# Calculating Line-to-Line Voltages from Line-to-Neutral Measurements with the NI 9225

## Overview

This document explains how to calculate line-to-line values on a three phase 240V system from line to neutral measurements.

## Contents

Assume you have a Y-connected 240Vac three-phase system as shown below in figure 1. The NI 9225 is connected between each phase and neutral: Figure 1. 240V 3-Phase System connected to NI 9225 300V AI module

## Meeting the NI 9225 Safety Specifications

Each channel in the NI 9225 will nominally have 240Vrms between the AI+ and AI- terminal. The NI 9225 is rated to 300Vrms from channel to earth, so the measurement is within the safety specifications.  Assuming a perfectly balanced system, each phase is separated by 120 degrees and the line-to-neutral voltage on each phase is the same. As long as these two assumptions hold true, then the magnitude of the line to line voltage is 1.732 * Vp , where Vp is the line-to-neutral voltage. If the line-to-neutral voltage is 240Vrms, then the line-to-line voltage will be 416Vrms. The NI 9225 is rated for 600V ch-ch isolation, so this voltage is within the safety specifications.

## Measuring line-to-line Voltages - The Most General Case

The connection scheme shown above in figure 1 can be used to indirectly measure the line-to-line voltages on a Y-connected systems. The line-to-line voltage is the vector sum of the line to neutral voltages. So, for example, if we know Van, Vbn, and Vcn (defined as in the diagram below), we can find the line to line voltages as follows: The following phasor diagram (diagram 1) will help visualize the phasor math required to convert from line-to-neutral voltages to line-to-line voltages. Diagram 1. Line-to-line calculation phasors

We will start with an example showing how to measure Vab.  As has been described above, the line-to-line voltage is the phasor sum of the line-to-neutral voltages, so Nominally, each phase in a three phase system is spaced exactly 120 degrees apart. However, since our intent is to perform measurements in the system, we cannot assume it to be ideal. Because of this, instead of assuming 120 degrees of separation between phases, we will simply use Θxx to denote the angle between two phases. Van as defined above has zero phase, and Vbn has a phase of Θab.

Breaking up Vbn into rectangular components results in the following: Subtracting the in-phase component of Vbn from Van: From the rectangular components, we can calculate the magnitude of the line to line voltage: To calculate the phase angle Θab, we now find the phase of the rectangular vector calculated above: To illustrate the above, we assume the line-to-neutral voltages are 240V, and the phase to phase angle is 240V.

Vab = 415.7V (as referenced in the first paragraph of this document) Θab = 30 IMPORTANT: The assumption that the line-to-line voltage is sqrt(3) times the line-to-neutral voltage is only valid when the magnitudes of the line-to-neutral voltages are all the same.

## Measuring Line-to-Line Voltages When the Line-to-Line Separation is Exactly 120 Degrees

If we can guarantee that the line-to-line phase is exactly 120 degrees, then the equations above can be somewhat simplified. This may be the case when the power source is a generator with three rotor windings physically 120 degrees apart. If we again look at the example of finding Vab based on Van and Vbn, the magnitude and phase can be calculated as follows: This eliminates the need to calculate sine and cosine functions, although the square root and arctangent operations are still needed. 