The Square Root of Three (√3) in Electrical Calculations

PDH Course E431 (2 PDH)

The Square Root of Three (3) in Electrical Calculations

David A. Snyder, PE

2014

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PDH Course E431



The Square Root of Three (3) in Electrical Calculations

David A. Snyder, PE

Table of Contents

Introduction..................................................................................................................................... 3 3 Relationship of Three-Phase Voltages....................................................................................... 4

Figure 1 ? 480Y/277V Wye-Delta Voltage Relationship ....................................................... 4 Figure 2 ?Wye-Delta Voltage Relationship ? Right Triangle Geometry ............................... 5 Figure 3 ? 208Y/120V Wye-Delta Voltage Relationship ....................................................... 6 3 In Three-Phase Line Current Calculations ................................................................................ 6 Table 1 ? Types of Loads for Line Current Examples............................................................ 7 Delta-Connected, Three-Phase, Resistance Loads ..................................................................... 7 Balanced, Delta-Connected, Three-Phase, Resistance Loads................................................. 7 Figure 4 ? 30 KW, Balanced, Delta-Connected, Three-Phase, Resistance Load ................... 7 Figure 5 ? Voltage Vab and Current Iab in Delta-Connected, Resistance Load .................... 8 Figure 6 ? Relationships between Three-Phase Voltages and Currents for Resistance Load 8 Figure 7 ? 30 KW, Balanced, Delta-Connected, Three-Phase, Resistance Load at 480 V .... 9 Unbalanced, Delta-Connected, Three-Phase, Resistance Loads........................................... 10 Figure 8 ? 30 KW, Unbalanced, Delta-Connected, Three-Phase, Resistance Load ............. 10 Figure 9 ? 30 KW, Unbalanced Three-Phase Resistance Load at 480 V ............................. 11 Balanced Three-Phase Load as Three Single-Phase Loads ...................................................... 12 Unbalanced Three-Phase Load as Three Single-Phase Loads .................................................. 13 Wye-Connected, Three-Phase, Resistance Loads..................................................................... 13 Balanced, Wye-Connected, Three-Phase, Resistance Loads without Neutral Connection .. 13 Figure 10 ? 10 KW, Balanced, Wye-Connected, Three-Phase, Resistance Load, without Neutral Connection ............................................................................................................... 14 Balanced, Wye-Connected, Three-Phase, Resistance Loads with Neutral Connection ....... 15 Figure 11 ? 10 KW, Balanced, Wye-Connected, Three-Phase, Resistance Load, with Neutral Connection ............................................................................................................... 15 Figure 12 ? 10 KW, Balanced, Wye-Connected, Three-Phase Load at 480 V..................... 16 Unbalanced, Wye-Connected, Three-Phase, Resistance Loads with Neutral Connection ... 16 Figure 13 ? 10 KW, Unbalanced, Wye-Connected, Three-Phase, Resistance Load, with Neutral Connection ............................................................................................................... 17 Figure 14 ? 10 KW, Unbalanced, Wye-Connected, Three-Phase, Resistance Load at 480 V ............................................................................................................................................... 18 3 In Three-Phase Transformer Banks ......................................................................................... 18 Closed-Delta Transformer Banks ............................................................................................. 18 Figure 15 ? Three-Phase Delta-Connected Transformer Banks at 240 V ............................ 19 Figure 16 ? Currents in Three-Phase Delta-Connected Transformer Banks at 240 V ......... 20 Open-Delta Transformer Banks................................................................................................ 21 3 In Three-Phase Voltage-Drop Calculations............................................................................. 22 Figure 17 ? The Square Root of Three in Balanced Three-Phase Voltage-Drop Calculations ............................................................................................................................................... 23 In Closing...................................................................................................................................... 24 Abbreviations ................................................................................................................................ 24

? 2014 David A. Snyder

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PDH Course E431



Additional Reading ....................................................................................................................... 24

Introduction

Everyone knows that we divide the KW or KVA of a balanced three-phase load by the square root of three (3) in order to get the full-load amps required by that load, but many people don't realize why we do that. The square root of three is also used in voltage drop calculations for balanced three-phase loads. The square root of three is the relationship between the two voltages in a 480Y/277V system (Figure 1 on page 4) and between the two voltages in a 208Y/120V system (Figure 3 on page 6). It is the 120? separation between each of the three-phase voltages that is the driving force behind our use of the square root of three in electrical calculations for three-phase systems.

