Unlocking Transformer Secrets: OC & SC Tests for Performance Analysis

Demonstrative Video

Equivalent Circuit

image
\[\begin{aligned} I_1^2R_2'& =I_2^2R_2 ~ \Leftarrow~\text{same power loss} \\ \Rightarrow~R_2'& =\left(\frac{I_2}{I_1}\right)^2R_2=\frac{R_2}{K^2} \\ &\\ &\boxed{R_2' = \frac{R_2}{K^2}} ~\qquad ~\boxed{X_2' = \frac{X_2}{K^2}} \end{aligned}\]

image
Equivalent parameters referred to primary:

\[\begin{aligned} R_{01} & =R_1+R_2'=R_1+\dfrac{R_2}{K^2} \\ X_{01} & =X_1+X_2'=X_1+\dfrac{X_2}{K^2} \\ Z_{01} & =\sqrt{R_{01}^2+X_{01}^2} \end{aligned}\]
image

Equivalent parameters referred to secondary:

image

\[\begin{aligned} R_{02} & =R_2+R_1'=R_2+K^2 \cdot R_1 \\ X_{02} & =X_2+X_1'=X_2+ K^2 \cdot X_1 \\ Z_{02} & =\sqrt{R_{02}^2+X_{02}^2} = K^2 \cdot Z_{01} \end{aligned}\]

Remember

Voltage Regulation

\[\begin{aligned} \text{Total voltage drop} & =E_2-V_2\\ &=O C-O A\\ &=O M-O A\\ &=A M=A N+N M\\ \end{aligned}\]
\[\begin{aligned} \text { Approximate voltage drop } & \approx A N \quad(\because N M \text { is very small) } \\ & =A D+D N \\ & =A D+B L \\ & =I_2 R_{02} \cos \phi+I_2 X_{02} \sin \phi \\ \% \text { regulation } & =\frac{I_2 R_{02} \cos \phi+I_2 X_{02} \sin \phi}{E_2} \times 100 \end{aligned}\]
For leading pf, Approximate voltage drop
\[\begin{aligned} &\boxed{\% \text { regulation } =\frac{I_2 R_{02} \cos \phi \pm I_2 X_{02} \sin \phi}{E_2} \times 100}\\ + &~~ \text{lagging pf} \qquad - ~~ \text{leading pf} \end{aligned}\]
\[\begin{aligned} \% {\color{red}{\textbf{regulation}}} & =\frac{I_1 R_{01} \cos \phi \pm I_1 X_{01} \sin \phi}{V_1} \times 100 \quad \Leftarrow~\text{primary side}\\ & \\ \% \text { regulation } & =\frac{100 I_2 R_{02}}{E_2} \cos \phi \pm \frac{100 I_2 X_{02}}{E_2} \sin \phi \\ & =\boxed{v_r \cos \phi \pm v_x \sin \phi }\\ &\boxed{v_r =\frac{ I_2 R_{02}}{E_2}\times 100}= {\color{magenta}{\textbf{percentage resistive drop }}}\\ & \boxed{v_x =\frac{I_2 X_{02}}{E_2}\times 100}= {\color{blue}{\textbf{percentage reactive drop }}} \end{aligned}\]

Efficiency

\[\begin{aligned} \eta & =\dfrac{\mbox{output power}}{\mbox{input power}}\\ &=\dfrac{\mbox{output power}}{\mbox{output power+losses}}\\ & =\dfrac{\mbox{output power}}{\mbox{output power+iron losses+copper losses}}\\ & \boxed{=\dfrac{V_{2}I_{2}\cos\Phi_{2}}{V_{2}I_{2}\cos\Phi_{2}+P_{i}+P_{c}}} \end{aligned}\]
where

If \(x\) is the fraction of the full-load, then efficiency is given as

\[\begin{aligned} \eta_{x} & =\boxed{\dfrac{x\times\mbox{Output}}{x\times\mbox{Output}+P_{i}+x^{2}P_{c}}}\\ &=\boxed{\dfrac{xV_{2}I_{2}\cos\Phi_{2}}{xV_{2}I_{2}\cos\Phi_{2}+P_{i}+x^{2}I_{2}^{2}R_{T}}} \end{aligned}\]

Condition for the maximum efficiency

\[\eta = \dfrac{V_2I_2\cos\Phi_2}{V_2I_2\cos\Phi_2+P_i+I_2^2R_T} = \dfrac{V_2\cos\Phi_2}{V_2\cos\Phi_2+P_i/I_2+I_2R_T}\]
\[I_2 = \sqrt{\dfrac{P_i}{R_T}}\]
For maximum efficiency . Then is the fraction of full load KVA at which If Value of output current for maximum efficiency

OC & SC Tests

Open-Circuit Test (OC)

\[\begin{aligned} & \text { Wattmeter reading }=W_i \\ & \text { Voltmeter reading }=V_1 \\ & \text { Ammeter reading }=I_0 \\ & W_i=V_1 I_0 \cos \phi_0 \\ & \boxed{\cos \phi_0=\frac{W_i}{V_1 I_0}} \\ & \boxed{I_w=I_0 \cos \phi_0} \qquad \boxed{I_\mu=I_0 \sin \phi_0} \\ & \boxed{R_0=\frac{V_1}{I_w}}\\ & \boxed{X_0=\frac{V_1}{I_\mu}} \\ & \end{aligned}\]
Calculation:

Short-Cicuit Test (SC)

\[\begin{aligned} & \text { Wattmeter reading }=W_{\mathrm{sc}} \\ & \text { Voltmeter reading }=V_{\mathrm{sc}} \\ & \text { Ammeter reading }=I_{\mathrm{sc}} \\ &\\ &\boxed{ Z_{sc}=\frac{V_{\mathrm{sc}}}{I_{\mathrm{sc}}}} \\ & \boxed{R_{sc}=\frac{W_{\mathrm{sc}}}{I_{\mathrm{sc}}^2}} \\ & \boxed{X_{sc}=\sqrt{\left(Z_{sc}\right)^2-\left(R_{sc}\right)^2}} \end{aligned}\]