Notes 5 - Waveguides part 2 parallel plate
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Transcript Notes 5 - Waveguides part 2 parallel plate
ECE 5317-6351
Microwave Engineering
Fall 2011
Prof. David R. Jackson
Dept. of ECE
Notes 5
Waveguides Part 2:
Parallel Plate Waveguide
1
Field Equations (from Notes 4)
Summary
Hx
Ez
j
c
kc2
y
kz
H z
x
These equations will be useful to us
in the present discussion.
Ez
H z
j
H y 2 c
kz
kc
x
y
Ex
Ez
H z
j
k
z
kc2
x
y
k 2 2 c
kc k k
2
j
Ey 2
kc
Ez
H z
kz
y
x
2 1/2
z
2
Parallel-Plate Waveguide
y
Both plates assumed PEC
w >> d,
0
x
, , s
d
Neglect x variation,
edge effects
x
w
z
The parallel-plate stricture is a good 1ST order model for a
microstrip line.
w
,,
d
3
TEM Mode
Parallel-plate waveguide
2 conductors 1 TEM mode
y
To solve for TEM mode:
for
t2 0
, , s
0 xw
0 yd
Boundary conditions:
( x,0) 0 ; ( x, d ) V0
2 2
2 2 0
x y
2
t
d
x
z
w
k z j k c k jk
k
k
c j
s
4
TEM Mode (cont.)
2
0
2
y
where
x,0 0 & x, d V0
( x, y ) A By
V
( x, y ) 0 y ; 0 x w
d
0 yd
et x, y t yˆ
@ y0
A0
@yd
Vo
ˆ
y
y
d
E ( x, y, z ) et ( x, y )e
jkz
V0
yˆ e
d
kz k c
B
V0
d
jkz
c j
s
5
TEM Mode (cont.)
V
E ( x, y, z ) yˆ 0 e
d
y
jkz
Recall
H
1
( zˆ E )
H x, y, z xˆ
d
V0
e
d
, , s
x
jkz
z
w
For a wave prop. in + z direction
y
Time-ave. power flow in + z direction:
V0
E
H
, , s
x
P
1 2 w 1 2 k z
V0 Re * e
2
d
*
1
P Re ( E H ) zˆ dS
2 s
2
wd
1 V0
2 k z
Re * 2 zˆ zˆ e dydx)
2 0 0 d
1
1 2 1
V0 2 wd Re * e2 k z
2
d
6
TEM Mode (cont.)
Transmission line voltage
y
0
V ( z ) E yˆ dy
I
k z k c
d
V ( z ) V0 e
jkz
V
d
C
Transmission line current
w
I ( z ) H x, d , z xˆ dx
0
I0
Note:
PEC : J s nˆ H
sz
J Hz
, , s
-
x
w
z
V w
I ( z) 0 e
d
I
+
Characteristic Impedance
V0 e jkz
Z0
I 0 e jkz
(Assume + z wave)
d
Z0
w
jkz
Phase Velocity (lossless case)
vp
c
r r
c = 2.99792458 108 m/s
7
TEM Mode (cont.)
For wave propagating in + z direction
Time-ave. power flow in +z direction:
(calculated using the voltage and current)
1
Re VI *
2
*
1
V0 w 2 k z
Re V0
e
2
d
P
P
Recall that we found from
the fields that:
1 2 w 1
P V0 Re * e2 k z
2
d
1
1
2 w
V0 Re * e 2 k z
2
d
same
This is expected, since a TEM mode is a
transmission-line type of mode, which is
described by voltage and current.
8
TEM Mode (cont.)
We can view the TEM mode in a parallel-plate waveguide as a
“piece” of a plane wave.
y
E
H
PEC
, ,s
PMC
PMC
x
PEC
The PEC and PMS walls do not disturb the fields of the plane wave.
PEC :
nˆ E 0
PMC :
nˆ H 0
9
TMz Modes (Hz = 0)
Recall Ez ( x, y, z) ez ( x, y) e
jkz z
y
where
2 2
2
k
2
c ez 0,
2
x y
d
1
2 2
z
, , s
x
kc [ k 2 k ]
z
w
subject to B.C.’s Ez = 0 @ y = 0, d
ez x, y A sin(kc y ) B cos(kc y )
@y0 B0
@ y d kc d n
n 0,1, 2,.... kc
n
d
10
TMz Modes (cont.)
n
ez x, y A sin
y
d
n 0,1, 2,...
n
Ez An sin
y e
d
k z k 2 kc2
jk z z
n
k
d
2
2
k 2 2 c
Recall:
j c Ez j c n
n
Hx 2
2 An
cos
kc y
kc
d
d
jk z Ez
jk z n
n
Ey
A
cos
n
kc2 y
kc2
d
d
Ex 0
Hy 0
No x variation
ye
ye
Hz 0
jk z z
jk z z
11
TMz Modes (cont.)
