DOI: 10.24352/UB.OVGU-2017-110 TECHNISCHE MECHANIK, 37, 2-5, (2017), 347-357 submitte: June 15, 2017 Simple Electromagnetic Motor Moel for Torsional Analysis of Variable Spee Drives with an Inuction Motor T. P. Holopainen, A. Arkkio Torsional vibrations must be consiere in the esign of all high-power rive-trains incluing an inuction motor. Electromagnetic (EM) fiel in the air gap of an inuction motor generates aitional magnetic stiffness an amping between the rotor an stator. The inclusion of these magnetic effects is limite by the availability of simple an portable motor moels. The main aim of this paper is to introuce a motor moel incluing the spee an torque variation. The presente moel is base on the linearization of the common space-vector moels of inuction motors. The parameters of this moel are ientifie for the rate operating conition. This motor moel can be extene to inclue variable spee an torque operation. The numerical results emonstrate that this moel escribes accurately the magnetic effects over the large spee an torque range. In aition, the numerical results emonstrate the significance of magnetic stiffness an amping in variable spee motor-riven compressors with a soft coupling. 1 Introuction Inuction motors rotate process machines by converting electric energy to mechanical work. The power is transmitte by a rive train incluing couplings an optional gears an branches. An essential part of the rive train esign is the torsional vibration analysis. This analysis requires the inertia an stiffness ata of all the rive train components with loaing an amping parameters. Increasing power ensity, together with increasing emans of reliability of inustrial systems, has le to the increase requirements of calculation accuracy. With motor riven reciprocating compressors, this has resulte in the inclusion of magnetic stiffness an amping in torsional analysis (Anon., 2015). Due to this nee, simple moels, base on the motor characteristic ata, have been presente for the evaluation of these magnetic parameters; see Knop (2012) an Hauptmann et al. (2013). These moels are base on the space-vector theory, evelope to escribe the steay-state an transient motor behaviour, an use for the evaluation of magnetic stiffness an amping, see Concoria (1952), Joran et al. (1979, 1980), Shaltou (1994) an Brunelli et al. (2015). Space-vector moels, or the simplifie versions of them, are well suite for the torsional analysis of rive trains. Using these moels, the parameters require to escribe the magnetic stiffness an amping can be calculate in avance by the motor manufacturer an submitte to the suppliers, e.g. compressor manufacturer, responsible for the rive train esign. Simultaneously, numerical methos have been increasingly applie for the analysis of inuction motors. This tren has been expane also to the etermination of magnetic stiffness an amping parameters (Repo 2008) an the effect of these has been evaluate in calculation examples of actual rive trains (Holopainen et al. 2010). The accuracy an moelling capability of numerical moels excees clearly the potential of space-vector moels. However, the accuracy is only one, though significant, requirement of feasible moels for torsional analyses. A remarkable shortcoming of numerical moels is the portability. The numerical moels are usually non-linear an the application requires the integration of particular coes for electromagnetic fiels. Thus, the portability of numerical motor moels is poor, or at least limite. By contrast, the space-vector moel is simple an the number of moel parameters is small, an thus, the portability excellent. The main aim of this paper is to introuce a simple linearize motor moel incluing the spee an toque variation. The secon aim is to show the significance of the magnetic stiffness an amping particularly in motor-riven reciprocating compressors with a soft coupling. 347
This paper is base strongly on the methos an finings of two previous papers, see Arkkio et al. (2016) an Holopainen et al. (2016). The paper starts by reviewing the linearization of the space-vector moel in the operating point. Next, the inclusion of the spee an torque variation is presente. These two parameters efine the steay-state operation of an inuction motor. After this, the evelope moels are applie for a 3.7 MW inuction motor. The obtaine magnetic stiffness an amping values are compare to the results calculate numerically by a refine finite element (FE) metho with time-stepping analysis. Finally, the motor moel is applie to evaluate the magnetic effects on a reciprocating compressor rive train. All the calculations are carrie out with a steay-state sinusoial voltage supply of the motor. Thus, all the effects inuce by the frequency converter control are neglecte an the scope is restricte purely to the motor. 