Two small industries located side-by-side are fed from a...

Two small industries located side-by-side are fed from a three-phase balanced $60 \mathrm{~Hz}$ source with a line voltage of $208 \mathrm{~V}$ rms. Load #1 consumes $24 \mathrm{~kW}$ at $0.75 \mathrm{PF}$ lagging and load #2 consumes $16 \mathrm{~kW}$ at $0.88 \mathrm{PF}$ lagging. Determine the current in the line that serves both loads.

Key Concepts

Three-Phase Power Systems

Three-phase power systems consist of three sinusoidal voltages of equal magnitude that are phase-shifted by 120 degrees. This configuration allows for a continuous and balanced power transfer, which is especially efficient for industrial and commercial power distribution. It simplifies the design and analysis of power systems through the use of symmetry, and the calculations for power, voltage, and current often involve relationships between line-to-line and phase values.

Balanced Loads

A balanced load in a three-phase system means that each phase draws equal power with equal impedance. This assumption greatly simplifies analysis since the behavior in one phase mirrors that of the others. Analyzing balanced loads allows the use of simplified single-phase equivalent calculations, making it easier to determine overall system performance and to size equipment correctly.

Power Factor

Power factor is the ratio of real power, which performs useful work, to apparent power, which is the product of the circuit’s voltage and current. It indicates the efficiency with which electrical power is converted into useful work. In AC circuits with inductive components, the power factor is less than one due to the phase difference between current and voltage, and a lagging power factor indicates that the current lags behind the voltage.

Complex Power

Complex power combines real (active) power and reactive power into a single quantity, typically represented as a complex number. The real component corresponds to work that is actually done or energy consumed, while the imaginary component represents energy that alternates between the source and reactive components in the circuit. This concept is crucial for accurately describing the power flow in AC systems.

Current Determination in Three-Phase Systems

Calculating the current in a three-phase system involves determining the total apparent power drawn by all connected loads and relating it to the line voltage. By summing the individual contributions of loads and using appropriate conversion factors (depending on whether the loads are connected in star or delta formation), one can compute the line current necessary for the system, ensuring proper equipment sizing and system safety.

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