On the theory of 3-phase squirrel-cage induction motors including space harmonics and mutual slotting

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ON THE THEORY OF 3-PHASE SQUIRREL-CAGE INDUCTION MOTORS INCLUDING SPACE HARMONICS AND MUTUAL SLOTTING

G . C. Paap Power Systems Laboratory Delft University of Technology P.O.Box 5031, 2600 CD Delft, The Netherlands

ABSTRACT - In this paper general equations for the asynchronous squirrel-cage motor which contain the influence of space harmonics and the mutual slotting are derived by using among others the power-invariant sym- metrical component transformation and a time-dependent transformation with which, under certain circumstances, the rotor-position angle can be removed from the coefficient matrix. The developed models implemented in a

machine-independent computer program form powerful tools, with which the influence of space harmonics in relation to the geometric data of specific motors can be analyzed for steady-state and transient perfor- mances. Simulations and measurements are presented in a companion paper.

Kewords - asynchronous machines, general theory, space harmonics, mutual slotting, transients.

1.INTRODUCTION

In general the most difficult problems in the theory and modeling of induction motors are saturation of the magnetic circuit and all "parasitic effects" caused by higher harmonics in the magnetic field in the air gap. Both phenomena become even more complicated if the influence of the slotting of stator and rotor sur- faces are considered.

Gradually, mathematical models of induction machines have been worked out which take more details of the real geometrical construction into account and enable increasingly closer insight into their influence. At first models were developed which included space harmonics but which ignored those combinations of harmonics which could cause the multiple armature reaction [1,2,3]. These models include only the asynchronous torques and do not include the synchronous and pulsating torques. In [5] the general equations for the squirrel-cage induction motor are derived by means of harmonic analysis. Although these equations describe the dynamic behavior, for numerical calculations, a relatively large computation time is required because of the problem of the rotor position angle. In [6] this model is extended by the addition of the influence of stator slotting. In [7] and [E] dynamic induction motor models have been developed where the squirrel cage and the 3-phase stator winding are represented by equivalent polyphase windings. In the model presented in [7] it is possible to simplify the set of equations when no more than two harmonics per phase group are taken into account, but it is not general for all squirrel-cage motors. In [9] a transformation has been developed which simplifies the general set of equations and under certain circumstances transforms the set of equations in such a way that the ever-present influence of the rotor position angle, when considering the multiple ar-

90

SI4

374-9

EC

the IEEE Electric Machinery Committee of the IEEE Power Engineering Society f o r presentation at the IEEE/ PES

1990

Summer Meeting, Minneapolis, Minnesota,

A paper recommended and approved by

mature reaction, can be removed from the parameters and only appears in the source voltage. This enables a particular solution of the differential equations. In [!], however, a smooth air gap which could have a significant influence on the results was assumed. Further, the zero-sequence component was not included in the equations.

In this paper the model presented in [9] is extended by the addition of the mutual slotting and the zero-sequence component. The influence of saturation due to the main field can, as a linear magnetic circuit is presupposed, be taken into account by making an increase in the air gap. The influence of the saturation due to the slot-leakage fluxes can be dealt with by ad-

ditional widening of the slot openings. 2.MATHEMATICAL MODEL

After applying the group transformation to the rotor equations and the symmetrical component transformation to both the stator and the rotor equations, the following set of matrix equations arises (6,101:

Te= (; I;*T

7

aL;S

where 0 is the rotor-position angle. A list of symbols is provided in section 6.

The elements of the voltage and current matrices are: 0

+

- T

u;= [us ,us ,us

1

; I; = [is0,is+,is-lT I;=

[ire,

ibl.. . . , irK.. . . ,

i;-"]

,subscript K-N/2z where N is the number of rotor slots and z the highest common factor of N and the number of pole pairs p. The influence of the rotor slotting on the self- inductance of the stator is only taken into account by the Carter factor [lo] and therefore no hlss-matrix appears in expression (2). In the rotor self-inductance and the mutual inductance the slotting is taken into account completely.

The transformed stator parameters are:

R;

-

Rs

-

diag[Rs,Rs,Rs] ; L;s

-

diag[zsO,Es,Es ] where the zero-sequence inductance

zso

contains the harmonics v=3,9,15,

. . . .

and the positive sequence inductance contains the harmonics U-1, -5,7, -11,

. .

.

For the rotor the transformed parameters are:

where AL;,-

C

AErn(ejdS Q + e - J d e Q*T) ; n-1,2,3,. . .

n

After these transformations the complex mutual inductance matrix becomes as given in equation ( 3 ) where July

15-19, 1990.

Manuscript submitted January.11, 1990;

made available f o r printing May 18, 1990.

g=O,-tl,f2,. . Y

-

1, -5,7:-11,13,.

.

-P'+

,

~-3,9,15,.

.

In each element of (3) only those harmonics appear that fulfill all the constraints that are given in this matrix. In [6,10] a comprehensive determination of the G885-8%9/91/o3W4076$01.00@199 lIEEE

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71

'kr=

separate parameters is provided. Appendix I gives a short overview of the resultant parameters.

*

M1O M1l' ''Ml(K-l) MIK M1(K-l) " . M1l M20 M21' "M2(K-1) M2K M3(K-1) ' ' ' M31 ( 4 )

*

M20 M31...'3(K-1) M2K M2(K-1)..' M21 -

. . .

"Y mY LLr- Y w-gN/z w-gN/z

+

1 w-

x

m z

c

m z Y W

. . . .

Lir-

1

w--gN/z v--(gN/z

+

1) v--

*

M1O 'MO(K-l) MIK MO(K-l) ' ' '

(K-1) M2K MZ0 M1

. .

.M MZ0 0

. . .

0

. . .

0 (10)

*

' ; K M;K-l) . " M; ~

*

I

1 . .

.

o

.

a2

.

b2 . . bl

.

al

.

A=

. .

. . . 1

. . . .

;B- The structure of the LLr-matrix

1 . .

.

o

.

c1

.

. c2

.

d2

.

' dl .

_.

.

_.

.

. . .

. . 1 . . . .

