Page:Elementary lectures on electric discharges, waves and impulses, and other transients (Steinmetz 1911).djvu/102

 In these equations (50), d and a′ are the maximum values of current and of voltage respectively, of the different harmonic waves. Between the maximum values of current, i₀, and of volt- age, e₀, of a stationary oscillation exists, however, the relation

where z is the natural impedance or surge impedance. That is

a'=dz 0) (51)

and substituting (51) into (50) gives

i e~ ut di sin  sin co + d$ sin 3  sin 3 co + d$ sin 5  sin 5 co e = z ~ ut I di cos cos co + d 3 cos 3 < cos 3 co -j- d 5 cos 5 cos 5 (52)

(d) If then the distribution of voltage e along the circuit is given at the moment of start of the transient, for instance, the voltage is constant and equals e throughout the entire circuit at the starting moment = of the transient, this gives the relation, by substituting in (52), ei = ZQ t~ ut di cos co + c? 3 cos 3 co + c? 5 cos 5 co +. . . }, (53) for all values of co. Herefrom then calculate the values of d, d 3 , d$, etc., in the manner as discussed in " Engineering Mathematics," Chapter III.