In an ideal three-phase system the phase voltages are equal in magnitude and exactly 120° apart. When that is not the case, the system is unbalanced — and the practical concern is the negative-sequence voltage this creates, which forces negative-sequence current through rotating machines.
\[ V_a = V_b = V_c \qquad\qquad \angle V_a-\angle V_b=\angle V_b-\angle V_c=\angle V_c-\angle V_a=120^\circ \]
- \(V_a,V_b,V_c\)
- the three phase-voltage magnitudes
- \(\angle V\)
- phase angle of each phase voltage
Key idea
- Unbalance is measured by the negative-sequence factor \(U_{\text{neg}}=\tfrac{U_2}{U_1}\times 100\).
- Use the fundamental only — remove harmonics before calculating unbalance.
- The usual index is the 10-minute, weekly 95% value; common objective ≈ 2%.
- It is a machine-duty issue: \(U_2\approx Z_2 I_2\), and a small \(U_2\) gives a large motor current.
Section 1
What voltage unbalance is
Voltage unbalance is a three-phase condition in which the phase voltages differ in magnitude and/or are not separated by exactly 120°. It matters because it produces negative-sequence voltage, and negative-sequence voltage produces negative-sequence current in motors, generators and other rotating machines — bringing additional heating, torque pulsation, vibration and loss of capability. The engineering concern is therefore a chain:
\[ \text{voltage unbalance}\;\rightarrow\;\text{negative-sequence current}\;\rightarrow\;\text{machine heating and derating} \]
Section 2
Symmetrical components and the unbalance factor
Unbalance is best described with symmetrical components: any unbalanced three-phase voltage set decomposes into a positive-sequence component (balanced, normal phase order), a negative-sequence component (balanced, reverse phase order) and a zero-sequence component (three equal, in-phase voltages). For power-quality assessment, unbalance is expressed as the ratio of negative-sequence to positive-sequence voltage:
\[ U_{\text{neg}}=\frac{U_2}{U_1}\times 100 \]
- \(U_1\)
- positive-sequence voltage component
- \(U_2\)
- negative-sequence voltage component
- \(U_{\text{neg}}\)
- negative-sequence voltage unbalance factor (per cent)
The same quantity is sometimes written as the voltage unbalance factor, \(VUF=\tfrac{V_2}{V_1}\times 100\). The zero-sequence component can also be calculated, but negative sequence is normally the most important for planning and equipment duty because of its effect on rotating machines.
Section 3
Use only the fundamental component
Voltage unbalance must be calculated from the fundamental-frequency components only; harmonics should be removed first, using a suitable method such as a DFT-based algorithm. Unbalance and harmonic distortion are different power-quality phenomena, so the correct path is:
\[ \text{fundamental voltage only}\;\rightarrow\;\text{symmetrical components}\;\rightarrow\;\frac{U_2}{U_1} \]
and not a calculation taken directly from a distorted waveform that still contains harmonics — which would give a misleading unbalance figure.
Section 4
The negative-sequence factor, with a worked example
The preferred index measures the negative-sequence voltage directly against the positive-sequence voltage. Take a simple case:
\[ U_1 = 1.0\ \text{pu},\quad U_2 = 0.02\ \text{pu}\qquad\Longrightarrow\qquad U_{\text{neg}}=\frac{0.02}{1.0}\times 100 = 2\% \]
So the negative-sequence voltage is 2% of the positive sequence. Two per cent may look insignificant, but for a motor it can be severe: the negative-sequence impedance is low, so a small negative-sequence voltage drives a relatively large negative-sequence current.
Section 5
The NEMA line-voltage approximation
Many equipment guides use an approximate formula based on line-to-line voltage magnitudes — the maximum deviation from the average line voltage:
\[ V_{\text{avg}}=\frac{V_{ab}+V_{bc}+V_{ca}}{3} \]
\[ \%VU=\frac{\max\!\big(|V_{ab}-V_{\text{avg}}|,\;|V_{bc}-V_{\text{avg}}|,\;|V_{ca}-V_{\text{avg}}|\big)}{V_{\text{avg}}}\times 100 \]
- \(\%VU\)
- NEMA voltage-unbalance factor (per cent)
- \(V_{ab},V_{bc},V_{ca}\)
- the three line-to-line voltage magnitudes
- \(V_{\text{avg}}\)
- average of the three line-to-line voltages
This is simple and widely used for motor checks, but it is an approximation — it does not compute the negative-sequence component directly, and it can be inaccurate where zero-sequence voltage, phase-angle displacement or the measurement configuration affects the result. For power-quality assessment the symmetrical-component factor \(U_{\text{neg}}=\tfrac{U_2}{U_1}\times 100\) is more accurate. Where possible, use line-to-line voltages, because that removes zero-sequence voltage from the assessment.
