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1 ,*)(48(&/83('3$$(7('(/)$&7:,',*6 G. Grandi *, D. Casadei *, A. Massarini ** * Dept. of Eletrial Engineering, viale Risorgimento, 436, Bologna - Italy ** Dept. of Engineering Sienes, via Campi, 3/B, 4, Modena - Italy Abstrat. The paper desribes a HF equivalent iruit for the stator winding of three-phase AC motors, valid in a wide frequeny range. The fitting is based on winding omplex impedane measurements. Both the phase-to-phase and the phase-to-ground impedanes are onsidered. The proposed an be utilized in theoretial and numerial HF analysis of inverter-fed AC motors. In this ase, the equivalent iruit allows to predit both differential- and ommon-mode onduted EMI. Furthermore, also transient effets an be predited when the analysis is performed in the time domain. The numerial results, obtained by means of PSpie, are ompared with the orresponding experimental tests in both frequeny and time domains. Keywords. HF, equivalent iruit, AC winding, onduted EMI,7'8&7, In reent years, inverter motor drives have beome the most popular systems to handle eletromehanial power onversion. The modern power onverters are based on swithing mode operation with stati omponents of the last generation. The stati swithes, suh as MOSFETs and IGBTs, are haraterized by very fast ommutations (i.e., frations of µs). The swithing frequeny is usually fixed in the range of tens of KHz to avoid aousti noise. As a onsequene, undesired harmoni voltage omponents an range from tens of KHz to several MHz. These High Frequeny (HF) voltage harmonis are responsible of HF urrents. The propagation path of the urrent harmonis is rather unexpeted owing to stray and parasiti motor parameters that take importane at high frequeny. Thus, a iruit of the motor windings is partiularly useful to predit HF urrent omponents and, in general, onduted Eletro- Magneti Interferenes (EMI) []. Sine the motor frame is usually grounded, both differential- and ommon-mode EMI must be onsidered. In the low and medium power range, the low voltage indution motor is the most frequently used type of motor. The windings of this type of motor are usually realized by a series onnetion of mush wound oils. A winding onsisting in a asade onnetion of single-oil s an be partiularly useful to determine the fast fronted voltage distribution among the oils []-[6]. Owing to the random distribution of the turns in eah oil, an analytial evaluation of the oil annot be based on single-turn s suh as in form wound oils. Hene, the lumped equivalent iruit of a oil an be defined in terms of equivalent impedane by a proper three-terminal iruit. Both the real and imaginary omponents of the impedane [7] or only the impedane magnitude [8] an be onsidered. In this paper the main results obtained in [7] are summarized and extended to the ase of multi-oil stator windings with referene to three-phase indution motors. 6,*/(&,/'(/ A detailed analysis of single-oil stator windings has been presented in [7] leading to the lumped parameter represented in Fig.. R g C L R P R C R L C L R P iron (ground) R C Fig. : HF equivalent iruit of a mush wound oil The oil is regarded as a series onnetion of N turns having a irular ross setion. The distribution of the N turns is random in both the slot region and the overhang region. The oil geometry and the oil ross setion in the slot region are given in Fig. A of the Appendix. %DFNJURXQG The HF iruit of a mush wound oil proposed in [7] onsiders both turn-to-turn and turn-to-iron apaitanes. Also additional dissipative phenomena suh as skin and proximity effets in the wires, dieletri and iron losses are taken into aount. The proposed approah is based on the equivalene between the oil and the iruit in terms of omplex impedane. Hene, both the real part (equivalent series resistane) and the imaginary part (equivalent series reatane) of are onsidered. Experimental tests on several oils have been arried out, showing in all ases a pair of more or less smoothed R L R g

