Steady State Analysis of Squirrel-Cage Induction Machine with Skin-Effect

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1 Steady State Analysis f Squiel-Cage Inductin Machine with Skin-Effect D.-Ing. O. I. Ok Depatment f Electical Engineeing Univesity f Nigeia, Nsukka Enugu State, Nigeia. gnnayak@htmail.cm ABSTACT The Steady state analysis f a ectangula shaped-t type squiel-cage inductin machine was examined numeically with the help f MATLAB. The esults f the ppsed Skin-effect mdel wee cmpaed with the cnventinal mdel. The esults f this cmpaisn shw that a lage e ccus when the influence f skin-effect is neglected in the analysis f a squiel-cage inductin machine at high t fequency. (Key wds: squiel-cage, inductin machine, steady state, skin effect, cnventinal mdel, ectangula t a) INTODUCTION The tansient state mdel the steady-state equivalent cicuit is usually emplyed t epesent an inductin machine. Geneally, the steady-state equivalent cicuit is used when the machine is unning at steady-state, and a tansient state mdel is used when the machine is unning at tansient peatins. Unftunately, the cnventinal mdel f an inductin machine, which cnsists f an equivalent cicuit with a single lp, is nt adequate t take int accunt the skin-effect in t as; which may nt e negligile when the machine is fed fm nnsinusidal vltage cuent suce [,]. etinently, f the study f skin-effect in t as, the t cicuit needs t e mdified in de t adequately take int accunt the wide ange f fequencies ccuing in the machine. Levy [3] and Cee [4] have ppsed a t mdel with thee t lps. It has een nted y [] that the mdel accuacy inceases when the nume f the t lps is inceased t fu me. In this pape, the t cicuit is mdeled as ppsed y []. This pape highlights the methd used f the identificatin f the t cicuit paametes. This pape als includes the develpment f the mathematical analysis f the steady-state cnventinal and skin-effect mdels. The pape cncludes with a cmpute simulatin and sme elevant cmments n the esults. OTO CICUIT AAMETE IDENTIFICATION The test machine used in this study is a KATT VDE 53, Class F insulatin, sufaced-cled Squiel-cage inductin mt. The ated pwe, speed, and cuent ae 7.5KW, 4pm, and 9.A, espectively. Figue shws the test machine. The slt gemety, shwing the stat and t shapes, is shwn in Figue. Figue : The 7.5KW test mt. The acific Junal f Science and Technlgy 56 Vlume 5. Nume. Octe 4 (Fall)

2 whee, µ = emeaility f fee space Nut = Width f the t a h sec = height f each a sectin χ cu = cnductivity f cppe cnduct L s = Length f t a It is imptant t nte, that equatins () and () ae mdified t accunt f all f the as and ae susequently efeed t the stat y using the tansfmatin fact, K. The values f the efeed t inductances and esistances ae defined mathematically y, L = k L (3a) Figue : Slt gemety f the test machine. T mdel the t a, a T-cnfiguatin netwk is used accding t the methd ppsed y Ba and Williams [5]. The t a is divided int five sectins as shwn in Figue 3. L = k L (3) L 3 = k L3 (3c) L 4 = k L4 (3d) L 5 = k L5 (3e) L=L+L/ (3f) e = k e (4a) = k (4) = k (4c) 3 = k 3 (4d) Figue 3: Equivalent T-Cicuit; Cnfiguatin f 5-sectin t a. 4 = k 4 (4e) 5 = k 5 (4f) The t a esistance and inductance f each sectin is, whee k is defined thus [6]: sec s = () χ h cu L sec Nut k = m k m k w w N N (5) L sec µ L h s sec = () Nut whee, m = nume f phases n the stat m = nume f phases n the t k w = stat winding fact The acific Junal f Science and Technlgy 57 Vlume 5. Nume. Octe 4 (Fall)

