A Transformer Thvenin Equivalent In the time domain, the (i-v) equations for a linear transformer are [

A Transformer Thévenin Equivalent In the time domain, the \(i-v\) equations for a linear transformer are

\[
\begin{aligned}
& v_{1}(t)=L_{1} \frac{d i_{1}(t)}{d t}+M \frac{d i_{2}(t)}{d t} \\
& v_{2}(t)=M \frac{d i_{1}(t)}{d t}+L_{2} \frac{d i_{2}(t)}{d t}
\end{aligned}
\]

Assuming zero initial conditions, transform these equations into the \(s\)-domain and show that the \(s\)-domain parameters of the Thévenin equivalent at the output are

\[
Z_{\mathrm{T}}=\left(1-k^{2}ight) L_{2} s \quad \text { and } \quad V_{\mathrm{T}}(s)=\left(k \sqrt{L_{2} / L_{1}}ight) V_{1}(s)
\]

where \(k\) is the coupling coefficient.