Power Quality & Harmonics

Measures of Harmonic Distortion

Once a distorted waveform is seen as a sum of harmonics, the next question is how to measure that distortion. This page sets out the engineering quantities used in harmonic studies — RMS, THD, the power quantities \(P\), \(Q\), \(S\) and distortion power, power factor, crest factor, telephone interference, the frequency response of \(R\), \(L\) and \(C\), and the network tools (frequency scans and harmonic power flow) that turn a waveform into a complete technical assessment.

Reading time ≈ 14 min · Part Three of the series

After understanding that a distorted waveform can be represented as the sum of sinusoidal harmonic components, the next important question is how this distortion can be measured. In practical power-system studies, it is not enough to say a waveform is distorted. The distortion must be quantified so that its effect on equipment, losses, voltage quality, power factor and resonance risk can be assessed.

Key idea

RMS gives the effective heating value; harmonic RMS values combine as the square root of the sum of squares.
THD is the overall level of distortion; pair it with the individual harmonic spectrum.
Under distortion, apparent power gains a distortion-power term, and true power factor falls below the displacement value.
Harmonic voltages depend on the network impedance at each frequency — so harmonics are both a waveform and a network problem.

Section 1

Representing a distorted waveform

A distorted current or voltage waveform can be expressed as a Fourier series. For current:

\[ i(t)=\sum_{h=1}^{\infty} I_h \cos(h\omega_0 t+\phi_h) \]

and for voltage:

\[ v(t)=\sum_{h=1}^{\infty} V_h \cos(h\omega_0 t+\theta_h) \]

\(h\): harmonic order
\(I_h,\ V_h\): peak values of the \(h^{\text{th}}\) harmonic current and voltage
\(\phi_h,\ \theta_h\): phase angles of the \(h^{\text{th}}\) harmonic current and voltage
\(\omega_0\): fundamental angular frequency, \(\omega_0=2\pi f_0\)

Here \(f_0\) is the fundamental frequency, which in most power systems is either 50 Hz or 60 Hz. This representation matters because a distorted waveform is not treated as one unknown shape: it is a combination of known sinusoidal components, each with a defined frequency, magnitude and phase angle. For example, a distorted current may contain:

\[ i(t)=I_1\cos(\omega_0 t+\phi_1)+I_5\cos(5\omega_0 t+\phi_5)+I_7\cos(7\omega_0 t+\phi_7)+I_{11}\cos(11\omega_0 t+\phi_{11}) \]

\(I_1,I_5,I_7,I_{11}\): peak amplitudes of the fundamental, 5th, 7th and 11th harmonic currents
\(\phi_1,\phi_5,\dots\): phase angles of each harmonic
\(\omega_0\): fundamental angular frequency

— a fundamental plus a 5th, 7th and 11th harmonic. The total waveform seen in the time domain is the result of adding these components together.

Section 2

RMS value

The RMS (root-mean-square) value is one of the most important quantities in harmonic analysis. It represents the effective value of a waveform; for current, the RMS value is directly related to heating effect, which is why it matters for cables, transformers, busbars, reactors and other current-carrying equipment.

For a waveform containing harmonics, the total RMS value combines the RMS value of each harmonic component — not arithmetically, but as the square root of the sum of squares. For voltage:

\[ V_{rms}=\sqrt{\sum_{h=1}^{\infty}V_{h,rms}^{2}} \]

\(V_{rms}\): total RMS voltage
\(V_{h,rms}\): RMS value of the \(h^{\text{th}}\) harmonic voltage
\(h\): harmonic order

and for current:

\[ I_{rms}=\sqrt{\sum_{h=1}^{\infty}I_{h,rms}^{2}} \]

\(I_{rms}\): total RMS current
\(I_{h,rms}\): RMS value of the \(h^{\text{th}}\) harmonic current
\(h\): harmonic order

So the total RMS value includes the fundamental and all harmonic components. For example, a current with a 100 A RMS fundamental and a 20 A RMS 5th harmonic does not give 120 A, but \(I_{rms}=\sqrt{100^2+20^2}=101.98\ \text{A}\). Any additional harmonics increase it further.

This matters because equipment heating depends on RMS current: even when harmonic currents are smaller than the fundamental, they still add to the total heating effect. In transformers and cables, high-frequency harmonic currents may also cause additional losses through skin effect, proximity effect and stray losses.

