Power Quality & Harmonics

Harmonic Sources and Their Signatures in Power Systems

Harmonics appear wherever the relationship between voltage and current is nonlinear. Each source — saturable magnetic equipment, arcing loads and power-electronic converters — leaves a different harmonic signature. This opening part of the series sets out the mechanism, surveys the main source families, and works through the characteristic harmonics of pulse converters and how distortion is finally judged at the point of common coupling.

Reading time ≈ 16 min · Part One of the series

Harmonics are produced when the relationship between voltage and current is not linear. In an ideal linear load a sinusoidal voltage produces a sinusoidal current of the same frequency. In a nonlinear device this no longer holds: the current can become distorted even when the applied voltage is sinusoidal, and a distorted current flowing through the system impedance produces a distorted voltage at the bus.

Key idea

Harmonics arise from nonlinearity — a sinusoidal voltage can still draw a non-sinusoidal current.
Each source has a distinct signature: magnetic saturation, arcing, or power-electronic switching.
A \(p\)-pulse converter produces characteristic harmonics of order \(h=pn\pm1\).
The distortion seen at the bus is the source spectrum acting on the network impedance, not the source alone.

Section 1

The harmonic mechanism

Sinusoidal voltage \(v(t)\) → non-sinusoidal current \(i(t)\) in a nonlinear device. That harmonic current, flowing through the network impedance, then produces a harmonic voltage at the bus.

The second half of the mechanism is the network: a harmonic current driven into the system impedance creates a harmonic voltage,

\[ i_h(t)\times Z_h \;\longrightarrow\; v_h(t) \]

\(i_h(t)\): harmonic current component at order \(h\)
\(Z_h\): system impedance at harmonic order \(h\)
\(v_h(t)\): resulting harmonic voltage at the bus

So harmonic distortion is not caused by the load alone: it is the interaction between the harmonic source and the frequency-dependent impedance of the network. A distorted current can be written as a fundamental plus a sum of harmonics:

\[ i(t)=I_1\sin(\omega_0 t+\phi_1)+\sum_{h=2}^{\infty} I_h\sin(h\omega_0 t+\phi_h) \]

\(I_1\): fundamental current
\(I_h\): current at harmonic order \(h\)
\(\omega_0\): fundamental angular frequency
\(\phi_h\): phase angle of the harmonic component

Each harmonic sits at an integer multiple of the fundamental frequency:

\[ f_h=h f_0 \]

\(f_h\): frequency of harmonic order \(h\)
\(f_0\): fundamental frequency (normally 50 or 60 Hz)

For a 50 Hz system the 5th harmonic is 250 Hz, the 7th is 350 Hz and the 11th is 550 Hz. The overall distortion is commonly expressed as total harmonic distortion:

\[ THD_I=\frac{\sqrt{\displaystyle\sum_{h=2}^{\infty}I_h^2}}{I_1} \qquad THD_V=\frac{\sqrt{\displaystyle\sum_{h=2}^{\infty}V_h^2}}{V_1} \]

\(I_1,V_1\): fundamental current and voltage
\(I_h,V_h\): current and voltage at harmonic order \(h\)

The fundamental is excluded from the numerator because it is not distortion. THD gives the total harmonic content but does not identify the dominant orders, so engineering assessment should use both THD and the individual harmonic spectrum.

Section 2

Main families of harmonic sources

Harmonic sources fall into three broad groups, each with its own mechanism:

Table 1 — The three main families of harmonic sources.

Source Group	Typical Examples	Main Harmonic Mechanism
Saturable magnetic equipment	Transformers, motors, generators	Nonlinear magnetisation
Arcing and discharge equipment	Arc furnaces, fluorescent and mercury lamps	Nonlinear arc voltage–current behaviour
Power-electronic equipment	Rectifiers, inverters, UPS, VSDs, HVDC, SVCs, EV chargers, switched-mode supplies	Switching and controlled conduction

In modern networks, power-electronic equipment is usually the dominant source of distortion. But classical sources — transformers and rotating machines — remain important, especially for triplen harmonics, saturation, ferroresonance and transformer-connection behaviour.

