Rotating Magnetic Fields

8/9/2019 Rotating Magnetic Fields

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Rotating Magnetic Fields

1 Physical Structure of Rotating Magnetic Field Machines

All rotating field machines have a stationary member, called the stator , which is a hollow circular

cylinder of laminated magnetic material, typically silicon steel. Usually, but not always, the inner

surface of the stator has slots into which the conductors forming the stator windings are placed. A round

rotor machine has a cylindrical rotor as in Fig.1-1(in most practical cases the rotor is also slotted).

Typically the effect of the slots on the shape of the main magnetic field is negligible and the machine of

Fig 1-1 is considered to be a uniform air gap machine. The length of the effective air gap is usually

increased from the physical air gap to compensate for the average effect of the slots. There are also

rotating field machines that have a salient pole rotor and a non-uniform air gap as in Fig 1.2 but the

following material will be confined to uniform air gap machines.

Slots

Teeth

Yoke

Winding

PhysicalAir Gap

NS

haft

otoroke

Fig. 1-1 Idealized Round-Rotor Rotating Magnetic Field Machine

2 Air Gap Magnetic Field Spatial Distribution

A major function of the stator winding is to create a rotating magnetic field in the air gap which

moves around the air gap at a uniform speed called synchronous speed. As a first step in understanding

how this is accomplished, consider the determination of the magnetic field created by a single N turn

winding where all N-turns are concentrated in two slots as shown in Fig. 2.1. The fundamental laws

governing the field are Ampere's Law and Gauss' Law. An appropriate integration path for Ampere's

Law is illustrated in Fig.2.1. The path crosses the air gap at two points 180˚ apart where Gauss' Law and

symmetry dictate that the magnetic field has the same amplitude but opposite sign (directed from rotor

to stator at one crossing and from stator to rotor at the other). In most practical cases the permeability


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ofthe iron is so much greater than air that the contribution of the portions of the integration path in the

iron are very small compared to the two air gaps. In addition the high permeability of the iron causes the

Slots

Teeth

Yoke

Winding

PhysicalAir Gap

+

otoroke

Shaft

Fig. 1-2 Idealized Salient Pole Rotating Magnetic Field Machine

+

.

+

.

Ampere’s LawIntegration Path

StatorYoke

RotorYoke

Air Gap Length = g

N-Conductors

N-Conductors

θ

Fig. 2-1 Idealized Salient Pole Rotating Magnetic Field Machine

field intensity H and flux density B to be perpendicular to the iron surfaces and hence parallel to the

integration path. Under these conditions Ampere's Law results in

Ni = 2H(θ)g = 2mmf g(θ) 2-1

where

i = winding current (may be time varying) (amp)

H(θ) = H-field in the air gap at the angle θ (amp-turns/meter)


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mmf g(θ) = air gap mmf at the angle θ (amp-turns)

The mmf is widely used in machines work since it is so directly related to the winding turns and current.

It is a spatially varying quantity and if the current is time varying it also varies with time. For the

concentrated coil winding in Fig 2-1 the spatial variation is readily determined by allowing theintegration path to move to different values of θ without changing shape. In this simple winding it is

clear that different values of θ all yield the same result and that the resulting spatial variation is the

rectangular wave shown in fig 2-2.

+

.

.

+N-Conductors

N-Conductors

Ni/2

θ

m m

f

π

Fig. 2-2 Spatial Distribution of MMF for Two Pole Concentrated Coil Winding

+

..

N/2-Conductors N/2-Conductors

Ni/2

+

..

N/3-ConductorsN/3-Conductors

Ni/2

+

.

N/3-Conductors

Ni/6

m m

f

m

m f

θ

θ

2π

2π

Fig. 2-3 Spatial Distribution of MMF for Two Pole Distributed Windings


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While concentrated coil windings are used in some machines, most rotating field machines use

distributed windings to reduce the spatial harmonic content of the resulting mmf wave. Two examples

are shown in Fig 2-3.

3 Multipole Windings

All of the windings in section 2 have only two changes in the direction of the mmf over the full 2π

radians of rotor circumference. Each of the two regions where the mmf has one polarity are called

magnetic poles, one a north pole and the other a south pole, and the windings are referred to as two pole

windings. It is possible to create windings that have larger numbers of poles with pole numbers as large

as 40 or more being useful in some types of machine. With respect to rotating fields, the pole number

(number of poles) is a primary quantity controlling the speed of rotation of the rotating magnetic field

with higher pole numbers creating slower rotation. Figure 3-1 illustrates two simple examples of 4-pole

windings. Several properties of the spatial mmf distribution of symmetric windings in uniform air gap

machines are illustrated by the examples in Figs 2-2, 2-3 and 3-1 and summarized below:

1) Windings with any even number of poles are possible by simply repeating the two pole windingpattern as many time as needed.

