In this paper induction squirrel cage motor is analyzed with respect to its dynamical behaviour. First analytical calculation of motor’s parameters and characteristics is performed by developing software program for their calculation. MATLAB is used for calculation of system of five differential equations with constant coefficients with Runge Kutta method of fourth order resulting in transient motor characteristics of speed and electromagnetic torque at different operational regimes. Transient characteristics are obtained as well as from simulation model developed in SIMULINK. Characteristics from both transient models are compared and conclusions are derived with respect to available data from experimental measurements and from manufacturer side

Sarac, Vasilija

Stefanov, Goce

UGD Academic Repository

NUMMERICAL AND SIMULATION METHODS FOR CALCULATION OF DYNAMICAL TRANSIENT CHARACTERISTICS OF SQUIRREL CAGE INDUCTION MOTOR  ЦИФРОВИ И СИМУЛАЦИОННИ МЕТОДИ ЗА ПРЕСМЯТАНЕ НА ДИНАМИЧНИТЕ ПРЕХОДНИ ХАРАКТЕРИСТИКИ НА АСИНХРОНЕН ДВИГАТЕЛ С НАКЪСО СЪЕДИНЕН РОТОР  Assist. prof. Dr. Eng. Sarac V.1, Assistant Msc. Eng. Stefaov G. 2   Electrotechnical Faculty – University ‘Goce Delcev’-Stip, Macedonia 1,2   Abstract: In this paper induction squirrel cage motor is analyzed with respect to its dynamical behaviour. First analytical calculation of motor’s parameters and characteristics is performed by developing software program for their calculation. MATLAB is used for calculation of system of five differential equations with constant coefficients with Runge Kutta method of fourth order resulting in transient motor characteristics of speed and electromagnetic torque at different operational regimes. Transient characteristics are obtained as well as from simulation model developed in SIMULINK. Characteristics from both transient models are compared and conclusions are derived with respect to available data from experimental measurements and from manufacturer side.  Keywords: INDUCTION MOTOR, TRANSIENT CHARACTERISTICS, NUMERICAL MODEL, SIMULATION MODEL  1. Introduction Squirrel cage induction motor has wide application in many industrial drive systems due to its simple construction, robust design and operational behaviour. Its application becomes even wider due to voltage inverters which enable very good speed-torque regulation characteristics. In this paper squirrel cage induction motor type 2AZ 155-4 from producer Rade Koncar is analyzed with respect to its dynamic behaviour. Motor has following rated data: voltage Un (Y/Δ)=220/380 V, number of poles 2p=4, rated current 8,7/5 A, power factor cosϕ=0,81, rated speed nn=1410 rpm, frequency 50 Hz. Motor cross section is presented on Fig.1   Fig.1 Motor’s cross section  Motor’s parameters and characteristics using analytical approach are calculated by developing a program in FORTRAN. On the base of obtained results motor operation characteristics are plotted, as well as no-load characteristics and electromagnetic characteristic. Motor’s dynamic behaviour is described with system of five linear differential equations with constant coefficients. Solution of this systems with numerical Runge Kutta method of IV order in MATLAB  leads to transient characteristics of speed and electromagnetic torque versus time. Accuracy of these characteristics is confirmed by comparison with transient characteristics obtained from simulation model developed in SIMULINK. Results from both models are compared with available data from manufacturer and from experimental measurements. 2. Development of analytical model On the base of the software calculation in FORTRAN motor operational characteristics are calculated : stator current-I1, efficiency factor-η, power factor-cosϕ, input power-P1, speed-n and output torque M2 versus output power: I1=f(P2), η=f(P2), cosϕ=(P2), P1=f(P2), n=f(P2), and M2=f(P2). Mechanical characteristic of electromagnetic torque Mem for different motor slips-s is calculated from motor T-equivalent circuit.  Operational characteristics are plotted on the base of results given in Table 1. Table 1. Motor operational characteristics I1 [A] η  [%] cosϕ [/] s  p.u. P1 [W] n [rpm] M2 [Nm] P2 [W] 2.17 0.128 0.119 0.00062 171.32 1499.07 0.21 22 2.22 0.59 0.255 0.00433 372.62 1493.5 1.48 220 2.32 0.73 0.392 0.00858 600.87 1487.13 2.9 440 2.49 0.79 0.508 0.01298 834.26 1480.53 4.34 660 2.71 0.82 0.601 0.01754 1073.14 1473.69 5.79 880 2.97 0.835 0.673 0.0223 1317.98 1466.56 7.25 1100 3.27 0.841 0.727 0.02727 1569.3 1459.1 8.73 1320 3.60 0.843 0.768 0.0325 1827.73 1451.26 10.23 1540 3.97 0.84 0.799 0.03802 2094.09 1442.97 11.74 1760 4.37 0.836 0.822 0.4389 2369.36 1434.16 13.28 1980 4.79 0.83 0.838 0.05019 2654.83 1424.72 14.85 2200   Fig.2 Operational characteristics η=f(P2), cosϕ=f(P2), s=f(P2)  Fig.3 Operational characteristics n=f(P2), M2=f(P2), I1=f(P2) 5 Fig.4 Mechanical characteristic Mem=f(s) 3. Development of numerical model System of n-differential equation of first order is represented by: (1) ( )( ) ( )( ) ( )nnnnnyyyxfdxxdyyyyxfdxxdyyyyxfdxxdy,........,.......,......,,),......,,(2121222111===  It can be solved by different numerical methods: Euller, Bulirsch-Stoer, Runge Kutta etc. Application of appropriate method is determined by boundary conditions and starting conditions which must be defined in order system of equations to be solvable. When it is used Euller method values of the function yi are determined for starting condition x=xp. Increasment of Δx leads to changes in all functions yi for adequate increasments Δyi.   (2) ( ) ( ) ipipi yxyxxy Δ+≈Δ+   Equation (2) can be adequately replaced with:  (3)  ( ) xyyyxfy ni Δ≅Δ ,....,,, 21 From equation (2) and (3) values of the function yi in new point xp+Δxp are known. By applying the same procedure value of the function yi(xp+2Δx) can be achieved. Consequently step by step value of the function yi in final point x=xk can be calculated. This is an Euller method which does not give an adequate accuracy. For practical calculation and for more accurate results Runge Kutta method with adaptive step is used.  If system (1) is solved with Euller method for step Δx from point x up to point x+Δx following solution is obtained:  (4)  ( ) ( ) ( ) errorxyyxfxyxxy niii +Δ+=Δ+ ,.....,, If this error is from order  method is from first order. A Euller method uses information about derivations from the beginning of the interval (x, x+Δx) for extrapolation of the starting values of yi on the end of the interval. As a result accuracy of the method is low. It is better to be used values of the derivation from the beginning, middle and end of the interval (x, x+Δx). On the base of these values better estimation of the value yi (x+Δx) on the end of the interval is obtained. This is the base for the whole family of Runge Kutta methods. Most frequently used is Runge Kutta of forth or fifth order where error is proportional with (Δx)5, i.e. (Δx)6.  ( )2xΔ Motor dynamic transient operational regime is determined by following system of linear differential equations: (5)   ( )( )αβαβαββββαααβββαααψψψψωψωψψψψωψψψψψωψψψωψsrrsemsemrrrrsrrrrsrrsnsrsnsaMJMMpdtdaadtdaadtdaatUdtdaatUdtd−=−=+−=−−=+−=+−=543432121/sin2cos2   where ω =314 [rad/s], J is motor moment of inertia and , , ,  and are coefficient which are determined by motor parameters stator and rotor resistance Rs and Rr stator and rotor inductance Ls and Lr and mutual inductance Lm: 1a 2a3a 4a 5amrsrsLLLLRa −=1 22 mrsmsLLLLRa −= 23 mrssrLLLLRa −=24mrsmrLLLLRa −= 255,1mrsmLLLpLa −=  Equation (5) gives the system of five differential equations with five unknown variables. They describe the motor stator and rotor circuit while the six equation calculates the electromagnetic torque on the base fluxes in stator and rotor circuits which are described with αβα ψψψ rss ,,  and βψ r  respectively. Solution of systems of equation (5) gives the transient characteristics of electromagnetic torque Mem [Nm] and rotor angular speed ωr [rad/s] when motor is supplied with three phase symmetrical power supply.  3. Development of simulation model Simulation of motor transient characteristics in SIMULINK is performed by building a simulation model. On Fig. 5 is presented a block diagram of simulation model consisted of following four main parts [1]: ? Power supply ? Transformation of a,b,c variables into variables in d,q system which rotates synchronously. ? Modelling of motor model and obtaining motor speed and electromagnetic torque as output variables. ? Transformation from d,q system into a,b,c system in order stator and rotor voltages and currents to be obtained.    Fig. 5 Block chart of Simulink motor model A,B,C reference system Modelling of d,q reference system Simulation  ωr  Mem Inverse transformation Three phase power supply  First motor power supply (three phase symmetrical power supply) is transformed from three phase system into d,q system which rotates with synchronous speed [2] as it is presented with equations (6) and (7).   6 (6) ( ) θθ sincos31acbds UUUU +−−=  (7) ( ) θθ cossin31acbqs UUUU +−=  where : dtdθω =  i.e.  ∫= t dt0ωθSince power supply is three phase symmetrical power supply in that case [ srade 314== ]ωω . Simulation model is based on following equations: (8)qsrsmsrtdsrrsmsrtdrrrsmrssrtdsrsmsrtdrrsmrstqrrsmrsqrLLLLdtLLLLdtiLLLLLLdtLLLLdtiLLLLLdtiLLLRLiψψωωψωω−+−−−−++−+−+−=∫∫∫∫∫202022020202 (9)( )dsrsmsrtqsrrsmsrtqrrrsmrssrtqsrsmsrtqrrsmrssrtdrrsmrsqrLLLLdtLLLLdtiLLLLLLdtLLLLdtiLLLLLLdtiLLLRLiψψωωψωω−+−+−−−−−−−−−−=∫∫∫∫∫2020220202202 In the equations (8) and (9) flux leakages are unknown variables and they can be expressed from: (10) dtdtiLRLdtLRdtUtdstqrsssrtqssstqsqs ∫∫∫∫ −+−=0000ψωψψ  (11) dtdtiLRLdtLRdtUtqstdrsssrtdssstdsds ∫∫∫∫ ++−=0000ψωψψ  Motor speed is calculated from: (12) ( ) 0226 MJMJiiiiJLdtd sqrdsdrqssrr −−−=ω  where  is load torque,  is a torque as a result of acceleration at no-load.  sM 0M4. Comparison of results In this section comparison between transient characteristics obtained from numerical and simulation model will be presented. On Fig.6 and 7 are presented transient characteristics of speed and electromagnetic torque  at no-load acceleration.    (a)    (b) Fig. 6   Transient characteristics of speed at no load  (a) simulation model (b) numerical model From Fig.6 can be concluded that motor acceleration by both methods is ended in approximately same time interval of 0,025 s. Transient characteristic ( )tfr =ω  from numerical model has more oscillations after motor acceleration is finished. In both motor models same speed at no-load operation condition is achieved of 313[rad/s].  (a)            (b)         Fig. 7 Transient characteristics of electromagnetic torque at no     load (a) simulation model (b) numerical model From Fig.7 can be concluded that in both models after motor acceleration is finished for acc. 0,025 s , electromagnetic torque drops up to value of 1.1 Nm i.e. up to the value of no-load torque.  On Fig. 8 and 9 are presented transient characteristics of speed and electromagnetic torque  at rated load condition Ms=14 Nm.         (a) 7 (b) Fig. 8 Transient characteristics of speed at Ms=14Nm  (a) simulation model (b) numerical model        t[s] (a)                      (b) Fig. 9 Transient characteristics of electromagnetic torque at Ms=14Nm  (a) simulation model (b) numerical model In Table 2 are presented results of different slips for different electromagnetic torques obtained by analytical model and simulation. Table 2. Motor slip from analytic calculation and simulation  Analytic Simulation Mem [Nm] s[/] s [/] 1 0.0036 0.0067 2 0.007 0.008 3 0.0105 0.01 4 0.0138 0.013 5 0.017 0.0167 6 0.0198 0.021 7 0.0232 0.024 8 0.0265 0.026 9 0.0301 0.0277 10 0.0332 0.031 11 0.0373 0.034 12 0.0406 0.036 13 0.0446 0.041 14 0.0485 0.043 00,010,020,030,040,050,060 5 10 15Mem [Nm]s [/]Analytic calculationSimulation Fig. 10 Comparison of motor slip from analytical and simulation model  From Fig.8 can be concluded that in numerical model motor acceleration is going faster approximately in 0,14 s while in Simulink model it rates up to 0,18 s. In both cases motor accelerates up to 300 [rad/s]. From Fig. 9 it is noticeable the similarity between both transient characteristics Mem=f(t) obtained by both motor models. In both cases after acceleration electromagnetic torque decreases up to the value of 15,2 Nm.   In Table 3 comparatively are presented results obtained by different methods.  Producer Analytical calculation Experim. Sim. model Num. model Pn[W] 2200 2200 2200   nn [rpm] 1410 1424,72 1410 1427,2 1430 In [A] 5 4,8 4,87 4,5 / sn[/] 0,06 0,058 0,06 0,053 0,051 cosϕ[/] 0,81 0,838 0,78 / / η [/] / 0,828 0,79 / / Mn [Nm] 14,9 14,67 13,8   Mem n[Nm] 16 16,58 13,88 15,2 15 5. Conclusion  Complex analysis of operational and transient characteristics of squirrel cage induction motor is performed. For that purpose three motor models are developed: analytical, numerical and simulation model. While the analytical model is mainly used for calculation of motor operational characteristics and mechanical characteristics, numerical and simulation models are used for calculation of transient dynamic characteristics of speed and electromagnetic torque versus time. At no load operating regime motor accelerates up to 313 [rad/s] for approximately the same time of 0,025 s in both models. After acceleration electromagnetic torque decreases up to 1,1 Nm. At rated load operating regime motor accelerates up to 300[rad/s] for 0,18s in simulation model and for 0,14s in numerical model. From the producer data rated speed is 295 [rad/s]. Agreement of the simulation results with producer data is satisfactory.  After acceleration electromagnetic torque decrease up to the value of 15,2 Nm in both motor models, which is expected result considering producer data for rated electromagnetic torque of 16 Nm.   From presented transient characteristics it is obvious the similarity of obtained results which confirms the accuracy of both models. Oscillations in transient characteristics are larger for the numerical model due to the strict conditions for stability of mathematical solution in numerical methods. 6. References [1] V.Sarac, L.Petkovska, M. Cundev: ‘Transient performance       characteristics of inverter fed induction motor’, 10-th EDPE-         Electrical Drives and Power Electronic Conference, 14-       16.X.1998.  [2] Krause, O. Wasynczuk, S.D. Sudhoff: ‘Analysis of electrical       machinery’, IEEE Press , 2nd Edition, New York, USA 1995.  8

Nummerical and simulation methods for calculation of dynamical transient characteristics of squirrel cage induction motor

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