Toward a fundamental model for steady point-plane corona discharges

<div class="line" id="line-9"> <span style="font-family: Lora, serif; font-size: 20px;"> A mathematical model expressed in prolate spheroidal coordinates is developed from fundamental&nbsp;equations&nbsp;for steady state&nbsp;corona discharges&nbsp;from needle&hyphen;to&hyphen;plane&nbsp;electrodes.&nbsp;The bases of the theory are the&nbsp;Poisson&nbsp;and current continuity&nbsp;equations&nbsp;coupled with a&nbsp;mobility&nbsp;model for the transport of the&nbsp;charge carriers.&nbsp;The potential and field are constrained to be consistent with&nbsp;boundary conditions&nbsp;at the plane&nbsp;electrode,&nbsp;the&nbsp;surface boundary&nbsp;of the&nbsp;glow discharge,&nbsp;and the&nbsp;surface&nbsp;delineating the finite bound of the distribution of space charge. In the fully developed&nbsp;corona&nbsp;glow&nbsp;the steady state current is assumed to be limited by space&hyphen;charge fields at the&nbsp;glow&nbsp;boundary&nbsp;surface&nbsp;and a potential&nbsp; </span> <i style="font-family: Lora, serif; font-size: 20px;"> V </i> <span style="font-family: Lora, serif; font-size: 12px;"> 0 </span> <span style="font-family: Lora, serif; font-size: 20px;"> &nbsp;across the&nbsp;glow&nbsp;region is required to sustain the&nbsp;discharge.&nbsp;An additional requirement is that the space&hyphen;charge region must satisfy a power&hyphen;balance&nbsp;equation.&nbsp;Utilizing current vector potential methods results in a current density exactly yielding Warburg&rsquo;s law at the plane&nbsp;electrode.&nbsp;A current&hyphen;potential characteristic&nbsp;equation&nbsp;varying as the square of the potential drop across the space&hyphen;charge region is predicted, but because the space&hyphen;charge swarm is finite the current is collected from a finite area of the plane&nbsp;electrode&nbsp;only. The calculated potentials and fields are given in terms of quadratures. All of the solutions are exact in that no approximations are used in the calculations, and the solutions predict the assumed boundary values. An appendix discusses possible empirical tests of the theory. </span></div>