11.4.2 | revision #18496 | Savchenko Solutions

Statement

11.4.2. How should the voltage change in an electrical circuit consisting of a series-connected inductor $$L$$ and resistance $$R$$ , so that the current in it: (a) increases linearly: $I = \alpha t$ ? (b) changed sinusoidally: $I = I_0\sin\omega t$ ?

Solution

From the Kirchoff's voltage law:

$iR+L\frac{di}{dt}=u,\tag{1}$

In the case (а):

$\begin{gather} \alpha tR+L\alpha=u, \end{gather}$

or equivalently

$\begin{gather} u=\alpha(Rt+L). \end{gather}$

In the case (b):

$\begin{gather} RI_0\sin\omega t + L\omega I_0\cos\omega t=u, \end{gather}$

or equivalently

$\begin{gather} u=I_0(R\sin\omega t + L\omega\cos\omega t). \end{gather}$

Answer

(а)

$u=\alpha(Rt+L);$

(b)

$u=I_0(R\sin\omega t + L\omega\cos\omega t).$