3. mathematical analysis of air-gap fields · 3. mathematical analysis of air-gap fields . ... =...

Institut für Elektrische

Energiewandlung • FB 18

TECHNISCHE

UNIVERSITÄT

DARMSTADT

Prof. A. Binder : Electrical Machines and Drives

3/1

3. Mathematical Analysis of Air-Gap

Fields



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3/2

Rotating waves - Travelling waves Rotating wave: x is stator circumference co-ordinate (rotating machine)

Travelling wave: x is stator linear co-ordinate (linear machine)

tf

xBtxB

p

2cosˆ),( 11

Wave velocity: Observer, who is moving with the

wave, sees constant argument of cos() = const.

pp

syn fftconstdt

d

dt

dxv

2)2.(

Wave in opposite direction: )2cos(ˆ),( 11 tfx

BtxBp

psyn fv 2

Example:

At frequency f = 50 Hz: vsyn in m/s is SAME number as pole pitch in cm:

e.g. 2-pole turbine generator (2p = 2) in thermal power plant: nsyn = 3000/min:

- stator bore diameter dsi = 1.2 m, pole pitch p = 1.2/2 = 1.88 m = 188 cm

- vsyn = 188 m/s = 676 km/h = rotor surface velocity, as rotor is spinning

synchronously with rotating stator magnetic field wave (synchronous machine !)

cmp

smsynv /



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3/3

Pulsating fields – standing waves

Standing wave is pulsating field: The location of wave nodes and wave amplitudes

is fixed, but the amplitudes are pulsating.

At time t = 0: Maximum of wave at x = 0 has amplitude

at t = T/8: ,

at t = T/4: amplitude is zero !

at t = T/2: and so on. )2cos(cosˆ),( 11 tf

xBtxB

p

2/ˆ1B

1ˆB

1ˆB



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3/4

FOURIER-series: A periodical function V() with period 2 may be described

by an infinite sum of sine & co-sine functions with decreasing wave length.

Ordinal numbers:

Amplitudes: ,

Average value:

Magnetomotive force (MMF) of air gap field:

a) NO UNIPOLAR flux: V0 = 0

b) MMF V symmetrical to abscissa: NO even ordinal numbers

c) By choosing origin so, that MMF V is even function V() = V(-):

NO sine-wave functions occur in FOURIER sum.

FOURIER-Analysis: Determining fundamental & harmonic waves

,...3,2,1

,,0 )sin(ˆ)cos(ˆ)(

ba VVVV

....,3,2,1

2

0

, )cos()(1ˆ dVV a

2

0

, )sin()(1ˆ dVV b

2

0

0 )(2

1dVV



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3/5

Magnetomotive force Vc(x) is block-shaped function, slot-Ampere-turns Q = 2Ncic

Circumference angle : :

Example: Four pole machine: Half circumference = in "mechanical degrees",

equals two pole pitches x = 2p , thus =2, in "electrical degrees"

FOURIER-Analysis of field of one full-pitched coil (q = 1)

px /

,...5,3,1

, )cos(ˆ)(

cc VV

, 2

sin4

2)cos()(

2ˆ

0

,

Q

cc

ΘdVV ....,5,3,1



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3/6

Compare MMF function´s FOURIER series with that of full-pitched coils: Amplitudes are

smaller by the ”pitch coefficient" kp, :

Example: W/p = 5/6 (q = 2): = 1: kp1 = 0.966, = 5: kp5 = 0.259

FOURIER-Analysis of field of one pitched coil (q = 1)

... ,5 ,3 ,1 ,2

sin4

2)cos()(

2ˆ

0

,

p

Q

cc

WΘdVV

2sin,

pp

Wk



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3/7

Magnetomotive force of field of full-pitched coil group: = 1, 3, 5, ...

”Distribution coefficient„

Example: q = 2

MMF amplitude: qQ/2 = Q

With relationship we get

FOURIER-Analysis of field of a full-pitched coil group q > 1

)2

sin(

)2

sin(

2sin

4

2)cos()(

2ˆ

0

,

mqq

mqdVV

Qgrgr

)2

sin(

)2

sin(

,

mqq

mkd

,,

2sin

4

2ˆ

22

2

2dgr

cccc

Qk

p

iNV

p

iN

p

ai

a

pqNiqN

qΘ



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3/8

FOURIER-analysis of MMF of one phase Vphase() :

Phase current

The MMF distribution (and hence the air gap field) is a sum of

pulsating, standing waves („Pulsating field“).

"Winding coefficient"

FOURIER-series of field of one phase with pitched coil groups

q > 1, W/p < 1

... ,5 ,3 ,1 , 4

2ˆ

,,,

dpphase kkp

iNV

)cos(2 tIi

... ,5 ,3 ,1 , 122ˆ

,,,

Ikkp

NV dpphase

,...5,3,1

, )cos()cos(ˆ),(

tVtV phasephase

,d,p,w kkk



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3/9

Pitch coefficient, distribution coefficient and winding coefficient for pitched

winding with 6, 12 and 18 slots per pole pair Q/p

Pitch and distribution coefficient REDUCE certain field harmonic amplitudes.

