rectangular Resonator (cont.)

Lp (Probe inductance)

Tank (RLC) circuit

R L C Circuit model

0

1 2 1RLC

RZ

j Q

0

1

LC

0

RQ

L

Q = quality factor of resonator0 ave

d

UQ

P

E HU U U energy stored

avedP average power dissipated

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Rectangular Resonator (cont.)

0

1 2 1RLC

RZ

j Q

f

RLCX

RLCZ

RLCR

0f

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Grounded Dielectric Slab

Assumption: There is no variation of the fields in the y direction,

and propagation is along the z direction.

x

z ,r r h

Derive a transcendental equation for wavenumber of the TMx surface waves by using the TRE.

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x

z

H ,r r E

TMx

Grounded Dielectric Slab

1 001 00

1 0

TM TMx xk kZ Z

( )TM z

y

xE

ZH

defined for a wave going in the direction

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TMx Surface-Wave Solution

TEN:

1 001 00

1 0

TM TMx xk kZ Z

2 21 1

12 2 2 22

0 0 0 0( )

x z

x z z x

k k k

k k k j k k j

R

00TMZ01

TMZ

x

h

01 1 00tan( )TM TMin inxZ jZ k h Z Z

�������������� �

The reference plane is chosen at the interface.

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TRE:

1 01

1 0

tan( )x xx

k kj k h

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0

tan( )xr x

x

kj k h

k

01 1 00tan( )TM TMxjZ k h Z

in inZ Z�������������� �

TMx Surface-Wave Solution (cont.)

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Letting

or

2 20 0 0 0,x x x zk j k k

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0

tan( )xr x

x

kk h

2 2 2 2 2 20 1 1tanr z z zk k k k h k k

TMx Surface-Wave Solution (cont.)

We have

Note: This method was a lot simpler than doing the EM analysis and applying the boundary conditions!