a computer application for teaching and learning on the induction ...

A COMPUTER APPLICATION FOR TEACHING AND LEARNING ON THE INDUCTION MOTOR DYNAMICS G. D. Marques Secção de Máquinas Eléctricas e de Electrónica de Potência Instituto Superior Técnico, Av. Rovisco Pais, 1096 Lisboa Codex, PORTUGAL FAX: 351 - 1 - 841 71 67; email: [email protected]

Abstract: The induction motor dynamics is modelled with five order nonlinear differential equations. As the equations are nonlinear and of high order, the traditional method used to teach this subject is difficult, boring and inefficient leading to use a great effort and to spend a considerable amount of time. To solve this problem a computer simulation for teaching and learning on the dynamics of the induction motor has been developed. The computer program has been designed as a complementary tool for the teaching of the induction motor dynamics and it is now used for student laboratory. It is supposed to work both as a teaching aid and a learning tool. The method was designed in a way that leads to easy understanding of the induction motor model as well as to obtain easily the transient response of the induction motor to small and great variations.

1. - INTRODUCTION Innovative learning environments that break the time and space barriers of traditional methods are becoming possible using new technologies. A great improvement can be obtained in the teaching of electrical machines where there are some themes difficult to teach and there is less time to use. This paper presents a computer program that was developed to use in the teaching of the induction motor dynamics in undergraduate courses [1] to [8]. The power of the proposed tool lies in the ability to study the dynamic behaviour of the induction machine in the absence of complicated mathematics. The program was designed to illustrate clearly the effects of the Park’s transformation. This is a subject that is normally very difficult to learn. By representing the model of the induction motor in a generic reference frame rotating with a angular speed ωt, and simulating some transients, with different speeds, the students learn quite well this concept. The model of the induction machine that was used has a simplified representation because the linkage fluxes were chosen as state variables in spite of the currents as is normally the case. Using the linkage fluxes, the model

becomes simpler and the representation by block diagrams becomes evident. MatLab/Simulink was used to solve the differential equations. MatLab is a well-known software package for computer simulation [9]. This software package is commonly used for teaching in undergraduate courses that enables the students to perform the simulation efficiently. Section 2 presents the model of the induction machine that will be represented in block diagrams in section 3. This block diagram representation is well suited for the integration of the equations with MatLab/Simulink. To illustrate the capabilities of the software proposed some results are presented in section 4. Section 5 presents the conclusion.

2. - INDUCTION MOTOR MODELING The traditional model [1] to [8] of the induction motor is used in this computer program. This model is obtained considering the induction motor as 6 magnetically coupled circuits. Figure 1 shows a representation of the induction motor. If no neutral connection is used, the sum of the three stator currents as well as the sum of the three rotor

d ψdr  u = rr idr + − (ωt − ωm )ψqr  dr dt  d ψqr  u qr = rr iqr + + (ωt − ωm )ψdr  dt

2

4

θ

ψqs   L s  = ψqr   M

5 1

6

(3.c)

M  iqs    Lr  iqr 

The electromagnetic torque is given by:

Mem = ψ dsiqs −ψ qsids 3 and the motion is described by: Fig. 1: Induction motor representation

J

currents are null. Only 4 independent variables are needed to model the induction machine. A transformation of variables can be used. The power invariant two-axes transformation is used and is defined as:

 x1     x2  =  x3 

  1  2 1 − 3 2  1 −  2

0 3 2 3 − 2

1   2 x  α 1   xβ 2  x 1  0  2

   

(1)

Where x denotes general variables, voltages, currents or linkage fluxes. Because xo is always null, only two variables are used. A new transformation is also used leading to the representation of the induction motor stator and rotor in the same reference frame that can be stationary or in movement with an arbitrary speed. This transformation is defined using equation 2.

xα  cos γ x  =   β   sin γ

− sin γ  xd    cos γ  x q 

(2)

dωm = M em − M c dt

3.- BLOCK DIAGRAM

The model of the induction motor in dq variables is represented by (3), (4) and (5). Currents or linkage fluxes can be used as state variables. Traditionally, the currents are used as state variables. In the computer program described in this paper, the linkage fluxes are chosen as state variables because this leads to a major understandable model and it was verified with the comparison with other models that this leads to faster programs. The MatLab/Simulink environment was used leading to a model written in understandable blocks. Figure 2 shows the general block diagram that represents the above equations. The variable ws=ωt represent the speed of the common reference frame used. This speed is an arbitrary speed.

