Power Quality & Harmonics
Harmonic studies rely on a few structural properties of harmonics: how waveform symmetry decides which Fourier terms survive, how each harmonic order maps to a phase sequence in three-phase systems, why triplen harmonics cancel in line voltages, and why each harmonic can be solved independently. This page sets out those characteristics and how they shape a harmonic study.
Harmonics in power systems have several important characteristics. These help engineers understand which harmonic components are likely to appear in a waveform, how they behave in three-phase systems, and how they can be studied in calculations.
One of the most useful characteristics is waveform symmetry. The symmetry of a waveform determines which terms appear in its Fourier series and which harmonic components may be cancelled.
Section 1
A waveform has odd symmetry when
The negative half of the waveform is the mirror image of the positive half, but with opposite sign. For this type of waveform the Fourier series contains only sine terms — the cosine terms are not present.
A waveform has even symmetry when
The waveform is mirrored around the vertical axis. For this type of waveform the Fourier series contains only cosine terms — the sine terms are not present.
Another important type is half-wave symmetry, where the second half of the cycle equals the negative of the first half:
Half-wave symmetry matters in power-system harmonic studies because it causes the DC component and the even-order harmonics to cancel. The 2nd, 4th, 6th, 8th and other even harmonics are not present in an ideal half-wave symmetrical waveform.
This is one reason even harmonics are often small or ignored in many balanced power-system harmonic studies: many power-system voltages and currents are approximately half-wave symmetrical, especially under normal balanced operating conditions.
Section 2
A typical example is the phase current of a six-pulse converter. The current waveform is not sinusoidal, but it has a regular repeating shape. If the waveform is odd and also has half-wave symmetry, its Fourier series will not contain cosine terms, will not contain a DC component, and will not contain even harmonics. The waveform can therefore be represented mainly by sine terms of selected odd harmonic orders:
The waveform can then be approximated by adding these harmonic components:
As more harmonic components are included, the reconstructed waveform becomes closer to the original rectangular waveform.
If the same rectangular waveform is shifted so that it has even symmetry instead of odd symmetry, the harmonic content changes in form. The waveform may still have half-wave symmetry, so the DC component and even harmonics are still cancelled; but because the waveform is now even, the Fourier series contains cosine terms instead of sine terms:
So the same physical waveform shape may be represented using sine or cosine terms depending on the chosen time reference. The harmonic orders remain similar, but the phase reference changes the mathematical form of the Fourier series.
Section 3
In three-phase power systems, harmonic phase sequence is also very important. In a balanced three-phase system, each harmonic order behaves as either a positive-sequence, negative-sequence or zero-sequence component, depending on the harmonic order.
Positive-sequence harmonics rotate in the same direction as the fundamental component; negative-sequence harmonics rotate in the opposite direction; and zero-sequence harmonics have the same phase angle in all three phases. The common pattern is:
| Harmonic Order | Sequence |
|---|---|
| 1st | Positive |
| 2nd | Negative |
| 3rd | Zero |
| 4th | Positive |
| 5th | Negative |
| 6th | Zero |
| 7th | Positive |
| 8th | Negative |
| 9th | Zero |
| 10th | Positive |
| 11th | Negative |
| 12th | Zero |
| 13th | Positive |
This pattern repeats every three harmonic orders. In general, \(h=3n+1\) gives positive-sequence harmonics, \(h=3n-1\) gives negative-sequence harmonics, and \(h=3n\) gives zero-sequence harmonics.
Section 4
Triplen harmonics are harmonics whose order is a multiple of three — the 3rd, 6th, 9th, 12th and 15th. In a balanced three-phase system they are zero-sequence components.
This has an important practical consequence. Balanced triplen harmonic voltages are in phase in the three phases, so when line-to-line voltages are calculated the triplen components cancel. For example:
Since the triplen harmonic components in \(V_a\) and \(V_b\) are equal and in phase, they subtract and disappear from the line-to-line voltage. This is why triplen harmonics may be present in phase voltages but absent from line voltages in balanced three-phase systems.
Sequence behaviour matters because it governs how harmonic currents flow in the network:
This also means balanced triplen harmonic currents cannot flow into a delta winding or through a three-wire system without a neutral or ground return path. In delta windings the triplen currents may circulate inside the delta, but they do not normally appear on the line side.
Section 5
Another important characteristic is independence. In a linear network, each harmonic frequency can be studied separately: the network can be analysed at the fundamental frequency, then at the 5th harmonic, then at the 7th, and so on. For each harmonic order the system impedance is different, because inductive and capacitive reactances depend on frequency.
Inductive reactance increases with harmonic order:
Capacitive reactance decreases with harmonic order:
So the network response at the 5th harmonic is not the same as at the fundamental, and the response at the 7th harmonic is different again. The usual harmonic-study approach is therefore:
| Step | Description |
|---|---|
| 1 | Identify the harmonic orders to be studied. |
| 2 | Build or calculate the network impedance at each harmonic frequency. |
| 3 | Solve the voltage and current response for each harmonic separately. |
| 4 | Combine the harmonic components if the time-domain waveform is required. |
This is possible because, in a linear system, the response to one harmonic does not affect the response to another. The total distorted waveform is obtained by adding the individual harmonic components together in the time domain.
The key point is that harmonic analysis separates a complex distorted waveform into simpler sinusoidal components. Each component has its own frequency, magnitude, phase angle and sequence behaviour — which makes it possible to understand and calculate the effect of harmonics in a structured way.
Waveform symmetry decides which Fourier terms survive: odd symmetry leaves only sine terms, even symmetry only cosine terms, and half-wave symmetry removes the DC component and all even harmonics. In three-phase systems each order is a positive-, negative- or zero-sequence component; triplen harmonics are zero-sequence and cancel in balanced line-to-line voltages; and in a linear network each harmonic can be solved independently and the results superposed.
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.
Waveform symmetry and Fourier terms, converter-current harmonics, three-phase phase sequence, triplen cancellation and superposition.
