1130 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



tions (277), which simplify somewhat if we write ix\ for T( , replace the 

 modified Bessel functions with ordinary Bessel functions according to 

 (292), and remember that H^, must vanish at the high-impedance 

 boundaries. We then get, approximately, 



H, = H,[ i(xi6)/i(xip) - ./l(xl&)A^l(xlP)]e""^ 



VI 



E,= j^'^. HolN.ixmiixw) - Ji(xih)N^{xip)]e~'% (302) 



E. = ^ Ho[Ni(xib)MxiP) - Jiixib)No{xip)]e~'% 

 9 



where Ho is an arbitrary amplitude factor. 



The potential difference between any two points in the same transverse 

 plane is the integral of —Ep between the points. Thus the total potential 

 difference between the core and the outer sheath is 



V = - [ Ep dp 



J a 



-Vl 



^' [iVi(xi&)/o(x.p) - J.i.xih)No{xip\y" (303) 



c Xi 



'p. 2Ho 



Mxib) 1 



e TTXi Lxia-^i(xia) Xi^J 

 after some transformations using equation (293) and the well-known 

 identity 



No(x)Jr(x) - Ni(x)Joix) = 2/t,x. (304) 



The average value of the conduction current density J z is 



7. = gEz = Hoxi[Ni(xih)Jo(xip) - Ji(xib)No(xipW''. (305) 



The current reverses at p = c, where a < c < b and c satisfies 



Ni(xib)Mxic) - Ji{xib)No(xic) = 0; (306) 



hence the value of c may be found with the aid of a table of Bessel func- 

 tions or from plotted curves. The total one-wa}' current in the outer 

 part of the stack is 



I = 2w Jzp dp 



= 2TcHo[Ji{xib)N^(xic) - Ni(xib)J,ixicW (307) 



^HoMxib) 



xiJoixic) 



e 



18 Reference 18, p. 208. 



