DYNAMIC BEHAVIOR OF A LINEAR INDUCTION MOTOR

Renato C. Creppe (*) Carlos R. de Souza (**) Gilio A. Simone (*) Paul0 J . ASerni (*)

(*) Unesp - Siio Paul0 State University

PO Box 473, 17033-360, Bauru, SP, Brazil Fax + 55 14 231 1718 creppe@bauru.unesp. br

FET - DEE

ABSTRACT

Although conventional rotating machines have been largely used to drive underground transportation systems, linear induction motors are also being considered for future applications owing to their indisputable advantages. A mathematical model for the transient behavior analysis of linear induction motors, when operating with constant r.m.s. currents, is presented in this paper.. Operating conditions, like phase short-circuit and input frequency variations and also some design characteristics, such as air-gap and secondary resistivity variations, can be considered by means of this modeling. The basis of the mathematical modeling is presented. Experimental results obtained in the laboratory are compared with the corresponding simulations and discussed in this paper.

1. INTRODUCTION

Linear induction machines are usually flat, with either long stator(primary) and short secondary or short stator and long secondary (the secondary is also referred to as linor). A schematic picture of a linear machine is shown in Figure 1.

However, when investigating the machine fundamental characteristics, laboratory prototypes use good conductivity discs made of aluminum or copper to get a very long secondary, which is not straightforward or cheap when using long flat motors. Sector and arc shaped motors are examples of linear- behaving machines which are, on the other hand, similar in construction to the rotating ones. As the stator iron arc does not cover all the secondary tubular surface, much of the flat linear machine

(**) Unicamp-University ofcampinas

PO Box 6101, 13081-970, Campinas, SP, Brazil Fax: +55 19 239 1395

c:hefinho@dsce. fee.unicamp .br

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behavior is present. Some of these machines characteristics are discussed in section 4.

d I E b .........................

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Figure 1: Linear machine basic configuration (a): front view ; (b) side view

2. THE, LINEAR MACHINE DEVELOPED THRUST CALCULATION

For regular simulations, the linear motor can be modeled 1-hrough the one-dimensional theory [ 11, also using the concept of equivalent circuit which allows the consideration of both the longitudinal and transversal end-effects that cannot be disregarded in this kind of machine. These effects are incorporated into the equivalent circuit by means of the required

corrections of the parameters. Otherwise the calculated force magnitudes can’t match the corresponding values which are obtained by experiment in the laboratory.

can be calculated [2] by means of: The linear machine developed force or thrust fxr

where jl , the stator current linear density, given by:

/ \

and b*(x,t) is the complex conjugate of the air-gap flux density b(x,t):

b(x,t) = BS.CO o.t - - - I - ~ s + 9 Y I (3)

Equation (3) is well known as the result of the application of the one-dimensional theory to the linear machine. The first of the three terms shown on the right hand side of equation (3) is the flux density fundamental wave bs(x,t); the second term - the wave bl(x,t) - represents the entry longitudinal end effect The linear machine behavior is modified by this wave as it deeply affects the machine developed thrust. The remaining term - the wave b2(x,t) - represents the exit longitudinal end eflect. Note that bs(x,t) and bl(x,t) are propagated in the same direction but b2(x,t) is traveling in the opposite direction and attenuates more than the wave b 1 (x,t). Therefore, for some specific calculations, equation (3) can be considered as having only the first two terms on the right hand side.

The thrust fxr can be considered as the sum of two components:

fxr = fx + fxe (4)

The component fx (Equation 5) , that is calculated as if the machine were a conventional rotating one, is given by:

T dL12 I, fx = I , - d X

where the expression of the mutual inductance is:

cos(n.x / tp) cos(n.x/ tp+2x /3) cos(n.x/ tp -2n /3)

COS(X.X/ t p + 2 ~ / 3)

cos(7t.x / tp)

COS(X. X/ tp -2n/ 3) COS(X.X / tp)

cos(x.x/ tp+2x / 3) cos(7t.x I tp -2n /3)

Appropriate factors [3-51 must be applied to the expression of fx used for calculating the developed thrust so that its simulated magnitude can agree with the actual value fxr of the thrust. This works as if another component (fxe) were introducing the end effects into the modeling. Therefore the realistic (actual) value of the thrust can be obtained.

3. SOLVING THE EQUATIONS

Rotary induction machine dynamic behavior is better obtained through d-q transformations. However, the equations could also be integrated directly. Of course, direct simulation demands more time than it would be necessary when this simulation is being carried out through the transformations. The use of transformations was introduced with the purpose of overcoming the difficulty of working with time function coefficients and became a convenient tool for analogue and digital simulation of electrical machines. In particular, the application of the Park transformation to the induction machine equations is a very interesting idea as the system of differential equations can be quickly solved by this means. The application of the Park transformation is better intended to symmetrical situations and

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sinusoidal-distributed stator windings, which is not the case of linear machines.

