SQUIRREL CAGE INDUCTION MACHINE MODEL FOR THE ANALYSIS OF SENSORLESS SPEED MEASUREMENT METHODS Jose M. Aller*, Jose A. RestrePo**, Alexander Bueno***, Maria I. Gimen#ez?*'*, Gaston Pesse* Universidad Simon Bolivar - Caracas - Venezuela * Dpto. de Conversion y Transporte de Energia ** Dpto. de Electronica y Circuitos ***Dpto. Tecnologia Industrial
[email protected]
Abstract In this work a squirrel cage induction machine model has been developed to include the stator current effects on the MMF space harmonics, dynamic and static eccentricity harmonics, and slot harmonics. The proposed model is intended to be used on testing the behavior of different sensorless speed measurement techniques. This dynamic model has been developed using the superposition principle and a matrix description of the machine inductance. The model has been validated using experimental data, and tested using a TimeFrequency-Distribution method for sensorless speed measurement [4].
Keywords: Induction Machine, Sensorless Speed Control, Measurements.
The flux density distribution in the airgap is given as the combination of the permeance and the MMF waves. The resulting flux density distrilbution has harmonics that are moving in relation to stator and rotor frames, therefore inducing corresponding electromotive harmonics in the stator and rotor windings. As a consequence, stator current harmonics appear when the machine is excited by a voltage source. The harmonics frequency, present in the stator current signal, is given by the following equation [I]:
where:
fi
supply frequency.
S
slip. number of pole pairs.
Introduction P
Modern AC machine vector control applications are widely used in today drive implementations. A growing number of researches are being done to perform the vector control technique without the use of speed sensor [ I ,2,3]. The sensorless speed measurement can be done using direct or indirect methods [4,5]. However, the indirect methods are in general dependent on the machine model parameters. This could lead to inaccuracy of the rotor speed estimation during normal machine operation. Direct sensorless speed measurement method used in this work takes the information present on the stator voltage or current spectrum. Some of the voltage and current harmonics depend on the machine's geometry. The more important geometric factors modifying the harmonic content are: The magnetomotive force spatial distribution, the presence of rotor and stator slots, and the dynamic and static eccentricities.
0-7803-4434-0 I 98 I $10.00 0 1998 IEEE
Qr %IS
number of rotor slots. stator time
harmonics. nrt nmr n'l
n sn
slot harmonics. rotor time harmonics. dynamic eccentricity. static eccentricity.
Recent works [7,8] have lead to conclude that static eccentricity gives non balance currents in the stator windings, but does not introduces harmonic currents. Dynamic eccentricity has a n important harmonic effect on the low order range. Rotor slots have the principal
harmonic influence in the high order range.
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In this work, a dynamic machine model of the induction machine has been developed to reproduce the principal stator current harmonics due to rotor and Stator slots, and several speed sensorless techniques have been used to prove and develop its performance with this model. Additionally the model can be used to produce design specification sets steered towards the improvement of the induction machine sensorless speed measurements.
The symmetric matrix [s,,~~] and the Cycle matrix [ q k q ] are uncoupled by the symmetric components transformation. The spatial vector transformation has the Same effect. Transforming to spatial vectors the Symmetric matrix and the Cycle matrix, we obtain:
I
_ _I -I [I
a
J][:$
1,2
__ __
Squirrel Cage Induction Machine Model
2
-j=ql 2
a
21
(7)
2
The dynamic machine's voltage equations in primitive coordinates are:
where:
Where:
[YI = [V"!
Vh\
VJ
[ i , ] = [in\ 4,
; [v.]
= [v.
v,
= [OI'
VJ
L]' ; [i, 1 = [L 4,
iJ
The inductance matrices are dependent of the angular rotor position 8. Each inductance parameter in equation (2) was obtained by superposition of the permeance fluctuation due to rotor and stator slots, and rotor eccentricity. The spatial MMF distribution can be also considered in the inductance expression. To simplify . - the model we can neglected the non-sinusoida' MMF distribution and the rotor eccentricity. Using only the principal harmonic due to slots' influence' the inductance and resistant matrices in equation ( 2 ) have the following structure:
(3 1
where:
PI = cosk0 cosk(O-'%) cosk(8-*%)
cosk(0- 2%) cosk0 cosk(0-4%)
cosk(0-4%)
cosk(0-'%) cos,40
sf
k = 3n *[l
q-
k = 3 n + ~-[I
If
=
3n
+
a
[,
a ' ] [C(kB)]= 0 a
a2][~(ke)]
a
a2]
[c(ke)l
- 2
Using the spatial vector transformation [9] to simplify the differential equation ( 2 ) , the induction machine model can be written as nine different models in function ofits stator and rotor slots per pole pair:
Case 1: q , = 3n; n E N
In this particular case, the rotor and stator slots have not influence in the stator or rotor current performance. This case is not possible in practice, because is needed that rotor and stator slots per pole (q, and q , ) have a minimum common multiplier equal one, to avoid reluctance torque problems. Most commercial induction motors have q, = 3 n , but the rotor bars number is chosen to make q, # 3n7.
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1.b) qr =3m+1; mczN
Case3: q\ = 3 n + 2 ; n
E N
three. 3.c) 4,. =3m+2; m E N 1.c) 4,. = 3 m + 2 ; m E N
Case2:
(I,=
3n+I;n~N
2.a) qr = 3m; m E N
Numerical Evaluation The principal problem in the evaluation of the previous nine differential equations is concerned with the inductance variations with the angular position 8. To In some machines the number of the stator slots per pole reduce the numerical work, a simpler description can be and phase are not an integer number [lo]. These performed using the flux leakage formulation in spatial induction motors show a more important harmonic effect vectors. Assuming linearity between fluxes and currents, in the stator currents due to the stator slot. the following relations can be obtained: 2.b) q,. = 3m+ 1; m E N
Using (1 8) in equation ( 2 ) :
[GI= [RI[F] + p [ [~(e)][7']] = [~ ~ [ r ([e i )] +]p [i] ( I9) Where:
[r(e)]:= [ ~ ( e ) ] - ' 2.c) q, = 3 m + 2 ; m E N
[::I=[: ;I[]+
The electric torque can be evaluated directly from the coenergy expression as:
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Only the real part of the equation (20) has a physical meaning. Numerical evaluation of expression (20) can be performed quickly, because the inductance matrix is order two. The machine model's canonical form can be expressed as:
I
$1
i'.. 1 -+
s=I--
f i
QT
-
I
(22)
= [
