Total harmonic distortion (THD): meaning, formula and limits

Total harmonic distortion (THD) measures how far a voltage or current waveform departs from a pure sine wave. It is the RMS value of all the harmonics, the components at whole multiples of the 50 or 60 Hz fundamental, divided by the RMS value of the fundamental, and it is usually given as a percentage. A clean supply voltage has a THD of a few percent; the current drawn by a variable-speed drive, a rectifier or a bank of LED drivers can reach 30% to 100% or more. In power systems, harmonics heat cables, neutrals and transformers, overload capacitors and lower the true power factor, so THD is one of the first figures a power quality survey looks at. (In audio, the same term describes amplifier distortion; this guide is about electrical power.)

The THD formula

For a voltage or current with a fundamental of RMS value X₁ and harmonics X₂, X₃, … Xₙ:

THD = √(X₂² + X₃² + … + Xₙ²) ÷ X₁ × 100%

The harmonics add as the square root of the sum of their squares, because harmonics of different frequencies do not add arithmetically. IEC standards count harmonics up to the 40th or 50th order; IEEE 519 counts to the 50th. The total RMS value follows from the same terms:

X_RMS = X₁ × √(1 + THD²)

So a current with 34.8% THD has an RMS value 5.9% higher than its fundamental, and the cable heats with the RMS value. The true RMS guide explains why a meter must measure that RMS value directly.

Worked example: current THD, TDD and true power factor

A feeder supplying variable-speed drives carries a 100 A fundamental with 30 A of 5th, 15 A of 7th, 8 A of 11th and 5 A of 13th harmonic, the pattern of a six-pulse rectifier:

The harmonic RMS is √(30² + 15² + 8² + 5²) = 34.8 A, so the current THD is 34.8% and the RMS current is 105.9 A. If the feeder's maximum demand current is 160 A, the same harmonics give a TDD of 34.8 ÷ 160 = 21.8%. The harmonics also lower the power factor: with a displacement power factor of 0.95 at the fundamental, the true power factor is 0.95 ÷ √(1 + 0.348²) = 0.90. Capacitors correct only the displacement part; the power factor guide explains the difference.

Voltage THD and current THD

Harmonic currents are produced by the load. They flow back through the impedance of the cables and the supply transformer, and the voltage drop they cause distorts the voltage that every other load on the network sees. Two consequences follow:

• Current THD describes a load or a feeder. It is high for electronic loads by nature and is limited where the site connects to the network.
• Voltage THD describes the supply at a point. It rises with the harmonic current and with the source impedance, so it is higher at the end of a long feeder or on a small transformer, and higher on a generator than on the grid.

A site with high current THD and low voltage THD has a stiff supply that absorbs the harmonics. A site where voltage THD is approaching its limit needs action whatever the current figures say.

Harmonic limits: IEEE 519 and EN 50160

In Great Britain, Engineering Recommendation G5/5 sets the harmonic limits for connecting a site to the distribution network. Higher voltages have tighter voltage limits in IEEE 519: 5% THD from 1 kV to 69 kV, for example. The power quality analyser guide explains EN 50160's week of 10-minute values and the instruments that measure to it.

Where harmonics come from and what they do

These ranges are indicative. The real figure depends on the load, the supply impedance and any filtering, so measure the load before you size a filter or derate equipment.

The effects show up as heat and nuisance:

• Neutral overload. Third-order (triplen) harmonics are in phase on all three phases and add in the neutral instead of cancelling. On a board of single-phase electronic loads, the neutral current can exceed the phase current. Measure it.
• Transformer heating. Eddy-current losses rise with the square of frequency, so harmonic current heats a transformer more than the same RMS current at 50 Hz. Derate the transformer or use a K-rated one.
• Capacitor overload and resonance. Power factor correction capacitors have low impedance at high frequency and can resonate with the supply inductance near the 5th or 7th harmonic. Use detuned (reactor-protected) banks on sites with drives.
• Measurement errors. Average-responding meters and some CTs misread distorted current. A drive output also runs at variable frequency, so measure on the supply side.

How THD is measured

IEC 61000-4-7 defines the measurement: the instrument takes a window of 10 cycles at 50 Hz (12 at 60 Hz), about 200 ms, applies a Fourier transform with 5 Hz resolution and groups the result into harmonic orders. IEC 61000-4-30 then aggregates these values over 3 seconds, 10 minutes and 2 hours. A Class A analyser follows both exactly, so two Class A instruments give the same answer. Many panel meters display THD from a simpler calculation, which is useful for trends but not for proving compliance with a limit.

To trend THD across a site, read it from the meters that already measure it. The Janitza UMG 96-S2 and Carlo Gavazzi WM15, for example, publish voltage and current THD per phase in their Modbus register maps. Both calculate it from the harmonics up to the 15th, not the 40th or 50th that the standards use, so on loads with strong high-order harmonics their figure reads lower than an analyser's. For a connection study or a dispute, use a Class A analyser over at least a week.

THD with EpiSensor

A ZEM electricity monitor measures true RMS voltage and current, energy and power factor at Class 0.5S, but it does not measure individual harmonics or THD. Edge on the ZGW-20 Gateway reads THD from Modbus meters and analysers that report it, alongside the ZEM readings, so harmonic trends sit next to the loads that cause them. The power quality monitoring guide shows the complete set-up for transformers and feeders, and the THD and TDD calculator works out the figures from a harmonic spectrum.