1

Ez = 0

From

Expanding for z-propagating field gets

where

2 2 2( ) 0xy z u zH H

2 22

2 2 ( , ) 0z zz

H Hh H x y

x y

( , ) j zz zH H x y e

2

In the x-direction

Since Ey = 0, then from

we have

2 2 2 2z z

yu u

H Ej jE

x y

0zHx

at x = 0 and x = a

3

In the y-direction

Since Ex = 0, then from

we have

0zHy

at y = 0 and y = b

2 2 2 2z z

xu u

H Ej jE

y x

4

Assume

then we have

( , )zH x y XY

1 2

3 4

( ) cos sin

( ) cos sin

x x

y y

X x c x c x

Y y c y c y

5

1. in the x-direction

at x = 0,

at x = a,

0zHx

1 2

( )sin cos 0x x x x

dX xc x c x

dx

2 0.c

0zHx

1

( )sin 0x x

dX xc x

dx

6

( 0,1,2,3,...) xa m m

. xma

7

2. in the y-direction

at y = 0,

at y = b,

0zHy

4 0c

0zHy

3 4

( )sin cos 0y y y y

dY yc y c y

dy

3

( )sin 0y y

dY yc y

dy

8

Properties of TE wave in y-direction of rectangular WGs (2)

( 0,1,2,3,...)yb n n

.y

nb

For lossless TE rectangular waveguides,

0 cos cos /j zz

m x n yH H e A m

a b

9

2 2

,

1

2 2

c mn

h m nf Hz

a b

, 2 2

2

c mn mm na b

10

For TE mode, either m or n can be zero, if a > b, is a smallest eigen value and fc is lowest when

m = 1 and n = 0 (dominant mode for a > b)

ha

10

1( )

22p

c TE

uf Hz

aa

10( ) 2c TE a m

11

For TM mode, neither m nor n can be zero, if a > b, fc is lowest when m = 1 and n = 1

11

2 21 1 1

( )2

c TMf Hza b

11 2 2

2( )

1 1c TM m

a b

12

13

14

General properties Radiation fields and patterns Antenna performance

15

A structure designed for radiating and receiving EM energy in a prescribed manner.

The importance of the shape and size of the structure› the efficiency of the radiation› the preferential direction of the radiation

16

Complex antenna impedance Zant needs to be matched to the system impedance.

17

Far field region (the distance where the receiving antenna is located far enough for the transmitter to appear as a point source)

22Lr

In the far field

where 0 = 120 .

Time-averaged power density:

0 rE a H

����������������������������or

0

1rH a E

����������������������������

1( , , ) Re( )

2P r E H

������������������������������������������W/m2.

18

Total power radiated by the antenna can be expressed as

2( , , ) ( , , ) sinradP P r dS P r r d d ����������������������������

W

19

The shape or pattern of the radiated field is independent of r in the far field.

Radiation patterns usually indicate either electric field intensity or power intensity.

A transmit-receive pair of antennas must share the same polarization for the most efficient communication.

Normalized power function or normalized radiation intensity

max

( , , )( , )n

P rP

P

20

The isotropic antenna radiates EM waves equally in all directions so that

( , ) 1n isoP

21

The directional antenna radiates and receives EM waves preferentially in some directions.

Normalized electric field pattern:

max

( , , )( , )n

E rE

E

22

E-field pattern is plotted as a function of for constant .

H-field pattern is plotted as a function of for = /2.

In decibels, E-field pattern and Power pattern are similar.

( , )( ) 20log ( , )n nE dB E and

( , )( ) 10log ( , )n nP dB P

23

The overall ability of an antenna to direct radiated power in a given direction.Pattern solid angle:

A steradian (sr) is defined by an area r2 at the surface. A differential solid angle d, in sr, is defined as

sind d d 24

The solid angle of a sphere is found by integrating d such that

An antenna’s pattern solid angle:

2

0 0

sin 4 ( )d d sr

( , )p nP d

Comparing p for two Radiation patterns.

25

Normalized power’s average value:

Directivity gain D(,) is defined as

The maximum directive gain is called Directivity Dmax:

( , )( , )

4pn

n ave

P dP

d

( , )( , ) .

( , )n

n ave

PD

P

maxmax max

( , ) 4( , )

( , )n

n ave p

PD D

P

26

Total radiated power can be written as

max max( ) 10logD dB D

therefore we have

max( , ) ( , ).nD D P

2max ( , )rad nP r P P d

or2max .rad pP r P

27

28

The antenna resistance Rant consists of the radiation resistance Rrad and a dissipative resistance Rdiss that arises from ohmic losses in the metal conductor.

Assume

so we can write

For maximum radiated power, Rrad must be as large as possible but still easy to match with the feed line.

ant ant antZ R jX

0jI I e

20

1.

2rad radP I R

29

Dissipated power Pdiss can be written as

Antenna efficiency e is measured as

The power gain can then be expressed as

20

1.

2diss dissP I R

.rad rad

rad diss rad diss

P Re

P P R R

( , ) ( , ).G eD

30