# Line to Line voltage calculation in a Three-Phase supply

In a three-phase electrical system, the voltage between two phase lines, often referred to as the line-to-line voltage or simply line voltage, is of significant importance. This article explains into the formula derivation and mathematical proof of how this voltage is derived.

## Understanding Three-Phase Systems

A three-phase system is a common type of polyphase system, widely used in power generation and transmission. It consists of three sinusoidal voltages of the same frequency but with a phase difference of 120 degrees.

Let’s denote the three phase voltages as:

\(V_a = V_m \cos(\omega t)\)

\(V_b = V_m \cos(\omega t – \frac{2\pi}{3})\)

\(V_c = V_m \cos(\omega t + \frac{2\pi}{3})\)

where:

- \(V_m\) is the maximum value (amplitude) of the phase voltage,
- \(\omega\) is the angular frequency, and
- \(t\) is time.

## Deriving the Line-to-Line Voltage

The line-to-line voltage is the voltage difference between any two phase voltages. Let’s calculate the line-to-line voltageV_{ab}between phasesaandb:

\(V_{ab} = V_a – V_b\)

\(V_{ab} = V_m \cos(\omega t) – V_m \cos(\omega t – \frac{2\pi}{3})\)

Applying the cosine difference identity, we get:

\(V_{ab} = V_m [\cos(\omega t)\cos(\frac{2\pi}{3}) + \sin(\omega t)\sin(\frac{2\pi}{3})]\)

Simplifying, we find:

\(V_{ab} = \sqrt{3} V_m \cos(\omega t – \frac{\pi}{6})\)

This shows that the line-to-line voltage is \( \sqrt{3} \) times the phase voltage and has a phase angle that is \( 30^\circ \) ahead of the corresponding phase voltage.

In a three-phase system, the line-to-line voltage is a crucial parameter for power transmission and distribution. Its value, \( \sqrt{3} \) times the phase voltage, is derived using basic trigonometric identities. This mathematical proof provides a deeper understanding of the relationships between voltages in a three-phase system.

Eg:- if the phase voltage, voltage between phase and neutral = 230V, then the line voltage = \( \sqrt{3} \) x230 =398.371V.