1 Spring 2011 Notes 6 ECE 6345 Prof. David R. Jackson ECE Dept.
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1
Spring 2011
Notes 6
ECE 6345ECE 6345
Prof. David R. JacksonECE Dept.
2
OverviewOverview
In this set of notes we look at two different models for calculating the radiation pattern of a microstrip antenna:
Electric current model Magnetic current model
We also look at two different substrate assumptions:
Infinite substrate Truncated substrate (truncated at the edge of the patch).
3
Review of Equivalence PrincipleReview of Equivalence Principle
new problem:
+- r
( , )E H
,out out
( , )E H
,out out (0,0)
n̂
esJ e
sM
S S
ˆ 0
ˆ 0
es
es
J n H
M n E
original problem:
PEC
,in in
4
Review of Equivalence PrincipleReview of Equivalence Principle
A common choice:
ˆ
ˆ
es
es
J n H
M n E
( , )E H
,out out
n̂
esJ e
sM
S
PEC
The electric surface current sitting on the PEC object does not radiate, and can be ignored.
Model of Patch and FeedModel of Patch and Feed
h rx
AS
Infinite substrate
(aperture)
patch
probe
coax feed
z
5
6
Model of Patch and FeedModel of Patch and Feed
ˆesM z E Aperture:
h rx
esM
Magnetic frill model:
hr
x
AS
S
( , )E H
zero fields
7
Electric Current Model: Electric Current Model: Infinite SubstrateInfinite Substrate
h rx
Sinfinite substrate
ˆ ˆ 0
ˆ
es t
es s
M n E n E
J n H J
Note: The direct radiation from the frill is ignored.
The surface S “hugs” the metal.
h rx
topsJ
probesJ
botsJ
( , )E H
zero fields
8
Let patch top bots s sJ J J
Electric Current Model: Electric Current Model: Infinite Substrate Infinite Substrate (cont.)(cont.)
probesJ
patchsJ
top view
h rx
patchsJ
probesJ
9
Magnetic Current Model: Magnetic Current Model: Infinite Substrate Infinite Substrate
h rx
Keep ground plane and substrate in the zero-field region. Remove patch, probe and frill current.
( , )E H
zero fieldsbS
hS
tS
ˆ 0, ,
ˆ 0,
0,
0,
es t b
es b
es t
es h
M n E r S S
J n H r S
J r S
J r S
(weak fields)
(approximate PMC)
esM
S
10
Exact model:
ˆesM n E
Approximate model:
Magnetic Current Model:Magnetic Current Model: Infinite Substrate Infinite Substrate (cont.)(cont.)
h rx
esJ
esMe
sJesJ
esM
h rx
esM e
sM
Note: The magnetic currents radiate inside an infinite substrate above a ground plane.
11
Magnetic Current Model: Magnetic Current Model: Truncated SubstrateTruncated Substrate
Note: The magnetic currents radiate in free space above a ground plane.
Approximate model:
esM
x
Now we remove the substrate inside the surface.
r
x
bS
12
Electric Current Model: Electric Current Model: Truncated SubstrateTruncated Substrate
rx
S The patch and probe are replaced by surface currents, as before.
Next, we replace the dielectric with polarization currents.
patchsJ
r
patch top bots s sJ J J
x
13
0 0
H j E
j E j E
0 1polrJ j E
Electric Current Model: Electric Current Model: Truncated SubstrateTruncated Substrate(cont.)(cont.)
0x
patchsJ
probesJ
polJ
In this model we have three separate electric currents.
14
Comments on ModelsComments on Models
Infinite Substrate
The electric current model is exact (if we neglect the frill), but it requires knowledge of the exact patch and probe currents.
The magnetic current model is approximate, but fairly simple.
For a rectangular patch, both models are fairly simple if only the (1,0) mode is assumed.
For a circular patch, the magnetic current model is much simpler (it does not involve Bessel functions).
15
Comments on Models (cont.)Comments on Models (cont.)
Truncated Substrate
The electric current model is exact (if we neglect the frill), but it requires knowledge of the exact patch and probe currents, as well as the field inside the patch cavity (to get the polarization currents). It is a complicated model.
The magnetic current model is approximate, but very simple. This is the recommended model.
For the magnetic current model the same formulation applies as for the infinite substrate – the substrate is simply taken to be air.
