The resonant frequency for the given circuit will be

A three phase balanced star connected voltage source with frequency $$\omega \,\,rad/s$$ is connected to a star connected balanced load which is purely inductive. The instantaneous line currents and phase to neutral voltages are denoted by $$\left( {{i_a},{i_b},{i_c}} \right)$$ and $$\left( {{V_{an}},\,\,{V_{bn}},\,\,{V_{cn}}} \right)$$ respectively and their $$rms$$ values are denoted by $$V$$ and $$1.$$ If $$$R = \left[ {{V_{an}}\,\,{V_{bn}}\,\,{V_{cn}}} \right]\left[ {\matrix{ 0 & {{1 \over {\sqrt 3 }}} & { - {1 \over {\sqrt 3 }}} \cr { - {1 \over {\sqrt 3 }}} & 0 & {{1 \over {\sqrt 3 }}} \cr {{1 \over {\sqrt 3 }}} & { - {1 \over {\sqrt 3 }}} & 0 \cr } } \right]\left[ {\matrix{ {{i_a}} \cr {{i_b}} \cr {{i_c}} \cr } } \right],$$$
then the magnitude of $$R$$ is

A $$3$$ $$V$$ $$dc$$ supply with an internal resistance of $$2\,\,\Omega $$ supplies a passive non-linear resistance characterized by the relation $${V_{NL}} = {\rm I}_{NL}^2.$$ The power dissipated in the non-linear resistance is

The probes of a non $$-$$ isolated, two-channel oscilloscope are clipped to points $$A, B, $$ and $$C$$ in the circuit of the adjacent fig. $${V_{in}}$$ is a square wave of a suitable low frequency. The display on $$c{h_1}$$ and $$c{h_2}$$ are as shown on the right. Then the ''signal'' and ''Ground'' probes $${S_1},\,\,{G_1}$$ and $${S_2},\,\,{G_2}$$ of $$c{h_1}$$ and $$c{h_2}$$ respectively are connected to points.