2 Methos 2.1 Space-vector Moel of Inuction Motor A single-cage space-vector moel for an inuction motor written in a reference frame rotating at the synchronous angular spee is s = s s + s 0= r r + r +j s s +j( s ) r (1) e= Im s * s where s, s an s are the space vectors of stator voltage, stator current an stator flux linkage, r an r are the space vectors of rotor current an rotor flux linkage, s is the stator resistance, r is the rotor resistance, is the angular spee of the rotor, is the number of pole pairs, j is the imaginary unit, an asterisk enotes complex conjugation. The angular spees are given in electrical raians, i.e. =, where is the mechanical rotational spee. The linear relation between the flux linkages an currents is s = s s + m r r = m s + r r (2) where s an r are the self-inuctances of the stator an rotor winings an m is the mutual inuctance between them. 2.2 Linearization of Space-vector Moel in Operation Point The system of equations (1) is non-linear because of the prouct of angular spee an rotor flux linkage in the secon equation. In aition, the electromagnetic torque in the thir equation is non-linear ue to the prouct of stator flux linkage an stator current. However, the torsional vibrations are manifeste by small oscillations aroun the equilibrium point. Thus, the calculation of torsional vibrations can be carrie out by linearizing the equations at the operation point. The linearize system of equations in the synchronously rotating reference frame is (Arkkio et al. 2016) s = s s + s s 0 = r r + m s +j s s s + r m +j s m r +j( s ) m s + r r + +j( s ) r r j m s0 + r r0 e = mim * r0 s + s0 (3) 348
where the currents have been chosen as the free variables, enotes a small variation from the steay state value, i.e. a linearize variable, an s0, r0 an are the steay-state stator current, rotor current an angular spee. The resistance an inuctance parameters of the non-linear an linearize space-vector moels, (1) an (3), were obtaine from a time-harmonic FE analysis (Repo et al. 2006). A non-linear effective permeability of operating point was use to get the parameters to calculate the steay-state currents of equation (1). A ifferential permeability was use to get the linearize parameters for equation (3). 2.3 Improve Space-vector Moels The simple space-vector moel of equations (1) an (3) can be improve to inclue skin effect of rotor bars by increasing the number of rotor branches. Figure 1 shows a steay-state equivalent circuit having three rotor branches or cages. The corresponing ynamic space-vector moel is (Arkkio et al. 2016) s = s s + s 0 = r r + r 0 = qq + q 0 = pp + p +j k s +j( s ) r +j( s ) q +j( s ) p (4) e = Im s * s where subscripts q an p refer to the secon an thir rotor branches. A ouble- an triple-cage linearize moel can be constructe in a similar way as the single-cage moel above. Again, the resistance an inuctance parameters were obtaine from a time-harmonic FE analysis (Repo et al. 2006) u s i s R s jx sσ R c2 jx c2 R c3 jx c3 is++ ir iq+ ip ir+ iq+ ip iq+ i jx m jx rσ jx qσ jx pσ R r0 R q0 R p0 p i r i q i p Figure 1. Steay-state space-vector equivalent circuit with three rotor branches (Arkkio et al. 2016) The number of real-value parameters, i.e. original an linearize moel resistances an inuctances, is 10, 18 an 26 in single-cage, ouble-cage an triple-cage moels, respectively. The number of complex-value variables, i.e. electromagnetic egrees-of-freeom, is 2, 3 an 4 in single-cage, ouble-cage an triple-cage moels, respectively. Because the number of variables in a typical FE moel use for the ientification of parameters is thousans, the reuction grae is remarkable. 2.4 Variable Spee an Torque Operation A large share of the new motors is use in variable spee applications. In these applications the motor spee an torque is ajuste accoring to process requirements. A common approach is to keep the funamental flux of the machine constant inepenently of the actual spee an torque. This means that the supply voltage is irectly proportional to the supply frequency up to the fiel weakening point, which is often above the maximum spee. It can be mentione that moern frequency controllers may ajust the torque base on the feeback. However, all the calculations of this paper are carrie out with a steay-state sinusoial voltage supply of the motor, an thus, all the effects inuce by the frequency converter control are neglecte. 349
