PROBLEM $1[20$pts for each section$]$

In a balanced three$-$phase system, the instantaneous values of phase a voltage and phase a current are stated as:

$v_a(t)=311\mathrm{.}13cos(t)$ volts and $i_a(t)=16\mathrm{.}97cos(t-30)$ ampers.

a$.$ Express the instantaneous values of voltages and currents related to phase $b$ & $c.$

b$.$ Calculate the phasor values of three$-$phase currents.

c$.$ Express phasor values of three$-$phase voltages.

d$.$ Calculate the three phase complex power and power factor of the system.

e$.$ Find out the required kVAr to increase the power factor up to $0\mathrm{.}95.$