Final answer:
The phase angle of the total complex power consumed by the loads is found by converting horsepower to watts, calculating the individual reactive and real power components, and then using the arctangent of the ratio of total reactive power to total real power.
Step-by-step explanation:
The question involves finding the phase angle of the total complex power consumed by two loads connected in parallel across an AC voltage source. The two loads in question are a 6HP inductor motor with a power factor of 0.87 lagging, and a lighting load consuming 2.9 kW of real power.
To solve the mathematical problem completely, we must convert the inductor motor's power from horsepower to watts, calculate the reactive and real power components of each load, and then find the total complex power. The phase angle can then be deduced from the arctangent of the ratio of total reactive power to total real power.
For the inductor motor, the real power (P1) is converted to watts:
P1 = 6 HP × 745.7 W/HP = 4474.2 W. The reactive power (Q1) can be calculated from the real power and power factor (PF):
Q1 = P1 × tan(φ), where φ = cos^{-1}(PF).
For the lighting load, the real power (P2) is 2.9 kW, and it has no reactive power (purely resistive, Q2 = 0).
To find the total real power (PT) and total reactive power (QT), we add the individual powers:
PT = P1 + P2,
QT = Q1 + Q2.
Finally, the phase angle (φT) in degrees is calculated from the arctangent of the total reactive power to total real power ratio:
φT = atan(ׄQT/PT)°.