Section 3

Total harmonic distortion (THD)

The most common distortion index is total harmonic distortion, or THD. For voltage and current:

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

\(V_1,\ I_1\): fundamental voltage and current components

The summation starts at \(h=2\) because the fundamental is not considered distortion — distortion is caused by all components above the fundamental frequency. THD is usually expressed as a percentage, \(\%THD=THD\times100\). For example, if the combined RMS of all harmonic currents is 30 A and the fundamental is 100 A:

\[ THD_I=\frac{30}{100}=0.30=30\% \]

so the total harmonic current content is 30% of the fundamental. THD is compact, but it does not show which harmonic orders are present: two waveforms can share the same THD yet have very different spectra (one dominated by the 5th, another by the 11th and 13th). THD should therefore be used together with the individual harmonic spectrum.

The total RMS value can be related directly to THD. For voltage and current:

\[ V_{rms}=V_{1,rms}\sqrt{1+THD_V^2} \qquad I_{rms}=I_{1,rms}\sqrt{1+THD_I^2} \]

\(V_{1,rms},I_{1,rms}\): fundamental RMS voltage and current
\(THD_V,THD_I\): voltage and current total harmonic distortion (as a fraction)

If THD is zero, the total RMS equals the fundamental RMS; as THD rises, the total RMS grows. For example, a 100 A fundamental at 30% current THD gives \(I_{rms}=100\sqrt{1+0.3^2}=104.4\ \text{A}\). The harmonic content is 30% of the fundamental, yet the total RMS rises by only about 4.4% — small, but still important for thermal loading when equipment runs close to its rating.

Section 4

Power under distortion

Active power is the real power transferred to the load. When voltage and current contain harmonics, active power comes only from voltage and current components of the same harmonic order — a voltage at one frequency produces no average power with a current at a different frequency:

\[ P=\sum_{h=1}^{\infty}V_{h,rms}\,I_{h,rms}\cos(\theta_h-\phi_h) \]

\(\theta_h-\phi_h\): phase-angle difference between voltage and current at harmonic order \(h\)

So active power is not purely a fundamental-frequency concept: harmonic voltages and currents also contribute if both exist at the same order. Reactive power follows the same pattern:

\[ Q=\sum_{h=1}^{\infty}V_{h,rms}\,I_{h,rms}\sin(\theta_h-\phi_h) \]

\(Q\): total reactive power
\(V_{h,rms},I_{h,rms}\): RMS voltage and current at harmonic order \(h\)
\(\theta_h-\phi_h\): voltage–current phase difference at order \(h\)

Apparent power is the product of the RMS values, both of which grow when harmonics are present:

\[ S=V_{rms}\,I_{rms} \]

\(S\): apparent power
\(V_{rms},I_{rms}\): total RMS voltage and current

Using THD, this can be written from the fundamental apparent power \(S_1\):

\[ S=S_1\sqrt{1+THD_V^2}\,\sqrt{1+THD_I^2} \]

\(S\): total apparent power
\(S_1\): apparent power of the fundamental component
\(THD_V,THD_I\): voltage and current total harmonic distortion

so both voltage and current distortion raise apparent power. In practice current distortion is usually much higher than voltage distortion near nonlinear loads; voltage distortion becomes significant when harmonic currents flow through system impedance and create harmonic voltage drops.

In a sinusoidal system, \(S^2=P^2+Q^2\). Under distortion this is no longer sufficient: harmonics introduce an additional distortion power term:

\[ D=\sqrt{S^2-P^2-Q^2} \qquad\Longrightarrow\qquad S^2=P^2+Q^2+D^2 \]

\(D\): distortion power
\(S\): apparent power
\(P\): active power
\(Q\): reactive power

Distortion power \(D\) is not useful output power; it is associated with the presence of harmonic voltage and current components. This is one reason harmonic distortion reduces the efficiency of power delivery and increases the rating requirements of equipment.

