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  • 1. I. I. OKONKWO, S. S. N. OKEKE / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, July-August 2012, pp.321-328 Efficiency Estimation Of Electric Machine Using Magnetic Flux Circuits I. I. OKONKWO* S. S. N. OKEKE** P. G. Scholar In Dept. Of Electrical Engineering Anambra State University, Uli. Nigeria* Professor In Dept. Of Electrical Engineering Anambra State University, Uli. Nigeria**Abstract: Improving efficiency of electrical machine is of two decades, magnetic equivalent circuit modelingparamount importance to optimization of operational method for electrical machines has been furtherand resource management. Efficiency method which do developed and applied to modeling many other types ofnot account for iron loss could undermine the effective machines, apart from induction machines: synchronousefficiency estimation. This paper simulates the efficiency machines by Haydock [8], inverter–fed synchronousvariations of electric machine in both transient and motors by Carpenter [10], permanent magnet motors andsteady state, magnetic flux linkages are used instead of claw-pole alternators by Roisse [11], and 3-D Lundellthe traditional current method, though also in d-q axis. alternators, which are main source of electric energy inOnly magnetic flux linkages are used in an arbitrary internal combustion engine automobiles, by Ostovic [9].reference frame while machine efficiency in various Sewell developed Dynamic Reluctance Meshreference frames namely; rotor, stator and the software for simulating practical induction machinessynchronous reference are investigated. Results are based on the previous work using magnetic circuitcompared with measured quantities and excellent models for electrical machines [7]. This algorithm wasestimations are obtained. developed from rigorous field theory. Combined with prior known knowledge of electrical machine behaviorKeyword: electrical machine efficiency, machine losses, determined from practical experience, this model ismagnetic circuit, generalized machine equation simplified in a physically realistic manner and results in a procedure that can efficiently simulate 3D machines1 INTRODUCTION within an iterative design environment without requiring Estimation of electric machine performance extensive computational resources. Its computation timethrough efficiency estimation enables electric machine was minimized by direct computation of the rate ofengineers to device control that could lead to improvement change of flux, thus the evaluation of time and rotorin electric machine efficiency, for example in electric position dependent inductances are avoided. Yet there ismachine losses control [1]. Without a robust and good a similar dynamic mesh modeling method but with somesimulation method, neither good estimation of efficiency additional developments in modeling and improvementsnor deep insight about loses variations could be obtained. in the solution process [12].Many researches have been carried out using magnetic Many different efficiency methods have beencircuit simulation techniques, in 1968 carpenter [2] developed with a field suitably as a goal. Theseintroduced and explored the possibility of modeling eddy – estimation methods have been enlisted in a thoroughcurrent using magnetic equivalent circuits, which are study on efficiency estimation techniques performed byanalogous to electric equivalent circuits. Magnetic terminal J. S. Hsu et al [13] and [14], which divides efficiencyare introduced which provide a useful means of describing techniques into the following classes: nameplatethe parameters in electromagnetic devices, particularly method, slip method, current method, statistical method,those operation depends upon induced current behavior. equivalent circuit method, segregated loss method, airThe virtual – work [3 – 4] principle was suggested for gap torque method and shaft torque method. In thistorque calculations and exploited in [7]. Generally, the paper efficiency is estimated through simulation ofmagnetic equivalent circuit modeling method can be used to magnetic flux linkages with a generalized arbitraryanalyze many types of electrical machine in terms of reference equation transformed to all flux linkagephysically meaningful magnetic circuits. For example, variables which are used in formulae of lossOstovic evaluated the transient and steady state minimization techniques [1, 13 – 14].performance of squirrel cage induction machines by usingthis magnetic equivalent method [5] 2 All Flux Linkages Generalized Machine [6], taking into account the machine geometry, the type of Equation In An Arbitrary Reference Framewindings, rotor skew, the magnetizing curve etc. In the past 321 | P a g e
