Teaching guide for the module

Power factor and AC power: Teacher's Guide

Teaching guide for the Power Factor & AC Power simulator: explain in class the power triangle (active, reactive, apparent), the power factor $cos φ$ , capacitive power factor correction of industrial loads and the sizing of the compensation capacitor. Designed for electrical engineering teachers.

Module: Power Factor & AC Power · Three tabs: V & I Waveforms · Power Triangle · PF Correction

Physical phenomenon

In an AC circuit, voltage and current can be out of phase by an angle $φ$ due to reactive loads (inductors, capacitors). The phase shift has profound consequences on the power exchanged between source and load, which decomposes into three distinct quantities:

Active power $P$ (in watts, W), the "useful power" effectively transferred to the load on average, the one that produces work or heat. $P = V_{R M S} \cdot I_{R M S} \cdot cos φ$ .
Reactive power $Q$ (in volt-amperes reactive, VAR): the power that "swings back and forth" between source and load to feed the magnetic fields of inductors or the electric fields of capacitors, without doing net work. $Q = V_{R M S} \cdot I_{R M S} \cdot sin φ$ .
Apparent power $S$ (in volt-amperes, VA), the product of RMS V and I. The "size" the grid must dimension: $S = V_{R M S} \cdot I_{R M S} = P^{2} + Q^{2}$ .

The three quantities form the power triangle, where $P$ is the horizontal cathetus, $Q$ the vertical one (positive for inductive loads, negative for capacitive) and $S$ the hypotenuse. The angle between $P$ and $S$ is $φ$ , and $cos φ = P / S$ is the power factor: a dimensionless number between 0 and 1 measuring the load's energy efficiency.

Power factor correction consists of adding a capacitor in parallel to the inductive load, with a calculated capacitance, so as to locally provide the reactive power required by the load, relieving the grid from supplying it.

Key concepts

Phase shift $φ$ : the angle between V and I of the load. Positive for inductive, negative for capacitive, zero for purely resistive.
$cos φ$ : the power factor: 1 = ideal (all active), 0 = worst (all reactive).
Power triangle: vector representation linking $P$ , $Q$ , $S$ via Pythagoras.
Inductive load ( $Q > 0$ ), motors, transformers, power supplies, ballasts. Almost all industrial loads.
Capacitive correction: the capacitor "supplies" $Q$ to the inductive load: the grid sees only the difference, ideally zero.
Penalties on the bill: industrial customers pay surcharges when $cos φ$ drops below contractual thresholds (typically 0.9). Correction is economically worthwhile.
Required C: the correction capacitance is computed as $C = Q_{compensate} / (2 π f \cdot V^{2})$ .

How to use it in the classroom

Opening: V & I Waveforms tab. Start at $φ = 0°$ (resistive load): voltage and current are overlapping sinusoids. Move $φ$ to $90°$ (purely reactive load): the current is in quadrature, and in the product $v (t) \cdot i (t)$ the "instantaneous power" is positive half the time and negative the other half, no net work flows. Use the 0/30/45/60/90° presets to capture the phenomenon visually.

Development: Power Triangle tab. Show the three vectors $P$ , $Q$ , $S$ forming the right triangle. Vary $φ$ and observe how $P$ shrinks and $Q$ grows, while $S$ stays nearly constant (at the same $V_{R M S}$ and $I_{R M S}$ ). Have students compute $cos φ = P / S$ at a glance. Explain that the grid "sees" $S$ , but the load uses only $P$ .

Deeper exploration: PF Correction tab. Show the correction circuit: source, inductive load, capacitor in parallel. The module computes and shows the required C (C_required) value to bring the power factor to the target (typically $cos φ = 0.95$ ). The Installed C slider lets you move from 0 to $2 \cdot C_{req}$ . The badge shows OPTIMAL / UNDER / OVER:

UNDER-COMPENSATED: the installed $C$ is insufficient, $φ$ stays inductive.
OPTIMAL: $φ$ is exactly at the target.
OVER-COMPENSATED: too much $C$ , $φ$ has become capacitive (worsening). It is a real sizing mistake.

The two pre/post triangles graphically show the reduction of $S$ and $∣ Q ∣$ at the same $P$ .

Closing: the economic meaning. Ask: "Why does the utility company charge penalties for low cosφ instead of ignoring it?" Answer: because a low- $cos φ$ load draws a higher current $I = S / V$ to deliver the same useful $P$ , and this extra current heats cables, takes up transformer capacity and forces network investments that other customers would end up paying for. The penalties "internalise" this cost.

Real-world examples

Industrial asynchronous motors. Almost all motors induce a significant phase shift ( $cos φ$ typical 0.7-0.85 unloaded, improves at full load). Companies correct centrally at the substation or for groups of motors.
Automatic correction banks. Industrial cabinets with capacitors switched in steps and managed by a controller that measures $cos φ$ in real time and activates the right number of stages.
Traditional fluorescent lamps with inductive ballast. They had very low $cos φ$ , which is why the tubes often had a small built-in correction capacitor. With modern LEDs the issue has shrunk but not disappeared.
Industrial bills. Below $cos φ = 0.95$ reactive-energy penalties kick in. Typically a correction system pays for itself in 1-2 years.
Transmission grid. Substations and grid nodes include reactive compensation systems (capacitor banks or reactors) to stabilise voltage and reduce losses.

Classroom discussion questions

At the same $V$ and $I$ , two loads have $cos φ = 1$ and $cos φ = 0.5$ . Which produces more useful work? How much, in proportion?
A purely reactive load: current flows, yet no work is done. Where does the "in-transit" energy go?
What does it mean to correct a plant's power factor, and why is it done with capacitors instead of other components?
What happens if I install too much correction capacitance? Does it improve or worsen the situation?
A small workshop with $cos φ = 0.7$ and another with $cos φ = 0.97$ both draw 10 kW useful. Which pays a higher bill? Why?

Related modules

AC Behaviour (R, L, C): the conceptual prerequisite: phase shift $φ$ between V and I and the impedance of reactive loads are what generate reactive power.
Three-Phase AC Systems: the same power triangle, extended to the three-phase system. Industrial correction is almost always done in three-phase, at the substation or for groups of motors.