370 BP.IJ- SYSTEM TECHNICAL JOURNAL 



f{t)Yi(t); etc. The resultant cun-ents arc thus represented as l)uilL up 

 1)\- a lo-and-fro inierchanue of enerti\- l)et\veen i)riniar\- and secondary, 

 or by a series of successi\-e reactions. In the important case where 

 the appHed e.m.f. and the variation of mutual inductance are both 

 sinusoidal time functions, of frequency F and/ respectively, it is easy 

 to show that each component current becomes ultimately equal to 

 a set of periodic steady-state currents. Thus the component Xo is 

 ultimately single periodic, of frequency F; Fi is ultimately doubly 

 periodic, of frequencies F-{-f and F—f; X2 triply periodic, of fre- 

 quencies F-\-2f, F and F-2f; Yz quadruply periodic, of frequen- 

 cies F+3f, F-\-f,F-f,F-3f; etc. 



The Solution for the Steady-State Oscillations 



For the very important case of periodic applied forces and periodic 

 variations of circuit elements we are often concerned exclusively 

 with the ultimate steady-state of the system, and not at all with the 

 mode in which the steady-state is approached : that is, attention is 

 restricted to the periodic oscillations which the system executes after 

 transient disturbances have died away. In this case, if the periodic 

 variations of circuit elements are sutliciently small, the required steady- 

 state is obtained in the form of a series by replacing each term of the 

 complete series solution by its ultimate steady-state value; a process 

 which is \-ery simple in view of the physical significance of each term 

 of the latter series. The appropriate procedure will be briefly illus- 

 trated in connection with the \ariable resistance element. In view 

 of the fact that we are concerned onh* with the ultimate steady- 

 state oscillations, we can base the solutions on the symbolic equation 



I = I^-'Ml (293) 



Here r{t) is the variable resistance element; lo is the current which 

 would flow in the absence of the resistance variation; and Z is a gen- 

 eralized impedance of the network, as seen from the variable branch. 

 Its precise significance and functional form is given below. 

 We now suppose that To is given by 



/o = /oe'"' (real part) (294) 



= i (/„f"«'+ Jog -'""') (295) 



where the bar indicates the conjugate imaginary of the unbarred 



