In any three phase system, the line currents Ia, Ib and Ic may be expressed as the phasor sum of:
– a set of balanced positive phase sequence currents Ia1, Ib1 and Ic1 (phase sequence a-b-c),
– a set of balanced negative phase sequence currents Ia2, Ib2 and Ic2 (phase sequence a-c-b),
– a set of identical zero phase sequence currents Ia0, Ib0 and Ic0 (cophasal, no phase sequence).
The positive, negative and zero sequence currents are calculated from the line currents using:
Ia1 = (Ia + hIb + h2Ic) / 3
Ia2 = (Ia + h2Ib + hIc) / 3
Ia0 = (Ia + Ib + Ic) / 3
The positive, negative and zero sequence currents are combined to give the line currents using:
Ia = Ia1 + Ia2 + Ia0
Ib = Ib1 + Ib2 + Ib0 = h2Ia1 + hIa2 + Ia0
Ic = Ic1 + Ic2 + Ic0 = hIa1 + h2Ia2 + Ia0
The residual current Ir is equal to the total zero sequence current:
Ir = Ia0 + Ib0 + Ic0 = 3Ia0 = Ia + Ib + Ic = Ie
which is measured using three current transformers with parallel connected secondaries.
Ie is the earth fault current of the system.
Similarly, for phase-to-earth voltages Vae, Vbe and Vce, the residual voltage Vr is equal to the total zero sequence voltage:
Vr = Va0 + Vb0 + Vc0 = 3Va0 = Vae + Vbe + Vce = 3Vne
which is measured using an earthed-star / open-delta connected voltage transformer.
Vne is the neutral displacement voltage of the system.
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°
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