18 BELL SYSTEM TECHNICAL JOURNAL 



system and assume that while ground a is fixed, ground b "is rotated 

 about a in a circle of fixed radius ab. This will vary the mutual 

 impedance between the two grounded conductors. The reactance 

 component will vary as the cosine of the angle between ab and AB. 

 The resistance component will also vary as the cosine of an angle, 

 measured, in general, from a direction other than AB. The maxi- 

 mum resistance component will also differ, in general, from the maxi- 

 mum reactance component. The locus of the mutual impedance 

 obtained for all positions of ab will be an ellipse. The ellipse becomes 

 a straight line when ground a lies on the bisecting normal of AB, 

 for then the direction giving the maximum resistance component is 

 parallel to AB and thus the same as for the maximum reactance com- 

 ponent. The straight line limit is also obtained when ground a lies 

 on the prolongation of AB in either direction, for the resistance 

 component then has its maximum value in the direction opposite to 

 AB. The maximum resistance component will be perpendicular 

 to AB at the points on Fig. 1 where the C/I contour, if drawn, would 

 be vertical. The locus of these points is 



y* = ( X 2 -b 2 )W[(x + b)W+(x-b)W]. ( 10 ) 



At remote points on this locus y = ± V2 x and the maximum resist- 

 ance is negligible compared with the maximum reactance since the 

 circuits are widely separated, while in the neighborhood of A and B 

 it is the maximum reactance which is negligible compared with the 

 maximum resistance. At some intermediate point the two maxima 

 are equal, and, since they occur for directions differing by 90°, the 

 elliptical impedance locus becomes a circle; and the mutual impedance 

 between the two grounded circuits does not change in magnitude 

 as ground b is rotated about ground a. For the assumed constants 

 (1 mile, 1 kilocycle and 1 megohm) this point lies at distances of 

 1.562 and 0.939 miles from the two terminals A and B, and its four 

 possible locations are shown by the four small crosses on Fig. 1. 



If a is rotated counter-clockwise about AB, the direction giving the 

 maximum resistance component also rotates counter-clockwise, making 

 two complete revolutions, while the ground a makes one revolution 

 about AB. 



7. Equivalent Ground Plane 



To a first approximation the direct-current mutual inductance 

 between two straight conductors AB and ab, forming opposite sides 

 of a rectangle on the surface of the earth, at a separation A a which 



