MUTUAL IMPEDANCES OF GROUNDED CIRCUITS 17 



6. Mutual Impedances for Conductors Lying on the Surface 



of the Earth 



In order to arrive as directly as possible at a concrete numerical 

 idea of the magnitudes and angles occurring in the mutual impedances 

 encountered in engineering work, we may advantageously start with 

 the following specific constants : 



Base Length (AB, OA or A a) 1 Mile, 



Frequency of the Alternating Current 1 Kilocycle, 



Resistivity of the Earth per Centimeter Cube . 1 Megohm, 



which are of the right order of magnitude and make the factors 



p/(2ir AB) = 10 6 /(2tt 0.1609 X 10 6 ) ^0.989, 

 27r/Xv4BXl0- 9 = 27r 0.1609 = 1.011, 



which are both equal to unity within about 1 per cent., so that the 

 approximate resistance and reactance components of mutual im- 

 pedances may be read directly from Figs. 1-8 without applying multi- 

 pliers. On the other hand, when mutual impedances are required 

 for other lengths, frequencies and specific resistances, the correcting 

 factors are readily applied. The tangent of the angle of the mutual 

 impedance is proportional to the frequency, to the square of the 

 linear dimensions of the circuits and to the reciprocal of the earth's 

 resistivity. 



Grounded circuits separated by a distance large compared with 

 the dimensions of the circuits have a mutual impedance with negligible 

 resistance component since ultimately this component varies in- 

 versely as the cube of the distance by (3), whereas ultimately the 

 mutual reactance varies inversely with the distance. 



For two parallel grounded conductors separated by one mile and 

 forming opposite sides of a rectangle, the two components of the 

 mutual impedance are shown by Fig. 5 for the assumed constants 

 (approximately a kilocycle and a megohm). The resistance com- 

 ponent of the mutual impedance is then always less than the reactance 

 component; when the rectangle becomes a square, the mutual im- 

 pedance angle is tan _1 l. 595 = 57.9°. Reducing the frequency to 

 0.627 kilocycles or reducing the side of the square to 0.792 miles or 

 increasing the resistivity to 1.595 megohms would reduce this angle 

 to 45°. 



Consider two straight grounded conductors AB and ab, the latter 

 distance being small compared with the other dimensions of the 



