Capacitors and Inductors: Storing Electrical Energy

Capacitors and Inductors: The Hidden Energy Stores

Imagine filling a water tank and then letting it drain slowly — that is essentially what a capacitor does with electric charge. Now imagine a flywheel storing kinetic energy and releasing it gradually — that is what an inductor does with current. Capacitors and inductors are the two fundamental components that store energy in entirely different ways, and they appear in every electronic circuit, motor drive, and power network.

The Capacitor: Storing Energy in an Electric Field

Physical Principle

In its simplest form, a capacitor is two parallel metal plates separated by an insulating material (dielectric). When voltage is applied, positive charges accumulate on one plate and negative charges on the other — creating an electric field that stores energy.

Capacitance

Capacitance measures the ability to store charge:

C = Q / V

where C = capacitance in farads, Q = charge in coulombs, V = voltage in volts.

The farad is enormous — practical values use:

• Microfarads: 1 µF = 10⁻⁶ F
• Nanofarads: 1 nF = 10⁻⁹ F
• Picofarads: 1 pF = 10⁻¹² F

Modern supercapacitors, however, reach 3000F and beyond.

Parallel-Plate Capacitance

C = ε₀ × εᵣ × A / d

where:

• ε₀ = permittivity of free space (8.854 × 10⁻¹² F/m)
• εᵣ = relative permittivity of the dielectric
• A = plate area
• d = distance between plates

Larger area or smaller gap means greater capacitance.

Stored Energy

E = ½ × C × V²

Energy scales with the square of voltage — doubling the voltage quadruples the stored energy.

Industrial Capacitor Types

The Inductor: Storing Energy in a Magnetic Field

Physical Principle

An inductor is a wire wound around a core (air or iron). When current flows, a magnetic field builds up and stores energy. The inductor resists changes in current — like a heavy object resists changes in velocity (inertia).

Inductance

V = L × (dI/dt)

where L = inductance in henrys, dI/dt = rate of current change.

If you try to change the current through an inductor suddenly, it produces a very high voltage opposing the change. This is why you see sparks when disconnecting an inductive circuit.

Solenoid Inductance

L = µ₀ × µᵣ × N² × A / l

where:

• µ₀ = permeability of free space (4π × 10⁻⁷ H/m)
• µᵣ = relative permeability of the core (iron: 1000 - 5000)
• N = number of turns
• A = cross-sectional area
• l = length of the coil

Stored Energy

E = ½ × L × I²

Energy scales with the square of current, and an iron core increases inductance by thousands of times.

The Fundamental Comparison

RC Circuits: Capacitor with Resistance

Charging

When an empty capacitor is connected to a voltage source through a resistor:

V(t) = V₀ × (1 - e^(-t/τ))

where the time constant τ = R × C.

After 1τ the voltage reaches 63%, after 3τ it reaches 95%, and after 5τ the capacitor is practically fully charged.

Practical example: a 100µF capacitor with a 10kΩ resistor:

τ = 10,000 × 0.0001 = 1 second

Full charge takes roughly 5 seconds.

Industrial RC Applications

• Simple timers: setting a pump start delay
• Noise filtering: removing interference from sensor signals
• Snubber circuits: protecting contactor contacts from arcing

RL Circuits: Inductor with Resistance

Current Growth

When an inductor is connected to a voltage source through a resistor:

I(t) = I₀ × (1 - e^(-t/τ))

where τ = L / R.

Example: a 0.5H inductor with a 100Ω resistor:

τ = 0.5 / 100 = 5 milliseconds

Current reaches its final value in about 25 ms.

Industrial RL Applications

• Contactor and relay coils: natural response delay
• Current transformers (CTs): converting high currents to measurable values
• Line reactors: protecting VFD inputs from harmonics

LC Circuits: Resonance

When a capacitor and inductor are connected together, resonance occurs — energy oscillates between them like a pendulum:

f₀ = 1 / (2π × √(L × C))

At the resonant frequency, impedance approaches zero (or infinity, depending on the configuration). This principle is used in:

• Filter circuits: separating wanted frequencies from unwanted ones
• Harmonic filters: removing distortion from factory power networks
• Communication circuits: tuning to a specific radio frequency

Power Factor Correction

The Problem

Motors and coils in factories draw current that lags behind voltage (inductive load). This means:

• Higher current for the same real power
• Greater cable losses
• Utility penalties when the power factor drops below 0.9

The Solution: Capacitor Banks

A capacitor bank installed on the main busbars produces leading current that compensates for the inductive lag, bringing the power factor close to 1.0.

Calculating the required capacitor bank:

Q_c = P × (tan(φ₁) - tan(φ₂))

where P = real power, φ₁ = original angle, φ₂ = target angle.

Example: a 500 kW factory with a power factor of 0.75 wants to improve it to 0.95:

Q_c = 500 × (tan(41.4°) - tan(18.2°)) = 500 × (0.882 - 0.329) = 276.5 kVAR

Advanced Industrial Applications

• VFD DC bus capacitors: large electrolytic capacitors smooth the rectified voltage inside variable frequency drives
• Chokes: inductors that limit inrush currents during motor starting
• EMI filters: small capacitors and inductors that prevent electromagnetic interference from spreading
• UPS systems: large capacitors and inductors store energy and smooth the output waveform

Summary

A capacitor stores energy in an electric field and resists voltage changes. An inductor stores energy in a magnetic field and resists current changes. These two simple principles underpin every filter, transformer, power factor correction unit, and motor drive in a factory. Mastering them opens the door to industrial electrical circuit design.