Page:Elementary lectures on electric discharges, waves and impulses, and other transients (Steinmetz 1911).djvu/21

the second case, of current $i'$, at the moment $t_{1}$; that is, from $t$ onward, curve $i$ is the same as curve $i'$ from time $t_{1}$ onward. Since

Fig. 4. — Curve of Simple Transient: Decay of Current.

$i'$ is proportional to $i$ from any point $t$ onward, curve $i'$ is proportional to the same curve $i$ from $t_{1}$ onward. Hence, at time $t_{1}$, it is

${\frac {di_{2}}{dt_{1}}}={\frac {di_{1}}{dt_{1}}}\times {\frac {i_{2}}{i_{1}}}$.

But since ${\frac {di_{2}}{dt_{1}}}$ and $i_{2}$ at $t_{1}$ are the same as ${\frac {di}{dt}}$ and $i$ at time $t$, it follows:

${\frac {di}{dt}}={\frac {di_{1}}{dt_{1}}}{\frac {i}{i_{1}}}$,

or,

${\frac {di}{dt}}=-ci$,

where $c=-{\frac {1}{i_{1}}}{\frac {di_{1}}{dt}}={\mbox{constant}}$, and the minus sign is chosen, as ${\frac {di}{dt}}$ is negative.

As in Fig. 4:

${\begin{array}{rcl}{\mbox{tan}}\ \phi &=&-\displaystyle {\frac {di_{1}}{dt}},\\{\overline {a_{1}t_{1}}}&=&i_{1},\\c=-\displaystyle {\frac {1}{i_{1}}}{\frac {di_{1}}{dt}}&=&\displaystyle {\frac {{\mbox{tan}}\ \phi }{\overline {a_{1}t_{1}}}}=\displaystyle {\frac {1}{\overline {t_{1}t_{2}}}};\end{array}}$