16
TheoremTheorem
Assumptions:
1)The electric and magnetic current models are based on the fields of a single cavity mode corresponding to an ideal cavity with PMC walls.
2)The probe current is neglected in the electric current model.
Then the electric and magnetic models yield identical results
at the resonant frequency of the cavity mode.
(This is true for either infinite or truncated substrates).
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Electric-current model:
ˆesM n E
Magnetic-current model:
h rx
esJ
h rx
esM e
sM
Theorem (cont.)Theorem (cont.)
ˆesJ z H
(E, H) = fields of resonant cavity mode
18
Theorem (cont.)Theorem (cont.)
Proof:
0 , 0,0f f E H At
ideal cavity
rx
PMC
PEC
( , )E H
We start with an ideal cavity having PMC walls on the sides. This cavity will support a valid non-zero set of fields at the resonance frequency f0 of the mode.
19
ProofProof
Equivalence principle:
Put (0, 0) outside S
keep (E, H) inside S
xS ( , )E H 0,0
PEC
x
S PMC( , )E H
The PEC and PMC walls have been removed in the zero field region. We keep the substrate and ground plane.
Infinite substrate
20
Proof (cont.)Proof (cont.)
x
( , )E HesM
esJ
0,0
ˆes iJ n H
ˆes iM n E
Note: The electric current on the ground is neglected (it does not radiate).
Note the inward pointing normal ˆin
21
Proof (cont.)Proof (cont.)
Exterior Fields:0e e
s sE J E M
ˆe patch Js s siJ n H J J
(The equivalent electric current is the same as the electric current in the electric current model.)
(note the inward pointing normal)
ˆ
ˆ
ˆ
es i
Ms
M n E
n E
n E
M
(The equivalent current is the negative of the magnetic current in the magnetic current model.)
22
Proof (cont.)Proof (cont.)
Hence
or
0J Ms sE J E M
J Ms sE J E M
23
Proof for truncated model:
Theorem for Truncated SubstrateTheorem for Truncated Substrate
x
S PMC( , )E H
PEC
x
S PMC( , )E H
PEC
polJ
Replace the dielectric with polarization current:
24
e patchs s
e Ms s
J J
M M
0patch pol MssE J E J E M
J MssE J E M
Proof (cont.)Proof (cont.)
Hence
x
( , )E H esM 0,0
polJ
esJ
patch pol MssE J E J E M or
25
Extension of TheoremExtension of Theorem
h rx
The electric and magnetic models yield identical results at any frequency if we include the probe current and the currents from the higher-order modes
h rx
patchsJ
probesJ
Electric current model Magnetic current model
h rx
esM e
sM
26
Extension of Theorem (cont.)Extension of Theorem (cont.)
Proof:
( , )E H
h rx
PMCPEC
(0, 0)
The frill source excites fields in the ideal cavity at any frequency. In general, all (infinite) number of modes are excited.
h rx
(0, 0)
esJ
esM
27
Extension of Theorem (cont.)Extension of Theorem (cont.)
The probe is then replaced by its surface current:
h rx
(0, 0)
esJ
esM
0patch probe MssE J E J E M
These currents now account for all modes.
28
h rx
patchsJ
probesJ
Electric current model Magnetic current model
Extension of Theorem (cont.)Extension of Theorem (cont.)
J MssE J E M
Hence, we have
h rx
esM e
sM
29
Rectangular PatchRectangular Patch
Ideal cavity model:
022 zz EkE
( , ) ( ) ( )zE x y X x Y y
2
2
2
0
0
X Y XY k XY
X Yk
X YX Y
kX Y
Let
0z
C
E
n
PMC
C
L
W
x
y
Divide by X(x)Y(y):
so
30
Rectangular Patch (cont.)Rectangular Patch (cont.)
Hence2( )
constant( ) x
X xk
X x
( ) sin cosx xX x A k x B k x
( ) cos( )xX x k x
Choose 1B
(0) sin( 0) 0x x x xX k A k B k k A
( ) sin 0x x
x
X L k k L
mk
L
General solution:
Boundary condition:
0A
Boundary condition:
31
Rectangular Patch (cont.)Rectangular Patch (cont.)
so
so
( ) cosm x
X xL
2 2 0x
Yk k
Y
2 2 2constant y xY
k k kY
( ) cosn y
Y yW
( , ) ( , ) cos cosm nz
m x n yE x y
L W
Returning to the Helmholtz equation,
Following the same procedure as for the X(x) function, we have
Hence
32
Rectangular Patch (cont.)Rectangular Patch (cont.)
where
0222 kkk yx
22
W
n
L
mkmn
mnmnk
221
W
n
L
mmn
2 2
mnr
c m n
L W
Using
we have
Hence
33
Rectangular Patch (cont.)Rectangular Patch (cont.)
