ANALYSIS OF OPERATING CONDITIONS OF ELECTRIC MACHINES WITH NONLINEAR MAGNETIC CHARACTERISTICS 1. INTRODUCTION

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1 Prace Naukowe Instytutu Maszyn, Napędów i Pomiarów Elektrycznych Nr 50 Politechniki Wrocławskiej Nr 50 Studia i Materiały Nr Boris SIDELNIKOV*, Galina ROGACHEVSKAJA** mathematical models, electric machines, transient process, saturation effect ANALYSIS OF OPERATING CONDITIONS OF ELECTRIC MACHINES WITH NONLINEAR MAGNETIC CHARACTERISTICS The effect of saturation on operating conditions of electric machines of different types is analyzed, both numerically and experimentally. It is shown that saturation may be accompanied by some phenomena that affect the occurring processes in both a quantitative and qualitative sense. However, this factor may be approximately allowed for in a number of practical important cases. 1. INTRODUCTION One of the most serious assumptions made in the classical theory of electric machines is the neglect of saturation. This assumption has made it possible to simplify the analysis of performance, and led to development of the fundamental method of two reactions in the theory of synchronous machines that has been in successful use up to the present time when solving practical problems in electromechanics. Along with this, the role of saturation in the modern highly utilized electrical machines is rather essential, specifically, at dynamic operating conditions. Due to this fact a great deal of various methods have been suggested by now to take into account the phenomenon involved [1 5]. However, the specialists have not agreed upon the required level of theoretical study of this question. The paper presents and systemizes the results of an analysis of transient performance of electric machines of various types with a due allowance for saturation of magnetic circuits, for the major flux and for leakage fluxes. The employed mathematical models are based on the simultaneous solution of nonlinear equations for electric and magnetic circuits, and describe the properties of real objects with fair details. Results of calculations are compared with experimental data and relationships derived from the simplified models. * Instytut Elektrotechniki, Politechnika Szczecińska, ul. Sikorskiego 37, Szczecin. ** St. Petersburg, Russia.

2 METHODS OF ANALYSIS Mathematical models employed for analysis are based on equations of a generalized electromechanical converter with the effect of saturation allowed for. Eddy currents in magnetic circuits, and also current displacement in windings are allowed for by a system of shorted contours. Their number was determined by required accuracy of process reconstruction, and the parameters were found with the help of methods, known from the literature. The equations of model in the matrix form and in relative units are given conventionally as: [ ψ] d = dτ dω = dτ [ u] ω[ c] [ ψ] [ r] [ i] ([ i] [ c] [ ψ] m T ) t H j 1 To calculate currents, with magnetic flux linkage known at each step of the integration, two approaches are used yielding identical results in the end. The first of them is based on employing magnetic equivalent circuits [4]. The task boils down to a simultaneous numerical solving of differential and algebraic nonlinear equations. It should be noted that it is possible to reflect completely enough all the complex of occurring phenomena, provided full information about the geometry of the magnetic circuit and the magnetizing characteristics is specified for the circuit components. The second approach is based on application of the no-load characteristic, with the help of which the B H curves required for analysis are determined. Such a way gives a quite simple solving algorithm, which excludes iteration procedures in the course of calculations. However, its most accessible realization method is possible when neglecting the interaction between the main flux and the leakage fluxes of the windings. The results obtained for individual modes of operation of commutator machines, as well as asynchronous and synchronous generators and motors, are compared with the experimental ones obtained in laboratory conditions. 3. SATURATION EFFECT IN ELECTRIC MACHINES OF VARIOUS TYPES It is known that due to saturation not only the flux is changed, but also a mutual inductive coupling appears between orthogonal loops. In DC commutator machines the quadrature armature reaction becomes noticeable in this case. Because of disturbance in normal commutating process an armature commutation reaction (i d ) occurs [6], transient voltage drop between the brushes and the commutator (U k ) considerably increases, an additional electromagnetic torque owing to an interaction between the transverse field and currents of commutated sections appears. The above-mentioned is illustrated in Fig.1, which gives a comparison between results calculated (full lines) from the nonlinear model and those experimental (dotted lines).

