### Formulas

In any three phase system, the line currents **I _{a}**,

**I**and

_{b}**I**may be expressed as the phasor sum of:

_{c}– a set of balanced positive phase sequence currents **I _{a1}**,

**I**and

_{b1}**I**(phase sequence a-b-c),

_{c1}– a set of balanced negative phase sequence currents **I _{a2}**,

**I**and

_{b2}**I**(phase sequence a-c-b),

_{c2}– a set of identical zero phase sequence currents **I _{a0}**,

**I**and

_{b0}**I**(cophasal, no phase sequence).

_{c0}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**and

_{be}**V**, the residual voltage

_{ce}**V**is equal to the total zero sequence voltage:

_{r}**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°**

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. | ||||||

BCEfGhIjLPQ | 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] | QRStVWXYZfw | 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] |