2 parallel resonanes in the onsidered frequeny range. Furthermore, very large dissipative phenomena assoiated with the resonanes have been observed. The equivalent iruit shown in Fig. allows the frequeny response of the oil to be represented in a satisfatory manner when the iruit parameters are properly evaluated. RGHOSDUDPHWHUHYDOXDWLRQ To make the fit with the experimental results an identifiation problem has to be solved. The identifiation proedure an be arried out by a simple trial and error method or more sophistiated numerial tehniques suh as the least squares method. The omplex admittanes between the two oil terminals ( ) and between a oil terminal and the stator frame ( g ) an be expressed as a funtion of the parameters and the exitation angular frequeny (ω) as follows g g where and are given by R g Mω (), Rg M ωc Rg g MωC g R R MωL p L R R MωL p L R C R MωC (). MωC An initial estimate of the main iruit parameters an be performed using the proedure presented in [7]. A brief desription of this proedure is given below. YHUDOOFRLOLQGXFWDQFHAs it is shown in Fig. A, the oil onsists of two semi-irular or retangular parts in the overhang region, onneted by two parallel straight parts in the slot region. We assume the oil indutane L as the sum of the indutane L DLU of the wires in the overhang region and the indutane L LURQ of the wires in the slot region. This assumption is satisfied with a good approximation. In fat, the flux lines produed by the overhang ondutors mainly lie in planes perpendiular to the iron lamination. These flux lines do not penetrate the iron ore owing to the shielding effet of the eddy urrents whih an freely irulate in the iron sheets. On the basis of the previous onsiderations, we an alulate the indutane L DLU by the image method related to a urrent in presene of a planar magneti shield [9]. In the ase of semi-irular overhangs, the indutane an be alulated as the indutane of a single irular loop of diameter D h L DLU D C 8D 7 ln N. (3) D 4 µ o h h In the ase of retangular overhangs, the method of partial indutane an be adopted [] to alulate the indutane L DLU. With referene to the indutane L LURQ, we an note that this ontribution is strongly affeted by the frequeny. In this ase the lamination opposes the eddy urrents so that the shielding effet beomes signifiant only at high frequenies. It has been verified that only for frequenies above several MHz the magneti field is ompletely rejeted outside the laminated iron ore. For this reason, the determination of the oil indutane requires an aurate and omplex field analysis taking eddy urrent and iron lamination into aount. However, negleting the magneti field penetration inside the laminated ironore, the indutane L LURQ an be evaluated as the indutane of a group of N parallel straight ondutors having length d ir, and surrounded by a ylindrial shield. This assumption leads to LURQ µ L o D ln s d ir N. (4) π 4 D The indutane measurements an be performed only up to few tens of khz due the parallel resonanes aused by turn-to-turn and turn-to-iron parasiti apaitanes. A method to extend the indutane measurements to the high frequeny range is desribed in [7]. It onsists of indutane measurements on oils with a redued number of turns. In this way the parasiti apaitive effets are redued and the parallel resonanes are shifted at higher frequenies. Then, the oil indutane an be alulated by an extrapolation proedure. It has been verified that the sum of the values alulated by (3) and (4) represents the asymptoti lower limit of the extrapolated values. &RLOFDSDFLWDQFHV An analytial evaluation of the oil apaitanes, suh as in [], is prevented by the random distribution of the turns in mush wound oils. The overall oil-to-ground apaitane C tg an be evaluated by short-iruiting the oil terminals and measuring the apaitane between the resulting node and the motor frame. In the equivalent iruit of Fig., the apaitive oupling to ground is represented by the two apaitanes C tg / onneted at the beginning and the end of the oil. &RLO ORVVHV The oil losses are taken into aount by series and parallel resistanes. The sum of R L and R L represents the AC wire resistane. The values of these resistanes are funtion of the exitation frequeny owing to skin and proximity effets. However, in order to obtain an equivalent iruit whih an be readily utilized for both frequeny and time domain analyses, these resistanes are fixed at average values (hundreds of Ohm). R p and R p take aount of the AC iron losses and the orresponding values are in the order of KΩ or tens of KΩ. R C,R C,and R g are introdued to the dissipative phenomena due to HF apaitive urrents, and dieletri losses. Usually their values are in the order of tens of Ohm.

