788 ELECTRICAL METHODS IChai'. 10 



A -y/g^ gQg2 Q _ Y^. Squaring and dividing by A^B^ and cos^ 6 give 



'B^ sin^ d + A^^ 



Z?(i.) + 2Z.Yf-^^) + Y^f 



A2B2 cos2 



= 1. (10-53c) 



This equation has the standard form of an inchned elUpse (see eq. [10-24c]), 

 so thatL = 1/A'; M = tan d/A^; N = (A' + B' sin' 9)/A'B' cos' 6. The 

 tilt angle \p from vertical is then 



+ o/ 2M 2 tan 9 B' sin 2^ .,„ „,. 



N - L A^ + B=^ sin'' 6 

 B2 cos2 d 



- 1 



A2 - B2 cos 25 ■ 



Fig. 10-109. Elliptical polarization resulting from a quadrature field (produced 

 by a conductor at C), which is equal to the in-phase (loop) field above C (after Edge 

 and Laby). 



The squares of the major and minor semiaxes of the ellipse are, in accord- 

 ance with eq. (10-256), 



2 ,2 



a , = 



L + N + V4M2 + (L - Ny 

 2A'B' cos' d 



A' + B' + V[A2 - B2 cos 2e]2 + [B2 sin 2dY' 



(10-53e) 



For two fields of equal maximum amplitude, A = B and tan 2\{/ = 

 sin 20/(1 - cos 20) = tan (x/2 - 6), so that i^ = 7r/4 - d/2. Eq. (10-53e) 

 is then a, h = B cos d/\/i ip ,sin d- 



Fig. 10-109 shows the variation of compression and tilt angle of the 

 polarization ellipse with distance for a conductor carrying a quadrature 

 current at depth d. If the out-of-phase field is directly above the con- 

 ductor, it is equal to the in-phase field (A = B), and if T declines from 

 there in proportion to d/r (since Tmax. - 21' /d and T = 21' /r), it is seen 



