Lorentz Reciprocity Theorem
Page 1
Lorentz Reciprocity Theorem
Reciprocity
the role of
Reciprocity
which each
is an important concept in antennas because it produced implications when we reverse
transmitting antennas and receiving antennas. A formal derivation of the Lorentz
Theorem begins by considering a volume containing two sets of sources, J 1 and J 2 ,
produce fields E 1 , H 1 and E 2 , H 2 , respectively, as shown in Figure 1.
Figure 1: Volume containing two electric sources
Consider the quantity
∇ · (E 1 × H 2 − E 2 × H 1 ),
(1)
which is expandable using a vector identity as
(∇ × E 1 ) · H 2 − (∇ × H 2 ) · E 1 − (∇ × E 2 ) · H 1 + (∇ × H 1 ) · E 2 .
(2)
From Maxwell’s curl equations,
∇ × E1
∇ × H1
∇ × E2
∇ × H2
=
=
=
=
−jωµH 1
jωE 1 + J 1
−jωµH 2
jωE 2 + J 2 .
(3)
(4)
(5)
(6)
Therefore,
∇ · (E 1 × H 2 − E 2 × H 1 ) = −jωµH 1 · H 2 − jωE 2 · E 1 − J 2 · E 1
+jωµH 2 · H 1 + jωE 1 · E 2 + J 1 · E 2
= J 1 · E 2 − J 2 · E 1.
(7)
(8)
Since we took the divergence of a quantity in (1), let us now integrate the divergence over the
volume of interest:
˚
˚
0
∇ · (E 1 × H 2 − E 2 × H 1 )dv =
(J 1 · E 2 − J 2 · E 1 )dv 0
(9)
V
V
Applying the Divergence Theorem to the left hand side:
"
˚
0
(E 1 × H 2 − E 2 × H 1 ) · ds =
(J 1 · E 2 − J 2 · E 1 )dv 0 .
S
Prof. Sean Victor Hum
(10)
V
ECE422: Radio and Microwave Wireless Systems
Lorentz Reciprocity Theorem
Page 2
A more useful form of this theorem, applicable to antennas, is found by noticing that for electric
and magnetic fields observed a large distance from a source (e.g., a sphere of infinite radius
surrounding an antenna),
• E × H points in the radial direction normal to the sphere, n̂.
• E and H are related through H = (n̂ × E)/η.
Using the latter relation, the integrand on the left hand side of (10) can be re-written as
(E 1 × H 2 − E 2 × H 1 ) · n̂dS = (n̂ × E 1 ) · H 2 − (n̂ × E 2 ) · H 1
= ηH 1 · H 2 − ηH 2 · H 1
= 0
Hence,
˚
˚
J 2 · E 1 dv 0
0
J 1 · E 2 dv =
V
(11)
(12)
(13)
(14)
V
This is the form of the Reciprocity Theorem that is used in the analysis of receiving antennas.
Prof. Sean Victor Hum
ECE422: Radio and Microwave Wireless Systems