PHYSICS REFINED CALCULATION OF INDUCTION MOTOR EQUIVALENT CIRCUIT NONLINEAR PARAMETERS BY AN ENERGY

The topicality of the research aim is caused by the analysis of the processes of energy conversion taking into account the induction motor particular nonlinearities that reveal the physical properties and phenomena in structural materials under the action of electrical and electromagnetic impacts. Taking into consideration the nonlinearities of the induction motor equivalent circuit influences the accuracy of determination of the electric machine operating characteristics. Most conventional methods for parameter identification do not enable assessment of the induction motor nonlinear characteristics and properties. It is proposed to use resistive impedance and inductance dependences on the rotor current to take into account the rotor nonlinear parameters. To form identification equations, the instantaneous power components for the rotor nonlinear resistive impedance and nonlinear inductance have been obtained. The solution of the identification equations resulted in determination of the equivalent circuit electromagnetic parameters taking into account the rotor nonlinear parameters and the amplitudes of the harmonics of the current cosine and sine components of the rotor and magnetization circuit. The results of identification of the induction motor equivalent circuit parameters taking into account the rotor nonlinear parameters have been obtained with sufficient accuracy. This is confirmed by a low error of determination of the electromagnetic parameters. The adequacy of the identified parameters is determined by comparison of the stator current experimental and calculated curves


Introduction
At the stage of improvement of mathematical description of electric machines, the importance of accurate determination of their characteristics with the use of various methods and means of implementation grows. At present, the processes of energy conversion taking into account the induction motor (IM) particular nonlinear parameters are the object of the research. There is also a necessity for the research of electrical energy processes regulation and its conversion into mechanical one when IM operates at constant values of voltage and frequency [1,2]. As a rule, IM nonlinear parameters represent physical properties of structural materials and phenomena in them. These phenomena occur under the impact of electromagnetic actions that can be observed in
Thus, development of the method for identification of IM parameters, taking into account their nonlinearity, is a topical task.

Literature review and problem statement
The interest in the problem of determination of induction motors electromagnetic parameters essentially grew in connection with the development of static frequency converters for the purposes of alternating current controlled electric drive [5]. At that time, there appeared a great number of papers dealing with determination of resistive impedance and inductance concerning simplified and detailed equivalent circuits [6]. The simplest solutions were obtained due to the use of L-shaped equivalent circuits with the magnetizing circuit taken to the terminals. More accurate results were obtained when the T-shaped equivalent circuit was used.
The analysis of the equivalent circuit with linear (constant) values of its components and resistances with taking into account the effect of current displacement in the rotor is common for the papers related to the considered problem [7]. The problem of taking into account the current displacement effect in the rotor is further developed by division of the rotor into several layers [8]. However, the drawback of this approach consists in impossibility to know the IM real geometric parameters and low accuracy of the obtained results. Elimination of the drawbacks concerning the knowledge of the geometric parameters is described in the paper [9], where the parameters of rotor resistance and leakage inductance are changed in order to take into account the effect of current displacement. The method of IM parameters identification [10] based on the integral calculation in the quiescent state for the admitted assumptions concerning IM parameters constancy is rather accurate. However, constancy of parameters in [9] and [10] can only be taken with possible big errors.
In [11], it is shown that creation of IM mathematical model is one of the ways of obtaining information about it. However, the information of the real state of the motor and current displacement effect influence on the rotor parameters is not taken into consideration in [11]. The use of genetic algorithms is rather topical but quite a lot of experiments are required to create a genetic model [12]. The paper [13] contains description of a method developed for the online assessment of the stator and rotor resistance with the use of artificial neural networks. In the mentioned approach, the assessment of the rotor resistance is insensitive to the change of the stator resistance, which is inadmissible.
Thus, there are a great number of the IM parameter determination methods based on the use of mathematical models or neural networks. However, they do not enable the assessment of the nonlinear characteristics and properties of the electric machines of this class.
The energy method is one of the modern methods for IM EMP determination [14]. The paper [14] contains the structure of creation of identification equations and special features of their use for IM EMP determination. Improvement of the energy method is shown in [15] in the form of introduction of the rotor nonlinear resistive impedance into the equivalent circuit; it takes into account the effect of current displacement in the rotor.
The analysis reveals that there are some factors due to which the interest in the considered problem shifts towards the IM characteristics detailing. Here the accent is based on the known propositions concerning IM nonlinear properties. This means that the possibilities of the energy method can be extended due to refinement of the expressions for the nonlinear parameters of the equivalent circuit.

