the induction machine handbook chuong (11)

Author: Ion Boldea, S.A.Nasar………… ……… Chapter 11 LOSSES IN INDUCTION MACHINES Losses in induction machines occur in windings, magnetic cores, besides mechanical friction and windage losses. They determine the efficiency of energy conversion in the machine and the cooling system that is required to keep the temperatures under control. In the design stages, it is natural to try to calculate the various types of losses as precisely as possible. After the machine is manufactured, the losses have to be determined by tests. Loss segregation has become a standard method to determine the various components of losses, because such an approach does not require shaft-loading the machine. Consequently, the labor and energy costs for testings are low. On the other hand, when prototyping or for more demanding applications, it is required to validate the design calculations and the loss segregation method. The input-output method has become standard for the scope. It is argued that, for high efficiency machines, measuring of the input and output P in , P out to determine losses Σp on load outin PPp −=Σ (11.1) requires high precision measurements. This is true, as for a 90% efficiency machine a 1% error in P in and P out leads to a 10% error in the losses. However, by now, less than (0.1 to 0.2)% error in power measurements is available so this objection is reduced considerably. On the other hand, shaft-loading the IM requires a dynamometer, takes time, and energy. Still, as soon as 1912 [1] it was noticed that there was a notable difference between the total losses determined from the loss segregation method (no-load + short – circuit tests) and from direct load tests. This difference is called “stray load losses.” The dispute on the origin of “stray load losses” and how to measure them correctly is still on today after numerous attempts made so far. [2 - 8] To reconcile such differences, standards have been proposed. Even today, only in the USA (IEEE Standard 112B) the combined loss segregation and input-output tests are used to calculate aposteriori the “stray load losses” for each motor type and thus guarantee the efficiency. In most other standards, the “stray load losses” are assigned 0.5 or 1% of rated power despite the fact that all measurements suggest much higher values. The use of static power converters to feed IMs for variable speed drives complicates the situation even more, as the voltage time harmonics are producing additional winding and core losses in the IM. Faced with such a situation, we decided to retain only the components of losses which proved notable (greater than (3 to 5%) of total losses) and explore their computation one by one by analytical methods. © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… Further on numerical, finite element, loss calculation results are given. 11.1. LOSS CLASSIFICATIONS The first classification of losses, based on their location in the IM, includes: • Winding losses – stator and rotor • Core losses – stator and rotor • Friction & windage losses – rotor Electromagnetic losses include only winding and core losses. A classification of electromagnetic losses by origin would include • Fundamental losses • Fundamental winding losses (in the stator and rotor) • Fundamental core losses (in the stator) • Space harmonics losses • Space harmonics winding losses (in the rotor) • Space harmonic core losses (stator and rotor) • Time harmonic losses • Time harmonics winding losses (stator and rotor) • Time harmonic core losses (stator and rotor) Time harmonics are to be considered only when the IM is static converter fed, and thus the voltage time harmonics content depends on the type of the converter and the pulse width modulation used with it. The space harmonics in the stator (rotor) mmf and in the airgap field are related directly to mmf space harmonics, airgap permeance harmonics due to slot openings, leakage, or main path saturation. All these harmonics produce additional (stray) core and winding losses called • Surface core losses (mainly on the rotor) • Tooth flux pulsation core losses (in the stator and rotor teeth) • Tooth flux pulsation cage current losses (in the rotor cage) Load, coil chording, and the rotor bar-tooth contact electric resistance (interbar currents) influence all these stray losses. No-load tests include all the above components, but at zero fundamental rotor current. These components will be calculated first; then corrections will be applied to compute some components on