Page 174 - Maxwell House

154 Chapter 3

Alternatively, in scalar form

∗ ∗

∬ � + � ≈ 4( + ) (3.95)

In case of one mode regime, we can simplify the integrand in (3.95) using the relationship (3.85)
between the aperture tangential component electric and magnetic field

= (3.96)

Then

∗ ∗ ∗
∗ 2
+ = (+ ) + � � (3.97)

Since this one-hole cavity in Figure 3.4.1 is assumed to be practically lossless, the input
impedance = + must be purely imaginary, i.e. in (3.97) 2 = ( + ) ≈ 0. Pay
∗
attention to the fact how seamless the symbiosis of Maxwell’s equations and circuit theory stats
working. Therefore, (3.97) can be rewritten as

�
2 (3.98)
≈ 4( + )/ ∬ �

Meanwhile, the stored EM energy can never be negative. If so, the derivative on the right-hand
side of (3.98) is always positive meaning that

()
> 0 (3.99)

If so, Foster’s reactance theorem tells us that the slope of the reactance curve as a function of
frequency is always positive for a no-loss or relatively small loss circuit. It is worthwhile to
point out that Foster’s theorem and following publications facilitated the fast development of
not only the network analysis but its synthesis. It is out of the scope of this book to go into
details about the network synthesis problem. There are excellent references in the literature [5],
and readers are encouraged to refer to them.

Hint: As often as possible try to build a circuit equivalent of your EM model before
starting the numerical simulation using your engineering experience and after using
Poynting’s theorem. Then put the numerical analysis results on Smith chart and check
Foster’s reactance theorem. Such analysis often reveals weirdness like negative
resistance or wrong frequency response occurring due to some instability of numerical
algorithm. Besides, such circuit analysis might detects abnormal and virtual resonances
due to Maxwell’s equations are able to provide some solutions that does exist in reality
(see Section 3.2.3).