1. 1
Chapter 3
Radiation Integrals
ECE 5318/6352
Antenna Engineering
Dr. Stuart Long

2. 2

VECTOR POTENTIALS
Mathematical tool used to simplify calculation of
radiated fields and
E H
Radiation Fields
,E H
Vector Potentials
,
FA
Sources
,J M

3. 3
Coordinate system for computing radiating fields
Fig. 3.2 (b) Coordinate system for computing
fields radiated by sources. Source not at origin
source
points
field
points




4. 4
Electric Field Strength [V/m]
Magnetic Field Strength [A/m]
Electric Volume Current Density [A/m²]
Magnetic Volume Current Density [V/m²]
J HE M

DEFINITIONS

5. 5

DEFINITIONS
(CONT)
Electric Vector Potential [Wb/m]
Electrical Scalar Potential [V]
Magnetic Vector Potential [A-sec/m]
Magnetic Scalar Potential [A]
A F m e

6. 6

Vector Potential
A for an
electric current source
J

Vector Potential
F for a
magnetic current source
M
1ejj       AAAAABHAEAHJE    1mjj       FFFFEFHFEMH   

7. 7

Vector Potential
A for an
electric current source
J

Vector Potential
F for a
magnetic current source
M
22' ejkRvjkedvR         AAAJAJ4     22' mjkRvjkedvR         FFFMFM4    

8. 8

The total fields are
FHEEEAFA    11j
(solutions derived in Sect. 3.5)
(also for surface and linear currents)
FFAEAHHH   j11
[3-29a]
[3-30a]

9. 9

FAR FIELD RADIATION
 22Dr r
Fields are essentially TEM to rAAEjAEjA         FFHjFHjF        

10. 10

FAR FIELDS RADIATION
(CONT)
 22Dr r
Fields are essentially TEM to rFFFFEHEH         AAAAEHEH          

11. 11

DUALITY
Identical equations for fields due to A and F
for J ≠ 0
M0


for J 0
M ≠ 0


EA HA J A k 
HF -EF M F k  










12. 12

RECIPROCITY

Assume LINEAR, ISOTROPIC
(but not necessarily homogeneous) materials
give sources J1 , M1 E1 , H1
and J2 , M2 E2 , H2

13. 13

RECIPROCITY
(CONT)

Lorentz Reciprocity Theorem
(also valid for fields away from finite sources – far fields)
each integral is called a “REACTION”
of fields (E, H ) to the sources (J, M )
' 1212' 2121dvdvvv MHJEMHJE
[3-66]

14. 14

For two antennas
equivalence of power transferred in
both directions for two antennas (p.147)

RECIPROCITY
(CONT)

15. 15
equivalence of transmit
and receive mode radiation
patterns (p.148)

For radiation patterns

RECIPROCITY
(CONT)