  • 2. I. I. OKONKWO, S. S. N. OKEKE / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, July-August 2012, pp.321-328 In developing the conventional machine model fortransient equation, the assumptions made were stated in 2.2Model Equation[15] as thus The generalized 4th order dq model equation of electrical1. The machine is symmetrical with a linear air-gap and machine [20] expressed in an arbitrary reference framemagnetic circuit rotating with arbitrary velocity ωc are the following,2. Saturation effect is neglected with the stator and rotor fluxes as state variables:3. Skin-effect and temperature effect are neglected Vqs  rs i qs  pλqs  ωc λds4. Harmonic content of the mmf wave is neglected5. The stator voltages are balanced Vds  rs i ds  pλds  ωc λqs 4  Vqr  rr i qr  pλqr  c  ω r )λdr (ωThe differential equations governing the transientperformance of the induction machine can be described in Vdr  rr i dr  pλdr  c  ω r )λqr (ωseveral ways; they only differ in minor detail and in their where flux equationssuitability for use in a given application. The conventional λds  Ls i ds  Lm i drmachine model is developed using the traditional method of λqs  Ls i qs  Lm i qrreducing the machine to a two-axis coil (d-q axis) model on 5 both the stator and the rotor as described by Krause and λdr  Lr i dr  LLm i dsThomas [16]. λqr  Lr i qr  Lm i qs2.1 D-Q Axis Transformation Theory Stator variables (currents, voltages and flux 3 Model equations in all flux variableslinkages) of a synchronous machine were first related with When dq current variables ids, idr, iqs and iqr arevariables of a fictitious windings rotating with rotor by park solved for in equation (5) and substituted in equation (4)[17]. This method was further extended by Keyhani [18] accordingly, a generalized model equation in an all fluxand Lipo [19] to the dynamic analysis of induction linkages of arbitrary reference is realized as follows.machines. By these methods, a polyphase winding can be For current variables as function of linkage fluxesreduced to a set of two phase-windings with their magnetic L λ  Lm λdraxes aligned in quadrature as shown in Figure 1 i ds  r ds Lr Ls  Lm 2 Ls λdr  Lm λds i dr  Lr Ls  Lm 2 Lr λqs  Lm λqr 6  i qs  Lr Ls  Lm 2 Ls λqr  Lm λqs i qr  Lr Ls  Lm 2Figure 1: Polyphase winding and d-q equivalent. The d-q axis transformation eliminates the mutual When equation (6) is substituted in equation (4) andmagnetic coupling of the phase-windings, thereby making simplified accordingly, a generalized electric machinethe magnetic flux linkage of one winding independent of magnetic flux based equation is obtained asthe current in the other winding. For three phases the  λ  aλ  cλ  ω λ  V qs qs qr c ds qstransformations are follows  λ  aλds  cλdr  ωc λqs  Vds i qd0  Tabc i abc  1 ds 7 also  λqr  bλqr  dλqs  fλdr  Vqr Vqd0  TabcV abc 2   λdr  bλdr  dλds  fλqr  VdrEquations (1-2) can be applied in any reference frame by wheremaking a suitable choice for theta (θ) in equation (3). rs Lr rr Ls a 2 , b ,If theta equals θr, then equations (1-2) lead to an expression Lr Ls  Lm Lr Ls  Lm 2for voltage in the rotor reference frame. Also, if θ is equal rs Lm r r Lm c  2 , d  , 8  Lr Ls  Lm 2to zero, then equations (1-2) apply to a frame of reference Lr Ls  Lmrigidly fixed in the stator (i.e. stationary reference frame).Otherwise, for θ equal to ωt in equations (1-2), a synchr- f  c  ωr ) (ωonously rotating reference frame results 322 | P a g e
