Rev. Roum. Sci. Techn.– Électrotechn. et Énerg.
Vol. 61, 1, pp. 8–12, Bucarest, 2016
ANALYSIS OF ELECTROMAGNETIC CHARACTERISTIC OF A NOVEL
DOUBLY SALIENT BRUSHLESS GENERATOR
LIWEI SHI1, BO ZHOU
Key words: Doubly salient electromagnetic generator, Fault tolerant, Four-phase, Electromagnetic characteristic,
Inductance model.
To develop a fault tolerant machine with the least redundant phases, the inductance model and a nonlinear radial
air gap flux density model of a new four-phase doubly salient electromagnetic generator (DSEG) is presented and
verified by the finite element analysis (FEA) and experiments result in this paper. With the proposed formulas, a
new four-phase 24-slot 18-pole DSEG without cogging torque was designed. It can be concluded from the models
that the magnetic circuit model is very helpful to find the most critical structural parameters of the machine. The
FEA and experiments of the DSEG show that the proposed models can calculate the radial air gap flux density and
the winding inductances.
1. INTRODUCTION
The electromagnetic generator is the most popular
generator in the world, since it doesn't have the big problem
of the uncontrollable flux of the permanent magnet (PM)
generator [1]. Doubly salient electromagnetic generator
(DSEG) is a new brushless dc generator that came from
switched reluctance generator. Both of the field winding
and phase windings are wound in its stator. Therefore, it
keeps the main merits both with wound-field synchronous
generator and switched reluctance generator, such as low
cost, simplicity, reliability and the output voltage can be
easily controlled if it is used as a generator [2].
Multiphase machines with more than three phases can be
applied because they can continue to run even with one or
two open-circuited phases [3, 4].
The elementary machine of four-phase DSEG has a
similar structure with 8-slot 6-pole switched reluctance
machine (SRM) [5].
As a typical machine with the least redundant phases, it
was reported that the 8/6-pole four-phase fault-tolerant
doubly salient permanent magnetic machine (DSPM) can
offer high torque capability with fault-tolerance [6].
Because parameter model of the magnetic circuit is very
helpful to master the basic principles and find the most
critical structural parameters that affect the performance of
the machine.
So it is very important to deduce the parameter model of
the magnetic circuit of a new machine [7].
This paper presents the magnetomotive force model, the
inductance model and the nonlinear radial air gap flux
density model of a new DSEG, which are developed to
predict the electromagnetic performance of the machine.
And the models will be verified by finite-element analysis
and experiments of a 24-slot 18-pole DSEG.
phase asymmetry. What is more, compared to the traditional
three-phase 24-pole 16-slot DSEG, the motor has fourphase winding, which will improve the fault tolerant
performance.
The new DSEG differs from the traditional DSPM [6] or
traditional four-phase DSEG (Fig.1) that comes from [6],
which stator pole arc coefficient αs = 0.5 . In order to
minimize the cogging torque ripple and the self-EMF of the
field winding of the DSEG, the inductance of the field
winding should be constant when the rotor rotates. In other
words, while one phase inductance starts increasing, there
should be and only be another phase inductance begins
decreasing [8]. A four-phase DSEG with new poles and
pole arcs combination is proposed, which pole arc coefficients
comply with:
pr 3
= , αs = 0.667, α r = 0.5.
ps 4
(1)
(a)
2. MACHINE STRUCTURE AND PARAMETERS
A sketch of the new DSEG with 24-slot 18-pole is
presented in Fig. 1(a). It has 24 coils which are divided into
four phases.
Each coil of the same phase is located in a different
place. One of them is in the middle of the field coil, and the
others are near by the field coil. This solves the problem of
1
(b)
Fig. 1 – Structure of (a) new 24-slot 18-pole and (b) traditional 8-pole
6-slot four-phase DSEG.
Nanjing University of Aeronautics and Astronautics, 29 Yudao, 210016, Nanjing, China, E-mail: liwei10@nuaa.edu.cn.
2
Analysis of a novel doubly salient brushless generator
3. THE ANALYTICAL MODELS OF THE WOUNDFIELD DOUBLY SALIENT MACHINE
3.1. EQUIVALENT AIRGAP LENGTH MODEL
By assuming that there is no saturation in the stator and
rotor core, the magnetomotive force (MMF) in the iron core
is ignored. And the end effects are not considered [9].
Similar to other machines [10], the typical magnetic field
lines at different locations of the doubly salient machines
can be drawn in Fig. 2. In the area with no iron core, the air
length of the magnetic path is different with its position.
