Page 255 - Electromagnetics

ˇ
Taking the complex conjugate of the phasor-domain Ampere’s law and dotting with E,
we have
ˇ
ˇ
ˇ ∗
ˇ ∗
ˇ ˇ ∗
E · (∇× H ) = E · J − j ˇωE · D .
We subtract these expressions and use (B.44) to write
ˇ
ˇ
ˇ
ˇ ∗
ˇ ˇ ∗
ˇ ∗
ˇ ∗
−E · J =∇ · (E × H ) − j ˇω[E · D − B · H ].
Finally, integrating over the volume region V and dividing by two, we have
1 1 1 1
ˇ
ˇ
ˇ
ˇ ∗
ˇ ˇ ∗
ˇ ∗
ˇ ∗
− E · J dV = (E × H ) · dS − 2 j ˇω E · D − B · H dV. (4.155)
2 V 2 S V 4 4
This is known as the complex Poynting theorem, and is an expression that must be obeyed
by the phasor fields.
As a power balance theorem, the complex Poynting theorem has meaning only for
c
i
dispersionless materials. If we let J = J +J and assume no dispersion, (4.155) becomes
1 1 1
ˇ
ˇ ˇ i∗
ˇ ∗
ˇ ˇ c∗
− E · J dV = E · J dV + (E × H ) · dS −
2 V 2 V 2 S

− 2 jω [ w e − w m ] dV (4.156)
V
where w e and w m are the time-average stored electric and magnetic energy densities
as described in (4.62)–(4.63). Selection of the real part now gives
1 1 1
ˇ
ˇ ∗
ˇ ˇ c∗
ˇ ˇ i∗
− Re E · J dV = E · J dV + Re E × H · dS, (4.157)
2 V 2 V 2 S
which is identical to (4.147). Thus the real part of the complex Poynting theorem gives
the balance of time-average power for a dispersionless material.
Selection of the imaginary part of (4.156) gives the balance of imaginary, or reactive
power:
1 1
ˇ
ˇ ∗
ˇ ˇ i∗
− Im E · J dV = Im E × H · dS − 2 ˇω [ w e − w m ] dV. (4.158)
2 V 2 S V
In general, the reactive power balance does not have a simple physical interpretation (it
is not the balance of the oscillating terms in (4.139)). However, an interesting concept
can be gleaned from it. If the source current and electric field are in phase, and there is
no reactive power leaving S, then the time-average stored electric energy is equal to the
time-average stored magnetic energy:

w e dV = w m dV.
V V
This is the condition for “resonance.” An example is a series RLC circuit with the source
current and voltage in phase. Here the stored energy in the capacitor is equal to the
stored energy in the inductor and the input impedance (ratio of voltage to current) is
real. Such a resonance occurs at only one value of frequency. In more complicated
electromagnetic systems resonance may occur at many discrete eigenfrequencies.
© 2001 by CRC Press LLC