## Symmetrical comp.

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} + h^{2}I_{c})
/ 3

I_{a2} = (I_{a} + h^{2}I_{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} = h^{2}I_{a1}
+ hI_{a2} + I_{a0}
I_{c} = I_{c1} + I_{c2} + I_{c0} = hI_{a1}
+ h^{2}I_{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°
h**

^{2}= - 1 / 2 - jÖ3 / 2 = 1Ð240° = 1Ð-120°

Some useful properties of h are:

1 + h + h^{2} = 0

h + h^{2} = - 1 = 1Ð180°

h - h^{2} = jÖ3
= Ö3Ð90°

h^{2} - h = - jÖ3
= Ö3Ð-90°