Where γ is the transformation angle. For the induction machine in an arbitrary reference frame with γs=ωtt+θo and γr=γs-θ. The model of the machine is given by:  d ψds u = rsids + − ωtψqs  ds dt  d ψqs  u qs = rsiqs + + ωtψqs dt  ψds   L s ψ  =  M  dr  

M  ids  L r  idr 

(3.a)

(3.b)

Fig. 2: Block diagram of the induction motor.

The “abcDQ” and “Dqabc” blocks realise the transformations of variables defined by (1) and (2). The block “Indução” represents the induction motor in dq variables in a common general reference frame. This block is described in (fig. 3).

Fig. 5: “ModEstator” block.

Fig. 3: DQ model of the induction machine.

The DQ model of the induction machine is represented with 4 sub-blocks and the representation of equation 5. The “ModEstator” and “ModRotor” are similar blocks that represent the stator and rotor equations respectively, (3.a) and (3.b). These blocks are represented in (fig. 5). The “Momento” block determines the electromagnetic torque using (4), fig 6, and the block “Fluxos-Correntes”, (fig. 4), computes the four dq currents using the four linkage fluxes and the inverse inductance matrix.

Fig. 6: “Momento” block.

4.- EXAMPLES 4.1.- Starting transient To show the possibilities of the computer program described above, several examples are presented on this paper. In this sub-section, a starting transient is presented. A Induction motor of de 3HP rated power is used. Simulation in real time is obtained with a pentium 120 MHz computer. 50 40 30

Fig. 4: “Fluxos-correntes” block. Stator current [A]

20 10 0 -10 -20 -30 -40 -50 0

0.2

0.4 0.6 Time [sec]

(a)

0.8

1

60 150

40

ids dq currents [A]

100

Rotor current [A]

50

0

20

0

iqs

-20 -50

-40 -100

-150 0

-60 0 0.2

0.4 0.6 Time [sec]

0.8

1

0.4

0.6

0.8 1 Time [sec.]

1.2

1.4

1.6

Fig. 9: dq currents.

(b) Fig. 7: Stator and rotor currents during the starting transient. 80

60

40

Torque [Nm]

0.2

20

0

Figures 7, 8 and 9 show the transient performance obtained. The starting transient is realised in 0.5 seconds. In these figures the starting current and the oscillation on the transient torque are clearly visible. The transformation of the frequencies realised by Park’s transformation is visible comparing (fig. 7.a and 9). In this last figure the dq currents in a reference frame synchronous with the rotating field are presented. The figure shows also the application of a mechanical load to the shaft that will be described in the next sub-section.

-20

-40 0

0.2

0.4 0.6 Time [sec]

0.8

1

In this transient a step on the load torque in the time instant t=1.2 sec was applied. Figures 10 to 13 show the performance obtained. In this case, it is clear that no big currents arise on the windings. A small decreasing on the speed is shown in fig. 11. This is a typical behaviour of the induction motor. Figures 12 and 13 show clearly the different frequencies arising on the stator and on the rotor of the machine.

160 140

Angular speed [rad/sec]

120 100 80 60 40 20 0 -20 0

4.2.- Step on the load torque

0.2

0.4 0.6 Time [sec]

0.8

1

Fig. 8: Electromagnetic torque and speed during the starting transient.