load. 11.2. FUNDAMENTAL ELECTROMAGNETIC LOSSES Fundamental electromagnetic losses refer to core loss due to space fundamental airgap flux density– essentially stator based–and time fundamental conductor losses in the stator and in the rotor cage or winding. The fundamental core losses contain the hysteresis and eddy current losses in the stator teeth and core, © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ………               +       + +               +       ≈ core 2 cs1 teeth 2 ts1 2 1e core n cs1 teeth n ts1 1h1Fe G 1 B G 1 B fC G 1 B G 1 B fCP (11.2) where C h [W/kg], C e [W/kg] n = (1.7 – 2.0) are material coefficients for hysteresis and eddy currents dependent on the lamination hysteresis cycle shape, the electrical resistivity, and lamination thickness; G teeth and G core , the teeth and back core weights, and B 1ts , B 1cs the teeth and core fundamental flux density values. At any instant in time, the flux density is different in different locations and, in some regions around tooth bottom, the flux density changes direction, that is, it becomes rotating. Hysteresis losses are known to be different with alternative and rotating, respectively, fields. In rotating fields, hysteresis losses peak at around 1.4 to 1.6 T, while they increase steadily for alternative fields. Moreover, the mechanical machining of stator bore (when stamping is used to produce slots) is known to increase core losses by, sometimes, 40 to 60%. The above remarks show that the calculation of fundamental core losses is not a trivial task. Even when FEM is used to obtain the flux distribution, formulas like (11.2) are used to calculate core losses in each element, so some of the errors listed above still hold. The winding (conductor) fundamental losses are 2 r1r 2 s1sco IR3IR3P += (11.3) The stator and rotor resistances R s and R r ’ are dependent on skin effect. In this sense R s (f 1 ) and R r ’(Sf 1 ) depend on f 1 and S. The depth of field penetration in copper δ Co (f 1 ) is () m f 60 1094.0 60 f 60210256.1 2 f2 2 f 1 2 1 6 Co10 1Co − − ⋅=       π⋅ = σπµ =δ (11.4) If either the elementary conductor height d Co is large or the fundamental frequency f 1 is large, whenever 2 d Co Co >δ (11.5) the skin effect is to be considered. As the stator has many layers of conductors in slot even for δ Co ≈ d Co /2, there may be some skin effect (resistance increase). This phenomenon was treated in detail in Chapter 9. In a similar way, the situation stands for the wound rotor at large values of slip S. The rotor cage is a particular case for one conductor per slot. Again, © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… Chapter 9 treated this phenomenon in detail. For rated slip, however, skin effect in the rotor cage is, in general, negligible. In skewed cage rotors with uninsulated bars (cast aluminum bars), there are interbar currents (Figure 11.1a). The treatment of various space harmonics losses at no-load follows. R /n q R /n b R /n q R /n b R /n q R /n b R /n q R /n b R /n q R /n b R /n q R er R er C (skewing) I (y+ y) ∆ m I m R - bar resistance R - end ring segment resistance R - bar to core resistance er b q B r g y stack straight bars skewed bars S=S n Figure 11.1 Interbar currents (I m (Y)) in a skewed cage rotor a.) and the resultant airgap flux density along stack length b.) Depending on the relative value of the transverse (contact) resistance R q and skewing c, the influence of interbar currents on fundamental rotor conductor losses will be notable or small. The interbar currents influence also depends on the fundamental frequency as for f 1 = 500 Hz, for example, the skin effect notably increases the rotor cage resistance even at rated slip. On the other hand, skewing leaves an uncompensated rotor mmf (under load) which modifies the airgap flux density distribution along stack length (Figure 11.1b). As the flux density squared enters the core loss formula, it is obvious that the total fundamental core loss will change due to skewing. As skewing and interbar currents also influence the space harmonics losses, we will treat their influence once for all harmonics. The fundamental will then become a particular case. Fundamental core losses seem impossible to segregate in a special test which would hold the right flux distribution and frequency. However a standstill © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… test at rated rotor frequency f 2 = Sf 1 would, in fact, yield the actual value of R r ’(S n f 1 ). The same test at various frequencies would produce precise results on conductor fundamental losses in the rotor. The stator fundamental conductor losses might be segregated from a standstill a.c. single-phase test with all phases in series at rated frequency as, in this case, the core fundamental and additional losses may be neglected by comparison. 