Phenomenon of Standing Waves on Uniform Single Layer Coils - Revisited and Extended

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Accurate knowledge of the natural frequencies and shapes of corresponding standing waves are essential for gaining deeper insight into the nature of response of coils to impulse excitations. Most of the previous analytical studies on coils assumed shape of standing waves as sinusoidal but numerical circuit analysis and measurements suggest otherwise. Hence, this paper revisits the classical standing wave phenomenon in coils to ascertain reasons for this discrepancy and thereafter extends it by analytically deriving the exact mode shape of standing waves for both neutral open/short conditions. For this, the coil is modeled as a distributed network of elemental inductances and capacitances while spatial variation of mutual inductance between turns is described by an exponential function. Initially, an elegant derivation of the governing partial differential equation for surge distribution is presented which is then analytically solved, perhaps for the first time, by the variable-separable method to find the complete solution (sum of time and spatial terms). Hyperbolic terms in spatial part of solution have always been neglected but are included here, thus, yielding the exact mode shapes. Voltage standing waves gotten from analytical solution are plotted and compared with simulation results on a 100-section ladder network. The same is measured on a large-sized single layer coil. So, it emerges that, even in single layer coils, shape of standing waves deviates considerably from being sinusoidal and this deviation depends on spatial variation of mutual inductance, capacitive coupling, and order of standing waves.