Nomenclature and Symbols
\(V_p, V_s\) - Primary/Secondary Voltage
\(I_p, I_s\) - Primary/Secondary Current
\(N_p, N_s\) - Primary/Secondary Turns
\(R_p, R_s\) - Primary/Secondary Resistance
\(X_p, X_s\) - Primary/Secondary Reactance
\(Z_p, Z_s\) - Primary/Secondary Impedance
\(E_p, E_s\) - Primary/Secondary Induced EMF
\(P_p, P_s\) - Primary/Secondary Power
\(S_p, S_s\) - Primary/Secondary Apparent Power
\(P_i, P_c\) - Iron Loss/Copper Loss
\(\Phi_m\) - Maximum Flux
\(K\) - Turns Ratio
\(f\) - Frequency (Hz)
\(B_m\) - Maximum Flux Density
\(A\) - Core Cross-sectional Area
\(\eta\) - Efficiency
Transformer Basics
$\text{Turns Ratio: } K = \frac{N_s}{N_p} = \frac{V_s}{V_p} = \frac{I_p}{I_s}$
$P_p = P_s \text{ (Ideal)}$
$S_p = S_s$
$P = V \cdot I \cdot \cos \phi$
$\text{KVA} = \frac{V \cdot I}{1000}$
Transformer EMF Equation
$E = 4.44 f N \Phi_m = 4.44 f N B_m A$
$E_1 = 4.44 f N_1 \Phi_m$
Primary EMF
$E_2 = 4.44 f N_2 \Phi_m$
Secondary EMF
Where:
\(f\) = Frequency (Hz), \(N\) = Number of turns, \(\Phi_m\) = Maximum flux (Wb)
\(B_m\) = Maximum flux density (T), \(A\) = Core area (m²)
\(f\) = Frequency (Hz), \(N\) = Number of turns, \(\Phi_m\) = Maximum flux (Wb)
\(B_m\) = Maximum flux density (T), \(A\) = Core area (m²)
Transformer on No-Load
$W_0 = V_1 I_0 \cos(\Phi_0)$
$I_w = I_0 \cos(\Phi_0) \quad \text{(Working component)}$
$I_m = I_0 \sin(\Phi_0) \quad \text{(Magnetizing component)}$
$I_0 = \sqrt{I_w^2 + I_m^2}$
Transformer on Load
$I_p' = \frac{N_2}{N_1} I_s = K I_s$
$\vec{I_p} = \vec{I_0} + \vec{I_p'}$
Note: \(I_p'\) is the load component of primary current that balances the secondary current effect.
Equivalent Circuit Parameters
Impedance Referred to Primary Side:
$R_{01} = R_1 + R_2' = R_1 + \frac{R_2}{K^2}$
$X_{01} = X_1 + X_2' = X_1 + \frac{X_2}{K^2}$
$Z_{01} = \sqrt{R_{01}^2 + X_{01}^2}$
Impedance Referred to Secondary Side:
$R_{02} = R_2 + R_1' = R_2 + K^2 R_1$
$X_{02} = X_2 + X_1' = X_2 + K^2 X_1$
$Z_{02} = \sqrt{R_{02}^2 + X_{02}^2}$
Voltage Regulation
$\text{Voltage Regulation (\%)} = \frac{E_2 - V_2}{V_2} \times 100$
$= \frac{I_2 R_{02} \cos\theta_2 \pm I_2 X_{02} \sin\theta_2}{V_2} \times 100$
$= v_r \cos\phi \pm v_x \sin\phi$
Sign Convention:
(+) for lagging power factor
(−) for leading power factor
(+) for lagging power factor
(−) for leading power factor
$v_r = \frac{I_2 R_{02}}{V_2} \times 100$
Resistive Drop %
$v_x = \frac{I_2 X_{02}}{V_2} \times 100$
Reactive Drop %
Open Circuit Test (OC Test)
$\cos\Phi_0 = \frac{W_0}{V_1 I_0}$
$I_w = I_0 \cos\Phi_0 \quad ; \quad I_m = I_0 \sin\Phi_0$
$R_0 = \frac{V_1}{I_w} \quad ; \quad X_m = \frac{V_1}{I_m}$
Purpose: Determines iron losses (\(P_i\)) and magnetizing parameters
Short Circuit Test (SC Test)
$Z_{sc} = \frac{V_{sc}}{I_{sc}}$
$R_T = \frac{W_{sc}}{I_{sc}^2}$
$X_T = \sqrt{Z_{sc}^2 - R_T^2}$
Purpose: Determines copper losses (\(P_c\)) and equivalent impedance
Transformer Losses
Iron/Core Losses (Constant):
$P_i = P_h + P_e$
- \(P_h\) = Hysteresis loss
- \(P_e\) = Eddy current loss
