RELAYS 101

In-Depth Learn to Calculate Negative Sequence

Negative 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. There are 3 symmetrical components, (Positive Sequence, Negative Sequence and Zero Sequence). After considerable experience and analysis, it was found that the Negative Sequence component appears in phase-to-ground and phase-to-phase faults. Knowing this is useful because if analysis shows that there are Negative sequence components but no Zero Sequence then the fault must be a phase-to-phase fault. The relay can then move to the phase-to-phase measuring units or phase-to-phase calculations. Negative Sequence is calculated by applying a numerical angular shift to the second and third phases then summing the three phases and dividing by 3. 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: NegSeq = (A + a2B + aC)/3; 
for Voltage: V2 = (VA + a2VB + aVC)/3;
for Current: I2 = (IA + a2IB +aIC)/3;
for Impedance: Z2 = (ZA + a2ZB + aZC)/3;
Where "a" is the "A-Operator" and represents a shift of 120 degrees.
Where "a2" is the "A squared-Operator" and represents a shift of 240 degrees.
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: Shift the Phase 2 Angle by the "A squared-Operator" which is done by adding 240 degrees to Phase 2 Angle

and shift the phase 3 Angle by the "A-Operator" which is done by adding 120 degrees to Phase 3 Angle.

Angle 2 shifted : Angle 3 shifted :

Step 3: 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 (shifted) phase 2 angle.

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

Phase 2 (shifted) Horiz: Phase 2 (shifted) Vert:

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

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

Phase 3 (shifted) Horiz: Phase 3 (shifted) Vert:

Step 4: Sum the three horizontal components and sum the three vertical components.

Horiz Comp Sum : Vert Comp Sum :

Step 5: 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 6: Using Trigonometry on the final calculated triangle (arctan = vertical / horizontal), find resultant vector angle.

Vertical : / Horizontal : = Tangent of Negative Sequence Angle

Step 7: Divide Resultant Vector Magnitude By 3 and find the ArcTangent of calculated Tangent value.

Negative Sequence Magnitude : Negative Sequence Angle (deg) :

The following is a snapshot in time of the calculated symmetrical components during a B-C Fault. In this example, Phase-A is colored black, Phase-B is colored green, Phase-C is colored red.
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 there is Positive and Negative but no Zero Sequence symmetrical components then this must be a phase-phase 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 phase-phase fault by the existence of Negative 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 B-C Fault