Section 3

Transformers as harmonic sources

A transformer produces harmonic current because its magnetic core is nonlinear. At no load it draws magnetising current to establish core flux; if the applied voltage is sinusoidal the flux is approximately sinusoidal, but the magnetising current is not, because the magnetising inductance changes with flux density. The voltage–flux relationship is:

\[ v(t)=N\frac{d\phi(t)}{dt} \]

\(v(t)\): applied winding voltage
\(N\): number of turns
\(\phi(t)\): core flux

For sinusoidal voltage the flux \(\phi(t)\) is approximately sinusoidal, but because the core \(B\text{–}H\) curve is nonlinear, the current needed to produce that flux is distorted. The magnetising current normally contains odd harmonics, with the 3rd often significant; under overvoltage or saturation the harmonic content rises.

Winding connection then decides whether triplen harmonic currents can flow. Triplen harmonics are the orders that are multiples of three, \(h=3,6,9,12,\ldots\) In a balanced three-phase system they are zero-sequence components — in phase in all three phases — and need a return path. A delta winding provides a closed path in which they circulate without appearing strongly in the line currents; an ungrounded wye gives no neutral return, so they are restricted; a grounded wye provides a return through the neutral or ground. For line-to-line voltages, balanced triplen components cancel:

\[ V_{ab}=V_a-V_b \]

\(V_{ab}\): line-to-line voltage between phases a and b
\(V_a,V_b\): phase-to-neutral voltages of phases a and b

If the 3rd-harmonic components in \(V_a\) and \(V_b\) are equal and in phase, their difference is zero — so triplen harmonics may appear in the phase-to-neutral voltage but not in a balanced line-to-line voltage.

Section 4

Rotating machines

Rotating machines produce harmonics because their windings sit in slots and cannot be perfectly sinusoidally distributed in space, creating a non-sinusoidal magnetomotive force. Slotting, winding distribution, saturation and air-gap effects all introduce harmonic components.

In three-phase machines, design techniques such as coil-pitch selection and distributed windings reduce specific harmonics, particularly the 5th and 7th. Large generators are usually connected to the grid through transformers, often with delta windings that block or circulate triplen components. In most practical studies the harmonics from rotating machines are smaller than those from converters, but they should not be ignored where machine saturation, generator neutral voltage or special winding arrangements are relevant.

Section 5

Arcing and discharge loads

Arcing loads are nonlinear because the arc voltage–current characteristic changes during operation. Arc furnaces are the major example: their current is irregular, fluctuating and rich in harmonics and interharmonics, and they can also produce flicker through rapid variation of reactive power and voltage.

Fluorescent and discharge lamps also draw distorted current. In older lamps the magnetic ballast is nonlinear and the lamp behaves as a discharge device; in modern lighting, electronic ballasts and switched-mode circuits give harmonic patterns different from traditional ballasts. The common point is that arcing and discharge loads do not behave as fixed sinusoidal impedances — their behaviour changes within each cycle, which produces distortion.

Section 6

Static VAR compensators

Static VAR compensators (SVCs) control voltage and reactive power, and may include thyristor-controlled reactors, thyristor-switched capacitors and harmonic filters. A thyristor-controlled reactor does not draw a smooth sinusoidal current: the thyristors control the conduction interval within each half-cycle, producing a chopped waveform that contains harmonics. The firing angle sets the current waveform and therefore the spectrum, so practical SVCs include filters to absorb the characteristic harmonics.

Controlled conduction → non-sinusoidal current → harmonic injection.

Section 7

Rectifiers, inverters and controlled converters

Power converters are among the most important harmonic sources in modern networks — used in variable-speed drives, battery chargers, DC supplies, UPS, HVDC, renewable generation, electrolysers, EV charging and industrial equipment. A rectifier converts AC to DC, an inverter converts DC to AC, and a controlled converter can do either depending on firing angle and DC-side conditions.

