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)