Digital Signal Processing Reference
In-Depth Information
This allows us to define the
Poynting vector
, which represents the flow of power
per unit area through a surface
S
at an instant in time.
S
=
E
×
H
W
/
m
2
(2-115)
E
and
H
.
Note that the direction of the power flow is perpendicular to both
2.6.1 Time-Averaged Values
When considering the electromagnetic power delivered by a sinusoidal
time-varying field, practical measurement considerations tend to favor the
time-averaged value of the power rather than the instantaneous value described
in (2-115). This is because the time-averaged power entering a passive network,
as measured with a watt-meter, is a measure of the power dissipated by heat
in all the resistive circuit elements. In the laboratory, the time average of
a time-harmonic function is taken over an interval of many periods. For a
steady-state sinusoidal function, the average of one period will be the same
as the average over many periods, since each period looks identical. The
time
average
of the Poynting vector is defined as the area under the function for one
period, divided by the duration of the cycle:
T
A
period
T
period
1
T
S
ave
S(x, y,z,t)dt
=
=
(2-116)
0
Although the math is not shown here, Jackson [1999] shows the derivation of
the time average value for the Poynting vector:
S
ave
2
Re
( E
H
∗
)
1
W
/
m
2
=
×
(2-117)
where the star represents the complex conjugate.
It is sometimes useful to represent the magnitude of the Poynting vector in
terms of the electric or magnetic fields and the intrinsic impedance defined in
equation (2-53). Integrating sinusoidal functions as described in (2-116) will yield
the time-averaged Poytning vector in terms of the
E
and
H
fields [Johnk, 1988]:
E
=
a
x
E
+
cos
(ωt
−
βz)
(2-118)
H
=
a
y
H
+
η
cos
(ωt
−
βz)
(2-119)
cos
(ωt
−
βz)
a
y
E
+
η
S
=
E
×
H
a
x
E
+
cos
(ωt
−
βz)
]
=
[
×
2
=
a
z
(E
+
)
η
cos
2
(ωt
−
βz)
(2-120)
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