Parameters of transformer: open-circuit test and short-circuit test
The closed circuit experiment will yield two equations instead of one:

$$\begin{align}
u_1(t) &amp;= r_1i_1(t) + L_{11}\frac{di_1}{dt} - L_M\frac{di_2}{dt}\\
u_2(t) &amp;= r_2i_2(t) - L_M\frac{di_1}{dt} + L_{22}\frac{di_2}{dt}
\end{align}$$

(Note that the signs may need to be changed)

Meaning it will have 2 equations and 2 unknowns (\$i_1(t)\$ and \$i_2(t)\$), which can be solved.

[Edit]

For your short-circuit experiment and when using phasors, this would result into something like:

$$\begin{align}
\underline{U_1} &amp;= \left(r_1 + j\omega L_{11}\right) \underline{I_1} - j\omega L_M \underline{I_2}\\
0 &amp;= -j\omega L_M \underline{I_1} + \left(r_2 + j\omega L_{22}\right) \underline{I_2}
\end{align}$$

The solution to this set of equations will be two complex values for \$\underline{I_1}\$ and \$\underline{I_2}\$. You can take the absolute value like with any phasor.

$$\underline{U_1} = U_{RMS}\cdot \sqrt{2}$$

$$I_{RMS} = \sqrt{\frac{\text{re}[\underline{I}]^2+\text{im}[\underline{I}]^2}{2}}$$

And you can calculate the complex power as well

$$\underline{P} = \underline{U_1}\cdot\underline{I_1}^*$$