  • 3. I. I. OKONKWO, S. S. N. OKEKE / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, July-August 2012, pp.321-3284 The Efficiency Of Electric Machine In Dq – FluxLinkage Variable Pcr  3rr i r 2  3 2  r r i dr 2  i qr 2  2 Three major losses that are accounted for in this 3  1 analysis are stator winding losses, rotor winding losses and  r   2 L L L 2 iron losses. The various expressions of all these losses are  r s m given [1] as functions of dq – currents, also the stator inputpower is the total power. To find the efficiency in dq – flux  2 2 (L s λdr  Lm λds )  s λqr  Lm λqs ) (L  16  2linkage variables all the dq – current variables must beeliminated as follows Pi  3ri i i 2  3  i m ωs ri  2  ri    i dm 2  i qm 2    Stator power losses 2  2     3 r  i m ωs   1Pcs  3rs i s 2 3   rs i ds 2  i qs 2 2  9  i     2  r i   Lr Ls  Lm 2           Rotor power losses    [λdr m  Ls ) λds m  Lr )] 2  (L  (L 17   2 Pcr  3rr i r 2  3 2  rr i dr 2  i qr 2    10  [λqr m  Ls ) λqs m  Lr )]   (L  (L     Iron losses 3 2 Pin  [Vqs i qs V ds i ds ]Pi  3ri i i 2 3 i ω  ri  m s 2  ri    i dm 2  i qm 2   11  2   3   L λ  Lm λqr    V qs  r qs  Power input  2   Lr Ls  Lm 2     3  real     18Pin  [Vqs i qs Vds i ds ]   12   L λ  Lm λdr     2  ds  r ds V  2     Lr Ls  Lm      where but efficiency ηi s  i ds  jiqs Pin   losses η 19 i r  i dr  jiqr Pini m  i dm  jiqm   13 Pin  Power inputi dm  i ds  i dr  losses  Pcs  Pcr  Pi Total Lossesi qm  i qs  i qr the values of Pcs, Pcr, Pi and Pin are found in equation [15 – 18]hencei s 2  i ds 2  i qs 2 5 Transient Simulation Of Electric Machine Dq – Flux Linkagesi r 2  i dr 2  i qr 2   14 In this paper transient dq – flux linkages of 2 2 2im  i dm  i qm electric machine are simulated by transforming the arbitrary reference frame electric machine equation in insubstituting equation (6) in (9 – 12) and simplifying its magnetic flux equivalent (7) from time domain toaccordingly to get laplace frequency domain, and hence rational function of the flux linkages in s – domain are the result of such   transformation. The inverse transforms of these flux 3Pcs  3rs i s 2  rs i ds 2  i qs 2 linkages (rational functions in s – domain) are close – 2 form transient time domain solution of electric machine 2 3  1  flux linkages. Discretization can now be achieved at  rs    desired time point from the close – form continuous 2  Lr Ls  Lm 2    time function of flux linkages obtained above.  2 (L 2 (L r λds  Lm λdr )  s λqs  Lm λqr )  15  The transformation of the magnetic flux equivalent of the generalized electric machine (7) to laplace frequency domain may be achieved as follows 323 | P a g e
  • 4. I. I. OKONKWO, S. S. N. OKEKE / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, July-August 2012, pp.321-328From equation (7), taking the laplace transform of (7(a)) to effects are not as sensitive as that of stator, rotor and theget magnetizing inductances. The iron loss is still the minimum of all the losses, though estimation of the iron λqs  λqs  aλqs  cλqr  ωc λds Vqs (s) (0) (s) (s) (s) (s) losses does not include the iron resistance in the (s) (0) 20  generalized machine equation that calculated the flux(s  a)λqs  ωc λds  cλqr  0 Vqs  λqs (s) (s) (s) linkages. Also The transient variation of the flux time is  ωc λqs (s)(s  a)λds  (s) 0  cλdr  (s) Vds  (0)21  (s) λds shown in fig (2) which shows that the stator flux is  dλqs (s) 0   b)λqr  (s (s) fλdr  (s) Vqr (s) λqr  (0) 22  always higher than that of the rotor flux as expected. The steady state flux linkages in the stator and the rotor0  dλds (s) fλqr  (s)(s  b)λdr  (s) Vdr (s) λdr  (0) 23  windings increase as the slip increase, that is reduced rotor speed. The results also shows that when the statorEquation (20 – 23) may be rearranged in compact form to and the rotor flux linkages increase more than theget optimal values , the operation of the motor results in s  a ωc c 0  λqs  Vqs  λqs  (s) (s) (0) reduced efficiency.  