Recent Developments of Electrical Drives - Part 32 pptx

Left: basic structure of a PM synchronous machine, with tooth coil armature winding.Right: coil winding senses around teeth.. rit can be shown that the winding factor kwof a three-phase

Figure 1 Left: basic structure of a PM synchronous machine, with tooth coil armature winding.

Right: coil winding senses around teeth

rcycle-phase: referring to a layer, portion of one cycle including adjacent coils belonging to the same phase; parent coil: in each layer, the first coil of every cycle-phase; its succession assignment defines the winding;

rthe no of teeth/cycle Ntcand the no of coils/cycle Ncc must be multiple of the no of phases Nph;

rlinks about no of teeth/cycle-phase Ntcph and no of coils/cycle-phase Nccph: Ntc=

NphNtcph; Ncc= NphNccph;

rin case of controverse coils, the no of coils/cycle-phase Nccphcoincides with the no of

teeth/cycle-phase Ntcph;

rthe optimal no of PMs/cycle Nmcdiffers by one with respect to Ntc: Nmc= Ntc± 1 (→

highest winding factor);

rthe optimal displacement among layers equals a no of teeth Ntsnearest to Nccph/2 (low harmonic distortion);

rthe no of cycles Ncequals the maximum no of parallel paths “a” of each phase;

rthe total no of PMs Nm= NmcNc of a rotating machine must be even; thus, if Nmcis even, the no of cycles Nccan be any integer; if Nmcis odd, Ncmust be even;

rthe no of coils/cycle-phase Nccphcan be any integer;

Figure 2 Double layer winding (two coils/tooth), with controverse tooth coils: Ntc= 12; Ncc= 12;

Nph= 3; Ntcph= Nccph= 4; Nts= 2

rit can be shown that the winding factor kwof a three-phase tooth coil machine (with two-layer windings) equals the product of a distribution factor kd times a displacement factor ks;

rfor the phase winding e.m.f of the jth order harmonic (j= 1, 3, 5, ), we have:

kdj = sin(j· π/6)

Nccph· sin[(j/Nccph)· π/6] , (2)

ksj = cos(j · (Nsp/Ndcf)· π/6; (3)

a traditional machine, with two-layer distributed windings, q slots/(pole-phase) and coil pitch shortening of caslots, exhibits a winding factor faequal to the product of a distribution factor fdtimes a pitch factor fp:

fdj= sin(j· π/6)

these expressions and the previous ones are exactly corresponding each other, provided that

we associate Nccphwith q and Ntswith ca: the difference lies in the fact that, with a 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;

rthe 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 Ntand Np(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 surface 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

Table 1 Combinations of Ntand Np(i= inferior; s = superior) of

three-phase controverse windings, for some values of Nccphand Nc

(Ncmin= 2); Scph= 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

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 Bt 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 fluxϕt0can be expressed as follows:

ϕt0= ϕr· ηPM= (Br· bm· ) · 1

1+ (1 + ε )· μrPM· g/hm

whereϕr= Br× bm×  is the PM residual flux, ηPMthe air-gap magnetization efficiency

of the PM,ε ,μrPM, and hm the PM leakage, the relative reversible permeability and the

PM height respectively, g the air-gap width

We adopted a NdFeB PM material (MPN40H: Br= 1.2 T; HcB= 700 kA/m at 80◦C),

choosing Nc= 2, Ntcph= 6, Nm= 34, bm= 10 mm, central air-gap g = 0.65 mm: with these values, hm= 3 mm is suited to gain an acceptable no-load magnetization (in fact, withε  ≈ 0.15, it follows: ηPM≈ 0.75; Bt= 1.32 T; tooth flux ϕt0= 0.761 mWb); FEM

simulations [14] confirmed (7) (ϕtanalytical= 1.012 × ϕtFEM)

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 the stator yoke, also the rotor yoke results definitely oversized (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