3 Figure 1 shows sudden short circuit mode at run-out for two DC machines rated at V n = 440 V, I n = 300 A, n n = 1000 rpm. Note that in neglecting the saturation, the current in the armature circuits (i a ) would be much greater than that depicted in Fig. 1. Its maximum value would attain 12.2 p.u. instead of 3.5 p.u., with allowance made for nonlinear properties of the magnetic circuit. Despite an evident influence of saturation, it is possible to obtain sufficiently accurate data employing simplified equations, assuming, for example, the equivalent commutated section resistance, the overall inductance of the armature circuit and leakage inductance of the exciting winding to be constant. An assumption also may be made of linear changing of mmf of the quadrature armature reaction and the voltage drop (U k ) in function of current i a. Alongside this, we are faced with the problem of choosing the averaged parameters corresponding to each individual operating condition. 231 Fig. 1. Sudden short circuit across D.C. machine terminals P n = 132 kw t k Complete allowance for saturation Allowance for saturation in main flux path Neglect of saturation effect second I max M max M min I max M max M min I max M max M min Table For high-power synchronous generators of special interest is the estimation of the effect of saturation in extreme conditions, such as a sudden shorting with its subsequent clearance and recovery of parallel operation with the network. Depending on the moment of shorting clearance, the aperiodical component of the armature winding flux linkages with the decay neglected may reach 2 p.u. Therefore, the overall flux linkage equal to the sum of periodical and aperiodical components approaches under these conditions of neglecting the results of calculations of maximum currents of the armature circuit (I max ) and of electromagnetic torque (M max ) following the clearance of a short-time 3-phase short circuit across a turbogenerator (P n = 500 MW, n n = 3000 rpm) terminals connected to a power system. From the Table 1 one can see that saturation along the main flux path does not practically influence the major indices. Saturation along the leakage flux paths results in an increase of splashes of currents up to 20% and of electromagnetic torques up to 10 15%.

4 232 In cases when short circuit occurred behind reactance x c = 0.1 p.u. the error of calculations with neglect of saturation becomes considerably lower and does not exceed 3 5% for currents and torques. At the same time there can be observed an essential influence of the phenomenon under consideration on the nature of the rotor moving. Fig. 2. Sudden putting the impedance load on a single operating turbogenerator The influence of saturation due to the main flux is more greatly revealed in Fig. 2 which depicts the envelopes of curves of change in the armature winding voltage (U), field winding voltage (U fd ), field current (i fd ) and load angle (θ) at sudden switching-in the impedance load (cos φ = 0.919) on a single operating turbogenerator (P n = 2000 kva, V n = 230 V, n n = 6000 rpm) using self-excitation scheme with compounding. Here you can find comparison of the calculated data, the relations obtained on a complete model being plotted by full line, and those found without allowance for mutual effect of the direct and quadrature magnetic fields by dotted line. A similar approximate approach extensively described in the literature does not introduce any essential error in determining the armature voltages and currents, even in case of a very saturated magnetic circuit. However, the calculated load angle θ in this case may differ from the actual one by 25 30%. The effect of saturation proves sometimes to be very great in small electric machines. The next figure makes it possible to estimate this factor in synchronous reluctance motors. Ohm running under steady state symmetrical operating conditions. Fig. 3. Calculated angular speed during motor starting Fig. 4. Self-controlled inverter-fed synchronous motor circuit

5 It can be seen that only for nonsaturated systems the circle is the locus of the end of current vector. This circle is elongated and is turned into a figure resembling an ellipse with increase in saturation degree. At the same time electromagnetic torque is slightly decreased, and its maximum is shifted to the side of lesser load angles. Fig. 3 shows the starting process of a two-phase motor (V n = 220 V, I n = 0.4 A, x d = 2.02 p.u., x q = 0.88 p.u.) with a capacitive phase-shifting element from a single-phase network. Here the conditional saturation factor is taken as a parameter that permits the degree of magnetic circuit non-linearity to be estimated. This factor is defined at rated voltage as a ratio of magneto-motive forces corresponding to real and straightened no-load characteristics. With increase in this factor within the limits of K µ = a reduction in maximum angular speed (ω) and more intensive damping of shaft speed oscillations at the final stage of the process is observed. The abnormal saturation (K µ >1.4), however, results in increase of the starting time, mainly due to a considerable reduction in synchronous component of electromagnetic torque. With nonsymmetrical power supply, the oscillations of double (regarding to power supply source) speed of rotation are greatly reduced and uniformity of running is improved as K µ increases. Inconstancy of permeability becomes especially appreciable with essential changes of the flux in ferromagnetic portion of magnetic circuit. Just such conditions occur in the self-controlled inverter-fed synchronous motors which present a synchronous machine connected to a D.C. source via dependent inverter (Fig. 4) controlled as a function of the rotor position by a respective sensor [7]. Fig. 5 gives comparison of the calculated and experimental (plotted by dots) relations for starting process of a self-controlled inverter-fed synchronous motor rated at P n = 24 kw, V n = 230 V, n n = 1000 rpm. The results obtained with full allowance for saturation are plotted by full lines and those with no allowance by dotted lines. Despite a profound effect of nonlinear properties of the magnetic circuit, numerical calculations have proved the possibility of using the linear models. 233 Fig. 5. Starting process of a self controlled inverter-fed synchronous motor Fig. 6. Two-phase induction variable-speed motor The error in values of the main variables may be reduced down to 10% by operating with the saturated values of synchronous reactances and with proper choice of the latter. However, there is no procedure for the unique choice of these parameters. In a number of cases specific phenomena occur in electric machines due to non-linear relation between magnetic field induction and magnetizing force of the windings. Thus, in two-phase induction variable-speed motors (Fig. 6) with a cylindrical rotor flattening of magnetic

