Capacitance & Inductance

While resistors immediately dissipate electrical energy as heat, capacitors and inductors are passive components capable of storing energy. Capacitors store energy in an electric field, while inductors store energy in a magnetic field. Because they store and release energy over time, their behavior depends on the rate of change of voltage or current, making them essential for timers, filters, and dynamic electrical systems.

Capacitance ($C$)

Capacitance is a measure of a component's ability to store electric charge. A capacitor typically consists of two conductive plates separated by a dielectric (insulating) material.

The ideal constitutive relation for a capacitor states that the current flowing through it is proportional to the rate of change of the voltage across it:

$I = C \frac{dV}{dt}$

where:

• $I$ = current through the capacitor [Amperes, A]
• $C$ = capacitance [Farads, F]
• $\frac{dV}{dt}$ = rate of change of voltage with respect to time [V/s]

The unit of capacitance is the Farad (F). One Farad is extremely large; in practical engineering, you will frequently see microfarads ($\mu$F), nanofarads (nF), or picofarads (pF).

Energy Stored in a Capacitor

The electrical energy ($W_c$) stored in the electric field of a charged capacitor is given by:

$W_c = \frac{1}{2} C V^2$

where $W_c$ is the energy in Joules [J].

Inductance ($L$)

Inductance is a measure of a component's opposition to a change in current. An inductor is typically constructed by coiling a wire, which concentrates the magnetic field generated by the flowing current.

The ideal constitutive relation for an inductor states that the voltage induced across it is proportional to the rate of change of the current flowing through it:

$V = L \frac{dI}{dt}$

where:

• $V$ = voltage across the inductor [Volts, V]
• $L$ = inductance [Henrys, H]
• $\frac{dI}{dt}$ = rate of change of current with respect to time [A/s]

The unit of inductance is the Henry (H). Practical inductors are often measured in millihenrys (mH) or microhenrys ($\mu$H).

Energy Stored in an Inductor

The electrical energy ($W_l$) stored in the magnetic field of a current-carrying inductor is given by:

$W_l = \frac{1}{2} L I^2$

where $W_l$ is the energy in Joules [J].

Simple DC Steady-State Behavior

A deep analysis of capacitors and inductors requires differential equations or frequency-domain techniques (phasors). However, their steady-state behavior in a purely Direct Current (DC) circuit is straightforward once all voltages and currents have stopped changing (i.e., when derivatives are zero).

• For a Capacitor: If the DC voltage is constant, $\frac{dV}{dt} = 0$, meaning the current $I = 0$. Thus, a capacitor acts as an open circuit to steady DC.
• For an Inductor: If the DC current is constant, $\frac{dI}{dt} = 0$, meaning the voltage $V = 0$. Thus, an inductor acts as a short circuit (a simple wire) to steady DC.

Engineering Check

Real-world capacitors and inductors are not ideal. A real capacitor has some small "leakage resistance" across its plates, and a real inductor is made of wire that has internal resistance. When designing sensitive circuits, these non-ideal parasitic properties must be included in the model.