machines basics
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Creating rotational
motionInInelectricelectricmachinesmachines
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DefinitionIn the following we deal with
electromechanical converters.
In electric machines we use rotating
magnetic field to create rotational (orlinear) motion.
In this part we summarize knowledgethat is required to understand theconnection of rotating magnetic fieldand rotational motion.
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Topology of rotating machines 2 magnetic field sources, interaction
stator and rotor
windings (coils, electromagnets): excitation airgap
stator
rotor
axleairgap
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Creating torque with magnetsThe effect between two permanent
magnet (or two excited coils) ifthe magneticpolesare thesame, see picture:
This arrangement is heteropolar(alternating poles along the
perimeter).The torquebetween the two
magnets is:
proportional to sinus of theangle between the twomagnetic axis.
sinM
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Creating torqueConclusion:
to reach M = constant we need = constant.
This can be reached if the twomagnets standor rotate withthe same angular frequency.
The later case means that themagnets are relative standstill.
(Common solution:current (and magnetic field) ofone (standing or rotating) coil-system is generated by theother (standing or rotating)
coil-system. )
D
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AC windingAC windings are always
heteropolar.Winding (coil) is placed like:
Winding sides are placed alongthe surface of the stator orthe rotor.
Winding sides are connectedwith end windings and socreate multiturn winding.
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AC windingSymbol of winding:
Axis of winding is the same as axis of the magneticfield inside (generated by its current).
The dimensional relation of the current and themagnetic field can be determined by the method ofright handed helix.
axis
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AC windingWinding sides are affixed by teethand slots:
If we put windings into slots we can reduce the airgapbetween stator and rotor
tooth
slot
airgap
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Meaning of induction vectorWe want to create induction, voltage and current as close to
sinusoidal as possible in rotationg machines.
We need sinusoidal signals because we use multiphasesinusoidal, symmetric system for whichelectric power is
constant.
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Meaning of induction vectorHow can we represent a rotating sinusoidal magnetic wave with one
induction vector?
Conclusion: If the magnetic field (induction) is sinusoidal along thesurface, and if the multiphase currents are sinusoidal, then themultiphase electric power and the torque on the axis of the machine(the mechanical power) is constant in time.
Axis of
machine
Rotating
induction
vector
Endpointcircle of
vector
const
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Field type: constant (unvarying) fielda) A coil excited by direct
current.
b) Coils arranged in 3-phase inspace (120 degrees) andexcited by DC.
Bresultant = 1,5 BA,is also constant
const
const
Vector sum(resultant)
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Field type: pulsating fieldOne coil excited by 1-phase alternating current.
Pulsating field is a standing wave which is the sum of two rotatingfield (in case of linearity)
FERRARIS THEOREM:pulsating field can be broken into tworotating field with same angular frequency and rotating inopponent direction to each other. The amplitude of the rotatingfields is half of the pulsating field.
project
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Poles
p
D
p= 2Is calledPolar pitch
gv p =
Relation betweenelectric (v) andgeometric (g) angle
electric
geometric
p: number of pole pairs
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Field type: rotating fieldTo create rotating field we need:
multiphase coil system
and
multiphase current (excitation) system}
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Field type: rotating fieldGeometric axis currents of coilsin space in time
ahead in space back in time
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Mathematical descriptionInduction wave of phase i:
where Bi(t) is timefunction,Ti(x)is geometricfunction.
Geometric distribution:
B B x t B t T xi i i i= = ( , ) ( ) ( )
T x x
T x x
a
p
b
p
( ) cos
( ) cos
=
=
0 2
3
1 2
3
T x x
c
p
( ) cos=
2
2
3
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Mathematical descriptionTime functions:
EULER theorem:
Transferring cosine functions into exponential functions, forexample for phase b:
=
=
=
3
22cos)(
3
21cos)(
3
20cos)(
1
1
1
tBtB
tBtB
tBtB
mc
mb
ma
cos
= +
e ej j
2
2)
3
22cos(
2
1
aeeaeet
jtjjtj
+=
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Mathematical descriptionFinally we get:
IN CASE OF THREE PHASE:
In case of mphase:
Amplitude of resultant field in general:
),(),(),(),(total txBtxBtxBtxB cba ++=
)(cos
2
3),( 13total
p
m
xtBtxB =
)(cos2
),( 1total
p
mm
xtB
mtxB =
B m
Be f,max , max=2
1
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Properties of rotating fieldI. Wave with constant angular speed along the surface.
If1= const, then the maximum point is:
1t x
p
= 1network angular frequency
0angular freq of the fieldx R t
R
p
R
pp
=
= =
0
2
2
R radius of stator or rotor
1
0t R t
R
p
=
p
p
10
01
=
= 00==
synchronoussynchronous
angularangularfreqencyfreqency
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Properties of rotating fieldIIII 00 is indepentent of phase number
m and n phase systems are equivalent if
In linear case
max,max,22
nm B
nB
m=
m N I n N I m m n n=
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Properties of rotating fieldIII. If we want torque we need the same pole number for
stator and rotor.
The two fields, Bstand Brothave to be in relative standstill:
0,st0,st== 0,r0,r
ppstst== pprotrotThis is required to create torque in excited coil systems.
IV.
where is stator and is rotor field angularfrequency relative to stator or rotor.
st B rot B mech= + So calledfrequency-requirement
Bst Brot
(1=const)
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Creating sinusoidal fieldTo simplify calculation we lay the surface of stator or rotor to a flat plain.
Let us use the law of excition!
IdUk
k
i
imag ===
=
lHH
,
rot
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Creating sinusoidal fieldLet us use the law of excition!
Integral path goes through the
airgap, stator and rotor.If we neglect magnetomotive
force inside iron (iron=) weget the excitation along thesurface, in this case astepped curve.
IdUk
k
i
imag === lH,
x
Induction in airgap:
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Creating sinusoidal fieldWhat happens if we use more
winding turns?
FIELDmore sinusoidal distribution
Advantageous
INDUCED VOLTAGE:
Lower, because there is phaseshift of induced voltage in turnsand the resultant voltage (vectorsum) is lower than thearithmetic sum of the inducedvoltages.
Disadvantageous
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Winding factorOn the left: resultant and induced voltage in case of q=3 turns.
Central and right: if we assume infinite number of turns q=. On the centralpicture the coil of a phase is along 120of the surface (we do not use
it), on the right 60.
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Winding factor
So the winding factor indicates how (the first harmonic of) theinduced voltage is reduced because of the distribution of the
winding
eeU
U= 1 arc
chord=e
3
3
1==e
Winding factor:
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Calculating induced voltageN
z1
1
2=
1)()( zvlxbtu ii =
1max
max,
,2
1
2zvlB
UU i
i
effi ==
!sinusoidalif2
maxaverage BB
=
v Rp
pp p p
= = = =
0 0 0 1
( ) max111111max,
, 44,422
422
22
1
2NflBNfNflB
UU piave
p
iave
i
effi ==
==
zh
N1 = turn ration/phasez1=total number of conductors/phase,zh=z1/Z
Ifq 1max111, 44,4 NfU effi =
0 p
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Frequency-requirement
st B rot B mech= +
This is the basis for all type of electric machines
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