2) To repeat the winding pattern requires shortening the pitch (the distance between the two sides of

a winding coil) of the winding. The angular span of one pole, called the pole pitch, is

pole pitch =2πP

(radians)

where P is the pole number.

+N-Conductors

N-Conductors

i/4 m m

f

2π

+

.

.

N/3-Conductors N/3-Conductors

Ni/4

+

.

N/3-Conductors

Ni/12

m m

f

2π

..

+

..

.

+ ++


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Fig. 3-1 Spatial Distribution of MMF for Four Pole Windings

3) The spatial peak of the mmf wave is equal to Ni/P, the total number of turns times the current/

total number of poles.

4 Winding Representation using Fourier SeriesAll of the mmf distributions in rotating machines are periodic functions, at the very least repeating

themselves over 2π radians at where one has returned to the starting point. In higher pole number

machines the mmf wave repeats each two poles and thus has a period given by

period =2πP/2

=4πP

= two pole pitches 4-1

Most often when dealing with periodic functions it is useful to utilize the concept of the Fourier Series,

which, simply stated, says that any periodic function can be expressed as an infinite sum of sinusoids.The period of the first term, called the fundamental component , is equal to the period of the wave being

represented. The remaining terms, called the harmonics have shorter periods, falling off inversely as the

order of the harmonic (i.e. the second term, called the second harmonic, has a period one half that of the

fundamental). Figure 4-1 illustrates the waveforms resulting from adding in the first five terms of the

Fourier series of a rectangular wave of period 2π.

Fundamental

Fundamental + 3rd

Fundamental + 3rd + 5th


+ 7th


+ 7th + 9th


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Fig. 4-1 Truncated Fourier series for Rectangular Wave

The equation for the five term series in the figure is

mmf(θ) = sin θ +

1

3 sin 3θ +

1

5 sin 5θ +

1

7 sin 7θ +

1

9 sin 9θ 4-2

and the amplitude of the rectangular wave being approximated is π /4 = 0.785. Note how the addition of

each harmonic carries the sum more and more toward a rectangular shape and how for some purposes

one might be able terminate the series with little loss of information.

The calculation of the amplitude and phase of the fundamental and harmonics of a specific wave is a

process outlined in most basic EE textbooks and will not be explained here. In fact, this calculation

process is not of direct concern for our goal of understanding rotating fields. For our immediate

purposes it is sufficient to simply recognize that the winding mmf of any winding can be expressed interms of the fundamental and harmonics of a Fourier series.

5 Sinusoidal Excitation of a Single Phase Winding

To initiate the concept of a rotating mmf wave, consider a single winding excited by a sinusoidally

time varying current. To allow generally applicable results, the winding will be characterized by the

fundamental Fourier series component allowing one to write the expression

mmf(θ,t) = Nf Im cosP2θ cos ωet 5-1

where

Nf Im = peak value of fundamental component of winding mmf

Im = maximum value of winding current

P = pole number

ωe = 2πf = electrical radian frequency of source

In this form the mmf is a stationary, spatially sinusoidally distributed wave with an amplitude varying

sinusoidally in time. Or in short, a stationary, pulsating wave.

This stationary wave can be reinterpreted as two oppositely traveling waves by using the following

identity for the product of two cosine waves

cos A cos B =12 cos (A – B) +

12

cos (A +B) 5-2

In engineering terms this expression states that a sine wave of frequency ωA, sinusoidally


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amplitude modulated at a frequency ωB gives rise to two sine waves of frequencies ωA + ωB and ωA –

ωB, often called the upper and lower sideband frequencies. Applying 5-2 using the symbols of 5-1

yields

mmf(θ,t) =Nf Im

2 cos(

P2θ + ωet) +

Nf Im2

cos(P2θ – ωet) 5-3

Viewed as spatial waves (i.e., as functions of θ) the two components of 5-3 are sinusoidal waves

with fixed amplitudes (Nf Im

2 ) and which have steadily increasing phase angles (ωet). A sine wave

with a fixed amplitude and an increasing phase angle is a traveling wave or, in our case a rotating

magnetic field. Clearly the two components move in opposite directions since the sign of the terms is

opposite for the two components. The speed of rotation can be determined from the observation that at

the peak of the sinusoid the argument of the is zero. Thus, at the peak of the second term

(P2θ – ωet) = 0 5-4

Differentiating, to find the speed dθ /dt

(P2

dθdt

– ωe) = 0 5-5

and solving for the speed

dθdt

=2P

ωe 5-6

implying that the wave described by the second term rotates forward. The first term yields the same

result except for a minus sign and represents a backward rotating wave.

