notes 13 ece 5317-6351 microwave engineering fall 2011 transverse resonance method prof. david r....
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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!