This course presents solutions to three-phase line current problems by using easily-visualized vector addition and subtraction, with some simple mathematics and formulas. For those Readers who prefer more-mathematical solutions, or who want an alternative method for balanced and unbalanced three-phase circuits, the Author suggests PDH Online Course E336 Calculating Currents in Balanced and Unbalanced Three Phase Circuits, listed in the Additional Reading section beginning on page 24.

The use of the square root of three will be discussed in the following applications: relationship of three-phase voltages, three-phase line current calculations, three-phase transformer bank ratings, and three-phase voltage-drop calculations. In this course, phase rotation is assumed to be A-B-C, in the counter-clockwise direction in vector space. Unless otherwise indicated, it is assumed that the line conductors from the power source to the loads have no resistance or reactance. Power factor will be ignored wholesale in this course, such as when calculating available current based on a transformer's KVA rating.

This course has several scaled drawings or figures. When printing a scaled PDF, choose "Actual Size" or a "Custom Scale" of 100% for accurate results. The Reader is encouraged to use a decimal scale or ruler (the decimal edge of a framing square will do, in a pinch) to measure the results illustrated in the scaled figures

? 2014 David A. Snyder

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PDH Course E431



3 Relationship of Three-Phase Voltages

The square root of three is the ratio of the line-to-line (phase-to-phase) voltage (480 V) to the line-to-neutral (phase-to-neutral) voltage (277 V) in three-phase power systems. Figure 1 below illustrates this relationship.

?C

Vbc = 480V Vcn = 277V

Vca = 480V

Neutral

Vbn = 277V

Van = 277V

?B

Vab = 480V

?A

SCALE: 1" = 100 V

0

1"

2"

3"

4"

5"

Figure 1 ? 480Y/277V Wye-Delta Voltage Relationship

Wye-Delta Voltage Relationships

Tsihmepvleolrtiagghet-rterliaatnigolneshgiepobmeetwtrye,enw2h7e7reVtFhweigyheuyparnoedtexn4yu8s0ze0Vis6d2e7lt7a

from Figure 1 can be thought of as V and the adjacent side to the 30?

angle is half of 480 V, or 240 V. The length of the short side opposite the 30? angle is of no

0 concern for1this exercise. 2Figure 2 below3is the bottom p4ortion of Figur5e 1 above. 6

7

? 2014 David A. Snyder

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PDH Course E431



cos(30?) = 0.866 = 240V / 277V Neutral

277V

60? 277V

?B

240V

240V

30? ?A

SCALE: 1" = 100 V

0

1"

2"

3"

4"

5"

FigurWe 2y?eW-Dyee-lDtaelVtaoVltoalgtaegeRReelalattioionnsshhiipps? -RiRghigthTtriTarnagnlegGleesometry As shown in Figure 2, the length of theF2ig40uVresixdye zof0b7oth right triangles is related to the length

of the 277 V sides by the cosine of 30?, which is 0.866. Alternatively, we could have used the 0 60? corner1 for reference 2in Figure 2 an3d stated that th4e relationship 5between 240 V6 and 277 V 7

was defined by: sin(60?) = 0.866 = 240 V / 277 V to get the same result. The value of 0.866 is actually 3 / 2, as one might expect from the voltages shown in Figure 2.

The square root of three also comes into play for the voltage applied to a wye-delta motor. See PDH Online Course E413 Wye-Delta Motor Starters, listed in the Additional Reading section beginning on page 24.

The same square root of three (3) relationship between the line-to-line voltage and line-toneutral voltage holds true for 280Y/120V as shown in Figure 3 below

? 2014 David A. Snyder

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