Winding coefficient is a PERIODICAL function of ordinal number .

Example of typical three phase windings A, B, C

kw,

kd, kp, kw, kd, kp, kw, kd, kp,

0.866 1

-0.866 1

0.866 1

-0.866 1

0.866 1

-0.902 0.960 -0.067 0.259

0.067 -0.259

0.933 -0.966

-0.933 -0.966

-0.067 -0.259

0.902 0.960

-0.038 0.218

-0.136 -0.177

-0.136 -0.177

-0.038 0.218

0.902 0.960 0.933 0.966

0.067 0.259 -0.866 1

0.866 1

-0.940 -0.259 0.866 19

0.940 -0.259 -0.866 17

-0.174 -0.966 0.866 13

0.766 0.966 -0.866 11

0.766 0.259 0.866 7

-0.174 0.259 -0.866 5

0.940 0.966 0.866 1

Q/p = 18 Q/p = 12 Q/p = 6

q = 3, W/p = 7/9 q = 2, W/p = 5/6 q = 1, W/p = 2/3

C B A



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3/10

Inserting form-wound two-layer winding in

induction generator stator

72436

2

qmpQ

Source:

Winergy

Germany

Stator iron

stack

coil ends

Winding

overhang

Q = 72 slots

72 coils

m = 3 phases

2p = 4 poles

q = 6 slots per

pole and phase

72634

2

qmpQ



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3/11

Inserting form-wound two-layer winding in stator

slots

Source:

ABB, Switzerland

Stator iron

stack

60 slots for a

4-pole

induction

machine

q = 5:

Winding

overhang

60534

2

qmpQ



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3/12

Stator three phase two-layer winding of induction

generator

Source:

Winergy

Germany

Q = 72 slots

72 coils

m = 3 phases

2p = 4 poles

q = 6 slots per

pole and phase

72634

2

qmpQ

- Completed two-

layer winding after

the impregnation with

resin

- Some resin “noses”

are removed after the

impregnation to get a

smooth air gap



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3/13

FOURIER-analysis of field of a 3-phase winding

Pulsating fields per phase are shifted spatially by 2p/3 as phases U, V, W. These

3 phases are fed by sinusoidal AC current with phase shift of T/3.

Pulsating field per phase is decomposed in negative and positive sequence

rotating field waves (clockwise and counter-clockwise rotation) by use of

trigonometric summation law:

)cos()cos(ˆ),( , tVtV phaseU

)3/2cos())3/2(cos(ˆ),( , tVtV phaseV

)3/4cos())3/4(cos(ˆ),( , tVtV phaseW

)cos()cos(2

1coscos

)cos(2

ˆ)cos(

2

ˆ),(

,,t

Vt

VtV

phasephaseU

)3

2

3

2cos(

2

ˆ)

3

2

3

2cos(

2

ˆ),(

,,

t

Vt

VtV

phasephaseV

)3

4

3

4cos(

2

ˆ)

3

4

3

4cos(

2

ˆ),(

,,

t

Vt

VtV

phasephaseW



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3/14

Fundamental of air gap MMF (and field)

Resulting effect (summation) of all 3 pulsating phase fields U, V, W:

Summation for ordinal number = 1 (“Fundamental”):

Result: rotating wave:

,...5,3,1 ,...5,3,1

),()),(),(),((),(

tVtVtVtVtV WVU

)cos(2

ˆ)cos(

2

ˆ),(

1,1,1 t

Vt

VtV

phasephaseU

)cos(2

ˆ)

3

4cos(

2

ˆ),(

1,1,1 t

Vt

VtV

phasephaseV

:get we0)3

8cos()

3

4cos()cos(With

)cos(ˆ2

3),( 1,1 tVtxV phase

)cos(2

ˆ)

3

8cos(

2

ˆ),(

1,1,1 t

Vt

VtV

phasephaseW

/ 0VB



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3/15

Harmonic waves of air gap MMF and field

Summation of pulsating phase fields for > 1:

For ordinal numbers , which are divisible by 3, the summation of harmonic phase fields

U, V, W is zero: The 3 positive and negative sequence phase waves cancel.

For = 7, 13, 19, ... cancel the 3 negative sequence phase waves. The 3 positive

sequence phase waves add with the SAME phase angle and rotate WITH the fundamental

wave.

For = 5, 11, 17, .... cancel the 3 positive sequence phase waves. The 3 negative

sequence phase waves add with the SAME phase angle, rotating inverse to fundamental.

Facit: A 3-phase winding, fed by symmetrical 3-phase sinusoidal current system, excites a

step-like MMF distribution in air gap V(x,t), which may be decomposed into fundamental

and harmonic rotating waves. Wave lengths are determined by ordinal numbers. Only odd

ordinal numbers exist = 1, 5, 7, 11, 13, 17, 19, ..., which are indivisible by 3.

