Page 153 - Maxwell House

POYNTING's THEOREM 133

numerical solution. Poynting’s theorem lets remove the wall between Maxwell’s equations and
much more transparent circuit theory. It allows updating the lumped elements equivalent
circuits and including in analysis mutual coupling effects between elements. In general, the
output of such hybrid analysis is the equivalent circuit model so it can be easily integrated with
other active and passive circuit models for a conventional circuit simulation.

We know that the electric and magnetic fields can exist not only in capacitors or inductances
but within any region of free space or space filled with matter. If so, we show in the next chapter
how to build an equivalent circuit of material free or filled up space using lumped elements.

3.1.16 Poynting’s Theorem in Space-Frequency Domain

In general, the direct measurement of instantaneous power considered above is very challenging
and difficult procedure requiring an ultra-fast EM field sensor in assembly with highly
sophisticated and expensive instruments. One way around is to measure, as an alternative, the
“time-average” in-phase and quadrature magnitude of power defined in Chapter 1 (see (1.80)).

The differential form of energy conservation law in the space-frequency domain. Following
(1.83) we can express the average power through the dot (∘) or cross (x) product of two phasor-
vectors (, )exp () and (, )exp () as

1 1
∗
∗
(, ) exp() ∘ (, ) exp(−) = (, ) ∘ (, )
2 2
1 1
∗
∗
(, ) exp() x (, ) exp(−) = (, ) x (, )
2 2
where (, ) = |(, )| () ,(, ) = |(, )| () . The values () and

() is the phase of (, ) and (, ), respectively. Both products belong to the space-

frequency domain and, evidently, are the time-independent values. Then (3.2) can be rewritten
as
∗
∗
∗
/2 ∘ x = − + + ( ∗
)
� � (3.44)
∗
/2 ∘ − x =
Here the factor replaces the time derivatives (see Table 1.9) and the complex conjugate
operator applied to the right-hand side of the first equation in (3.44) revises the factor
to – . Taking scalar product or dotting the top equation in (3.44) with and bottom one
with , as shown in (3.44), we get summing all the products
∗
1 ∗ 1 ∗ 1 ∗ ∗ 1 ∗ 1 ∗
∘ x − ∘ x = − ( ∘ − ∘ ) + ∘ + ( ) ∘ (3.45)
2 2 2 2 2
∗ ∗
Applying the vector identity (A.35) from Appendix, ∘ x − ∘ x = − ∘
( x ), we can transform (3.45) and obtain after regrouping the terms
∗
) ∘ = ∘ ( x ) − ( ∘ − ∘ ) + ∘
1 1 1 1
− ( ∗ ∗ ∗ ∗ ∗ (3.46)
2 2 2 2