MUTUAL IMPEDANCES OF PARALLEL WIRES 513 



to the total, or resultant, voltage along the filament; that fs i?/// = Wf 

 = F/ + Uf. Hence Vf = Rflf — Uf, which is the most convenient 

 form in many applications, particularly those involving inductances. 

 Before taking up (in the next section) the more complicated subject 

 of the mutual and self impedances of "thick" wires, some of the fore- 

 going principles will be illustrated by applying them to the simple 

 system represented by Fig. 2, which comprises two filamentary wires 



POSITIVE DIRECTIONS 



H 



ooooo(^ ]G V 



K 



Fig. 2 — An illustrative circuit of two "filamentary" wires, H and K. 



II and K forming a loop. The dotted line G is merely a geometrical 

 path traced directly between the initial terminals of the two wires. 



The first form of principle "1," when applied to the two separate 

 paths G and IIK between the initial terminals, gives: 



Vg = Vh + (- Vk). (1) 



The following two equations result from the last form of principle 

 "4," when supplemented by the definitions of the self and mutual 

 inductances of filamentary wires, which enable the U's to be expressed 

 in terms of the /'s : 



Vh = RhIh — Uh ^ RhIh + i^Lnln + ioiLnKlK, (la) 

 Vk = RkIk — Uk = RkIk + icoLkIk + icoLkhIh, (lb) 



where Lh denotes the self inductance of wire H, Lhk the mutual in- 

 ductance '' between // and K, co = It times the frequency, and 

 i = ^| — 1. Further, on account of the choice of positive directions 

 shown in Fig. 2, Ik = — Ih- Accordingly, replacing Ik by — In and 

 substituting the resulting values of Vh and Vk into equation (1) 

 gives : 



Vg = {Zhh + Zkk - 2Zhk)Ih, (Ic) 



where Zhk = io^LnK = iccLKii = Zkh, Zhh = Rh + iooLn, etc. It 

 will be observed that while the "current voltages" have been elimi- 

 nated (through the self and mutual inductances and the currents), 

 the "charge voltages" remain and play the role of "applied voltages." 

 For wire H (Fig. 2), the equation (la), when written in the form 



RhIh = Wh = Vh -\- Uh = Vh - MLh - Lhk)Ih, (2) 



'The first subscript designates the "disturbed" wire, the second the "disturb- 

 ing" wire ("inducing" wire). 



