VARIABLE FREQUENCY ELECTRIC CIRCUIT THEORY 515 



Instead of dealing with an arbitrary instantaneous frequency ^{t) 

 we shall suppose that 



Q{t) = CO + mW, (3) 



where co is a constant and ij.(t) is the variable part of the instantaneous 

 frequency. In practical applications yu(/) will be written as \s(t) where 

 X is a real parameter and the mean square value s^ of s{t) is taken as 

 equal to 1/2. Other restrictions on ^t(/) will be imposed in the course 

 of the theory to be developed in this paper. Fortunately these 

 restrictions do not interfere with the application of the theory to 

 important problems. 



The steady-state current as given by (1) varies with time in precisely 

 the same way as the impressed e.m.f. When the frequency is variable 

 this is no longer true. On the other hand, formula (1) suggests a 

 "quasi-stationary" or "quasi-steady-state current" component, Igss, 

 defined by the formula 



I,ss = E Y(i^) -expli C' ndt \ , (4) 



which corresponds exactly to (1) with the distinction that the ad- 

 mittance is now an explicit function of time. We are thus led to 

 examine the significance of I^ss as defined above and the conditions 

 under which it is a valid approximate representation of the actual 

 response of the network to a variable frequency electromotive force, 

 as given by (2). 



We start with the fundamental formula of electric circuit theory.^ 

 Let an e.m.f. F(t) be impressed at time / = 0, on a network of indicial 

 admittance A{t); then the current /(/) in the network is given by 



/(/) = f^Fit - r)-A'{T)dT. (5) 



X 



Here A'(t) = dldt-A(t) and it is supposed that ^(0) = 0. (This 

 restriction does not limit our subsequent conclusions and is introduced 

 merely to simplify the formulas. Furthermore ^(0) is actually zero 

 in all physically realizable networks.) 



Omitting the superfluous amplitude constant E we have 



F{1) = exp(i rndt) 



= exp ( iwt + i I iJLdt J , (6) 



^ See J. R. Carson, "Electric Circuit Theory and Operational Calculus," p. 16. 



