Page:Elementary lectures on electric discharges, waves and impulses, and other transients (Steinmetz 1911).djvu/22

that is, $c$ is the reciprocal of the projection $T={\overline {t_{1}t_{2}}}$ on the zero line of the tangent at the starting moment of the transient.

Since	$c={\frac {1}{T'}}$,
	${\frac {di}{i}}=-cdt$;

that is, the percentual change of current is constant, or in other words, in the same time, the current always decreases by the same fraction of its value, no matter what this value is.

Integrated, this equation gives:

	${\begin{array}{rcl}log\ i&=&-ct+C,\\i&=&Ae^{-ct},\end{array}}$
or	$i\ \ \ =\ \ \ Ae^{-{\frac {t}{T}}}$;

that is, the curve is the exponential.

The exponential curve thus is the expression of the simplest form of transient. This explains its common occurrence in electrical and other transients. Consider, for instance, the decay of radioactive substances: the radiation, which represents the decay, is proportional to the amount of radiating material; it is ${\frac {dm}{dt}}=cm$, which leads to the same exponential function.

Not all transients, however, are of this simplest form. For instance, the deceleration of a ship does not follow the exponential, but at high velocities the decrease of speed is a greater fraction of the speed than during the same time interval at lower velocities, and the speed-time curves for different initial speeds are not proportional to each other, but are as shown in Fig. 5. The reason is, that the frictional resistance is not proportional to the speed, but to the square of the speed.

5. Two classes of transients may occur:

1. Energy may be stored in one form only, and the only energy change which can occur thus is an increase or a decrease of the stored energy.

2. Energy may be stored in two or more different forms, and the possible energy changes thus are an increase or decrease of the total stored energy, or a change of the stored energy from one form to another. Usually both occur simultaneously.

An instance of the first case is the acceleration or deceleration