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
I_{e} = earth fault current
E_{a} = normal phase voltage at the fault location
Z_{1} = positive phase sequence network impedance to the fault
Z_{2} = negative phase sequence network impedance to the fault
Z_{0} = zero phase sequence network impedance to the fault
Single phase to earth – fault from phase ‘a’ to earth:
V_{a} = 0I_{b} = I_{c} = 0 I_{ f} = I_{a} = I_{e} | I_{a1} = I_{a2} = I_{a0} = I_{a} / 3V_{a1} + V_{a2} + V_{a0} = 0 _{ } | I_{a1} = E_{a} / (Z_{1} + Z_{2} + Z_{0})I_{a2} = I_{a1} I_{a0} = I_{a1} | I_{ f} = 3I_{a0} = 3E_{a} / (Z_{1} + Z_{2} + Z_{0}) = I_{e}I_{a} = I_{ f} = 3E_{a} / (Z_{1} + Z_{2} + Z_{0}) _{ } |
Double phase – fault from phase ‘b’ to phase ‘c’:
V_{b} = V_{c}I_{a} = 0 I_{ f} = I_{b} = – I_{c} | I_{a1} + I_{a2} = 0I_{a0} = 0 V_{a1} = V_{a2} | I_{a1} = E_{a} / (Z_{1} + Z_{2})I_{a2} = – I_{a1} I_{a0} = 0 | I_{ f} = – jÖ3I_{a1} = – jÖ3E_{a} / (Z_{1} + Z_{2})I_{b} = I_{ f} = – jÖ3E_{a} / (Z_{1} + Z_{2}) I_{c} = – I_{ f} = jÖ3E_{a} / (Z_{1} + Z_{2}) |
Double phase to earth – fault from phase ‘b’ to phase ‘c’ to earth:
V_{b} = V_{c} = 0I_{a} = 0 I_{ f} = I_{b} + I_{c} = I_{e} | I_{a1} + I_{a2} + I_{a0} = 0V_{a1} = V_{a2} = V_{a0} _{ } | I_{a1} = E_{a} / Z_{net}I_{a2} = – I_{a1}Z_{0} / (Z_{2} + Z_{0}) I_{a0} = – I_{a1}Z_{2} / (Z_{2} + Z_{0}) | I_{ f} = 3I_{a0} = – 3E_{a}Z_{2} / S_{zz} = I_{e}I_{b} = I_{ f} / 2 – jÖ3E_{a}(Z_{2} / 2 + Z_{0}) / S_{zz} I_{c} = I_{ f} / 2 + jÖ3E_{a}(Z_{2} / 2 + Z_{0}) / S_{zz} |
Z_{net} = Z_{1} + Z_{2}Z_{0} / (Z_{2} + Z_{0}) and S_{zz} = Z_{1}Z_{2} + Z_{2}Z_{0} + Z_{0}Z_{1} = (Z_{2} + Z_{0})Z_{net}
Three phase (and three phase to earth) – fault from phase ‘a’ to phase ‘b’ to phase ‘c’ (to earth):
V_{a} = V_{b} = V_{c} (= 0)I_{a} + I_{b} + I_{c} = 0 (= I_{e}) I_{ f} = I_{a} = hI_{b} = h^{2}I_{c} | V_{a0} = V_{a} (= 0)V_{a1} = V_{a2} = 0 _{ } | I_{a1} = E_{a} / Z_{1}I_{a2} = 0 I_{a0} = 0 | I_{ f} = I_{a1} = E_{a} / Z_{1} = I_{a}I_{b} = E_{b} / Z_{1} I_{c} = E_{c} / Z_{1} |
The values of Z_{1}, Z_{2} and Z_{0} 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 Z_{0} is less than (2Z_{1} – Z_{2}).
Note also that if the system is earthed through an impedance Z_{n} (carrying current 3I_{0}) then an impedance 3Z_{n} (carrying current I_{0}) 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] |
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??