Transients in Parallel Grounded Circuits, One of Which is of Infinite Length

HE problem of mutual impedance between grounded circuits of infinite length for steady state sinusoidal currents has been treated by a number of authors, and the solution of this problem is now well established. 1,2> 3 In addition to the steady state voltages induced the transient voltages are also of importance. Riidenberg 4 and Ollendorf 5 have given approximate solutions for transient voltages due either to d.-c. switching or the sudden flow of a sinusoidal current on the assumption of circular s y m m e t r y and for circuits one of which is of infinite length. Since the assumption of circular symmetry holds only for a limited set of conditions it is desirable to develop formulas for the transient induced voltages based on the exact solution for steady state conditions referred to above. The discussion in this paper will be limited to the case of parallel wires, one of which is of infinite length, and both located on the surface of the earth but insulated from it except at their ends. Disturbing currents of the forms I = sin «/ and I = will be assumed. A more general case with both wires above the earth's surface leads to complicated expressions for the induced voltage not well adapted for engineering use. The restriction to wires on the earth's surface results in appreciable simplification and does not introduce a serious departure from actual conditions. With these assumptions, the following formulas holding for small and large values of time, determine the induced voltage per unit length on a secondary wire 2 due to the sudden flow of a current /(/) = sin cot in a primary wire 1 infinite in length, separated from wire 2 by a distance x centimeters.