Figure P15:34 is an equivalent circuit of a perfectly coupled transformer. This model is the basis for the transformer equivalent circuits used in the analysis of power systems. The inductance \(L_{\mathrm{m}}\) is called the magnetizing inductance. The current through this inductance represents the current needed to establish the magnetic field in the transformer at no load ( \(i_{2}\) \(=0)\). Show that the \(i-v\) equations for this circuit are as follows:
\[
\begin{aligned}
& v_{1}(t)=L_{\mathrm{m}} \frac{d i_{1}(t)}{d t}+n L_{\mathrm{m}} \frac{d i_{2}(t)}{d t} \\
& v_{2}(t)=n L_{\mathrm{m}} \frac{d i_{1}(t)}{d t}+n^{2} L_{\mathrm{m}} \frac{d i_{2}(t)}{d t}
\end{aligned}
\]
Use these equations to show that \(k=1\).

Transcribed Image Text:

VI(1) i(t) ni(t) Lm 1:n Ideal i(t) V(1)