Copper Losses (Variable):
$P_c = I_1^2 R_1 + I_2^2 R_2$
$= I_1^2 R_{01} = I_2^2 R_{02}$
$\text{Total Losses} = P_i + P_c$
Efficiency
$\eta = \frac{\text{Output Power}}{\text{Input Power}} = \frac{\text{Output}}{\text{Output + Losses}}$
$\eta = \frac{V_2 I_2 \cos\phi_2}{V_2 I_2 \cos\phi_2 + P_i + P_c} \times 100\%$
$\eta = \frac{x \cdot \text{KVA}_{FL} \cdot \cos\phi}{x \cdot \text{KVA}_{FL} \cdot \cos\phi + P_i + x^2 P_c} \times 100\%$
Where: \(x\) = Fraction of full load, \(\text{KVA}_{FL}\) = Full load KVA rating
Maximum Efficiency Condition
Condition for Maximum Efficiency:
Iron Loss = Copper Loss
Iron Loss = Copper Loss
$P_i = x^2 P_c$
$x = \sqrt{\frac{P_i}{P_c}}$
$I_2 = \sqrt{\frac{P_i}{R_T}}$
$\text{KVA at } \eta_{max} = x \times \text{KVA}_{FL} = \text{KVA}_{FL} \sqrt{\frac{P_i}{P_c}}$
Maximum Efficiency:
$\eta_{max} = \frac{x \cdot \text{KVA}_{FL} \cdot \cos\phi}{x \cdot \text{KVA}_{FL} \cdot \cos\phi + 2P_i} \times 100\%$
All-Day Efficiency
$\eta_{all-day} = \frac{\text{Output in kWh}}{\text{Input in kWh}} \times 100\%$
$= \frac{\sum_{i=1}^{n} P_i t_i}{\sum_{i=1}^{n} P_i t_i + P_{iron} \times 24 + \sum_{i=1}^{n} P_{cu,i} t_i} \times 100\%$
Note: Used for distribution transformers that operate at varying loads throughout the day. Iron losses occur for full 24 hours.
Auto-Transformer Relations
$K = \frac{V_2}{V_1} = \frac{N_2}{N_1}$
$\text{Transformation Ratio: } a = \frac{V_2}{V_1} = \frac{N_1 + N_2}{N_1} = 1 + K$
$\text{Saving in Cu: } \frac{W_{auto}}{W_{two-winding}} = 1 - K$
$\text{Power transferred inductively} = K \times \text{Total Power}$
$\text{Power transferred conductively} = (1-K) \times \text{Total Power}$
Per Unit (p.u.) System
$\text{Per Unit Value} = \frac{\text{Actual Value}}{\text{Base Value}}$
$Z_{pu} = \frac{Z_{actual}}{Z_{base}}$
$Z_{base} = \frac{V_{base}^2}{S_{base}}$
$\%Z = Z_{pu} \times 100$
$Z_{pu} = \frac{I_{sc} Z}{V_{rated}}$
Parallel Operation of Transformers
Conditions for Parallel Operation:
- Same voltage ratio and polarity
- Same percentage impedance
- Same impedance angle (X/R ratio)
- Same phase sequence
$\text{Load shared by transformer A: } S_A = \frac{S_T \cdot Z_B}{Z_A + Z_B}$
$\text{Load shared by transformer B: } S_B = \frac{S_T \cdot Z_A}{Z_A + Z_B}$
Note: Load sharing is inversely proportional to impedances
Transformer Cooling Classifications
ONAN - Oil Natural Air Natural
ONAF - Oil Natural Air Forced
OFAF - Oil Forced Air Forced
ODAF - Oil Directed Air Forced
ODWF - Oil Directed Water Forced
AN - Air Natural (Dry type)
Key Points to Remember
- Step-up transformer: \(N_s > N_p\), \(V_s > V_p\), \(I_s < I_p\)
- Step-down transformer: \(N_s < N_p\), \(V_s < V_p\), \(I_s > I_p\)
- Ideal transformer: No losses, \(\eta = 100\%\), \(V_pI_p = V_sI_s\)
- OC test: Performed on LV side (convenient)
- SC test: Performed on HV side (convenient)
- Zero regulation: At leading power factor when \(I_2X_{02}\sin\phi = I_2R_{02}\cos\phi\)
- Transformer rating: Always in KVA (not kW) because losses independent of load pf