In this paper, the moel parameters, i.e. inuctances an resistances, are ientifie for the rate operating conition. These moel parameters are extene to an arbitrary spee an torque by ajusting first the supply voltage accoring to the constant flux approach s = s s, s,rat (5) where the subscript rat refers to the rate values. The rotational spee an torque are connecte by the slip of the rotor with respect to the rotating magnetic fiel = ( s ) s (6) This slip must be solve iteratively from the non-linear equation (1) using the pre-set values of spee an torque. In aition, the inuctances of the linearize moel, equation (3), must be ivie by the synchronous spee ratio s s,rat. All other parameters of the non-linear an linearize moels remain unchange. 2.5 Calculation of magnetic Stiffness an Damping The analytical expression for the frequency response function for the single-cage space-vector moel is obtaine from equation (3) by replacing the small variations by phasor variables of oscillation frequency an solving the relation between the torque an rotation angle of the rotor. The magnetic stiffness m an amping coefficients m can be associate with the real an imaginary parts of the frequency response function frf m = Re frf ( ) m = Im frf ( ) (7) where is the angular frequency of oscillation. 2.6 Reference results by Finite Element Analysis (FEA) In the 2D FEA moels, the magnetic fiel in the core region of the motor is assume to be two-imensional. En-wining impeances are ae to the circuit equations of the winings to approximatively moel the 3D en-wining fiels. The fiel an circuit equations are iscretize an solve together (Arkkio 1990). Movingban technique in the air gap of the machine is use for rotating the rotor (Davat et al. 1985). The torque is compute using Coulomb s metho (Coulomb 1983). The resistive losses of the winings were inclue in the moel when solving the fiel equations within FEA. The frequency response function was neee for the reference result. Two time-stepping simulations in steaystate are use. In the first one, the rotor is rotate at a constant spee. In the secon one, the rotation spee is force to oscillate at a frequency an small amplitue aroun the constant spee of the first simulation. The results of the two simulations, particularly the electromagnetic torque an the rotation angle of the rotor, are subtracte from each other. The component varying at frequency is extracte from the torque an rotation angle ifferences by complex Fourier analysis, an finally, the complex value of the frequency response function at frequency is obtaine by iviing the Fourier components of the torque an rotation angle. This process was repeate at about 30 ifferent excitation frequencies between 10 Hz an 100 Hz to collect ata for the comparison of the analytically an numerically obtaine frequency response functions. Another way to get the frequency response function numerically is to excite the machine by a single pulse in the rotation angle (Repo 2008). In this way, all the interesting frequencies can be obtaine from two simulations, one with the pulse an another one without it. This metho is applie to obtain the reference results for the variable spee an torque comparison. 350
3 Results 3.1 Magnetic stiffness an amping A 3.7 MW inuction motor is use in all the calculation examples. The main parameters of this motor are shown in Table 1. Table 1. Rate parameters of the example motor. Parameter Value Unit Power 3551 kw Frequency 60 Hz Spee 895.3 rpm Number or poles 8 Connection star Voltage 4000 V Current 620 A Rate torque 37.88 knm Breakown torque 82.71 knm Figure 2 shows the magnetic stiffness an amping calculate by an analytic formula presente by Hauptmann et al. (2013), by single- an triple-cage moels with parameters base only on the non-linear moels, an FEA results. The FEA results are assume to be most accurate an will be use here an later as reference values. In this case, the analytic equation unerestimates the EM stiffness an amping an neglects the effects close to the supply frequency. Similarly, all the cage moels unerestimate the EM stiffness an amping. The triple-cage moel gives the best preiction. Figure 2. Magnetic stiffness an amping in rate operating conition with non-linear moel parameters in cage moels. 351
Figure 3. Magnetic stiffness an amping in rate operating conition with linearize space-vector moel. Figure 3 shows the magnetic torsional stiffness an amping in the rate operating conition calculate by linearize space-vector moels an by FEA. The parameters of the space-vector moels are ientifie for the slip frequency 1.2 Hz (Holopainen et al. 2016). As can be seen the single-cage moel unerestimates the stiffness an amping. In the contrary, the ouble- an triple-cage moels overestimate somewhat the stiffness an preict the amping accurately. The ifference between the ouble- an triple-cage moels is small. 3.2 Effect of Spee an Torque The synchronous spee range of the example motor in the variable spee operation is 450 900 rpm. This correspons roughly to the supply frequency range 30 60 Hz. The fiel weakening point of this motor is at 60 Hz an the loa torque of reciprocating compressors epens on the process meium an pressure ratio, an is inepenent of the spee. Thus, it is assume that the torque varies between 50 100 % of the rate torque. Table 2 shows the calculation points. These points are the corner points of the spee-torque omain. However, for simplicity the calculation points are efine by the supply frequency an electric power. The slip, torque an spee values in table 2 are calculate by iteration uring the space-vector an FEA solutions. Figure 4 shows the