For a controlled rectifier the firing angle \(\alpha\) sets where in the AC cycle conduction begins, which controls the average DC output voltage. For a single-phase controlled rectifier with continuous current:

\[ V_d=\frac{2V_m}{\pi}\cos\alpha \]

\(V_d\): average DC output voltage
\(V_m\): peak AC voltage
\(\alpha\): firing (delay) angle

and for a three-phase six-pulse controlled bridge:

\[ V_d=\frac{3V_m}{\pi}\cos\alpha \]

\(V_d\): average DC output voltage
\(V_m\): relevant peak AC voltage
\(\alpha\): firing angle

With \(0^\circ<\alpha<90^\circ\) the converter rectifies and power flows from the AC system to the DC side; with \(90^\circ<\alpha<180^\circ\) it inverts and power flows from the DC side back to the AC system, provided the DC source and commutation conditions allow it. The firing angle also affects harmonic phase angles and waveform shape, so converter studies should consider operating mode, firing angle, commutation overlap, DC smoothing, transformer phase shift and network impedance.

Section 8

DC-side reactance and commutation

The DC-side inductance strongly shapes the converter current. With small DC inductance the DC current may be discontinuous — existing only for part of the cycle — giving a different, often broader harmonic spectrum. With large DC inductance the current becomes more continuous and the AC-side current tends towards a rectangular shape, the common approximation for line-commutated converters.

Commutation — the transfer of current from one valve to another — is also important. AC system reactance causes commutation overlap, during which two devices conduct together and the current transfers gradually rather than instantly. A simplified relationship is:

\[ \cos(\alpha+\mu)=\cos\alpha-\frac{\omega L_c I_d}{V_m} \]

\(\alpha\): firing angle
\(\mu\): commutation overlap angle
\(L_c\): commutating inductance
\(I_d\): DC current
\(\omega,V_m\): angular frequency and peak AC voltage

Higher commutating reactance and higher DC current increase the overlap angle. Overlap normally smooths some high-frequency content but also changes the voltage waveform, the reactive-power demand and the harmonic phase angles.

Section 9

Characteristic harmonics of pulse converters

A key rule for line-commutated converters is that a \(p\)-pulse converter produces characteristic harmonics of order:

\[ h=pn\pm1 \]

\(h\): characteristic harmonic order
\(p\): pulse number of the converter
\(n\): positive integer (1, 2, 3, …)

For a six-pulse converter \(h=6n\pm1\), giving orders \(5, 7, 11, 13, 17, 19, 23, 25, \ldots\) For a twelve-pulse converter \(h=12n\pm1\), giving \(11, 13, 23, 25, 35, 37, \ldots\) Increasing the pulse number therefore removes the lower-order characteristic harmonics. A twelve-pulse converter is normally two six-pulse bridges fed through transformer windings with a 30° phase shift; the shift cancels the 5th and 7th on the primary side, leaving the 11th and 13th as the first significant harmonics. Higher pulse numbers are built by combining six-pulse bridges through phase-shifting transformers, with an approximate shift between secondaries of:

\[ \Delta\theta=\frac{360^\circ}{p} \]

\(\Delta\theta\): phase shift between converter transformer secondaries
\(p\): total pulse number

Table 2 — Pulse number, phase shift and characteristic harmonics.

Converter Type	Approx. Phase Shift	Characteristic Harmonics
6-pulse	Not applicable	\(6n\pm1\)
12-pulse	\(30^\circ\)	\(12n\pm1\)
24-pulse	\(15^\circ\)	\(24n\pm1\)
36-pulse	\(10^\circ\)	\(36n\pm1\)
48-pulse	\(7.5^\circ\)	\(48n\pm1\)

The magnitude of each characteristic harmonic, for an ideal rectangular converter current, is often approximated by:

\[ I_h \approx \frac{I_1}{h} \]

\(I_h\): magnitude of harmonic order \(h\)
\(I_1\): fundamental current
\(h\): harmonic order

In practice the actual values are usually lower or different because of commutation overlap, DC smoothing, transformer impedance, firing-angle control, unbalance, background distortion and converter control.