Forward Wave = 1.0 cos(θ – ωet)Backward Wave = 1.0 cos(θ + ωet)

Total Wave = 2.0 cos θ cos ωet

Fig. 5-1 Forward and Backward Waves at ωωωωet ≈≈≈≈ 45˚


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Figure 5-1 gives a snapshot of the stationary pulsating wave and the forward and backward waves

at ωet equal to approximately 45˚. It is difficult to visualize how the two moving waves can sum up to

the stationary wave.

Start Movie

To assist in understanding this relationship a short movie entitled " Fwd-Bkwd Waves Movie"

has been prepared. Open the movie from the course website and use the cursor to change the value of

ωet) and observe the following points:

1) When the two traveling waves are aligned the pulsation is at its peak and equal to twice the traveling

wave peak..

2) When the two traveling waves have each moved 90˚ the pulsation is zero.

3) When the two traveling waves have each moved 180˚ the pulsation is at its negative peak and equal to

twice the traveling wave peak.

4) Run the movie in the automatic mode and observe how the two traveling waves model the pulsating

wave.

END

6 Influence of Pole Number on Rotation Speed

The speed of rotation of the forward wave is given in 5-6 and repeated here

dθdt

=2P

ωe (radians/sec) 6-1

Expressed in rpm the speed equation becomes (with ωe = 2πf e)

dθdt

=2P

2πf e602π

=120f e

P (rpm) 6-2

This are extremely important results illustrating that the speed of rotation of the magnetic field can be

controlled by varying the frequency of the excitation (f e) or by designing for different pole numbers.

Since the pole number must be an even integer only certain discrete speeds are attainable at any constant

frequency. For example, at 60 hz the attainable rotating field speeds are equal to 7200/P or 3600 rpm,

1800 rpm, 1200rpm, 900 rpm, 720 rpm, and so on.


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Start Movie

The mechanism by which the pole number influences the rotation speed is illustrated in a movie

called "Wave Speed Movie". The two waves depicted in the movie are

mmf 2pole = cos (θ – ωet) (blue) mmf 4pole = cos (2θ – ωet) (red) 6-3

Open the movie from the course website and use the cursor to change the value of ωet to observe and

study the following points:

1) Moving the two pole field marker (blue) from zero to 6.28 (2π), only moves the four pole field

marker (red) half as much to 3.14.

2) Note that moving the two pole field two pole pitches, also moves the four pole field two pole pitches.

Since the four pole, pole pitch is smaller (half as much), the four pole field moves a smaller distance.

3) Run the movie in the automatic mode and observe how the two pole field "outruns" the four pole

field.

END

7 Two Phase Rotating Field

Although single phase excitation producing both forward and backward rotating waves does have

application in single phase induction motor s, eliminating the backward rotating field to create a pure

forward rotating wave is far more effective. This is accomplished by adding one or more windingshaving their magnetic axes shifted in space and excited with currents shifted in time. The simplest (and

oldest) arrangement is the two phase winding consisting of two windings one half pole pitch apart in

space excited by two currents 90˚ apart in time.

mmf a(θ,t) = Nf Im cosP2θ cos ωet 7-1

mmf b(θ,t) = Nf Im sinP2θ sin ωet 7-2

Expanding each of these into forward and backward waves will show that the forward waves add and the

backward cancel to yield the rotating magnetic field

mmf a(θ,t) + mmf b(θ,t) = Nf Im cos (P2θ – ωet) 7-3


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Note that each winding has P-poles and that the sum of the two also has P-poles and that the two

windings carry currents which are 90˚ out of phase. Note that reversing either of the two currents will

cause the resultant wave to rotate in the opposite direction.

Start Movie

The mechanism by which the individual winding fields combine to produce the rotating field is

illustrated in a movie called "2 Ph Rot Fld Movie". The waves depicted in the movie are

mmf a = 1.5 cos θ cos ωet (blue) mmf b = 1.5 sin θ sins ωet (red) 7-4

mmf a + mmf b= 1.5 cos θ cos ωet + 1.5 sin θ sinsωet (black) 7-5



1) When the axis of the total mmf (black vertical line) lies on the axis of one of the windings, all of the

rotating field mmf comes entirely from that winding. The amplitude of the rotating wave is therefore

equal to the peak value of either one of the windings.

2) When the axis of the total mmf is in between the winding axes both windings contribute to the total

mmf.

3) Run the movie in the automatic mode and observe how the two pulsating fields combine to produce

the rotating wave.