Section 6
Time aggregation and common indices
Unbalance changes with load variation, single-phase loading, switching and transformer loading, so it is assessed statistically rather than from one instant. As with harmonics, the fundamental voltage is measured over a 10-cycle (50 Hz) or 12-cycle (60 Hz) window and aggregated upward:
\[ 10/12\ \text{cycle}\;\rightarrow\;3\ \text{s}\;\rightarrow\;10\ \text{min}\;\rightarrow\;2\ \text{h}\;\rightarrow\;\text{weekly statistic} \]
Table 1 — Common voltage unbalance indices.
| Index | Typical Meaning |
|---|
| \(U_{\text{neg},\text{vs}}\) | Very short-time (3-second) negative-sequence unbalance |
| \(U_{\text{neg},\text{sh}}\) | Short-time (10-minute) negative-sequence unbalance |
| \(U_{\text{neg},\text{lt}}\) | Long-time (2-hour) negative-sequence unbalance |
| \(U_{\text{neg},95\%,\text{weekly}}\) | 95% weekly value — the most common practical index |
| Maximum value | Worst measured value over the assessment period |
The 10-minute, weekly 95% value is the workhorse index: it means 95% of the week’s 10-minute values were at or below the reported figure. The compliance check is:
\[ U_{\text{neg},95\%,\text{weekly}}\;\le\;U_{\text{neg},\text{objective}} \]
- \(U_{\text{neg},95\%,\text{weekly}}\)
- 95th-percentile weekly negative-sequence unbalance
- \(U_{\text{neg},\text{objective}}\)
- applicable objective or limit
For rotating machines, the maximum or long-duration value can also matter, because thermal effects accumulate with time.
Section 7
Voltage unbalance objectives
Many standards use around 2% as the common MV objective. Up to 3% can occur or be accepted where there is significant single-phase loading, single-phase traction or single-phase distribution. HV and EHV objectives are often lower, in the 1–2% range, depending on the network and planning philosophy.
Table 2 — Typical voltage unbalance objectives.
| Voltage Level | Typical Objective |
|---|
| LV / MV | Around 2% |
| MV with significant single-phase loading | Up to 3% in some areas |
| HV / EHV | Often 1% to 2% |
| Motor terminals | Derating may apply above 1%, per motor guidance |
The exact value should always be confirmed from the applicable standard, grid code, connection agreement or utility planning criteria.
Section 8
Motor derating and equipment impact
Unbalance matters most for motors. The negative-sequence voltage drives a negative-sequence current whose field rotates against the rotor, producing extra rotor heating and torque pulsation. Because the motor’s negative-sequence impedance is low, that current is much larger than the voltage figure suggests — roughly:
1% voltage unbalance → several per cent negative-sequence current (depending on motor design and loading).
Motor standards commonly recommend derating above about 1% unbalance, and operation above 5% is generally not advised. The main effects are:
Table 3 — Effects of voltage unbalance on motors.
| Effect | Explanation |
|---|
| Additional heating | Negative-sequence current increases losses |
| Reduced output capability | The motor may need derating |
| Torque pulsation | The opposing field produces pulsating torque |
| Vibration | Mechanical stress increases |
| Reduced efficiency | Extra losses reduce performance |
| Reduced insulation life | Higher temperature accelerates ageing |
For motor-heavy installations, unbalance is therefore both a network-quality issue and an equipment-duty issue.
Section 9
Causes of voltage unbalance
Unbalance arises from network asymmetry, load asymmetry or a combination of the two:
Table 4 — Common causes of voltage unbalance.
| Cause | Explanation |
|---|
| Unequal single-phase loading | Different phase currents give unequal voltage drops |
| Single-phase traction loads | Large single-phase loads on a three-phase system |
| Untransposed overhead lines | Different phase impedances |
| Blown capacitor fuse | One phase of a capacitor bank disconnected |
| Unequal transformer loading | Different phase currents |
| Faulty / high-resistance connection | A poor joint on one phase |
| Open conductor or abnormal switching | Severe unbalance |
| Distributed-generation imbalance | Unequal single-phase generation |
The unbalance at a bus can be understood conceptually as a negative-sequence current flowing through the negative-sequence impedance:
\[ U_2 \approx Z_2\,I_2 \]
- \(Z_2\)
- negative-sequence system impedance
- \(I_2\)
- negative-sequence current from unbalanced loads
Unbalanced load current + system impedance = voltage unbalance — so a weaker network shows higher unbalance for the same load.
