Question #43348

A primary coil of a transformer is modeled as an ideal inductor of reactance $X_L = 20\ \Omega$ in series with a temperature-dependent resistor $R(T)$. The coil is connected to an AC voltage source of constant RMS value $V = 100\ \text{V}$. At the ambient temperature $T_0$, the initial resistance of the coil is $R_0 = 10\ \Omega$. The temperature coefficient of resistance of the coil material is $\alpha = 4 \times 10^{-3}\ \text{K}^{-1}$. During operation, the coil dissipates heat to the surroundings at a rate $H = K(T - T_0)$, where $K = 1\ \text{W/K}$ is the cooling constant and $T$ is the instantaneous temperature of the coil. Assuming the secondary winding is open-circuited, the inductive reactance remains constant, and the system reaches a steady thermal state, what is the steady-state RMS current drawn by the primary coil?