3 '(7$,/(':,',*'(/ For the onsidered AC motor size (i.e., from frations of kw up to few tens of kw), the stator windings are built as a series onnetion of mush wound oils. In order to obtain the iruit of the motor windings, a asade onnetion of oil s an be onsidered. In order to evaluate the indutive and apaitive ouplings among the oils, either numerial or experimental approahes an be employed.,qgxfwlyhfrxsolqjv The numerial evaluation of the mutual indutanes among the oils is a hard task due to the presene of a laminated iron ore. In fat HF eddy urrents an flow in the iron ore despite of the iron lamination. As a onsequene, the mutual indutanes hange as a funtion of the exitation frequeny. The alulation of the mutual indutanes an be performed by negleting the magneti penetration depth in the iron ore and onsidering the oupling in the overhang region only. This value orresponds to the asymptoti lower limit of the mutual indutane obtained for frequenies above MHz. With referene to retangular overhangs, an extension of the method of partial indutane [] an be usefully employed to evaluate the mutual ouplings. As in the ase of single-oil analysis, the parallel resonanes limit the maximum frequeny at whih the mutual indutane measurements an be performed. Although there are impedane bridges that an measure the mutual indutane, a ommon method to determine M is based on measuring the equivalent indutane with different oil onnetions. This proedure leads to M L L () 4 where L and L represent the measured indutanes when the two oils are onneted in series with positive and negative mutual oupling, respetively. &DSDFLWLYHFRXSOLQJV The random distribution of the oil turns prevents either analytial or numerial alulations of oil-to-oil and oil-to-ground apaitanes. A detailed measurement of these apaitanes requires partiular tehnique to be adopted owing to the omplexity of the parasiti apaitane network. In order to deouple the oil-to-oil and oil-to-ground apaitanes, the oils terminals must be properly onneted. As an example, in order to evaluate the oil-to-ground apaitane, it is onvenient to onnet all the oil terminals at the same node and measure the overall apaitane between the node and the ground. In this way both the oil-to-oil and the inner oil apaitanes are short-iruited. Thus, the oil-toground apaitane of a single oil an be readily obtained dividing the measured apaitane by the total number of oils. The equivalent iruit of the motor windings based on series onnetions and ouplings among single-oil s results in a very omplex eletrial network. Furthermore, the high number of the iruit parameters makes burdensome the fitting. On the other hand, this detailed an be usefully employed when the transient voltage distribution among the oils has to be predited or analyzed. In partiular, the oil voltage drop an be onsidered as the worst ase in terms of maximum voltage between adjaent turns. In this way, the maximum eletrial stress in the winding interturn insulation an be predited. This is the ase when the enameled wires at the beginning and at the end of a oil touh one another []. The effets of fast-fronted eletrial fields in the stator winding insulation have been disussed in terms of existene of partial disharges in [] and [3]. 6,3/,),('7((3$6(:,',*'(/ When the analysis of the voltage distribution among the oils is not required, a simplified equivalent iruit for the motor winding an be defined. In this ase the AC motor an be regarded as a blak box with one terminal for eah motor phase and a terminal for the motor frame (ground). Thus, the equivalene is formulated in terms of phase-to-phase and phase-to-ground omplex impedanes. Some impedane measurements on different stator windings have shown that the same topology of the single-oil equivalent iruit (Fig. ) an be adopted also for eah phase of the AC motor. In this ase, the mutual oupling among the phases should be introdued. In this paper the indutive ouplings among the phases are taken into aount by mutual oeffiients (M and M ). The diret phase-to-phase apaitive ouplings have been negleted being the orresponding apaitanes muh lower then the phase-to-ground apaitanes. The resulting valid for three-phase motor windings is represented in Fig.. $ % & C R P R C C L M R L L M (M, M ) M Fig. : Sheme of the three-phase winding M R P M R C R L 3 In order to make the three-phase winding fit with the experimental data, an identifiation problem has to be solved. The problem is equivalent to that of the single-oil disussed at the beginning of this paper and in [7].