The behavior of the squirrel-cage induction motor is strongly determined by the magnetic coupling between stator and rotor [9]. The presence of a harmonic in an element of the mutual inductance matrix is directly connected to the number of rotor bars N and by z. On the grounds of the present symmetry it is possible to write the L' -matrix as follows:

sr

n A jwp8

and in general kiejrl

-

X

my

-

C mwe ,k-1,2,3 ( 5 )

Y Y

Time-dependent transformations

The rotor and stator variables are transformed with complex, power-invariant, time-dependent transformations, as presented in [9]. These transformations change the LLr-matrix in such a way that the mutual

which satisfy the condition: AT.B*= U or A-'= B*T,where U is the unity matrix, and where

ai- DiiZi/E: e bi- Dii3i/E: e

c

-

M /D. e di- -M3i/Di e Di=J(MZi+ M3i) for i

-

l,Z,..,K-l. j (-ri-ai) -j (ii-Pi) j -j (yi-Bi) A A i 2i 1 ^ 2 * 2 E ~ - J ( M ~ ~ ^ 2

-

M ~ : ) A if ( M ~ ~ + M ~ ~ ) A *

If MZi- M3i= 0 then a i = ci- 1, bi- di-0

The relations between the old and the new variables are given by

U; = A.U; ; I ; = B . I ;

1;

-

[ i rO, it, rl, 1" r2,. . . ,irK,.

. .

,i;2,i;ll

( 6 ) Because of the structure of the transformation matrix B the transformed currents can be written as follows:

*

The equations are further developed by transforming the stator variables, using the following relations between the old and new quantities:

U; -C U; and I ' =C I ; where C -diag[ 1, e", e-"] ( 7 )

The new stator variables are defined as:

U:= [ UsO,Usf,Usb~T and

11-

[ isO,isf,isbl T

*

where U sb- usf and

The introduced angles y , and y l,..., yK-l will later be determined in relation to specific motor data. The transformation of equations (1) and ( 2 ) using relations ( 6 ) and (7) yields:

isb= isf. = R89I" + C-' dc(LM I n 8 + LV I t , ) +

h(

_I88 ₊_LVt 1 8 8 ) s s s dt s s s sr r dt s s s sr r (9) and L" = A L' B and R" = A RLrB. rr rr rr

In the derivation of the electromagnetic torque equation the fact that AL" = AL;r*T was used.

The mutual inductance matrix becomes: rr

A j(ai-r)

for i-0 or K 2i- M2ie

where Mli= 2M cosri and _li M

A j(ri-r)

Mi- Mie for i=1,

. . . .

K - 1

A 2

Mi- E /Di

A A

(if MZi= Mji= 0 then Mi-O).

A A j(y.-a +e.) A A j(ri-pi-ei)

Moi= ,[M 11 .M 2ie ' - M . M . ~ 11 31

1

for i-1, . . . . K - 1

if M2i-M3i=0, M

A A jei

0 iXMl ie

In the second and third row, the mutual inductance matrix now contains inductances which consist of a real part M and a complex exponential function. M depends, in general, on the rotor position angle 8 , The angles yi and y in the complex functions are still to be

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determined. The free angles Ti also appear in the sequence components in the first row.

The new rotor matrices become:

zero L;r'

.

-

.

...

LrK

.. .

+AL;r ; R;

i

.

0 =O-

.

Rrl

.

kliirl

.

-

.

...

RrK..

.

A A where

-

2 M2iM3i ej(ai+fii-27i) c. A

, if M2i- Mji-O: ki- 0 (12) ki- Df

Matrix AL"

where

can be written as: AL;,(i,k)

-X

AErnAn(i,k) rr

n

In

* *

-In -jn

* *

jn

+(ckdie + v i e )qN/z-k, i+(ckdie + v i e lqi,N/z-k n * nNse.

For AL;r it holds that:

AL" rr (i,k)- AL;:(k,i) and ALir(i,k)

-

AL;r(N/z-k,N/z-i) The factors A-'

a

and A-l

E

in equations (8) and (9) can be written as:

dt

T o . .

.

. O

.

el.

.

fl

. .

e2

.

f2.

*

.

A - u 4

dt

I

. . . . .

0

. . . .

. . . :

where

.

A - l U

ae

A A

If Mpi- M3i- 0 then e i- fi- gi- hi- 0.

The matrices dt and %-IC' are: 1 1s

dt -j gdiag[ O,l,-l] and

g-

C--j diag[ O,l,-l] As follows from equations (10) and (ll), mutual connections between positive and negative sequence stator currents are replaced by additional mutual connections between positive and negative sequence rotor currents. From the general equations in matrix form it appears that in general:

Lir, L;r, and R" depend on 8 .

A A

rr

L;, and R" are functions of M2i.M3i. (ai+Bi-2ri).

A A

rr

A-'dA/dt depends on M2i.M3i.ej(ui+fii-27i) , d-ri/dt, dui/dt and dfii/dt.

A A

A-'aA/aO depends on Mgi .Mji. ej (ai+Bi-27i) I

ari/ae,

aai/ar and

agi/ae.

AL;, contains exponential functions with N 8 , ai, ak, pip pk9 yiand yk'

C-'dC/dt and C-laC/ae depend on dY/dt and 87/88 respec- t ively

.

In order to see the advantages of these transformations the structures of the mutual inductance matrix and the rotor self-inductance matrix have to be examined in order to develop simpler equations. In [9] it is shown that 3-phase induction motors can be divided into two types, each with a specific kind of multiple armature reaction. Both types can be recognized by the structure of their mutual-inductance matrix. This also holds when the mutual slotting is taken into account. In Table 1. representatives of these two types are given.

3 . p

The first type of armature reaction occurs in 3 - phase induction motors where N/z

-

k.3 and this can be seen in the structure of the L' -matrix.

An example of this type is given in Table 1.a from which the following properties can be derived:

sr 0 . .

.

. o

.

gl.

.

hl

. .

g2

.

h2.

. . . . .

0

. . . .

. . .

_.

. . .

' *

. .

0 h;.

.

gl

either Mpi-O or Mji-O or M2i-M3i-0 and M20-M2K-0.

A A

From this it follows that A

-

B and that A itself be- comes unitary. The transformation is then only an angle transformation and does not influence the amplitudes. The elements in the matrices containing M2iM3i disappear. Further, it is easy to see in equations (12) and

(13) for this type ki- fi- hi-0. This means that

all exponential terms in the general equations (8) and (9) disappear, except in the mutual inductances Mi in expression (10). The exponential functions in the second and third row of the mutual inductance matrix can now be removed by introducing the following constraint: that 7i

-

7 . . .. 1 1

er-

a +fi -27 i i i'

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79

stator

matrix. This means that the zero-sequence component does not have any influence on the electro-mechanical behavior of the motor and therefore yields the conclusion:

are linked neither by the Lir- nor by the

Motors with the first type of armature reaction do not display any difference in electro-mechanical behavior for a star-connected or delta-connected stator winding.

