Recent Developments of Electrical Drives - Part 32 pptx

308 Di Gerlando et al. Figure 1. Left: basic structure of a PM synchronous machine, with tooth coil armature winding. Right: coil winding senses around teeth. r cycle-phase:referring to alayer,portion ofonecycleincludingadjacent coils belongingto the samephase; parent coil:in eachlayer, thefirst coil ofevery cycle-phase; its succession assignment defines the winding; r the no. of teeth/cycle N tc and the no. of coils/cycle N cc must be multiple of the no. of phases N ph ; r links about no. of teeth/cycle-phase N tcph and no. of coils/cycle-phase N ccph :N tc = N ph N tcph ;N cc = N ph N ccph ; r in case of controverse coils, the no. of coils/cycle-phase N ccph coincides with the no. of teeth/cycle-phase N tcph ; r the optimal no. of PMs/cycle N mc differs by one with respect to N tc :N mc = N tc ± 1(→ highest winding factor); r the optimal displacement among layers equals a no. of teeth N ts nearest to N ccph /2 (low harmonic distortion); r the no. of cycles N c equals the maximum no. of parallel paths “a” of each phase; r the total no. of PMs N m = N mc N c of a rotating machine must be even; thus, if N mc is even, the no. of cycles N c can be any integer; if N mc is odd, N c must be even; r the no. of coils/cycle-phase N ccph can be any integer; Figure 2. Double layer winding (two coils/tooth), with controverse tooth coils: N tc = 12; N cc = 12; N ph = 3; N tcph = N ccph = 4; N ts = 2. III-1.2. High Pole Number, PM Synchronous Motor 309 r it can be shown that the winding factor k w of a three-phase tooth coil machine (with two-layer windings) equals the product of a distribution factor k d times a displacement factor k s ; r for the phase winding e.m.f. of the jth order harmonic (j = 1, 3, 5, ), we have: k w j = k d j · k s j with (1) k d j = sin(j · π/6) N ccph · sin[(j/N ccph ) · π/6] , (2) k s j = cos(j · (N sp /N dcf ) · π/6; (3) a traditional machine, with two-layer distributed windings, q slots/(pole-phase) and coil pitch shortening of c a slots, exhibits a winding factor f a equal tothe productof adistribution factor f d times a pitch factor f p : f a j = f d j ·f p j , with (4) f d j = sin(j · π/6) q · sin[(j/q) ·π/6] , (5) f p j = cos(j · (c a /q) · π/6); (6) these expressions and theprevious ones are exactly corresponding each other,provided that we associate N ccph with q andN ts with c a : the differencelies in thefact that,witha traditional machine, good quality performances (high winding factor and good e.m.f. waveform, no cogging, teeth harmonics, magnetic noise, and vibrations) can be obtained by adopting structures with q ≈ 5–6, while a tooth coil machine (with the described features) exhibits similar performance quality with q values practically equal to 0.33: thus, machines with a given no. of poles can be realized with armature structures with a very low no. of slots; r the other main advantages of these machines are: –the stator assembly is simplified: no skewing is required; only concentrated coils are used, that can be prepared separately (no endwindings overlapping; reduced copper mass; and armature losses); –the torque is high at low speed, allowing to eliminate any gears. Table 1 shows some combinations of N t and N p (i =inferior; s =superior), for three-phase windings. Design analysis of a basic prototype In order to study the basic features of this kind of machine, we have decided to modify an existing induction motor, by re-winding its stator according to the previous theory and designing a new rotor, equipped with surf ace mounted PMs: of course, this choice has prevented from obtaining an optimized stator core, but, besides to easily provide a first test motor, it has also allowed to evaluate the suitability of existing laminations for the new machine. The main data of the used stator core are given in Table 2. 