The aim and objectives of the study
The aim of the research consists in the development of the methods for taking into account IM equivalent circuit nonlinear parameters when the energy method is used.
To achieve this aim, the following tasks of the research were solved: -refinement of the expressions describing nonlinear inductance in the function of the rotor current; -determination of the instantaneous power components for nonlinear inductance; -assessment of the efficiency of the induction motor equivalent circuit parameter identification taking into account the refined expressions for the calculation of nonlinear resistive impedance and inductance with the use of the energy method.

The materials and methods of the research
Taking into account IM nonlinear properties refers to the sphere of their parameters identification during post-repair tests and current monitoring of the electric machines condition in the process of their operation.
Literature [1,15] review revealed that the rotor resistive impedance is characterized by nonlinear variation due to the influence of current displacement effect and can be approximated by an even-power polynomial. In this case, the polynomial for the description of the rotor nonlinear resistive impedance can be limited to the second power with sufficient accuracy [15,16]: where R 20 -the rotor resistive impedance without taking into consideration its variation due to the displacement effect; k R -the coefficient of approximation of nonlinear resistive impedance; i 2 (t) -rotor current flowing through the nonlinear resistive impedance.
In the general form, the rotor current can be presented by the dependence: where m -the number of the current harmonic; Mthe quantity of the analyzed current harmonics; I 2m -the value of the rotor current of the m-th harmonic; Ω -the circular frequency of the network; y m -the angle of current harmonics phases shift; I 2am =I 2m cosy m -the cosine component of the rotor current of the m-th harmonic; I 2bm = =I 2m siny m -the sine component of the rotor current of the m-th harmonic.
The effect of current displacement in the rotor at the start also influences the value of inductive reactance that can also be expressed by a nonlinear dependence on current [14]: where L 20 -rotor inductance without taking into account its variation caused by the displacement effect; k L -the coefficient of nonlinear inductance approximation. IM EMP determination taking into consideration dependences (2) and (3) is possible with the use of the energy method. This method is based on the use of complete equations of balance of instantaneous power harmonics components of the power supply and all the consumers [15,17].
When the energy method is used, the system of equations is based on equality of the components of instantaneous power p(t) of the power supply and the sums of powers of all the consumers [14,15]: where j -the index of the corresponding consumer; G -the number of the consumers. In relation to IM, each element of the equivalent circuit is considered to be a consumer (Fig. 1), i.e. consumers include stator resistive impedance R 1 and inductance L 1 , rotor resistive impedance R 2 (I 2 ) and inductance L 2 (I 2 ) and magnetization circuit inductance L µ [6]. The method for calculation of instantaneous power components for linear elements of the equivalent circuit is described in [14,17]. Determination of instantaneous power components for nonlinear resistive impedance is given in [15]. That is why it is necessary to additionally obtain instantaneous power components for the rotor nonlinear inductance.
Instantaneous powers for nonlinear elements of the equivalent circuit are determined as follows: Instantaneous power components for the rotor nonlinear inductance: where k -the number of the instantaneous power harmonic; P 0L2 , P kaL2 , P kbL2 -constant, cosine and sine components of instantaneous power for nonlinear inductance; -in the calculation of P kaL2 , the following is taken into consideration: -for the third summand k=n 2 -n 1 at n 1 <n 2 ; -for the fourth summand k=n 1 -n 2 at n 1 >n 2 ; -for the fifth summand k=n 1 +n 2 ; -for the sixth summand k=2n 1 at n 1 =n 2 ; -in the calculation of P kbL2 , the following is taken into consideration: -for the third summand k=|n 1 ±n 2 |; -for the fourth summand k=n 1 -n 2 at n 1 >n 2 ; -for the fifth summand k=n 1 +n 2 ; -for the sixth summand k=2n 1 at n 1 =n 2 ; ( ) where n -the number of the cosine or sine harmonic component of the square of the rotor current i 2 2 (t); I 2am1 , I 2am2 -cosine components of the rotor current for harmonics m 1 , m 2 ; I 2bm1 , I 2bm2 -sine components of the rotor current for harmonics m 1 , m 2 ; in the calculation of A 0 , m 1 =m 2 is taken into account; in the calculation of A n , B n , the following is taken into account: -for the first summand n=m 2 ±m 1 at m 1 <m 2 ; -for the second summand n=m 1 +m 2 ; -for the third summand n=m 2 -m 1 at m 1 <m 2 ; -for the fourth summand n=m 1 . For a T-shaped equivalent circuit (Fig. 1), the resistive impedances and inductances of the stator, rotor and magnetization circuit are the unknown parameters. The rotor and magnetization circuit current harmonics are not known either. The total number of the unknown parameters is 17.
To get the required number of identification equations, it is necessary to analyze three voltage and current harmonics at a time. Additional equations are obtained due to the use of equations created according to the first Kirchhoff's law. The specific features of the choice of voltage and current harmonics and generation of instantaneous power harmonic components are given in [14].
Thus, with the use of three harmonics of voltage and current (the first, the third and the fifth harmonics), the equations of the balance of instantaneous power harmonic components will be of the form:  I  I  I ; I  I  I ;   I  I  I ; I  I  I ;   I  I  I ; I  I  I where P 0s , P 0R1 , P 0R2 -constant components of instantaneous power of the voltage source, stator and rotor resistive impedances; P kas , P kaR1 , P kaR2 , P kaL1 , P kaLµ , P kaL2 -cosine components of instantaneous power of the voltage source, stator and rotor resistive impedances, inductances of the stator, magnetization circuit and rotor; P kbs , P kbR1 , P kbR2 , P kbL1 , P kbLµ , P kbL2 -sine components of instantaneous power of the voltage source, stator and rotor resistive impedances, inductances of the stator, magnetization circuit and rotor; I 1a1 , I 1a3 , I 1a5 , I 1b1 , I 1b3 , I 1b5 -cosine and sine components of the first, third and fifth harmonics of the stator current; I µa1 , I µa3 , I µa5 , I µb1 , I µb3 , I µb5 -cosine and sine components of the first, third and fifth harmonics of the magnetization current; I 2a1 , I 2a3 , I 2a5 , I 2b1 , I 2b3 , I 2b5 -cosine and sine components of the first, third and fifth harmonics of the rotor current.
To confirm the obtained theoretical provisions, an experimental research was carried out.

