Equivalent parameters of induction machines windings in permanent nonsinusoidal regime. Theoretical and experimental determination

Transcription

1 Proceedings of the 9th WSEAS International Conference on POWE SYSTEMS Equivalent parameters of induction machines windings in permanent nonsinusoidal regime Theoretical and experimental determination SOIN MUSUOI, VALEIU-NICOLA OLAESCU, DOU VATAU, CIPIAN SOANDAU Faculty of Electrical Engineering POLITEHNICA University of Timişoara Bd Vasile Pârvan, O-3003, Timişoara OMANIA Abstract: - In the present paper is carried out a study of theoretical and experimental induction motor behavior in the presence of distorting regime (non-sinusoidal) The main aims of the relations of calculating the parameters of the equivalent winding machine fueled by converter and the practice of verifying them Also proposed is a comparative study with case management sinus, with emphasis on the influence of the presence converter on the machine parameters Issue is limited to three phase range of induction machine-cage of small and medium-sized powers, to 45 [W], the most currently used in the act with speed adjustable usual Index Terms: induction machine, equivalent parameters, non-sinusoidal regime Introduction In the present, the efficient changes of the induction machine speed, with eeping the technical-economic performance, is made trough different control strategies witch involve the feeding of the machines trough static convertors In this non-sinusoidal feeding regime of the machine appears a change in parameters compared with the sinusoidal feeding case, as a presence of the superior frequency harmonics in voltage wave on the input of the machine In this way, if is used PWM modulation, the current and voltage harmonics are νjm f ± type, where m f is the modulation frequency factor As the harmonics spectrum contain only ν odd type differing of 3 and multiples of 3, because (Jm f ±) to be odd, an odd J establish a even and vice versa The positives harmonics goes in direct sense (+) and the negatives harmonics goes in opposite sense (-) of the magnetic field corresponding to the fundamental This drives for a deformant regime in the induction machine For operation study of the machine in this case, the non-sinusoidal voltage is decomposed into a lot of harmonics, and the influence of each harmonic is separately analyzed, and after that is applied the effects superposition principles The behavior of the induction machine could be analyzed regarding a series of motors, having the same shaft, with have the n speed, motors feedings from different voltages U (ν) and frequencies f (ν) networs In this paper is performed an experimental and theoretical study regarding the determination of the parameters of the medium and small power of the induction machine windings, when is fed from a frequency converter (nonsinusoidal regime) In future reasoning, will be considerate the follow simplificant hypothesis: is neglected the repression phenomena of the current in rotor pole pitch, respectively will be taen into account only fundamentals harmonics of the winding On these two hypothesis is added another one: the neglecting of the saturation considering the parameters of the machine (resistance and inductance) are not influenced by the saturation phenomenon in function by the load, so, are not time parameters trought this This hypothesis is practically valid on the load changes, in limits imposed by the normal running regime The determination of the equivalent parameters of the stator winding For this case, of the medium and small power induction machine the values of the statoric winding parameters are not practically affected by the sin effect, this affirmation is valid for fundamentals and also for the time superior harmonics ν Can be written []: () n () () an () ( ν ) ( ) n (3) ν ν νa (4) ( ) n ISSN: ISBN:

2 Proceedings of the 9th WSEAS International Conference on POWE SYSTEMS The above mentioned equations allow the determination of the parameters of the statoric windings in the situation when in the feeding voltage of the machine is present only the fundamental (the parameters have the index (), see equations () and (), or just a superior ν order time harmonic (the parameters are identified through the index ν, see equations (3) and (4) In the following, in the feeding voltage of the motor are considered present the foundamental and also the superior time harmonics Is noted with p Cu(CSF) the losses that are in the statoric winding when feeding the machine through a converter These losses are covered by a part of the power absorbed by the motor through the converter, from the networ, P (CSF) According to the principle of the superposition of the effect, can be considered: pcu ( CSF ) pcu() + pcu( ν ) 3() I() + 3 ( ν) I( ν) (5) ν The resistance of the stator winding corresponding to the fundamental, (), and also the resistances of the stator windings corresponding to all superior time harmonics, (ν), are replaced through a single equivalent resistance, (CSF), corresponding to all harmonics, including the fundamental The equivalence is made through the condition that in this resistance should be the same losses p Cu(CSF), given by the equation (5), as in the ν case of considering of ν resistances (ν), the current I (ν) going through each of them This equivalent resistance (CSF) is determined at the fundamentals frequency and is covered by the I (CSF) current (effective value) I CSF I + (6) Thus, ( ) ( ) ( ν) I ν ( ) ( ) ( ) ( ) ( ) ( ) pcu CSF 3 CSF I CSF 3 CSF I + I ν (7) ν Maing the relations (5) and (7) equal, is obtained: ( ) ( ) ( ) 3 CSF I + ν ( ) ( ) + ( ν) I 3 I 3 I( ν), (8) ν ν or, taing in consideration relation (3): ( ) () ( ) () () ( ) ( ) () ( ) + 3 CSF I + + I ν 3 I I ν 3 ν I I ν, (9) ν ν ν from which results: ( CSF ) ( ) ( ν ) n (0) ( CSF ), () Is noted with the report between (CSF) and This represents a factor that highlights the which is normal regarding the conductors with modifications that happen to a stator phase which the stator winding is realized, having a resistance in the case of feeding of the machine diameter of (usually) d [mm] through a converter compared to a sinusoidal Applying the principle of superposition of the feeding It can be easily seen that: effects also for the reactive power absorbed by the stator winding (Q Cu(CSF) ), is obtained: As in the previous case, the stator winding reactance corresponding to the fundamental () (determined at the fundamentals frequency f () ) and the stator winding s reactance corresponding to all superior time harmonics (ν) (determined at the f (ν) ν f frequency, where νjm f ±), are replaced with an equivalent reactance (CSF), determined at Q Cu ( CSF ) QCu( ) + QCu( ν ) 3 ( ) I( ) + 3 ( ν) I( ν) () ν ν the fundamentals frequency This equivalent reactance traveled by the current I (CSF), gives the same reactive power Q Cu(CSF), as in the case of considering of (ν) reactance (each being determined at f (ν) frequency and being traveled by the current I (ν) ) After the equivalence can be written: ISSN: ISBN:

3 Proceedings of the 9th WSEAS International Conference on POWE SYSTEMS ( ) ( ) ( ) ( ) ( ) ( ) QCu CSF 3 CSF I CSF 3 CSF I + I ν (3) ν Maing relations () and (3) equal is obtained: ( ) () ( ) 3 CSF I ν () () + + I 3 I 3 ( ν) I( ν), (4) ν ν or: ( ) () ( ) () () ( ) () ( ) + ν CSF I + + ν I ν I I ν I I ν (5) ν ν ν ( CSF ) nominal operating regime For establishing the Is noted with - factor that shows the relations that define these parameters, considering modifications of the reactance value of a stator the sin effect, the expression of the rotoric phase phase in the case of feeding the machine through a impedance is used, that is reduced to the stator For converter, compared to the sinusoidal feeding, both this, the rotor with multiple bars is replaced by a being determined at the fundamental frequency