Section 5

Power factor

Power factor is the ratio of active power to apparent power:

\[ pf=\frac{P}{S} \]

\(pf\): (true) power factor
\(P\): active power
\(S\): apparent power

In a pure sinusoidal system, power factor is set mainly by the phase-angle difference between voltage and current — the displacement power factor:

\[ pf_{disp}=\cos(\theta_1-\phi_1) \]

\(pf_{disp}\): displacement power factor
\(\theta_1-\phi_1\): phase difference between the fundamental voltage and current

When harmonics are present, the true power factor is affected by both displacement and distortion, and can be written as their product:

\[ pf=pf_{disp}\times pf_{dist} \]

\(pf\): true power factor
\(pf_{disp}\): displacement power factor (fundamental phase shift)
\(pf_{dist}\): distortion power factor (effect of harmonics)

This is an important practical point: a load may have a displacement power factor close to unity yet a lower true power factor because its current is highly distorted. Many power-electronic loads draw current in pulses — the fundamental current is almost in phase with the voltage (good displacement power factor), but the significant harmonic content lowers the true power factor. Assessing a nonlinear load by displacement power factor alone is therefore misleading; harmonic distortion must also be considered.

Section 6

Crest factor

Crest factor is the ratio of the peak value of a waveform to its RMS value:

\[ CF_I=\frac{I_{peak}}{I_{rms}} \qquad CF_V=\frac{V_{peak}}{V_{rms}} \]

\(CF_I,CF_V\): current and voltage crest factor
\(I_{peak},V_{peak}\): peak values of the waveform
\(I_{rms},V_{rms}\): RMS values of the waveform

For a pure sine wave the peak is \(\sqrt{2}\) times the RMS, so \(I_{peak}/I_{rms}=\sqrt{2}\) and \(V_{peak}/V_{rms}=\sqrt{2}\). This holds only for a sine wave: when harmonics are present the waveform shape changes and the peak may no longer have the normal sinusoidal relationship with the RMS.

A distorted waveform may have a higher crest factor than a sine wave if the harmonics increase the peak, and its peak shape depends on the harmonic phase angles. This matters for insulation stress, current- and voltage-transformer performance, metering accuracy, protection-relay measurement and the input stages of electronic equipment.

Section 7

Telephone interference

Harmonic currents can interfere with nearby communication circuits — for example, telephone interference caused by inductive coupling between power and communication circuits. The effect depends not only on the magnitude of the harmonic current but also on its frequency, since some frequencies are more likely to cause interference than others. A telephone interference weighting factor accounts for this. The \(I\cdot T\) product is one method used to estimate it:

\[ I\cdot T=\sqrt{\sum_{h=1}^{\infty}(I_h T_h)^2} \]

\(I_h\): current at harmonic order \(h\)
\(T_h\): telephone interference weighting factor at that harmonic

This gives more weight to harmonics that are more likely to interfere with communication systems, so a small current at a highly sensitive frequency may matter more than a larger current at a less sensitive frequency.

Section 8

Harmonics and passive elements

Harmonics affect passive elements differently depending on whether they are resistive, inductive or capacitive. In a pure resistance, the power loss depends on the square of the RMS current:

\[ P_R=I_{rms}^{2}R \]

\(P_R\): resistive power loss
\(I_{rms}\): total RMS current
\(R\): resistance

For a constant resistance this can be written from the fundamental loss \(P_{R1}\):

\[ P_R=P_{R1}\left(1+THD_I^2\right) \]

\(P_R\): total resistive loss including harmonics
\(P_{R1}\): resistive loss from the fundamental current alone
\(THD_I\): current total harmonic distortion

so harmonic currents increase resistive heating. At 30% current THD, \(P_R=P_{R1}(1+0.3^2)=1.09\,P_{R1}\) — a 9% increase over the fundamental loss alone. In practice the increase may be larger because resistance rises with frequency through skin and proximity effects, which is important in large conductors, busbars, transformer windings and cables.

In a pure inductance, reactance increases with frequency:

\[ X_L=h\,\omega_0 L \]

\(X_L\): inductive reactance at harmonic order \(h\)
\(h\): harmonic order
\(\omega_0\): fundamental angular frequency
\(L\): inductance

so the inductor presents higher impedance to higher orders — the 5th-harmonic reactance is \(X_{L5}=5X_{L1}\) and the 7th is \(X_{L7}=7X_{L1}\). The response of inductors, transformers, motors and reactors therefore depends strongly on harmonic order. In a pure capacitance, reactance decreases with frequency:

\[ X_C=\frac{1}{h\,\omega_0 C} \]

\(X_C\): capacitive reactance at harmonic order \(h\)
\(h\): harmonic order
\(\omega_0\): fundamental angular frequency
\(C\): capacitance

so the capacitor presents lower impedance to higher orders — \(X_{C5}=X_{C1}/5\) and \(X_{C7}=X_{C1}/7\). This is important because capacitor banks may attract harmonic currents, and capacitors can interact with network inductance to create resonance. It is one reason harmonic studies are often required before installing power-factor-correction capacitors or harmonic filters.

