Current in Electric Circuit/L, R in Series/Sinusoidal EMF
Theorem
Consider the electric circuit $K$ consisting of:
- a resistance $R$
- an inductance $L$
in series with a source of electromotive force $E$ which is a function of time $t$.
Let the electric current flowing in $K$ at time $t = 0$ be $I_0$.
Let an EMF $E$ be imposed upon $K$ at time $t = 0$ defined by the equation:
- $E = E_0 \sin \omega t$
The electric current $I$ in $K$ is given by the equation:
- $I = \dfrac {E_0} {\sqrt {R^2 - L^2 \omega^2} } \map \sin {\omega t - \alpha} + \paren {I_0 - \dfrac {E_0 L \omega} {R^2 + L^2 \omega^2} } e^{-R t / L}$
where $\tan \alpha = \dfrac {L \omega} R$.
Proof
From Electric Current in Electric Circuit: L, R in Series:
- $L \dfrac {\d I} {\d t} + R I = E_0 \sin \omega t$
defines the behaviour of $I$.
This can be written as:
- $(1): \quad \dfrac {\d I} {\d t} + \dfrac R L I = \dfrac {E_0} L \sin \omega t$
$(1)$ is a linear first order ODE in the form:
- $\dfrac {\d I} {\d t} + \map P t I = \map Q t$
where:
- $\map P t = \dfrac R L$
- $\map Q t = \dfrac {E_0} L e^{-k t}$
Thus:
\(\ds \int \map P t \rd t\) | \(=\) | \(\ds \int \dfrac R L \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {R t} L\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds e^{\int P \rd t}\) | \(=\) | \(\ds e^{R t / L}\) |
Thus from Solution by Integrating Factor, $(1)$ can be rewritten as:
- $\map {\dfrac \d {\d t} } {e^{R t / L} I} = e^{R t / L} \dfrac {E_0} L \sin \omega t$
and so the general solution becomes:
\(\ds e^{R t / L} I\) | \(=\) | \(\ds \frac {E_0} L \int e^{R t / L} \sin \omega t \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {E_0} L \frac {e^{R t / L} \paren {\frac R L \sin \omega t - \omega \cos \omega t} } {\paren {\frac R L}^2 + \omega^2} + C\) | Primitive of $e^{a x} \sin b x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {E_0 \paren {R \sin \omega t - L \omega \cos \omega t} } {R^2 + \omega^2 L^2} e^{R t / L} + C\) |
When $t = 0, I = I_0$.
\(\ds I_0\) | \(=\) | \(\ds E_0 \, {\frac {-L \omega} {R^2 + \omega^2 L^2} } + C\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds C\) | \(=\) | \(\ds I_0 + \frac {E_0 L \omega} {R^2 + \omega^2 L^2}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds I\) | \(=\) | \(\ds \frac {E_0 \paren {R \sin \omega t - L \omega \cos \omega t} } {R^2 + \omega^2 L^2} + \paren {I_0 + \frac {E_0 L \omega} {R^2 + \omega^2 L^2} } e^{-R t / L}\) |
The result follows by application of Multiple of Sine plus Multiple of Cosine: Sine Form.
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.13$: Simple Electric Circuits: Problem $1 \ \text{(b)}$