Section 10
Assessment location
The assessment location must be stated, because a site can comply at the PCC yet still show unacceptable unbalance at an internal motor terminal due to unequal internal loading or voltage drops:
Table 5 — Where unbalance is assessed, and why.
| Location | Purpose |
|---|
| PCC | Check supply quality or connection compliance |
| Main switchboard | Check internal site voltage quality |
| Motor terminals | Check motor duty and derating |
| Transformer secondary | Check phase loading and supply balance |
| MV busbar | Check the network planning objective |
| HV / EHV busbar | Check the transmission coordination objective |
\(U_{\text{neg},95\%,\text{weekly}} = 1.4\%\) at the PCC — or \(U_{\text{neg}} = 2.3\%\) at the motor terminals: very different statements.
Section 11
Unbalance versus harmonics and flicker, and the reference levels
Unbalance is a fundamental-frequency phenomenon, distinct from harmonics (integer multiples of the fundamental, \(h f_0\) — for example \(5f_0,\,7f_0,\,11f_0\)) and from flicker (slow voltage fluctuation over time). Keeping the three apart is essential when reading power-quality measurements:
Table 6 — Three power-quality phenomena, three indices.
| Phenomenon | Frequency Content | Main Index |
|---|
| Voltage unbalance | Fundamental only | \(U_2/U_1\) |
| Harmonic distortion | Integer multiples of the fundamental | \(\text{THD},\;U_h\) |
| Flicker | Voltage fluctuation over time | \(P_{st},\;P_{lt}\) |
As with harmonics and flicker, three reference terms must be kept distinct:
Compatibility level = the disturbance level equipment and users should generally tolerate (for unbalance, around 2% at MV).
Voltage characteristic = the supply quality a user can expect at the terminals (often \(U_{\text{neg},95\%,\text{weekly}}\le 2\%\)).
Planning level = the operator’s internal target for allocating unbalance among users (e.g. 2% at MV, 1% at HV).
The exact values depend on national standards, network structure, voltage level, existing background unbalance and the type of connected load.
Section 12
Mitigation measures
Unbalance is reduced by correcting the source of unbalanced current or by strengthening the network:
Table 7 — Common unbalance mitigation measures.
| Mitigation Method | Purpose |
|---|
| Rebalance / redistribute single-phase loads | Reduce negative-sequence current |
| Repair faulty connections | Remove high-resistance phase conditions |
| Phase-balancing equipment | Dynamically compensate unbalance |
| Connect large single-phase loads at stronger points | Reduce voltage-drop imbalance |
| Dedicated transformers or feeders | Isolate disturbing loads |
| Improve line transposition | Reduce impedance asymmetry |
| Check capacitor-bank phase condition | Avoid single-phase capacitor loss |
| Increase short-circuit strength | Reduce unbalance sensitivity |
Cut the cause and the index follows: \(I_2\downarrow\;\Rightarrow\;U_2\downarrow\;\Rightarrow\;U_{\text{neg}}\downarrow\).
Section 13
Interpretation and key message
The practical process runs from the phase voltages to the comparison:
\[ V_a,V_b,V_c\;\rightarrow\;V_1,V_2,V_0\;\rightarrow\;U_{\text{neg}}\;\rightarrow\;10\text{-min values}\;\rightarrow\;95\%\ \text{weekly}\;\rightarrow\;\text{comparison with objective} \]
An unbalance statement should pin down six things — the assessment location, the measurement period, the time aggregation, the statistical index, the calculation method and the objective. A bare “voltage unbalance = 2%” is incomplete: it does not say whether it is \(U_2/U_1\) or a NEMA-style magnitude approximation, nor whether it is instantaneous, a 10-minute value, a weekly 95% value, a maximum, or a motor-terminal reading. A defensible statement looks like:
\(U_{\text{neg},95\%,\text{weekly}} = 1.8\%\) at the PCC, based on 10-minute values.
Key message
Voltage unbalance is measured by the negative-sequence factor \(U_{\text{neg}}=\tfrac{U_2}{U_1}\times 100\), calculated from the fundamental only with harmonics removed. The usual index is the 10-minute, weekly 95% value, with objectives around 2% at MV and 1–2% at HV/EHV. It is above all a machine-duty issue, because \(U_2 \approx Z_2 I_2\) and a small negative-sequence voltage drives a large negative-sequence current. A robust assessment must specify the fundamental-only measurement + calculation method + time interval + statistical index + assessment location + objective. Only then can voltage unbalance be compared fairly and technically.
