Symmetrical components: split unbalance into three balanced sets

Drag I_a and read the three components

Of the three currents on the left, I_b and I_c are fixed in their balanced positions, and only the tip of I_a can be dragged. The three bars on the right are, from the top, the magnitudes of the positive sequence I_1, the negative sequence I_2 and the zero sequence I_0. When the tip of I_a sits where the unit circle meets the horizontal axis on the right, the set is balanced: only the positive sequence remains and the negative and zero sequences are 0. Since only I_a moves in this figure, the moment it leaves balance the two lower bars always grow together at the same length. The zero sequence is 1/3 of the sum of the three currents you saw in A5.

A sum of three balanced sets

By the symmetrical-component theorem, any three phasors, whatever their shape, can be written in exactly one way as the sum of three balanced sets. The positive sequence is a balanced three-phase set with the same phase order as the system (a-b-c), the negative sequence is a balanced set with the order reversed (a-c-b), and the zero sequence is a set of three phasors equal in both magnitude and phase. So I_a = I_1 + I_2 + I_0, and I_b and I_c are the same three components rotated by the operator a (1∠120°) and added.

Why splitting makes it easier

In a balanced system the positive-, negative- and zero-sequence circuits do not interact; they separate cleanly. So every generator and line is given its own positive-, negative- and zero-sequence impedance, and an unbalanced fault is solved by connecting these three circuits in the way that matches the fault type. One hard unbalanced problem becomes three familiar balanced ones. In normal operation only the positive sequence flows; negative-sequence current overheats a generator's rotor, and zero-sequence current is the signal that lets protective relays detect a ground fault.

Zero sequence · the neutral current of A5

The zero sequence I_0 = (I_a + I_b + I_c)/3 is exactly 1/3 of the neutral current from A5. The three zero-sequence currents are equal in magnitude and phase, so instead of cancelling they add up and flow into the neutral or the earth. That requires a return path such as a grounded neutral, so zero-sequence current cannot flow in the lines of an ungrounded Y connection or a Δ connection (inside a Δ it can circulate around the loop). This zero sequence, 0 while the system is balanced, shoots up in a ground fault where one phase touches earth and becomes the clue that protection relies on.

Back to the first screen

On the first screen, placing I_a at the balanced spot shrank the negative- and zero-sequence bars to 0 and left only the positive sequence, and dragging it away made the two bars grow together at the same length. Unbalance comes down to how much negative and zero sequence sits on top of the positive sequence. The zero sequence among them is the neutral current of A5 shared equally among the three phases, and once the unbalanced phases are split into three balanced sets like this, the next unit can solve every kind of fault as a connection of familiar balanced circuits.

With the three sequence circuits in hand, the next unit (PW-D3) takes up fault types and fault current. A three-phase short circuit is solved with the positive-sequence circuit alone, a fault in which one line touches the ground connects the positive, negative and zero circuits in series, and a line-to-line short connects the positive and negative circuits in parallel. Only the way the three circuits are joined changes from fault to fault, so the same tool works both for the three-phase short, which usually draws the largest current, and for the most common ground fault, which shows itself through the zero sequence.