3 k w = t winding fact N = nume f seies-cnnected tuns pe phase f the stat N = nume f seies-cnnected tuns pe phase f the t But, m = (nume f t as)/(nume f pais f ples) Q m = (6) Seinsch [7] defines the elatinship etween the t a esistance and the t esistance as: = a (7) with the equivalent t efeed esistance as, = k a (8) whee, k k = (9) By using the ptimizatin technique epted in [], the values f the t cicuit paametes can e fund at 4KHz t fequency. The esult f the ptimizatin, as shwn in Figue 4, clsely matches with the actual t a chaacteistics f the machine. At appximately 4KHz fequency, the e in the develped t mdel is aut 6%. Tale shws the estimated t cicuit paametes. CONVENTIONAL AND SKIN EFFECT STEADY STATE ANALYSIS F the pupse f digital simulatin, the electical mdel f the squiel-cage inductin machine can e epesented in state vaiale fm with cuents as state vaiales y equatin (). [] d dt i = [ L] ([ ] + [ G] )[] i + [ L] [ V] ω () Figue 4: Mdel lts f Ba Sectins () and Mdel Sectins (5). Tale: Cmputed Mdel Impedance at 4KHz. esistance [mω] Inductance [µh].338 L 6.5e-.656 L 9.94e- 3.3 L L L In cnsideatin f the steady-state equatin f the inductin machine with skin-effect, the time deivatives f cuent in equatin () ae set t ze, with the t speed cnstant, and the machine d- and q-vltages and cuents ae efeed t the synchnusly tating efeence fame. By s ding, the esultant algeaic equatin can e expessed in cmpact fm as: [ ] [ Z ][ I ] V = () and, [ ] [ ] I = Z V () F the cnventinal steady- state mdel: The acific Junal f Science and Technlgy 58 Vlume 5. Nume. Octe 4 (Fall)

4 [ ] [ ] t I = I I I I (3) qs ds q d whee, [ V ] = [ V V ] and, [ Z ] qs s Ls = ds ( ω ) Lm ( ω ) ( ω ) L ( ω ) L m ωl s s t L m V qs = Steady-state q-axis stat vltage (4) V ds = Steady-state d-axis stat vltage i qs,i ds = Steady-state q- and d-axis stat cuents, espectively i q,i d = Steady-state q- and d-axis t cuents, espectively Lm I q, I d = Steady-state q- and d-axis t cuents, espectively I Q, I Q = Steady-state q-axis t cuents L in lp and lp, espectively I D, I D = Steady-state d-axis t cuents in lp and lp, espectively (5) f = ated fequency ω F the skin-effect steady-state mdel, we have: The electmagnetic ehavi f the machine unde steady-state cnditin is descied y the equatin: I [ s iqs = id id iq id iq id3 iq3 4 iq4 id5 iq ] t (6) T =. 5L ( i i i i ) id 5 e m qs d ds q () [ ] = [ Vd ] t V (7) and, s Vq s s Lsω Lmω Lsω s Lmω ( ω ) ( ) Lm ω L ω ) Lm ( ( ω ) L [ ] Z = (8) = e + (9a) = + (9) 33 = + 3 (9c) 44 = (9d) 55 = (9e) ω = πf (9f) whee, T e L m = Steady-state electmagnetic tque = Magnetizing inductance = Nume f ple pais. ESULTS AND DISCUSSION MATLAB m-files [8] wee develped and used t slve equatins () and (), f th the cnventinal and skin-effect steady-state analysis. The develped pgam accepts cnstant stat and t quantities, such as esistances and inductances as inputs. These cnstant quantities wee fund thugh the pen-cicuit test, lcked-t test, and D.C. measuement test, caied ut n the 7.5KW Squiel-cage inductin machine. The machine data is shwn in Appendix A. By supplying these input paametes, the steady-state ehaviu f the machine can e pedicted. The gaphical epesentatins f stat cuent, input pwe, utput pwe, tque, efficiency, and pwe fact as a functin f mechanical t speed ae shwn in Figue 5 and Figue 6 f the cnventinal and skin-effect analysis espectively. The acific Junal f Science and Technlgy 59 Vlume 5. Nume. Octe 4 (Fall)

5 Figue 5: Steady State efmance Cuves (Cnventinal Mdel) Figue 6: Steady State efmance Cuves (Skin-Effect Mdel) The acific Junal f Science and Technlgy 6 Vlume 5. Nume. Octe 4 (Fall)

