Phase Angle Wheel Model

Three Phase Diagram

Phase Angle (Voltage Angle)

Am

plitu

de

• Three phase electrical power is the most common and economical method for the transmission of alternating current electric power. A three phase electrical system, three circuit conductors carry (alternating currents of the same frequency) reaching their instantaneous peak values at independent and discrete times.

• With one conductor as a reference, the other two currents are delayed by one-third and two-thirds of one cycle of the electrical current. The benefit of this delayed current is that the voltage remains at a constant power transfer over each cycle of the current.

Phase AngleTransmission systems operate at a phase

angle somewhere between the extremes of being purely resistive or purely capacitive

Wheel Model 3D LayoutY

XZ

D1

 

r

Point of Perspective

zyxzyxD

zyxzyxD2220002

1110001

,,point to,,point from Distance

,,point to,,point from Distance

Wheel Model

Unique Properties of Three-phase Systems

The phase currents tend to cancel out one another, summing to zero in the case of a linear balanced load. As a result, it is possible to remove or reduce the size of the neutral conductor. Also, all the phase conductors carry the same current and so can be the same size for a balanced load.

Power transfer into a linear balanced load is constant, which helps to reduce generator and motor vibrations.

Three-phase systems can produce a magnetic field that rotates in a specified direction, which simplifies the design of electric motors.

Phase Angle • The angle by which the sine curve

of the voltage in a circuit element(or a combination of elements) leads or lags the sine curve of the current in that circuit element (or elements) is called the phase angle theta.

• It is useful to think of the phase angle as the angle by which the peak voltage leads or lags the current.

- If theta is positive the voltage leads the current and the circuit is inductive. - If theta is negative the voltage lags the current and the circuit is capacitive.• The phase angle can also be

thought of as the variable

;22

;

fT

RTAN xx cL

Capacitive isCircuit ;0inductive isCircuit ;0

Voltage Wave Lagging Current Wave

When the voltage wave lags the current sin wave, the circuit is more capacitive.

The voltage wave is at its peak or limit, when the current is zero.

The current wave is at its maximum when the voltage wave crosses the X-axis, where the voltage is at maximum change.

Capacitive isCircuit ;0

Current Angle Leading Voltage Angle

When the current cycle leads the voltage cycle the circuit is more capacitive. In a purely capacitive circuit, the current leads the voltage sin wave by 90 degrees.

Voltage Wave Leading Current Wave

When the voltage wave leads the current wave, the circuit is more inductive.

Unity Power Factor

In a purely resistive system that does not have inductive load (such as motors) the current and voltage cycles are in phase and the load would have unity power factor

Voltage Cycle Leading the Current CycleWhen the voltage

cycle leads the current cycle the circuit is more inductive.

In a purely inductive circuit the voltage sin wave will lead the current sin wave at 90 degrees.

Wheel ModelPoint of PerspectiveY

X

Z

D1

D2

YX

Z

• When the wheel rotates the band from 𝜃= 0° to 𝜃= 90°, the band is stretched causing a pulling force on the wheel in the counterclockwise direction.

• The model will show that when the wheel is rotated more than 𝜃=90° the force resisting in the counterclockwise direction will decrease. We can calculate how much force is pulling the wheel counterclockwise if we know the dimensions and we know the elasticity of the bands.

• Similarly, the voltage of a Three Phase AC system will increase from 𝜃=0° and peak at 𝜃=90° and will decrease after 𝜃=90° and reach it’s lowest at 𝜃=0°

Elastic Band

Wheel ModelPoint of PerspectiveY

X

Z

YX

Z

In this example, the wheel is turned 35 degrees in the clockwise direction causing the three bands to stretch and pull on the wheel. The symmetry of the model predicts that the elastic force of each band will be equal. I am focusing on one band for simplicity.

r

Y

X

Z

r

zzyyxxd 121212 :Formula Distance22

zyx 000,,

Distance Formula

zyx 111,,

D1

D2

The Distance formula is used to find how much the band is stretched as the wheel is rotated. The distance formula is useful for finding distances between points in the XYZ Axis.

zyx 222,,

YX

Z

 

Y

X

y2

x2

Pythagorean TheoremYX

Z

The bands rotation through the Z1 Plane travels around the circumference of the circle. The X1 and Y1 parameters can be found using the principles of the Pythagorean Theorem as the wheel is rotated.

Y

X

Z

Fx

Fy

zyx 222,,

)point aat wheelon the band one of torque(the wheelon the Force Tangential

Point of Perspective

Formulas for Stress and Strain of an Elastic Band

Mpa

A

L; L

LL

ΔLε

AFσ

rπ 01.0 Elasticity of Modulus

2Band Elastic of Area Sectional Cross

n)compressioor elongationafter (length length present theis

andlength initial theis whereStrain

Stress

2

Distance FormulaThe Distance

Formula is used to calculate the initial length of the band from point A to point B.

Distance D2 is the distance from point A to a point along the circumference of Wheel # 2 from zero to 90 degrees.

Wheels theof Radius

,,point to,,point from Distance

,,point to,,point from Distance

121212 :Formula Distance

2220002

1110001

222

r

d

zyxzyxDzyxzyxD

zzyyxx

Wheels theof Radius

,,point to,,point from Distance

,,point to,,point from Distance

121212 :Formula Distance

2220002

1110001

222

r

d

zyxzyxDzyxzyxD

zzyyxx

Distance Formula

Horizontal ViewF Y

F T

y2

N

Top View

F X

1x2

) N(Length #1Position at Band

Ft #2Position at Band

Newtons 6901.0

12855.042.14 42.14

42.145.0

12855.0ARCTAN

5.012855.0TAN

TAN

Rotation 40at

2

2

Finding

FFF

x

x

T

TT

Fx

Fx

Fx

Fx

Fx

mSINSIN

mm

N

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 1800.00

5.00

10.00

15.00

20.00

25.00

0.00 0.180.72

1.602.77

4.19

5.81

7.56

9.38

11.23

13.03

14.75

16.33

17.7318.93

19.8920.59 21.02 21.16

0.00 0.06 0.24 0.53 0.92 1.401.94

2.523.13

3.744.34

4.925.44 5.91 6.31 6.63 6.86 7.01 7.05

0.00 0.01 0.10 0.31 0.691.23

1.902.66

3.444.17

4.78 5.19 5.34 5.204.71

3.902.79

1.46

0.000.00 0.00 0.02 0.09 0.26 0.591.14

1.922.95

4.17

5.54

6.98

8.40

9.7410.92

11.9012.62 13.07 13.22

ELASTIC FORCE VS ANGLE OF ROTATIONTotal-

BandForce (N)Force of One Band (N)Fx (Resultant Force)Fy Resultant Force

ANGLE OF ROTATION (DEGREES)

ELA

STIC

FO

RC

E (N

EWTO

NS)

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 1800.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.00 0.020.06

0.14

0.25

0.39

0.54

0.70

0.83

0.920.96

0.92

0.82

0.68

0.51

0.34

0.17

0.00

TORQUE VS ANGLE OF ROTATION

TORQUE (NEWTONS * METERS)

ANGLE OF ROTATION (DEGREES)

TOR

QU

E (N

EWTO

NS

X M

ETER

S)