RLC Circuits, Resonance, and Power Factor

RLC Circuits, Resonance, and Power Factor

What Is an RLC Circuit?

A series RLC circuit contains a resistor (R), an inductor (L), and a capacitor (C) connected end-to-end in a single loop. When AC voltage is applied, all three components oppose current flow, but they do so in fundamentally different ways. The resistor dissipates energy as heat. The inductor stores energy in a magnetic field and causes current to lag voltage by up to 90°. The capacitor stores energy in an electric field and causes current to lead voltage by up to 90°. Because inductive and capacitive effects are opposite in phase, they partially cancel each other.

Impedance in a Series RLC Circuit

Impedance (Z) is the total opposition to AC current, measured in ohms. It combines resistance and reactance using the Pythagorean relationship:

Z = √(R² + (X_L − X_C)²)

Where:

• X_L = inductive reactance = 2πfL (ohms)
• X_C = capacitive reactance = 1 / (2πfC) (ohms)
• f = frequency in hertz
• L = inductance in henries
• C = capacitance in farads

The difference (X_L − X_C) is called the net reactance (X). When X is positive the circuit is net inductive; when negative it is net capacitive.

Resonance

Resonance occurs at the frequency where X_L exactly equals X_C, so net reactance equals zero. At that point impedance equals R alone — its minimum possible value — and current reaches its maximum. The resonant frequency is:

f_r = 1 / (2π√(LC))

Below resonance the circuit behaves capacitively (current leads). Above resonance it behaves inductively (current lags). At resonance voltage across L and voltage across C are equal in magnitude and opposite in phase, so they cancel; only the resistor drop appears across the source.

The Power Triangle: P, Q, and S

AC circuits with reactive components require three power quantities:

• True power (P) — real power actually consumed and converted to work or heat. Unit: watts (W). P = I²R = V·I·cos θ
• Reactive power (Q) — power shuttled back and forth between source and reactive components; does no useful work. Unit: volt-amperes reactive (VAR). Q = I²X = V·I·sin θ
• Apparent power (S) — the total power the source must supply, combining P and Q. Unit: volt-amperes (VA). S = V·I = √(P² + Q²)

These three form a right triangle with S as the hypotenuse, P as the base, and Q as the vertical leg. This is called the power triangle.

Power Factor

Power factor (PF) is the ratio of true power to apparent power:

PF = P / S = cos θ

where θ is the phase angle between voltage and current. PF ranges from 0 to 1. A PF of 1.0 (unity) means all power is doing useful work. Low power factor means the source must supply excess current to deliver the same real power, increasing losses in conductors and transformers. Utilities often penalize commercial customers for low power factor.

• Lagging PF: inductive load (motors, transformers)
• Leading PF: capacitive load

Power-Factor Correction

Adding capacitance in parallel with an inductive load reduces the net reactive power drawn from the source. The capacitor supplies reactive power locally, so the line current drops even though the load's true power stays the same. This is power-factor correction — a practice you will encounter whenever you work on commercial motor installations or large HVAC equipment.

Lab Safety Reminders

Before energizing any circuit on the bench, observe the following:

• Lockout/Tagout is not required on low-voltage bench supplies, but you must treat every energized circuit as a shock hazard. De-energize the supply before adding or removing components.
• Wear safety glasses when working with capacitors; a charged capacitor can arc and eject material.
• Discharge capacitors before handling them after power-down. Use the discharge resistor provided on the trainer, or allow the circuit to dissipate through the load.
• Keep the bench supply output within the rated component voltage to avoid overheating or component failure.
• Use a true-RMS DMM for AC measurements — average-responding meters read incorrectly on non-sinusoidal waveforms from function generators.
• Report any component that is hot to the touch, discolored, or has an unusual smell to the instructor immediately.

Connecting Theory to the Trade

Every motor, transformer, and fluorescent ballast on a jobsite is an inductive load with a lagging power factor. Electricians size conductors and overcurrent protection based on apparent power (VA), not just watts. An undersized conductor in a low-PF installation overheats even if the kilowatt load seems modest. Understanding the power triangle helps you explain to a building owner why corrective capacitors save money on their utility bill and prevent nuisance tripping.