An electrical network is fed by two $$ac$$ sources, as shown in Fig. Given that
$${Z_1} = \left( {1 - j} \right)\Omega ,\,\,{Z_2} = \left( {1 + j} \right)\Omega $$ and $${Z_L} = \left( {1 + j0} \right)\Omega .$$ Obtain the Thevenin equivalent circuit (Thevenin voltage and impedance) across terminals $$X$$ and $$Y$$, and determine the current $${{\rm I}_L}$$ through the load $${Z_L}.$$

The graph of an electrical network has $$N$$ nodes and $$B$$ branches. The number of links, $$L$$, with respect to the choice of a tree, is given by

In the resistor network shown in Fig. all resistor values are $$1\,\,\Omega .$$ $$A$$ current of $$1A$$ passes from terminal $$a$$ to terminal $$b,$$ as shown in the figure. Calculate the voltage between terminals $$a$$ and $$b.$$ [Hint: You may exploit the symmetry of the circuit].

Consider the circuit shown in Fig. If the frequency of the source is $$50$$ $$Hz,$$ then a value of $${t_0}$$ which results in a transient free response is