310 Di Gerlando et al. Table 1. Combinations of N t and N p (i = inferior; s = superior) of three-phase controverse windings, for some values of N ccph and N c (N cmin = 2); S cph = sequence of the parent coils within two cycles N ccph N tc N c N t N pci N pi S cph.i N pcs N ps S cph.s 2 6 2 12 5 10 AcBaCb 7 14 AbCaBc 3 9 3 27 8 24 ACBACB 10 30 ABCABC 4 12 2 24 11 22 AcBaCb 13 26 AbCaBc 5 15 3 45 14 42 ACBACB 16 48 ABCABC 6 18 2 36 17 34 AcBaCb 19 38 AbCaBc About the rotor design, the available degrees of freedom are air-gap width and PM sizes and material: their choice is made by considering the operating point of the PM and the flux density B t in the stator teeth. Considering the alignment condition between the PM axis and the tooth axis, from the analysis of the equivalent magnetic circuit concerning a zone extended to a tooth pitch, the no-load peak tooth flux ϕ t0 can be expressed as follows: ϕ t0 = ϕ r · η PM = (B r · b m · ) · 1 1 + (1 +ε  ) · μ rPM · g/h m , (7) where ϕ r = B r × b m ×  is the PM residual flux, η PM the air-gap magnetization efficiency of the PM, ε  , μ rPM , and h m the PM leakage, the relative reversible permeability and the PM height respectively, g the air-gap width. We adopted a NdFeB PM material (MPN40H: B r = 1.2T;H cB = 700 kA/m at 80 ◦ C), choosing N c = 2, N tcph = 6, N m = 34, b m = 10 mm, central air-gap g = 0.65 mm: with these values, h m = 3 mm is suited to gain an acceptable no-load magnetization (in fact, with ε  ≈ 0.15, it follows: η PM ≈ 0.75; B t = 1.32 T; tooth flux ϕ t0 = 0.761 mWb); FEM simulations [14] confirmed (7) (ϕ tanalytical = 1.012 × ϕ tFEM ). Fig. 3 shows the designed rotor during the construction process: the PMs are glued on the steel surface, inserted in suited slots for their correct and accurate positioning. As thestator yoke, also the rotor yokeresults definitelyoversized (in fact, it was designed for a four pole motor). Table 2. Main constructional data of the stator magnetic core used for the PM machine (obtained from an available standard induction machine lamination); main PM data Stator internal diameter, D i 140 mm Stator external diameter, D e 220 mm Stator yoke width, h y 19.5 mm Lamination stack length, ι 85 mm No. of stator teeth, N t 36 No. of PMs, N m 34 Slot opening width, b a 2.7 mm Slot opening height, h a 0.55 mm Tooth body width, b t 6.7 mm Tooth body height, h t 20.00 mm Tooth head width, b e 9.5 mm PM polar arc, α m 0.77 pu III-1.2. High Pole Number, PM Synchronous Motor 311 Figure 3. Picture of thePMrotor,duringtheassembling process: just some PMsaregluedontherotor surface; small slots (0.3 mm deep) allow a precise and reliable PM positioning, without appreciable increase of the flux leakage among adjacent PMs. The complete cross section of the machine is represented in Fig. 4, that shows also the adopted winding disposition (in it, a layer displacement N ts = N ccph /2 = 3 has been adopted). The FEM evaluated distribution [14] of the no-load flux densityamplitude in the toothed zone (at half stator tooth height) is shown in Fig. 5; the following remarks are valid: r the FEM peak value B t confirms the analytical result; r the peripheral amplitude distribution of |B t0 | appears substantially sinusoidal, thanks to the gradual displacement among PMs and teeth within each cycle. Figure 4. Top: magnetic structure and winding arrangement ofthe analyzed and constructed concen- trated coil PM motor. Bottom: disposition conventions of coils and PMs. 312 Di Gerlando et al. Figure 5. Peripheral amplitude distributionoftheno-loadfluxdensityB t0 in thestatorteeth(evaluated by FEM simulation, at half the tooth height) for the machine described in Table 2. This sinusoidal distribution allows to express the r.m.s. no-load fundamental flux linkage  0 as follows:  0 = (k w1 · N c · 2 ·N tcph · ϕ t0 / √ 2) · N tuc =  0 1 · N tuc , (8) where the dependence on the no. of turns of each coil (N tuc ) is evidenced. In a two-layer winding, the no. of turns around each tooth N tut is even: in fact, N tut =2× N tuc occurs. The no-load fluxlinkage 0 can be evaluatedalso byFEM: some simulationshave shown the accuracy of (8). Of course, N tuc is included also in the expressions of the equivalent resistance R and synchronous inductance L: R = R 1 · N 2 tuc (9) L = L 1 · N 2 tuc . (10)  01 ,R 1 , and L 1 are the corresponding parameters of a phase winding consisting of one-turn series connected coils, being the same the coil total copper cross section: R 1 = 2 2 · N tcph ·  N c  a 2  · ρ cu · [ tu /(α cu · (A s /2))], (11) L 1 = 2 2 ·  N c  a 2  · N tcph ·  e , (12) with: a = no. of winding parallel paths, equal to N c , or sub-multiple of it (here a = 1 has been chosen);  tu = average turn length; A s = slot cross section; α cu = slot filling factor;  e = “per tooth” equivalent permeance. While R 1 is simple