30

25

20

20

10

Rotor current [A]

Torque [Nm]

30

15

10

5

0

-10

-20

0 1

1.1

1.2

1.3 Time [sec]

1.4

1.5

-30 1

1.6

1.1

Fig. 10: Electromagnetic torque.

1.2

1.3 Time [sec]

1.4

1.5

1.6

Fig. 13: Rotor current..

200

4.2- Starting of a 2250 HP motor

180

Angular speed [rad/sec]

160 140

For the correct understanding of this subject it is important to know the differences between low-power and high-power machines. Figures 14 to 16 show the starting transient of a 2250 HP motor. Figure 16 shows the same transient in dq variables in a reference frame synchronous with the rotating field.

120 100 80 60 40 20 0 1

1.1

1.2

1.3 Time [sec]

1.4

1.5

1.6 5000 4000

Fig. 11: Angular speed.

3000 2000

Stator current [A]

15

Stator current [A]

10

5

1000 0 -1000 -2000 -3000

0

-4000 -5

-5000 0

-10

-15 1

0.5

1

1.5 Time [sec]

(a) 1.1

1.2

1.3 Time [sec]

1.4

Fig. 12: Stator current.

1.5

1.6

2

2.5

3

A comprehensive tool for the teaching and learning of the induction machine dynamics was developed. The computer program uses linkage fluxes as state variables. A simple and efficient model was obtained using MatLab/Simulink environment. This tool was used to teach leading to excellent results. Because the model is very simple and flexible, several transients in different conditions can be studied. This theme becomes very interesting for the students and the results obtained are very good.

5000 4000 3000

Rotor current [A]

2000 1000 0 -1000 -2000 -3000 -4000 -5000 0

0.5

1

1.5 Time [sec]

2

2.5

3

Appendix

(b) Parameters of the 3.2kW induction machine p=2; Uds=380;Uqs=0;ws=2*pi*50; rs=1.5;rr=0.22;Ls.14;Lr=0.017;Mo=0.0446;Jin=0.1;

Fig. 14: Stator and rotor currents.

200

Parameters of the 2250HP induction machine p=2; Uds=2300;Uqs=0;ws=2*pi*60;

180

Angular speed [rad/sec]

160 140

rs=.029;rr=0.022;Ls=(13.04+.226)/ws;Lr=Ls;Mo=13.04/ws; Jin=63.87;

120 100 80 60

Acknowledgement: Part of the costs related with the publication of this paper was supported by CAUTL.

40 20 0 0

0.5

1

1.5 Time [sec]

2

2.5

3

References

Fig. 15: Angular speed. [1] G. J. Retter, “Matrix and Space-phasor theory of electrical Machines”, Akadémiai Kiadó, Budapest, 1987.

5000

[2] D. O’Kelly and Simons, “Introduction to Generalised

dq currents [A]

ids

Electrical Machine Theory”, McGraw-Hill, 1968.

0

[3] David C. White and Herbert H. Woodson “Electromechanical Energy Conversion” John Wiley & Sons, Inc. 1959

iqs -5000

[4] Charles V. Jones “The Unified Theory of Electrical Machines” Plenum Press 1967 -10000 0

0.5

1

1.5 Time [sec]

2

2.5

3

[5] Adkins, B. “The General Theory of Electrical Machines” Chapman & Hall Ltd, 1962.

Fig. 16: dq currents. [6] Concordia, Ch. “Synchronous Machines” Wiley, New York, 1951.

5.- CONCLUSION

[7] Hancock, N.N., “Matrix Analysis of Machinery” 2 nd Ed., Pergamon, Oxford 1974.

Electric

[8] Krause, “Analysis of Electric Machinery”, McGrawHill, 1986. [9] Matlab Users Guide. Mathworks, 1993

[10] Kwok-Tong Chau “A software tool for Learning the Dynamic Behavior of Power Electronics Circuits” IEEE Trans. on Education, vol. 39, Nº1, Feb. 1996.