### Formulas

For an ideal two-winding transformer with primary voltage **V _{1}** applied across

**N**primary turns and secondary voltage

_{1}**V**appearing across

_{2}**N**secondary turns:

_{2}**V**

_{1}/ V_{2}= N_{1}/ N_{2}The primary current

**I**and secondary current

_{1}**I**are related by:

_{2}**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**primary turns and secondary voltage

_{1}+ N_{2})**V**appearing across

_{2}**N**secondary turns:

_{2}**V**

_{1}/ V_{2}= (N_{1}+ N_{2}) / N_{2}The primary (input) current

**I**and secondary (output) current

_{1}**I**are related by:

_{2}**I**

_{1}/ I_{2}= N_{2}/ (N_{1}+ N_{2}) = V_{2}/ V_{1}Note that the winding current is

**I**through the

_{1}**N**section and

_{1}**(I**through the

_{2}– I_{1})**N**section.

_{2}For a single-phase transformer with rated primary voltage **V _{1}**, rated primary current

**I**, rated secondary voltage

_{1}**V**and rated secondary current

_{2}**I**, the voltampere rating

_{2}**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**, rated secondary phase voltage

_{1}**V**and rated secondary current

_{2}**I**, the voltampere rating

_{2}**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**primary turns and

_{1}**N**secondary turns is:

_{2}**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**primary turns and

_{1}**N**secondary turns is:

_{2}**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**unloaded and

_{2}**V**at rated load, the per-unit voltage regulation

_{2}**DV**is:

_{2pu}**DV**

_{2pu}= (E_{2}– V_{2}) / V_{2}Note that the per-unit base voltage is usually

**V**and not

_{2}**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**) of the transformer:

_{Fe}**P**

_{oc}= P_{Fe}The per-phase star values of the shunt magnetising admittance **Y _{m}**, conductance

**G**and susceptance

_{m}**B**of an

_{m}**m**-phase transformer are calculated from the open-circuit test results for the per-phase primary voltage

**V**, per-phase primary current

_{1oc}**I**and input power

_{1oc}**P**using:

_{oc}**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**, secondary copper loss

_{1Cu}**P**and stray loss

_{2Cu}**P**) of the transformer:

_{stray}**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**and stray loss

_{2Cu}**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**and reactance

_{s1}**X**of an

_{s1}**m**-phase transformer are calculated from the short-circuit test results for the per-phase primary voltage

**V**, per-phase primary current

_{1sc}**I**and input power

_{1sc}**P**using:

_{sc}**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**and

_{1}**X**are primary values and

_{1}**Z**,

_{2}**R**and

_{2}**X**are secondary values

_{2}*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**to flow, then the resistance

_{dc}**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**or

_{star}**R**are:

_{delta}**R**

_{star}= R_{test}/ 2**R**

_{delta}= 3R_{test}/ 2The per-phase star primary and secondary winding resistances **R _{1star}** and

**R**of an

_{2star}**m**-phase transformer may be used to calculate the separate primary and secondary copper losses

**P**and

_{1Cu}**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**and

_{1star}**R**are related by:

_{2star}**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**may also be used to check the effect of stray loss on the total series resistance referred to the primary,

_{2}**R**, calculated from the short circuit test results:

_{s1}**R**

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

BE 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] | SsTVXYZdFfhqw | 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] |