The recent development of better and fast computers allows the direct integration of electric machine f i ”en ta1 equations with comparatively low processing times.

4. ESPECIAL MACHINES FOR LABORATORY USE

Two specific machine configurations are very convenient for laboratory use -- the disc-secondary and the arc-stator motors -- as they are very similar to the actual linear machine but cost less. The basic configuration of the disc-secondary machine is presented in Figure 2

3

4

1

1: aluminum disc 2: stator (double-sided) 3: stator winding 4: shaft

Figure 2: The disc-secondary motor hdamentals.

This machine has a double-sided stator and the disc is made of aluminum. The end effects of a linear machine are present in this configuration.

As for the sector-motor, it is an induction machine having a three-phase stator winding, where the stator is not covering all the rotor surface. The rotor is the same of a regular squirrel-cage rotating induction machine. The basic configuration is shown in Figure 3.

Accurate simulations depend on precise calculations of the machine parameters. Several well known procedures are applied for the determination

of rotating induction machines parameters, depending on the desired accuracy.

End

I

Arc-shaped stator

Figure 3: The sector-motor fundamentals.

However, for the linear machine, the traditional no-load running and locked-secondary tests may not produce accurate parameters owing to the presence of end effects. For example, no-load conditions of linear machines may lead to comparatively high (or even negative) values of the slip. That is why the determination of the machine parameters in this work is mainly based on the specific design procedure equations;.

5. APPLICATION RESULTS

In order to certify that the simulated and corresponding experiment results can agree with each other, the flat linear motor, whose main characteristics were shown in figure 2, is considered for the analysis. Consider that this machine, which is at first sit standstill, is subjected to the following transients;:

(a).at t=Os the starting transient consists of a sudden application of the rated stator currents (6A) and the machine has a mechanical load torque of 5N. These conditions are kept until a steady state condition is attained. (b) at %6s, assuming that the machine is already in steadly state, an additional mechanical torque of 3N is a.pplied and kept until another steady state Condition is obtained.

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(c) at t=9s the additional mechanical load is disconnected and another steady state condition is attained.

The simulated variations of the developed force with time, during the described transient, is presented in figures 4-10.

The linear machines were tested in the laboratory and the characteristic of the force as a finction of the slip were measured for comparison with the mathematical simulations. This result is shown in figure 10.

5.1 Graphics

L

a d

0

0

F

r

e 0

0

C

time(s) Figure 4 - Load force (N) x time (s)

I I I I I I I I I I

-2 I I I I , 0 2 4 6 8 1 0 1 2

time (s) Figure 5 - Developed force (N) x time (s)

F

r

e 0

0

C

time(s) Figure 6 - Developed force (N) x time (s)

time(s) Figure 7 - Slip x time (s)

I I I I I -1 I I I I I 1

0 2 4 6 8 1 0 1 2

time (s) Figure 8 - Secondary current (A) x time (s)

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1 5 * 5

. I

-1 5 0 2 4 6 8 10 12

time(s) Figure 9 - Voltage (V) x time (s)

F

r

e 0

0

C

I I I I

0.8 0 0.2 0.4 0.6 1 I I I I -2

(1-slip) - Simulation o o Measured

Figure 10 - Developed force (N) x slip (1-s) - and measured points

As shown in figure 10, the measured and simulated results agree quite well not only for high values of the slip but also for other values of the machine speed. Similar results were also obtained for the sector motor.

6. CONCLUSIONS

of the lorlgitudinal end effect was also an‘interesting decision because it could represent the machine behavior in the most critical range of high speeds. Some simplification was also possible when analyzing the machine by dealing directly with the fundamental equations instead of applying transformations, which can be a good alternative.

7. REFERENCES

[ l ] S. ’Ya”ura “Theory of Linear Induction Motors”. University of Tokyo Press, 235 p., 2. ed., T~okyo, 1972.

[2] T. Hirasa, S . Ishikawa, T. Yamamuro, Equivalent Circuit of Linear Induction Motors with End Effect Taken Into Account, Trans. IEE Japan, vol. 100, no 2, 1980.

[3] R. C. Creppe “Linear Induction Motors Dynamic Modelling (in Portuguese)”; PhD. Thesis; University ofcampinas - Brazil, 1997.

[4] J. I:. Gieras, “Linear Induction Drives” - Clareridon Press/Oxford University Press, 297 p., Nova York, 1994.

[ 5 ] E. L. Russell, K. H. Norsworthy “Eddy Currents and Wall Losses in Motors” Proceedings 175,1958.

Screened-Rotor Induction IEE, Vol. 105A, p. 163-

The application of the proposed mathematical modeling for the simulation of the machine behavior proved to give good results for different operational conditions. The use of the factors for the correction

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