Transients in Parallel Grounded Circuits, One of 

 Which is of Infinite Length 



By LISS C. PETERSON 



This paper deals with a mathematical discussion of induction due to tran- 

 sient currents of the forms / = sin ut and / = e~^K Formulas and curves 

 are developed for the calculation of the induced voltage in exposed telephone 

 lines due to currents of the above types. 



Part I 



THE 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.^' ^'^ In addition to the steady state voltages 

 induced the transient voltages are also of importance. Riidenberg ■* 

 and Ollendorf ^ 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 symmetry 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. Dis- 

 turbing currents of the forms / = sin oot and / = e~^'' 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 wt 

 in a primary wire 1 infinite in length, separated from wire 2 by a dis- 

 tance X centimeters. 



1 PoUaczek, F., E. N. T., Vol. 3, 1926. 



- Carson, J. R., Bell System Techtiical Journal, Vol. 5, 1926. 

 ^Haberland G., Z. ang. Math. U. Mech., Vol. 6, No. 5, 1926. 

 * Wiss. Veroff. a. d. Siemens-Konzern, Vol. 5, No. 3, 1927. 

 5 E. N. T., Vol. 5, No. 3, 1928. 



760 



