MUTUAL IMPEDANCES OF GROUNDED CIRCUITS 21 



face of the earth. Any flow of current through this system of con- 

 ductors may be divided into elementary filaments each of which is 

 made up of segments beginning and ending at the earth's surface 

 and not crossing the earth's surface between these terminals. Ground 

 each segment at both ends. Let 2^ and 5/ designate segments having 

 terminals A and B, W (for example, an open wire circuit) never 

 going below the surface of the earth and 2/ (for example, an under- 

 ground cable circuit) never going above the surface of the earth. Add 

 the underground flow & to Table II at the foot of the left-hand 

 column and, proceeding as before, 



= N(ty- JT){*-a) = N&*-N2/*+%A = NT/sr+NT/o-^, 

 = N(2S- JT) ( A - a) = Nt/a - N?/a - \ A = Nt/a + Nt/ - f A , 

 or Nt/o = § A - N&* = f A - Nta , 

 and similarly for the flow &/ above ground, 



Nw, = ±A + Nw„ = Nwa - |A. 

 Hence the three cases which may occur give 



N(te/- £)(u,- ) = Nwi* - N&/„ - Nt/u, 



(17) 

 = Nw», - Nwa - Nj? w + 2 A , 



N( 7/- S ) ( u - ,) = Nt/u + N2/x + Njr u 



f (18) 



= Nt/u -f Nt/a + Nj( u - 2A , 



N(2£/-€)( u -*) = NWu - Nw„ + Nj? u 



(19) 

 = Nw u - Nwa + Njf u . 



The importance of these equations lies in the fact that the earth 

 return flows are replaced by the simpler nadir, zenith and horizon 

 return flows. // the conductors comprise only broken straight filaments, 

 making any angles with each other and the earth, the required Neumann 

 integrals, if we use the expressions involving the nadir and zenith returns, 

 are the known expressions between straight filaments. If the conductors 

 lie in horizontal planes with vertical ground connections, it is con- 

 venient to employ the expressions involving the horizon return flows, 

 since the required integral is (4a) derived above. Formulas (33)-(41) 

 of the appendix are the resulting formulas for the three general cases 

 and for a number of important special cases. 



In general we may say that the effect of changing the height of one 

 or both conductors by an amount which is small compared with the 

 length of the conductors will be relatively small, since the effective 