Summary
n
Ez An sin
ye
d
y
jk z z
jk z
n
Ey
An cos
ye
kc
d
j c
n
Hx
An cos
ye
kc
d
Ex H y H z 0
kc
n
;
d
n 0,1, 2,...
n
kz k 2
d
k 2 2 c
, , s
d
jk z z
x
w
z
jk z z
Each value of n corresponds to a unique
TM field solution or “mode.”
TMn mode
2
Note:
n 0 kz k
TM 0 TEM
(In this case, we absorb the An coefficient
with the kc term.)
12
TMz Modes (cont.)
Lossless Case
kc
2
n
2
kz k
d
2
k k
2
2
c
1
2
c
1
2
y
n 0,1, 2,...
d
x
k
2
2
for k 2 kc2
k z k 2 kc2
propagating mode
, , s
z
w
for k 2 kc2
k z j kc2 k 2 j
e jk z z e z
Fields decay exponentially
evanescent fields
“cutoff” mode
13
TMz Modes (cont.)
Frequency that defines border between cutoff and propagation
(lossless case):
f cutoff frequency
c
@ f fcn
c
k kc cn
n
f cn
2d
1
cutoff frequency for TMn mode
TEM
prop.
single
mode
prop.
n
d
TM1
2
modes
prop
TM2
TM3
3
mode
prop.
….
f
cuttoff
0
f c1
fc2
fc3
14
TMz Modes (cont.)
Time average power flow in z direction (lossless case):
TMn
P
wd
1
Re E H * zˆ dydx
2 0 0
c
wd
1
Re E y H x*dydx
2 0 0
PTMn
2
2 n
Re{
k
}
A
w
cos
y
z
n
0 d dy
2kc2
d
y
d
, , s
x
z
w
d
;
n
0
2
2 Re{k z } An w 2
2kc
d ; n 0
n 0,1, 2,...
Real for f > fc
Imaginary for f < fc
15
TEz Modes
Recall
H z ( x, y, z) hz ( x, y) e
y
jkz z
, , s
d
where
x
2 2
2
2 2 kc hz x, y 0,
x y
subject to B.C.’s Ex = 0
1
2 2
z
kc [ k 2 k ]
z
w
@ y=0, d
1 H z H y
Ex
j c y
z
hz A sin(kc y ) B cos(kc y )
PEC : H nˆ 0
@y 0 A0
n
@ y d kc d n , n 1, 2,3,... kc
d
16
TEz Modes (cont.)
n
hz x, y Bn cos
y
d
n
H z Bn cos
ye
d
n 1, 2,3, ...
k z k 2 kc2
n
k
d
jk z z
Recall:
j H z j n n jkz z
Ex
2 Bn
ye
sin
2
kc
y
kc
d d
jk z H z
jk z n n jkz z
Hy
Bn
y e
sin
2
2
kc y
kc
d d
Hx 0
Ey 0
No x variation
2
2
k 2 2 c
Ez 0
17
TEz Modes (cont.)
Summary
n
H z Bn cos
ye
d
y
jk z z
j
n jk z z
Bn sin
ye
kc
d
jk z
n jk z z
Hy
Bn sin
ye
kc
d
d
Ex
H x E y Ez 0
kc
n
; n 1, 2,...
d
n
kz k 2
d
k 2 2 c
2
, , s
x
z
w
Each value of n corresponds to a
unique TE field solution or “mode.”
TEn mode
Cutoff frequency
n 1
f cn
2d
18
All Modes
For all the modes of a parallel-plate waveguide, we have
n 1
f cn
2d
TEM
prop.
single
mode
prop.
c
TE1
TM1
TE2
TM2
TE3
TM3
3
modes
prop
5
mode
prop.
….
f
cuttoff
0
f c1
fc2
fc3
The mode with lowest cutoff frequency is called the “dominant”
mode of the wave guide.
19
Power in TEz Mode
Time average power flow in z direction (lossless case):
TEn
P
c
wd
1
*
Re E H zˆ dydx
2 0 0
wd
1
Re Ex H *y dydx
2 0 0
d
P
d
2
2
c
4k
Re k z Bn
, , s
x
n
2 Re{k z } Bn W sin
y dy
2kc
d
0
TEn
y
2
2
z
w
Wd
n = 1,2,…..
Real for f > fc
Imaginary for f < fc
20
Field Plots
y
TEM
x
y
x
TM1
y
TE1
x
21