The results of the experimental research of the induction motor parameter identification taking into account the nonlinear elements
The experimental research was carried out for IM switched to a transformer via a block of voltage and current sensors. The experiments were performed for the following voltage levels: 220 V, 180 V, 120 V, 40 V, recorded by the voltage and current sensors. Then the required levels of voltage and current harmonics were singled out.
During the experiments, 4АP100L4U3 IM was used: -rated power of 4 kW; -rated voltage of 220 V; -rated current of 8.7 А; -stator resistive impedance of R 1 =1.35 Ohm; -stator inductance of L 1 =0.00676 Hn; -magnetization circuit inductance of L µ =0.246 Hn; -rotor resistive impedance of R 2 =1.38 Ohm; -rotor inductance of L 2 =0.00678 Hn. The equivalent circuit parameters were calculated with the use of the system of identification equations (8), (9) the measured values of voltage and current, as well as instantaneous power harmonic components for the elements. Tables 1, 2 contain the measured values of the stator current harmonics for the following voltage levels 220 V, 180 V, 120 V, 40 V, and also the calculated components of the rotor current harmonics and the magnetization circuit. Table 2 contains the calculated electromagnetic parameters of IM equivalent circuit taking into consideration their nonlinearity. Table 2 also contains the coefficients of approximation of the rotor nonlinear resistive impedance and nonlinear inductance. Besides, these Tables contain the rotor current cosine and sine components I 2a1 , I 2a3 , I 2a5 , I 2b1 , I 2b3 , I 2b5 , resistive impedance R 20 , inductance L 20 and coefficients k R and k L describing the rotor nonlinear resistive impedance and nonlinear inductance.
As a result of the analysis of the obtained data (Tables 1, 2), errors of determination of IM equivalent circuit EMP are calculated; they are shown in Table 3. Maximum error of determination of the parameters of the analyzed motor is 4.07 %. Verification based on comparison of the stator current experimental and calculated curves (Fig. 2) revealed the adequacy of the obtained results as the determination coefficient R 2 equals 0.995. Table 2 Results of determination of induction motor parameters taking into account the equivalent circuit nonlinear elements Coefficient k L , Hn/А 2 0.9853·10 -6 0.9525·10 -6 1.03·10 -6 1.162·10 -6 Table 3 Errors of induction motor parameter determination taking into account the nonlinear elements of the equivalent circuit  For practical purposes, it is convenient to use resistive impedance and inductance dependences on supply voltage for refinement of the start and energy characteristics, creation of models, etc. Dependences on the supply voltage of the efficient value of the rotor current first harmonic, rotor resistive impedance and rotor inductance are shown in Fig. 3-5. It can be seen from the analysis of the mentioned dependences that when the supply voltage changes, the rotor current values change (Fig. 3), the values of nonlinear resistive impedance (Fig. 4) and the rotor nonlinear inductance (Fig. 5)   Thus, the analysis of the obtained results revealed that the energy method allows determination of IM parameters taking into account their nonlinearity in equivalent circuits. Results shown in Tables 1, 2 make it possible to come to the conclusion as to the expediency of the use of the energy method for identification of nonlinearities parameters.
The performed research made it possible to specify the calculation of IM parameters due to the use of the energy method. The essential advantage consists in additional introduction of nonlinear resistive impedance and inductance of the rotor into the equivalent circuit; they take into account the effect of current displacement in the rotor.
The performed research and the analysis of the obtained results revealed the efficiency and expediency of the use of the proposed method for the calculation of the nonlinear character of resistive impedance and inductance by the energy method.
The proposed method of induction motor nonlinear parameter determination with the use of the energy method is recommended for application at electrical-repair enterprises for identification of the parameters of motors with defects or breakages. The described method can be practically realized in creation of the software for determination of induction motor parameters in the problems of outgoing inspection at electrical-repair enterprises and enterprises-manufacturers.
In the future, the proposed method can be used for the analysis of the processes in IM as a nonlinear electromechanical converter. In this case, the number of the identified parameters in the equivalent circuit can be extended.

