Part 4 - Capacitance and Inductance

In this part we’ll cover two very crucial components that are often used in electrical circuits.

Capacitors and inductors, we will see why these are so important later on as well.

Let’s start with looking at capacitors.

Capacitors

Let’s start by properly defining what a capacitor is:

Definition 1 (Capacitor)

A capacitor is a device that stores electric charge by creating an electric field between two conductive plates separated by an insulating material.

So, essentially, it is a component that stores electric energy.

The ideal capacitor is a passive component with the following circuit symbol:

Figure 1: Circuit symbol of a capacitor.

The capacitance is measured in farads ( $\mathrm{F}$ ) - typically $\mathrm{pF}$ to $\mathrm{\mu F}$ .

Let’s jump right into some formulas that we’ll need to understand:

Charge is proportional to the voltage,

q = C \cdot V.

Further, the current-voltage relation is therefore,

I = \frac{dq}{dt} = C \frac{dV}{dt}.

Let’s now look at the power & energy when using capacitors:

Power & Energy in Capacitors

\begin{align*} I & = \frac{dq}{dt} = C \frac{dV}{dt}, \newline P & = I \cdot V = CV \frac{dV}{dt}. \end{align*}

We know that,

\begin{align*} W(t) & = \int_{t_0}^{t} P(t) \, dt \newline & = \int_{t_0}^{t} CV \frac{dV}{dt} \, dt \newline & = C \int_{v(t_0)}^{v(t)} V \, dV \newline & = \frac{C}{2} [v(t)^2 - v(t_0)^2]. \end{align*}

If $V = 0$ at $t_0$ then:

\begin{align*} W(t) & = \frac{C v(t)^2}{2} & & \| \quad q = CV \newline & = \frac{v(t) q(t)}{2} & & \| \quad C = \frac{q}{V} \newline & = \frac{q(t)^2}{2C} & & \| \quad V = \frac{q}{C} \end{align*}

Let’s now take a look at charge and voltage in terms of current in capacitors.

Charge and Voltage in Capacitors

Charge in terms of current,

q(t) = \int_{t_0}^{t} i(t) \, dt + q(t_0) \quad \| \quad q = Cv.

Voltage in terms of current,

\begin{align*} CV(t) & = \int_{t_0}^{t} i(t) \, dt + CV(t_0), \newline V(t) & = \frac{1}{C} \int_{t_0}^{t} i(t) \, dt + V(t_0). \end{align*}

Capacitors in parallel

If we now take a look at a circuit which has three capacitors in parallel like such:

We can derive the following,

\begin{align*} i & = i_1 + i_2 + i_3, \newline i & = C \frac{dV}{dt}, \newline i & = C_1 \frac{dV}{dt} + C_2 \frac{dV}{dt} + C_3 \frac{dV}{dt} \newline & = (C_1 + C_2 + C_3) \frac{dV}{dt}, \end{align*}

\boxed{C_{eq} = C_1 + C_2 + C_3}.

Capacitors in series

In series, this becomes,

\begin{align*} V & = v_1 + v_2 + v_3 \newline & = \frac{Q}{C_1} + \frac{Q}{C_2} + \frac{Q}{C_3} \newline & = \left(\frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}\right) Q, \end{align*}

\boxed{\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}}.

So, let’s try to summarize capacitors.

Capacitors summary

Summary

Capacitors are open circuits to DC voltage (If $V$ is constant, then $I = 0$ ).
The voltage on a capacitor cannot jump (Change instantaneously, since then we would have infinite current).
Capacitors store energy ( $I \cdot V > 0$ ), or, deliver energy ( $I \cdot V < 0$ ).

Inductors

Let’s first properly define what an inductor is:

Definition 2 (Inductor)

An inductor is a component in an electrical circuit that utilizes electromagnetic induction to resist changes in current flow by generating a voltage that opposes the change.

An ideal inductor is a passive element with the following circuit symbol:

Figure 3: Circuit symbol of an inductor.

Let’s dive into the formulas now!

The current-voltage relation is,

V(t) = L \frac{dI}{dt}.

The unit of inductance, $L$ , is henry ( $\mathrm{H}$ ).

Now with the above formula we can derive,

\begin{align*} dI & = \frac{1}{L} V(t) \, dt, \newline \int_{I(t_0)}^{I(t)} dI & = \frac{1}{L} \int_{t_0}^{t} V(t) \, dt. \end{align*}

Which means,

I(t) = \frac{1}{L} \int_{t_0}^{t} V(t) \, dt + I(t_0).

Power & Energy in Inductors

P(t) = I(t) V(t) = I \left(L \frac{dI}{dt}\right) = \frac{dW}{dt}.

Which means,

W = \frac{LI^2}{2}.

As you can see, the formulas for inductors and capacitors are nearly identical.

Inductors in series

If we have the following circuit:

We can derive,

\begin{align*} V_s & = V_1 + V_2 + \ldots + V_N \newline & = L_1 \frac{dI}{dt} + L_2 \frac{dI}{dt} + \ldots + L_N \frac{dI}{dt} \newline & = (L_1 + L_2 + \ldots + L_N) \frac{dI}{dt}, \end{align*}

\boxed{L_{eq} = L_1 + L_2 + \ldots + L_N}.

Inductors in parallel

If we have the following circuit:

We can derive,

\begin{align*} V & = L \frac{d}{dt}(I_1 + I_2 + \ldots + I_N) \newline & = L \left(\frac{dI_1}{dt} + \frac{dI_2}{dt} + \ldots + \frac{dI_N}{dt}\right) \newline & = L \left(\frac{V}{L_1} + \frac{V}{L_2} + \ldots + \frac{V}{L_N}\right). \end{align*}

Which means,

\boxed{L_{eq} = \frac{1}{\frac{1}{L_1} + \frac{1}{L_2} + \ldots + \frac{1}{L_N}}}.

Inductors summary

Summary

Inductors are short circuits to DC voltages (If $I$ is constant, then $V = 0$ ).
The current through an inductor cannot jump (change instantaneously, otherwise we would have infinite voltage).
Inductors store energy ( $I \cdot V > 0$ ), or, deliver energy ( $I \cdot V < 0$ ).

Time-Varying Circuits

Figure 6: A charged capacitor discharging through a resistor.

When we’re dealing with circuits like this, meaning, we have a time aspect to consider. We need new ways to describe our circuits.

So let’s introduce our new formulas,

V_C(t) = V_i e^{-t / \tau},

where $\tau$ is,

\tau = RC.

For the inductor case:

Formulas:

\begin{align*} I(t) & = I_0 e^{-Rt / L}, \newline \tau & = \frac{L}{R}. \end{align*}