Capacitance & Inductance

While resistors dissipate electrical energy instantly as heat, Capacitors and Inductors are passive components that can "store energy". Capacitors store energy in an electric field, while inductors store energy in a magnetic field. Because storing and releasing energy takes time, the behavior of these components depends on the rate of change of voltage or current, making them essential for timing circuits, signal filters, and dynamic electrical systems.

Capacitance ($C$)

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

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

The unit of capacitance is the Farad (F). One Farad is a massive value; in practical engineering, you will usually encounter 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 can be calculated by:

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

Where $W_c$ is energy in Joules [J].

Inductance ($L$)

Inductance is a measure of opposition to the change in electrical current. An inductor is usually formed by coiling a wire, which concentrates the magnetic field strength generated by the current passing through it.

The ideal governing equation for an inductor states that the voltage induced across it is directly proportional to the rate of change of the current passing through it:

The unit of inductance is the Henry (H). Common inductors are typically in the millihenry (mH) or microhenry ($\mu$H) range.

Energy Stored in an Inductor

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

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

Where $W_l$ is energy in Joules [J].

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 pure DC circuits is very straightforward. When voltage and current stop changing (i.e., when their derivatives are zero):

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

Engineering Check

Real-world capacitors and inductors are not ideal. Real capacitors have a small leakage resistance between the plates, and real inductors are made of wire that possesses internal resistance. When designing highly sensitive circuits, these non-ideal parasitic properties must be included in the calculation models.