magnetic stiffness an amping for these four operating points calculate by the linearize triple-cage space-vector moel. It can be clearly seen that both the spee an torque affects the magnetic parameters. Figure 5 shows the same information in the form of the complex frequency response functions (FRFs) between the torque an rotor oscillation, see equation (5). In this figure, the FRFs are plotte as functions of the relative oscillation frequency. This presentation format gives a better overview of the spee an torque effect on the magnetic stiffness an amping. It can be seen that the effect of spee (50% reuction) is slightly larger than the effect of torque (50% reuction). In aition, it can be seen that the change of torque changes the real part of the FRF, i.e. magnetic stiffness, but leaves the imaginary part of FRF, i.e. magnetic amping, almost intact. 352
Table 2. Calculate operation points of the example motor use in FEA. Case Freq. Power Space-Vector Moel FEA Slip Torque Spee Slip Torque Spee [Hz] [kw] [%] [knm] [rpm] [%] [knm] [rpm] 1 60 3729 0.546 39.78 895.1 0.543 39.78 895.1 2 60 1864 0.257 19.83 897.7 0.259 19.83 897.7 3 30 1864 1.114 40.01 445.0 1.108 40.00 445.0 4 30 932 0.518 19.88 447.7 0.521 19.88 447.7 Figure 4. Magnetic stiffness an amping calculate by the linearize triple-cage space-vector moel. Figure 6 shows the corresponing frequency response functions for the example motor calculate by the FEM in time omain an the pulse metho (Repo 2008). It can be conclue that the behaviour of the linearize spacevector moel (Figure 5) an the FE moel (Figure 6) correlates well over the whole spee range. 3.3 Motor riven reciprocating Compressor The example motor rives a reciprocating compressor. The rive train consists of the following components: Motor, flexible coupling, flywheel an reciprocating four-cyliner compressor. This compressor can be use in irect-on-line operation with constant spee 895 rpm an in variable spee operation (450 900 rpm) supplie by a frequency converter. The mechanical rive train was moelle with 26 inertias with connecting torsional stiffness elements. The viscous amping was ae to the motor an compressor cyliner locations. The amping inuce by the flexible coupling was neglecte ue to the missing input ata. 353
Figure 5. Frequency response function between the magnetic torque an rotor oscillation calculate by the linearize space-vector moel. The real parts of the complex value functions are below an the imaginary parts are up. Figure 6. Frequency response function between the magnetic torque an rotor oscillation calculate by the FEA. The real parts of the complex value functions are below an the imaginary parts are up. Table 3 shows the natural frequency an amping ratio for the lowest moes without an with magnetic effects. The calculations are carrie out by aing the magnetic stiffness an amping to the mechanical moel. The calculation was carrie out iteratively in orer to use the magnetic stiffness an amping values corresponing to the natural frequencies. The first moe without magnetic effects is the rigi boy moe. The main eformation of the secon moe occurs in the flexible coupling. The thir moe is an internal moe of the coupling, an in the fourth moe the flywheel an the compressor line are in the opposite phase without angular isplacement of the motor. Table 3 shows that the electromagnetic interaction increases clearly the natural frequency an amping ratio of the two first moes. The effect on moes 3 an 4 is negligible. This is logical ue to the moal amplitues of the moes 1 an 2 in contrast to the amplitues of the moes 3 an 4. Due to the variation of supply frequency, the magnetic stiffness an amping changes though the torque is assume to be constant. Figure 7 shows the natural frequency an amping ratio of the two lowest moes as a function of the motor spee. In aition, the first an secon orer excitations are shown in this Campbell iagram. It can be seen that the natural frequency an amping ratios are only slightly epenent on the spee above 300 rpm. Below that spee the effects of the supply frequency on the secon moe are clearly visible. Actually, the amping factor is negative in the spee rage 140 190 rpm. The secon moe crosses the first orer excitation frequency at about 600 rpm. The response of the system on this torsional critical spee epens on the first orer excitation amplitue an the total amping. In this case the amping factor is preicte to be 354
