By Dragan Poljak

This article combines the basics of electromagnetics with numerical modeling to take on a wide variety of present electromagnetic compatibility (EMC) difficulties, together with issues of lightning, transmission strains, and grounding structures. It units forth a superior beginning within the fundamentals earlier than advancing to really good subject matters, and permits readers to boost their very own EMC computational types for functions in either learn and undefined.

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171). 13 GENERAL BOUNDARY CONDITIONS AND UNIQUENESS THEOREM For the realistic problem under consideration, the boundary qV of a problem domain V is generally composed of three distinct boundary segments qD, qN, and qC. Now, let the solution u (representing electric scalar potential, magnetic vector potential, electric ﬁeld, or magnetic ﬁeld) satisfy the following conditions on the three segments: u ¼ u0 on qD ðDirichletÞ ð2:172Þ qu ¼ q0 qn on qN ðNeumannÞ ð2:173Þ on qCðCauchyÞ ð2:174Þ qu þ au ¼ p0 qn The Dirichlet conditions are called essential or principal boundary conditions, whereas the Neumann boundary conditions are called natural boundary conditions.

Finally, the fourth Maxwell equation states that all magnetic poles occur in pairs and are due to electric currents; no free poles can exist in the electromagnetic ﬁeld theory. This is expressed by the divergence Maxwell equation r~ B¼0 ð2:4Þ which implies that the magnetic ﬁeld is always solenoidal. 1, induces an electromotive force around the loop. 1) results in a following expression: Z ~ Z qB ~ dS ð2:5Þ rx~ Ed~ S¼À qt S S where d~ S ¼~ ndS. 1 The surface S with related contour C. The surface integral is taken over any surface S bounded by the loop c.

73), therefore, represents the sum of the Joule heat losses and the negative rate at which the electromotive forces are doing work. The last term on the right-hand side requires deeper consideration. Namely, the term Z 0 ~ E Á~ J dV ð2:74Þ V represents an energy which has been hitherto neglected, since for static and quasistatic ﬁelds, the integral tends to vanish by taking an arbitrarily large enclosing surface. 70) Z 1 ~ ~ Wel ¼ ðE Á DÞdV ð2:75Þ 2 V represents a total energy stored in the electric ﬁeld system.