T.Lehtla, Ph.D
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

1. Calculation of a characteristic as the approximate function for given experimental or catalogue data, for example, approximation with polynomials or with different modifications of Kloss formula.
2. Calculation of a characteristic for given electromechanical model or equivalent circuit with lumped parameters of an AC induction motor.
3. Calculation of a characteristic for a given field model or construction of a motor, for example, using FEM calculation method and distributed parameters.
The first method is described in [1,2]. The coefficients of the polynomial are not related to with any electromechanical phenomena of an electrical motor. Consequently, we can discuss this polynomial as a special kind of pure mathematical model of an induction motor.

The method, offered by E. Risthein [1], 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 [3]. The three link (four impedants) equivalent circuit (ladder network) is shown in Fig. 3c and is described in [3].

Figure 2. The equivalent circuits of AC induction motor

Figure 3. The equivalent circuits for considering the skin effect of the AC induction motor


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 [3].

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
Efficiency, hnom
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

. (1)
The nominal torque
. (2)
The nominal stator current
. (3)
The summary impedance, resistance and reactance of the equivalent circuit

, (4)


The summary resistance and reactance of the rotor circuit can be approximately determined as:

. (5)

Flux leakages coefficient
. (6)
The main inductance of the AC induction motor
. (7)
The stator and rotor leakage inductances for the first approximation of the equivalent circuit (Fig 5a):
. (8)
The rotor resistances

, (9)


The stator resistance can be measured or proposed approximately as
The parameters above must be regarded as the starting point, to begin the fuzzy process of the equivalent circuit parameter approximation. Another starting point is the speed-torque characteristic of the squirrel cage AC induction motor.

Determination of the speed-torque characteristics

The methods of the determination of the speed-torque characteristics can be divided into three groups.

1. The use of electromechanical AC motor - DC generator system and the torque calculation as the function of the measured DC generator current.
2. The direct measurement of the torque by special torque sensors.
3. The speed transient measurement of an unloaded motor and the torque calculation as a function of acceleration.
The use of DC generator as the integrated loading device and load sensor is the most simplest method for speed-torque characteristic determination. However, in practice the exact torque calculation is relatively complicated due to nonlinearities and friction effects.

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.

[missing figure]

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).

Table 1

Nom. and max. torque
Min. and starting torque

Table 2

LL R1 R2 R3 R4 R5 R6 R7 R8 R9
NM             R i    
MM                 R81


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.

Table 3

Parameter Type0735M


Type 0940M


Type 4A132


Power, W 5 516 11 030 11 000
Nom. speed  1 800 1 800 1 500
Rs, W 0.793 0.350 0.332
Rr, W 0.155 0.086 0.089
Rr1, W 2.24 0.931 1.249
Rr2, W 0.533 0.261 0.178
Lm 0.164 0.117 0.121
Ls 0.00639 0.00393 0.0033
Lr1 0.00393 0.00190 0.0009
Lr2 0.00655 0.00455 0.0041

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 automation).

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