Prof. David R. JacksonDept. of ECE

Fall 2013

Notes 23

ECE 6340 Intermediate EM Waves

1

Duality

( )

( )

ic

ic

v

mv

E M j H

H J j E

E

H

Assume a region of space with sources,and allow for all types of sources and charges:

/

/c

c m

j

j

Notes:

(1) The electric charge density will only exist inside the source region of impressed electric current, since it must be zero in a source-free region.

(2) A magnetic conductivity m is introduced for generality, though it is always zero in practice.

(3) The magnetic charge density will only exist inside the source region of impressed magnetic current, since it must be zero away from the impressed source (it does not exist in nature).

2

Maxwell’s equations now have a completely

symmetric form.

Duality (cont.)

The substitutions shown below leave Maxwell’s equations unaffected.

, ,

, ,c c

c c

E H

H E

i i

i i

mv v

mv v

J M

M J

3

icE M j H

icH J j E

Example:

icH J j E

icH J j E

icE M j H

icE M j H

Duality (cont.)

4

Duality (cont.)

Duality allows us to find the radiation from a magnetic current.

Steps:

1) Start with the given magnetic current source of interest.

2) Consider the dual problem that has an electric current source with the same shape or form (Case A).

3) Find the radiation from the electric current source.

4) Apply the dual substitutions to find the radiation from the original magnetic current (Case B).

Note: When we apply the duality substitution, the numerical values of the permittivity and permeability switch. However, once we have the formula for radiation from the magnetic current, we can let the values be replaced by their conventional ones.

5

Problem of interest:

iM

Radiation From Magnetic Current (cont.)

A magnetic current is radiating in free space.

Free-space is assumed for simplicity. To be more general, we can replace

0 c 0 c

, , , ,iM x y z f x y z

6

Case (A): J i onlyiJ

0

0 ( )4

jk r ri

V

eA J r dV

r r

0

1H A

Radiation From Magnetic Current

The shape of the electric current is the same as that of the original magnetic current that was given.

, , , ,iJ x y z f x y z

7

E j A

0 0

1( )E j A A

j

so

0 0A j

Radiation From Magnetic Current (cont.)

and

Also

8

Case (B): M i only

iM

0

0 ( )4

jk r ri

V

eF M r dV

r r

where0

1E F

From duality:

0

1H A

0

0 ( )4

jk r ri

V

eA J r dV

r r

Radiation From Magnetic Current (cont.)

Case A Case B

, , , ,iM x y z f x y z

9

where

0 0

1( )H j F F

j

0 0

1( )E j A A

j

Also,

Radiation From Magnetic Current (cont.)

Case A

Case B

10

Radiation from Magnetic Current Summary

0

1E F

0

0 ( )4

jk r ri

V

eF M r dV

r r

iM

0 0

1( )H j F F

j

11

General Radiation Formula

Both J i and M i

:

iM iJ

0 0 0

0 0 0

1 1( )

1 1( )

E F j A Aj

H A j F Fj

Note: Duality also holds with the vector potentials, If we include these:

A F

F A

iJ AiM A

Use superposition:

12

Far-Field

Case (A) (electric current only):

0

0

~ ( , , )

~ ( , )4

1ˆ~ ( )

tE j A r

A r a

H r E

( , ) ( )i jk r

V

a J r e dV

where

0jk re

rr

13

Far-Field (cont.)

Case (B) (magnetic current only):

where

0

0

~ ( , , )

~ ( , )4

ˆ~ ( )

tH j F r

F r f

E r H

( , ) ( )i jk r

V

f M r e dV

14

Example

Radiation from a magnetic dipole

y

x

l

z

K magnetic Amps

Dual problem:

Case B

15

y

x

l

I electric Amps

Case A

z

Example (cont.)

Duality:

0 02

( ) 1ˆ sin4

jk rIl jkH e

r r

I K

H E

0 02

( ) 1ˆ sin4

jk rKl jkE e

r r

So, for magnetic dipole:

For electric dipole:

16

ExampleA small loop antenna is equivalent to a magnetic dipole antenna

17

x

y

z

I

a

y

x

z

Kl

0 0Kl j AI2A a

020 0 0 sin

4

jk rk a I eE

r

0 0( )sin

4jk rKl jk

E er

Far-field:

Far-field:

Boundary Conditions

n̂

sJ

1

2(1) (2)ˆ ( ) sn H H J

Duality:n̂

sM

1

2(1) (2)ˆ ( ) sn E E M

Electric surface current:

18

Boundary Conditions (cont.)

n̂

sM

1

2

(1) (2)ˆ ( ) sn E E M

19

Example: Magnetic Current on PEC

sM PEC

(1) (2)ˆ sn E E M

A magnetic surface current flows on top of a PEC.

PEC

1

2

sMn̂

20

Example (cont.)

(1) (1)ˆ ˆ ˆ ˆs t sn E M n n E n M

1 ˆt sE n M or

ˆt sE n M

Dropping the superscript, we have

Note: The relation between the direction of the electric field and the

direction of the magnetic current obeys a "left-hand rule."

21

PEC

0tE

n̂

sM

Example (cont.)

ˆt sE n M

22

PEC

0tE

n̂

sM

ˆ ˆ ˆt sn E n n M

ˆ t sn E M

ˆsM n E Hence

This form allows us to find the magnetic surface current necessary to produce any desired tangential electric field.

We then have

Note: The tangential subscript can be dropped.

Modeling of Slot Antenna Problem

ˆsM n E

sMn̂0tE

0tE 0tE

Side view

tEPEC

0tE

0tE 0tE

23