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# Per-unit System

### Formulas

For each system parameter, per-unit value is equal to the actual value divided by a base value:

Epu = E / Ebase

Ipu = I / Ibase

Zpu = Z / Zbase

Select rated values as base values, usually rated power in MVA and rated phase voltage in kV:

Sbase = Srated = Ö3ElineIline

Ebase = Ephase = Eline/ Ö3

The base values for line current in kA and per-phase star impedance in ohms/phase are:

Ibase = Sbase / 3Ebase ( = Sbase / Ö3Eline)

Zbase = Ebase / Ibase = 3Ebase2 / Sbase ( = Eline2 / Sbase)

Note that selecting the base values for any two of Sbase, Ebase, Ibase or Zbase fixes the base values of all four. Note also that Ohm’s Law is satisfied by each of the sets of actual, base and per-unit values for voltage, current and impedance.

#### Transformers

The primary and secondary MVA ratings of a transformer are equal, but the voltages and currents in the primary (subscript 1) and the secondary (subscript 2) are usually different:

Ö3E1lineI1line = S = Ö3E2lineI2line

Converting to base (per-phase star) values:

3E1baseI1base = Sbase = 3E2baseI2base

E1base / E2base = I2base / I1base

Z1base / Z2base = (E1base / E2base)2

The impedance Z21pu referred to the primary side, equivalent to an impedance Z2pu on the secondary side, is:

Z21pu = Z2pu(E1base / E2base)2

The impedance Z12pu referred to the secondary side, equivalent to an impedance Z1pu on the primary side, is:

Z12pu = Z1pu(E2base / E1base)2

Note that per-unit and percentage values are related by:

Zpu = Z% / 100

 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]