2.1i.1 Inductor

Inductor

An inductor is a circuit element that stores energy in a magnetic field. It consists of a solenoidal wire wrapped around in loops. Its inductance $L=\frac{N^2\mu A}{l}$ is measured in the SI unit Henry (H) and depends on the number of the turns ($N$) of the wire, the solenoid core’s magnetic permeability ($\mu$) and length ($l$), and the area ($A$) of the loops.

\usepackage{circuitikz}
\begin{document}
\begin{circuitikz}[american]
 \node at (0,3.75) {Symbol};
 \draw
   (0,0) to [inductor, v<=$V_L(t)$, i<=$I_L(t)$, l=$L$] (0,3);

\end{circuitikz}
\hspace{1cm}
\begin{tikzpicture}
\node at (0,2.25) {$V_L(t)=L\frac{d{I}_l(t)}{dt}$};
\node at (0,1.75) {$I_L(t)=\frac{1}{L}{\int }_{t_0}^t{V}_L(\tau )d\tau +I_L(t_0)=\frac{1}{L}{\varphi }_B(t)+I_L(t_0)$};
\node at (0,0){};
\end{tikzpicture}
\end{document}

The inductor has two main properties:
1. The current through an inductor cannot change instantaneously.
2. Immediately after  a current is passed through the inductor, it acts as an open circuit, but as $t \to \infty$, it starts acting as a short.

Behavior

The inductor’s behavior can be explained using Lenz’s Law. At $t=0$, or whenever there is a sudden change in voltage, the inductor tries to resist the current flow by generating an electromotive force (EMF) opposite to the direction of the current. This is because as current flows through the inductor, the loops of the solenoid merge their own magnetic fields into a relatively uniform one.

Since the EMF generated by the inductor at $t_0$ creates a repulsive voltage, no current can pass through the inductor at that time and due to the potential difference between the inductor’s terminals we can view it as an open circuit. However, as $t\to \infty$, the current flow through the inductor becomes constant and therefore $\frac{d{I}_L(t)}{dt}=0$.

Example

RL Circuit

Similar to the RC circuit, the RL circuit consists of a resistor and an inductor.

\usepackage[american]{circuitikz}
\begin{document}
\begin{circuitikz}
\draw
(-1,-1) to[V, v^=$V_s(t)$] (-1,2)
to [R, R=$R$, i=$i$] (3,2)
to [L, L=$L$, v=$V_L(t)$, i=$i$] (3, -1)
to [short] (-1, -1);
\end{circuitikz}
\end{document}

Here we perform KVL to find out that
$$V_s - V_L = iR.$$
Knowing that $V_L = L \frac{dI_t}{dt}$, we write out the function explicitly as
$$\frac{dI(t)}{dt}=\frac{V_s(t)}{L}-\frac{R}{L}i(t).$$
The above can easily be distinguished as a first-order differential equation with a constant non-homogeneous term, solving it as
$$I(t)=(I(t_0)-\frac{V_s}{R})e^{-\frac{R}{L}(t-t_0)}+\frac{V_s}{R}.$$
Assuming that before t = 0 there was no current flow,
$$I(t)=\frac{V_s}{R}(1-e^{-\frac{R}{L}(t-t_0)}).$$
From this equation, we can derive $V_L(t) = RI(t)$ as
$$V_L(t)=V_se^{-\frac{R}{L}(t-t_0)}.$$
Let us pass a step-response to our circuit. We can plot the relationships of voltage and current with respect to time after assuming $R=1$ and $L=1$ as
```tikz
\begin{document}

\begin{tikzpicture}[domain=0:4]
\draw[<->] (-1,0)—(10,0) node[below]{$t$};
\draw[<->] (0,-1)—(0,3) node[left, color=red]{$V_s$} node[right, color=green]{$V_L(t)$} node[right, yshift=-15, color=orange]{$I_L(t)$};
\draw[-, color=red] (0,0)—(2,0);
\draw[-, color=red] (2,2.5)—(2,0) node[below, color=black]{$t=2$};
\draw[-, color=red] (10,2.5)—(2,2.5) node[above]{$5V$};
\draw[color=green] (2,2.5)—(2,0);
\draw[color=green] plot (\x + 2,{2.5exp(-\x)});
\draw[color=green] (6,0.05)—(10,0.05);
\draw[color=orange] plot (\x + 2,{2.5 - 2.5exp(-\x)});
\draw[color=orange] (6,2.45)—(10,2.45);
\end{tikzpicture}
\end{document}

We can see that at $t=2$ seconds the voltage increases to $5V$, but the inductor behaves like an open circuit since $I \approx 0$ and the potential difference is $V_L=5V$. The circuit reaches equilibrium (or steady-state) at around $t=6$ seconds. By that time, the voltage $V_L$ has decayed to $\approx 0$, indicating the steady-state short property of the inductor.