A s the positive-sequence and negative-sequence rotor currents are not linked, the final equations only contain the forward-sequence component of the stator variables and the positive-sequence components of the rotor variables. This leads to the following voltage equations: x x x x x x x x x x x x x x x column number 2 x x x x x a ) F i r s t type: N = 30, z = 2, N/z = 15 b ) Second type: N = 28, z = 2 , N/z = 14 c ) Second type: N = 22, z = 2 , N/z = 11

Table 1. Distribution of harmonics in the Lkr-matrix.

W30. z=2 W28. z=2 W22. z=2

0 1 2 3 4 5 6 7 8 Q 1011121314 0 1 2 3 4 5 6 7 B Q 10111213 0 1 2 3 4 5 6 7 8 9 30

LY' LY' LY'

L4." LST" LSr" - -E D - -0 2 3 4 5 6 7 8 Q 0 1 1

jI_r

-~ -

--;Em:

B

deita Lrr" delta Lrr" delta w

a) bl d

Table 2. The structure of Lkr, LJr and for both types of armature reaction. The italic symbols mean complex conjugate values.

From Table 2.a, which gives a general impression of the structure of the LLr-,Lir- and A.LL;=-matrices for the first type of armature reaction, it can be concluded that for this type the sequence components of the

the flux-current relations: K-1

K-1 Y;k- %isf+ rk rk i-l rr i" +

X

AL" (k,i)i''i

and the electromagnetic torque equation:

1

*J

K-1 K-1 aAL;"(i,k)

+

,Z

X

[,

a s

+

g AL" (i,k)]i;ki;i i rr i-1 k-1 where

M.= M2i, ei= j(w

-

dai/dt) , gi= -j aai/aO when Mgi= 0 Mi= -Mgi, ei- j(w

-

dpi/dt), gi

-

-j

api/aO

when M2i= 0 For this type it holds that no equation for a 0-column or a middle-column K , as far as it exists, can be present. A s the maximum number of voltage equations can be counted:

For the stator: 1

For the rotor : K-1, where K -N/2z when N/z is even K -N/2z+1/2 when N/z is odd. A further reduction in the number of equations is achieved when the mutual inductance matrix contains columns which are not filled with inductances. These potential equations yield trivial solutions and can therefore be omitted. If the motor is connected to a symmetrical power source then the transformed voltage becomes

for the star connection: usf=

ji

U N and

for the delta connection: usf- i j 6 UNej(Ot +

'

-')

where

6'-

4 - r / 6

Upon taking the remaining free angle y

-

wt

+

4 ' (or wt

+

4) ,

2 -

w and usf becomes a DC voltage.

A A

+

"

- 7 )

and UN is the terminal voltage (rms).

This possibility to transform usf into a DC voltage here enables the formulation of the conclusions [ 9 1 :

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*

The starting characteristics of this type of motor do not depend on the moment of switching, but only on the rotor position angle at s-1 as Mi and AL" (i,k) still remain functions of 8.

This type has only one synchronous torque which ap- pears at s-1. When the rotor moves, only pulsating

torqyes exist. rr

4.THE SECOND TYPE OF ARMATURE REACTION Three-phase sauirrel-cage I motors for which N/z is not a multiple of three display the multiple-armature reaction of the second type. This type possesses the following characteristic geometrical properties (see figures 1.b and c, 2.b and c):

In the mutual inductance matrix both Mpi and Mgi can 2K have values not equal to zero. Further, M20 and M never occur together. From Table 2.b and c, which gives a general impression of the structures of the L' -,L;r- and AL,;r-matrices for the second type of armature reaction, it can be concluded that for this type the sequence componentq of the stator and the rotor are linked by the Llr- and by the AL;r-matrix respectively. Therefore, for the second type, the equations in their most general form haye to be used when all harmonics have to be taken into account. These equations (8) and

( 9 ) can be written as:

sr

d

-

RsisO + %($so)

Usf

-

Rsisf +

!psf)

+ j$ $sf 0

-

Rrmirm

+

$,($&)

0 -grk(i;k+ %i;E)+ e k $" rk

+

fk$"E+

tt($ik),

k-1.2,.

.

,K-1 with the flux-current relations:

$so

-

EsOisO

+

Mlmim

+

X

[Moii;!

+

Moii;i] $sf

-

Es

isf

+

M2mirm

+

C

Mi i;i

i-1

$&

-

MlmisO + M;misf + MpmiZf +

(Erm+

Am)irm +

, m

-

0 or K

*

K- 1 i-1 K- 1 K- 1 i-1 + C [Amii;,

+

Am-ii;;]

$gk

-

M;lkiso + Qsf + crk(i;k + \i;i) +

wrm+

K- 1

+

x

[%ii;i

+

%-ii;;~ i-1

and the electromagnetic torque equation:

K- 1 K- 1

i-1 k-1 Ik

+2 C [(g.A. +h A* im m-i)irm+

X

[(g.A. +h.%-i)i;k+

*

In these equations the following short notations are used:

However for normally constructed machines, usually a characteristic set of harmonics can be selected which represents the most significant influence on the behavior of the machine.

When in every column in the mutual inductance matrix this set of harmonics fulfills the following con-

hki-ALgr(k, i) ; %-i-AL;r(k,N/z-i)

the exponential functions in the second and third row of the LLr-matrix, as well as in the factors ki, hi and fi disappear. Constraint (14) can only be fulfilled if every element of the L' -matrix contains at the most only one inductance.

The inductances in the first row remain functions of 8 , however, in consequence of the constraint (14) the exponential functions in Moi in expression (10) become each other's complex conjugate:

sr

A j c A - j e

i Moi-

n

M 1l .(M 21 .e -M3ie ) where e--yi-ai+ei---yi+Bi+C Normally this results in one single angle for all columns (f37 for combinations of the lower zero- sequence harmonics and kg.3-y for the higher, where g=0,1s2,

. . .

) .

