Decompose an unbalanced three-phase voltage set into three balanced symmetrical components using Fortescue's method. Slide V_a, V_b, V_c magnitudes and V_b phase to see how each sequence and the NEMA voltage unbalance factor change in real time.
NEMA voltage unbalance factor: $$\mathrm{VUF}=\frac{|V_2|}{|V_1|}\times 100\,\%$$
$a=e^{j120^\circ}$ is the 120° rotation operator and $a^2=e^{j240^\circ}$. Under perfect balance $V_0=V_2=0$ and only $V_1$ remains.
What is the Symmetrical Components Simulator?
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I keep seeing "symmetrical components" in power-system textbooks. Why bother transforming the three voltages? Can't we just analyse V_a, V_b, V_c directly?
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Great question! Roughly speaking, the trick is to express an unbalanced three-phase set as the sum of three balanced sub-sets — that makes fault analysis (single-line-to-ground, line-to-line, etc.) tractable. Fortescue introduced it in 1918. In this simulator, drop V_c from 200 V to 180 V and you'll see the original set is unbalanced, but the three small phasor plots show it as a sum of a positive, a negative and a zero sequence.
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Positive vs negative sequence — both look like three arrows 120° apart. What's the difference?
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The rotation direction is reversed. Positive sequence rotates a→b→c (the normal supply direction). Negative sequence rotates a→c→b. The latter is deadly for induction motors: a reverse-rotating field appears in the airgap, causing braking torque and excess heating. That's why NEMA caps VUF at 2%. Losses roughly double for each 1% of VUF.
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And zero sequence has all three arrows pointing the same way. When is that relevant?
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Zero sequence is the in-phase part that arises when Va + Vb + Vc ≠ 0. In a three-wire system there is no return path so zero-sequence current stays near zero even under unbalance. But in a grounded-wye system, a single-line-to-ground fault drives a large zero-sequence current through the ground path — and that's exactly what ground-fault relays detect. Try dropping V_b in the slider: V_0 grows and the VUF readout climbs.
Physical Model and Key Equations
Fortescue's transform expresses any unbalanced three-phase set (V_a, V_b, V_c) as the sum of zero $V_0$, positive $V_1$ and negative $V_2$ sequence components, using the 120° rotation operator $a = e^{j120^\circ} = -0.5 + j0.866$ and $a^2 = e^{j240^\circ}$:
The inverse transform $V_a = V_0 + V_1 + V_2$ etc. rebuilds the original unbalanced phase voltages exactly. The NEMA voltage unbalance factor is $\mathrm{VUF}=|V_2|/|V_1|\times 100\,\%$.
Real-world Applications
Induction motor protection: Negative-sequence voltage $V_2$ creates braking torque and excess current. NEMA MG-1 requires derating once VUF exceeds 1%, and operation is discouraged above 5%. This tool gives a live VUF readout to gauge severity.
Power-system fault analysis: Unbalanced faults (single-line-to-ground, line-line, double-line-to-ground) are routinely solved by combining three independent sequence networks. This visualisation shows the voltage-side decomposition that underlies that workflow.
Protective relaying: Ground-fault overcurrent (51N), negative-sequence overcurrent (46), and voltage-based residual and negative-sequence elements all rely on quantities computed exactly like the ones shown here.
Power electronics & rectifiers: Voltage-source converters, STATCOMs and rectifiers experience unbalanced input as negative-sequence components, which cause unwanted 2nd-harmonic ripple on the DC link unless controlled actively.
Common Pitfalls
First, symmetrical components are not measurable directly. They are a mathematical decomposition computed from V_a, V_b, V_c. A voltmeter on the bus reads only the physical phase voltages; what you see in this simulator's right-hand plots is the result of applying Fortescue's formula in software.
Second, beware the NEMA-VUF vs IEEE LVUR / PVUR distinction. This tool uses the rigorous |V_2|/|V_1| definition. The simpler max-deviation-over-average versions (IEEE LVUR, NEMA PVUR) agree only when unbalance is small; at larger unbalance the two metrics can differ by a factor of two or more.
Third, understand where zero-sequence current actually flows. A delta winding traps zero-sequence current internally and prevents it from appearing on the line side. So even if V_0 is non-zero on a bus, the zero-sequence current path depends on transformer connection and neutral grounding — a critical design constraint for protection schemes.
Frequently Asked Questions
Yes. With V_a=200 V∠0°, V_b=200 V∠−120°, V_c=180 V∠+120°, the analytical values are V_0=V_2≈6.67 V and V_1≈193.3 V, giving VUF = 6.67/193.3 × 100 ≈ 3.45%. NEMA MG-1 recommends staying below 2% for continuous motor operation, so 3.45% requires derating.
When the three phasors have equal magnitude and are 120° apart, both V_0 and V_2 vanish and V_1 equals the common magnitude. Setting V_a=V_b=V_c=200 V with V_b phase = −120° gives V_1=200 V and VUF=0%.
Yes. The Fortescue transform is identical for currents: $I_0=(I_a+I_b+I_c)/3$, and $I_1$, $I_2$ follow the same pattern as $V_1$, $V_2$. In a three-wire system $I_a+I_b+I_c=0$ so $I_0=0$. This tool focuses on voltage visualisation but the math is the same.
A reduction in one phase magnitude or a phase shift away from 120° produces an a-c-b reverse-rotating component. Slide the V_b phase angle from −120° toward −110° and watch V_2 grow — that is the unbalance an induction motor will see as a brake.