How to Calculate Line to Line Voltage

The myriad uses for electricity means it can take different forms. You may wonder how the electricity supplied to your house differs from the electricity of power plants. Studying the properties underlying electrical signals lets you figure out how features like line to line voltage emerge. This can help you better understand the forms electricity takes throughout the world.

Three-phase voltage

While single-phase power sources are much more prevalent around the world, electrical power sources that take the form of three phases can be found in electrical generators. This lets power stations produce three times as much electricity than they would otherwise as they send electricity across three wires instead of two.

Though you won't be using it in your home, industrial purposes include motors and other devices that take advantage of the smooth nature of 3 phase voltage.

The 3 phase voltage calculation formula shows you how to quantify this voltage. For three wires, a, b and c, the line to line voltages are ​vab​, ​vbc​ and ​v​​ca​ to represent the changes across the wires from the first subscript to the second. For example, ​vab​ is the difference from wire a to b.

The line to line voltage is the voltage or potential between two wires. For two voltage values that share a common wire, you can compare them as


or, adding the two voltages as


The notation for these differences in voltage can let you calculate phase to earth voltage. This is the voltage difference between a certain phase of the 3 phase voltage power source and the earth, or ground. If you know the voltage between one phase a and the earth as well as between wire b and wire a, you can denote the former as ​vae and the latter as vba​. You can use that to calculate the phase difference of another wire b and the earth as


Thyristor Rectifier Example

A ​thyristor rectifier​ may have input line to line voltages of

\begin{aligned} &v_{ab}=\sin{(\omega t)}\\ &v_{bc}=\sin{(\omega t-120)}\\ &v_{ca}=\sin{(\omega t-240)}\end{aligned}

for angular frequency "omega" ω = 2πf and frequency f across time t. Frequency measures how many waveforms of the input electrical power source pass over a given point each second. These rectifiers are used when switching between power sources of heavy electric loads.

The circuit diagram of six thyristor devices shows their arrangement in two rows of three to switch between each of the three wires in one direction or the other. The differences of 120​°​ indicate each wire is out of phase with the other wires by 120​°​ in one direction and 120​°​ in the other direction.

Line to Line Current Formula

Just as you can write the voltage drops across various parts of three-phase voltage devices, use ​Ohm's Law​ ​V = IR​ for voltage ​V​, current ​I​ and resistance ​R​ to rewrite the voltages and currents. In the case of three-phase voltage circuits, however, you measure impedance instead of resistance. This means you can rewrite a certain voltage drop between two points x and y as ​vxy​. This is, then, equal to ​Ixy x Zxy​ for current between and impedance of the the two points.

Using three-phase voltage sources mean you should be aware of and take into account the phase of the voltage for different elements of an electrical circuit. You can use line to line voltage to illustrate these relationships.



About the Author

S. Hussain Ather is a Master's student in Science Communications the University of California, Santa Cruz. After studying physics and philosophy as an undergraduate at Indiana University-Bloomington, he worked as a scientist at the National Institutes of Health for two years. He primarily performs research in and write about neuroscience and philosophy, however, his interests span ethics, policy, and other areas relevant to science.