rotor with a single bar on the pole pitch [] Initially From the equation (5) results: is considered only the fundamental present in the feeding of the motor The rotor impedance reduced ( ν) + I I() + νi( ) ν to the stator has the equation: ν ( ) ν I CSF ν () () (6) () + j () (0) I () + I( ν) ( ν) ν + I s() Knowing that the induced emf by the ν I () fundamental component of the main magnetic field According []: from the machine in the pole pitch bars is: I( ν) U( ν), (7) U e () I () (), () I * () U() νfr x sc where, for the general case of multiple cages is * sc valid the relation []: x sc - is the short circuit impedance () reported, corresponding to the frequency f f n (usually f n 50 [Hz]) and f r is the reported frequency With this, equation (7) becomes: U( ν) + * ( ) () CSF ν ν f U r xsc (8) U( ν) + * ν ν f x U r sc () It can be observed that : > In these conditions, the equivalent impedance of the stator winding (CSF ), corresponding to all harmonics and defined at the fundamentals frequency is: + j (9) ) ( CSF ) ( CSF ) (CSF 3 Determining the equivalent parameters of the rotor winding considering the repressions of the current The rotor winding s parameters are affected by the sin effect, at the start of the motor and also at the c U e() I () I cδ() δ Δ () c Δδ() () δ In the relation (), the number of the cages and respectively the rotor bars/ pole pitch is equal with c In the case of motors with the power up to 45 [W], that are the subject of this paper, c (simple cage or high bars) or c (double cage) Δ () is the determinant corresponding to the equation system [], [3]: c U e() δε() I cε(), ε,,, c (3) ε having the expression [], [3]: () n() Δ () (4) () nn() n Δ δ() is the determinant corresponding to the fundamental obtained from Δ (), where the column δ is replaced by a column of [], [3]: ISSN: ISBN:

4 Proceedings of the 9th WSEAS International Conference on POWE SYSTEMS Δ () δ (), δ (), δ+ () n() () n, δ () n, δ+ () nn() n (5) Because in the first phase the stabilized regime is under focus, the phenomenon from the rotor corresponding to the fundamental are presenting the poulsation ω () s ω, where s is the slip of the motor corresponding to the sinusoidal feeding, in the stationary condition If the relation () is introduced in (), the expression of the equivalent impedance of the phase rotor is obtained reduced to the stator, corresponding to the fundamental, valid when considering the sin effect: Δ() (6) () c Δ δ δ With this can be written the calculation of the relations for the resistance, respectivelly the reactance of the rotoric phase reduced to the stator, corresponding to the fundamental, both affected by the sin effect: () () e () s() () Im () [ ], (7) [ ] (8) By considering in the feeding of the motor only of the ν harmonic, similar expressions are obtained for the corresponding rotoric parameters Thus: ( ν) c δ Δ ( ν), (9) Δ δ( ν) ( ν) e[ ( ν) s( ν) ( ν) Im ( ν) ], (30) [ ] (3) pδ( CSF ) r( CSF ) pδ( CSF ) ~ I cδ( ν) () ( ) rδ + rδ ν ν I cδ() I δ( ν) + c ν I cδ() δ We continue to consider the case of real power converter and the machine is to determine the parameters equivalent (appropriate for all harmonics, including fundamentals) of rotoric winding To top it will review the case of induction machines with simple cage, and high bar Thus, the resistance of a rotoric phase close to the fundamentals, corresponding higher harmonics, (), and resistance of the rotoric phase (ν) (νjm f ±), is replaced by a equivalent resistance (CSF), which gives a part of the same active power as ν resistance This resistance is equivalent to define the frequency fundamentals and is crossed by I (CSF) current: I CSF I + (3) ( ) ( ) ( ν) I ν Whereas factor in determining global equivalent of a change in resistance that r (CSF), has used the principle of conservation of active power is sufficient for the expression of equivalent rotor resistance deduction to start from the global factor expression [] Thus, the equivalent phase of rotor resistance reduced to the stator, all appropriate harmonics, of the frequency fundamentals, can be written: ( CSF ) r( CSF ) c + i, (33) were: c is the resistance at fundamental frequency, a part from rotor phase winding from pole pitch, reported to the stator; i is the resistance considered at frequency fundamentals, part of rotoric winding with neglecting the sin effect, reported to the stator; r(csf) is the global factor change in resistance winding rotoric, having given expression to the relation [] I ( ) ( ) ( ) ( ) cδ rδ + I cδ ν rδ ν ν I δ() + δ( ν) c I c ν (34) ISSN: ISBN:

5 Proceedings of the 9th WSEAS International Conference on POWE SYSTEMS In relation (34), p δ(csf)~ represents the total loses in ac in δ bars (tacing into account the sin effects for all the harmonics), and p δ(csf) are the total loses without considering the repression of the current in δ bars; rδ () and rδ (ν) are the grooving resistance factors in ac corresponding to the fundamental harmonics, respectively to the order ν harmonics; and I cδ() and I cδ(ν) is the current that flows through δ bars, corresponding to the fundamentals, respectively to the order ν harmonics For simple cage, respective high bars, c is considered To trac the changes that appear on the resistance of the rotor winding when the machine is feeding trough a static converter, compared when the machine is fed in sinusoidal regime, is introduced the factor: ( CSF ), (35) where is the resistance of the rotor winding, reported to the stator, when the machine is fed sinusoidal: r c + i, (36) in witch r is the change of the rotoric resistance in ac factor, when the machine is fed sinusoidal r r() If the relations (33) and (36) are inserted in (35), is obtained: r( CSF ) c + + r c i i (37) If is divided, also the nominator and the denominator of second member on the relation (37) with r and then with c, is obtained: r( CSF ) i + r c r i + r r, (38) + r r r c i c + r r const, (39) constant for a single motor unit, at a given fundamental frequency For c, r >, results >, witch means: (CSF) > Similar is for the reactance The reactance of the rotoric phase, corresponding to the fundamentals, (), and also the reactance corresponding to the superior harmonics, (ν), are replaced with an equivalent reactance (CSF) As in case of rotoric resistance, we can write: ( CSF ), (40) where (CSF) is the equivalent reactance of the rotoric phase, reduced to the stator, corresponding to all harmonics, including the fundamentals, on the fundamentals frequency: i ( CSF ) ( CSF ) c +, (4) and is the reactance of the statoric phase reduced to the stator, witch characterized the machine when is fed in sinusoidal regime: c + i (4) In relation (4) and (4), was noted: c -the reactance of the pole pitch rotoric winding part, reduced to the stator, in witch the sin effect is present ; i - the reactance of the rotoric winding phase, part, where the sin effect can be neglecting (CSF) - is defined with (43) relation, in witch is considered c [] qδ( CSF ) x( CSF ) qδ( CSF ) xδ ~ () + + ν ν ω L δnσ ω L xδ δn I cδ( ν) ν δ() I c I cδ( ν) ν δ() I c () I () ( ) ( ) cδ + ν xδ ν I cδ ν ν σ I δ() + ν δ( ν) c I c ν xδ ( ν) (43) Taing account of the relations (4) and (4), relationship (40) becomes: ISSN: ISBN:

6 Proceedings of the 9th WSEAS International Conference on POWE SYSTEMS i c ( CSF ) + i x ( CSF ) c + + i c, (44) c i + i + + x c x, (45) is a constant for one and the same motor at a time of increased fundamental <, with the consequences < şi (CSF) < [] With this, the impedance of a rotor phase reported to the stator in the case of a machine supplied by a power converter, receives the form: ( CSF ) ( CSF ) + j ( CSF ) s( CSF ) s( CSF ), (46) ( CSF ) I ( CSF ) U e( CSF ) (47) e e ν U (48) ( CSF ) U e( ) + U ( ν) In the case of induction machines with double cage, the rotor parameters are necessary to be determined for both cages The calculation of them