Q$7.$

a$.$ For the circuit shown in Figure Q$7$

i$.$ Find the value of the total impedance $Z_T$ and the current I.

ii$.$ Write the voltage and current in time$-$domain representation and the sketch the waveforms for a half$-$cycle indicating the peak value and the phase angle?

iii. Calculate the complex power and the power factor, indicating its type.

Figure Q$7$

b$.$ A single phase motor operating at $230V,50Hz,15kW$ with efficiency $88\%$ and $0\mathrm{.}7$ power factor lagging.

i$.$ Determine the input active, reactive and apparent powers

ii$.$ Sketch the power triangle

iii. What is the value of parallel capacitor needed to correct the motor's power factor to $0\mathrm{.}95.$

c$.$ A balanced RYB$-$ sequence Y$-$connected three phase, $50$ Hz $,$ source with $V_{RN}=\frac{120}{{?}_{15}}V,$ is connected to a $-$connected balanced load $(10+j4)$ per phase.

i$.$ Calculate the phase and line currents

ii$.$ Determine the total power supplied to the load

iii. Sketch the phasor diagram illustrating various phase and line currents.