Page 254 - Electromagnetics

| ˜σ ij | E σ E
+ cos( ˇωt + ξ + ξ ) cos( ˇωt + ξ ) +
i
j
ij
ˇ ω
3

H µ H
+ ˇω |H i ||H j | −| ˜µ ij | sin( ˇωt + ξ + ξ ) cos( ˇωt + ξ ) .
j ij i
i, j=1
Using the angle-sum formulas and discarding the time-varying quantities, we may obtain
the time-average input power density:
3
ˇ ω
E E | ˜σ ij | E E σ
p in (r) =− |E i ||E j | |˜ ij | sin(ξ − ξ + ξ ) − cos(ξ − ξ + ξ ) −
j
ij
ij
j
i
i
2 ˇ ω
i, j=1
3
ˇ ω
H H µ
− |H i ||H j || ˜µ ij | sin(ξ j − ξ i + ξ ).
ij
2
i, j=1
The reader can easily verify that the conditions that make this quantity vanish, thus
describing a lossless material, are

|˜ ij |=|˜ ji |, ξ =−ξ , (4.150)
ij ji
σ
σ
| ˜σ ij |=| ˜σ ji |, ξ =−ξ + π, (4.151)
ij ji
µ µ
| ˜µ ij |=| ˜µ ji |, ξ =−ξ . (4.152)
ij ji
µ
σ

Note that this requires ξ = ξ = ξ = 0.
ii ii ii
The condition (4.152) is easily written in dyadic form as
†
˜ ¯ µ(r, ˇω) = ˜ ¯µ(r, ˇω) (4.153)
where “†” stands for the conjugate-transpose operation. The dyadic permeability ˜ ¯µ is
hermitian. The set of conditions (4.150)–(4.151) can also be written quite simply using
the complex permittivity dyadic (4.24):
c c
†
˜ ¯ (r, ˇω) = ˜ ¯ (r, ˇω). (4.154)
Thus, an anisotropic material is lossless when the both the dyadic permeability and the
complex dyadic permittivity are hermitian. Since ˇω is arbitrary, these results are exactly
those obtained in § 4.5.1. Note that in the special case of an isotropic material the
conditions (4.153) and (4.154) can only hold if ˜ and ˜µ are real and ˜σ is zero, agreeing
with our earlier conclusions.
4.9 The complex Poynting theorem
An equation having a striking resemblance to Poynting’s theorem can be obtained
by direct manipulation of the phasor-domain Maxwell equations. The result, although
certainly satisfied by the phasor fields, does not replace Poynting’s theorem as the power-
balance equation for time-harmonic fields. We shall be careful to contrast the interpre-
tation of the phasor expression with the actual time-harmonic Poynting theorem.
We begin by dotting both sides of the phasor-domain Faraday’s law with H to obtain
ˇ ∗
ˇ
ˇ ∗ ˇ
H · (∇× E) =− j ˇωH · B.
ˇ ∗

© 2001 by CRC Press LLC