After having analyzed the behavior of the squirrel-cage motor with the second type of armature reaction, from the general model some simplified models can be derived. In this paper only the equations for a first- order approximation for the star connection with ungrounded neutral will be given which provides a system of differential equations with constant coefficients. The 8-dependent elements in the first row of the Lkr-matrix disappear from the equations. (iso-O). The selected set of harmonics has to fulfill constraint (14). The AL.;r-matrix contains, in principle, inductances which depend on 8 , however, in this case, most elements become constant and, in anal- ogy with the elements in the first row of the Lkr-

matrix, the few remaining 8-dependent elements in the ALgr-matrix acquire the same angle dependence: fg.6-y.

The contribution of these 8-dependent terms is most of- ten a second-order effect and will be ignored in this model. This results in the following system of differential equations with constant coefficients.

u sf -R s i sf

+L(*sf)+jg

dt

aSf

0 -Rrmirm+ !t(@;m)

0

-E

(i"

+

\i;E) +ekPk+ fkQ;t+ $Wk, k-1,2, and the flux-current relations:

,

m- 0 or K rk rk

-

K - 1 Psf- L i +M i

+

C

M i" s sf 2m rm i-l i ri ,K-1

@im-

i2m(isf+izf)

+(Erm+

Am)irm + K-

X

1 [A .in + Am-ii;21

i-l mi ri

The complete L-matrix is real and constant. In calculations it has to be inverted only once, which saves much computer application time.

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81 5.CONCLUSIONS

The torque equation becomes:

K - 1 ,

*

* *

[gii;i+hii;i]Miisf- K - 1

+ X

i-1 wher K- 1 k-1 Ik

(giAim+hiAm-i)i

+ X

((g.A. +hiAk-i)i;k+

the factors ei, gi, f. and h. become: i ; . ]

- I

Note that when not more then one harmonic inductance is present in every element of the mutual inductance matrix LLr, see expression ( 4 ) , the factors gi and hi are constant too.

As this system has constant coefficients a particular solution is possible when the slip s is constant [lo]. Upon assuming a symmetric voltage system between the terminals of the star-connected stator winding the transformed stator voltage can be written as:

Usf=

,/?

e

For a constant value of the slip s , 7 can be written as: 7 = 7'pB

-

y'(1-s)wt and after introducing

s

-

l-y'(l-s), the transformed voltage becomes: j(ot+d'-r)

v

j ( Syot+d

'

) ₁

, where

usf=

,/?

UN and Usf- U-sf e

U is the rms-value of the terminal voltage.

From here it is clear that, because of the presence of groups of harmonics that result in different ~ ' s , the synchronous torques which change into pulsating torques with a frequency of 2 s ut outside their synchronous speeds always cause synchronous torques to appear simultaneously with pulsating torques. This contrasts with the first type where only one synchronous torque exists at stand-still.

For the delta connection similar models can be derived. In these models the Llr- matrix remains 0 dependent because of the elements in row l.

One difficulty in this development of a system with a maximum of constant elements is constituted by the practical fact that for column 1 it may be necessary to consider more inductances in element 3 in the interests of obtaining more accurate results [9].

The complete set of equations, which describes the transient state includes the mechanical equation of mo- tion:

N

where T is the load torque, J the inertia of the complete rotor mass and d

me

the damping coefficient. me

In this paper general equations for squirrel-cage induction motors have been derived based on the real geometry of the motor. The squirrel cage has been described by its meshes; no equivalent windings have been used. By means of complex time-dependent transformations free angles are introduced which are very helpfully simplifying the set of equations when the specific geometrical properties of the both types, in which the asynchronous machines fundamentally can be divided, are taken into account. The equations derived are general in the sense that all space harmonics are taken into account, due to the MMF as well as to the double slotting. This provides a better calculation of the synchronous, pulsating and asynchronous torques. The final equations enable the formulation of some specific properties of both types in connection to their electromechanical behavior. Further they are valid for star and delta connection and for any ar- bitrary source voltage.

6.ABBREVIATIONS AND SYMBOLS

i(t), u(t) instantaneous values (complex or real) i*(t), u*(t) conjugate complex values

I,U,L matrices

IT, UT, etc transposed matrices

I ,U rms values (complex or real) L, M, etc magnitudes (real)

E,

etc inductances after group- and

symmetrical-component transformation. i',I',LLr,etc quantities after group- and symmetrical-

component transformation.

i" , I " , L:;, etc quantities after the time-dependent transformation.

A *

Svmbols ( in SI-units )

coefficient of mechanical damping current

complex number j-(O,l) inertia of rotor mass

integer number K=-N/2z or N/2z+1/2

inductance mutual inductance mutual inductance number of rotor slots number of stator slots number of pole pairs resistance slip electromagnetic torque load torque voltage unit matrix

greatest common divisor of N and p angle in composed inductance of one element in the mutual inductance matrix. angle introduced by transformation rotor position angle

integer numbers

phase angle of stator phase voltage phase angle of stator terminal voltage flux linkage

angular frequency of the source voltage short notation for ALrr(k,i)

rotor inductance matrix due to stator slotting

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Subscripts i

i

forward component of stator current backward component of stator current sf sb i 1 2 3 4 5 6 7 8 9

rotor mesh current 7.REFERENCES

Nasar, S.A.: "Electromechanical energy conversion in nm-winding double cylindrical structures in presence of space harmonics.", IEEE Trans. on PAS, vol. PAS- 87, No.4, pp.1099-1106, April 1968

Willems, J.L.: "The analysis of induction machines with space harmonics.", Elektrotechnische Zeit- schrift-A, vo1.93, No.7, pp.415-418 , 1972

Dunfield, J.C.: Barton, T.H.: "Axis transformations for practical primitive machines.",IEEE Trans. on PAS, vol. PAS-87, No.5, pp.1346-1354, may 1968. Barton, T.H.; Pouloujadoff, M.A.: "A generalized symmetrical component* transformation for cylindrical electrical machines.", IEEE Trans. on PAS, vol. PAS-91, pp.1781-1786, September 1972.

