## Transformer

**
For
an ideal two-winding transformer with primary voltage V _{1} applied
across N_{1} primary turns and secondary voltage V_{2}
appearing across N_{2} secondary turns:**

V_{1} / V_{2} = N_{1} / N_{2}

The primary current I_{1} and secondary current I_{2} are related by:

I_{1} / I_{2} = N_{2} / N_{1} = V_{2} / V_{1}

For
an ideal step-down auto-transformer with primary voltage V_{1}
applied across (N_{1} + N_{2}) primary turns and secondary
voltage V_{2} appearing across N_{2} secondary turns:

V_{1} / V_{2} = (N_{1} + N_{2}) / N_{2}

The primary (input) current I_{1} and secondary (output) current I_{2}
are related by:

I_{1} / I_{2} = N_{2} / (N_{1} + N_{2})
= V_{2} / V_{1}

Note that the winding current is I_{1} through the N_{1}
section and (I_{2} - I_{1}) through the N_{2}
section.

**
For
a single-phase transformer with rated primary voltage V _{1}, rated
primary current I_{1}, rated secondary voltage V_{2} and
rated secondary current I_{2}, the voltampere rating S is:**

S = V_{1}I_{1} = V_{2}I_{2}

**
For
a balanced m-phase transformer with rated primary phase voltage V _{1},
rated primary current I_{1}, rated secondary phase voltage V_{2}
and rated secondary current I_{2}, the voltampere rating S is:**

S = mV_{1}I_{1} = mV_{2}I_{2}

**
The
primary circuit impedance Z _{1} referred to the secondary circuit
for an ideal transformer with N_{1} primary turns and N_{2}
secondary turns is:**

Z_{12} = Z_{1}(N_{2} / N_{1})^{2}

**
The
secondary circuit impedance Z _{2} referred to the primary circuit
for an ideal transformer with N_{1} primary turns and N_{2}
secondary turns is:**

Z_{21} = Z_{2}(N_{1} / N_{2})^{2}

The
voltage regulation
DV_{2}
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 E_{2} unloaded
and V_{2} at rated load, the per-unit voltage regulation
DV_{2pu}
is:

DV_{2pu}
= (E_{2} - V_{2}) / V_{2}

Note that the per-unit base voltage is usually V_{2} and not E_{2}.

**
Open
Circuit Test
If a transformer with its secondary open-circuited is energised at rated
primary voltage, then the input power P _{oc} represents the core
loss (iron loss P_{Fe}) of the transformer:
P_{oc} = P_{Fe} **

**
The
per-phase star values of the shunt magnetising admittance Y _{m},
conductance G_{m} and susceptance B_{m} of an m-phase
transformer are calculated from the open-circuit test results for the
per-phase primary voltage V_{1oc}, per-phase primary current I_{1oc}
and input power P_{oc} using:**

Y_{m} = I_{1oc} / V_{1oc}

G_{m} = mV_{1oc}^{2} / P_{oc}

B_{m} = (Y_{m}^{2} - G_{m}^{2})^{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 P_{sc} represents the load
loss (primary copper loss P_{1Cu}, secondary copper loss P_{2Cu}
and stray loss P_{stray}) of the transformer:

P_{sc} = P_{1Cu} + P_{2Cu} + P_{stray}

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 P _{1Cu}, secondary
copper loss P_{2Cu} and stray loss P_{stray}:**

P_{1Cu} = mI_{1sc}^{2}R_{1star}

P_{2Cu} = mI_{2sc}^{2}R_{2star}

P_{stray} = P_{sc} - P_{1Cu} - P_{2Cu}

If
the stray loss is neglected, the per-phase star values referred to the
primary of the total series impedance Z_{s1}, resistance R_{s1}
and reactance X_{s1} of an m-phase transformer are calculated from
the short-circuit test results for the per-phase primary voltage V_{1sc},
per-phase primary current I_{1sc} and input power P_{sc}
using:

Z_{s1} = V_{1sc} / I_{1sc} = Z_{1} + Z_{2}(N_{1}^{2}
/ N_{2}^{2})

R_{s1} = P_{sc} / mI_{1sc}^{2} = R_{1}
+ R_{2}(N_{1}^{2} / N_{2}^{2})

X_{s1} = (Z_{s1}^{2} - R_{s1}^{2})^{½}
= X_{1} + X_{2}(N_{1}^{2} / N_{2}^{2})

where Z_{1}, R_{1} and X_{1} are primary values and
Z_{2}, R_{2} and X_{2} 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 V _{dc} causes
current I_{dc} to flow, then the resistance R is:
R = V_{dc} / I_{dc} **

If
the winding under test is a fully connected balanced star or delta and the
resistance measured between any two phases is R_{test}, then the
equivalent winding resistances R_{star} or R_{delta} are:

R_{star} = R_{test} / 2

R_{delta} = 3R_{test} / 2

**
The
per-phase star primary and secondary winding resistances R _{1star}
and R_{2star} of an m-phase transformer may be used to calculate the
separate primary and secondary copper losses P_{1Cu} and P_{2Cu}:**

P_{1Cu} = mI_{1}^{2}R_{1star}

P_{2Cu} = mI_{2}^{2}R_{2star}

Note that if the primary and secondary copper losses are equal, then the primary and secondary resistances R_{1star} and R_{2star} are related by:

R_{1star} / R_{2star} = I_{2}^{2} / I_{1}^{2} = N_{1}^{2} / N_{2}^{2}

The
primary and secondary winding resistances R_{1} and R_{2}
may also be used to check the effect of stray loss on the total series
resistance referred to the primary, R_{s1}, calculated from the
short circuit test results:

R_{s1} = R_{1} + R_{2}(N_{1}^{2} / N_{2}^{2})