Section 9

Network response and frequency scans

Harmonic distortion is calculated by analysing the system at each harmonic frequency. In simple systems this uses equivalent circuits; in larger systems, harmonic-analysis software. Because impedance depends on frequency, the network impedance at the fundamental is not the same as at the 5th, 7th, 11th or 13th harmonic. For each harmonic order:

\[ V(h)=Z(h)\,I(h) \]

\(V(h)\): vector of harmonic bus voltages
\(Z(h)\): bus impedance matrix at harmonic order \(h\)
\(I(h)\): vector of harmonic current injections

Harmonic bus voltages are produced when harmonic currents flow through the system impedance at the corresponding frequency. For example, a nonlinear load injecting 5th-harmonic current produces a 5th-harmonic bus voltage that depends on that current and the network impedance at the 5th harmonic; the same applies to other orders.

A frequency scan (or impedance scan) plots the driving-point impedance at a selected bus against frequency or harmonic order. Where the impedance becomes very high, parallel resonance may occur, and a small harmonic current can produce a large harmonic voltage. Where the impedance becomes very low, series resonance may occur, and harmonic current can be amplified through the resonant path.

Section 10

Harmonic power-flow studies

A harmonic power-flow study is more detailed than a frequency scan: it calculates harmonic voltages and currents throughout the network from the harmonic sources and the system impedance. The general process is:

Table 1 — General harmonic power-flow calculation process.

Step	Description
1	Identify nonlinear loads or harmonic sources.
2	Define the harmonic current spectrum of each source.
3	Build the network model at each harmonic frequency.
4	Calculate the bus impedance or admittance matrix.
5	Solve the harmonic voltage at each bus.
6	Calculate harmonic currents in branches.
7	Calculate RMS values, THD, power factor and distortion indices.
8	Check whether harmonic limits or equipment ratings are exceeded.
9	Study mitigation options if required.

Nonlinear devices such as variable-speed drives, rectifiers, converters, inverters, arc furnaces and power-electronic loads are usually represented as harmonic current sources, with a spectrum that defines which orders are injected and how large they are relative to the fundamental. A six-pulse converter, for example, produces characteristic harmonics of the form:

\[ h=6k\pm1,\qquad k=1,2,3,\dots \]

\(h\): characteristic harmonic order of a six-pulse converter
\(k\): any positive integer (1, 2, 3, …)

giving orders \(5, 7, 11, 13, 17, 19, 23, 25, \ldots\) These harmonic currents flow through the network impedance and create harmonic voltage distortion.

The important point is that harmonic distortion is both a waveform issue and a network issue: the source produces harmonic currents, but the resulting harmonic voltages depend on the network impedance. A weak or resonant network may experience higher voltage distortion for the same harmonic current injection, so harmonic assessment normally needs both the source spectrum and the frequency-dependent network impedance.

The key message is that harmonic distortion must be quantified with engineering measures: RMS gives the effective heating value, THD the overall distortion level, the individual harmonic magnitudes show which orders are present, the power quantities \(P\), \(Q\), \(S\) and \(D\) explain the effect on power, crest factor relates peak to RMS, and frequency scans and harmonic power-flow studies show how the network responds at each frequency. Together they take an engineer from a visual waveform to a complete technical assessment of harmonic impact.

Key message

Harmonic distortion must be quantified with engineering measures. RMS gives the effective heating value and THD the overall distortion level, while the individual harmonic magnitudes show which orders are present. The power quantities P, Q, S and distortion power D explain the effect on power, crest factor relates peak to RMS, and frequency scans and harmonic power-flow studies show how the network responds at each frequency — together turning a visual waveform into a complete technical assessment.

Part Three Reading now

Measures of Harmonic Distortion

RMS and THD, the power quantities and distortion power, power factor, crest factor, frequency scans and harmonic power flow.

Series progress 3 of 5

Part Four

Resonance in Harmonic Studies

Series and parallel resonance, the resonant harmonic order, quality factor and damping, and reading resonance from a frequency scan.