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ECE 8830 - Electric Drives
Topic 2: Fundamentals of Electric Motors
Spring 2004
Generic Electric Motor
Below figure shows cartoon of induction
motor. Most (but not all) machines have
this structure.
Ref: J.L. Kirtley, Jr. MIT Course 6.11s (June 2003)Course Notes
Generic Electric Motor (cont’d)

Rotor - mounted on shaft supported by
bearings (usually rotor on inside
- but not always)
- shown with conductors but may
have permanent magnets instead
- sometimes just an oddly shaped
piece of steel (variable reluctance
machines)
Generic Electric Motor (cont’d)

Stator - armature (electrical input) on
stator (opposite to DC brush motor)
- on outside with windings


In most electric motors, rotor and stator
are made of highly magnetically permeable
materials - steel or iron.
In many common motors, rotor and stator
are made of thin sheets of silicon steel
(laminations). Punched into these sheets are
slots which contain rotor and stator
conductors.
Generic Electric Motor (cont’d)




Time-varying magnetic fields passing
through ferromagnetic materials
(iron/steel) -> eddy currents to flow
-> energy loss (power dissipation).
Laminations (thin sheets of steel) are
used to minimize eddy current losses.
Windings - many turns of Al/Cu conductor
concentrically wound about a common
axis.
Field winding carries excitation flux.
Armature winding carries electrical power.
Basic Principles of Operation of
Electric Motors



Changes in flux linkage between
rotor and stator creates torque and
therefore relative motion between
rotor and stator.
F=q(vxB)
F= l(ixB)
Basic Principles of Operation of
Electric Motors (cont’d)
Electrical Radians and
Synchronous Speed
Electrical Radians and
Synchronous Speed (cont’d)
frequency of
induced voltage
P
 e   m  p m
2
electrical rads.
P
 e   m  p m
2
electrical rads./sec.
m
P N
N
f 
p
p
2 60
60
2
Hz
where P= # of poles
p=# of pole pairs and
N=synchronous speed of rotor (rpm)
Flux per Pole
Consider a sinusoidally distributed flux
density, B(e)=Bpkcos e. The flux per pole
is given by:
e /2
4
 pole  
B pk cos elRd  m  B pk lR
 e / 2
P
Induced Voltage
Full-pitched coil w/N turns moving
laterally w.r.t. sinusoidal flux density.
Induced Voltage (cont’d)
At t=0 coil’s axis coincides w/flux density
wave peak. Thus, at time t, flux linked by
coil is given by:
 (t )  N pole cos( et )
 induced voltage in full-pitch coil is
given by:
d pole
d
e
N
cos  et   e N pole sin  et
dt
dt
transformer voltage
speed voltage
RMS Value of Induced Voltage
RMS value of sinusoidally varying speed
voltage term is:
Erms  4.44 fN pole
In high power ac machines may have
distributed or short-pitch windings. Use
distribution and pitch factors (kd and kp
respectively) to account for these designs.
The rms value of the induced voltage
under these conditions becomes:
Erms  4.44 fkw N pole
where kw=kdkp is the winding factor.
Distribution Factor
Phase windings may have series/parallel
coils under a different pole-pair. Within
each pole-pair region, the coils of a
distributed winding are spread out over
several pairs of slots.
Distribution Factor (cont’d)
The voltages induced in component
coils for a single phase winding
occupying adjacent slots will be
separated by the slot angle separating
them se (electrical angle subtended by
arc between two adjacent slots.)
Distribution Factor (cont’d)
The distribution factor can be defined as the
ratio of the resultant voltage with coils
distributed to resultant voltage if coils were in
one location, i.e.
kd =
Resultant voltage of coils under one pole-pair |Epole|
Arithmetic sum of coil voltages i |Eci|
If a phase winding has q coils/phase/pole,
|Epole| = 2REsin(qse/2) and |Eci|=2REsin(se/2),
and
e
sin( q s / 2)
kd 
q sin(  se / 2)
Pitch Factor
Short-pitching is when coils with
less than one pole-pitch are used.
Pitch Factor (cont’d)
Short-pitching is used in machines with
fractional-slot windings (non-integral
slots/pole or slots/pole/phase) in a doublelayer winding arrangement. Allows for a
finite set of stampings with a fixed number
of slots to be used for different speed
machines.
Also, short-pitching can be used to
suppress certain harmonics in the phase
emfs. Although short-pitching also offers
shorter end connections, the resultant
fundamental phase emf is reduced.
Pitch Factor (cont’d)
The pitch factor kp is defined by:
kp =
Resultant voltage in short-pitch coil
Arithmetic sum of voltages induced in full coil
With sinusoidal voltages, each coil voltage
is the phasor sum of its two coil-side
voltages. Thus, for coil a, Eca= Ea+E-a
=>
~
Eca
e
kp 
~  cos
2
2 | Ea |
Spatial MMF Distribution of a Winding
A current i flowing through a single coil
of nc turns creates a quasi-square wave
mmf of amplitude F1 given by F1 = nci/2.
Spatial MMF Distribution of a
Winding (cont’d)
The fundamental component of this quasisquare wave mmf distribution is given by:
4 nc
Fa1 
i cos  e
 2
The fundamental component of the airgap
flux density in a uniform airgap machine is:
Fa1 4  0 nc
B1   0

i cos e
g  g 2
where g is the airgap.
Effect of Distributing Phase Coils
Consider a three-phase distributed
winding with 4 coils each per phase in a
2-layer arrangement in a 2-pole stator.
Effect of Distributing Phase Coils (cont’d)
If each coil has nc turns, the sum of the
fundamental mmf components produced
by coils a1 and a2 is given by:
s 