END

8 Three Phase Rotating Field

The two phase winding is quite adequate for production of a rotating field and was widely used in

the distant past. However, because of its superiority for power transmission, the three phase system is

nearly universally used. Using the same concept of arranging windings to cancel the backward fields, a

three phase winding employs three windings spaced two thirds pole pitch apart in space and three

currents 120˚ apart in time as follows

mmf a(θ,t) = Nf Im cosP2θ cos ωet 8-1

mmf b(θ,t) = Nf Im cos (P2θ – 120˚ ) cos (ωet – 120˚ ) 8-2

mmf c(θ,t) = Nf Im cos (P2θ + 120˚ ) cos (ωet + 120˚ ) 8-3


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The trigonometry is more complicated but the result is similar to the two phase case

mmf a(θ,t) + mmf b(θ,t) + mmf c(θ,t) =32 Nf Im cos (

P2θ – ωet) 8-4

the only difference being the 3/2 multiplier in the amplitude. Again reversing one current will reverse

the direction of the resultant rotating field.

Start Movie

The mechanism by which the individual winding fields combine to produce the rotating field is

illustrated in a movie called "3 Ph Rot Fld Movie". The waves depicted in the movie are

mmf a = cosθ

cosω

et (blue) 8-5

mmf b = cos (θ – 120˚ ) cos (ωet – 120˚ ) (red) 8.6

mmf c = cos (θ – 120˚ ) cos (ωet – 120˚ ) (green) 8.7

mmf a + mmf b + mmf c = sum (black) 8-8



1) When the axis of the total mmf (black vertical line) lies on the axis of one of the windings, there are

still contributions from all three phases. Verify that the amplitude of the rotating wave is equal to 3/2

times the peak value of either one of the windings.

2) Find a condition where only two phases contribute to the total.

3) Run the movie in the automatic mode and observe how the three pulsating fields combine to produce

the rotating wave.

END

9 Influence of Harmonics on the Rotating Field

The constant amplitude rotating fields of the preceding sections result from sinusoidally distributed

magnetic fields and are generally thought of as representing the fundamental component of the magnetic

field of real windings. To obtain the total magnetic field of a two phase or three phase the harmonics

must be added in. Because the harmonics are also sinusoids (of higher pole number) they also create


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rotating fields or, in the case of the third and other triples in three phase windings, add to zero. The

rotation direction can be the same or opposite to the fundamental component.

Start Movie

An example of a three phase rotating field with fundamental, fifth and seventh harmonics is shown inthe movie called "ND +5th + 7th Movie". The amplitudes are 1.0, 0.1, and 0.075 for the fundamental,

5th and 7th harmonics.


study the influence of harmonics on the rotating field. Run the movie in the automatic mode and

observe the motion of the distorted wave .

END

In the design of rotating field machines the harmonic content of the individual phase windings is

controlled to produce an acceptable rotating field. In general the space harmonics have undesirable

effects on machine performance. However, the winding changes required to reduce the space harmonic

content also produce undesirable effects. This forces a compromise in the design process, a situation

that occurs in almost all types of engineering design.

10 Summary

1) The mmf spatial distribution of a symmetric winding on a uniform air gap machine:a) can be found using Ampere's Law and symmetry,

b) has a waveform that depends on the conductor distribution,

c) has a peak value equal to the turns/pole times the winding current,

d) has a pole pitch =2πP

(radians)

e) is usually represented by its Fourier Series components - fundamental plus harmonics.

2) A single sinusoidal winding excited with sinusoidal current can be viewed as

a) as a stationary, pulsating mmf

mmf(θ,t) = Nf Im cosP2θ cos ωet 5-1

b) or as two counter rotating mmf's of half amplitude

mmf(θ,t) =Nf Im

2 cos(

P2θ + ωet) +

Nf Im2

cos(P2θ – ωet) 5-3


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3) The speed of a rotating mmf wave is

dθdt

=2P

ωe (radians/sec) =120f e

P (rpm) 6-1,2

and depends

a) directly on the excitation frequency

b) inversely on the pole number

c) at 60 hz the attainable rotating field speeds are equal to 7200/P or 3600 rpm, 1800 rpm,

1200rpm, 900 rpm, 720 rpm, and so on.

4) A pure forward rotating field can be attained using

a) a two phase winding consisting of two windings with their axes one half pole pitch apart

and excited with two currents 90˚ apart,

b) a two phase winding consisting of three windings with their axes two thirds pole pitch

apart and excited with three currents 120˚ apart,c) in either case the rotation direction can be reversed by reversing one current.

5) The winding mmf harmonics also create rotating fields which distort the fundamental

component rotating field. The extent to which the harmonics must controlled is a basic design

issue.

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Rotating Magnetic Fields