Exercise 1.3.63

Question

The Heaviside function $H$ is defined by
$$
H(t)= \begin{cases}0 & \mathrm { if } \ t<0 \ 1 & \mathrm { if } \ t \geqslant 0\end{cases}
$$

It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage when a switch is instantaneously turned on.
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of the Heaviside function.
\item Sketch the graph of the voltage $V(t)$ in a circuit if the switch is turned on at time $t=0$ and 120 volts are applied instantaneously to the circuit. Write a formula for $V(t)$ in terms of $H(t)$.
\item Sketch the graph of the voltage $V(t)$ in a circuit if the switch is turned on at time $t=5$ seconds and 240 volts are applied instantaneously to the circuit. Write a formula for $V(t)$ in terms of $H(t)$. (Note that starting at $t=5$ corresponds to a translation.)
\end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}[label=(\alph*)]
\item \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, xlabel={$x$}, ylabel={$y$}, ymax=2, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=-3.2:0, samples=2, mark=none, color=violet, ultra thick] {0};
         \addplot[domain=0:3.2, samples=2, mark=none, color=violet, ultra thick] {1};
         \addplot[mark=*, mark options={scale=1, fill=white}] coordinates {(0,0)};
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
\item \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, xlabel={$x$}, ylabel={$y$}, ymax=130, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=-3.2:0, samples=2, mark=none, color=violet, ultra thick] {0};
         \addplot[domain=0:3.2, samples=2, mark=none, color=violet, ultra thick] {120};
         \addplot[mark=*, mark options={scale=1, fill=white}] coordinates {(0,0)};
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
\item \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, xlabel={$x$}, ylabel={$y$}, ymax=250, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm}, ytick={0, 40, 80, 120,160,200,240}]
         \addplot[domain=-3.2:5, samples=2, mark=none, color=violet, ultra thick] {0};
         \addplot[domain=5:9.2, samples=2, mark=none, color=violet, ultra thick] {240};
         \addplot[mark=*, mark options={scale=1, fill=white}] coordinates {(5,0)};
         \draw[dashed] (axis cs:5, 0) -- (axis cs:5, 240);
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
\end{enumerate}

Exercise 1.3.63

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Exercise 1.3.63

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