Consider a 1 F capacitor.The current across the capacitor is given by iC(t) = { 1, 0

Question

Consider a 1 F capacitor. The current across the capacitor is given by

[latex]i_{C}(t)= \begin{cases}1, & 0

Find and plot the voltage across the capacitor if [latex]v_{C}(0)=0[/latex].

Accepted Answer

First, we note that [latex]v_{C}(t)=0[/latex] for [latex]t<0[/latex], since there is no current in this period. In addition, for [latex]0

[latex]v_{C}(t)=v_{C}(0)+\int_{0}^{t} 1 d t^{\prime}=(t) \mathrm{V},[/latex]

leading to [latex]v_{C}(1)=1 \mathrm{~V}[/latex]. Finally, for [latex]t>1 \mathrm{~s}[/latex], we get

[latex]v_{C}(t)=v_{C}(1)+\int_{1}^{t} 0 d t^{\prime}=1 \mathrm{~V} .[/latex]

Overall, the voltage across the capacitor can be written as

[latex]v_{C}(t)= \begin{cases}0, & t<0 \ t, & 01 \mathrm{~s}\end{cases}[/latex]

The current and voltage across the capacitor can be plotted as follows.

Once again, we note that the current across an ideal capacitor can be discontinuous, while the voltage must be continuous for finite values of the current.

Question 6.99: Consider a 1 F capacitor.The current across the capacitor is...

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