In a balanced three-phase system, each of the three phase voltages has the same magnitude but is separated from the others by 120 degrees. We can represent these voltages as vectors, often called phasors, on a complex plane. Using Euler's formula allows us to express these sinusoidal waves in a convenient exponential form.

While phase voltage ($V_{an}$) measures the potential between a single phase and the neutral line, the line-to-line voltage ($V_{ab}$) measures the potential between two different phases. This is the voltage that powers heavy machinery in Wye-connected systems and is found by calculating the vector difference between two phase voltages.

To prove this relationship mathematically, we perform the subtraction using the exponential forms of the phasors.

Now, we convert this rectangular form back to polar form to find its magnitude and phase angle.

This derivation explains the standard voltages found in industrial settings. A system with a 120V phase-to-neutral voltage will have a line-to-line voltage of $120V \times \sqrt{3} \approx 208V$. This is written as 120/208V.

Similarly, a system with a 277V phase voltage (common in large commercial buildings for lighting) provides a line-to-line voltage of $277V \times \sqrt{3} \approx 480V$. This powerful 480V supply is used for heavy-duty motors, HVAC systems, and other large loads.

Let's review the key relationships we've uncovered.

Now, test your understanding of these concepts.

Understanding this core mathematical relationship is fundamental to designing and analyzing any three-phase power system.