A load operates at 20 kW, 0.8 pf lagging. The load voltage is 220 \underline{/ 0^{\circ}} V rms at 60 Hz. The impedance of the line is 0.09 + j0.3 Ω. We wish to determine the voltage and power factor at the input to the line.
HINT
1. Use the given P_{L}, cos θ, and V_{L} rms to obtain S_{L} and I_{L} based on Eqs. (9.33) P=\operatorname{Re}( S )=V_{ rms } I_{ rms } \cos \left(\theta_{v}-\theta_{i}\right) and (9.29) S = V _{ rms } I _{ rms }^{*}, respectively.
2. Use I_{L} and Z_{line} to obtain S_{line} using Eq. (9.35) S = V _{ rms } I _{ rms }^{*}=\left( I _{ rms } Z \right) I _{ rms }^{*}= I _{ rms } I _{ rms }^{*} Z =I_{ rms }^{2} Z =I_{ rms }^{2}(R+j X)=P+j Q.
3. Use S _{S}= S _{\text {line }}+ S _{L}.
4. V _{S}= S _{S} / I _{L}^{*} yields V_{S} \text { and } \theta_{v}. Since V _{S}=V_{S} \underline{/ \theta_{v}} and \theta_{i} is the phase of I _{L}, pf =\cos \left(\theta_{v}-\theta_{j}\right).
