Understanding 3-Phase Load Power

In electrical engineering, understanding the power consumed by a 3-phase load is crucial. A 3-phase system involves three alternating currents that are out of phase with each other. The power consumed by such a load depends on the voltage, current, and the phase difference between them.

Formula for 3-Phase Power

The total active power (P) consumed by a balanced 3-phase load can be calculated using the line voltage ($V_L$), the line current ($I_L$), and the power factor ($\cos \varphi$). The power factor represents the cosine of the angle ($\varphi$) between the voltage and current phasors.

The standard formula for the active power in a 3-phase system, when using line voltage and line current, is:

$$P = \sqrt{3} \times V_L \times I_L \times \cos \varphi$$

Where:

• $P$ is the active power in Watts (W).
• $V_L$ is the Line-to-Line voltage in Volts (V).
• $I_L$ is the Line current in Amperes (A).
• $\cos \varphi$ is the power factor of the load.

Analyzing the Options

Let's examine the given options in light of the correct formula:

• Option 1:

  3 $V_L$$I_L$ $\cos \varphi$

  - This formula is typically associated with the sum of powers in a single-phase system or relates differently to line/phase quantities in 3-phase systems depending on context, but it's not the standard expression for total 3-phase power using line values.
• Option 2:

  $\sqrt{3}$ $V_L$$I_L$ $\cos \varphi$

  - This matches the standard formula for calculating the total active power in a balanced 3-phase load using line voltage and line current.
• Option 3:

  3 $V_L$$I_L$ $\sin \varphi$

  - This uses the sine of the angle instead of the cosine, which is incorrect for active power calculation.
• Option 4:

  $\sqrt{3}$ $V_L$$I_L$ $\sin \varphi$

  - This uses the sine term, making it incorrect for active power.

Therefore, the correct expression for the power taken by a 3-phase load using line voltage ($V_L$) and line current ($I_L$) is $\sqrt{3} V_L I_L \cos \varphi$.