1 Spring 2011 Notes 6 ECE 6345 Prof. David R. Jackson ECE Dept.

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

êsM 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:

êsM 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).

17

Electric-current model:

êsM n E

Magnetic-current model:

h rx

esJ

h rx

esM e

sM

Theorem (cont.)Theorem (cont.)

êsJ 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

ês iJ n H

ês iM n E

Note: The electric current on the ground is neglected (it does not radiate).

Note the inward pointing normal în

21


Exterior Fields:0e e

s sE J E M

ê 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


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


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


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


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



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


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


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


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


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


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

1 Spring 2011 Notes 6 ECE 6345 Prof. David R. Jackson ECE Dept.

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