Harmonics Formulas: THD and True Power Factor

Harmonics formulas for current THD, RMS current, true power factor, the resonant harmonic of a capacitor bank and detuned reactor tuning, with worked examples.

The formulas

I1 is the fundamental current and I2, I3 and so on the harmonic currents, all in the same unit. THD is a fraction here (multiply by 100 for percent). Ssc is the short-circuit level at the bus and Qc the capacitor bank size in the same unit. The true power factor formula applies when the voltage is nearly sinusoidal.

Worked example: THD and RMS current

A load draws 100 A at the fundamental, with 30 A of 3rd, 20 A of 5th and 10 A of 7th harmonic.

1. THD = √(30² + 20² + 10²) / 100 = 37.42%
2. Irms = 100 × √(1 + 0.14) = 106.8 A
3. With a displacement power factor of 0.95 the true power factor is 0.95 / √(1 + 0.14) = 0.8898

Worked example: capacitor bank resonance

A 400 kVAR bank on a bus with a short-circuit level of 20 MVA (20,000 kVA).

1. h = √(20000 / 400) = 7.071, close to the 7th harmonic, so there is a risk of resonance
2. A 7% detuned reactor tunes the bank to 50 / √0.07 = 189 Hz, below the 5th harmonic at 250 Hz, which avoids the resonance

True power factor at a displacement power factor of 0.95

Frequently asked questions

Why do third harmonics matter in a four-wire system?

The third harmonic currents of the three phases are in step, so they add in the neutral instead of cancelling. The neutral current can exceed the phase current.

What causes harmonics?

Non-linear loads such as drives, rectifiers, switch-mode power supplies and LED drivers draw current in pulses rather than as a sine wave.

Why can a capacitor bank be a problem?

The bank and the supply inductance form a resonant circuit. If it resonates near a harmonic that the loads produce, the current and voltage can become very large.

What is a good THD limit?

It depends on the standard and the point of measurement. Many standards compare the voltage THD at the point of common coupling with a limit near 5%. Check the one that applies to you.

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