Discussion of the results of identification of induction motor electromagnetic parameters taking into account their nonlinearities
The expressions for taking into account the nonlinear character of the resistive impedance and inductance of the rotor made it possible to improve the energy method for identification of the induction motor electromagnetic parameters. The performed experimental research and the analysis of the obtained data revealed the efficiency and expediency of the use of the proposed method for identification of the induction motor electromagnetic parameters.
The improvement of the energy method consists in introduction of additional components into the system of identification equations. These components take into account the nonlinear character of the resistive impedance and inductance of the rotor. This allowed enhancement of the accuracy of identification of the induction motor electromagnetic parameters when the energy method is used.
In the future, the proposed method can be used to improve the accuracy of calculation of the induction motor electromagnetic parameters and published data in the problems of outgoing inspection at electrical-repair enterprises and enterprises-manufacturers.

Conclusions
1. Expressions enabling improvement of the method for the calculation of the induction motor electromagnetic parameters due to the introduction of nonlinear elements into the equivalent circuit to take into account the effect of current displacement in the rotor have been obtained.
2. Expressions for the instantaneous power components at the nonlinear resistive impedance and inductance, making it possible to take into consideration the effect of current displacement in the rotor during identification of the electromagnetic parameters by the energy method, have been obtained.
3. The use of the proposed method enabled improvement of the efficiency of the energy method application taking into account the effect of current displacement in the rotor, in particular, it allowed the decrease of the error of the induction motor electromagnetic parameters identification to 4.07 %. The adequacy of the obtained results has been assessed by the determination coefficient that makes 0.995.