Page 162 - Maxwell House

142 Chapter 3

leaving the resonator through the elements coupling it with external devices. Since the cavity
resonator is just the different realization of the well-known parallel ℒC circuit, the Q-factor can
be defined as usual via the measured resonance bandwidth 2∆ (see Figure as 3.2.1b) as

0 0
( ) = = (3.69)
0
2∆ 2∆
If the resonator disconnected from the external circuitry ( ) = 0 and the ratio
Σ
0
( 0 )
( ) = (3.70)
0 0
( 0 )
defines the self or unloaded resonance quality.

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. In general, the comparison allows to identify the energy consumed
in electrical elements as well as the energy stored in the model elements. Such analysis
often reveals weirdness like negative resistance or wrong frequency response occurring
due to some instability of numerical algorithm and let check the numerical convergence.
Besides, such circuit analysis might detects abnormal and virtual resonances due to
Maxwell’s equations might provide some solutions not to exist in reality. Equivalent
circuit based on Poynting’s theorem is often a good tool to verify assumptions inherent
in the model. Regardless of the outcome, look at your computer model and scrutinize
your expectation for a solution, not necessarily know what the answer will be for a
simulation. That's where your good knowledge of circuit theory is critical.

3.3 UNIQUENESS THEOREM FOR EXTERIOR
ELECTROMAGNETICS PROBLEMS

3.3.1 Radiation Condition

The typical exterior or radiation problem for Maxwell’s equations can be formulated as the
asymptotic limit of an internal problem when the closed boundary is extended to infinity
and is the whole 3-D space. We assume that all conductive and non-conductive objects, all
current and charge sources are contained within a bounded region. How to get the unique
solution without the designation of boundary conditions on this far-far away artificial surface?
Idealizing the problem as much as possible let us consider a point-like electric current source
of ℜ( ()) = 1W power inducing the electric and magnetic field in surrounding loss-free