RL Circuits — Question 10

Question 10

A repeated on-off drive reaches a periodic current rather than resetting to zero

A series RL circuit is driven by +V for time T and then zero volts for time T, repeatedly. The zero-voltage interval leaves the RL loop closed. Consider the settled periodic state and define $a=e^{-T/\tau}$, $\tau=L/R$.

1. Find the minimum and maximum currents during a cycle.

2. Find the ripple and the mean current over a complete period.

Question 10 - Solution

A repeated on-off drive reaches a periodic current rather than resetting to zero

(a) Method and calculation

$$i_{\max}=V/R+(i_{\min}-V/R)a,\quad i_{\min}=ai_{\max}.$$$$\boxed{i_{\max}=\frac{V/R}{1+a},\quad i_{\min}=\frac{aV/R}{1+a}}.$$

(b) Interpretation and further calculation

$$\boxed{\Delta i=\frac VR\frac{1-a}{1+a}}.$$$$\langle L\dot i\rangle=0\Rightarrow R\langle i\rangle=\langle V\rangle=V/2\Rightarrow\boxed{\langle i\rangle=V/(2R)}.$$

Physical check

The inductor preserves current between pulses; the mean follows from periodicity even though the waveform is exponential rather than rectangular.