A six-pulse converter produces strong 5th and 7th harmonic currents — important because they are low-order and likely to interact with power-factor capacitors or network resonance. By sequence, \(h=5=3n-1\) is a negative-sequence harmonic and \(h=7=3n+1\) is a positive-sequence harmonic. Negative-sequence currents matter for rotating machines, as they produce a field rotating opposite to the fundamental, increasing heating and torque pulsation. A twelve-pulse converter cancels the 5th and 7th by phase shift, leaving the 11th and 13th and a generally lower THD — though the cancellation is imperfect if the two bridges are unequally loaded, the phase shift is inaccurate, or the AC system is unbalanced.

Section 10

Switched-mode supplies and modern loads

Switched-mode power supplies, UPS, LED drivers, computer supplies and many low-voltage electronic devices can produce significant current distortion. Their input current is often highly peaked, because current is drawn only near the voltage peaks — especially in simple diode-capacitor input stages. Unlike ideal six-pulse industrial converters, they produce a broad spectrum, including triplen harmonics in single-phase systems.

In low-voltage networks the 3rd harmonic is particularly important: single-phase nonlinear loads connected phase-to-neutral inject triplen currents that, being zero-sequence, do not cancel in the neutral — they add arithmetically. Where many single-phase nonlinear loads are present, this can overheat the neutral conductor.

Section 11

Harmonic assessment at the connection point

For practical studies, harmonics are assessed where the user installation meets the supply — the point of common coupling (PCC). The assessment should normally consider:

Table 3 — What a PCC harmonic assessment should consider.

Assessment Item	Purpose
Harmonic source spectrum	Identifies injected harmonic orders and magnitudes
Background voltage distortion	Determines pre-existing harmonic voltage at the PCC
Network impedance versus frequency	Identifies resonance and amplification risk
Transformer connection	Determines triplen propagation and sequence behaviour
Capacitor banks and filters	May shift resonance or absorb harmonics
Operating scenarios	Captures different load, generation and switching conditions
Individual harmonics	Identifies dominant problematic orders
THD	Gives the total distortion level
Compliance limits	Checks whether planning or compatibility levels are exceeded

The simplified harmonic voltage calculation is:

\[ V_h = Z_h I_h \]

\(V_h\): harmonic voltage at order \(h\)
\(I_h\): injected harmonic current
\(Z_h\): network impedance at harmonic order \(h\)

This is why the same nonlinear load causes different voltage distortion in different networks: a strong network with low harmonic impedance sees limited distortion, while a weak or resonant network can see high distortion from the same current injection.

The practical message is that the source of harmonics is not simply “nonlinear load” — each source has a distinct signature. Transformers mainly give magnetising-current harmonics; rotating machines give smaller components from winding distribution, slotting and saturation; arcing and discharge loads give irregular, fluctuating content; and power-electronic converters give characteristic harmonics tied to switching, firing angle, pulse number, phase shift and commutation. For converter equipment the first check is usually the pulse number, \(h=pn\pm1\).

Key message

Harmonics come from nonlinearity, and every source leaves a different signature — saturable magnetic equipment, arcing loads and power-electronic switching. For converters the dominant orders follow \(h=pn\pm1\) (a six-pulse bridge gives 5, 7, 11, 13, …; a twelve-pulse bridge gives 11, 13, 23, 25, …). But the distortion that finally appears at the bus depends on the network as well: harmonic source + network impedance = harmonic distortion. A sound assessment must therefore identify both the harmonic-producing equipment and the system conditions that amplify or attenuate it.

Part One Reading now

Harmonic Sources and Their Signatures

Where harmonics come from — transformers, machines, arcing loads and converters — and the characteristic harmonics of pulse converters.

Series progress 1 of 2

Part Two

Effects on Power System Equipment

How harmonics heat and stress equipment — capacitors, transformers, cables, neutrals, machines, protection and metering — and how impact is assessed and mitigated.