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 DV_2pu is:
DV_2pu = (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 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_1oc² / P_oc
B_m = (Y_m² – G_m²)^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_1sc²R_1star
P_2Cu = mI_2sc²R_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 = Z₁ + Z₂(N₁² / N₂²)
R_s1 = P_sc / mI_1sc² = R₁ + R₂(N₁² / N₂²)
X_s1 = (Z_s1² – R_s1²)^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 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 = mI₁²R_1star
P_2Cu = mI₂²R_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 = 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, R_s1, calculated from the short circuit test results:
R_s1 = R₁ + R₂(N₁² / N₂²)