to be evaluated, L 1 can be analytically evaluated only with some approximation; on the otherhand, it can beobtained with energy calculationsby amagneto- staticFEMsimulation,substitutingthePMswithpassiveobjects,withthesamepermeability of the PMs. For the machine of Table 2, Fig. 4, the values of Table 3 have been obtained. III-1.2. High Pole Number, PM Synchronous Motor 313 Table 3. Calculated parameters of a PM motor with the data of Table 2, Fig. 4, equipped with “single turn per coil” windings Flux linkage,  01 (equation 8) 11.5 mWb rms Resistance, R 1 (equation 11) 8.03 m Inductance, L 1 (equation 12) 51.5 μH The choice of N tuc is a key design issue, greatly affecting the performances. In the following, just the Joule losses will be taken into account, neglecting the core P c and mechanical losses P m , that can be considered separately. To evaluate the influence of N tuc , the phasor diagram of Fig. 6 must be considered, analyzing the machine operation under sinusoidal feeding, at voltage V. It is useful to define the quantities ρ E and I k as follows: ρ E = E V = ω ·  0 V = ω ·  0 1 V · N tuc (13) I k = V Z = V  R 2 + (X) 2 = V N 2 tuc ·  R 2 1 + (ω · L 1 ) 2 : (14) they represent the e.m.f./voltage ratio and the locked rotor current respectively, and depend on the number N tuc . The input current in loaded operation is given by: I = I k ·  1 + ρ 2 E − 2 ·ρ E · cos (δ), (15) where δ is the load angle (see Fig. 6). Called p = N m the no. of poles, the torque T is given by: T = 3 ·  0 · (p/2) ·I k · [cos (ϕ z − δ) − ρ E · cos (ϕ z )], (16) where ϕ z = atan(X / R) = atan(ω · L 1 /R 1 ) (17) is the characteristic angle of the motor internal impedance (independent on N tuc ) and δ the load angle (see Fig. 6). From (16), the load angle δ in loaded operation follows: δ = ϕ z − acos{T/[3· 0 · (p/2) ·I k ] + ρ E · cos(ϕ z )}. (18) Figure 6. Phasor diagram for the analysis of the tooth coil synchronous motor, in sinusoidal feeding operation, at voltage V. 314 Di Gerlando et al. Moreover, (16) shows that the max. torque T max (pull-out torque) occurs for the static stability limit angle δ max : δ max = ϕ z , (19) T max = 3 ·  0 · (p/2) ·I k · [1 −ρ E · cos(ϕ z )]. (20) Imposing the condition T = 0 in (18) leads to evaluate the no-load angle δ 0 and the corresponding no-load current I 0 : δ 0 = ϕ z − acos(ρ E · cos(ϕ z )), (21) I 0 = I k ·  1 + ρ 2 E − 2 ·ρ E · cos(δ 0 ). (22) Assuming a suited value of the rated current density S n , the rated current I n can be expressed as follows: I n = S n · [(α cu · A s )/(4 · N tuc )] (23) (in our motor, thermal status suggested: S n = 6.5 A/mm 2 ). Substituting (23) in (15) gives the rated load angle: δ n = acos  1 + ρ 2 E − (I n /I k ) 2  /(2.ρE)  , (24) and inserting (24) in (16) gives the rated torque T n . The reactive power absorbed by the motor is expressed by: Q = 3 · V ·I k · [sin(ϕ z ) − ρ E · sin(ϕ z + δ)]; (25) while the ideal input power P i equals (P c ,P m neglected): P i = T ·  +3 ·R ·I 2 . (26) From (25) and (26), the power factor: cos ϕ = 1   1 + (Q/P i ) 2 . (27) is a function of ρ E and N tuc , by (9), (15), and (16). As concerns the transient model, the differential equations in terms of Park vectors are as follows: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dθ dt =  L · d  i P dt =v P − R ·  i P − j · p 2 ·  · √ 3 ·  0 · e j ·θ ·p/2 J tot · d dt = p 2 · √ 3 ·  0 · Im   i P · e −j ·θ ·p/2  − T load : (28) θ is the mechanical angle between PM and phase “a” axes; J tot =J rot +J load the total inertia, T load the load torque. In the following, the diagrams in Figs. 7–12 will show the effect of N tuc changes on the previously defined quantities: all the curves refer to steady state operation under sinusoidal feeding (V = 380 Vrms, f = 50 Hz). III-1.2. High Pole Number, PM Synchronous Motor 315 Figure 7. Input current I of the motor of Table 2 and Fig. 4, as a function of the torque T, in sinusoidal operation under V =380 Vrms, f = 50 Hz, for different values of the no. of turns/coil N tuc . Figure 8. Ratio ρ E as a function of N tuc , together with the curves of the ratios δ 0 /ϕ z and δ n /ϕ z (see equations (13), (21), and (24)), in sinusoidal operation under V =380 Vrms, f =50 Hz, for different values of the no. of turns/coil N tuc . Figure 9. Locked rotor (I k ), rated (I n ), and no-load (I 0 ) input currents of the motor of Table 2 and Fig. 4, as a function of the no. of turns/coil N tuc (sinusoidal feeding: V = 380 Vrms, f = 50 Hz). 