Taegen,F und E.Hommes : "Das allgemeine Gleichungssystem des Kafiglaufermotors unter Berucksichtigung der Oberfelder. Teil I : Allgemeine Theorie.", Archiv fur Elektrotechnik, vo1.55, pp.21- 31. 1972

Taegen,F und E.Hommes : "Die Theorie des Kafiglaufermotors unter Berucksichtigung der stander- und Laufernutung.", Archiv fur Elektrotechnik, vo1.56, pp.331-339, 1974

Rogers, G.J.: "A dynamic induction motor model including space harmonics.", Electric Machines and Electromechanics, vo1.3, pp. 177-190, 1979

Fudeh, H.R.; Ong, C.M.: "Modeling and analysis of induction machines containing space harmonics. Part I,II,III.", IEEE Trans. on PAS, vol.PAS-102, pp.2608-2628, august 1983

Hommes,E and G.C.Paap : "The analysis of the 3-phase squirrel-cage induction motor with space harmonics, Part I : Equations developed by a new time-dependent transformation.", Archiv fur Elektrotechnik, vo1.67, pp. 217-226, 1984

10 Paap,G.C. : "The influence of space harmonics on the electromechanical behavior of asynchronous machines.", Dissertation Lodz, 1988.

BIOGRAPHY

Gerardus Chr. Paap was born in Rotterdam, the Netherlands on February 2, 1946. He received his M.SC. from Delft University of

Technology in 1972 and his Ph.D. from the Technical University of Lodz in 1988. From 1973 he has been with the Faculty of Electrical Engineering of the Delft University of Technology. First, from 1973 to 1985, with the Electrical Machines and Drives Division where he lectured on the fundamen- tals of electrical machines and from 1985 with the Power Systems Laboratory as Assistant Professor. His research interests include power systems transients and the dynamics of electrical machines.

APPENDIX I

The resultant parameters which appear in the equations after group and symmetrical component transformation are :

the zero seauence inductance:

the positive sequence inductance :

L SY

x p L 6 v U

where 6 the effective air gap, w the number of turns per stator phase in series, D diameter of stator bore and 1 the effective iron length.

E

and

is,

are the zero-sequence and positive-sequence leakage inductances.

For the rotor the transformed parameters are: grk= 2%+ Rrb( 2.sin )2 , k = 0,1,.

.

. . . , (N/z-1)

E

r

,

-

2$,+ Lrbu( 2.sin )2 k = 0,1,.

. .

. . , (N/z-1) rb where

the resistance of a bar. $,and Lrbo are the leakage inductances of a ring segment and a bar respectively.

,9 is the rate between the conductance of the unipolar flux path with cross-section A. and the effective air gap A and one rotor tooth with cross-section A

the resistance of a rotor-ring segment and R

t'

For the n-th harmonic of the stator-conductance wave it holds for the elements in the matrix Q[q(i,k)] :

q(i,k) = 0 i f i = k

q(i,k) + 0 if n.N

+

(i-k)p

-

gN ; g=O,1,2,.. q(i,k)

-

,9/(N+,9) if i = 1 or k = 1

q(i,k) = 1 if i z k a n d k z l

The elements in the resultant mutual inductance in expression (5) are: n.N /N z 1,2,3,

. . . .

n A n - l n A

iij

=

J[

c

"k2+ 2

c

x

ml %cos((" 1- U k k=l 1-1 k-l+l n n,.

q = arctan[ k-1 C %sin(vkp8)/ k=l C %COS(Y k PO)] and the original inductances in expression (3):

where v = v , p is the is the E,, is the is the is the Esu rsu E," 'srn

EAskn

is the

winding factor of the stator stator slot factor.

skew factor rotor slot factor factor of mutual slotting skew factor in AL

(8)

IEEE

Transactions on %rgy Conversion, Vol. 6, No. 1, March 1991 ₈₃

PERFORMANCE ANALYSIS

OF

PERMANENT MAGNRT SYNCHRONOUS

MOTORS

PART:II OPJZRATION

FROM

VARIABLE SOURCE AND TRANSIENT CHARACTERISTICS

S.

M.

0sheba.MIEEE.

F. M.

Abdel-Kader,MIEEE,

Faculty

of

mineering, Faculty

of

E M .

,

and Tech.

,

Menoufia University,

Shebin El-Kom, FSYF'T. Port Said, EGYPT.

Suez Canal University.

Abstract

A comprehensive analysis of transient

performance of a permanent magnet

(PM)

synchronous motor, operating from a variable

source is described. Internal damping is

modeled and optimum values of design

parameters which improve transient performance

are obtained. The effects of varying the

supply voltage and frequency on the optimum

values

of

thasc: parameters are demonstrated.

Simulated nonlinear response to a load change and starting performance are discussed. The

results illustrate that significant

improvement in transient performance can be achieved, over a wide range of voltages and frequencies, when some parameters are properly chosen.

Keywords: permanent-magnet synchronous motors INTRODUCTION

Modeling of permanent magnet and steady

state performance of PM motors have been a

subject of growing interest since the early

8 0 ' s [l-31. The effects of varying supply frequency on motor parameters and steady state

performance have also been studied C41.

Moreover, the characteristics of this motor, when operated from an inverter with phase

controller, has been considered

[SI.

Starting

performance was firstly discussed by Honsinger

[ 6 ] . He investigated the effects of the rotor

cage parameters on the run-up characteristics and valuable conclusions were reached. This study has been followed up by Rahman et a1 who examined the effect of saturation on

starting performance [ 7 1 . Miller [ 8 1 developed

a pull-in criterion and studied the run-up characteristics of line start PM motors.

motors results from Synchronization of PM

90 SM 377-5 EC

by the IEEE Electric Machinery Committee of the IEEE Power Engineering Society for presentation at the IEEE /PES 1990 Summer Meeting, Minneapolis, Minnesota, July 15-19, 1990.

1989; made available for printing May 28, 1990. A paper recommended and approved

Manuscript submitted August 28,

+,he action of synchronizing torque. This

torque is attributable to both the magnet and the difference between the axes reactances.

Also, the hunting oscillations, following a load change, are damped by the action of

damping torque. This torque component is

significantly affected by the motor design

parameters. The effects of all design

parameters on dynamic performance are

extensively examined using the damping and

synchronizing torque technique [9]. The

effective parameters which increase motor

damping are identified and their optimum

values, that improve dynamic performance, are obtained.

The object of this paper is to examine the effects of supply voltage and.frequency on the

optimum values of the effective design

parameters and also on the starting

performance. The results give physical

interpretation to transient performance when the motor operates from a variable source. They also show that with a proper choice of some parameters, the transient performance can be significantly improved over a wide range of operating voltages and frequencies.

