MODULE 3 — 1-Phase Transformer
3.1 Transformer Principle
Transformation Ratio
$$ \frac{V_1}{V_2} = \frac{N_1}{N_2} = \frac{I_2}{I_1} = a $$
EMF Equation
$$ E_1 = 4.44 f N_1 \Phi_m $$
$$ E_2 = 4.44 f N_2 \Phi_m $$
Input $V_1$: 240V
Output $V_2$: 120V
3.2 Ideal vs Real
Ideal Transformer
$$ V_1 I_1 = V_2 I_2 \quad \text{(100% eff)} $$
$$ \Phi_1 = \Phi_2, \quad R = 0, \quad X_m = \infty $$
3.3 Equivalent Circuit
$$ R_{eq} = R_1 + a^2 R_2 \quad X_{eq} = X_1
+ a^2 X_2 $$
$$ Z_{eq} = R_{eq} + jX_{eq} $$
3.4 Voltage Regulation
$$ VR = \frac{V_{2NL} - V_{2FL}}{V_{2FL}} \times 100\% $$
$$ VR \approx \varepsilon_R \cos\phi \pm \varepsilon_X
\sin\phi $$
(+) for lagging, (-) for leading PF
Power Factor Toggle
VR > 0 (Drops)
3.5 Efficiency & Max Eff.
$$ \eta = \frac{x S \cos\phi}{x S \cos\phi + P_i + x^2 P_{cu}}
$$
$$ \eta_{max} \text{ occurs when } P_i = P_{cu} $$
3.6 Transformer Tests
Open Circuit (Core Loss)
$$ \cos\phi_0 = \frac{W_0}{V_1 I_0} $$
Short Circuit (Cu Loss)
$$ Z_{eq} = \frac{V_{sc}}{I_{sc}}, \quad R_{eq} =
\frac{W_{sc}}{I_{sc}^2} $$