Power Quality & Harmonics
In an ideal power system every voltage and current is a pure sine wave at a single frequency. Real networks are not so clean: nonlinear loads distort the waveform, and that distortion can be understood as a sum of sinusoids — the fundamental plus a family of harmonics. This page builds that idea from first principles, through the Fourier series, to the time- and frequency-domain views used in every harmonic study.
In an ideal power system, voltage and current waveforms are expected to be sinusoidal. A pure sinusoidal waveform contains only one frequency component, called the fundamental component. In a 50 Hz power system, this fundamental component has a frequency of 50 Hz.
In practice, however, voltage and current waveforms are not always perfectly sinusoidal. Many electrical devices draw current in a nonlinear way, and this can distort the waveform. A distorted waveform may look different from a pure sine wave, but it is not random — it can be understood as the combination of several sinusoidal waveforms added together.
Section 1
The main sinusoidal component is the fundamental component. The additional sinusoidal components are called harmonics. Harmonics have frequencies that are integer multiples of the fundamental frequency. So when a waveform contains harmonics, it means the waveform contains frequency components other than the fundamental frequency.
| Component | Frequency |
|---|---|
| Fundamental | 50 Hz |
| 2nd harmonic | 100 Hz |
| 3rd harmonic | 150 Hz |
| 4th harmonic | 200 Hz |
| 5th harmonic | 250 Hz |
Each component is itself a simple sinusoid. The fundamental is \(\sin(\omega t)\); the second, third and fifth harmonics are \(\sin(2\omega t)\), \(\sin(3\omega t)\) and \(\sin(5\omega t)\), where \(\omega\) is the angular frequency of the fundamental component.
Section 2
When harmonic components are added to the fundamental component, the resulting waveform becomes distorted. For example, the fundamental plus a one-third third harmonic:
This waveform contains the fundamental component and the third harmonic component. Adding a one-fifth fifth harmonic as well:
This waveform contains the fundamental, third harmonic and fifth harmonic components. Each added sinusoid is still a sine wave, but their sum is no longer a simple sinusoid: the more harmonic components are added, the more the waveform shape changes.
Section 3
A distorted periodic waveform can be represented mathematically using the Fourier series. The Fourier series is important because it allows a complex periodic waveform to be separated into its individual sinusoidal components. In general, a periodic waveform can be written as:
The same waveform can also be written in amplitude and phase form, in which each harmonic component has an amplitude, a frequency and a phase angle:
This is useful because a distorted waveform can be studied by identifying which harmonic components are present and how large they are. Instead of only looking at the distorted waveform in the time domain, the Fourier series allows the waveform to be examined in terms of its frequency components.
Section 4
A square waveform is a useful example. A square waveform is not sinusoidal, but it is periodic — and because it is periodic, it can be represented by a Fourier series. For a symmetrical square waveform, the main components are the odd harmonics: the fundamental, third, fifth, seventh and higher odd harmonics. For example:
This already gives an approximate square-shaped waveform. If more odd harmonic components are added, the waveform becomes closer to an ideal square waveform:
This shows that a non-sinusoidal periodic waveform can be built from sinusoidal components. The fundamental component gives the main shape of the waveform, and the harmonic components modify the shape.
Section 5
The key point is that harmonic distortion is the result of additional sinusoidal components being present in a waveform. A pure sine wave contains only the fundamental component; a distorted waveform contains the fundamental component plus harmonic components. Harmonics can therefore be understood in two complementary ways:
| View | Meaning |
|---|---|
| Time-domain view | Shows the actual waveform shape |
| Frequency-domain view | Shows the harmonic components inside the waveform |
The time-domain view shows how the waveform looks; the frequency-domain view shows what the waveform is made of. For harmonic studies this distinction is important: a waveform may look distorted in the time domain, but the reason for that distortion can only be properly understood by identifying its harmonic components.
A pure sine wave contains only the fundamental component; a distorted waveform is simply the sum of the fundamental and its harmonics, which sit at integer multiples of the fundamental frequency. The Fourier series lets any periodic waveform — even a non-sinusoidal one such as a square wave — be separated into those sinusoidal components, so distortion can be examined in the frequency domain as well as the time domain.
Five-Part Technical Series
A five-part guide to harmonics in power systems — from what harmonics are and how distortion is measured, through resonance, to capacitor banks, power-factor correction and transformer winding effects.
What harmonics are, building distorted waveforms from sinusoids, the Fourier series, and the time- and frequency-domain views.
