A single - phase voltage source inverter shown in figure is feeding power to load. The triggering pulses of the devices are also shown in the figure.

If the load current is sinusoidal and is zero at $$0,\,\,\pi ,2\pi ....,$$ the node voltage $${V_{AO}}$$ has the waveform

For a $$400$$ $$km$$ long transmission line, the series impedance is $$(0.0 + j0.5)$$ $$\Omega /km$$ and the shunt admittance is $$(0.0 + j5.0)$$ $$\mu \,mho/km.$$ The magnitude of the series impedance (in $$\Omega $$) of the equivalent $$\pi $$ circuit of the transmission line is __________.

The figure shows the single line diagram of a single machine infinite bus system.

The inertia constant of the synchronous generator $$𝐻 = 5$$ $$MW-s/MVA.$$ Frequency is $$50Hz.$$ Mechanical power is $$1$$ pu. The system is operating at the stable equilibrium point with rotor angle $$\delta $$ equal to $${30^ \circ }$$. A three phase short circuit fault occurs at a certain location on one of the circuits of the double circuit transmission line. During fault, electrical power in pu is $$\,{P_{\max }}\,\,\sin \delta .\,\,\,$$. If the values of $$\delta $$ and $$d$$$$\delta $$$$/dt$$ at the instant of fault clearing are $${45^ \circ }$$ and $$3.762$$ radian/s respectively, then $$\,{P_{\max }}$$ (in pu) is _________.

In a long transmission line with $$r, l, g$$ and $$𝑐$$ are the resistance, inductance, shunt conductance and capacitance per unit length respectively, the condition for distortionless transmission is