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POYNTING's THEOREM 149

1. Ignore the singularities. They are all local and according to (3.73) and (3.74) carry
relatively small energy. Eventually, the overall simulation error depends on the slice of
reactive energy concentrated around the singularity relative to the total energy balance.
Subsequently, too accurate simulation of EM fields around vertexes and tips typically is
not required. This advice is impractical when the central purpose of the numerical
simulation is the estimation of such phenomena as corona impact, lightning strike or
breakdown probability.
2. Give the edge a finite radius of curvature as shown in Figure 3.3.3a smoothing the
conductive surface and keeping close to real the fast field variations around the areas of
high curvature. The central question is how far we can proceed with such smoothing. In
general, this approach is efficient, widely used in engineering practice, but requires the
dynamic mesh generator that automatically or interactively put more small cells in such
areas. The total mesh count goes up drastically and can reach many millions for the model
like depicted in Figure 3.3.8. It means more cumbersome and irritating for a user
discretization algorithm, much longer computational time for numerical model preparation,
time-consuming numerical simulations and following post-processing animation of results.
3. Incorporate into the numerical algorithm the a priori known analytical as well as an
empirical description of field singularities restricted by the finite energy principle. In
general, such approach leads to vast improvement in numerical accuracy. The problem is
that the data, required for such description are not always available. Besides, the
implementation complexity of algorithm increases and the numerical algorithm
universality shrinks.

Hint: Always use in the numerical simulations a variable mesh with the increasing
density near the sharp edges and check the limitedness of stored energy.

3.4 REFLECTION CONCEPT. LORENTZ’s RECIPROCITY
THEOREM

3.4.1 Concept of Reflection and Impedance

The considered below Foster theorem, named after the American mathematician Ronald Martin
Foster (1896-1998) and published in 1924, gives us a universal tool to check the correctness of
the numerical or analytical frequency domain analysis of high complexity circuit with the
minimal knowledge about the internal structure of RF circuit. To demonstrate this, we need
first to introduce the concept of complex reflection coefficient Γ and characteristic
impedance .

At low frequencies including DC, all time delay effects imposed by the finite speed of light are
so tiny that the circuit elements and the entire circuitry can be treated as lumped. If so, we can