machines basics

8/10/2019 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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