A 3 ϕ, four-pole, Y-connected synchronous motor is operating...

A $3 \phi$, four-pole, $Y$-connected synchronous motor is operating at a rated voltage of $550 \mathrm{~V}$ and the input current is $\mathbf{I}_s=45 /-25^{\circ} \mathrm{A}$. Find the three-phase power developed by the machine.

Key Concepts

Power Factor and Phase Angle

The power factor is defined as the cosine of the phase angle (?) between the voltage and the current. It represents the portion of the power that is being effectively converted into work. A phase angle indicating a lag (or lead) in current relative to voltage is critical in power calculations because it directly affects the real (active) power; a lower power factor means less of the apparent power is being used for useful work.

Synchronous Motor Operation

Synchronous motors operate such that their rotor speed is synchronized with the frequency of the supply current. This synchronization is essential for applications requiring constant speed operation. Understanding the operating principles of synchronous motors is important for analyzing how electrical power is converted to mechanical power and how system specifications, like input current and voltage, relate to the performance of the motor.

Three-Phase Power Calculation

In balanced three-phase systems, the total real power is given by the formula P = ?3 × V_L × I_L × cos(?), where V_L is the line voltage, I_L is the line current, and ? is the phase difference between voltage and current. This relationship is central to AC power computations as it accounts for the vector nature of the currents and voltages in each phase, enabling the determination of power by taking into account both magnitude and phase relationships.

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