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# Transformers

### Formulas

For an ideal two-winding transformer with primary voltage V1 applied across N1 primary turns and secondary voltage V2 appearing across N2 secondary turns:
V1 / V2 = N1 / N2
The primary current I1 and secondary current I2 are related by:
I1 / I2 = N2 / N1 = V2 / V1

For an ideal step-down auto-transformer with primary voltage V1 applied across (N1 + N2) primary turns and secondary voltage V2 appearing across N2 secondary turns:
V1 / V2 = (N1 + N2) / N2
The primary (input) current I1 and secondary (output) current I2 are related by:
I1 / I2 = N2 / (N1 + N2) = V2 / V1
Note that the winding current is I1 through the N1 section and (I2 – I1) through the N2 section.

For a single-phase transformer with rated primary voltage V1, rated primary current I1, rated secondary voltage V2 and rated secondary current I2, the voltampere rating S is:
S = V1I1 = V2I2

For a balanced m-phase transformer with rated primary phase voltage V1, rated primary current I1, rated secondary phase voltage V2 and rated secondary current I2, the voltampere rating S is:
S = mV1I1 = mV2I2

The primary circuit impedance Z1 referred to the secondary circuit for an ideal transformer with N1 primary turns and N2 secondary turns is:
Z12 = Z1(N2 / N1)2

The secondary circuit impedance Z2 referred to the primary circuit for an ideal transformer with N1 primary turns and N2 secondary turns is:
Z21 = Z2(N1 / N2)2

The voltage regulation DV2 of a transformer is the rise in secondary voltage which occurs when rated load is disconnected from the secondary with rated voltage applied to the primary. For a transformer with a secondary voltage E2 unloaded and V2 at rated load, the per-unit voltage regulation DV2pu is:
DV2pu = (E2 – V2) / V2
Note that the per-unit base voltage is usually V2 and not E2.

Open Circuit Test
If a transformer with its secondary open-circuited is energised at rated primary voltage, then the input power Poc represents the core loss (iron loss PFe) of the transformer:
Poc = PFe

The per-phase star values of the shunt magnetising admittance Ym, conductance Gm and susceptance Bm of an m-phase transformer are calculated from the open-circuit test results for the per-phase primary voltage V1oc, per-phase primary current I1oc and input power Poc using:
Ym = I1oc / V1oc
Gm = mV1oc2 / Poc
Bm = (Ym2 – Gm2)½

Short Circuit Test
If a transformer with its secondary short-circuited is energised at a reduced primary voltage which causes rated secondary current to flow through the short-circuit, then the input power Psc represents the load loss (primary copper loss P1Cu, secondary copper loss P2Cu and stray loss Pstray) of the transformer:
Psc = P1Cu + P2Cu + Pstray
Note that the temperature rise should be allowed to stabilise because conductor resistance varies with temperature.

If the resistance of each winding is determined by winding resistance tests immediately after the short circuit test, then the load loss of an m-phase transformer may be split into primary copper loss P1Cu, secondary copper loss P2Cu and stray loss Pstray:
P1Cu = mI1sc2R1star
P2Cu = mI2sc2R2star
Pstray = Psc – P1Cu – P2Cu

If the stray loss is neglected, the per-phase star values referred to the primary of the total series impedance Zs1, resistance Rs1 and reactance Xs1 of an m-phase transformer are calculated from the short-circuit test results for the per-phase primary voltage V1sc, per-phase primary current I1sc and input power Psc using:
Zs1 = V1sc / I1sc = Z1 + Z2(N12 / N22)
Rs1 = Psc / mI1sc2 = R1 + R2(N12 / N22)
Xs1 = (Zs12 – Rs12)½ = X1 + X2(N12 / N22)
where Z1, R1 and X1 are primary values and Z2, R2 and X2 are secondary values

Winding Resistance Test
The resistance of each winding is measured using a small direct current to avoid thermal and inductive effects. If a voltage Vdc causes current Idc to flow, then the resistance R is:
R = Vdc / Idc

If the winding under test is a fully connected balanced star or delta and the resistance measured between any two phases is Rtest, then the equivalent winding resistances Rstar or Rdelta are:
Rstar = Rtest / 2
Rdelta = 3Rtest / 2

The per-phase star primary and secondary winding resistances R1star and R2star of an m-phase transformer may be used to calculate the separate primary and secondary copper losses P1Cu and P2Cu:
P1Cu = mI12R1star
P2Cu = mI22R2star
Note that if the primary and secondary copper losses are equal, then the primary and secondary resistances R1star and R2star are related by:
R1star / R2star = I22 / I12 = N12 / N22

The primary and secondary winding resistances R1 and R2 may also be used to check the effect of stray loss on the total series resistance referred to the primary, Rs1, calculated from the short circuit test results:
Rs1 = R1 + R2(N12 / N22)

 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 E f G I j k m N n P p R susceptance induced voltage frequency conductance current j-operator coefficient number of phases number of turns rotational speed power pole pairs resistance [siemens, S] [volts, V] [hertz, Hz] [siemens, S] [amps, A] [1Ð90°] [number] [number] [number] [revs/min] [watts, W] [number] [ohms, W] S s T V X Y Z d F f h q w voltamperes slip torque terminal voltage reactance admittance impedance loss angle magnetic flux phase angle efficiency temperature angular speed [volt-amps, VA] [per-unit] [newton-metres, Nm] [volts, V] [ohms, W] [siemens, S] [ohms, W] [degrees, °] [webers, Wb] [degrees, °] [per-unit] [centigrade, °C] [radians/sec]