Tallinn Technical University, Estonia
Abstract. The paper describes a method of parameter identification of an equivalent circuit of an induction motor using fuzzy logic controller. The method is based on the step-by-step approach in which the parameters are calculated from an equivalent circuit and real measured speed-torque characteristic. The displacement of two characteristics as a complex input variable for a fuzzy logic controller is used. In order to demonstrate the reliability of the proposed methods, an example of speed-torque characteristic of induction motors and parameter determination of an equivalent circuit is discussed.
Keywords. AC induction motor, speed-torque characteristic,
equivalent circuit parameters, fuzzy logic controller
During the last years, the significance of the squirrel
cage induction motors in speed and position controlled drives have grown
drastically. The reason is the large-scale exploitation of the AC induction
motors in the technolgies which traditionally used DC motors. Futhermore,
to achieve high efficiency of the technology, many non-controlled AC drives
are reconstructed by adding frequency converters, and they are used now
as speed controlled drives. To attach perfect static and dynamic qualities
of these drives, control engineers need more information about the control
object. Therefore, the importance of the characteristics and parameter
determination of the squirrel cage induction motor has markedly grown.
The most important means of the drive design and control is a perfect model
of a motor. Moreover the model-based control methods of the induction motor
drives are most effective and usable.
Many different models available for squirrel cage induction motors are sucessfully used for drive design and control. Generally, these models can be classified as static or dynamic, linear or nonlinear, partial or integral (Fig. 1). In addition, models with lumped or distributed parameters exist. A single-phase equivalent circuit with lumped parameters is the most traditional model of an AC induction motor. Several modifications of the Kloss formula are derived from this circuit for the calculation of speed-torque characteristics. However, for a squirrel cage induction motor, this model does not satisfy the conditions of exact calculations.
Parameter determination of an equivalent circuit for a squirrel cage induction motor is a complex problem, because no reliable theory exists and no methods of direct measurements in a rotor circuit are available. The skin effect in the rotor winding and the iron core saturation effect lead to complications in the modelling process of a squirrel cage motor. Therefore indirect measurement methods and calculations must be used for the parameter determination from the data given by reference or by experimentally measured speed-torque characteristics.
It is relatively simple to calculate speed-torque characteristics from the data of an equivalent circuit, but it is much more complicated to solve the inverse problem - to calculate parameters from a referred characteristic, because an equivalent circuit can be varied and many different parameters can be chosen.
There are many kinds of equivalent circuits, used to calculate the speed-torque characteristics (Fig. 2). The equivalent circuits a and b could be sucessfully used for wound rotor induction machine, but due to skin effect in the case of a squirrel cage motor, the calculated speed-torque characteristic differs markedly from the real characteristic.
The scheme c could be used for induction motors with large airgap, such as the magnetohydrodynamical (MHD) pumps and some types of linear motors. In special cases, also the schemes d and e could be used. The scheme e is useful for the analysis of vector controlled AC induction motor drive, where the electromotive force is observed and the AC motor like a DC motor can be accepted. There are many ways to calculate speed-torque characteristic of the squirrel cage induction motor.
Figure 1. The classification of models
The method, offered by E. Risthein , allows us to calculate the torque in a large range of slip variation s = 0...2. The torque can be calculated as a sum of two or three components, each of which is similar to the Kloss formula. The basic advantage of the given method is the possibility to calculate the speed-torque characteristic in a large range of rotor slip variation and the fact, that the calculations take into consideration the real physical effects. Due to the skin effect and variation of the rotor resistance during the starting process there are two torque maximums. The drawback is that it does not enable the calculation of the speed-torque characteristic as a function of equivalent circuit parameters.
The second way to calculate the speed-torque characteristic for the equivalent circuits given in Fig. 2, is most traditional, but it is possible only for the small slip s < smax condition. Due to the skin effect, the calculated torque on the slip range 2 > s > smax differs considerably from the real torque.
The third method can be characterized as an exact, but a very complex one. Therefore any simplest method of calculation of the speed-torque characteristic, based on the physical model of the induction motor and giving satisfactory results, is useful.
We can combine the second and the third method, and the skin effect in rotor winding can be considered if we use a special class of equivalent circuits. The Figure 3 shows the three versions of equivalent circuits, which allow the calculation of the speed-torque characteristics for a large-range of slip variation.
The rotor winding equivalent circuit consists of a chain
of RL links with lumped parameters. The most suitable equivalent circuits
are shown in Fig. 3a and 3b. The number of rotor circuit
resistance and inductance components may be varied. Furthermore, the rotor
inductance can be segregated into frequency dependent and saturation dependent
components . The three link (four impedants) equivalent circuit (ladder
network) is shown in Fig. 3c and is described in .
Figure 2. The equivalent circuits of AC induction motor
Figure 3. The equivalent circuits for considering the skin effect of the AC induction motor
EQUIVALENT CIRCUIT PARAMETER DETERMINATION
The usual approach is to divide the deep bars of rotor winding into selections and form two or more separate notional "cages". The parameters of partial cages are then calculated from the proposed dimensions of these selections. The multi-link (ladder network) equivalent circuit parameters can be computed by using single slot finite element method and by approximative choosing of parameter values, which give best agreement between the impedance of the bar and the input impedance of the lader network, over the desired frequency range .
An alternative rotor impedance approximation by choosing
of component values can be realised, if we use the method, based on the
step-by-step approach of the calculated from equivalent circuit and real
measured speed-torque characteristics. The displacement of two characteristics
as a complex input variable for the fuzzy logic controller is used.
