Vector Control for Linear Induction Motor

Ezio Femandes da Silva' Euler Bueno dos Santosz Paulo C. M. Machado' Marco A. A. de Oliveira'

'Centro Federal de Educaciio Tecnol6gica de Goiis Rua 75 no 46, Centro

74055-1 10 - Goiinia - Goiis - Brazil

'Universidade Federal de Goitis Escola de Engenharia Ektrica, sin, Setor Universitario

76605-220 - Goiinia - Goias - Brazil efs@cefetgo.br ebs@eee.ufg.br

Abstract- In this paper, a methodology for the vector control for linear induction motor by considering end effects is presented. A comparison with the traditional induction motor model is also performed.

1 Introduction

In the last years the vector control technique has been widely developed and comprehensively applied to speed control of conventional rotary induction motor drivers, resulting in high performance, such as that in the direct current. The basic idea behind the vector control is uncoupling the flux and the torque of an induction motor, such as a torque response similar of a direct current machine is obtained. The flux orientation can be obtained by aligning the rotor flux vector with the reference d-axis, therefore becoming a highly coupled non-linear system dynamic control in a linear and uncoupled one. The aim of the control oriented by the field is maintaining constant the d-axis rotor flux and making null the q-axis rotor flux. In most of the actual applications, the field orientation is performed on the linor (secondary) flux space vector, requiring an accurate knowledge of the linor resistance. The dynamic model of the linear induction motor (LIM) is analyzed by using the dq model of the equivalent electrical circuit with end effects included. A speed inverse function factor is determined to express the effects that the linear induction motor speed cause in the magnetization branch of the equivalent electrical circuit [I]. For a LIM with short primary and infinite linor, where the primary is movable and the linor is fixed, the primary will be continuously entering in a new linoric region. This new linoric region tends to oppose to the sudden increase in the penetration of the magnetization flux allowing a gradual accumulation of the magnetization field density in the air gap. Fig. 1 shows the LIM used in this research.

Rim-

The arising of a new linoric region and its influence in the magnetic field modifies the LIM performance when compared to the traditional induction motor [2]. The spatial distribution of the magnetic flux density along the width of the primary depends on the relative speed between primary and linor [5-8]. For null speed of the primary, the LIM can be considered as having an infmite primary and in this case the end effects can be neglected.

2 Mathematical Model

The q-axis equivalent circuit of the linear induction motor is identical to the q-axis equivalent circuit of the traditional induction motor, i.e. the parameters do not vary with the end effects. However, the d-axis entry linoric currents affect the air gap flux by decreasing h:, . Therefore, the d-axis equivalent circuit of the traditional induction motor cannot be used in the linear induction motor analysis when the end effects are considered.

). i-

Fig. I -Linear induction motor double face with linor in aluminum disc.

0-7803-7852-0/03/$17.00 02003 IEEE

Fig. 2 - dq equivalent circuit ofthe LIM including the end effects

Fig.Z(a) shows the d-axis equivalent circuit which magnetization branch is different from the traditional induction motor. In Fig. 2 (b) the equivalent circuit is the same that the traditional induction motor. From the dq equivalent circuit of the LIM (Fig. 2), the

51 8 IClT 2003 - Maribor. S l o v e n i a

primary and linor voltage equations in a synchronous reference system (superscript “ e ”) aligned with the linor flux are given by [I]:

dh‘ dt

v;s = R i‘ +‘+w,h‘, 5 c

where v, = v B + jv, is the primary voltage,

v, = vd, + jv, the linor voltage, L, = L, + Lm the

primary inductance, L, = L,, + Lm the linor inductance,

L,3 the primary leakage inductance, L,, the linor leakage

inductance, and L, the magnetizing inductance. Subscripts ‘Y’ and “I” denote the primary and linor values, respectively. Subscripts d and q denote d-axis and q-axis values, respectively. vd,, Y,+ are the primary voltages;

vd,, vq, the linor voltages; i, , i, the primary electrical

currents; i,, i,, the linor electrical currents;

h, = hd, + jh, , 1, = hd, + j h , the primary and liner

linkage flux; R, , R, the primary and linor resistances;

Ws, the slip speed; P the pole number; v the linear speed in d s ; D the primaq length in meters; Q is a factor related to the primary length, which quantifies the end effects as a function of the speed (” ). It can be seen in (IO) that the Q factor depends on the inverse of the speed, i.e. for null speed the primary length can be considered infinite and, therefore, the end effects can be neglected. One can see that the primary length decreases with increasing speed, therefore increasing the end effects. This will reduces the magnetization current of the LIM. This effect can be quantified by modifying the magnetization inductance. The resistance inserted in series with the inductance L,(I-f(Q)) in the magnetization branch of the d-axis equivalent electrical circuit is determined 6-om the

increases of the loss with increasing linor inducted electrical cuments in the entry and exit. These losses can be represented by the linor resistance times the f(Q) factor, i.e. R,f(Q). The thrust force is given by [I]:

3 Vector Control for LIM

The vector control for LIM can be analyzed in the same way that the traditional induction motor. One problem in the LIM case is that a new resistance and inductance, speed dependents, are included in the magnetization branch. This will be difficult the flux and thrust force uncouplement. In this paper the parameter variations are neglected. However, new researches will be carried out in the way to develop new techniques to include parameter variations [4].

3.1 Slip speed

In order to determine the slip angular speed, the linor q- axis flux must be considered zero (A;, =o). Then, the linor q-axis electrical current can he determined from (8) giving the following equation:

L, From (4) one can obtain the slip frequency :

hir Replacing (12) in (13) it is obtained:

where T, = L, /R , is the linor time constant.

The slip frequency equation Os, is the same that the conventional induction motor. The main difference is the characteristic h:r.

ii, can be obtained from (3):

Replacing (15) in (7) one can obtain:

(16) h‘, is obtained from (16):

where p is the differential operator representation dldi

519

If we consider constant linor flux in (17), we can obtain:

Replacing (18) in (14), we obtain:

0 , I = i,, (1 + f ( Q ) ) (19)

r, .ids 1 - f(Q)l

For a LIM in low speed operation, i.e. for f(Q) approximately zero in (19), one can obtain an equation to the slip frequency similar to the used for the vector control of the traditional induction motor. The slip frequency w,, , which depends on the rotor time constant and f@), can be determined from (19). However, linor resistance variations can turn difficult to obtain the linor and thrust uncouplement. Several control strategies have been elaborated in order to obtain a vector control that does not depend on the parameter variations. The primary angular frequency can be determined by adding the linor angular speed with the slip frequency. This will give us:

The transformation of the linear speed of the LIM to an angular speed is given by:

( 3

we =w, +CO,, (20)

(21) T P

where: W, represents the linor angular speed and T~ is the step polar, which is given by T p = D/P

From the knowledge that ,,,,=&c/dr, the linor flux position angle related to the primary can be determined by:

lt 0, =-v

8, = JW,.df+ JW,,.dt (22)

I - ”’ 5

Fig. 3 . Block diagram ofthe proposed vector control for LIM

Replacing (24) in (23) one can obtain (25) after some mathematical manipulations.

L:,l = L& (25 )

From (26), it can be seen that the linor d-axis flux (index “2”) is a function of id3,, idr2 and f(Q), where idrI is a control variable.

In order to obtain the control current (;;2,), by maintaining constant the linor reference flux ($), the equation (27) has to be considered:

L”,

The primary reference current (i:,) is applied t o a PI controller as seen in Fig. 3.

The linor flux instantaneous oosition anele have to be used’ in the transformation of the stationary reference systems to synchronous and vice versa, as can be seen in the block diagram of the vector control in Fig. 3.

3.2 Determination of the primary reference electrical current.

The aim of the vector control is obtaining the linor flux and thrust uncouplement. Therefore it is necessary a constant linor flux in steady-state. The linor flux can be separated in two parts as can be seen in the appendix. The “ I ” index parts are independent from the extremities effects and the “2” index parts are dependent from the end effects. The linor flux have to be considered constant, therefore (17) can be rewritten as:

The LIM parameters used in the dynamics simulation are given in table 1.

TABLE I Linear Induction Motor Parameters Used

Paramrtrrs Values (Units) Primly length - D 2 1 0 m

Primary Width 45 mm Number of Poles - P 2

Pale Pitch - IO5 mm

Linor thickness 4,5 m Number of Slots 12

Primly resistance- R, 5.348 n Liner resistance - R, 11,603 n

Primay Inductance - r, 0.1073 mH

Linor inductance - 0.094618 mH Maenetizine inductance - 0.09213 mH

7 -

Air Gap lenglh 8 mm

Inertia Moment - I 0.00241 Kgm‘

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The SIMULINKm dynamics model used for simulation of vector control is shown in fig. 3. Fig. 10 shows a primary current of the LIM when the vector control is applied by using the speed reference and linor flux that have been presented before.

SlMULlNKD

1 : : : : : : : : . I

Fig. 5 -Thrust force simulation Fe

Fig. 6 - Linor d-axis flux (solid line) and linor flux reference (dashed-line) of O S Wb.

