MUTUAL IMPEDANCES OF PARALLEL WIRES 529 



(3) Determine An^'^^ and Bn^^^ in terms of C„^^^ and ZP„(') by con- 

 ditions at Tx = a. 



Then, we have, for example, 



Z ( - ^j C„ = E ( - ^ j (Cn^o) + C„(i) +• C„(2) + • • •) (35) 



and similar expressions for the other summations. Now, putting 

 (7„(o) = i;)„(o) = 0, the first approximation to the proximity effect 

 is given by 



^,(0) ^ (0) 



A/»)/i + A,.(o)/o = ^ + ^ . (36) 



Next, since, for example, ^2^^7(2^)^ is of the same order of magnitude 

 as ^1^^7(2^), the increment due to C„^^^ and Dn^^^ will be 



yl/') 5/^) P/i) 

 a, ix ^ ^m 11 2c ^ 2c 2c 



HoW) Cp^') D9<1) 



(2c)2 (2c)2^ (2c)2- ^^'^ 

 Then a second approximation to A^/i + A,„7'2 will be 



(A,(o) + A,(i))/i + (A^(°> + A„.(i))/2 

 and, in general, 



AJi + A„,/2 = E (A.(»)/i + A«(")/2). (38) 



Applying this method we assume unit permeability for wires and 

 dielectric. Then putting the first approximation in equation (34) 

 gives equation (13) above for M^^'^ . Neglecting terms containing 

 \\ we find Ci(2)/2c and i^i(2)/2c ignorable. In B2^'^l{2c)\ C2^'^/{2cy 

 and D^^^^ jilcy we require the first terms. We then have 



^,^'Ul + Aj'^h = - 2io:h(ki - 6kx^ + • • • + 1^2 + • • •) 



- 2io}h{kx - Iki" + • • • + 4/C2 + • • •). (39) 

 Hence 



Z„. = Ua>i\og V2 - 2y^i + Jki" - •■' - 4^2 + • • •) (40) 



and 



Z, = 2Zi + 4ico C log ^ - Ski + 6ki'' - ••• -^ko-\- ■■■\ (41) 



