RELAYS 101

In-Depth Learn to Calculate Zero Sequence

Zero Sequence is one of three symmetrical components used in analysis of 3-phase electrical systems. Symmetrical Components were invented by a GE electrical engineer (named Charles Fortiscue) circa 1921. He originally put together the mathematical concept so he could try to understand the unbalanced behavior of the early GE 3-phase motors. Of the 3 symmetrical components, (Positive Sequence, Negative Sequence and Zero Sequence), Zero Sequence is the easiest to calculate. After considerable experience and analysis, it was found that the Zero Sequence component will only show up in the 3-phase system when there is a fault to ground. All three sequence components appear in a phase-to-ground fault, but of the various fault types, Zero Sequence will only appear in the ground fault. This is very useful in determining fault-types; if zero sequence appears then it is a ground fault thus switch over to the phase-ground measuring units or to the phase-ground calculations. Zero Sequence is calculated by summing the three phases and dividing by 3. Just like if you were finding the "average" of the three phases. Remember when adding vectors, first find the horizontal (cosine) components and the vertical (sine) components of the three phases. Then add together all of the horizontal components, then add together all of the vertical components, finally find the resultant vector from the two compoent sums. 
Mathematically it looks like this: ZerSeq = (A + B + C)/3; 
for Voltage: V0 = (VA + VB + VC)/3;
for Current: I0 = (IA + IB +IC)/3;
for Impedance: Z0 = (ZA + ZB + ZC)/3;
Any phase value is equal to the sum of its components.
Mathematically: 
VA = VA0 + VA1 + VA2; VB = VB0 + VB1 + VB2; VC = VC0 + VC1 + VC2
IA = IA0 + IA1 + IA2; IB = IB0 + IB1 + IB2; IC = IC0 + IC1 + IC2
ZA = ZA0 + ZA1 + ZA2; ZB = ZB0 + ZB1 + ZB2; ZC = ZC0 + ZC1 + ZC2

Step 1: Enter the three phases of data (note that the vectors/phasors can be voltage, current or impedance).

Phase 1 Magnitude : Phase 1 Angle (deg) :

Phase 2 Magnitude : Phase 2 Angle (deg) :

Phase 3 Magnitude : Phase 3 Angle (deg) :

Step 2: Find the horizontal and vertical components of each of the three vectors.

Phase 1 Horizontal Component is found by multiplying phase 1 magnitude by the cosine of phase 1 angle.

Phase 1 Vertical Component is found by multiplying phase 1 magnitude by the sine of phase 1 angle.

Phase 1 Horiz: Phase 1 Vert:

Phase 2 Horizontal Component is found by multiplying phase 2 magnitude by the cosine of phase 2 angle.

Phase 2 Vertical Component is found by multiplying phase 2 magnitude by the sine of phase 2 angle.

Phase 2 Horiz: Phase 2 Vert:

Phase 3 Horizontal Component is found by multiplying phase 3 magnitude by the cosine of phase 3 angle.

Phase 3 Vertical Component is found by multiplying phase 3 magnitude by the sine of phase 3 angle.

Phase 3 Horiz: Phase 3 Vert:

Step 3: Sum the three horizontal components and sum the three vertical components.

Horiz Comp Sum : Vert Comp Sum :

Step 4: Using Pythagorean Theorem on the horizontal and vertical component sums, find resultant vector magnitude.

Horiz Comp Sum (squared) : + Vert Comp Sum (squared) : =

Find square root of the sum of the two squares : = Resultant Vector

Step 5: Using Trigonometry on the final calculated triangle (arctan = vertical / horizontal), find resultant vector angle.

Vertical : / Horizontal : = Tangent of Zero Sequence Angle

Step 6: Divide Resultant Vector Magnitude By 3 and find the angle associated with the calculated tangent value.

Zero Sequence Magnitude : Zero Sequence Angle (deg) :

The following is a snapshot in time of the calculated symmetrical components during an A-G Fault. In this example, Phase-A is colored black, Phase-B is colored green, Phase-C is colored red and zero sequence is standing out by being colored blue. As zero sequence only appears during a ground fault then this picture is showing Ground (phase) as blue.
One can visualize that "Positive" Sequence has the original phase sequence. (One can visualize a counter-clockwise rotation of the positive sequence set of vectors as having a "phase-sequence" of black-green-red (A-B-C). Looking at the set of Negative Sequence phasors one can visualize a "reverse phase sequence". (That is if you visualize the negative sequence set with a counter-clockwise rotation then the sequence is black-red-green (A-C-B).
Note that since all three symmetrical components are present then this must be a ground fault.
Note that one can visualize that the "NEGATIVE SEQUENCE VOLTAGE LAGS THE NEGATIVE SEQUENCE CURRENT". This occurs during a fault that is in the forward direction.
So here are two excellent examples of how symmetrical components assist in the analysis of 3-phase system faults.
A protective relay can distinguish a ground fault by the existence of Zero Sequence and the relay can determine direction by comparing the phase relationship of the negative sequence voltage to the negative sequence current.

Symmetrical Components during a sampled A-G Fault