Current:
ˆ ˆpatchsJ n H z H
1
1ˆ
1ˆ ˆ
z
z z
H Ej
z Ej
z E z Ej
( )1zˆH z E
jωμ
-= ´ Ñ
so
34
Rectangular Patch (cont.)Rectangular Patch (cont.)
Hence1patch
S zJ Ejωμ
=- Ñ
ˆ ˆsin cos cos sinzm m x n y n m x n y
E x yL L W W L W
Dominant (1,0) Mode:
1ˆ( , ) sins
xJ x y x
j L L
( , ) coszx
E x yL
35
Rectangular Patch (cont.)Rectangular Patch (cont.)
00
( , ) 1
0
( , ) 0
z
s
E x y
J x y
Static (0,0) mode:
This is a “static capacitor” mode.
A patch operating in this mode can be made resonant if the patch is fed by an inductive probe (a good way to make a miniaturized patch).
36
Radiation Model for Radiation Model for (1,0)(1,0) Mode Mode
Electric-current model:
ˆ sinpatchs
xJ x
j L L
L
W
x
y
hr
x
patchsJ
37
Radiation Model for Radiation Model for (1,0)(1,0) Mode (cont.) Mode (cont.)
Magnetic-current model:
ˆ
ˆ ˆ cos
MsM n E
xn z
L
0ˆ
ˆ
0ˆ
ˆ
ˆ
yy
Wyy
xx
Lxx
nx
y
L
W
38
Hence
ˆ ˆcos
ˆ ˆcos0 0
ˆ cos
ˆ cos 0
Ms
y y x L
y y x
Mx
x y WL
xx y
L
radiating edges
MsM
L
W
x
y
Radiation Model for Radiation Model for (1,0)(1,0) Mode (cont.) Mode (cont.)
The non-radiating edges do not contribute to the far-field pattern in the principal planes.
39
Circular PatchCircular Patch
set
a
022 zz EkE
( ) cos( )cos
( ) sin( )n
zn
J k n m zE
Y k n h
1/ 22 2
1/ 222
zk k k
mk
h
k
0m
PMC
40
Circular Patch (cont.)Circular Patch (cont.)Note: cos and sin modes are degenerate (same resonance frequency).
cosz nE n J k Choose cos :
( ) 0nJ ka ( )nJ x
x
1nx 2nx
0z
a
E
41
Circular Patch (cont.)Circular Patch (cont.)
Hence
so
npka x
np np
r
cx
Dominant mode (lowest frequency)
11
( , ) (1,1)
1.841
n p
x
(1,1)1( , ) cos ( )zE J k
42
Circular Patch (cont.)Circular Patch (cont.)Electric current model:
set
1 1ˆˆ
1 1ˆˆ cos ( ) ( )sin ( )
J z zs z
n n
E EJ E
j
k n J k n n J kj
1 1
1 1ˆˆ cos ( ) sin ( )JsJ k J k J k
j
1, 1n p
x
y Very complicated!
43
Circular Patch (cont.)Circular Patch (cont.)Magnetic current model:
so
ˆ
ˆ ˆ
ˆ
Ms
z
z
M n E
z E
E
ˆ cos ( )Ms nM n J ka
set
1ˆ cos ( )M
sM J ka
1, 1n p
44
Circular Patch (cont.)Circular Patch (cont.)
Note:
At
so
1
( ) ( )
cos ( )
z aV hE
h J ka
0
0ˆ cosMs
VM
h
0 1(0) ( )V V h J ka
Hence 01
ˆ ˆcos ( ) cosMs
VM J ka
h
45
Circular Patch (cont.)Circular Patch (cont.)Ring approximation:
000
cos cosh M M
s s
VK M dz hM h V
h
0( ) cosK V
x
y
( )K
h rx
esM
h rx
K
ˆK K
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