Line-to-Line vs Line-to-Neutral Voltage: Formula and Examples

The short answer: identify both measurement points

Line-to-line voltage is measured between two line conductors. Line-to-neutral voltage is measured between one line and the neutral. In an ideal balanced star system, line-to-line voltage is √3 times line-to-neutral voltage. “The voltage is 415 V” is incomplete unless the measurement points are understood.

For an ideal balanced system:

VLL = √3 × VLN

If line-to-line voltage is 415 V, the ideal line-to-neutral magnitude is about 415 / 1.732 = 239.6 V. At 400 V line-to-line, the same calculation gives about 230.9 V. Check the 415 V star example in the line/phase converter.

Label	Measurement points	Example in an ideal balanced 400/230 V system
VLL	One line to another line	400 V
VLN	One line to neutral	400 / √3 ≈ 230.94 V
VLG	One line to protective earth or ground reference	Requires its own measurement and system context

Line-to-ground is not automatically interchangeable with line-to-neutral. They can be close in some correctly functioning grounded systems, but grounding arrangement, conductor impedance and faults affect what is measured. The simple √3 relationship is between balanced line-to-line and phase-to-neutral magnitudes, not a rule for an arbitrary ground reading.

Which value belongs in the current formula?

The common balanced three-phase expression is:

P = √3 × VLL × IL × PF

It expects line-to-line voltage. If you enter line-to-neutral voltage, the current result will be too high by a factor of √3.

A single-phase line-to-neutral load uses its actual terminal voltage in P = V × I × PF; no extra √3 factor is added.

Voltage across the equipment phase

Connection changes the voltage experienced by each load phase. In star, each phase is connected line to star point, so its phase voltage is the line-to-neutral magnitude for a balanced system. In delta, each phase is connected across two lines and sees line-to-line voltage.

That distinction matters for motor windings, heater elements and transformer connections. The supply description alone does not show how the internal phase is connected.

Nominal and measured voltage

Nominal voltage is a system designation. The voltage at a load can differ because of source regulation, conductor drop, transformer tap position and changing load. Use nominal values for a clearly labelled preliminary estimate. Use measured values from the same operating condition when assessing actual performance.

Do not combine measured current from one phase with a voltage taken at a different time or point and expect a high-quality power estimate.

A checklist before calculation

1. Identify the two points between which voltage is defined.
2. Confirm whether the load is single phase or balanced three phase.
3. Confirm star, delta or another equipment connection where phase quantities matter.
4. Label the value VLL or VLN in notes and schedules.
5. Use the formula written for that voltage definition.

The arithmetic is short. Most errors happen one step earlier, when the correct number is attached to the wrong voltage label.

Measurement and documentation habits

Before measuring, use appropriate procedures, equipment ratings and competent personnel. Record the two points, not merely the number: “L1–L2 = 412 V” or “L1–N = 238 V”. On a three-phase supply, record all relevant line-to-line and line-to-neutral values if balance matters.

In drawings and calculation notes, use explicit labels such as 415 V VLL and 240 V VLN. That small addition makes calculator inputs reviewable and prevents a factor-of-√3 mistake from surviving into cable, transformer or protection calculations.

Why 415 V divided by √3 is not 230 V

The balanced relationship gives 415 / √3 = 239.60 V, approximately 240 V. Starting instead with 400 V gives about 230.94 V. Do not mix the 415/240 and 400/230 nominal descriptions in one calculation and expect exact equality. Use the actual voltage reference relevant to the equipment and operating condition.

In a balanced delta-connected load, each winding is across two lines, so its phase voltage equals VLL. Delta does not create a neutral point simply because the word “phase” appears in the calculation.

Try the line/phase converter. It also covers line and winding current without confusing the two quantities.

For the related current and neutral questions, follow the three-phase learning route.