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Formulas

The different types of short-circuit fault which occur on a power system are:

  • single phase to earth,
  • double phase,
  • double phase to earth,
  • three phase,
  • three phase to earth.

For each type of short-circuit fault occurring on an unloaded system:

  • the first column states the phase voltage and line current conditions at the fault,
  • the second column states the phase ‘a’ sequence current and voltage conditions at the fault,
  • the third column provides formulae for the phase ‘a’ sequence currents at the fault,
  • the fourth column provides formulae for the fault current and the resulting line currents.

By convention, the faulted phases are selected for fault symmetry with respect to reference phase ‘a’.

I f = fault current

Ie = earth fault current

Ea = normal phase voltage at the fault location

Z1 = positive phase sequence network impedance to the fault

Z2 = negative phase sequence network impedance to the fault

Z0 = zero phase sequence network impedance to the fault

Single phase to earth – fault from phase ‘a’ to earth:

Va = 0Ib = Ic = 0

I f = Ia = Ie

Ia1 = Ia2 = Ia0 = Ia / 3Va1 + Va2 + Va0 = 0

Ia1 = Ea / (Z1 + Z2 + Z0)Ia2 = Ia1

Ia0 = Ia1

I f = 3Ia0 = 3Ea / (Z1 + Z2 + Z0) = IeIa = I f = 3Ea / (Z1 + Z2 + Z0)

Double phase – fault from phase ‘b’ to phase ‘c’:

Vb = VcIa = 0

I f = Ib = – Ic

Ia1 + Ia2 = 0Ia0 = 0

Va1 = Va2

Ia1 = Ea / (Z1 + Z2)Ia2 = – Ia1

Ia0 = 0

I f = – jÖ3Ia1 = – jÖ3Ea / (Z1 + Z2)Ib = I f = – jÖ3Ea / (Z1 + Z2)

Ic = – I f = jÖ3Ea / (Z1 + Z2)

Double phase to earth – fault from phase ‘b’ to phase ‘c’ to earth:

Vb = Vc = 0Ia = 0

I f = Ib + Ic = Ie

Ia1 + Ia2 + Ia0 = 0Va1 = Va2 = Va0

Ia1 = Ea / ZnetIa2 = – Ia1Z0 / (Z2 + Z0)

Ia0 = – Ia1Z2 / (Z2 + Z0)

I f = 3Ia0 = – 3EaZ2 / Szz = IeIb = I f / 2 – jÖ3Ea(Z2 / 2 + Z0) / Szz

Ic = I f / 2 + jÖ3Ea(Z2 / 2 + Z0) / Szz

Znet = Z1 + Z2Z0 / (Z2 + Z0) and Szz = Z1Z2 + Z2Z0 + Z0Z1 = (Z2 + Z0)Znet

Three phase (and three phase to earth) – fault from phase ‘a’ to phase ‘b’ to phase ‘c’ (to earth):

Va = Vb = Vc (= 0)Ia + Ib + Ic = 0 (= Ie)

I f = Ia = hIb = h2Ic

Va0 = Va (= 0)Va1 = Va2 = 0

Ia1 = Ea / Z1Ia2 = 0

Ia0 = 0

I f = Ia1 = Ea / Z1 = IaIb = Eb / Z1

Ic = Ec / Z1

The values of Z1, Z2 and Z0 are each determined from the respective positive, negative and zero sequence impedance networks by network reduction to a single impedance.

Note that the single phase fault current is greater than the three phase fault current if Z0 is less than (2Z1 – Z2).

Note also that if the system is earthed through an impedance Zn (carrying current 3I0) then an impedance 3Zn (carrying current I0) must be included in the zero sequence impedance network.

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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Page edited by E.C. (Google).

One Comment


  1. Vishnu
    Mar 10, 2015

    In a single phase to ground fault calculation, we find the source impedance (using the formula kV*kV/MVA) and use it directly as zero sequence impedance. Why so? Actually this impedance will be offered to +ve as well as to -ve sequence components. Then why do we consider the resultant impedance as only zero sequence impedance??

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