Assessment of filtering properties of asynchronous electric drive with pulse width modulation

. The process of pulse-width modulation in the converter-motor system is considered. It is noted that the main task of pulse width modulation of voltage formed by the electronic key converter in the converter-motor system is to reduce current pulsations during the modulation period. It is established, that the dynamic process of current change in the electric machine during the period of pulse-width modulation can be approximated by R, L filter. It is shown, that the most important task of pulse-width modulation is minimization of modulation error by current in electric motor. The analysis of possible criteria of estimation of modulation process quality in control systems of electric drive is carried out. It is noted that as a criterion of optimality to assess the quality of modulation in the converter-electromotor system, it is advisable to use the integral quadratic criterion of relative error, called the local variance of the current, as well as the average value of the local variance of the current at the period of the modulating function, called the integral variance of current. The process of estimating the filtering ability of electric machines as part of the electric drive is considered. A system of basic values based on nominal values of electric motor stator voltage, its nominal angular speed and nominal stator current has been selected. The index of filtering properties of the load considering introduction of basic values has been proposed. The equations of an asynchronous electric machine with constant coefficients in the complex form of writing are given. The dependency of the real part of roots of the characteristic equation for an asynchronous machine as a function of angular speed of the rotor rotation is constructed. An analytical expression for the estimation of the filtering properties of asynchronous machines is obtained. The comparison of different types of electric machines according to this parameter is made.


Introduction
Control of electric power flows with the development of power electronics by means of pulse width modulation (PWM) has found wide application in various fields of technology and, in particular, for electric motor control in automated electric drive systems.In practice E3S Web of Conferences 363, 01025 (2022) https://doi.org/10.1051/e3sconf/202236301025INTERAGROMASH 2022 PWM is most widely used in asynchronous electric drive for implementation of frequency and vector control systems.The major task of PWM voltage generated by electronic-key converter in converter -electric motor system is reduction of current ripple on modulation period [1][2][3][4][5].In this case, PWM refers to the process of approximating the voltage pulses to the desired smooth voltage, which is necessary for motor control.The measure of discrepancy between the desired voltage and its pulse approximation is a characteristic of the modulation quality.The smaller this discrepancy, the higher the modulation quality.PWM quality is significantly affected by the pulse repetition rate.At the same time, an increase in modulation frequency leads to an increase in dynamic losses in the electronic keys of the frequency converter, which is an essential element of the electric drive.Therefore, improving the modulation quality by increasing the modulation frequency alone does not give the desired effect and leads to additional losses.It is necessary to use all possibilities to improve the modulation quality.Thus, improvement of quality of modulation of voltage on a load objectively leads to reduction of losses of power and decrease in vibration-noise characteristics of electric drives.Hence the importance of tasks aimed at improvement of quality of modulation in electric drive system.
The important question of providing of qualitative processes of modulation is a choice of criterion estimating this process.Nowadays a sufficiently large number of methods realising PWM is known, for comparison of which quality criteria of modulated voltage [6], [7] and current [8,9] are used.The harmonic coefficient [6], [10] is quite often used as a measure of PWM quality.The standard deviation of the modulated voltage from the modulating voltage [6] as well as the power loss criterion in the motor [11] are also used.
It should be noted that, on the one hand, minimization of modulating power loss is achieved with a minimum of current harmonic coefficient.On the other hand, current harmonic coefficient as PWM optimization criterion is equivalent to current dispersion.It follows that the current dispersion is proportional to the modulation power loss in the motor winding.In [12][13][14] issues of optimal pulse width modulation in inverter-motor system according to the criterion of minimum current dispersion in the load are considered.At the decision of these tasks the necessary condition is the estimation of filtering properties of drive motors as a part of electric drive.
As it is known, at the output of any electronic-key system implementing PWM, a lowfrequency filter is required which provides the required quality of currents flowing through the load.As a load of electronic switched-mode electric energy converters synthesizing voltage at the PWM output, in this paper we consider an asynchronous electric motor, which itself is a low-frequency filter.Therefore, in order to estimate the current dispersion, which should be considered as a criterion for PWM optimization [2], it is necessary to estimate the filtering properties of the electrical machines.In [15] a characterisation of the filtering properties of the load ε is introduced.It represents the ratio of modulation period to time constant loads L, R.
To estimate the filtering properties of any dynamical system whose behaviour is described by differential equations, it is necessary to investigate the relation of its parametric space with state variables.To carry out such a study in the general case seems to be rather problematic.For dynamical systems, whose behaviour is described by linear differential equations, this relationship is defined by a characteristic equation.It is known that negative real parts of roots are characteristics of decay rate of dynamical processes.Inverse values of real parts of roots are time constants.Maximum root of the characteristic equation pмах can be used to assess the filtering properties of the load ε = T0•pмах.
For the problem under consideration, the dynamic behaviour of an induction electric machine is described by differential equations with periodic coefficients.In this case it becomes problematic to write the characteristic equation directly from the original differential equations.In this case it is expedient to subject the equations of asynchronous machine to Lyapunov transformation [3] which allows reducing the initial equations to linear differential equations with constant coefficients.In this case problem of finding roots of the characteristic equation is simplified significantly.
In investigations of filtering properties of asynchronous electric motor, it is assumed that speed of electromagnetic processes considerably, by an order of magnitude, exceeds speed of mechanical processes.Therefore, in the analysis of electromagnetic processes we will assume that the rotor speed of the electric motor does not change.

