Symmetrical Components

Formulas

In any three phase system, the line currents I_a, I_b and I_c may be expressed as the phasor sum of:

– a set of balanced positive phase sequence currents I_a1, I_b1 and I_c1 (phase sequence a-b-c),

– a set of balanced negative phase sequence currents I_a2, I_b2 and I_c2 (phase sequence a-c-b),

– a set of identical zero phase sequence currents I_a0, I_b0 and I_c0 (cophasal, no phase sequence).

The positive, negative and zero sequence currents are calculated from the line currents using:

I_a1 = (I_a + hI_b + h²I_c) / 3

I_a2 = (I_a + h²I_b + hI_c) / 3

I_a0 = (I_a + I_b + I_c) / 3

The positive, negative and zero sequence currents are combined to give the line currents using:

I_a = I_a1 + I_a2 + I_a0

I_b = I_b1 + I_b2 + I_b0 = h²I_a1 + hI_a2 + I_a0

I_c = I_c1 + I_c2 + I_c0 = hI_a1 + h²I_a2 + I_a0

The residual current I_r is equal to the total zero sequence current:

I_r = I_a0 + I_b0 + I_c0 = 3I_a0 = I_a + I_b + I_c = I_e

which is measured using three current transformers with parallel connected secondaries.

I_e is the earth fault current of the system.

Similarly, for phase-to-earth voltages V_ae, V_be and V_ce, the residual voltage V_r is equal to the total zero sequence voltage:

V_r = V_a0 + V_b0 + V_c0 = 3V_a0 = V_ae + V_be + V_ce = 3V_ne

which is measured using an earthed-star / open-delta connected voltage transformer.

V_ne is the neutral displacement voltage of the system.

The h-operator

The h-operator (1Ð120°) is the complex cube root of unity:

h = – 1 / 2 + jÖ3 / 2 = 1Ð120° = 1Ð-240°

h² = – 1 / 2 – jÖ3 / 2 = 1Ð240° = 1Ð-120°

Some useful properties of h are:

1 + h + h² = 0

h + h² = – 1 = 1Ð180°

h – h² = jÖ3 = Ö3Ð90°

h² – h = – jÖ3 = Ö3Ð-90°