Computation of the Effect of Space Harmonics on Starting Process of Induction Motors Using TSFEM
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1 Journal of elecrical sysems Special Issue N 01 : November 2009 pp: Compuaion of he Effec of Space Harmonics on Saring Process of Inducion Moors Using TSFEM Youcef Ouazir USTHB Laboraoire des sysèmes élecriques indusriels (LSEI) BP 32 El-Alia Alger, Algérie The ime sepping finie elemens (TSFE) mehod is used o invesigae he effec of space harmonics on no load and full load direc line saring of inducion moor. The mehod used in his compuaion leads o consider only he firs or jus some significan space harmonics in ime sepping finie elemens model. The ransien resuls of saring process are validaed wih he experimenal measuremens, and he effec of space harmonics on curren, speed and orque are clearly shown. Index Terms Inducion moor, Saring process, Space harmonics, Inerface coupling, Time sepping finie elemen mehod. T I. INTRODUCTION HE INSTANTANEOUS magneic field in inducion moors include boh space and ime harmonics. They are caused respecively by he geomerical effec of winding and slos and by he sauraion and saic converers. If he moor is driven by sinusoidal volage sources, he phase-bel harmonics and slo harmonics of he saor and roor cause relaively large harmonic copper and core losses and breaking forces, which may decrease he orque of he moor [1]. Various works presened in he lieraure are based on equivalen circui modeling, where he sauraion and eddy curren effecs are negleced. When using he Finie Elemens mehod, ha is he mos precise way for modeling he coupled field circuis and moion, we can no analyze he fundamenal and harmonic fields separaely. This paper presens he ransien magneic field modeling of cage inducion moors using he air-gap inerface mehod [2], [3]. In addiion o considering he movemen of he roor, his mehod offers he possibiliy o consider only he firs space harmonic, in he case of sinusoidal disribuion winding, or some significan space harmonic order. The effec of space harmonics on he orque, curren and speed are compued wih his echnique where some resuls are presened. II. TIME STEPPING FINITE ELEMENT MODEL OF INDUCTION MACHINE When using a 2D FEM modeling of inducion moors, i is assumed ha he magneic vecor poenial has only one componen in he axial direcion and he magneic problem is solved in wo reference frames relaed o he saor and roor. In order o accoun for he movemen of he roor by using he air-gap inerface coupling mehod, he disconneced domains of he saor and roor ( Ωs and Ω r ), shown in Fig. 1, are coupled on an arificial inerface Γ, locaed a radii R in he air gap. The finie elemens form of he field coupled o elecrical circui equaions leads o he following sysem [4]: QX + T X = P where he homogenous Newman condiions, are considered in he inerface coupling. In equaion (1), Q is he siffness and elecrical circui coefficien marix of saor and roor, T is he conduciviy and elecrical circui coefficien marix, P is a vecor associaed wih inpu volages, and X is he unknown vecor associaed wih magneic vecor poenial in he saor and roor domains (A s and A r ), he saor curren and he volage beween he wo ends of cage bar. The coupled field-circui model described above (1) is closely dependen on mechanical movemen of he roor. Then, he mechanical equaion governing he roor moion mus be inroduced in his model. I is given by: Ω J m + f Ω + Cch = C em where J m is he momen of ineria, f he fricion coefficien, C ch is he load orque. C em is he elecromagneic orque. In erm of angular posiion he mechanical speed of he roor is given by: θ Ω = III. COMPUTATION METHOD To solve equaions (1), (2) and (3), he backward Euler s mehod is used for he ime discreizaion. They we can wrie: 1 +Δ 1 Q + T X = T X + P Δ Δ Δ Ω + Δ = Ω + ( Cch + f Ω + Cem J ) (5) (1) (2) (3) (4) Digial Objec Idenifier insered by JES +Δ +Δ θ = Ω Δ + θ (6)
2 Journal of elecrical sysems Special Issue N 01 : November 2009 pp: Ω r Γ Ω s To validae he saring process resuls, he simulaion are down under he same measured volage shown in Fig. 2. The ime harmonics of his volage affecs direcly he waveform of he absorbed curren. The compued and measured resuls for ransien curren and speed are depiced in Fig.3. We noice ha heir shapes are similar. Fig.1. Decoupled saor and roor domains on one pole of moor In his compuaion he global field-circui problem and mechanical moion are solved separaely. The meri of his mehod is is simpliciy [5]. The sysem equaion (4) aking ino accoun sauraion, fundamenal wihou or wih space harmonics field and exernal circui equaions is solved by Newon Raphson mehod. A his sep we can obain he elecromagneic orque necessary o solve he mechanical equaion by Runge-Kua mehod of fourh order. The elecromagneic orque is compued by he Maxwell sress ensor on conour Γ surrounding he inerface coupling of saor and roor. I is given by: (a) C em R = 2 L μ0 2π b b dθ n (7) where b and b n are respecively he angenial and normal componens of field and L is he effecive lengh of he moor. In hese condiions he ransien phenomena of inducion moor are hen due o magneic, elecric circui and mechanical behaviors. IV. MODEL VALIDATION A sandard inducion moor of large use in indusry (wih 4 poles, 5kW, 380 V, [2]) has been modeled, using he developed echnique. In such resoluion, he ime sep Δ is chosen in order o compue he insananeous variaions of all he quaniies as well as possible. (b) (c) Fig. 2.Measured inpu neural-phase volage. Fig.3. Measured and compued curren and speed. (a)- Transien curren. (b)- Curren a seady saes (c) - Speed.