000),(3 tV )5cos(ˆ2

3),( 5,5 tVtV phase

)7cos(ˆ2

3),( 7,7 tVtV phase 000),(9 tV



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3/16

Fundamental and harmonic air gap field waves

= 1, -5, 7, -11, 13, -17,... Phase number m: is usually 3

Positive and negative ordinal numbers:

Velocity of harmonic waves decreases with 1/ :

Wave amplitudes (% of fundamental): (%) Underlined: “slot harmonics” !

)cos(

2),(),(

,t

xI

kN

p

mtxVtxV

p

w

-5.3 0.38 5.3 19

5.9 -0.4 -5.6 -17

-0.3 7.7 7.7 13

-1.4 -9.1 -9.1 -11

-2.2 -1.0 14.3 7

-0.8 1.4 -20 -5

100 100 100 1

q = 3, W/p = 7/9, Q/p = 18 q = 2, W/p = 5/6, Q/p = 12 q = 1, W/p = 2/3, Q/p = 6

...,3,2,1,021 gmg

/fv p,syn 2

1 B̂/B̂



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3/17

Magnetomotive force and air gap field of squirrel cage winding

a) Rotor: Squirrel cage: Qr conductive bars (copper, aluminium) in Qr slots. Bars are

short-circuited by 2 conductive rings an the front ends.

b) Symmetrical rotor current system: In each bar flows a sinusoidal bar current

with a constant phase shift to the current of the adjacent bar. Thus each bar is a

phase of a Qr-phase system.

Example: Qr = 28 bars, 2p = 4: Bar current system repeats after Qr/p = 14 bars. Phase

shift is ”slot angle” Q = 2p/Qr = /7.



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3/18

FOURIER-Analysis of MMF (and field) of squirrel cage

Each bar = 1 phase AND each bar = half turn of a coil. Hence:

Pitch and distribution coefficient = 1.

: Ordinal number of harmonic waves of rotor field of cage winding

Relationships:

FOURIER-series:

with ordinal numbers

Example: Qr = 28 bars, 2p = 4: Ordinal numbers = 1, -13, 15, -27, 29, ...

Wave amplitudes decrease with 1/ !

,,1,,2/1 , barwr IIkQmN

,...1

)2cos(1

2

12),(

tf

xI

p

QtxV r

pbar

r

,...2,1,01 rrr gg

p

Q



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3/19

Copper cage rotor of an induction machines

Source:

Breuer Motors,

Germany



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3/20

Salient pole air gap field, excited by DC current Air gap field H : e.g. 2-pole rotor of salient pole synchronous machine: Rotor

winding ampere-turns NfPol.If , (NfPol : number of turns per pole)

Air gap enlarged at inter-pole gaps, hence air gap is function of circumference co-

ordinate (x).

Calculation of radial component B of rotor air gap field by AMPERE´s law:

(Iron: Fe )

)(22)(222 xHHxHVINsdH FeFefffPol

C

)()()( 00

x

VxHxB

f



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3/21

Completed salient pole before mounting

Source:

Andritz Hydro,

Austria

Pump storage

hydro power plant

Vianden/Belgium

Refurbishment

Three-fold hammer

head fixation

“Cooling fins” by

increased copper

width

Damper ring

segments



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3/22

Completed “big hydro” salient pole synchronous rotor for high

centrifugal force at over-speed, 14 poles

Source:

Andritz Hydro,

Austria

Dove tail fixation of

rotor poles

“Cooling fins” by

increased copper

width

Damper ring

Damper retaining

bolts

Rotor back iron

Rotor spider

Generator shaft



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3/23

Field curve is symmetrical to abscissa:

only harmonics with odd ordinal numbers

Field curve as FOURIER-Cosine-series:

Shape of field curve is determined by air gap function (x) ( = contour of pole shoe);

hence also ordinal numbers, divisible by 3, are existing !

Poles rotate with constant rotational speed n: m = 2n = 2vm/dsi Hence all rotor waves

(fundamental and harmonics) B move with same speed vm with respect to stator winding.

Stator induced voltage of -th harmonic has -times higher frequency than that of

fundamental = 1.

Facit: In stator winding sinusoidal voltage due to fundamental rotor wave ( = 1) is induced,

but ALSO harmonic voltages with higher frequencies, whose harmonic stator currents,

excite magnetic fields, which may interfere from power line to parallel telephone cables.

FOURIER-Analysis of air gap field of DC-excited salient poles

)()( BB

... ,9 ,7 ,5 ,3 ,1 )cos(ˆ)(,...5,3,1

BB

npp

f m

22tptt mrrs )()(

)cos(ˆ)cos(ˆ)cos(ˆ)( tx

BtpBBBp

msrr

3. mathematical analysis of air-gap fields · 3. mathematical analysis of air-gap fields . ... =...

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