Self-Induction — Question 6

Question 6

Corners in a continuous current waveform cause voltage jumps rather than impulses

An ideal inductor L carries a triangular current: it rises from zero to I0 in time T, then falls to zero in time 2T. The total pulse duration is 3T.

1. Find the passive voltage in each interval.

2. Explain whether the corner at t=T requires an infinite voltage impulse and evaluate the net voltage-time area.

Question 6 - Solution

Corners in a continuous current waveform cause voltage jumps rather than impulses

(a) Method and calculation

$$0<t<T:\quad\boxed{v=LI_0/T}.$$$$T<t<3T:\quad\boxed{v=-LI_0/(2T)}.$$

(b) Interpretation and further calculation

$$\int_0^{3T}vdt=LI_0-LI_0=\boxed0.$$$$i(T^-)=i(T^+)\Rightarrow\boxed{\text{no impulse is required; only a finite voltage jump}}.$$

Physical check

An actual current discontinuity would require nonzero voltage impulse area; a discontinuity in slope does not.