8.6%. The amount of amping provie by the electromagnetic system is significant compare to the mechanical amping (Table 3). Table 3. Natural frequency an amping ratio for the four lowest moes without an with electromagnetic coupling at rate operation. Moe Magnetic Effects Without With f [Hz] ζ [-] f [Hz] ζ [-] 1 0.00 0.00 4.14 10.09 2 6.35 1.60 10.03 8.63 3 37.27 0.01 37.29 0.02 4 130.29 0.92 130.29 0.92 Figure 7. Natural frequency an amping ratio of the motor compressor train. 4 Discussion an Conclusions The obtaine results show that the linearize space-vector moels can be use to preict the magnetic stiffness an amping of inuction motors. Particularly, the preiction capability of the ouble- an triple-cage moels in the rate operating conition is goo. However, there seems to be a ifference in magnetic stiffness compare to the FEA results. The triple-cage moel yiels about 5% higher stiffness values than the FEA. The origin of this iscrepancy is not known. The parameters of the presente non-linear an linearize space-vector moels are obtaine from the timeharmonic FEA. The number of parameters is 10, 18 an 26 in single-cage, ouble-cage an triple-cage moels, 355
respectively. Because the number of variables in a typical FE moel, use for the ientification of parameters, is thousans, the reuction grae is remarkable. More importantly, the space-vector moel is portable to be a part of stanar torsional analyses without irect coupling to solvers of electromagnetic fiels. The calculation results inicate that the space-vector moel can be extene to variable spee an torque operation. The results of the simple three-cage moel are well compatible to the results obtaine by the nonlinear FEA in the time omain. The calculation example for a reciprocating compressor train shows that the inclusion of magnetic effects is significant with flexible coupling. This follows from the two effects: The magnetic stiffness increases the natural frequencies an the magnetic amping ecreases the oscillation amplitues of torsional moes. The magnetic amping is particularly avantageous an can be exploite in the esign of torsional rive trains. The calculations were carrie out with a steay-state sinusoial voltage supply of the motor. Thus, all the effects inuce by the frequency converter control are neglecte an the scope is restricte purely to the motor. References Anon.: Guieline an recommene practice for control of torsional vibrations in irect-riven separable reciprocating compressors. Gas Machinery Research Council, ACI Services, (2015). Arkkio, A.: Finite element analysis of cage inuction motors fe by static frequency converters. IEEE Trans. Magnetics, 26, (1990), 551-554. Arkkio, A.; Holopainen, T. P.: Space-vector moels for torsional vibration of cage inuction motors. IEEE Trans. on Inustry Applications, 52, (2016), 2988 2995. Brunelli, M.; Fusi, A.; Grasso, F.; Pasteur, F.; Ussi, A.: Torsional vibration analysis of reciprocating compressor trains riven by inuction motors. In Proc. 9 th Int. Conf. on Compressors an their Systems, Lonon, UK, Sept.7-9, (2015), 10 p. Concoria, C.: Inuction motor amping an synchronizing torques. AIEE Trans. Power Apparatus an Systems, 71, (1952), 364-366. Coulomb, J. L.: A methoology for the etermination of global electromechanical quantities from a finite element analysis an its application to the evaluation of magnetic forces, torques an stiffness. IEEE Trans. Magnetics, 19, (1983), 2514-2519. Davat, B.; Ren, Z.; Lajoie-Mazenc, M.: The movement in fiel moeling, IEEE Trans. Magnetics, 21, (1985), 2296-2298. Hauptmann, E.; Howes, B.; Eckert, B.: The influence on torsional vibration analysis of electromagnetic effects across an inuction motor air gap. In Proc. GMRC Gas Machinery Conf. Albuquerque, New Mexico, Oct. 6-9, (2013), 11 p. Holopainen, T. P.; Repo, A.-K.; Järvinen, J.: Electromechanical interaction in torsional vibrations of rive train systems incluing an electrical machine. In Proc. 8th IFToMM Int. Conf. Rotorynamics. Seoul, Korea, Sept 12-15, (2010) 8 p. Holopainen, T. P.; Liukkonen, O. J.; Arkkio, A.: Portable inuction motor moel for torsional vibration analysis of rive train systems. In Proc. 11th Int. Conf. on Vibrations in Rotating Machinery. Manchester, UK, Sept. 13-15, (2016), 141-151. Joran, H.; Müller, J.; Seinsch, H. O.: Über elektromagnetische un mechanische Ausgleichsvorgänge bei Drehstromantrieben. Wiss. Ber. AEGTELEFUNKEN, 52, (1979), 263-270. Joran, H.; Müller, J.; Seinsch, H. O.: Über as Verhalten von Drehstromasynchronmotoren in rehelastischen Antrieben. Wiss. Ber. AEGTELEFUNKEN, 53, (1980), 102-110. 356
Knop, G.: The importance of motor ynamics in reciprocating compressor rives. In Proc. 8th Conf. EFRC. Düsselorf, Germany, Sept. 27-28, (2012), 10 p. Repo, A.-K.: Numerical impulse response tests to ientify ynamic inuction-machine moels. Dissertation, Helsinki University of Technology, Finlan, (2008). Repo, A.-K.; Niemenmaa, A.; Arkkio, A.: Estimating circuit moels for a eep-bar inuction motor using time harmonic finite element analysis. In Proc 17 th Int. Conf. on Electrical Machines, Chania, Greece, Sept. 2-5, (2006), 6 p. Shaltout, A. A.: Analysis of torsional torques in starting of large squirrel cage inuction motors. IEEE Trans. Energy Convers. 9, (1994), 135-142. Aress: Timo P. Holopainen, ABB Oy, P.O. Box 186, FI-00381 Helsinki, Finlan email: timo.holopainen@fi.abb.com Antero Arkkio, Aalto University, P.O. Box 15500, FI-00076 Aalto, Finlan antero.arkkio@aalto.fi 357