FILE (SILVER) D:\TEACHING\L5EMT&RFID10\PART9(OTHERNOTESTEXT)\
FORMULAESUPPLIEDINEXAMS.DOC
SUMMARY OF FORMULAE
1.
Dipole Fields
1.1
Electric dipole
In spherical polar coordinates at a point P(r,) the non-zero field components of an oscillating
small electric dipole of length L carrying a current I and of moment P where jP = IL are
 2 j P  2
2 j   j r
Er 

cos
(1)

e
2
4
( r )3 
 ( r )
1.2
E 
 2 j P  j
1
j   j r


sin 

e
2
4
( r )3 
 ( r ) ( r )
( 2)
H 
 2 j P  j
1   j r

e

sin 
4  (  r ) (  r ) 2 
(3)
Magnetic Dipole
In spherical polar coordinates at a point P(r,) the non-zero field components of an oscillating
small magnetic dipole of moment M = IA are:
Hr 
j 3 M  2
2 j   j r

e

cos
2
4  (  r )
( r )3 
H 
E  
2.
Radiation Resistances
2.1
Electric dipole
j 3 M
4
 j
1
j   j r

e


sin 
2
(  r )3 
 ( r ) ( r )
j 3 M
4
 j
1   j r

e

sin
2 
( r ) ( r ) 
(4)
(5)
(6)
The radiation resistance of a short electric dipole of length L, operating at a frequency for which
the free space propagation constant has magnitude , is given in Ohms by
Rr  20(L) 2
2.2
(7)
Magnetic dipole
The radiation resistance of a small current loop of radius a, operating at a frequency for which
the free space propagation constant has magnitude , is given in Ohms by
7052 Electromagnetic Theory & RFID Applications 2010 Formulae Supplied in Exams Page 2
Rr  20 2 (a) 4
(8)
Small loops of other shapes but the same area have the same radiation resistance.
3.
The Fourier Series
The Fourier series
x(t ) 

X
n  
n
e j 2fnt
for the rectangular pulse train x(t) of unit amplitude, pulse duration  and pulse repetition period
T as shown in the diagram below
x(t)
1
-
O
T
t
A rectangular pulse train.
is given by
   sin(f n )
Xn   
 T  f n
(9)
We see the d.c. component is /T, and the spectral lines have a (sinx)/x envelope, all as
illustrated in magnitude form in the diagram below.
|Xn|
T
13.5 dB
O
1
2
n
f= T
1.5
Spectrum of a rectangular pulse train.
We note that the first side lobes are located at a frequency of 1.5 times the frequency of the first
null, and are 13.5 dB down on the central peak which is at zero frequency. The spectral lines are
at frequencies fn = n/T, and do not have to coincide with nulls of the response defined by the
(sinx)/x envelope.
7052 Electromagnetic Theory & RFID Applications 2010 Formulae Supplied in Exams Page 3
4.
Inductance Calculations
4.1
Planar Circular Coil
The self inductance of a single-turn circular coil of diameter D made from wire of diameter d is,
when the currents flow on the surface, given by
oD 
8D
log e ( )  2
(10)

2 
d

An empirical but useful formula for the self inductance of a thin wire solenoidal coil of N turns
wound over a length of l on a former of diameter of 2r as shown in the diagram below is
L 
l  0.9 r
.
(11)
2r
L
 0r 2 N 2
N turns
l
4.2
Twin Wire Line
The self inductance L of a twin-wire line in which the conductors have diameter d and
separation s is given by

s
L  o arc cosh  
(12)
 d

o
2s

log  
when s  d
(13)
d

5.
Axial Field of a Circular Coil
In the magnetostatic approximation, the field at a point at a distance z along the axis of a single
turn circular coil of radius a carrying a current I is given by
H z (0, 0, z) 
Ia 2
2 a  z
2
6.
3
2 2

(14)
Skin effect
Skin depth  in a metal at an angular frequency  is given by

1


2

The surface resistivity Rs per square due to skin effect is
1

Rs 


2
(15)
(16)
7052 Electromagnetic Theory & RFID Applications 2010 Formulae Supplied in Exams Page 4
7.
Waveguide Propagation
The dominant mode field configuration for forward wave in rectangular waveguide of interior
dimensions a and b along the x and y axes respectively is given by
Ex  0
E y   j
0
x
H sin( )e  jz
c
a
Ez  0
(16)

x
H x  j H sin( )e  jz
c
a
Hy  0
x
H z  H cos( )e  jz
a
The propagation constant  for either the TE or TM modes of a wave guide satisfies the relation
  
   c  
 c   c
2
2
2
(17)
The cut-off angular frequency c of the TElm or TMlm mode in rectangular waveguide is given
by
c 
 l 
 m 

     
a
 b 
 c 
2
2
2
(18)
The transverse field wave impedances ZTE and ZTE for rectangular waveguide are given by
 
Z TE    0 
  
  

Z TM   
 0 
(19)
The dimensions of standard X-band waveguide are a = 22.86 mm and b = 10.16 mm.
8.
Reciprocity
The integral form of the Lorenz reciprocity relation for two solutions E1, H1 and E2, H2 of
Maxwell’s equations in the same region and at the same angular frequency  is under
appropriate conditions
 E  H
S
1
2
 E 2  H1   ds 
 J
v
1
 E 2  J 2  E1  dv
7052 Electromagnetic Theory & RFID Applications 2010 Formulae Supplied in Exams Page 5
9.
Vector Calculus
In spherical polar co-ordinates the components of curl A where A is an arbitrary vector are
given by
1   (sin A ) A 

r sin   
 
1  1  ( Ar )  (rA ) 
(  A)  

r  sin  
r 
1   (rA )  ( Ar ) 
(  A)  

r  r
 
(  A) r 
10.
Power Transfer
When a source of impedance ZS feeds a load of impedance ZL, the ratio of the power transferred
to the load to the available source power PA is given by
PL
| Z  Z L* |2
.
 1 s
PA
| Z s  Z L |2
11.
Scattering matrices
The canonical form of the scattering matrix for a lossless reciprocal two-port is

r
S
2
  j 1  r
 j 1 r2 

r

where r is the reflection factor of the junction at a reference plane at which r is real and positive.
END OF FORMULAE SUPPLIED