Changing the speed reference to 5 mis , one can verify that the d-axis linor flux becomes less than the speed reference

Fig. 8 - Linor speed (dashed-line) and speed reference (solid line) of 2misec.

Fig. 9 - d-anis current (solid line) and q-axis primary current (dashed-line).

Fig. 10 - Voltage of a LIM primly phase ''8''

52 1

Fig. 1 I - Linar d-axis flux for 2 d s (solid line) and S d s (dashed-line) with reference of 0,5Wb.

5 Conclusions

In a first step in studying the vector control for LIM by using vector control techniques, we have obtained good results in comparison with what has been predefined. In order to have a good vector control is necessary that the d-axis linor flux graphics be constant in relation to the preset reference. However, the q-axis linor flux would he zero. Small variations in the linor flux were observed as shown in Fig. 6 and 7. These variations come from the end effects. One can observe that the end effects become more accentuated with increasing speed, as can be seen in Fig.11. The equations (19) and (27) were used in the LIM vector control and they have worked very well in relation to the speed reference and strength load reference as can be seen in Figure 8 and 5 , respectively. It can be seen that in spite of the small d- and q-axis linor flux variation, the responses have maintained without suffering from oscillations maintaining the high performance proposed by the vector control. We point out that new methodologies are been studied in order to consider the parameter variations and minimize the linor flux variations. The vector control technique has been successfully applied to the simulation the speed control of a LIM.

Appendix

A. Separating the dq Primary Currents from the Linor Current The d- and q-axis primary currents and linor current of the LIM can be determined after some mathematical manipulation of the equations ( I ) to (8), and obtained[3]:

The dq primary currents and linor current of the LIM can be separated in two parts, where the fust one is independent from the end effects and the second one is dependent from this effect. The first part behaves as a traditional induction motor current and the second part represents a function of the attenuation that exists in the LIM due to the end effects. In order to obtain these currents, it has to he considered that the linkage flux are separated in two parts: the fust one refers to the linkage flux that is independent from the end effects and it will be denoted by the "1" index. The second part refers to the linkage flux that is dependent from the end effects and it will he denoted by the "2" index. These flux are obtained from the following equations:

x; = ?.;, + ?,L2 qs = qxl + q s 2 h', = vm + G 7 z

(A.3)

(A.4)

(A.5)

= &;.I + q . 2 ( A 4

Inserting the equations (A.3 to A.6) in (A.l to A.2), and after some mathematical manipulations one can obtain the following expressions:

where:

4 id,2 = A , + A , + A ,

4 = A, + A , + A ,

04.7)

( A 4

(A.9)

(A.lO)

(A.11)

(A.12)

(A.13)

(A.14)

(A. 15)

(A.16)

(A.17)

(A.18)

(A.19)

(A.20)

(A.21)

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(A.24)

L, = L,L, - L: (A.25)

4 = + (A.26)

B. Separating the Linkage Flux. The linkage flux can be obtained from the following differential equations:

References

[ l ] K.Nam, J. H Sung, “A New Approach to Vector Control for Linear Induction Motor Considering End Effects,” IEEE IAS annual meeting, 3-7 Oct., in Phoenix, Arizona, pp. 2284-2289, 1999.

[2] J . Duncan, “Linear Induction Motor - Equivalent Circuit Model,”. in IEE Proc., Vol. 130, pt. B, no 1, pp. 51-57, January, 1983.

[3] E.F. Silva, E. B. Santos, M. A. A. Oliveira, “A Novel Dynamic For Linear Induction Motors,” in 7”COBEP, Fortaleza-CE, Brasil, 2003, pp. 713-719.

[4] E.F. Silva, E. B. Santos, P C. M. Machado, “Vector Control For Linear Induction Motors Considering End Effects,” in 7”COBEP, Fortaleza-CE, Brasil, 2003, pp. 958-963.

[5] J.F. Gieras, Linear Induction Drives, University of Tokyo Press, Japan, 1994.

[6] R.C. Creppe, Uma contribuigio B modelagem de Miquinas de InduFBo Lineares, Ph.D. Thesis, UNICAMP, Campinas, S l o Paulo, Brasil, 1997.

[7] E.B. Santos, J.R. Camacho, A.A. Paula, “The Linear Induction Motor (LIM) Power Factor, Efficiency and Finite Element Considerations.”, in ICEM 2002, CD- ROM.

[SI L.M. Neto, E.B. Santos, “Metodos para Determina@o de Induthcias de um Motor de InduFlo Linear”, in Annals of the Congr. Brasil. de Automritica - Uberlhdia-MG, v. 1, pp. 231-236,2000,

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