Current in Electric Circuit/L, R, C in Series
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Theorem
Consider the electric circuit $K$ consisting of:
- a resistance $R$
- an inductance $L$
- a capacitance $C$
in series with a source of electromotive force $E$ which is a function of time $t$.
The electric current $I$ in $K$ is given by the linear second order ODE:
- $L \dfrac {\d^2 I} {\d t^2} + R \dfrac {\d I} {\d t} + \dfrac 1 C I = \dfrac {\d E} {\d t}$
Proof
Let:
- $E_L$ be the drop in electromotive force across $L$
- $E_R$ be the drop in electromotive force across $R$
- $E_C$ be the drop in electromotive force across $C$.
From Kirchhoff's Voltage Law:
- $E - E_L - E_R - E_C = 0$
From Ohm's Law:
- $E_R = R I$
From Drop in EMF caused by Inductance is proportional to Rate of Change of Current:
- $E_L = L \dfrac {\d I} {\d t}$
From Drop in EMF caused by Capacitance is proportional to Accumulated Charge:
- $E_C = \dfrac 1 C Q$
where $Q$ is the electric charge $Q$ that has accumulated on $C$.
Thus:
- $E - L \dfrac {\d I} {\d t} - R I - \dfrac 1 C Q = 0$
which can be rewritten:
- $L \dfrac {\d I} {\d t} + R I + \dfrac 1 C Q = E$
Differentiating with respect to $t$ gives:
- $L \dfrac {\d^2 I} {\d t^2} + R \dfrac {\d I} {\d t} + \dfrac 1 C I = \dfrac {\d E} {\d t}$
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.13$: Simple Electric Circuits