3 Journal of elecrical sysems Special Issue N 01 : November 2009 pp:48-52 V. EFFECT OF SPACE HARMONICS ON TRANSIENT DIRECT LINE STARTING In his secion he sudy is focused on he compuaion of he influences of space harmonics on saring quaniies of inducion moor. In doing so, he ransien curren, orque and speed of he roor waveforms are compued for he case where only he fundamenal space harmonic of field is considered and for he case where some significan space harmonics field are considered. In his compuaion of he ransien saring process, he sudied moor is supplied wih sinusoidal volage. A ime sep of 0.1 ms is sufficien o compue he ransien operaion when only he firs space harmonic is considered and 0.01 ms when he some space harmonics are considered. I is eviden ha he CPU ime is reduced in he firs case where only he firs space harmonic is considered. A. No load saring process The ransien curren, speed and orque resuls a no load are shown respecively in Fig. 4 o 6 for he fundamenal and muliharmonic cases. The effec of space harmonics on saring curren is clearly shown. In ransien region he peak values of curren is grea in he muliharmonic case, bu a seady sae relaively imporan ripples on saor curren are caused by he space harmonics. This effec on speed is relaively no imporan. The compued orque waveform shows a large peak values and ripples in ransien region. Fig. 4 Compued curren a no load saring B. Full load saring process The resuls for full load direc line saring are shown in Fig. 7 o 9 for curren, speed and orque respecively. The effec of space harmonics on saor curren (Fig. 7) is no imporan in ransien region, bu a seady sae he peak values of saor curren are increased and he corresponding waveform is deformed. The decreasing roor speed a seady sae in muliharmonic case is caused by he breaking harmonic orques generaed by he space harmonics. This effec can be explained by he large orque ripples observed in boh ransien and seady sae regions of orque waveform (Fig. 9). Fig.5 Compued speed a no load saring Transien region Fig.6 Compued orque a no load saring
4 Journal of elecrical sysems Special Issue N 01 : November 2009 pp: Fig. 8 Compued speed a full load saring Transien region Fig. 7 Compued saor curren a full load saring. Fig. 9 Compued orque waveforms a full load VI. CONCLUSION The saring process of inducion machine is compued wih nonlinear ime sepping finie elemen mehod aking ino accoun only he fundamenal field wihou or wih some significan space harmonics. The obained resuls show clearly he effec of space harmonics on he curren, speed and orque in boh ransien and seady sae regions. This echnique can be used o invesigae he effec of space harmonics on sray orque and losses.
5 Journal of elecrical sysems Special Issue N 01 : November 2009 pp: VII. REFERENCES [1] K. Yamazaki, Y. Haruishi, and T. Ara, Calculaion of negaive orque caused by slo ripples of inducion moors, IEEE Trans. Mag., vol. 40, No. 2, pp , March [2] Y. Ouazir, N. Takorabe, R. Ibiouen, S. Mezani, Consideraion of Space Harmonics in Complex Finie Elemens Analysis of Inducion Moor wih an Air-gap Inerface Coupling, IEEE Trans. Mag., Vol. 42, No. 4, pp , April [3] Y. Ouazir, N. Takorabe, R. Ibiouen, R. Benhadadi, Time Sepping FE Analysis of Cage Inducion Moor wih Air-Gap Inerface Coupling, Thireenh Biennial IEEE Conference on Elecromagneic Field Compuaion, CEFC2008, Ahens, Greece, May 11-15, 2008, pp. 43. [4] Y. Ouazir, N. Takorabe, R. Ibiouen, R. Benhadadi, Time Sepping FE Analysis of Cage Inducion Moor wih Air-Gap Inerface Coupling Taking Ino Accoun Phase-Bel Harmonics, IEEE Trans. Mag., Vol. 45, No. 3, Mars [5] S. L. Ho, H. L. Li, W. N. Fu, and H. C. Wong, A novel approach o circui-field-orque coupled ime sepping finie elemen modeling of elecric machines, IEEE Trans. Mag., Vol. 6, No. 4, pp , July 2000.
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