The characteristic equation of a $$3\,\, \times \,\,3$$ matrix $$P$$ is defined as
$$\alpha \left( \lambda \right) = \left| {\lambda {\rm I} - P} \right| = {\lambda ^3} + 2\lambda + {\lambda ^2} + 1 = 0.$$
If $${\rm I}$$ denotes identity matrix then the inverse of $$P$$ will be

A single phase fully controlled converter bridge is used for electrical braking of a separately excited $$dc$$ motor. The $$dc$$ motor load is represented by an equivalent circuit as shown in the figure.

Assume that the load inductance is sufficient to ensure continuous and ripple free load current. The firing angle of the bridge for a load current of $${{\rm I}_0} = 10\,\,A$$ will be

In the circuit shown in the figure, the switch is operated at a duty cycle of $$0.5.$$ A large capacitor current is connected across the load. The inductor current is assumed to be continuous.

The average voltage across the load and the average current through the diode will respectively be

In the single phase voltage controller circuit shown in the figure, for what range of triggering angle $$\left( \alpha \right)$$, the output voltage $$\left( {{V_0}} \right)$$ is not controllable?