Slot opening width, b a 2.7 mm Slot opening height, h a 0.55 mm

Tooth body height, h t 20.00 mm

Figure 3 Picture of the PM rotor, during the assembling process: just some PMs are glued on the rotor

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 Nts= Nccph/2 = 3 has been

adopted)

The FEM evaluated distribution [14] of the no-load flux density amplitude in the toothed zone (at half stator tooth height) is shown in Fig 5; the following remarks are valid:

rthe FEM peak value Btconfirms the analytical result;

rthe peripheral amplitude distribution of|Bt0| appears substantially sinusoidal, thanks to the gradual displacement among PMs and teeth within each cycle

Figure 4 Top: magnetic structure and winding arrangement of the analyzed and constructed

concen-trated coil PM motor Bottom: disposition conventions of coils and PMs

Figure 5 Peripheral amplitude distribution of the no-load flux density Bt0in the stator teeth (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

0as follows:

0= (kw1· Nc· 2 · Ntcph· ϕt0/√2)· Ntuc 01· Ntuc, (8) where the dependence on the no of turns of each coil (Ntuc) is evidenced In a two-layer winding, the no of turns around each tooth Ntutis even: in fact, Ntut= 2× Ntucoccurs The no-load flux linkage 0can be evaluated also by FEM: some simulations have shown the accuracy of (8)

Of course, Ntuc is included also in the expressions of the equivalent resistance R and synchronous inductance L:

R= R1· N2

L= L1· N2

01,R1, and L1are the corresponding parameters of a phase winding consisting of one-turn series connected coils, being the same the coil total copper cross section:

R1= 22· Ntcph·Nc



a2

· ρcu· [tu/(αcu· (As/2))], (11)

L1= 22·Nc

a2

with: a= no of winding parallel paths, equal to Nc, or sub-multiple of it (here a= 1 has been chosen);tu= average turn length; As= slot cross section; αcu= slot filling factor;

e= “per tooth” equivalent permeance

While R1 is simple to be evaluated, L1 can be analytically evaluated only with some approximation; on the other hand, it can be obtained with energy calculations by a magneto-static FEM simulation, substituting the PMs with passive objects, with the same permeability

of the PMs

For the machine of Table 2, Fig 4, the values of Table 3 have been obtained

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) 51.5μH

The choice of Ntuc is a key design issue, greatly affecting the performances In the following, just the Joule losses will be taken into account, neglecting the core Pc and mechanical losses Pm, that can be considered separately To evaluate the influence of Ntuc, 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 quantitiesρEand Ikas follows:

ρE= E

Ik= V

N2 tuc·R2

1+ (ω · L1)2

they represent the e.m.f./voltage ratio and the locked rotor current respectively, and depend

on the number Ntuc

The input current in loaded operation is given by:

I= Ik·1+ ρ2

whereδ is the load angle (see Fig 6).

Called p= Nmthe no of poles, the torque T is given by:

T 0· (p/2) · Ik· [cos (ϕz− δ) − ρE· cos (ϕz)], (16) where

ϕz= atan(X / R) = atan(ω · L1/ R1) (17)

is the characteristic angle of the motor internal impedance (independent on Ntuc) andδ the

load angle (see Fig 6)

From (16), the load angleδ in loaded operation follows:

Figure 6 Phasor diagram for the analysis of the tooth coil synchronous motor, in sinusoidal feeding

operation, at voltage V

Moreover, (16) shows that the max torque Tmax (pull-out torque) occurs for the static stability limit angleδmax:

Tmax 0· (p/2) · Ik· [1 − ρE· cos(ϕz)]. (20) Imposing the condition T = 0 in (18) leads to evaluate the no-load angle δ0 and the corresponding no-load current I0:

δ0 = ϕz− acos(ρE· cos(ϕz)), (21)

I0= Ik·1+ ρ2

E− 2 · ρE· cos(δ0). (22) Assuming a suited value of the rated current density Sn, the rated current In can be expressed as follows:

In= Sn· [(αcu· As)/(4 · Ntuc)] (23) (in our motor, thermal status suggested: Sn= 6.5 A/mm2) Substituting (23) in (15) gives the rated load angle:

δn = acos1+ ρ2

E− (In/Ik)2

and inserting (24) in (16) gives the rated torque Tn

The reactive power absorbed by the motor is expressed by:

Q= 3 · V · Ik· [sin(ϕz)− ρE· sin(ϕz+ δ)]; (25) while the ideal input power Piequals (Pc, Pmneglected):

From (25) and (26), the power factor:

cosϕ = 1

is a function ofρEand Ntuc, by (9), (15), and (16)

As concerns the transient model, the differential equations in terms of Park vectors are

as follows:

⎧

⎪

⎪

⎪

⎪

dθ

dt

L· diP

dt = vP− R · iP− j ·p

2

√

3 0· ej· θ · p/2

Jtot·d

dt = p

2·√3 0· ImiP· e−j · θ · p/2 − Tload

θ is the mechanical angle between PM and phase “a” axes; Jtot= Jrot+ Jloadthe total inertia,

Tloadthe load torque

In the following, the diagrams in Figs 7–12 will show the effect of Ntucchanges on the previously defined quantities: all the curves refer to steady state operation under sinusoidal feeding (V= 380 Vrms, f = 50 Hz)

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 Ntuc

Figure 8 RatioρEas a function of Ntuc, together with the curves of the ratiosδ0/ϕzandδn/ϕz(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 Ntuc

Figure 9 Locked rotor (Ik), rated (In), and no-load (I0) input currents of the motor of Table 2 and Fig 4, as a function of the no of turns/coil Ntuc(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 (Tn) and maximum torque (Tmax) of the motor of Table 2 and Fig 4, as a function of the no of turns/coil Ntuc(sinusoidal feeding: V= 380 Vrms, f = 50 Hz)

Figure 11 Rated torque (Tn) of the motor of Table 2 and Fig 4, as a function of Ntuc(sinusoidal feeding: V= 380 Vrms, f = 50 Hz)

Fig 7 shows the current-torque characteristics, for some Ntucvalues, traced by (15) and (16), forδ0≤ δ ≤ δmax= ϕz

The adoption of high Ntucvalues (Ntuc→ 61, corresponding to ρE→ 1) allows to reduce the no-load current, but reduces also the maximum torque and, thus, the motor overloading capability and the self-starting performances

Fig 8 showsρE as a function of Ntuc, together with the curves of the ratiosδ0/ϕzand

δn/ϕz(see equations (13), (21), and (24)), in sinusoidal feeding with V= 380 Vrms, f = 50 Hz: it is worth to observe thatδ0is negative, approaching unity whenρEapproaches unity too (E→ V)

Fig 9 confirms the remark concerning the no-load current I0as a function of Ntuc, also showing the change of the rated current Inand of the locked rotor current Ik

Fig 10 illustrates the decrease of the power factor cosϕ when lowering Ntuc, while the maximum torque shows a significant increase As the rated torque, it shows an almost flat maximum around Ntuc= 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 Ntuc= 48

As regards losses, rated torque and power factor, the best choice would be Ntuc= 48; considering also the importance of Tmax, a lower Ntucvalue can allow better overloading and self-starting features: for this reason, we have chosen Ntuc= 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 Ntuc, in sinusoidal feeding (V= 380 Vrms, f = 50 Hz)

Figure 13 Measured waveform of the no-load e.m.f at the terminals of a probe coil of Np= 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 a constructed 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 Np= 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 eph: the amplitude evaluated by (8) is confirmed; moreover, it is evident the great shape improvement 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 ephhas 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, Ntuc= 46 turns/coil)

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:... values of the no of turns/coil Ntuc

Figure Locked rotor (Ik), rated (In), and no-load (I0) input currents of the motor of. .. class="text_page_counter">Trang 10

Figure 13 Measured waveform of the no-load e.m.f at the terminals of a probe coil of Np= 10 turns, disposed around