ω s  a 0  λ (s) V (s) λ (0)   c   ds   ds  ds   c   24  8 Conclusion  d 0 s  b f   λqr  Vqr (s) (s) λqr   (0)      Simulation software has been formulated which  0 d  f s  b   λdr  Vdr (s) (s) λdr   (0) estimates the efficiency the electric motor. The optimalThe expression for a, b, c, d, e and f are given in equation parameters of the electric motor which are responsible(8) for maximum efficiency was able to be estimated through simulation and the results agreed with the6 Steady State Simulations Of Dq Flux Linkages designed parametric configurations of the machine. The For steady state simulation equation (24) is speed versus flux linkages of the machine are in themodified for use in the steady state solution of the dq flux agreement with the expected results. The formulatedlinkages. Modification is by replacing all the (s) in the simulation program has shown to be useful in accessingmatrix operator of equation (24) by (jω) and also the optimum configuration of electrical machinetransforming all the source voltages to time domain parameters especially as they affect the efficiency of theequivalent and all the initial values set to zeroes. Hence machine at no load. For example, other parameters of  jω  a ωc c 0  λqs  Vqs  (t) (t) the simulated induction motor are found to be fully  ω jω  a 0  λ (t) V (t)   c   ds   ds   optimized with only the stator winding resistance which  c  25  when change from 4.85 Ω to 5.4 Ω improves the no load  d 0 jω  b f   λqr  Vqr  (t) (t) efficiency only but little figure (6)       0 d f jω  b   λdr  Vdr  (t) (t) Improvement in the method of the estimation of the machine efficiency may include the inclusion of theThe expression for a, b, c, d, e and f are given in equation iron resistance Ri in the generalized machine equation(8) that simulated the d – q flux linkages of the magnetic circuit equation.7 Simulation Results Efficiency variations of the induction motor aresimulated in steady state while flux linkages in the windings TABLE 1 MACHINEPARAMETERare also simulated during transient. All the simulations are Machine Parameter Actual Estimateddone with MATLAB 7.40 mathematical tools, the results of Stator resistance 4.85 Ω 5.4 Ωthe simulation of the electrical induction motor are shown Rotor resistance 3.805 Ω 3.8 Ωin fig. (2) thru fig. (13). Efficiency of the motor is simulated Iron loss resistance 500 Ω 1000+ Ωat various motor parameter variations, optimal machine Mutual inductance 0.258 H 0.259 Hparameters are obtained and compared with the designed Stator inductance 0.274 H 0.272 Hparameters, and complete results are summarized in table Rotor inductance 0.274 H 0.272 H(1). These results are of that of the rotor reference frame, Rotor inertia 0.031 Kg/m2 – –there were no much differences observed in the results from Friction coefficient 0.008 Nm.s/rd – –other reference frames Output power 1.5 Kw – – The optimal slip of the motor (0.26) with Poles 2x2 – –maximum efficiency of (97 %) is the result of the Voltage 220/380V 220 Vsimulation fig (4). The efficiency varies as other machine’s Current 3.64/6.31 A – –parameter varies with exemption of the terminal voltage Rated speed 1420 rev/min – –which shows no significant efficiency changes at variations Frequency 50 Hz 50 Hzat no – load. Though, stator and the rotor resistance Maximum efficiency – – 97.04 %variations affects the efficiency of the induction motor, their 324 | P a g e
  • 5. I. I. OKONKWO, S. S. N. OKEKE / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, July-August 2012, pp.321-328Nomenclatures, r : stator and rotor indicesxdq : components in the synchronous reference framedq axesRs, R r : stator and rotor resistancesRi: Iron loss resistancesLs, Lr: stator and rotor inductances Figure 3: Steady State Stator Andls, lr: stator and rotor leakage inductances Rotor Slip Versus Flux ResponseLm: magnetizing fluxs, r: stator and rotor flux Steady State Efficiency Response Simulation Graphsm: magnetizing fluxΩ s: stator pulsation (rd/s)Ωr: rotor speed (rd/s)p: number of pole pairsPcs, Pcr: stator and rotor copper lossesPi: Iron lossesSteady And Transient State Flux ResponseSimulation Graphs Figure 4: Slip Versus Efficiency ResponseFigure 2: Transient State Stator And Rotor Time Versus Flux Response 325 | P a g e
  • 6. I. I. OKONKWO, S. S. N. OKEKE / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, July-August 2012, pp.321-328Figure 5: Stator Voltage Versus Efficiency Response Figure 9: Magnetizing Inductance Versus Efficiency Response Quick surveyFigure 6: Stator Resistance Versus Efficiency Response Figure 10: Stator Inductance VersusFigure 7:Rotor Resistance Versus Efficiency Response Efficiency Response Optimal point surveyFigure 8: Iron Resistance Versus Figure 11: Stator Inductance Versus Efficiency Response Efficiency Response 326 | P a g e