Calculation of initial reference parameters
Commonly, the AC induction motor is defined by the output
parameters, given in the catalogues:
Nominal output power, Pnom
Nominal frequency, fnom
Nominal voltage, Unom
Nominal speed or slip, snom
Nominal power factor cosjnom
Relative maximum torque Tmax / Tnom
Relative starting torque Tst / Tnom
Relative minimum torque Tmin / Tnom
Relative starting current Ist /Inom
These output values can be used for the calculations, to determine the parameters of any proposed equivalent circuits. First, the rated parameters for initial reference must be calculated.
The nominal speed
Determination of the speed-torque characteristics
The methods of the determination of the speed-torque characteristics can be divided into three groups.
The second method, providing the direct measurement of torque, is most accurate. However, torque sensors are usually relatively expensive. Therefore this method may be used mainly in large stationary laboratories.
The third method is the simplest and the cheapest one. Futhermore, it allows us to determine the characteristic in dynamic conditions. Thus, this method was realised in the laboratory.
The squirrel cage induction motor was loaded dynamically by inertial mass to slow down the starting process. The signal of the speed sensor (in this case DC tachogenerator) was registered by digital oscilloscope. Next, the PC data processing was realized. Data processing included: 1) the numerical filtering of the measured signal; 2) the approximation of the transients with polynomial functions; 3) the function differentiation and calculation of the acceleration transient process, and 4) the calculation of the speed-torque characteristic. The results of this data processing are shown in Fig. 4.
Figure 4. The determination of the speed-torque characteristic of the AC induction motor by the calculation of the derivation of the measured speed curve
The first curve (a) illustrates the real starting
process of an induction motor, which is registered as the sugnal of a DC
tachogenerator, connected to a motor shaft. The next curve (b) is
the numerically filtered signal. The third curve (c) presents the
eighth order polynomial function, which is the best approximate function
to describe the starting transient process of a motor. The fourth curve
(d) is the derivation of the polynomial function. It is proportional
to the torque of the unloaded motor. The last curve (e) is the dynamic
speed-torque characteristic of the induction motor. If a large moment of
inertia occurs, for example, when the high inertia wheel is connected to
the motor shaft, the found curve is very close to a real static speed-torque
characteristic. To calculate the equivalent circuit parameters, the speed-torque
characteristic may be determined experimentally, as we described it or
given by reference (by motor vendors). For the parameter determination,
a special algorithm of step-by-step approach and a fuzzy logic controller
were developed and used.
Fuzzy parameter determination system
The block diagram of the equivalent circuit parameter determination is shown in Fig. 5. The first operations of this algorithm, as the speed-torque characteristic determination was described in the preceding section of this paper. The following operation is the step-by-step approximation of the speed-torque curve, to estimate the equivalent circuit parameters.
The input values for a fuzzy logic controller are obtained by comparing the speed-torque characteristics calculated from an equivalent circuit and the reference. The displacements between two characteristics are evaluated in four points. These are: the rated slip point, the maximum torque point, the minimum torque point and the starting torque point (Fig. 6). These data may be used as referent inputs for calculations. These data, compared with the results of the calculations, yielded 24 = 16 error functions. After the linguistic evaluation of the error functions, the decision making logic of the fuzzy controller is used to approach the characteristics.
To evaluate of the linguistic values, terms less and more were used. The minimal rule base with 16 rules is shown in Table 1. A more exact solution can be achieved, if there are three linguistic terms (less, normal and more) for every variable used (Table 2).
The dimension of the rule base grows markedly with the number of used terms, because 34 = 81. The approximation process will befinished, when the value of the error functions becomes smaller, than the allowed reference value, and the determined parameters of an equivalent circuit will be printed out.
Figure 5. The algorithm of computer controlled determination
of induction motor characteristics and parameters
Figure 6. Four-point error functions of the characteristic
The rule base consists of IF x AND y THEN z kind of sentences. For example, in the case of the state MLML (Table 1) and an equivalent circuit in Fig. 3a, the rule R11 can be realized as:
IF Tmin > T'min AND Tst < T'st AND Tn < T'n AND Tmax < T'max THEN values (Rs, Ls, Lr1, Lr2, Rr1, Rr2, Rr) = new values (kiRs, kiLs, kiLr1, kiLr2, kiRr1, kiRr2, kiRr),
where the coefficient ki is a variable, which considers the value of error (displacement between two characteristics) and the effect of different parameters to the speed-torque curve. The initial reference values of ki are relatively large, but during the calculation process, if the approximate curve becomes closer to the referred curve, the values of ki will also diminish.
The result of the computer controlled step-by-step approach of the speed-torque characteristics and the parameters of an equivalent circuit for a VOLTA motor in comparision with BALDOR motors are shown in Table 3 and Fig. 7.
|Power, W||5 516||11 030||11 000|
|Nom. speed||1 800||1 800||1 500|
Figure 7. The computer controlled process of the step-by-step approach of the parameter determination
The further applications in the field of electrical drives are characterized with: 1) more complex control systems; 2) indefinite conditions of function environment; 3) multi-dimensionality (MIMO - multi-input, multi-output systems) and 4) knowledge-based approach to the control and identification problems.
Consequently, it may be concluded that the computer-aided
fuzzy logic controller is a most suitable means for several tasks of parameter
determination in an electrical equipment diagnostics system (in industrial