MUTUAL IMPEDANCES OF PARALLEL WIRES 



523 



cularly symmetrical. Physically this is equivalent to assuming the 

 concentration of the current on the axes of the secondary as if it were 

 filamentary so that the proximity effect in this pair is eliminated. 

 Thus equation (13) formulates the solution of the case which is 

 represented in Fig. 5. Regarded as an approximation to the solution 

 when both pairs of wires are of large cross-section, it will be seen 

 that these values account for about 50 per cent of the departure 

 of the final results from the d.c. value. (This is 0.261 microhenry for 

 the square arrangement.) The second approximation (formula (14)) 

 takes into account the circularly unsymmetrical components of the 

 field due to the unsymmetrical distribution of current density in the 

 wires of the secondary as well as of the primary and so adds the prox- 

 imity effect due to the thickness of the secondary. A summary of the 

 formulas for mutual inductance follows: 



Formulas 

 With the notation 



X = ajlc 



a = radius of wires in centimeters 



2c = diagonal interaxial separation of wires in centimeters 

 <T = conductivity of wires in emu. 

 / = u/2ir = frequency 



and denoting by M'-^'' the complex mutual inductance per unit length 

 of two circuits, one of wires of large cross-section and one filamentary 

 or, from the other point of view, a first approximation to the mutual 

 inductance of two circuits of wires of large cross-section and, by AI, 

 a second approximation to the latter, we have 



ilf(o) = 4(iog^ V2 - ^i), (13) 



M = M(«> - 4{h - 7ki' + ^h), (14) 



where 



h 



Tl 



T2 



- 2X2ti 



- 12X^2 



2 ber' x -\- i bei' x 



-\-t 



X ber X -\- i bei x 

 V -\- iv when x ^ « 



. 8 4 ber x + i bei x 

 .r^ X ber' x -\- iheV x 



- 2v -\- i2v when x 

 aV47ro-co = V2/1', 

 /(27raVjV). 