6 In de t visualise the effects f skin effect n the steady and tansient states pefmances f the inductin machine, the esults f the simulatin invlving skin-effect have t e cmpaed gaphically with that fm the cnventinal mdel in Figue 7. The steady-state magnitude f the t a cuents f each sectin, as a functin f mechanical t speed, is shwn in Figue 8. Fm Figue 8, it can e seen that as the a depth inceases, the magnitude f the t a cuent deceases. This explains why highe cuent flws thugh the fist t a sectin as ppsed t the the sectins at stating. CONCULSIONS The pape has systematically pesented the steady-state analysis f Squiel-cage inductin machine with skin-effect. The simulated esults indicate that thee exists a significant diffeence etween the cnventinal steady-state mdel, and the steady-state mdel with skin-effect. The develped tque in the skin-effect machine mdel was aut thee times geate than that f the cnventinal machine mdel at stating. Additinally, at stating, the magnitude f the stat cuent, pwe fact, and input pwe f the skin-effect mdel was seved t e highe than that f the cnventinal mdel. Fm the esults pesented in this pape, it can e seen that at high t fequency, the cnventinal machine mdel fails t pedict accuately the steady-state ehaviu f the machine. The esults pesented in this pape will theefe, fm a veitale tl f an industial enginee in pedicting the actual ehaviu f a deep-a Squiel-cage inductin machine pi t design. ACKNOWLEDGEMENT The auth wishes t expess his thanks t DAAD f thei financial suppt and t f. D.- Ing. B. Weidemann f the Univesity f Kassel, Gemany f his assistance and advice in pducing this pape. Figue 7: Cnventinal and Skin-Effect Mdels Cmpaisn The acific Junal f Science and Technlgy 6 Vlume 5. Nume. Octe 4 (Fall)

7 [3.] Levy, W., C.F. Landy, and M.D. McCullch. 99. Impved Mdels f the Simulatin f Deep Ba Inductin Mts. IEEE Tansactins n Enegy Cnvesin. 5(): [4.] Cee, W. H., D.W. Nvtny, and T.A. Lip Deteminatin f Equivalent Cicuits f Inductin Machines with Skin- Effect Using Teminal Chaacteistics. Electic Machines and we Systems. (5-6): [5.] Ba, D.S. and J.E. Williams. 95. Netwk Analysis f A-C Machine Cnducts. AIEE Tansactins. 7:- 5. Figue 8: t Ba Cuents f each Sectin with Skin-Effect AENDIX A: Output we 7.5KW ated vltage 34V Winding cnnectin Delta Nume f les 4 ated speed 4pm ated fequency 5Hz Stat esistance.595hm Stat leakage eactance.9545hm t esistance.9769hm t leakage eactance.9945hm Magnetizing eactance hm Mechanical shaft tque 5.636N.m Estimated t inetia mment.7393kgm^ ated cuent 9.A Mment f inetia f the D.C. mt.958kgm^ Shaft stiffness cnstant 43Nm/ad EFEENCES [.] DellA quila, A and L. Salvate Mdeling f Invete-fed Inductin Mts y FFT cessing f Vltage and Cuent Signals. IMACS [.] Ok, O.I.. Dynamic and Themal Mdeling f Inductin Machine with Nn- Linea Effects. Dissetatin. Univesity f Kassel, Gemany. [6.] Nasa, S.A. and L.E. Unneweh Electmechanics and Electic Machines. Jhn Wiley and Sns: New Yk. [7.] Seinsch, H. O Gundlagen Elektische Maschinen und Antiee. Teune: Stuttgat, Gemany. [8.] Mathwks, Inc MATLAB Use s Guide. The Mathwks, Inc: Natick, New Yk. ABOUT THE AUTHO D. Ing. O.I. Ok hlds a h.d. in Electical Machines fm the Univesity f Kassel, Gemany whee he cnducted his eseach unde a DAAD Schlaship. He cuently lectues in the Depatment f Electical Engineeing at the Univesity f Nigeia, Nsukka. His eseach inteests ae in the aeas f dynamic simulatin and cntl f inductin machines as well as in the themal and dynamic analysis f AC machines. He is a meme f the IEEE. SUGGESTED CITATION Ok, O.I. 4. Steady State Analysis f Squiel-Cage Inductin Machine with Skin- Effect. acific Junal f Science and Technlgy. 5():56-6. acific Junal f Science and Technlgy The acific Junal f Science and Technlgy 6 Vlume 5. Nume. Octe 4 (Fall)

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