4 )UHTXHQF\HVSRQVH 3KDVHWRSKDVH 3KDVHWRJURXQG meas. Re[Z] meas. Im[Z] - Fig. 3: Real and Imaginary impedane omponents meas. Re[Z] meas. Im[Z] - Fig. : Real and Imaginary impedane omponents meas. Z meas. Z >]@ Fig. 4: Impedane magnitude An initial estimate of the reative parameters of the three-phase an be obtained by the following proedure. 6HOIDQGPXWXDOSKDVHLQGXFWDQFHV The overall phase indutane L p L L an be evaluated by impedane measurements at the phase terminals for frequenies lower than the first resonant frequeny. This parameter is frequeny-dependent but an average value an be extrapolated and fixed in the onsidered frequeny range. The mutual indutane M between two motor phases an be evaluated by () using the proedure desribed above. 3KDVHWRJURXQG FDSDFLWDQFHV The apaitive oupling between eah motor phase and the ground is represented by two lumped apaitanes at the beginning and the end of eah phase (Fig. ). These apaitanes an be evaluated by onneting the six terminals of the phases at the same node and measuring the overall nodeto-ground apaitane C tg : C tg /6. C tg orresponds to the apaitane between one phase terminal and the ground measured at a low frequeny (e.g., below the resonanes). In fat, for either delta or star onnetions, the phase indutive reatanes are negligible with respet to the apaitive parallel reatanes.,qqhuskdvhfdsdflwdqfhv With referene to Fig., the input-to-output inner phase apaitane C p is represented, for high exitation frequenies (e.g., above the resonanes), by the series of the apaitanes C and C of the two resonators. The evaluation of this apaitane an be arried out by measuring the apaitane C tp between two phase terminals at a high frequeny. Assuming star onneted motor phases yields >]@ ( ) Fig. 6: Impedane magnitude Ctp Cg Cp C p Ctp Cg. Both L p and C p must be splitted into L,L,and C,C, respetively, aording to the equivalent iruit of Fig.. As a first order approximation, this an be done by imposing the frequeny of the two parallel resonanes. (;3(,(7$/7(676 In order to verify the proposed for three-phase motor windings, frequeny and time domain measurements have been performed on a test motor. The rated harateristis of the motor are given in Table of the Appendix. The behavior of both phase-to-phase and phase-to-ground impedanes has been investigated. )UHTXHQF\GRPDLQDQDO\VLV The measurements have been performed by a programmable RLC meter HP 49 with the frequeny ranging from KHz to MHz. The parameter values obtained by the fitting proedure are given in Table of the Appendix. The orresponding alulated and measured impedanes are represented from Fig. 3 to Fig. 6. The solid lines represent the numerial results whereas the markers indiate the measured values. Figs. 3 and 4 are related to phase-to-phase impedane (A-B). Figs. and 6 illustrates the results of phase-to-ground impedane (A-g). It an be noted that the agreement between numerial and experimental data is good in a wide frequeny range despite of the fitting whih has been performed by a simple trial and error proedure. In this ase, the auray ould be further improved by numerial tehniques.

5 DXPHULFDOUHVXOWV 7UDQVLHQW$QDO\VLV E([SHULPHQWDOUHVXOWV Current [ ma/div] Time [ µs/div] -.8 (a) Fig. 7: Phase-to-phase retangular voltage (b) Current [ ma/div] Time [ µs/div] -.8 (a) Fig. 8: Phase-to-phase triangular voltage (b) Current [ ma/div] Time [ µs/div] -.8 (a) Fig. 9: Phase-to-ground retangular voltage (b) Current [ ma/div] Time [ µs/div] -.8 (a) Fig. : Phase-to-ground triangular voltage (b)

