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Formulas

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

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°

NOTATION
The symbol font is used for some notation and formulae. If the Greek symbols for alpha beta delta do not appear here [ a b d ] the symbol font needs to be installed for correct display of notation and formulae.
B
C
E
f
G
h
I
j
L
P
Q
susceptance
capacitance
voltage source
frequency
conductance
h-operator
current
j-operator
inductance
active power
reactive power
[siemens, S]
[farads, F]
[volts, V]
[hertz, Hz]
[siemens, S]
[1Ð120°]
[amps, A]
[1Ð90°]
[henrys, H]
[watts, W]
[VAreactive, VArs]
Q
R
S
t
V
W
X
Y
Z
f
w
quality factor
resistance
apparent power
time
voltage drop
energy
reactance
admittance
impedance
phase angle
angular frequency
[number]
[ohms, W]
[volt-amps, VA]
[seconds, s]
[volts, V]
[joules, J]
[ohms, W]
[siemens, S]
[ohms, W]
[degrees, °]
[rad/sec]

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