Fa1 a1  Fa 2 a 2  nc i cos e  

2 

where e is the angle measured from the
a-phase winding and s is the angle
between the center lines of adjacent slots.
Similarly for coils a3 and a4,
s 
4

Fa 3 a 3  Fa 4 a 4  nc i cos e  

2

4
Effect of Distributing Phase Coils (cont’d)
Therefore distribution factor is:
s 
s 
s  
s 
1

2
2
kd 
cos  e    cos  e    2 cos e   cos e  
2
2
2
2 
2



 cos
s
2
Effect of Short-Pitching
Consider a layout of windings that are shortpitched by one slot angle. Let’s consider this
to be made of four fictitious full-pitch coils:
(a1,-a3), (a4,-a2), (a2,-a1) and (a3,-a4).
Effect of Short-Pitching (cont’d)
The fundamental mmf component from
these 4 fictitious full-pitch coils is given by:
Fa  Fa1,  a 3  Fa 4,  a 2  Fa 3,  a 4  Fa 2,  a1

1
1


nc i cos e  cos e   s   cos e   s 
 
2
2


4
4

nc i cos e (1  cos  s ) 
4

nc i cos e cos
2
s
2
Effect of Short-Pitching (cont’d)
Comparing this expression to:
Fa 
4

2nc ik d k p cos  e
and allowing for the distribution of the four
full-pitch coils by a kd of cos(s/2), the
factor due to the short-pitching by one slot
angle is given by:
k p  cos
s
2
Effect of Short-Pitching (cont’d)
In general, the expression for the
fundamental mmf component of a
distributed winding with winding factor
kw and a total of npole turns over a twopole region is given by:
4 n pole
Fa1 
ik w cos e
 2
Effect of Short-Pitching (cont’d)
Assuming total phase turns Nph in the Ppole machine are divided equally among
P/2 pole-pair regions, number of turns per
pole-pair = Nph/P. In terms of Nph,
fundamental mmf is:
4  N phk w 
i cos e
Fa1  
 P 
The effective number of full-pitch concentric
coils per pole-pair to achieve this same
fundamental mmf is:
N eff 
2 N ph k w
P
Winding Inductances
Here we derive expressions for self- and
mutual winding inductances for the
elementary machine shown below.
Winding Inductances (cont’d)
Self-inductance of the stator winding,Lss,
with Neffs turns per pole-pair linking pole
(ignoring leakage inductances) is given by:
Lss 
( P / 2) N effs pole
i
4 0 2

N effslR
 g
Similarly, the self-inductance of the rotor
winding, Lrr, with Neffr turns per pole-pair is
given by:
4 0 2
Lrr 
N effrlR
 g
Winding Inductances (cont’d)
An expression for the mutual inductance
between the stator and rotor windings can
be obtained by considering the flux linking
the windings, rs which is given by:
   / 2 4 0 N effs
P
 2 
rs    N effr l 
is cos e Rd  e  
2
 P 
   / 2  g 2
4 0

N effr N effslRi s cos 
 g

rs
4 0
Lrs 

N effr N effslR cos 
is  g
Rotating Fields
The fundamental component of space mmf
for a single-phase winding carrying a
sinusoidal current i=Iacost is given by:
Fa1  Fm1 cos t cos a
 4   N ph 
 I a is the peak value
where Fm1   kw 
   P 
of the fundamental mmf and a is the
electrical angle measured in the counterclockwise direction from the winding axis.
Rotating Fields (cont’d)
This equation may be rewritten as:
1
1
Fa1  Fm1 cos( a  t )  Fm1 cos( a  t )
2
2
Two interpretations:
1) pulsating standing wave
2) two counter-revolving mmf waves of
half the amplitude of the resultant;
forward component rotates counterclockwise, reverse component rotates
clockwise.
Rotating Fields (cont’d)
Rotating Fields (cont’d)
In a three-phase machine the axes of the
windings are spaced 2/3 apart. Assuming
balanced operation (phase currents are of
same magnitude) the currents are given by:
ia  I m cos t
2
ib  I m cos(t  )
3
2
ic  I m cos(t 
)
3
Rotating Fields (cont’d)
The fundamental airgap mmfs of the three
phases are given (in terms of a) by:
1
1
Fa1  Fm1 cos( a  t )  Fm1 cos( a  t )
2
2
1
1
4
Fb1  Fm1 cos( a  t )  Fm1 cos( a  t  )
2
2
3
1
1
4
Fc1  Fm1 cos( a  t )  Fm1 cos( a  t  )
2
2
3
Rotating Fields (cont’d)
The sum of these three winding mmfs is:
3
Fa1  Fb1  Fc1  Fm1 cos( a  t )
2
Therefore the resulting airgap mmfs is a
constant amplitude sinusoidal wave
rotating wave whose peak coincides
with the magnetic axis of the a-phase
winding at t=0 and rotates with a speed
 in a direction corresponding to the
sequence of peaking of the phase
currents.
Torque in a Uniform Airgap Machine
From basic energy conversion principles,
the torque developed in a machine is given
by:
'
T
W fld
 m
|i cons tan t
The co-energy is the complement of the
field energy:
Wfld’ = i - Wfld
Torque in a Uniform Airgap Machine
(cont’d)
Example of three-phase machine
(see text for other approaches/results)