364 BELL SYSTEM TECHNICAL JOURNAL 



While this proposition is precisely analogous to the corresponding 

 proposition in the case of a sudden short circuit, it does not explicitly 

 determine the voltage V{t), which must be calculated as follows: 



Let the driving point indicial admittance of the network, as seen 

 from branch ah be denoted by Aah{t). Then, from the preceding propo- 

 sition, it follows at once that V{t) is given by 



~ rViT)Aab{t-r)dT^-I(t), t>0. 

 at ,/o 



This is a Poisson integral equation in V{t), from which V(t) is calcul- 

 able. With V(t) determined, the currents in any part of the network 

 are calculable by usual methods, and consist of two components: — 



(1) The current distribution in the network due to the impressed 

 source E{t) in the absence of the open circuit. 



(2) The current distribution due to the electromotive force V{t) 

 inserted in branch ah at time t = 0. 



As in the case of the sudden short circuit, we have thus reduced the 

 problem of a sudden open circuit to the determination of the current 

 distribution in an invariable network in response to a suddenly im- 

 pressed electromotive force. 



Variable Circuit Elements 



In the preceding cases of sudden open and short circuits it will be 

 observed that the network changes discontinuously from one invariable 

 state to another. A more general case, and one which includes the pre- 

 ceding as limiting cases, is presented by a network which includes a 

 variable circuit element : that is, a circuit element which varies, con- 

 tinuously or discontinuously, with time. A network which includes 

 such a variable circuit element will be called a variable network. Vari- 

 able circuit elements of practical importance are the microphone trans- 

 mitter, which consists of a variable resistance, varied by some source of 

 energy outside the system; the condenser transmitter, which consists of 

 a condenser of variable capacity; and the induction generator, in 

 which the mutual inductance between primary and secondary, or 

 stator and rotor, is varied by the motion of the latter. The case of a 

 variable resistance will serve as an introduction to the general theory 

 of such variable networks. 



Consider a network, energized by a source E{t) in branch 1, and 

 containing a variable resistance element r{t) in branch n. The func- 

 tional notation r{t) indicates that the resistance r varies with time. 

 Let /«(/) denote the current in branch n, and assume that the network 