Methods and materials
To evaluate the filtering properties of an induction motor, the first step is to draw up a mathematical description of the machine, which is presented in the form of differential equations.When investigating differential equations, it is convenient to use relative units of different physical quantities.This convenience is due to the fact that the results of investigation of different electric drives, characterized by significant differences in power, become universal and comparable.In addition, the introduction of relative units, based on the theorems of similarity theory, leads to a reduction in the number of parameters involved in the study, and as a consequence, simplifies the process of analysis and interpretation of the results obtained.
To introduce relative units of variables used in the mathematical model of the electrical machine, it is necessary to identify a set of basic quantities.Let us divide these quantities into basic and derived ones.As a set of basic quantities, we will take a minimum set of constants with different units of measurement, which allows to express the remaining derived basic quantities with other units through the set of basic basic basic quantities.
When selecting a basic set of basic quantities, it is advisable to proceed from the nominal data of the electric machine: nominal stator voltage of the electric motor Un; of rated stator current In; the rated angular frequency of the stator voltage ωн.Thus, the basic reference values will be defined as follows: The derived basic values are synthesised from the basic values.The result is: Rb = Ub/Ib = Un/In -resistance; Lb = U/bIb/ωb is inductivity; Mb = р⋅(m/2)⋅Ub⋅Ib/ωb -electromagnetic momentum; Тb = 1/ωb -time; Pb = (m/2)⋅Ub⋅Ib -power, m is the number of phases; p is the number of pole pairs.
A variable expressed in relative units is equal to the ratio of the variable to the corresponding reference value.Variables expressed in relative units will be marked with an upper index * .
The filtering properties of the load, taking into account the introduction of reference values, can be written in the following form: here p * мах is the minimum relative value of the real part of the roots of the characteristic equation; f * -is the relative modulation frequency.Note that a smaller value of the ε corresponds to the higher filtering properties of the electric machine.
In order to evaluate the filtering properties of asynchronous electric motor, let's consider the behavior of roots of the characteristic equation as a function of rotor speed of the machine.
Equations of electric equilibrium, relating voltages and currents on stator and rotor windings of the induction machine, written in natural coordinate system, have the following form: here R1 is stator winding resistance; R2 is rotor winding resistance to stator; L1 is stator winding leakage inductance; L2 is the stator-current leakage inductance of the rotor winding; US is vector of stator phase voltages; IS is vector of reduced stator currents; IR is rotor current vector; LSS = (2/m) L0⋅DS T (0)⋅DS(0) defines stator winding inductance matrix; LRR = (2/m) L0⋅DR T (0)⋅DR(0) shows rotor winding inductance matrix; LSR(γ) = LRS Т (γ) = (2/m) L0⋅DS T (γ)⋅DR(0) is the matrix of mutual inductances of stator and rotor windings; L0 is the main winding inductance of the machine; DS(γ) = DR(γ) is a phase matrix, which is a Lyapunov matrix.The phase matrix is the projections of the magnetic axes of the stator phase windings onto the d, q coordinate axes in the cross-sectional plane of the machine's rotor: It is assumed that the coordinate axis d coincides with the direction of the highest magnetic conductivity of the rotor, and the coordinate axis q coincides with the direction of the lowest magnetic conductivity of the rotor.
Stress equations ( 2) have periodic coefficients.Their direct use for control system synthesis is problematic.Therefore, it is expedient to transform them to equations with constant coefficients.Considering that γ=(ω1⋅t) for the stator and γ=(ω⋅t) for rotor, we get: After transformations, the system of equations ( 2) for a constant-coefficient induction machine will be where L01=L0+L1; L02=L0+L2; ω1 is an angular frequency of stator currents; ω2 is angular frequency of the rotor currents.
The equations written for two-dimensional vectors in the d, q coordinate axes can be rewritten in the complex form of notation by formal replacement E per imaginary unit j, and vectors U1, I1, I2 to complex variables u1= u1d+j⋅u1q, i1= i1d+j⋅i1q, i2= i2d+j⋅i2q.
Then the equations of equilibrium of the electric machine in the complex form of recording will take the following form: Equation ( 5) corresponds to the characteristic polynomial: Y(p) = a2⋅p 2 + a1⋅p + a0, where: Since the characteristic polynomial is of second order, find the roots of the characteristic equation Y(p) = 0 can be calculated with the formula: We will assume that the basic inductance is sufficiently large and has no significant influence on the dynamics of the transients and, therefore, on the values of the roots of the characteristic equation.In addition, we will assume that the active resistances of stator and rotor windings of the induction motor are the same: R1=R2.Based on these assumptions, let us simplify the expression for the roots of the characteristic equation: ( where sk = R2 * /(L1 * +L2 * ) is critical slip; ω2 * =ω2/ ωb is the relative magnitude of the angular frequency of the rotor currents; ω * =ω/ ωb is the relative value of the angular electric speed of the rotor.Error due to assumption L0 → ∞, for the real parts of the roots, the error is significant only in the vicinity of ω * =0.
The subordinate expression in the formula for the roots of the characteristic equation can be either real or imaginary.It is negative if ω * ∉[-2⋅sk,2⋅sk].In this case, the real parts of the relative values of the roots of the equation are the same and are defined by the expression When controlling an electric motor, an approximate constant ratio is maintained to keep the magnetisation of the magnetic core constant ω * /f * .Therefore, based on the expression (1), we can conclude that the lower values of the filtering property index of the asynchronous electric motor have at the speed ω * =1.Consequently, it is appropriate to evaluate the filtering properties of asynchronous electric machines by the parameter: The relative parameter values of the different types of asynchronous machines take the following values:

