CIKCCIT IHIiOKy .1X1) Ofl R.ilKKWll. C.ll.CCl.CS IM 



roiuparinji tlicso expressions and remenilicrii»K that .!/»,= . U,», it 

 follows that the ciirretit in the A-tli l>ranrh corresponding to an ex|)o- 

 lunlial impressed e.ni.f. in the ./'ih branch, is efpial to the current in 

 the /'th branch corresponding; to the same e.m.f. in the ^ih br.mch. 

 Ihis relation is of tlic greatest technical importance. 



In many important technical problems we arc interested oidy in 

 two accessil)le branches, such as the sentlini; and receiNing. In such 

 (asc-s, where we are not concerned with the currents in the other 

 meshes or branches, it is often convenient to eliminate them from 

 the equation. Thus suppose that we have electromotive forces £i 

 ,inil £2 in meshes 1 and 2 and arc concerned only with the currents 

 in these meshes. If we soke ec)uations 3, 4, . . n, « — 2 in number, 

 for I3 . . In in terms of /i ami /: and then substitute in (1) and (2) 

 we t;et 



ZnIi-\-Z\zIi = E\. 



The Steady Stale Solutions 



The steady state solution, on which the wiiole theor\-of alternating 

 currents depends, is immediately deri\able from the exponential 

 solution. Let us suppose that £2 = £3= . . . =£„ = o and that £1 = 

 F cos {(jit — 6). Now by virtue of the well known formula in the 

 theory of the complex variable, cos .v= Je'^+jc""', we can write 



£1 = 4 £*'("'-<" +\Fe-'^"'-^\ 



= ] (cos e-i sin e)Ft»"' + i(ccs 6 + i sin 6) Fe" '"'1 (9) 



= J £V"-'+i£"e "'<*''. 



Now, by \irtue of this formula, the applied electromotive force £1 

 consists of tw'o exponential forces, one varying as e'"' and the other 

 as e"'"'. Hence it is easy to see that the currents are made up of 

 two compf)nents, thus 



/, = y/f '■"' + y/v- '■"' (j=i:2 . . n) ( lO) 



and we have merely to use the exjionential solution given abo\e, 

 substituting for X,/to and — iw respectively. That is, 



Jj = o 7 ,: , and Ji = \ 



2 Zji(iu) ' -Zji(-iw} 



, 1 Fe-'^ i^t, 1 Fe'" -ia 



2 Zj,{iwf 2 Zj,{-io,r 