  • 7. I. I. OKONKWO, S. S. N. OKEKE / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, July-August 2012, pp.321-328 Quick survey 4 J. Mizia, K. Adamiak, A. R. Eastham, And G. E. Dawson, Finite Element Force Calculation: Comparison Of Methods For Electric Machines, IEEE Transactions On Magnetics, (1988), Vol. 24, No. 1, Pp.447-450. 5 V. Ostovic, A Method For Evaluation Of Transient And Steady State Performance In Saturated Squirrel Cage Induction Machines, IEEE Transactions On Energy Conversion, (1986), Vol. 1, No. 3, Pp. 190-197. 6 V. Ostovic, A Simplified Approach To Magnetic Equivalent-Circuit Modeling Of Induction Machines, IEEE Transactions On Industry Applications,(1988), Vol.24, No. 2,Figure 12: Rotor Inductance Versus Pp. 308-316. Efficiency Response 7 P. Sewell, K.J. Bradley, J.C. Clare, P.W. Wheeler, A. Ferrah, R. Magill, Efficient Dynamic Models For Induction Optimal point survey Machines, International Journal OfNumerical Modeling: Electronic Networks, Devices And Fields, (1999)Vol. 12, Pp. 449-464. 8 L. Haydock, The Systematic Development Of Equivalent Circuits For Synchronous Machines, Ph.D Thesis, University Of London, 1986. 9 V. Ostovic, J.M. Miller, V.K. Garg, R.D. Schultz, S.H. Swales, A Magnetic-Equivalent- Circuit-Based Performance Computation Of A LundellAlternator, IEEE Transactions On Industry Applications, (1999), Vol. 35, No. 4, Pp. 825-830. 10 M.J. Carpenter, D.C. Macdonald, CircuitFigure 13: Rotor Inductance Versus Representation Of Inverter-Fed Synchronous Efficiency Response Motors, IEEE Transactions On Energy Conversion, (1989), Vol. 4, No. 3, Pp. 531- 537. 11 H. Roisse, M. Hecquet, P. Brochet, Simulation Of Synchronous Machines Using A Electric-Refereences Magnetic Coupled Network Model, IEEE1 Mehdi Dhaoui And Lassa Sbita, A New Method Transactions On Magnetics, (1998), Vol. 34, For Losses Minimization In IFOC Induction No. 5, Pp. 3656-3659. Motor Drives. International Journal Of Systems 12 Li Yao, Magnetic Field Modelling Of Machine Control, Volume 1-2010/Iss.2 Pp 93-99 And Multiple Machine Systems Using2 M.J. Carpenter, Magnetic Equivalent Circuits, Dynamic Reluctance Mesh Modelling, Ph.D Proceedings IEE, Vol. 115 (10), 1968, Pp.1503- Thesis, University Of Nottingham 2006. 1511. 13 Lim S. , Nam K. ( 2004). Loss Minimization3 J.L. Coulomb, G. Meunier, Finite Element Control Scheme For Induction Motors. IEE Implementation Of Virtual Work Principle For Proc. Electr. Power Applications, Vol.(151), Magnetic Or Electric Force And Torque No. 4, Pp. 385-397. Computation, IEEE Transactions On Magnetics, 14 Geng Y. Et Al., (2004). A Novel Control (1984), Vol. 20, No. 5, Pp. 1894-1896. Strategy Of Induction Motors For The 327 | P a g e
  • 8. I. I. OKONKWO, S. S. N. OKEKE / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, July-August 2012, pp.321-328 Optimization Of Both Efficiency And Torque Response. In Proc. IEEE Conf. Ind. Electron. Society, Pp. 1405-1410.15 O. I. Okoro, Matllab Simulation Of Induction Machine With Saturable Leakade And Magnetizing Inductances. IPJST Transactions Volume 5. Number 1, 2003.16 Krause, P.C. And Thomas, C.H.; Simulation Of Symmetrical Induction Machinery. Transactions IEEE, PAS – 84, Vol. 11, Pp. 1038-1053, 1965.17. He, Yi-Kang And Lipo, T.A.: Computer Simulation Of An Induction Machine With Spatially Dependent Saturation. IEEE Transactions On Power Apparatus And Systems, Vol. PAS-103, No.4, PP.707-714, April 1984.18. Keyhani, A. And Tsai, H.: IGSPICE Simulation Of Induction Machines With Saturable Inductances. IEEE Transactions On Energy Conversion, Vol. 4, No.1, PP.118-125, March 1989.19. Lipo, T.A. and Consoli, A.: Modeling and simulation of induction motors with saturable leakage reactances. IEEE Transactions on Industry Applications, Vol. IA-20, No.1, PP. 180- 189, Jan/Feb, 1984.20 Navrapescu, V., Craciunescu, A., And Popescu, M.: - “Discrete-Time Induction Machine Mathematical Model For A Dsp Implementation“ – In Conf.Rec. Pcim’98, May 1998, Nuremberg, Germany, Pp. 433-438 328 | P a g e