TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 5 



is of great importance in understanding and interpreting the behavior 

 of selective networks to transient disturbances. 4 



The indicial admittance A (/) is calculable from and may be regarded 

 as defined by the very compact formula II. 5 In this equation Z{p) 

 is the operational impedance of the network. It is derived from the 

 differential equations of the problem by replacing the differential 

 operator d/dl by the symbol p, thus formally reducing the equations 

 to an algebraic form from which the ratio 1/Z(p) of the current 

 to electromotive force is gotten by ordinary algebraic processes. 

 Z{p) will involve the constants and connections of the network 

 and will depend, of course, on the mesh or branch in which the 

 electromotive force is inserted and that in which the required cur- 

 rent is measured. 



The procedure in formulating the transient behavior of networks 

 is as follows. Derive the operational impedance Z(p) as stated above. 

 With Z(p) formulated, the corresponding indicial admittance A{t) 

 is determined by the integral equation II. The appropriate methods 

 of solution of the integral equation are briefly discussed in "The 

 Heaviside Operational Calculus." Sometimes the solution can be 

 recognized by inspection as in the case of the low pass wave-filter. 

 Otherwise the procedure in general is to expand \/Z{p) in such a form 

 that the individual terms of the expansion are recognizable as identical 

 with infinite integrals of the required type. Two expansions of this 

 kind lead to the Heaviside Expansion and power series solution, re- 

 spectively. The appropriate form of expansion depends on the par- 

 ticular problem in hand and often calls for considerable ingenuity 

 and experience. An excellent illustration of the appropriate process 

 is furnished by the detailed derivation 6 of the indicial admittance of 

 the band pass filter which is rather intricate. 



In connection with the problem of the energy absorbed from forces 

 of finite duration, and from random interference, the following formulas 

 are required, of which VIII and IX are original and hitherto unpub- 

 lished. Formula X, which is a special case of VIII and IX was derived 

 by Rayleigh {Phil. Mag., Vol. 27, 1889, p. 466), in connection with 

 an investigation of the spectrum of complete radiation. 



If an applied force /(/) exists only in the finite time interval 

 0^/^r, during which it has a finite number of discontinuities and a 



* It may be noted in passing that these formulas show the futility of attempting, 

 as so many inventors have done in connection with the problem of protection from 

 "static" disturbances, to design a circuit, which, in the language of patent specifi- 

 cations, shall be unresponsive to shock excitation while at the same time shall be 

 sharply responsive to sustained forces. 



6 The Heaviside Operational Calculus, J. R. Carson, B. S. T. J., Nov., 1922. 



6 See Appendix I. 