1 0.9 0.8 0.7 0.5 0.4 0.6 Figure 10. Power factor (cosϕ), rated (T n ) and maximum torque (T max ) of the motor of Table 2 and Fig. 4, as a function of the no. of turns/coil N tuc (sinusoidal feeding: V = 380 Vrms, f = 50 Hz). 316 Di Gerlando et al. Figure 11. Rated torque (T n ) of the motor of Table 2 and Fig. 4, as a function of N tuc (sinusoidal feeding: V = 380 Vrms, f = 50 Hz). Fig. 7 shows the current-torque characteristics, for some N tuc values, traced by (15) and (16), for δ 0 ≤ δ ≤ δ max = ϕ z . The adoptionof highN tuc values (N tuc →61,corresponding toρ E →1)allows toreduce the no-load current, but reduces also the maximum torque and, thus, the motor overloading capability and the self-starting performances. Fig. 8 shows ρ E as a function of N tuc , together with the curves of the ratios δ 0 /ϕ z and δ n /ϕ z (see equations (13), (21), and (24)), in sinusoidal feeding with V =380 Vrms, f =50 Hz: it is worth to obser ve that δ 0 is negative, approaching unity when ρ E approaches unity too (E → V). Fig. 9 confirms the remark concerning the no-load current I 0 as a function of N tuc , also showing the change of the rated current I n and of the locked rotor current I k . Fig. 10 illustrates the decrease of the power factor cosϕ when lowering N tuc , while the maximum torque shows a significant increase. As the rated torque, it shows an almost flat maximum around N tuc = 48, as better visible in Fig. 11. On the other hand, a correlative property is shown in Fig. 12, showing that the ratio among the Joule losses and the output power has a minimum for N tuc = 48. As regards losses, rated torque and power factor, the best choice would be N tuc = 48; considering also the importance of T max ,alowerN tuc value can allow better overloading and self-starting features: for this reason, we have chosen N tuc = 46 (→wire diameter: 0.63 mm). Figure 12. Ratio between stator Joule losses and output power of the motor of Table 2 and Fig. 4, as a function of N tuc , in sinusoidal feeding (V = 380 Vrms, f = 50 Hz). III-1.2. High Pole Number, PM Synchronous Motor 317 Figure 13. Measured waveform of the no-load e.m.f. at the terminals of a probe coil of N p = 10 turns, disposed around one stator tooth: the typical trapezoidal shape can be observed. Simulation and experimental results Several simulations and experimental tests have been performed on aconstructed prototype based on the previous data, in order to validate the design and operation models and to verify the achievable performance levels. Fig. 13 shows the measured waveform of a “tooth” e.m.f., i.e. the no-load e.m.f. at the terminals of a probe coil of N p = 10 turns, disposed around one stator tooth: even if a certain distortion can be observed, the amplitude estimable from (8) is fairly confirmed. Fig. 14 shows the measured waveform of the no-load phase-to-neutral e.m.f. e ph : the amplitude evaluated by(8) is confirmed;moreover,it is evidentthegreat shapeimprovement compared with the tooth e.m.f. It is particularly noticeable the absolute absence of slotting effects, in spite of the very low no. of slots/(pole-phase). The phase-to-neutral e.m.f. is almost sinusoidal: in fact, the harmonic analysis e ph has evidenced limited harmonics, except for an appreciable, even if low, third harmonic e.m.f.; but, as known, this component is cancelled in the line-to-line voltage, while the actual lowest order harmonics (fifth, seventh order) are reduced by the layer displacement (see (3)). Figure 14. No-load phase-to-neutral measured e.m.f., for the constructed motor (data of Table 2, Fig. 4, N tuc = 46 turns/coil) . N ph ; r links about no. of teeth/cycle-phase N tcph and no. of coils/cycle-phase N ccph :N tc = N ph N tcph ;N cc = N ph N ccph ; r in case of controverse coils, the no. of coils/cycle-phase N ccph coincides. eachlayer, thefirst coil ofevery cycle-phase; its succession assignment defines the winding; r the no. of teeth/cycle N tc and the no. of coils/cycle N cc must be multiple of the no. of phases N ph ; r links. no. of teeth N ts nearest to N ccph /2 (low harmonic distortion); r the no. of cycles N c equals the maximum no. of parallel paths “a” of each phase; r the total no. of PMs N m = N mc N c of a