11.3. NO-LOAD SPACE HARMONICS (STRAY NO-LOAD) LOSSES IN NONSKEWED IMs Let us remember that airgap field space harmonics produce on no-load in nonskewed IMs the following types of losses: • Surface core losses (rotor and stator) • Tooth flux pulsation core losses (rotor and stator) • Tooth flux pulsation cage losses (rotor) The interbar currents produced by the space harmonics are negligible in nonskewed machines if the rotor end ring (bar) resistance is very small (R cr /R b < 0.15). 11.3.1. No-load surface core losses As already documented in Chapter 10, dedicated to airgap field harmonics, the stator mmf space harmonics (due to the very placement of coils in slots) as well the slot openings produce airgap flux density harmonics. Further on, main flux path heavy saturation may create third flux harmonics in the airgap. It has been shown in Chapter 10 that the mmf harmonics and the first slot opening harmonics with a number of pole pairs ν s = N s ± p 1 are attenuating and augmenting each other, respectively, in the airgap flux density. For these mmf harmonics, the winding factor is equal to that of the fundamental. This is why they are the most important, especially in windings with chorded coils where the 5th, 7th, 11th, and 17th stator mmf harmonics are reduced considerably. pole pitch t s b os stator slotting θ m Figure 11.2 First slot opening (airgap permeance) airgap flux density harmonics Let us now consider the fundamental stator mmf airgap field as modulated by stator slotting openings (Figure 11.2). © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… () () msm11 0 1m10 mpN Ncosptcos a aF t,B 1s θθ−ω µ =θ ± (11.6) This represents two waves, one with N s + p 1 pole pairs and one with N s – p 1 pole pairs, that is, exactly, the first airgap magnetic conductance harmonic. Let us consider that the rotor slot openings are small or that closed rotor slots are used. In this case, the rotor surface is flat. The rotor laminations have a certain electrical conductivity, but they are axially insulated from each other by an adequate coating. For frequencies of 1,200 Hz, characteristic to N s ± p 1 harmonics, the depth of field penetration in silicon steal for µ Fe = 200 µ 0 is (from 11.4) δ ≈ 0.4 mm, for 0.6 mm thick laminations. This means that the skin effect is not significant. Therefore we may neglect the rotor lamination-induced current reaction to the stator field. That is, the airgap field harmonics (11.6) penetrate the rotor without being disturbed by induced rotor surface eddy currents. The eddy currents along the axial direction are neglected. But now let us consider a general harmonic of stator mmf produced field B ν g .       ω− ν =         ω− τ νπ = ννννν tS R x cosBtS p x cosBB m 1 m g (11.7) R–rotor radius; τ–pole pitch of the fundamental. The slip for the ν th mmf harmonic, S ν , is () () 1 0S 1 0S p 1S1 p 1S ν −=         − ν −= = = ν (11.8) For constant iron permeability µ, the field equations in the rotor iron are 0B ,0Bdiv ,0rotB z === νν (11.9) This leads to 0 y B x B 0 y B x B 2 y 2 2 y 2 2 x 2 2 x 2 = ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ (11.10) In iron these flux density components decrease along y (inside the rotor) direction ()       ω− ν = ν tS R x cosyfB m yy (11.11) According to 0 y B x B y x = ∂ ∂ + ∂ ∂ . © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… ()       ω− ν               ν ∂ ∂ = ν tS R x sin R y yf B m y x (11.12) From (11.10), () () 0yf R y yf y 2 2 y 2 =       ν − ∂ ∂ (11.13) or () R y y eByf ν ν = (11.14) Equation (11.14) retains one term because y < 0 inside the rotor and f y (y) should reach zero for y ≈ −∞. The resultant flux density amplitude in the rotor iron B r ν is R y r eBB ν νν = (11.15) But the Faraday law yields dt B J 1 rotErot Fe ∂ −=         σ = (11.16) In our case, the flux density in iron has two components B x and B y , so t B z J t B z J y Fe x x Fe y ∂ ∂ σ−= ∂ ∂ ∂ ∂ σ−= ∂ ∂ (11.17) The induced current components J x and J y are thus         ω− ν ωσ−=         ω− ν ωσ−= ν ν νν ν ν νν tS R x sinzeBSJ tS R x coszeBSJ m R y Fey m R y Fex (11.18) The resultant current density amplitude J r ν is zeBSJJJ R y Fe 2 y 2 xr ν ννν ωσ=+= (11.19) The losses per one lamination (thickness d Fe ) is © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… ∫∫∫ − −∞= = π = νν σ = 2/d 2/d y 0y R2 0x 2 r Fe lam Fe Fe dzdydxJ 1 P (11.20) For the complete rotor of axial length l stack , () stack 2 Fe 22 Fe 0 Rl2 R dSB 24 P π ν ω σ = ννν (11.21) Now, if we consider the first and second airgap magnetic conductance harmonic (inversed airgap function) with a 1,2 (Chapter 10),         β = s os 2,12,1 t b F g a (11.22) β(b os,r /g) and F 1,2 (b os,r /t s,r ) are to be found from Table 10.1 and (10.14) in Chapter 10 and gK 1 a ; a a BB c 0 0 2,1 1g == ν (11.23) where B g1 is the fundamental airgap flux density with () 1 s 1 1s 1s p N p pN 1S and pN ≈ ± −=±=ν ν (11.24) The no-load rotor surface losses P o ν are thus () ()         + ππ         σ ≈ ± ν 2 0 2 2 2 1 stack s 1 2 Fe 2 1 2 1 s 2 1g Fe PN 0 a a2a Rl2 N Rp df2 p N B 24 2P 1s (11.25) Example 11.1. Let us consider an induction machine with open stator slots and 2R = 0.38 m, N s = 48 slots, t s = 25 mm, b os = 14 mm, g = 1.2 mm, d Fe = 0.5 mm, B g1 = 0.69 T, 2p 1 = 4, n 0 = 1500 rpm, N r = 72, t r = 16.6 mm, b or = 6 mm, σ Fe = 10 8 /45 (Ωm) -1 . To determine the rotor surface losses per unit area, we first have to determine a 1 , a 2 , a 0 from (11.22). For b os /g = 14/1.2 = 11.66 from table 10.1 β = 0.4. Also, from Equation (10.14), F 1 (14/25) = 1.02, F 2 (0.56) = 0.10. Also, gK 1 a 2,1c 0 = ; K c (from Equations (5.3 – 5.5)) is K c1,2 = 1.85. Now the rotor surface losses can be calculated from (11.25). () ()() 2 2 22 62 2 2 6 stack 0 m/W68.8245 85.1 1.04.0202.14.0 19.0105.0602 2 48 69.0 24 1022.22 Rl2 P =         ⋅+⋅ ⋅ ⋅⋅⋅⋅π       ⋅⋅ ⋅⋅ = π − ν © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… As expected, with semiclosed slots, a 1 and a 2 become much smaller; also, the Carter coefficient decreases. Consequently, the rotor surface losses will be much smaller. Also, increasing the airgap has the same effect. However, at the price of larger no-load current and lower power factor of the machine. The stator surface losses produced by the rotor slotting may be calculated in a similar way by replacing F 1 (b os /t s ), F 2 (b os /t s ), β(b os /g) with β(b or /g), F 1 (b or /t s ), F 2 (b or /t s ), and N s /p 1 with N r /p 1 . As the rotor slots are semiclosed, b or << b os , the stator surface losses are notably smaller than those of the rotor, They are, in general, neglected. 11.3.2. No-load tooth flux pulsation losses As already documented in the previous paragraph, the stator (and rotor) slot openings produce variation in the airgap flux density distribution (Figure 11.3). t s g b os max stator tooth flux B gmax minimum stator tooth flux B gmax a.) φ tooth s φ 0 φ max φ min π N r 2π N r 3π N r θ m b.) Figure 11.3 Airgap flux density as influenced by rotor and stator slotting a.) and stator tooth flux versus rotor position b.) In essence, the total flux in a stator and rotor tooth varies with rotor position due to stator and rotor slot openings only in the case where the number of stator and rotor slots are different from each other. This is, however, the case, as N s ≠ N r at least to avoid large synchronous parasitic torques at zero speed (as demonstrated in Chapter 10). The stator tooth flux pulsates due to rotor slot openings with the frequency () 1 r 1 rPS p f N p S1 fNf = − = , for S = 0 (no-load). The flux variation coefficient K φ is © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… 0 minmax 2 K φ φ−φ = φ (11.26) The coefficient K φ , as derived when the Carter coefficient was calculated in [6] s 2 t2 g K γ = φ (11.27) with γ 2 as g b 5 g b or 2 or 2 +         =γ (11.28) Now, by denoting the average flux density in a stator tooth with B ots , the flux density pulsation B p1 in the stator tooth is s 2 otsPs t2 g BB γ = (11.29) oss s 0gots bt t BB − = (11.30) B g0 –airgap flux density fundamental. We are now in the classical case of an iron region with an a.c. magnetic flux density B Ps , at frequency f Ps = N r f/p 1 . As N r f/p 1 is a rather high frequency, eddy current losses prevail, so ght teeth weistatorG ;G 50 f 1 B CP steethsteeth 2 Ps 2 Ps ep0Ps −             = (11.29) In (11.31), C ep represents the core losses at 1T and 50Hz. It could have been for 1T and 60 Hz as well. Intuitively, the magnetic saturation of main flux path places the pulsation flux on a local hysteresis loop with lower (differential) permeability, so saturation is expected to reduce the flux pulsation in the teeth. However, Equation (11.31) proves satisfactory even in the presence of saturation. Similar tooth flux pulsation core losses occur in the rotor due to stator slotting. Similar formulas as above are valid. 