Discussion
The research carried out makes it possible to compare the filtering properties of different electric machines.Thus, the obtained expression for the index ε allows actuators with different electric machines to be operated according to the filtering capacity criterion.At a rated winding voltage frequency of 50 Hz, the relative switching frequency f, * is generally greater than a value of 20.Consequently, the filtering property parameter for the different types of electric machines will be determined by the following values: reactive machines with toothed rotor and stator ε=(0.06÷2.4);Reactive machines with anisotropic magnetic conductivity of the rotor (cross charging) ε=(1.35÷4.0);reactive machines with anisotropic magnetic rotor conductivity (longitudinal charging) ε=(2.0÷7.0);asynchronous machines ε=(3.1÷12);synchronous machines ε=(1.6÷14).Large values for the indicator ε refer to smaller machines.From the values given, it follows that reactive machines with a toothed rotor and stator have the best filtering properties.However, their power factor is rather low.Reactive machines with anisotropic rotor magnetic conductivity are superior to asynchronous and synchronous motors in their filtering properties.

Conclusion
The PWM process is considered as the problem of optimal approximation of a given smooth function by a pulse function.The local current dispersion in the load is chosen as the optimization criterion.If modulating functions are periodic, the integral current variance, which is defined as an average value of the local current variance per period of the modulating function, is proposed for estimation of modulation quality.As a result of fulfilled research, the following results have been obtained.
1.It is concluded that at the present time the most widely used frequency-controlled electric drive, in order to improve the quality of which it is necessary to conduct research in order to optimize the process of PWM.
2. The integral quadratic criterion of relative error in the form of local current variance and integral current variance in the form of mean value of local current variance per period of modulating function has been suggested to estimate the quality of modulation in the converter-motor system.
3. Filtering properties of electric machines on modulation interval can be represented by R, L-chain.
4. Analytical expression for estimation of filtering ability of asynchronous electric drive is obtained.
5. Comparison of filtering ability of different types of electric motors is performed.

Fig. 1 .
Fig.1.Dependencies of the effective parts of the roots of the characteristic equation as a function of the rotor angular velocity of an induction motor.