This document discusses vector potentials and radiation integrals used in antenna engineering. It defines key terms like electric and magnetic vector potentials, electric and magnetic fields, and electric and magnetic current densities. It describes how vector potentials are used to simplify calculating radiated fields from sources. It also summarizes concepts like far field radiation patterns, duality between electric and magnetic fields, and reciprocity theorems relating fields and sources.

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)