MODELING OF TORQUE COMFWNENTS Algorithm

Modeling and accurate prediction of damping and synchronizing torques provide a

quantitative assessment and physical

realization of motor performance. Damping

torque is defined as the torque component in-phase with the rotor speed, while the

synchronizing torque is that in-phase

component with rotor angle. This indicates that, following a small load change, the deviation in motor torque can be written as

C9l :

A T M ( t ) - K , * A w ( t ) + K , O A B ( t ) ( 1 )

Where

K,

and

K S

are damping and synchronizing

torque coefficients, respectively. These

coefficients must be positive for the stable

motor. Using minimization procedures to

minimize squared errors, the following

algorithm is obtained:

(9)

0.04

0.03

0.0 2

The above equations give the values of the

torque coefficients K D and K, which can be

solved numerically. This solution can be

obtained either on-line, in experimental

applications or off-line with digital

simulation

,

over a period of time t=

N.T,

where

T

is the integration step and

N

is the

number of iterations. The advantage

of

using

this technique is that it gives the torque components directly from the motor nonlinear response without approximations.

Optimum Parameters

The PM motor used in this investigation,

and also in Part

I [91,

is a 4 h.p., 2 pole,

1-phase, 230 Volt. The per-unit values of motor parameters’ are given in the Appendix. The damping and synchronizing torque algorithm has been applied to assess the effects of all design parameters on the dynamic performance

of

PM motorsC91. To examine the effect of a

specific parameter, a 50% load disturbance was

applied for 20 ms. The corresponding

deviations in rotor angle

,

rotor speed and

electrical torque were obtained over a period of 3 seconds. This is required to solve the decomposition algorithm, Eqns.2-3, and compute

the time invariant torque coefficients, K D and

K,. It has been concluded that a proper choice

of the cage resistances,

R,

and R , , increases

damping and improves dynamic performance. Each of these parameters has an optimum value at which maximum damping occurs. These values are designated as R,Opt and RDOpt. The question now is to what extent the supply voltage and

frequency affects both RDOpt and R,Opt and

how

these parameters affect starting performance. This is described subsequently.

OPERATION FROM

VARIABLE

SOURCE

Sensitivity

Of

Optimum Parameters

At each operating voltage and frequency, a 50% load disturbance was applied for 20 ms and

the corresponding deviations in rotor angle

,

rotor speed and electrical torque were

obtained. While varying either R , or R, the

torque decomposition algorithm has been

applied to define their optimum values at this voltage and frequency. This has been repeated to define both RQOpt and RDOpt versus supply voltage at different frequencies as shown in Figs. 1-2. These results illustrate that both

opt

and RoOpt vary within a limited region

Over the whole operating range of voltages and

frequencies. Moreover, a study was made

concerning the internal damping, over a wide range of voltages and frequencies, using:

(i) one set of optimum R D and R Q values,

determined at the nominal Voltage and

frequency, and

(ii) the optimum values of R, and R ~ ,

sponding to

frequency

.

each operating voitage and

Figs.3 & 4 . show the results of this study.

This demonstrates clearly that the optimum R,and R, may be chosen at the nominal voltage and frequency with only a slight reduction in the internal motor damping. Therefore, these optimum values of parameters are used in com- puting the results presented subsequently.

0 04 f -30 Hz 0 02 2 1 - 5 0 Hz 3 f -75 Hz v P.U. 0.5 1.0 1.5 0

L,

Fig.l Effects of voltage and frequency on optimum value of R . ~ R Q optimum 1 f =30 Hz 2 1 - 5 0 Hz 3 f =75 HZ v p.u. 0 0.5 1.0 1.5

Fig.2 Effects of voltage and frequency on

$0’

optimum value of R ,

Figure 3 Torque coefficients Versus V

-

opt at nominal v & f

_ _ _ _

opt at every V & f

100 2 BO 3 60 LO 20 0 0.5 1.0 1.5

Figure 4 Torque coefficien1.s versus V

-

o opt at nominal v & f

(10)

Damping Characteristics

The objects of this section are to:

1. examine the effects of operating voltage

and frequency on damping and

synchronizing torques.

2 . study the effects of using the optimum

parameters on torque coefficients in

comparison with the nominal parameters. The results of this investigationareconcluded

in Fig. 5. Concerning the first object, the

results illustrate that motor damping

increases with either increasing the supply

voltage or decreasing the frequency.

Deterioration in dynamic performance at low voltage may be attributed to lack of damping torque. Moreover, there is an optimum value of supply voltage at which maximum synchronous torque occurs. This optimum voltage is related

directly to supply frequency. Failure in

synchronization at low-frequencies and high voltages may be due to lack of synchronizing torque. Therefore, at each synchronous speed, the optimum operating volhgc: is that value which provides maximum synchronizing torque.

A comparison between damping and

synchronizing torque coefficients obtained

using either the nominal or the optimum

parameters, over a wide range of voltage and frequencies, is demonstrated in Fig.5. It is clear that the use of the optimum parameters increases motor damping at all voltages and

frequencies. As an example, the damping

torque is increased by about 100% at V=O.50 p.u. and f= 30 Hz which represents a signifi- cant improvement in motor dynamics. However, the obtained synchronizing torques, computed using both nominal and optimum parameters, are similar. Therefore, the solid lines in Fig. 5 represent the synchronizing torques for both cases.

The synchronizing torque is significantly affected by field strength over the whole range of operating voltages and frequencies. To demonstrate this, the optimum parameters are considered and the torque components are

obtained versus I , as shown in Fig. 6. These

results illustrate that increasing the magnet strength increases the synchronizing torque

without significant variations in damping

coefficient. Time Response

To further elucidate the above important

remarks, the motor is subjected to a pulse

load increase, and the time response is shown in Fig.7. This represents a comparison between motor responses at different voltages and

frequencies, using either the nominal or the

optimum parameters. The results illustrate well-damped oscillations and a reduction in

the rotor first swing, at all operating

voltages and frequencies, when the optimum parameters are used. Comparing Figs.7.a and 7.b shows that at a given supply frequency, the increase in supply voltage increases motor damping and reduces rotor first swing which indicates an increase in stability reserve.

These results confirm those Previously

predicted from Fig. 5.

Fig.8 shows a comparison between motor

30 20' 10 85 4 . ~ ~ ~ 1 6 3 1 I = 30 Hz v=o.5 P.". --- v10.9 P.". v .l.2 P.".

