(i) Obtaining expression for impedance & phase angle Condition of current being in phase with voltage Naming of circuit condition (ii) Calculation of $\frac{P_1}{P_2}$ (i) From Figure $\vec{V} = \vec{V}_L + \vec{V}_R + \vec{V}_C$ where $|\vec{V}_R| = i_m R$ $|\vec{V}_L + \vec{V}_C| = V_{C m} - V_{L m}$ $= i_m (X_C - X_L)$ $\Rightarrow V_m^2 = V_{R m}^2 + (V_{C m} - V_{L m})^2$ $l_m^2 Z^2 = l_m^2 R^2 + I_m^2 (X_C - X_L)^2$ $Z = \sqrt{R^2 + (X_C - X_L)^2}$ From Figure $\tan \phi = \frac{V_{C m} - V_{L m}}{V_{R m}}$ $= \frac{i_m (X_C - X_L)}{i_m R}$ $\phi = \tan^{-1}\left( \frac{X_C - X_L}{R} \right)^{\frac{1}{2}}$ Condition for current and voltage are in phase: $V_L = V_C \text{ or } X_L = X_C$ Circuit is called Resonant circuit. (ii) Power factor $P_1 = \frac{R}{Z} = \frac{R}{\sqrt{R^2 + R^2}} = \frac{1}{\sqrt{2}}$ $\text{ (as } X_L = R)^{1/2}$ Power factor when capacitor $C$ of reactance $X_C = X_L$ is put in series in the circuit $P_2 = \frac{R}{Z} = \frac{R}{R} = 1$ $\text{ as } Z = R \text{ at resonance}$ ∴ $\frac{P_1}{P_2} = \frac{1}{\frac{\sqrt{2}}{1}} = \frac{1}{\sqrt{2}}$ OR (i) Function of transformerWorking principle and diagramVarious energy losses (two) (ii) Calculation of part (a), (b), (c), (d) & (e) (i) Conversion of ac of low voltage into ac of high voltage & vice versa. Mutual induction : When alternating voltage is applied to primary windings, emf is induced in the secondary windings. Energy losses:(a) Leakage of magnetic flux(b) Eddy currents(c) Hysteresis loss(d) Copper loss (ii) $N_p = 100$Transformation ratio $= 100$(a) Number of turns in secondary coil $N_S = 100 \times 100 = 10000$(b) Input Power $=$ Input voltage $\times$ current in primary $1100 = 220 \times I_p$ $\Rightarrow I_p = 5 \mathrm{A}$ (c) $\frac{V_S}{V_p} = \frac{N_S}{N_p}$ $\frac{V_S}{220} = 100$ $\Rightarrow V_S = 2.2 \times 10^4 \text{ volts}$ (d) $\frac{I_p}{I_S} = \frac{N_s}{N_p}$ $\frac{5}{I_s} = 100$$\Rightarrow I_s = \frac{5}{100} = 0.05 \mathrm{A}$ (e) Power in secondary = Power in Primary $= 1100 \mathrm{W}$