onsider the network shown in Figure $2$ where a single a$.$c$.$ supply is used to energize an interconnection of multiple load models. The absolute values of the impedance decompositions $($i$.$e$.,R,x_L$ and $x_C)$ of all loads are shown.

Figure $3.$ An a$.$c$.$ network of series$-$parallel connected load models

Using classical network theory,

$($a$)$ Expressing your solutions in the frequency domain, determine the three $(3)$ branch currents $I_1,I_2$ and $I_3.$

$($b$)$ Hence, or otherwise, calculate $|S_T|,P_T$ and $Q_T,$ the respective total active, reactive and apparent powers supplied to the network.

$($c$)$ Calculate the overall power factor of the circuit.