In a three-phase system, the active power (P) is given by which of the following formulas?

Explanation:

Active Power in a Three-Phase System

Definition: Active power (P), also known as real power, is the actual power consumed by electrical devices to perform useful work, such as lighting, heating, or rotating a motor. In a three-phase electrical system, active power is calculated based on the voltage, current, and the power factor of the system.

Correct Formula: The active power (P) in a balanced three-phase system is given by:

P = √3 × VL × IL × cosΦ

Where:

• √3: A constant derived from the relationship between line-to-line and line-to-neutral voltages in a three-phase system.
• VL: Line-to-line voltage (in volts).
• IL: Line current (in amperes).
• cosΦ: Power factor, which represents the phase angle between voltage and current.

Explanation:

The formula P = √3 × VL × IL × cosΦ is derived based on the principles of three-phase power systems. In a balanced three-phase system, the power delivered by each phase is equal. The total active power is the sum of the active power contributions from all three phases. Using trigonometric relationships and electrical theory, the factor √3 is introduced to account for the phase difference between the line voltages and currents.

Why Option 1 is Correct:

Option 1 correctly states the formula for calculating active power in a three-phase system. The inclusion of the constant √3 accounts for the relationship between the line-to-line and line-to-neutral voltages in a three-phase system, ensuring accurate calculation of the total active power. The power factor (cosΦ) is a critical component, as it reflects the efficiency of power usage in the system. This formula is widely used in electrical engineering for power calculations in industrial and commercial applications.

Applications:

• Used for calculating power consumption in industrial machinery and equipment.
• Applicable in power system analysis for load flow studies.
• Essential for designing energy-efficient systems by optimizing the power factor.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: VL × IL

This formula is incomplete and incorrect for a three-phase system. While it may represent apparent power in a single-phase system, it does not account for the phase relationships and power factor in a three-phase system. Therefore, this option cannot be used to calculate active power.

Option 3: 3 × VL × IL × cosΦ

This formula is incorrect for a three-phase system. The factor "3" implies direct addition of the power contributions from each phase without considering the √3 constant, which is necessary to relate line-to-line voltage and line-to-neutral voltage. Thus, this formula leads to incorrect results for active power calculation.

Option 4: VL × IL × cosΦ

This formula represents the active power calculation for a single-phase system, not a three-phase system. It does not incorporate the √3 constant required for three-phase systems, nor does it account for the contributions from all three phases. Hence, it is unsuitable for three-phase power calculations.

Conclusion:

The correct formula for calculating active power in a balanced three-phase system is P = √3 × VL × IL × cosΦ, as stated in Option 1. This formula accurately accounts for the contributions of all three phases, the relationship between line-to-line and line-to-neutral voltages, and the power factor. Understanding the correct formula is essential for efficient power system analysis and energy management in industrial and commercial applications.