Symmetrical components calculator

Phase phasors to zero, positive and negative sequence components and back, drawn as phasors.

Result

V0, zero sequence: 5.349∠−22.85°
V1, positive sequence: 89.68∠6.299°
V2, negative sequence: 9.77∠−52.62°

0a=0b=0c: 5.35∠−22.9°

1a: 89.68∠6.3°
1b: 89.68∠−113.7°
1c: 89.68∠126.3°

2a: 9.77∠−52.6°
2b: 9.77∠67.4°
2c: 9.77∠−172.6°

Zero sequence is one phasor shared by all three phases. Positive sequence turns counterclockwise, negative sequence clockwise. Each diagram is scaled to its own largest phasor.

Va: 100.00∠0.0°
Vb: 80.00∠−110.0°
Vc: 90.00∠130.0°

What it calculates

Any unbalanced set of three phasors can be split into three balanced sets: a zero-sequence set, a positive-sequence set and a negative-sequence set. The split uses the operator a, a rotation of 120°:

a = 1\angle 120^\circ = -\tfrac{1}{2} + j\tfrac{\sqrt{3}}{2}

The zero, positive and negative components of phases a, b and c are:

\begin{bmatrix} V_0 \\ V_1 \\ V_2 \end{bmatrix} = \dfrac{1}{3}\begin{bmatrix} 1 & 1 & 1 \\ 1 & a & a^{2} \\ 1 & a^{2} & a \end{bmatrix}\begin{bmatrix} V_a \\ V_b \\ V_c \end{bmatrix}

And the phases are rebuilt from the components with the inverse:

\begin{bmatrix} V_a \\ V_b \\ V_c \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & a^{2} & a \\ 1 & a & a^{2} \end{bmatrix}\begin{bmatrix} V_0 \\ V_1 \\ V_2 \end{bmatrix}

Reading the three sets

Zero sequence is three equal phasors in phase with each other. Nothing rotates between phases, and the current needs a return path through ground or a neutral.
Positive sequence is three equal phasors 120° apart, in the order a, b, c. It is the healthy, forward-turning system, and it turns counterclockwise in the diagram.
Negative sequence is three equal phasors 120° apart in the order a, c, b. It turns the other way, clockwise, and it is what an unbalanced load adds to a rotating machine.

The phase phasors are the sum of the three sets, taken phase by phase. The diagrams animate the positive and negative sets to show the direction each turns; the phases themselves are drawn still.

Worked example

Start with the simplest unbalanced case: phase a carries 1∠0°, and phases b and c carry nothing. Each sequence component is one third of it:

V_a = 1\angle 0^\circ,\ V_b = V_c = 0

With 100∠0°, 80∠−110° and 90∠130° on phases a, b and c, the calculator opens on this case. The zero-sequence component is 5.349∠−22.85°, the positive is 89.68∠6.299° and the negative is 9.770∠−52.62°. A perfectly balanced set gives exactly zero for the zero and negative components.

Assumptions

The phase sequence is a-b-c. A system that rotates a-c-b swaps the positive and negative sequence.
The operator a is 1∠120°, and phase a is the reference for all three sets.
The transform is linear and works for any three phasors, voltages or currents, in any unit.

Questions

What does each sequence set mean physically?

Positive sequence rotates A-B-C like a healthy system. Negative sequence rotates A-C-B and heats rotating machines. Zero sequence is three equal, in-phase phasors that can only flow where there is a return path through ground or a neutral.

Why is the zero-sequence component zero in a three-wire system?

Zero-sequence current is one third of the sum of the three line currents. Without a neutral or ground path, the three line currents must sum to zero, so no zero-sequence current can flow.

Does it matter whether I enter voltages or currents?

No. The transform is the same for any set of three phasors. The units you enter are the units you get back.

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For learning, not for design

PhasorPrep is an independent study resource. It is not affiliated with, endorsed by, or sponsored by NCEES. Questions are original and written for practice; they are not actual exam questions. Calculators are for learning, not for engineering design.

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