We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists. International authors and editors

Transcription

1 We are IntechOen, the world s leading ublisher of Oen Access boos Built by scientists, for scientists 4,000 6,000 0M Oen access boos available International authors and editors Downloads Our authors are among the 54 Countries delivered to TOP % most cited scientists.% Contributors from to 500 universities Selection of our boos indexed in the Boo Citation Index in Web of Science Core Collection (BKCI) Interested in ublishing with us? Contact boo.deartment@intechoen.com Numbers dislayed above are based on latest data collected. For more information visit

2 Chater The Behavior in Stationary egime of an Induction Motor Powered by Static Frequency Converters Sorin Muşuroi Additional information is available at the end of the chater htt://dx.doi.org/0.577/ Introduction Generally, the electric induction motors are designed for suly conditions from energy sources in which the suly voltage is a sinusoidal wave. The arameters and the functional sizes of the electric motors are guaranteed by designers only for it. If the electric motor is owered through an inverter, due to the resence in the inut voltage waveform of suerior time harmonics, both its arameters and its functional characteristic sizes will be more or less different from those in the case of the sinusoidal suly. The resence of these harmonics will result in the aearance of a deforming regime in the machine, generally with adverse effects in its oeration. Under loading and seed conditions similar to those in the case of the sinusoidal suly, it is registered an amlification of the losses of the machine, of the electric ower absorbed and thus a reduction in efficiency. There is also a greater heating of the machine and an electromagnetic torque that at a given load is not invariable, but ulsating, in raort with the average value corresonding to the load. The occurrence of the deforming regime in the machine is inevitable, because any inverter roduces voltages or rinted currents containing, in addition to the fundamental harmonic, suerior time harmonics of odd order. The deforming regime in the electric machine is unfortunately reflected in the suly ower grid that owers the inverter. Generalizing, the outut voltage harmonics are groued into families centered on frequencies: f Jm f Jm f J,, 3,..., () j f c f and the various harmonic frequencies in a family are: f f f (Jm )f Jm f, () j c f c f 0 Muşuroi, licensee InTech. This is an oen access chater distributed under the terms of the Creative Commons Attribution License (htt://creativecommons.org/licenses/by/3.0), which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original wor is roerly cited.

3 46 Induction Motors Modelling and Control with Jm (3) In the above relations, mf reresents the frequency modulation factor, f is the fundamental s frequency and fc is the frequency of the control modulating signal. Whereas the harmonic sectrum contains only ν order odd harmonics, in order that (Jmf±) is odd, an odd J determines an even and vice versa. The resent chater aims to analyze the behavior of the induction motor when it is sulied through an inverter. The urose of this study is to develo the theory of three-hase induction machine with a squirrel cage, under the conditions of the non-sinusoidal suly regime to serve as a starting oint in imroving the methodology of its constructive-technological design as advantageous economically as ossible. f. The mathematical model of the three-hase induction motor in the case of non-sinusoidal suly In the literature there are nown various mathematical models associated to induction machines fed by static frequency and voltage converters. The majority of these models are based on the association between an induction machine and an equivalent scheme corresonding to the fundamental and a lot of schemes corresonding to the various ν frequencies, corresonding to the Fourier series decomosition of the motor inut voltage - see Fig. (Murhy & Turnbull, 988). In this model the sin effect is not considered. I () I () () X () X () I () () X () I () X () I 0() I 0() U () U e() m() s U () U e() m() s X m() X m() a) b) Figure. Equivalent scheme of the machine sulied through frequency converter: a) for the case of fundamental; b) for the order harmonics (ositive or negative sequence). For the equivalent scheme in Fig..a, corresonding to the fundamental, the electrical arameters are defined as: ; X X ax ; n n ; X X ax ; n n a ; X X ax ; (4) m m mn m m mn

4 The Behavior in Stationary egime of an Induction Motor Powered by Static Frequency Converters 47 a s s c In relations (4), n, Xn, n, Xn, mn, Xmn reresents the values of the arameters, X,, X, m and Xm in nominal oerating conditions (fed from a sinusoidal ower suly, rated voltage frequency and load) and n f n a ; f n n n n n n n n n c sa n n n n n (5) In the relations (5), f and fn are random frequencies of the rotating magnetic field, and the nominal frequency of the rotating magnetic field resectively. For order harmonics, the scheme from Fig..b is alicable. The sli s( ), corresonding to the order harmonic is: n n n c s n n a, (6) where sign (-) (from the first equality) corresonds to the wave that rotates within the sense of the main wave and the sign (+) in the oosite one. For the case studied in this chater - that of small and medium ower machines the resistances ( ) and reactances X( ) values are not ractically affected by the sin effect. In this case we can write:, (7) n X L L, (8) where Lσ( ) is the stator disersion inductance corresonding to the order harmonic. If it is agreed that the machine cores are linear media (the machine is unsaturated), it results that the inductance can be considered constant, indeendently of the load (current) and flux, one can say that: L L L (9) By relacing the inductance Lσ( ) exression from relation (9) in relation (8), we obtain: X L X ax (0) n For the rotor resistance and rotor leaage reactance, corresonding to the order harmonic, both reduced to the stator the following exressions were established:, () n X X ax () n

5 48 Induction Motors Modelling and Control The magnetization resistance corresonding to the order harmonic, m, is given by the relation: a (3) m " K mn K is a coefficient deendent on iron losses and on the magnetic field variation. The magnetization reluctance corresonding to the magnetic field roduced by the order harmonic is: X a X (4) m K mn Further the author intends to establish a single mathematical model associated to induction motors, sulied by static voltage and frequency converter, which consists of a single equivalent scheme and which describes the machine oeration, according to the resence in the inut ower voltage of higher time harmonics. For this, the following simlifying assumtions are taen into account: - the ermeability of the magnetic core is considered infinitely large comaring to the air ermeability and the magnetic field lines are straight erendicular to the slot axis; - both the ferromagnetic core and rotor cage (bar + short circuit rings) are homogeneous and isotroic media; - the marginal effects are neglected, the slot is considered very long on the axial direction. The electromagnetic fields are considered, in this case lane-arallels; - the sin effect is taen into account in the calculations only in bars that are in the transverse magnetic field of the slot. For the bar ortions outside the slot and in short circuit rings, current density is considered as constant throughout the cross section of the bar; - the assing from the constant density zone into the variable density zone occurs abrutly; - in the real electric machines the sin effect is often influenced by the degree of saturation but the simultaneous coverage of both henomena in a mathematical relationshis, easily to be alied in ractice is very difficult, even recarious. Therefore, the simlifying assumtion of neglecting the effects of saturation is allowed as valid in establishing the relationshis for equivalent arameters; - the local variation of the magnetic induction and of current density is considered sinusoidal in time, both for the fundamental and for each harmonic; - one should tae into account only the fundamental sace harmonic of the EMF. Under these conditions of non-sinusoidal suly, the asynchronous motor may be associated to an equivalent scheme, corresonding to all harmonics. The scheme oerates in the fundamental frequency f() and it is reresented in Fig.. According to this scheme, it can be formally considered that the motors, in the case of sulying through the ower frequency converter (the corresonding arameters and the dimensions of this situation are mared with index "CSF") behave as if they were fed in sinusoidal regime at fundamental s frequency, f() with the following voltages system:

6 The Behavior in Stationary egime of an Induction Motor Powered by Static Frequency Converters 49 u U sin t;u U sin t ; u U sin t A (CSF) B (CSF) C (CSF), (5) 3 3 where, U U U (6) (CSF) () ( ) U is the hase voltage suly corresonding to the order harmonic. Corresonding to ( ) the system suly voltages, the current system which go through the stator hases is as follows: where I(CSF) is given by: i I sin t A CSF CSF i I sin t, (7) B CSF CSF 3 4 i I sin t C CSF CSF 3 I I I (8) (CSF) () ( ) Figure. The equivalent scheme of the asynchronous motor owered by a static frequency converter. Power factor in the deforming regime is defined as the ratio between the active ower and the aarent ower, as follows: P P CSF CSF CSF S U CSF CSF ICSF (9) If we consider the non-sinusoidal regime, the active ower absorbed by the machine P(CSF) is defined, as in the sinusoidal regime, as the average in a eriod of the instantaneous ower. The following exression is obtained:

7 50 Induction Motors Modelling and Control T P dt U I cos U I cos U I cos CSF T 0 (0) Therefore, the active ower absorbed by the motor when it is sulied through a ower static converter is equal to the sum of the active owers, corresonding to each harmonic (the rincile of suerosition effects is found). In relation (0), cos() is the ower factor corresonding to the order harmonic having the exression: cos s X X s () The aarent ower can be defined in the non-sinusoidal regime also as the roduct of the rated values of the alied voltage and current: S U I CSF CSF CSF, () Taen into account the relations (0), () and (), the relation (9) becomes: UI cos U I cos U U I I CSF (3) Because Δ(CSF), formally (the hase angle has meaning only in harmonic values) an angle (CSF) can be associated to the ower factor Δ(CSF), as: cos CSF CSF. With this, the relation (3) can be written: cos cos U I cos U I CSF U I U I (4) If one taes into account the relation (Murhy&Turnbull, 988): I I U * f x U r sc, (5) where x * sc is the reorted short-circuit imedance, measured at the frequency f = fn, relation (4) becomes:

8 The Behavior in Stationary egime of an Induction Motor Powered by Static Frequency Converters 5 cos CSF U cos cos * f x U r sc U U * U f x U r sc (6) 3. The determination of the equivalent arameters of the stator winding The equivalent arameters of the scheme have been calculated at the fundamental s frequency, under the resence of all harmonics in the suly voltage. Under these conditions, we note by Cu(CSF) the losses that occur in the stator winding when the motor is sulied through a ower frequency converter. These losses are in fact covered by some active ower absorbed by the machine from the networ, through the converter, P(CSF). According to the rincile of the suerosition effects, it can be considered: 3 I 3 Cu CSF Cu Cu I (7) Further, the stator winding resistance corresonding to the fundamental, () and stator winding resistances corresonding to the all higher time harmonics ( ), are relaced by a single equivalent resistance (CSF), corresonding to all harmonics, including the fundamental. The equalization is achieved under the condition that in this resistance the same loss Cu(CSF) occurs, given by relation (7), as if considering the resistances ( ), each of them crossed by the current I( ). This equivalent resistance, (CSF), determined at the fundamental s frequency, is traversed by the current I(CSF), with the exression given by (8). Therefore: 3 I 3 I I Cu CSF CSF CSF CSF (8) Maing the relations (7) and (8) equal, it results: from which: 3 I I 3 I I 3 I I CSF, (9) (CSF) = () =. (30) Alying the rincile of the suerosition effects to the reactive ower absorbed by the stator winding QCu (CSF), the following exression is obtained: Q Q Q 3 X I 3 X Cu CSF Cu Cu I (3)

9 5 Induction Motors Modelling and Control As in the revious case, the stator winding reactance corresonding to the fundamental, X() (determined at the fundamental s frequency f()) and the stator winding reactances, corresonding to all higher time harmonics X( ) (determined at frequencies f( )=f where Jmf±) are relaced by an equivalent reactance, X(CSF), determined at fundamental s frequency. This equivalent reactance, traversed by the current I(CSF), conveys the same reactive ower, QCu(CSF) as in the case of considering reactances X( ), (each of them determined at f( ) frequency and traversed by the current I( )). Following the equalization, the following exression can be written: Q 3X I 3X I I CuCSF CSF CSF CSF (3) Maing the relations (3) and (3) equal, it results: One can notice the following: X I I X I X I X CSF I I (33) X X X the factor that highlights the changes that the reactants of the stator hase value suffer in the case of a machine sulied through a ower frequency converter, comared to sinusoidal suly, both calculated at the fundamental s frequency. From relations (5) and (33) it follows: CSF U U * * X f x U f x U CSF r sc r sc X U U * * f x U f x U r sc r sc X (34) where: X * sc X Z - is the short circuit imedance reorted, corresonding to the frequency f=fn and fr is the reorted frequency. One can notice that: X>. With the equivalent resistance given by (30) and the equivalent reactance resulting from the relationshi (34) we can now write the relation for the equivalent imedance of the stator winding, Z(CSF) covering all frequency harmonics and including the fundamental: sc () Z CSF jx j X (35) CSF CSF CSF X

10 The Behavior in Stationary egime of an Induction Motor Powered by Static Frequency Converters Determining the equivalent global change arameters for the ower rotor fed by the static frequency converter Further, it is considered a winding with multile cages whose bars (in number of "c") are laced in the same notch of any form, electrically searated from each other (see Fig. 3). These bars are connected at the front by short-circuiting rings (one ring may corresond to several bars notch). This "generalized" aroach, ure theoretically in fact, has the advantage that by its alying the relations of the two equivalent factors r(csf) and x(csf), valid for any notch tye and multile cages, are obtained. The rotor notch shown in Fig. 3 is the height hc and it is divided into "n" layers (stris), each stri having a height hs = hc/n. The number of layers "n" is chosen so that the current density of each band should be considered constant throughout the height hs (and therefore not manifesting the sin effect in the stri). The notch bars are numbered from to c, from the bottom of the notch. The lower layer of each bar is identified by the index "i" and the to layer by the index "s. Thus, for a bar with index characterized by a secific resistance and an absolute magnetic ermeability, the lower layer is noted with N i and the extremely high layer with N s. The current that flows through the bar is noted with ic (Ic - rated value). The length of the bar, over which the sin effect occurs, is L. For the beginning, let us consider only the resence of the fundamental in the ower suly, which corresonds to the suly ulsation, ω()=ω=πf. In this case: L L n ~ N s ~ I Ic N b i r N s N i N s e n b Ni s b Lh I b b s c N b i Ni Ni 3 x N n b, (36) where b and bε are the width of and ε order stris and Ψδnσ() is the bar flux corresonding to the fundamental of the own magnetic field, assuming that for the order stri, the magnetic linage corresonds to a constant reartition of the fundamental current density on the stri. (37) c hs hs Ncs Nci hs hs N s N i hc hs hs Ns N i Figure 3. Notch generalized for multile cages.

11 54 Induction Motors Modelling and Control If in the motor ower suly one considers only the order harmonic which corresonds to the suly ulsation ( )=, the relations (36) and (37) remain valid with the following considerations: index "" is relaced by index "" and the rotor henomena are with the ulsation ( ) given by the relation: s s, (38) Subsequently we shall consider the real case, where in the bar both the fundamental and order time harmonics are resent. For this, the equivalent d.c. global factor of the bar resistance modification is calculated with the relation: r CSF CSF ~ CSF ~ CSF CSF, (39) where (CSF)~ reresents the total a.c. losses in bar (considering the aroriate sin effect for all harmonics) and (CSF)- reresents the bar total losses, without considering the reression henomenon. The a.c. total losses in the bar are obtained by alying the effects suerosition rincile by adding all the bar a.c. losses caused by each order time, including the fundamental. Therefore one can obtain: CSF ~, (40) ~ The a.c. loss in bar, corresonding to the fundamental, ()~, is calculated with the following relation: I (4) ~ c r In the same way, the exression of the bar a.c. losses roduced by some order time harmonic is obtained: I I (4) ~ c ~ c r By relacing the relations (4) and (4) in relation (40), it results: I I I I CSF ~ c r c r c r c r. (43) The bar losses without considering the reression henomenon in the bar are calculated using the following relationshi: where: P I, (44) CSF c CSF

12 The Behavior in Stationary egime of an Induction Motor Powered by Static Frequency Converters 55 I I I c CSF c c (45) is the rated value of the current which runs through the bar, in the case of a motor sulied by a frequency converter. By relacing the relation (45) in relation (44): CSF c c I I (46) By relacing the relations (43) and (46) in (39) one obtains the exression for the global equivalent factor of the a.c. increasing resistance in the bar, r (CSF), in case of the resence of all harmonics in the motor ower: I c r r I I c r c r I CSF ~ c CSF I I I c c c I c r CSF (47) The global equivalent change of a.c. bar inductance modification has the exression: x CSF q q CSF ~ CSF, (48) where q (CSF)~ is the a.c. total reactive ower, in the bar, and q (CSF)- is the total reactive ower for a uniform current distribution in the bar. Alying the suerosition in the case of a.c. total reactive ower, the following relationshi is obtained: q q q CSF ~ ~ ~, (49) A.c. reactive ower corresonding to the fundamental is calculated using the following relation: q L I ~ x n c (50) In the same way, the exression of the a.c. reactive ower in the bar corresonding to the order harmonic is obtained: q L I L I ~ n ~ c x n c (5) By relacing the relations (50) and (5) in the relation (8), the exression for calculating the total a.c. reactive ower in the bar is obtained:

13 56 Induction Motors Modelling and Control q L I I I CSF ~ x n c x c x c L I I n x c x c (5) The total reactive ower for an uniform current reartition in the bar, in the case of a motor sulied through a frequency converter, is calculated by the relation: q q q CSF, (53) where q ()- is the reactive ower corresonding to the fundamental, in case of an uniform current distribution Ic () in the bar, while q ( )- is the reactive ower corresonding to the harmonic in case of a uniform current distribution Ic ( ) in the bar: q L I L I n c n c. (54) Similarly, for the reactive ower corresonding to the harmonic, in the case of an uniform current Ic ( ) reartition in the bar, the following relation is obtained: q L I L I n c n c (55) By relacing the relations (54) and (55) in relation (53), the exression for the total reactive ower for a uniform current distribution in the bar becomes: q L I L I L I I CSF n c n c n c c (56) By relacing the relations (5) and (56) in relation (48), the exression for the global equivalent factor of the a.c. modifying inductance is obtained: I c x x q L I I n I x c x c c CSF ~ (57) q CSF L I I I n c c c I c x CSF 5. Determining the equivalent arameters of the winding rotor, considering the sin effect The rotor winding s arameters are affected by the sin effect, at the start of the motor and also at the nominal oerating regime. For establishing the relations that define these arameters, considering the sin effect, the exression of the rotor hase imedance

14 The Behavior in Stationary egime of an Induction Motor Powered by Static Frequency Converters 57 reduced to the stator is used. For this, the rotor with multile bars is relaced by a rotor with a single bar on the ole itch. Initially only the fundamental resent in the ower suly of the motor is considered. The rotor imedance reduced to the stator has the equation: Z jx s (58) Knowing that the induced EMF by the fundamental comonent of the main magnetic field from the machine in the ole itch bars is: where, for the general case of multile cages is valid the relation: Ue I Z, (59) c c Ue I I c (60) In the relation (60), the number of the cages and resectively the rotor bars/ ole itch is equal to c. In the case of motors with the ower u to 45 [W], c= (simle cage or high bars) or c= (double cage). Δ() is the determinant corresonding to the equation system: having the exression: c c Ue I, =,,, c, (6)... n n nn (6) Δδ() is the determinant corresonding to the fundamental obtained from Δ(), where column δ is relaced by a column of :...,,... n.... n... n, n,... nn (63) Because in the first hase the steady-state regime is under focus, the henomenon in the rotor corresonding to the fundamental has the ulsation ω()=sω, where s is the motor sli for the sinusoidal ower suly in the steady-state regime. If the relation (63) is introduced

15 58 Induction Motors Modelling and Control in (60), the exression of the equivalent imedance of the rotor hase reduced to the stator, corresonding to the fundamental valid when considering the sin effect is obtained: Z (64) c Thus, the exressions for the rotor hase resistance and inductance reduced to the stator, corresonding to the fundamental, both affected by the sin effect can be written. s ez, (65) X m Z (66) By considering in the motor ower suly the ν harmonic only, similar exressions are obtained for the corresonding rotor arameters. Thus: Z, (67) c s ez, (68) X m Z (69) Further on we consider the real case of an electric induction machine fed by a frequency converter. For the beginning, the case of simle cage resectively high bars induction motors will be analyzed. Thus, a rotor hase resistance corresonding to the fundamental, (), and rotor hase resistance corresonding to higher order harmonics (ν) are relaced by an equivalent resistance (CSF), which dissiates the same art of active ower as in the case of ν resistances. This equivalent resistance is defined at the fundamental s frequency and it is traversed by the I (CSF) current: I I I CSF (70) For the rotor hase equivalent resistance reduced to the stator, corresonding to all harmonics, defined at the fundamental s frequency, one can write:, (7) CSF r CSF c i

16 The Behavior in Stationary egime of an Induction Motor Powered by Static Frequency Converters 59 where: c is the resistance, considered at the fundamental s frequency of a art from the rotor hase winding from notches and reorted to the stator, i is the resistance of a art of the rotoric winding, neglecting sin effect reorted to the stator, r(csf) is the global modification factor of the rotor winding resistance, having the exression given by the relation (47). To trac the changes that aear on the resistance of the rotor winding when the machine is sulied through a frequency converter, comaring to the case when the machine is fed in the sinusoidal regime, the factor is introduced: CSF, (7) where is the rotor winding resistance reorted to the stator, when the machine is fed in the sinusoidal regime:, (73) r c i where r is the modification factor of the a.c. rotor resistance, in the case of sinusoidal: rr(). It is obtained: rcsf c r c i i (74) If both the nominator and the denominator of the second member on the relation (74) are divided by r and then by c, the following exression is obtained: rcsf i r r r c r r i r c r r, (75) where: r i c const., which is constant for the same motor, at a given fundamental s frequency. For c=, r>, it results that >, which means that (CSF)> also. The rocedure is similar for the reactance. The rotor hase reactance, corresonding to the fundamental, X (), and also the reactance corresonding to the higher harmonics, X ( ), are relaced by an equivalent reactance X (CSF). As in the case of the rotor resistance, we can write: X X X CSF, (76)

17 60 Induction Motors Modelling and Control where X (CSF) is the equivalent reactance of the rotor hase, reduced to the stator, corresonding to all harmonics, including the fundamental, on the fundamental s frequency: X X X, (77) CSF X CSF c i and X is the reactance of the rotor hase reduced to the stator which characterizes the machine when it is fed in the sinusoidal regime: X X X (78) X c i In relation (77) and (78), we noted: X c -the reactance of the rotor winding art from the notches, reduced to the stator, in which the sin effect is resent, X i- the reactance of the rotor winding hase where the sin effect can be neglected. X(CSF) is defined in relation (57), where c. Taing into account the relations (77) and (78), the relation (76) becomes: X XCSF Xi X X x X XCSF c i X X c X X X X X c i i x X c X X X, (79) where: x Xi X, c is a constant for the same motor at a given fundamental s frequency X<, with the consequences X < and X (CSF)<X. With this, the imedance of a rotor hase reorted to the stator in the case of a machine sulied by a ower converter, receives the form: Z CSF s CSF CSF jx CSF, (80) where: s CSF I CSF CSF Ue CSF (8) and: U U U e CSF e e (8) In the case of double cage induction motors, the rotor arameters are necessary to be determined for both cages. The rincile of calculation ees its validity from the above resented case,

18 The Behavior in Stationary egime of an Induction Motor Powered by Static Frequency Converters 6 the induction motors with simle cage, resectively cage with high bars, with one remar: in the relations for determining r(csf) resectively x(csf),it is considered that c= (for δ= the woring wor cage results and for δ=c= the startu cage results). The comlex structure of the used algorithm and its comonent comuting relations synthetically resented in the aer, request a very high volume of calculation. Therefore the resence of a comuter in solving this roblem is absolutely necessary. In the Laboratory of Systems dedicated to control the electrical servomotors from the Polytechnic University of Timişoara the software calculation CALCMOT has been designed. It allows the determination and the analysis of the factors r(csf), x(csf) and the arameters of the equivalent winding machine induction in the nonsinusoidal regime. Further on, the exressions of the equivalent arameters for the magnetic circuit will be set (corresonding to all harmonics). Thus, to determine the equivalent resistance of magnetization m(csf), we have to tae into account that this is determined only by the ferromagnetic stator core losses which are covered directly by the stator ower without maing the transition through the stereo-mechanical ower. By aroximating that I0(CSF) Iμ(CSF), for m(csf) it is obtained: m CSF 3ICSF z CSF j CSF, (83) where z(csf) and j(csf) are global losses occurring resectively in the stator teeth and in the yoe due to the sulying of the motor through the frequency converter. In determining the total magnetization current Iμ(CSF), the rincile of the suerosition effects is alied: I I I CSF (84) For the equivalent magnetizing reactance, corresonding to all harmonics, determined at the fundamental s magnetization frequency f(), we obtain: U CSF X m CSF CSF m CSF I CSF (85) For the equivalent imedance of the magnetization circuit it can be written: ZmCSF jx m CSF mcsf (86) Given these assumtions and considering that the equivalent arameters were calculated reduced to the fundamental s frequency (in the conditions of a sinusoidal regime), one may formally accet the calculation in comlex quantities. Corresonding to the unique scheme shown in Fig., the motor equations are: UCSF ZCSF ICSF U ecsf ;

19 6 Induction Motors Modelling and Control UeCSF ZCSF ICSF U ecsf; UeCSF ZmCSF I 0CSF ; (87) I0CSF ICSF I CSF 6. Exerimental validation The induction machines which have been tested are: MAS 0,37 [W] x 500 [rm] and MAS, [W] x 500 [rm]. To validate the exerimental studies of the theoretical wor, tests were made both for the oeration of motors sulied by a system of sinusoidal voltages, and for the oeration in case of static frequency converter suly. In Tables and are resented theoretical values (obtained by running the calculation rogram) and the results of measurements, for and X, factors, resectively the calculation errors of, for both motors tested. Nr. f() [Hz] (CSF) (calculated) (measured) ε [%] X X X (CSF) (calculated) X (measured). 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, ,0,079 6,8 0,975 0,94-6,5 Table. The theoretical and exerimental values of factors and X, resectively the errors of calculation, corresonding to 0.37 [W] x 500 [rm] MAS. εx [%] Nr. f() [Hz] (CSF) (calculated) (measured) ε [%] X X X (CSF) (calculated) X (measured). 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, ,08,08 6,8 0,966 0,87-9,83 Table. The theoretical and exerimental values of factors şi X, resectively the errors of calculation, corresonding to. [W] x 500 [rm] MAS. εx [%]

20 The Behavior in Stationary egime of an Induction Motor Powered by Static Frequency Converters 63 Parameters of the winding machine sulied by the ower converter can be calculated with errors less than 0 [%]. The main cause of errors is the assumtion of saturation neglect. Even in this case the results can be considered satisfactory, which leads to validate the theoretical study carried out in the aer. 7. Theoretical analysis of the magnetic losses 7.. Statoric iron losses 7... The main stator iron losses A. The main stator teeth losses In the teeth, the magnetic field is alternant and generates this tye of losses. In the case of the direct sulying system the total losses from the stator teeth zl are being comosed by the magnetic hysteresis losses, zlh and the eddy currents losses, zlw: f f B G, (88) z zh h zw w zm z where: h is a material constant deending on the thicness and the quality of the steel sheet, f is the sulying frequency, Bzlm reresents the magnetic induction in the middle of the stator tooth, Gzl reresent the weight of the stator teeth, w is a material constant similar to h, deending on the sheet thicness and quality and reresents the thicness of the sheet. zh and zw are two factors which have the mission of underlining resectively the hysteresis losses increment and the eddy currents losses increment due to the mechanical modifications of the stator s sheets. In the case of converters-mode sulying system, at the total losses from the stators teeth caused by the fundamental the losses induced by the higher time harmonics must be taen into account. For an exact analytic exression in the following it is roosed an analysis method of the iron losses based uon the equalization of the hysteresis losses with the eddy currents ones. For the start, only the fundamental is considered resent in the sulying system. Distinct from the sine-mode sulying system, when in most cases the sulying frequency is f=fn=50 [Hz], is the fact that in the case of the inverter based sulying system the fundamental frequency can tae values higher than 50 [Hz]. At very high magnetization frequencies the influence of the sin effect must be taen in consideration. In the following, the minimum value of the magnetization frequency is being determined and for that the sin effect must be considered. The comuting relation for the magnetization frequency f is the following: where ξ is the refulation factor. f, (89) The minimum magnetization frequency fmin, comuted with the relation (89), from which the sin effect must be considered is 40[Hz]. Consequently, in the fundamental - wave

21 64 Induction Motors Modelling and Control sulying mode, at which usually we have f 0 [Hz], the rincial losses from the stators teeth, can be written as following: f f B G, (90) z zh h zw w zm z where Bzm() reresents the magnetic induction from the middle of the tooth, B B. zm zm In order to be able to aly the rincile of over osition effects, the machine is being considered as being ideal; therefore we neglect the hysteresis henomenon. For this, we roosed the equalization of the hysteresis losses with the eddy current losses, an assumtion that allows the linearization of the machines equations. Through this equalization, the real machine that is ractically non-linear and in which the rincial losses are made of a sum of two comonents: the one of eddy currents losses and the one of hysteresis losses - is being relaced with a theoretical linear machine, characterized only by its eddy currents losses. Energetically seaing, the two machines must be equivalent. As a following, if we tae *zw() as the eddy currents losses corresonding to the fundamental, which aear in the theoretical model of the machine adoted, than these losses must be equal to the main losses from the stator teeth characteristic to the real machine, losses given through the relation: * zw z (9) We consider these equivalent losses, *zw(), equal to the real losses through the eddy currents corresonding to the fundamental, zw(), multilied with a ze() factor. This is an equalization factor of the real losses from the stators teeth with losses resulted only from zw() fundamental-mode sulying state: * zw ze zw (9) We consider that through this equalization factor a covering value of the rincial stator teeth losses is obtained. The relation (9) made exlicit becomes: zm ze zm f f B G f B G. (93) zh h zw w z zw w z Because of the fact that the usually used sheets have the thicness =0.5 [mm]=const, one can consider that: K z ze f (94) where we have K K / z z with K z h w zh zw In the following art we consider that only the order harmonic is resent in the sulying wave, characterized by the magnetization frequency f( )=f. Therefore, the rincial losses

22 The Behavior in Stationary egime of an Induction Motor Powered by Static Frequency Converters 65 in the stator teeth occurring in the real machine corresonding to the order time harmonic must be corrected through the two factors h( ) and w( ), which are a function of the reaction of the eddy currents: f f B G (95) z zh h h zw w w zm z In the relation (95), Bzm( ) reresents the magnetic induction according to the order time harmonic from the middle of the tooth. The factors h( ) and w( ) have the exressions: sh sin 3 sh sin ; ; h w ch cos ch cos (96) As in the case of the fundamental-wave sulying case, the real machine is relaced by a theoretical linear machine which has only losses given by the eddy currents. easoning as in the case of the fundamental, we obtain: Kz h K z h ze f f w w, (97) * f B G z zw ze zw ze zw w w zm z (98) where *zw(ν) are the equivalent losses corresonding to the ν harmonic. If we have z(csf) for the losses from the stators teeth with the machine sulied by inverters, by alying the rincile of over osition effects for the theoretical linear model of the machine, it will be written: B f B G z CSF zw w zm z ze ze w B zm zm (99) In order to analyze the modifications suffered by the main losses in the stators teeth while the motor is sulied by an inverter versus the sine-mode sulying system, we analyze the ratio between the relations (99) and (88). After maing the intermediary comutations in which the relations (93), (94) and (99) are taen into account we obtain: z CSF ze z w Bz, z ze, (00) where Bz(ν,) = Bzm(ν) / Bzm(). B. The rincial losses in the stator yoe In the case of the direct mode sulying system of the machine, the rincial yoe losses consist of the hysteresis losses, jh and eddy currents losses, jw:

23 66 Induction Motors Modelling and Control f f B G (0) j h jh w jw j j where: Bjl is the magnetic induction in the stator yoe, Gjl reresents the weight of the stator yoe,, where jw is a coefficient that corresonds to the non uniform jw jw jw reartition of the magnetic induction in the yoe and jw is a coefficient that corresonds to the currents closing erendicular to the sheets, through the laces with imerfections in the sheets isolation layer and also in the wholes made in the cutting rocess. In the case on an inverter sulying system at the total losses from the stator yoe caused by the fundamental, the suerior time harmonics losses must be added. In order to aly the rincile of over-osition effect the method is similar to the one used in the case of the rincial losses in the teeth. We equalize energetically the real machine with the linear theoretical one where we consider only the eddy currents losses. As a following, for the fundamental sulying mode, the rincial losses in the stator yoe for a real machine, j() are: f f B G (0) j h jh w jw j j If we have *jw() as losses in eddy currents, than these must be equalized with the rincial losses from the stator yoe described with the relation (0): * jw j (03) These equivalent losses, *jw() are considered equal to the real eddy currents losses jw(), multilied with an equalizing factor of the real yoe losses with jw() tye losses, je(): * jw je jw (04) Similarly to oint A, as a following of the equalization we obtain the relation: K K w je f f w, (05) where we have: K w h w jh jw and K K w w As a following we consider resent in the sulying system of the machine only the order suerior time harmonic. Because of the fact that the magnetization frequency f( ) is the fundamental one multilied with, the rincial losses from the stator yoe which aear in the fundamental must be adjusted with the two coefficients: h( ) and w( ). These factors tae into account resectively the sin effect and the eddy currents reaction. f f B G (06) j h h jh w w jw j j

24 The Behavior in Stationary egime of an Induction Motor Powered by Static Frequency Converters 67 In the relation (06), Bj( ) reresents the magnetic induction accordingly to the order harmonic. Through the energetically equalization realized from the relacement of the real machine with the linear model, we obtain the equalizing factor of the stator yoe losses, with the jw() tye losses: Kw h K w h je f f w w (07) In conclusion, the rincial losses in the stator yoe, corresonding to the order time harmonic can be written by equalizing as: where: * j jw je jw, (08) f B G (09) jw w w jw j j As a following we have considered the situation of the machine sulied by the fundamental and the suerior time harmonics as well. Taing j(csf) as the global losses occurring in the stator yoe due to the converter sulying mode, by alying the over osition effect rincile on the theoretical linear model we can write: f B G j CSF w jw j j je je w B j B j (0) In order to analyze the changes that the rincial losses from the stator yoe suffer when the machine is being sulied through an inverter versus the sine-mode sulying case, we divide the relation (0) at (0). After finishing the comutations we have: j CSF je j w Bj, j je, () where: Bj(ν,) = Bj(ν) / Bj() The sulementary stator iron losses A. Surface sulementary losses In the case of a networ sulying mode, the magnetic induction distribution curve over the olar ste is not very different from a sine-curve. The surface stator losses are given by the exression: b P l D N n B,5 c 4 o c c c ()

25 68 Induction Motors Modelling and Control In the relation () the significance of the sizes is the following: D is the inner diameter of the stator, c is the ste of the stator slot and c is the ste of the rotor slot, b4 is the oening of the stator slot, Nc is the number of stator slots, n is the rotation seed, is a factor deendent on the ratio b4/ (b4 is the oening of the rotor slot), is an air ga factor, o is an adjustment factor which deends on the materials resistivity and its magnetic ermeability. In the case of the inverter sulying method, due to the deforming state at the sulementary losses roduced by the fundamental, the surface losses roduced by the suerior time harmonics must be considered. Because of the fact that the surface losses in the olar ieces are treated as the eddy current losses develoed in the inductor sheets, we can aly the over osition effect rincile without any further arallelism. Therefore, the surface sulementary losses in the stator in the case of a machine sulied by inverters can be comuted with the relation: b,5 B c 4 P ld N n B CSF o c c B c (3) Dividing the sulementary losses in the stator surface when having an inverter sulying system for the machine, P (CSF), by the sulementary losses in the stator surface when we have the sine-mode sulying system for the machine, P, and maing the intermediary comutations we obtain the increment factor of the sulementary stator surface losses in the inverter versus the sine-mode sulying case, P, as following: P B CSF P P B, B, (4) where Bδ(ν,) = Bδ(ν) / Bδ(). By analyzing the relation (4) one can notice the fact that the P factor tends to because of the fact that the value is ractically very low. Consequently, the surface sulementary losses increase due to the inverter sulying system to an extent that is not to be taen into consideration. B. The ulsation sulementary losses In the case of the sine-mode sulying system, the ulsation sulementary losses in the stator, rovided that the magnetic field along the olar ste is not much different from a sine-wave, has the following exression: P N n G B c P w wp c z zm, (5) where wp is an increment coefficient of the stator losses by eddy currents due to rocessing, is the total air ga factor and is constant for the one and the same machine, deended on the oening of the stator slot and the air ga dimension. In the situation in which the

26 The Behavior in Stationary egime of an Induction Motor Powered by Static Frequency Converters 69 machine is sulied by inverters, by alying the over osition effect rincile, the following exression for the sulementary ulsation losses in the stator PP(CSF) is obtained: B zm P N n G B P CSF w wp c z zm B c zm (6) Dividing the ulsation stator losses in the case of the inverter sulying system PP(CSF), by the ulsation stator losses in the case of sine-mode sulying system PP, we obtain the increment factor of the sulementary ulsation losses in the inverter versus sine-wave sulying system, P: P B P CSF zm P Bz, P B P zm (7) By analyzing the relation (7) we can state that in the case of an inverter sulied machine we have not obtained a significant increment of the ulsation losses in the stator due to the small value of the Bz,. 7.. otor iron losses 7... Princial losses in the rotor iron A. The rincial losses in the rotor s teeth Firstly, only one suerior time harmonic is considered resent in the sulying system of the machine, of an average order. The real losses that this harmonic roduces in the rotor teeth have the exression: s f s f B G z zh h h zw w w zm z (8) In the relation (8), Bzm( ) reresents the magnetic induction corresonding to the order harmonic from the middle of the rotor tooth. In the theoretical model adoted, these losses given by the relation (8) are roduced only by eddy currents: * z zw ze zw, (9) where ze( ) is an equalizing factor of the real losses from the rotor teeth, only with the losses of zw() tye, corresonding to the order time harmonic. Develoing the relation (9) by using the relation (8), after finishing the intermediary comutations we obtain: Kz h K z h ze s f s f w w (0)

27 70 Induction Motors Modelling and Control Therefore, the rincial losses from the rotor teeth, corresonding to the order time harmonic can be written by equalization as it follows: s f B G () z ze zw w w zm z In the conditions in which in the sulying system of the machine all the suerior time harmonics are resent, the rincial losses in the rotor teeth can be written as: B. The rincial losses from the rotor s yoe z CSF z () In the hyotheses in which in the sulying system only the order harmonic is resent, the real rincial losses induced by it in the rotor yoe have the exression: s f s f B G (3) j h h jh w w jw j j Through the energetic equalization, due to the relacement of the real machine by a theoretical linear model we can obtain the equality: * j jw je jw (4) easoning as in the revious cases, we can determine the equalizing factor of the real losses in the rotor yoe, only with losses of the tye jw( ) tye as it follows: Kw h K w h je s f s f w w (5) Consequently, the rincial rotor yoe losses corresonding to the order harmonic can be written by equalization in the form: s f B G (6) j je jw w w j j Disregarding all these, in the case of the inverter sulying system the total rincial losses in the rotor yoe, j(csf), are comuted with the relation: j CSF j (7) 7... The sulementary losses in the rotor iron A. The surface sulementary losses If the machine is directly sulied from the ower suly, the surface sulementary rotor losses are calculated with the relation:

28 The Behavior in Stationary egime of an Induction Motor Powered by Static Frequency Converters 7 P l b c 4 c, (8) where the secific rotor surface losses have the exression:,5 N n B (9) o c c In the relations (8) and (9) we noted by b4 the oening of the rotor slot, Nc the number of rotor slots, a factor deendent on the b4/ ratio and the air ga factor. Proceeding similarly we can obtain the exression of the increment factor of the sulementary losses in the rotor surface while the machine is being sulied by inverters versus the sine-mode sulying system, P : P B CSF P B, P P B (30) B. The sulementary ulsation losses The sulementary ulsation rotor losses, in the sine-mode sulying system have the following exression: P N nb G P w wp c P z (3) BP reresents the ulsation induction in the rotor teeth. Consequently, taing into account the fact that: B B zm B B zm B,, (3) we obtain: PPCSF P B, P P (33) 8. Conclusions This aer aims to study the theoretical behavior of asynchronous three-hase motor in the case of sulying through a ower frequency converter. This study has aimed to develo the theory of the asynchronous three-hase motor in non-sinusoidal eriodic regime to serve as a starting oint in otimizing the design methodology. Given that the asynchronous threehase motor is fed through a static frequency converter, the machine oeration in the resence of higher time harmonics in the suly voltage can be described by a single mathematical model. The model consists of a single equivalent scheme corresonding to all harmonics and it is defined at the fundamental frequency.

29 7 Induction Motors Modelling and Control Author details Muşuroi Sorin Politehnica University of Timişoara, omania 9. eferences [] Muşuroi, S.; Vătău, D.; Andea, P.; Şurianu, F.D.; Frigură, F. & Bărbulescu, C. (007). Analysis of the Magnetic Losses from the Induction Machines Sulied by Inverters, Proceedings of EUOCON 007 International Conference on Comuter as a Tool, , ISBN , Warsaw, Poland, Setember 9-, 007 [] Muşuroi, S.; Olărescu, N.V.; Vătău, D. & Şorândaru, C. (009). Theoretical and Exerimental Determination of Equivalent Parameters of Three-Phase Induction Motor Windings in Case of Power Electronic Converters Suly, WSEAS Transactions on Systems, Vol.8, Issue0, (October 009),. 5-4, ISSN [3] Muşuroi, S.; Şorândaru, C.; Olărescu, N.V. & Svoboda, M. (009) Mathematical Model Associated to Three-Phase Induction Servomotors in the Case of Scalar Control, WSEAS Transactions on Systems, Vol.8, Issue 0, (October 009),. 5-34, ISSN [4] Mohan, N.; Undeland, T.M. & obbins, W.P. (995). Power Electronics: Converters, Alications and Design, ( nd edition), John Wiley & Sons, ISBN , Inc., New-Yor, USA [5] Murhy, J. & Turnbull, F. (988). Power Electronic Control of AC Motors, Pergarmon Press, ISBN , Oxford, 988 [6] Dordea, T. & Dordea, P.T. (984). Ersatzläuferimedanz einer indutionsmachine mit vielfachem äfig und in den selben nuten untergebrachten stäben, evue oumaine des Sciences Techniques, Vol.9, No., (December 984),. 5-59, ISSN