1 sprteeth 2 Pr 2 Pr ep0Pr p f Nf ;G 50 f 1 B CP =             = © 2002 by CRC Press LLC [...]... frequencies, the lamination skin depth δFe < d and thus the magnetic field is confined to a skin depth layer around the stator slot walls and on the rotor surface (Figure 11.10) The conventional picture of rotor leakage flux paths around the rotor slot bottom is not valid in this case The area of leakage flux is now, for the stator, Al = lf δFe, with lf the length of the meander zone around the stator... approximately, so the rotor conductor losses are Pconr ≈ 3Vν 2 (2πf ν K L )2 f ν −0.32 K R f ν 0.5 ~ Vh 2 f ν1.18 (11.98) The rotor conductor losses drop notably as the time harmonic frequency increases The situation in the stator is different as there are many conductors in every slot (at least in small power induction machines) So the skin effect for Rs will remain fν dependent in the initial stages... constant PFe is almost constant in these conditions Case 2 – Slight lamination skin effect, δFe ≈ d When δFe ≈ d, the frequency fν is already high and thus δAl < dAl and a severe skin effect in the rotor slot occurs Consequently, the rotor leakage flux is concentrated close to the rotor surface The “volume” where the core losses occur in the rotor decreases In general then, the core losses tend to decrease... converters Exploring the conductor and core losses up to such large frequencies becomes necessary High carrier frequencies tend to reduce the current harmonics and thus reduce the conductor losses associated with them, but the higher frequency flux harmonics may lead to larger core loss On the other hand, the commutation losses in the PWM converter increase with carrier frequency The optimum carrier...  2 ⋅ 50   The open slots in the stator produce large rotor tooth flux pulsation no-load specific losses (PPr) The values just obtained, even for straight rotor (and stator) slots, are too large Intuitively we feel that at least the rotor tooth flux pulsations will be notably reduced by the currents induced in the cage by them At the expense of no-load circulating cage-current losses, the rotor flux... important as their winding factor is the same as for the fundamental © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… The mmf harmonic Fν amplitude is Fν = F1 K wν p1 K w1 ν (11.49) where Kw1 and Kwν are the winding factors of the fundamental and of harmonic ν, respectively If the number of slots per pole and phase (q) is not very large, the first slot (opening) harmonics Ns ± p1 produce the largest... kHz or more), the skin effect in the stator conductors enters the fν0.5 domain and the stator conductor losses, for given harmonic voltage, behave like the rotor cage losses (decrease slightly with frequency (11.98)) This situation occurs when the penetration depth becomes smaller than conductor height 11.10.2 Core losses Predicting the core loss at high frequencies is difficult because the flux penetration... carrier frequency, the skin effect, both in the conductors and iron cores, may not be neglected 11.10.1 Conductor losses The variation of resistance R and leakage Ll inductance for conductors in slots with frequency, as studied in Chapter 9, is at first rapid, being proportional to f2 As the frequency increases further, the field penetration depth gets smaller than the conductor height and the rate of change... Measurements suggest that the cross-path or transverse impedance Zd is, in fact, a resistance Rd up to at least f = 1 kHz Also, the contact resistance between the rotor bar and rotor teeth is much larger than the cross-path iron core resistance This resistance tends to increase with the frequency of the harmonic considered and it depends on the manufacturing technology of the cast aluminum cage rotor... accounting for the first airgap magnetic conductance harmonic, the value of Bν is a1 ; N s = (N s ± p1 ) m p1 2a 0 B Ns = Bg1 (11.79) a1 and a0 from (11.22 and 11.23) The speed Vrν is Vrν = R with ν = N s ± p1 2πf  p1  1 ±  ν p1 60  Vrν ≈ R 2πf = VNs p1 60 (11.80) (11.81) The airgap reactance for the ν = Ns ± p1 harmonics, X0ν, is X 0 Ns ≈ X 0 p p1 Ns (11.82) X0p, the airgap reactance for the fundamental, . IN INDUCTION MACHINES Losses in induction machines occur in windings, magnetic cores, besides mechanical friction and windage losses. They determine the. each other, respectively, in the airgap flux density. For these mmf harmonics, the winding factor is equal to that of the fundamental. This is why they