-

0 v P.U. 0.5 1.0 1.5 2.0

Figure 5 Torque Coefficients verfius v

_ _ _ - - - 60 _ _ * - - - - _ _ - - -

_ _ _ _ _ _ _ _

- - -

4 0

0 0.5 1.0 1.5 2.0 2.5 3.0 Figure 6 Torque coefficients versus I ,

(11)

"

2.00

-

z

5

0.00 n Q,

2

1.00

-

a,

2

-1.00

-

a -2.00

-

Optimum Parameters

- - _ - _ _ _ _ .

Nominal Parameters : i

7:

i !-'ib : ; \

--

v = 1 . 2 p.u. I* tI IJ! f, = 1.5 .I t -3.00 0.60 0.40 0.b 0.do 0.Bo 1.h 3.00

1

b -3.00

'

* o.& 0.40 0 . b 0.60 0.40 1-60

4

_I _.

n

O - 7 - -1 .oo

A_\

\i /

!

v.1.2 p.u. f , = 0.6 ! i \ :

,.;

0.40 0 . k 0.60 0.b 1.60

Time,Sec

Figure 7 Comparison of time response

response at a constant frequency and variable voltage. These remarks confirm those extracted from Fig.5, indicating that either increasing supply voltage or reducing frequency, improves damping and increase motor stability reserve. This is indicated by the reduction in the first rotor swing in both cases.

-3.00 0.60

'

0.lO 0.LO 0.60 0.BO 1.b e Time ,Sec 4.00 Q I f , -1.5

-

V 0.5 P.U.

_______

v

z1.2 p.u. -3.00 \ I 0. O.!!O O h 0.60 D,hO 1.60 7 Time,Sec b

Figure 8 Response at Constant f m w e n c y

An important test has been carried out to

obtain the motor response at low and high voltage levels using three different values of supply frequency. These results are shown in

Fig. 9, which illustrates:

1. At low frequency (curve 1, in Fig.9.a

-

9.b 1, the performance is improved at low

voltage due to high damping torque and deteriorates at high voltage due to lack of the synchronizing torque.

2 . There

is

an optimum operating voltage corresponding to each synchronous speed. This voltage can be chosen to yields the maximum synchronizing torque.

STARTING CHARACTEBISTICS

E f f e c t s O f

Optimum

Parameters

During starting, the cage resistance must be large enough to compensate for the negative magnet torque. The object of this section is to demonstrate the effects of the optimum parameters on starting performance. It has been demonstratedC91, that damping is more

sensitive to changes in

R,

rather than in

R D .

Fig.10 shows the effects of varying

R D

with

different values of

R,

and loads. Examining

these results indicates that using

R,

at its

optimum value is necessary to :

1. achieve maximum improvements in dynamic

(12)

87 V ~ 0 . 5 P.U. 1.00

-

U a Ln

:

-1.00

-

-2.00 - p' I -3.00 \ I

0.b 0.lO 0.LO 0.80 0.dO 1.h

c Tirne,Sec a I 3.00

-

V = 1.5 P.U.

-

fr= 0.6 I1 1.16 0.40 0.60 0.BO 1.h

-

-2.00 Tirne,Sec b

Figure 9 Response at a constant voltage

-

T =0.3p.u. ----T =0.6 nu.

Y

i

'1.

1 RQ:0,005 P.U.

z

RQ.0.035 P.U. (optimum)

2 . reduce any adverse effects on motor

dynamics due to an increase in R , more

than its optimum value.

The above remarks are of significant importance to the starting performance. The

value of R , can be increased beyond its

optimum value to substitute for the negative magnet torque. This will improve the starting

performance without any adverse effects

following changes in load torque. A

comprehensive investigation has been

undertaken to determine the optimum value of

R , which improves both starting performance and response following small load changes. The

effects of these optimum parameters on

starting are shown in Fig.11. The results

illustrate poor starting, under different

loads, when either R , o r R , deviated from its

optimum value.

Starting

From

Variable Source

Fig.12 shows the effects of supply voltage and frequency on the starting performance. As explained in Fig.5, the loss of synchronism at

low-frequency, high-voltage (Fig, 12. a) is due

to lack of synchronous torque. Also there is

an optimum value of supply voltage which

yields good starting performance at each

synchronous speed. This optimum value may be determined as that value which increases the synchronizing torque coefficient

CONCLUSION

This paper defines the parameters which affect the transient performance of the PM motors over a wide range of operating voltages and frequencies. The results demonstrate that a proper choice of these effective parameters at the nominal voltage and frequency improve the motor performance over a wide range of supply voltages and frequencies. Increasing supply voltage and reducing frequency increase both motor damping and stability reserve. However, performance may deteriorate at low frequency with the increase of supply Voltage

due to lack of synchronous torque. The

starting performance is significantly improved with the use of the optimum parameters. There

is an optimum value of supply voltage

corresponding to each synchronous speed at which maximum synchronizing torque occurs. The choice of operating voltage at this value has no adverse effects on the damping torque. Moreover, increasing the supply voltage above this optimum value causes a failure of the motor to reach synchronous speed due to lack of synchronizing torque. These results are of practical importance, giving some valuable

insight in the PM motors dynamics when

operated from a variable voltage and frequency source. - - - _ - _ 4 1 -10 0.8 R D 1.0

- - _ _ _

- -

t 0 0.2 0.4 0.6

REFERENCES

Fig. 10 Effect of cage resistances

on internal damping

[ I ]

V.

B.

Honsinger, "Performance of

Polyphase Permanent Magnet Machines ' I ,

IEEE Transactions on Power Apparatus and

System, Vol. PAS-99,

No.

_ _ 4 , 1380,

(13)

( a )

-

Optimum R , , Optimum R ,

---

Optimum R , , R , <Optimum

---

Optimum R,. R , >Optimum 150 OG ( b )

:

mum

d

20C.W Y)

2

25oW Optimum R , , Optimum R , L

---

Optimum R , . R , <Optimum

---

Optimum R,. R , >Optimum ,50tc -U

g

l0OW v) 53 00 LW 25o.w - yi0.w - TimeSec 50.00 0.00 0.20 0.;0 0.60 0.l0 1. Time,Sec ( b ) 0 . h 0.h 0.60 ado t. Tirne,Sec F i g u r e 1 1 Effects of c a g e p a r a m e t e r s on s t a r t i n g c h a r a c t e r i s t i c s 120.0 100.0 U \ -0

I

w.O g 60.0 U-

g

40.0 cn 20.0 0.0 II)

.

f r 1 5 HZ. ...

-...-

._..._....C..,... ,.... ,,-. - . . . - V.0.3 P.U.

-

V.0.6 P.u. ,...

- -

v.1.0 p.u.

...

I 180.0

4

0.0 0.

Jd,

0 . h 0.40 0.& ado 1.b I 0.h 0 . 9 0.80 0.64 t f Time.Sec Time,Sec a b Time,Sec C d

(14)

89

K . J . B i n n s and T . M. Wong, " A n a l y s i s and p e r f o r m a n c e of a H i g h - F i e l d Permanent--Magnet Synchronous Machine", P r o c . I E E , Vol. 131, No. 6 , 1 9 8 4 ,

K . M i y a s h i t a . S . Y a n s h h i t a ,

S.

Tanbe, T .

Shimozu, and H . S e n t o , "Developed of a High Speed 2 - p o l e Permanent Magnet Synchronous M o t o r " , IEEE T r a n s a c t i o n s on Power A p p a r a t u s and S y s t e m s , V o l . PAS-99, B . J . C h a l m e r s , S . A . Hamed and G . D . B a i n e s , " P a r a m e t e r s and P e r f o r m a n c e of a H i g h - F i e l d Permanent-Magnet Synchronous Motor f o r V a r i a b l e - F r e q u e n c y O p e r a t i o n " , P r o c . I E E . Vol. 1 3 2 , No. 3 , 1985, P . C . K r a u s e ,

R.

R . Nucera, R . J . K r e f t a and 0. Wasynczuk, " A n a l y s i s o f a Permanent Magnet Synchronous Machine S u p p l i e d From a 180 I n v e r t e r With P h a s e C o n t r o l " , I E E E T r a n s a c t i o n s on Energy C o n v e r s i o n , Vol. EC-2. No. 3 , 1 9 8 7 , pp. 423.- 431 *

V . R . H o n s i n g e r , "Permanent. Magnet. Machines, Asynchronous O p e r a t i o n " , I E E E

T r a n s a c t i o n s on Power A p p a r a t u s and S y s t e m s , Vol. PAS-99, No. 4 , 1980. M. A. Rahman and T . A . L i t t l e , "Dynamic P e r f o r m a n c e A n a l y s i s o f Permanent Magnet Synchronous M o t o r s " , IEEE T r a n s a c t i o n s on Power A p p a r a t u s and Systems, Vol. T . J . E . M i l l e r , " S y n c h r n n i z a t i n n of 1 i n e - - S t a r t Permanent-Magnet AC M o t o r s " ,

I E E E T r a n s a c t i o n s on Power A p p a r a t u s and S y s t e m s , Vol. PAS-103, No. 7 , 1984,

E', M . Ahdel-Kader and S . M . Osheha,

" P e r f o r m a n c e A n a l y s i s of Permanent Magnet Synchronous Motors . P a r t I : Dynamic P e r f o r m a n c e " , IEEE W i n t e r M e e t i n g . E'etruarlr 4-8. 1990, At,lant,a, Gsorg i a . PP. 2 5 2 - 2 5 7 . N O . 6 . 1980, pp.2175-2183. p p . 117-124. p p . 1510--1518. PAS-103, N O . 6 , 1984. pp. 1277-1282. pp. 1822.-1828. 1 . i S T S Y M B O L S S u p p l y f r e q u e n c y Hz

.

Damping t o r q u e c o e f f i c i e n t , p . u / r a d / s e c S y n c h r o n i z i n g t o r q u e c o e f f .

.

p . u / r a d . P e r u n i t f r e q u e n c y ( f / f , ) S u p p l y v o l t a g e p . u . R e s i s t a n c e p . U . Torque p . U . E q u i v a l e n t f i e l d c u r r e n t R e a c t an c e Load a n g l e , r a d . A n g u l a r v e l o c i t y r a d . ; s e c . SUBSCRIPTS D D i r e c t , a x i s damper c i r c u i t .

Q

Q u a d r a t u r e a x i s damper c i r c u i t . R , -0.0173 R , -0.108

X,,

-0.543 Xu,, -0.478 X,, -0.608 H -0.2510sec

APPENDIX

Motor P a r a m e t e r s i n P e r - U n i t v a l u e 1 7 1 : R , -0.054 i,

-

1.817

X,,

= 1.086 xu,

-

1.021

X,,

-

1.151 Shaban M . Osheba w a s b o r n i n Egypt i n August 2 4 , 1 9 4 8 ( M ' 8 1 ) . H e r e c e i v e d h i s B. S c . and M. S c . Degrees i n E l e c t r i c a l E n g i n e e r i n g from Menoufia Univ. and Helwan Univ. i n 1971 and 1977 r e s p e c t i v e l y . From 1971 t o 1977 h e w a s working as a r e s e a r c h and t e a c h i n g a s s i s t a n t i n Menoufia Univ. P r o f e s s o r Osheba r e c e i v e d h i s Ph.D i n E l e c t r i c a l E n g i n e e r i n g from L i v e r p o o l U n i v e r s i t y , England, i n 1981. S i n c e 1981 h e i s a s t a f f member a t Menoufia Univ. where h e i s c u r r e n t l y p r o f e s s o r i n E l e c t r i c a l

e n g i n e e r i n g . H i s f i e l d o f i n t e r e s t i n c l u d e s , e l e c t r i c a l m a c h i n e s , power system c o n t r o l and modeling of system d y n a m i c s .

r e s e a r c h and t e a c h i n g

' a s s i s t a n t i n El-mansoura i Univ. P r o f e s s o r Abdel Kader r e c e i v e d h i s Ph. D i n E l e c t r i c a l E n g i n e e r i n g from Nottingham U n i v e r s i t y , England, i n 1979. S i n c e 1979 h e i s a s t a f f member a t Suez Canal Univ. where h e i s c u r r e n t l y p r o f e s s o r i n E l e c t r i c a l E n g i n e e r i n g . H i s f i e l d o f i n t e r e s t i n c l u d e s e l e c t r i c a l machines and modeling o f s y s t e m d y n a m i c s .