6.4 Energy storage in capacitors and inductors

Energy Storage in Capacitors and Inductors

Capacitors and inductors are the two fundamental energy-storing components in electrical circuits. Capacitors store energy in electric fields, while inductors store energy in magnetic fields. Understanding how they store, hold, and release energy is key to analyzing any circuit with time-varying signals, from simple RC timing circuits to power supplies and filters.

Energy Storage

Electric Field Energy

When you apply a voltage across a capacitor, charge accumulates on its plates, and an electric field forms between them. That field is where the energy actually lives.

The energy stored in a capacitor depends on its capacitance ($C$) and the voltage ($V$) across it:

$U_E = \frac{1}{2}CV^2$

Notice the voltage is squared. Doubling the voltage quadruples the stored energy, while doubling the capacitance only doubles it. For example, a $10 \, \mu F$ capacitor charged to $5 \, V$ stores $U_E = \frac{1}{2}(10 \times 10^{-6})(5)^2 = 125 \, \mu J$.

You can also describe how densely energy is packed into the electric field using energy density (energy per unit volume):

$u_E = \frac{1}{2}\epsilon E^2$

Here, $\epsilon$ is the permittivity of the dielectric material between the plates and $E$ is the electric field strength. Materials with higher permittivity (high-k dielectrics) allow more energy to be stored in the same physical space.

Magnetic Field Energy

When current flows through an inductor, a magnetic field builds up around its coils. That magnetic field stores energy.

The energy stored in an inductor depends on its inductance ($L$) and the current ($I$) through it:

$U_M = \frac{1}{2}LI^2$

The structure mirrors the capacitor formula, but with current instead of voltage. A $100 \, mH$ inductor carrying $2 \, A$ stores $U_M = \frac{1}{2}(0.1)(2)^2 = 0.2 \, J$.

The corresponding energy density in the magnetic field is:

$u_M = \frac{1}{2\mu}B^2$

Here, $\mu$ is the permeability of the core material and $B$ is the magnetic flux density.

Energy Units and Conversions

• The SI unit for energy is the joule (J).
• One joule equals the work done when a force of one newton acts over one meter.
• Energy density is typically expressed in $J/m^3$ (joules per cubic meter).
• In circuits, energy constantly converts between forms: electrical energy becomes electric field energy in a capacitor, magnetic field energy in an inductor, or thermal energy in a resistor.
• High energy density matters in applications where size and weight are constrained, such as portable electronics and electric vehicles.

Capacitor and Inductor Dynamics

Charging and Discharging Processes

Capacitors and inductors don't change state instantly. They store energy gradually during charging and release it gradually during discharging.

Capacitor behavior:

1. Charging: A voltage source is applied. Current flows into the capacitor, charge builds on the plates, and the voltage across the capacitor rises exponentially toward the source voltage.
2. Discharging: The capacitor is connected to a load. Current flows out, the plates lose charge, and the voltage decays exponentially toward zero.

Inductor behavior:

1. Charging: A voltage is applied across the inductor. Current ramps up exponentially as the magnetic field builds.
2. Discharging: The voltage source is removed. The collapsing magnetic field drives current through the circuit (in the same direction it was flowing), and the current decays exponentially toward zero.

A key difference: a capacitor resists sudden changes in voltage, while an inductor resists sudden changes in current. This is why inductors can produce voltage spikes when current is interrupted abruptly.

Time Constants and Transient Response

The time constant ($\tau$) tells you how fast a capacitor or inductor charges and discharges.

• For an RC circuit: $\tau = RC$
• For an RL circuit: $\tau = \frac{L}{R}$

What $\tau$ means in practice:

A larger time constant means a slower response. For example, an RC circuit with $R = 10 \, k\Omega$ and $C = 100 \, \mu F$ has $\tau = (10{,}000)(100 \times 10^{-6}) = 1 \, s$. It takes about 5 seconds to fully charge. Swap in a $1 \, k\Omega$ resistor and $\tau$ drops to $0.1 \, s$, so charging finishes in roughly half a second.

Power Dissipation and Energy Loss

Not all the energy you put into a capacitor or inductor comes back out. Some is lost as heat in the resistive parts of the circuit.

The power dissipated in a resistor is:

$P = I^2 R$

During charging, current flows through whatever series resistance exists (wire resistance, the resistor in an RC circuit, the inductor's winding resistance), and that current generates heat. This means the energy you get back during discharge is always less than what the source originally supplied.

To minimize these losses:

• Use components with low equivalent series resistance (ESR) for capacitors and low DC resistance (DCR) for inductors.
• Keep wiring resistance low.
• Choose circuit designs that limit the peak currents through resistive elements.

In an ideal (lossless) circuit, all energy stored in the capacitor or inductor would be recoverable. In real circuits, the gap between stored and recovered energy is determined by how much $I^2 R$ heating occurs during the charge/discharge cycle.