AC Power Calculations Exercises

These exercises practise AC power calculations for sinusoidal steady-state systems. The goal is to connect phasor quantities, impedance, apparent power, active power, reactive power, and power factor. The calculations are introductory, but the interpretation is practical: electrical equipment is rated by voltage, current, thermal loading, insulation limits, and apparent power as well as by useful real power.

Assume sinusoidal waveforms and RMS values unless stated otherwise. Harmonics, unbalance, transients, saturation, and nonlinear loads require additional analysis.

How to Use These Exercises

For each problem:

1. confirm whether voltage and current are RMS, peak, line-to-line, or phase values;
2. define the passive sign convention and whether reactive power is lagging or leading;
3. separate active power, reactive power, apparent power, and current rating;
4. check whether the result affects conductor sizing, transformer loading, protection, voltage drop, or power-factor correction;
5. state which operating measurement would validate the calculation.

The most common mistake is sizing equipment from kW alone. Conductors, switchgear, transformers, UPS systems, generators, and protection devices respond to RMS current, thermal loading, voltage stress, and apparent power, not only useful active power.

Exercise 1: RMS and peak voltage

A single-phase supply is rated at 230\ \text{V RMS}. Find the peak voltage for an ideal sine wave.

Solution

For a sine wave:

Therefore:

This is why insulation, rectifier, and capacitor voltage ratings must consider peak voltage, not only RMS voltage.

Exercise 2: Current through an impedance

A load has impedance:

It is connected to a 120\ \text{V RMS} source. Find current magnitude, current angle, and power factor.

Solution

Magnitude of impedance:

Impedance angle:

Current magnitude:

If voltage angle is taken as zero, current angle is:

The current lags because the reactance is inductive. Power factor is:

The load power factor is 0.80 lagging.

Exercise 3: Real, reactive, and apparent power

Using the load from Exercise 2, find apparent power S, active power P, and reactive power Q.

Solution

Apparent power magnitude:

Active power:

Reactive power:

Because the load is inductive, Q is positive in the usual passive sign convention.

The complex power can be written:

Exercise 4: Power factor correction

A single-phase load consumes P = 8.0\ \text{kW} at a power factor of 0.70 lagging. Find the reactive power before correction. Then find the capacitor reactive power required to improve power factor to 0.95 lagging.

Solution

Initial angle:

Initial reactive power:

Target angle:

Target reactive power:

Required capacitor reactive power:

The capacitor bank should supply approximately 5.5\ \text{kvar} at the operating voltage and frequency. In real systems, switching steps, harmonics, resonance, load variation, and utility requirements must be checked.

Exercise 5: Three-phase balanced load

A balanced three-phase load is supplied at 400\ \text{V} line-to-line. Line current is 32\ \text{A} and power factor is 0.85 lagging. Find total active power, apparent power, and reactive power.

Solution

For a balanced three-phase load:

Active power:

Reactive power:

The result means conductors and transformers must carry current corresponding to 22.2 kVA even though useful active power is 18.8 kW.

Exercise 6: Compare two loads with same real power

Load A consumes 10\ \text{kW} at unity power factor. Load B consumes 10\ \text{kW} at 0.65 lagging power factor. Both are supplied from a 230\ \text{V} single-phase circuit. Find current for each load.

Solution

For single phase:

Load A:

Load B:

Both loads consume the same active power, but Load B requires much more current. This increases conductor heating, voltage drop, transformer loading, and protective-device duty.

Engineering interpretation

AC power calculations are not bookkeeping. They explain equipment stress. Active power describes useful energy transfer. Reactive power affects current and voltage regulation. Apparent power describes the RMS voltage-current burden on equipment. Power factor connects these quantities.

When reviewing an AC power calculation, ask:

• Are values RMS or peak?
• Is the load single-phase or three-phase?
• Is the power factor lagging or leading?
• Are harmonics significant?
• Is the system balanced?
• Are conductors, transformers, and protection rated for current, not only kW?
• Are capacitor banks checked for resonance, switching transients, voltage rating, and harmonic current?
• Are voltage drop, neutral current, derating, duty cycle, and enclosure temperature relevant to the equipment rating?
• Can the calculation be validated from field measurements of voltage, current, power factor, total harmonic distortion, and operating load?

These checks prevent a common mistake: sizing equipment only by active power while ignoring apparent power and current.