6 7LPHGRPDLQDQDO\VLV The transient analysis has been performed by supplying the motor with a proper voltage soure and measuring the resulting phase urrent. As test signals, both retangular and triangular voltage waveforms have been adopted ( khz). The urrent signal is represented by the voltage aross a Ω shunt resistane. The orresponding numerial and experimental results are shown from Fig. 7 to Fig.. The agreement is good even if the dependene of all the parameters on the exitation frequeny has been negleted. This assumption allows numerial simulations to be ahieved by PSpie with a very low omputational time (few seonds). &&/86,6 The proposed HF lumped equivalent iruit for AC motor stator windings is valid in a wide frequeny range. The fitting is based on phase-to-phase and phaseto-ground (frame) impedane measurements. The real and imaginary impedane omponents have been separately onsidered. A method to ahieve an initial estimate of the reative parameters has been proposed. A simple trial and error proedure has been employed for the parameter identifiation leading to a good agreement between alulated and experimental results. More sophistiated numerial methods ould be also employed to improve the auray. By the equivalent iruit presented in this paper both frequeny and time domain analyses an be performed. The is partiularly useful to determine differential- and ommon-mode onduted EMI in the ase of inverter-fed AC motors. ()((&(6 [] E. Zhong, S. Chen, T.A. Lipo, Improvements in EMI Performane of Inverter-Fed Motor Drives, 3URF RI $3(&, Marh 994, pp [] M.T. Wright, S.J. Yang, K. MLeay, General Theory of Fast-Fronted Interturn Voltage Distribution in Eletrial Mahine Windings,,((3URF3DUW%, Vol. 3, No. 4, July 983, pp. 46. [3] J.L. Guardado, K.J. Cornik, A Computer Model for Calulating Steep-Fronted Surge Distribution in Mahine Windings,,(((7UDQV, Vol. EC-4, No., Marh 989, pp. 9-. [4] H. Oraee, P.G. MLaren, Surge Voltage Distribution in Line-End Coils of indution Motors,,(((7UDQV, Vol. PAS-4, No. 7, July 98, pp [] R.G. Rhudy, E.L. Owen, D.K. Sharma, Voltage Distribution Among the Coils and Turns of a Form Wound AC Rotating Mahine Exposed to Impulse Voltage,,(((7UDQV.,Vol. EC-, No., June 986, pp. -6. [6] L. Gubbala, A. von Jouanne, P. Enjeti, C. Singh, H. Toliyat, Voltage Distribution in the Windings of an AC Motor Subjeted to High dv/dt PWM Voltages, 3URF RI3(6&&RQIHUHQFH, June 99, Atlanta, pp [7] G. Grandi, D. Casadei, U. Reggiani, Equivalent Ciruit of Mush Wound AC Windings for High Frequeny Analysis, 3URF,6,(&RQI, July 997, Guimarães (Pt) [8] A. Consoli, G. Oriti, A. Testa, A.L. Julian, Indution Motor Modeling for Common Mode and Differential Mode Emission Evaluation, 3URF RI,$6 &RQIHUHQFH, Otober 996, Vol., pp [9] H.A. Haus and J.R. Melher: (OHFWURPDJQHWLF )LHOGV DQG(QHUJ\. Prentie Hall, New Jersey: 989. [] C.R. Paul: (OHFWURPDJQHWLF &RPSDWLELOLW\. John Wiley, New York: 99. [] G. Grandi, A. Massarini, M.K. Kazimierzuk, U. Reggiani, Capaitanes of Single-Layer Air-Core Indutors for High-Frequeny Appliations, 3URFRI,$6&RQIHU HQFH, Otober 996, Vol. 3, pp [] M. Kaufhold, G. Borner, M. Eberhardt, J. Spek, Failure Mehanism of the Interturn Insulation of Low Voltage Eletri Mahines Fed by Pulse-Controlled Inverters,,((((,DJD]LQH, Sept/Ot 996, pp [3] A. Mbaye, F. Grigoresu, T. Lebey, Bui Ai, Existene of Partial Disharges in Low-voltage Indution Mahines supplied by PWM Drives,,(((7UDQVRQ'(,, Vol. 3, No.4, August 996, pp. 46. $33(',; D h Ovehang (air) h Stator (iron) d ir / Slot Figure A: Coil geometry and details of a slot ross setion. 7$%/(7KUHHSKDVHLQGXFWLRQPRWRUGDWD rated power rated voltage rated frequeny. kw 38 V (Y) Hz d w D D s 7$%/(7KUHHSKDVHPRGHOSDUDPHWHUV R L Ω R p kω C 38 pf L 7.3 mh R L Ω R p kω C pf L 3 mh R C Ω R g Ω 83 pf M -.6 mh R C Ω M mh