\[\begin{aligned} \text{Turns Ratio} ~K&= \frac{N_s}{N_p} = \frac{V_s}{V_p} = \frac{I_p}{I_s} \\ P_p &= P_s \\ S_p &= S_s \\ P_p &= V_p \cdot I_p \cdot \cos \phi_p \\ P_s &= V_s \cdot I_s \cdot \cos \phi_s \\ \text{KVA} &= \frac{V_s \cdot I_s}{1000} = \frac{V_p \cdot I_p}{1000}\\ \end{aligned}\]

\[\begin{aligned} E_1 &= 4.44fN_1\Phi_m = 4.44fN_1B_mA \\ E_2 &= 4.44fN_2\Phi_m = 4.44fN_2B_mA \\ \end{aligned}\]

\[\begin{aligned} W_0 & =V_1 I_0 \cos \left(\Phi_0\right) \\ I_w & =I_0 \cos \left(\Phi_0\right) \\ I_m & =I_0 \sin \left(\Phi_0\right) \\ I_0 & =\sqrt{I_w^2+I_m^2} \end{aligned}\]

\[\begin{aligned}&I_{p}^{'}=\left(\frac{N_{2}}{N_{1}}\right)I_{s}=KI_{s}\\&\overset{\to}{I_{p}}=\overset{\to}{I_{0}}+\overset{\to}{I_{p}^{'}}\end{aligned}\]

\[\begin{aligned} Z_1 & =\sqrt{\left(R_1^2+X_1^2\right)} \\ Z_2 & =\sqrt{\left(R_2^2+X_2^2\right)} \\ R_{01} & =R_1+R_2^{\prime}=R_1+R_2 / K^2 \\ R_{02} & =R_2+R_1^{\prime}=R_2+K^2 R_1 \\ X_2^{\prime} & =X_2 / K^2 \quad \text { and } \quad X_1^{\prime}=K^2 X_1 \\ X_{01} & =X_1+X_2^{\prime} \quad \text { and } \quad X_{02}=X_2+X_1^{\prime} \\ Z_{01} & =\sqrt{\left(R_{01}^2+X_{01}^2\right)} \\ Z_{02} & =\sqrt{\left(R_{02}^2+X_{02}^2\right)} \end{aligned}\]

\[\begin{aligned} \% \text{voltage drop in secondary}=& \frac{I_{2}R_{02}\cos\Phi\pm I_{2}X_{02}\sin\Phi}{_0V_{2}}\times100 \\ &=v_{r}cos\Phi\pm v_{x}sin\Phi \\ & + \quad \text{lagging}~\qquad - \quad \text{leading}\\ v_{r}& =\frac{I_2R_{02}}{_0V_2}\times100=\frac{I_1R_{01}}{V_1}\times100=\text{percentage resistive drop} \\ v_{x}& =\frac{I_2X_{02}}{_0V_2}\times100=\frac{I_1X_{01}}{V_1}\times100=\text{percentage reactive drop} \end{aligned}\]

\[\begin{aligned} W_{0}& =V_{1}I_{0}\cos\Phi_{0} \\ \Rightarrow \cos\Phi_0 & =\frac{W_0}{V_1I_0} \\ \Rightarrow I_{w}& =I_0\cos\Phi_0 & \Rightarrow I_{m} & =I_0\sin\Phi_0 \\ \Rightarrow R_{0}& =V_1/I_w & \Rightarrow X_{m} & =V_1/I_m \end{aligned}\]

\[\begin{aligned}&Z_{sc}=\frac{V_{sc}}{I_{sc}}\\ & \Rightarrow R_T=\frac{W_{sc}}{I_{sc}^2} \\ &\Rightarrow X_T=\sqrt{Z_{sc}^2-R_T^2}\end{aligned}\]

\[\begin{aligned} \text{voltage regulation}~(\%)&=\frac{E_{2}-V_{2}}{V_{2}}\times100 \\ & = \dfrac{I_2R_2\cos\theta_2\pm I_2X_2\sin\theta_2}{V_{2}} \end{aligned}\]

\[\begin{aligned}\eta&=\frac{\text{output power}}{\text{input power}}=\frac{\text{output power}}{\text{output power}+\text{losses}}\\&=\frac{\text{output power}}{\text{output power+iron losses}+\text{copper losses}}\\&=\frac{V_2I_2\cos\Phi_2}{V_2I_2\cos\Phi_2+P_i+P_c}\end{aligned}\]

\[\begin{aligned} \text{Iron loss} & =P_i\\\text{Copper loss}&=x^2P_c\\ \text{For maximum efficiency }x^2P_c & =P_i\\ I_{2}&=\sqrt{\frac{P_{i}}{R_{T}}} \\ x&=\sqrt{\frac{P_{i}}{P_{c}}} \\ \eta_{m}& _{ax}=x\times KVA_{FL} \\ \Rightarrow\eta_{max}& =KVA_{FL}\times\sqrt{\frac{P_i}{P_c}} \end{aligned}\]