6 234 flux density is observed in the air gap with a sinusoidal space distribution of magnetizing force of the windings. Due to this fact higher harmonics are generated in the emf and currents of the windings. With the presence of the phase-shifting capacitances resonance overvoltages may appear across the stator circuit elements. Table 2 presents the calculated and experimental amplitude values of current of current harmonics of control winding at rated operating conditions for motors of two types rated at: I type: P n = 10 W, f n = 50 Hz, m en = 0.8 N m; II type: P n = 4 W, F n = 400 Hz, m en = 0.615N m. Table 2 Value Experiment Calculation Experiment Calculation Harmonic order Motor type I I II II Fig. 7. Starting process of a two-phase induction motor Fig. 7 presents comparison of starting characteristics of motor I obtained with no allowance for saturation (dotted line) and experimental relations (full line) which differ from the data calculated using the complete model no greater than by 3 4%. 4. CONCLUSIONS The experience gained in calculation of different operating conditions reflected partly by illustarions given has shown that: effect of saturation in the majority of practically important cases is essential; the neglect of this phenomenon may lead not only to quantitative but also to qualitative errors; in order to describe the processes in saturated electrical machines, mathematical models may be used to advantage, built on the basis of the equivalent circuits with nonlinear

7 elements. Characteristics of these models are found experimentally or by solving the problems of stationary magnetic field in non-linear medium, the complex problem of electrodynamics being divided into two partial sub-problems, the volume of computations work is reduced and physical understating of the materials is simplified; in a number of investigations it is quite permissible to use simplified relations and mathematical models with an approximate reflection of mutual influence of direct and quadrature fields, of magnetic circuit saturation along the paths of leakage fluxes, of increase in transient voltage drop between the brush and commutator as well as of commutating armature reaction in D.C. machines with distortion should be rigorously substantiated in each specific instance in accordance with the range of investigations; there still remains open the problem of allowance for influence of hysteresis with which the possibility of appearance of subharmonic oscillations in the system is associated. 235 REFERENCES [1] MAKKY A.M., Representation of Saturated Salient-Pole Synchronous Machine by a Single Magnetization Curve, Electric Machines and Electromechanics, Vol. 7, No 5, [2] CHIRICOZZI E., HONORATI O., Influence of Magnetic Saturation on the Dynamic Performance of Synchronous Generators Models, IEEE, PES, Winter Meeting, New York, [3] TAMURA J., TAKEDA I.A, New Model of Saturated Synchronous Machines for Power System Transient Stability Simulations, IEEE Trans. On Energy Conversion, June [4] СИДЕЛЬНИКОВ Б.В СИДЕЛЬНИКОВ А.В., Эффект насыщения в экстремальных режимах турбогенераторов, Электротехника, No 8, [5] МАКАРОВ И.В., СИДЕЛЬНИКОВ Б.В., Нелинейная математическая модель насыщенного вентильного двигателя постоянного тока, Электротехника, No 5, [6] СИДЕЛЬНИКОВ Б.В., РОГАЧЕВСКАЯ Г.С., Особенности формирования математических моделей для анализа динамических режимов работы коллекторных машин, Межвузовский сб.науч.тр. Задачи динамики электрических машин, Омск, [7] СИДЕЛЬНИКОВ Б.В., Коммутационная реакция якоря машин постоянного тока в переходных режимах. Известия вузов: Электромеханика, No 5, [8] CИДЕЛЬНИКОВ Б.В, ЧАГЛАСЯН Б.O., Метод учета пространственных гармоник магнитного поля насыщенных управляемых асинхронных двигателей, Электричество, No 4, ANALIZA STANÓW MASZYN ELEKTRYCZNYCH O NIELINIOWYCH CHARAKTERYSTYKACH MAGNETYCZNYCH W referacie przedstawiono wyniki analizy numerycznej i doświadczalnej dotyczącej wpływu nasycenia na pracę maszyn elektrycznych różnych typów. Wskazano, że nasyceniu mogą towarzyszyć zjawiska wpływające zarówno na jakościowy, jak i na ilościowy obraz zachodzących procesów. Wykazano, że istnieje możliwość przybliżonego uwzględnienia tego czynnika w wielu ważnych przypadkach praktycznych.