Asymmetrical Faults

This discussion has touched upon three topics that are each independent from the others.

1. Asymmetry due to only one or two phases being faulted
2. Asymmetry due to a DC offset of the sinusoidal fault
3. The decay of fault current with time due to differences in the reactance of generators with time (not really an asymmetry and not addressed by my comments)

The negative-sequence vectors do not rotate in the opposite direction of the positive-sequence vectors. All three (positive, negative, and zero) rotate in the same direction. The difference between the positive and negative sequences is the "phase rotation".

In the United States, the common convention is to name the three phases A, B, and C. If B-phase reaches peak after A-phase and before C-phase, the phase rotation is ABC. On the other hand if C-phase reaches peak after A-phase and before B-phase, the phase rotation is ACB. Using the convention of clockwise rotation, ABC rotation, and with nearly balanced conditions, A-phase is arbitrarily placed at zero degrees; B-phase is at nearly 240 degrees, and C-phase is nearly at 120 degrees.

The A-phase positive sequence is found by first rotating B-phase by 120 degrees to nearly align it with A-phase then rotating C-phase by 240 degrees to nearly align it with A-phase and finally taking the average of the three vectors (that is summing them vectorially and dividing by three. This results with A-phase positive sequence near zero degrees (assuming the system is nearly balanced). B and C-phase positive sequence vectors are equal to A-phase positive sequence, but rotated by 240 and 120 degrees, respectively. This results in the positive sequence being balanced (all three phase magnitudes being equal and 120 degrees apart) with ABC rotation.

The A-phase negative sequence is found by first rotating B-phase by 240 degrees to nearly align it with the actual C-phase then rotating C-phase by 120 degrees to nearly align it with the actual B-phase and finally taking the average of the three vectors (that is summing them vectorially and dividing by three). B and C-phase positive sequence vectors are equal to A-phase positive sequence, but rotated by 120 and 240 degrees, respectively. . This results in the negative sequence being balanced (all three phase magnitudes being equal and 120 degrees apart) with ACB rotation.

ACB rotation may be called "negative rotation" because a three-phase motor, connected to a system with ACB rotation, will rotate in the opposite direction than a three-phase motor connected to a system with ACB rotation, but in both cases, all vectors rotate in the same direction.

Finally, the zero sequence is simply the average of the three actual phases (again, summing them vectorially and dividing by three).

The zero sequence impedance being about 2 to 3.5 times the positive and negative impedance, but this is only for overhead four-wire systems (three phases and a neutral conductor).

It is common place to have the neutral conductor farther away than the phase conductors are from each other. This results in a higher zero sequence impedance, since the zero sequence is forced to return in the neutral conductor, which is farther away.

However, with underground cable systems with a full neutral, the zero sequence impedance is equal to the positive and negative impedance.

Finally, with a three wire system (no neutral conductor), there is no return path for zero sequence current and the zero sequence impedance is infinite.

The asymmetry due to a DC offset is a result of the power system being primarily inductive and the current not being able to change instantaneously.

With near unity power factor (typical of load current), the current at a voltage zero is nearly zero. Should a fault occur near a voltage zero, there is nearly a full half cycle of voltage of the same polarity. This drives the current in the same direction for nearly a full half cycle resulting in a peak current of nearly twice the sinusoidal peak and peaking at the next voltage zero (180 degrees after the first voltage zero, not 90 degrees as is the case with near unity power factor load). This is a fully offset sinusoidal current.

If the X/R (inductive impedance divided by the resistance) is infinite, the DC offset lasts forever. However, any resistance causes the DC offset to decay. The lower the X/R the faster the decay.

On the other hand, if the fault occurs at a voltage angle equal to the arctangent of the X/R (nearly 90 degrees for high X/Rs), the current is zero near the peak of the voltage and there is no DC offset in the fault current.