retain their validity of the case presented above, the induction motors with simple cage, respectively cage with high bars, with one observation: the relations of determination of r(csf) respectively x(csf), is considered c for δ produces the wor cage, and for δc is obtained the startup cage) The complex structure of the used algorithm, and its component computing relationships synthetic presented in the paper, request a very high volume of calculation This maes that the presence of the computer in solving this problem is absolutely necessary In the Laboratory of systems dedicated to control electrical servomotors from the Politehnica University of Timişoara has designed the software calculation CALCMOT, which allows the determination and analysis of factors r(csf) respectively x(csf), and the parameters of the equivalent winding machine induction in the nonsinusoidal regime This analysis allows conclusion in quantitative - qualitative influence non-sinusoidal regime due to power converter machine by the sin effect compared to the sinusoidal supply 4 Experimental validation Induction machines have been tested are: MAS 0,37 [W] x 500 [rpm] and MAS, [W] x 500 [rpm] To validate the experimental studies of the theoretical wor, tests were made both for the operation of motors supplied by a system of sinusoidal voltages, and for the operation in case of static frequency converter supply With test of the short cut at variable frequency were determined parameters of stator and rotor windings at different frequencies The values resulting from the tests are presented in tables (sinusoidal supply) and (supply through converter) -for 037 [W] machine) and tables 3 (the sinus) and 4 (the non-sinusoidal) for [W] machine) For the inductivity separation was considered the value of L, calculated at 50 Hz The value of L (CSF) was equal to the product of (factor calculated using the CALCMOT program) and L, and the result is also constant Table The results of the shortcut probe with variable frequency supply, for the induction motor with 037 [W] x 500 [rpm] Nr f f P sc Isc U sc + + L +L [Hz] [W] [A] [V] [H] 59,98 3,87,0 64,56 4,470 5,9 6,57 46,80 0,4 50,58 45,4,07 6,433 4,99 5,9 6,9 40,7 0,6 3 39,96 43,79,070 56,656 4,843 5,9 5,94 3,447 0,9 4 30,03 4,90 0,963 46,6 4,300 5,9 5,4 5,3 0,33 5 5,95 64,59,56 53,773 4,055 5,9 5,55,869 0,34 ISSN: ISBN:

7 Proceedings of the 9th WSEAS International Conference on POWE SYSTEMS Table The results of the shortcut probe with variable frequency, supplied through power converter, for the induction motor with 037 [W] x 500 [rpm] Nr f () f () P sc(csf) I sc(csf) U (CSF) + sc(csf) (CSF) (CSF) + L (CSF) + (CSF) (CSF) (CSF) L (CSF) [Hz] [W] [A] [V] [H] 5 60,869, 53,35 4, ,748 0,577 0, ,975,5 56,9 4,59 6 6,59 4,7 0, ,49, 64,53 4,94 6 6,94 3,667 0, ,79,8 68,908 43, ,505 38,955 0, ,38,7 74,699 43, ,889 46,369 0,3 Table 3 The results of the shortcut probe with variable frequency supply, for the induction motor with [W] x 500 [rpm] Nr f f P sc I sc U sc + + L +L [Hz] [W] [A] [V] [H] 60,4 70,55,4 3,30,8 5,53 6,9 8,646 0,049 56,38 67,55,384 9,63,686 5,53 6,56 7,534 0, ,67 64,673,368 6,93,59 5,53 5,989 5,957 0, ,65 63,63,368 4,84,330 5,53 5,8 4,90 0, ,09 77,460,55 4,58,0 5,53 5,57,686 0, ,8 68,33,44,587 0,95 5,53 5,4 0,08 0,05 7 9,96 6,084,383 7,687 0,64 5,53 5, 7,095 0, ,66 69,54,488 7,8 0,46 5,53 4,93 5,8 0, ,87 65,654,464 6,308 0, 5,53 4,68 4,454 0, ,9 59,76,46 4,887 9,93 5,53 4,4 3,453 0,079 Table 4 The results of the shortcut probe with variable frequency, supplied through power converter, for the induction motor with [W] x 500 [rpm] Nr f () f ( ) P sc(csf) [W] I sc(csf) [A] U sc(csf) [V] (CSF) + (CSF) (CSF) (CSF) (CSF) + (CSF) L (CSF) + L (CSF) [H] [Hz] 0 73,356,45 9,34,63 5,55 6,08 6,534 0, ,70,4,393,6 5,55 6,05 9,63 0, ,68,39 3,87,84 5,55 6,9,440 0, ,57,34 5,804,977 5,55 6,47 5,079 0, ,707,4 30,8,35 5,55 6,8 7, In Tables 5 and 6 are presented theoretical values (obtained by running the calculation program) and the results of measurements, for and, factors, respectively the calculation errors of, for both motors tested Table 5 The theoretical and experimental values of factors and, respectively the errors of calculation, corresponding to 037 [W] x 500 [rpm] MAS (CSF) (CSF) f Nr () ε ε [Hz] (measured) [%] (measured) [%] (calculated) (calculated) 5,048, 5,58 0,863 0,894 3,6 30,06,077 4,97 0,9 0,857-6, ,0,06 3,77 0,944 0,884-6, ,04,075 6,0 0,967 0,897-7,3 5 60,0,079 6,8 0,975 0,94-6,5 ISSN: ISBN:

8 Proceedings of the 9th WSEAS International Conference on POWE SYSTEMS Table 6 The theoretical and experimental values of factors şi, respectively the errors of calculation, corresponding to [W] x 500 [rpm] MAS Nr f () [Hz] (CSF) (measured) ε [%] (CSF) (calculated) (calculated) 0,098,85 7,9 0,8 0,8,08 30,04,0 7,58 0,886 0,96 3, ,034,06 6,96 0,96 0,89-3, ,03,089 6,45 0,956 0,863-9,7 5 60,08,08 6,8 0,966 0,87-9,83 (measured) ε [%] 5 Conclusions In the case of the machine supplied through a power converter it can consider that the motor presents an equivalent resistance of the stator winding (CSF), defined by the fundamental frequency, whose value does not differ from the case of sinusoidal supply Stator winding, so the resistance (CSF) which characterizes it, is crossed by current I (CSF) Differences in the table between and (CSF) is because during the tests where the machine was supplied through the converter, the motor has warmed more than in the case of supplying it direct from networ This supplemental heat is due to the presence in the wave of supply voltages of higher harmonics generated by the inverter (the current "harmonics") In the case of the machine supplied by the power converter it can consider that the stator winding is characterized by an equivalent reactance (CSF), crossed by current I (CSF) and determined at the fundamental frequency This reactance, compared to the sinusoidal supply suffers a growing, the factor which considers this increase quantitatively,, having a over unit value (the result appears as normal, taing into account that the stator winding for the considered powers in the paper, the phenomenon of repression is neglected) 3 Applying the principle of conservation of active and reactive powers, expressions for calculating the equivalent resistance (CSF), respectively equivalent reactance (CSF), corresponding to the rotor phase in the condition of repression and a non-sinusoidal supply were determined Both parameters were determined as function of fundamental frequency Comparing these equivalent parameters of the machine supplied by power converter with the corresponding parameters of a sinusoidal supply, it is found that resistance of the rotor phase suffer an increase, and the reactance suffers a decrease in comparison with the case of sinusoidal regime Changes are even more important as the motors power rating is higher The explanation resides in the fact that as the driving power increases, the sin effect which is present in rotor bars is more pronounced (the height of notches increase) 4 Parameters of the winding machine supplied by the power converter can be calculated with errors less than 0 [%] The main cause of errors is the assumption of saturation neglect Even in this case the results can be considered satisfactory, which leads to validate the theoretical study carried out in the paper eferences [] Murphy, J, Turnbull, F Power Electronic Control of AC Motors Pergamon Press, British Library, London, 988 [] Muşuroi, S, Bogoevici, Gh The Determination of the Equivalent Modification Factors of the rotoric Parameters while the Cage Induction Motors are Being Supplied by Inverters ICEM 00, Bruges, Belgium [3] Dordea, T, Dordea, PT Ersatzläuferimpedanz einer indutionsmachine mit vielfachem äfig und in den selben nuten untergebrachten stäben evue oumaine des Sciences Techniques, Académie de la épublique Socialiste oumanie, Tome 9, Bucureşti, 984 ISSN: ISBN: