Question 21.29: In Example 2.17, we examined the variation in voltage across......
In Example 2.17, we examined the variation in voltage across a capacitor, C, when it was switched in series with a resistor, R, at time t = 0. We stated a relationship for the time-varying voltage, v, across the capacitor. Prove this relationship. Refer to the example for details of the circuit.
First we must derive a differential equation for the circuit. Using Kirchhoffs voltage law and denoting the voltage across the resistor by v_R we obtain
v+v_{\mathrm{R}}=0Using Ohm’s law and denoting the current in the circuit by i we obtain
v+i R=0For the capacitor,
i=C \frac{\mathrm{d} v}{\mathrm{~d} t}Combining these equations gives
v+R C \frac{\mathrm{d} v}{\mathrm{~d} t}=0We now take the Laplace transform of this equation. Using \mathcal{L}\{v\}=V(s) we obtain
\begin{aligned} & V(s)+R C(s V(s)-v(0))=0 \\ & V(s)(1+R C s)=R C v(0) \\ & V(s)=\frac{R C v(0)}{1+R C s}=\frac{v(0)}{\frac{1}{R C}+s} \end{aligned}Taking the inverse Laplace transform of the equation yields
v=v(0) \mathrm{e}^{-t /(R C)} \quad t \geqslant 0This is equivalent to the relationship stated in Example 2.17.