TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 3 



restrictions are desirable on their own account, because the selective 

 properties, both in the transient and steady-state, are isolated and 

 exhibited in the clearest manner when the disturbing effects of dissi- 

 pation and reflections are absent. As regards dissipation, its effect is 

 usually small for filters of ordinary length and, as regards transient 

 phenomena, is always of such a character as to require no essential 

 modification of the conclusions reached from a study of the ideal non- 

 dissipative filter. In fact the conclusions reached in this paper re- 

 garding the inherent limitations of selective circuits in the transient 

 state are conservative. 



II. General Theory and Formulas 



Before taking up the investigation of wave-filters it is necessary 

 to write down the fundamental formulas of electric circuit theory, 

 which are required in the analysis, and briefly discuss their applica- 

 tion to the investigation of transient phenomena in networks in 

 general. The theory and calculation of electrical networks may be 

 approached in a number of ways, as for example, from the Fourier 

 integral. 3 Perhaps the simplest way, however, is to base the theory 

 on the fundamental formulas 



^)= Tt £^-y)^y)^ 



i 



and 



l/pZ(p)- £ e-> l A(t)dt. II 



In these formulas I(t) is the current (expressed as an explicit time 

 function) in any branch or mesh of an electric network which flows 

 in response to the electromotive force f(t) which is applied to the 

 network at time /^0 in the same or any other branch or mesh of the 

 network. The function A{t) is a characteristic function of the con- 

 stants and connections of the network only which may be termed the 

 indicial admittance or the Heaviside Function. Its physical sig- 

 nificance may be inferred by setting f(t) = 1, whence it follows that 

 /(/) = A(t). That is to say A(t) is equal to the current in response to a 

 " unit e.m.f." (zero before, unity after time t = 0). 



In the following we shall be principally concerned with the case 

 when the applied electromotive force is sinusoidal. To deal with 

 this case we set/(/) =sin (co/-f 9) and equation I becomes 



lit) =a(w, t) sin (a>/ + 9) + b(w, /) cos (co/+G) III 



' The Solution of Circuit Problems, T. C. Fry, Phys. Rev., Aug., 1919. 