9
To simplify the problem, δ s1 and δ s2 can be calculated
as two 90° arcs [9]. For the four-phase DSEM in this paper,
if we set up a coordinate system from the beginning of the
stator slot, as shown in Fig. 2:
πD ⎛ θ ⎞
sin ⎜ ⎟ ,
2
⎝2⎠
πD ⎛ π θ ⎞
sin ⎜ − ⎟ ,
=
2
⎝ 72 2 ⎠
δ s1 =
δ s2
(6)
(7)
where D is the air gap diameter. Since the air gap is much
narrow than the diameter of the air gap diameter, the rotor
outer diameter and the stator inner diameter are both replaced
with the air gap diameter D.
The air length of the stator slot is derived as:
⎧
⎛θ⎞ ⎛ π θ⎞
sin ⎜ ⎟sin ⎜
− ⎟
⎪
π
π
D
⎝ 2 ⎠ ⎝ 72 2 ⎠
⎪
,0<θ<
⎪ 4
π
π ⎞
36 .
⎛θ
δs = ⎨
sin
cos⎜ −
⎟
144
⎝ 2 144 ⎠
⎪
⎪
π
π
<θ<
⎪0,
36
12
⎩
(8)
Similarly, the air length of the rotor can be expressed as:
Fig. 2 − The air length of the magnetic path in different positions.
⎧
⎛θ⎞ ⎛ π θ⎞
⎪ π sin ⎜ ⎟sin ⎜ − ⎟
D
⎝ 2 ⎠ ⎝ 36 2 ⎠ , 0 < θ < π
⎪
⎪ 4
π
18 .
⎛θ π ⎞
δr = ⎨
sin cos⎜ − ⎟
72
2
72
⎝
⎠
⎪
⎪
π
π
<θ<
⎪0,
18
9
⎩
Figure 3 shows the air length of the stator slot and the
rotor slot on the basis of (8) and (9).
Because the pole arc of the stator pole is larger than the
rotor pole arc, and the stator slot is narrower than the rotor
slots, the maximum air length of the stator is smaller than
the rotor’s. Similarly, if the stator and the rotor slot is
narrow, the air length is small and the leakage magnetic
flux is large.
Because the air path length shown in Fig. 3 includes the
pole part and the slot part, in the areas I, II, and III, the
equivalent air length g e (θ) can be written as:
Fig. 3 − The equivalent air length of the stator and the rotor.
In area I:
g e (θ) = δ .
(2)
In area II:
g e (θ) = δ + δs or g e (θ) = δ + δ r .
(3)
In area III:
g e (θ) = δ + δ s + δ r ,
(4)
where δ is the air gap length, δs is the air length of the
stator slot, and δ r is the air length of the rotor.
When the point being calculated is located under the
stator poles, δs = 0. And when the point being calculated is
located under the stator slots, as shown in point A, the air
length of the magnetic path can be calculated as the parallel
magnetic path of δ s1 and δ s2 . It can be calculated as:
δs =
δ s1δ s2
.
δ s1 + δ s2
(9)
(5)
g e (θ) = δ + δs + δ r .
(10)
3.2. LINEAR INDUCTANCE MODEL
For a typical rotor-inside doubly salient machine, the
phase inductances increase with their pole overlap angle
β s and β r are shown in (11).
- β r + βs
β -β
⎧
≤θ< r s
⎪βs ,
2
2
⎪
β
β
β
β
β
+
β
⎪β + r s - θ,
r
s ≤θ< r
s
⎪⎪ s
2
2
2
, (11)
β=⎨
β r + βs
2π β r + βs
⎪0,
≤θ<
Nr
2
2
⎪
⎪
2π β r - βs
2π β r + βs
2π β r - βs
⎪βs +
+ θ,
≤θ<
⎪⎩
Nr
Nr
Nr
2
2
2
where βs and β r are the stator and rotor pole arcs, as
shown in Fig. 4.
Liwei Shi, Bo Zhou
10
When the rotor and stator poles have no overlap, the
phase inductance is equal to a minimum Lmin . And when the
poles are aligned, the phase inductance take the maximum
value Lmax . So a linear inductance model of phase A La′
can be described as:
β=0
⎧ Lmin ,
⎪
⎪⎪
La′ = ⎨ Lmax ,
β = βmax
⎪
− Lmin
L
⎪ Lmax + max
β, else
⎪⎩
βs
.
μ 0 β s Dl e N p2
2δ
(12)
(19)
where H (θ) is the magnetic field strength, F (θ) is the
MMF of the field windings.
To deduce F (θ) , the stator, the rotor and the coils are
peak value is N f if . So the MMF of the field windings can
be described as (20), where if is the field coil current.
(13)
,
μ F (θ)
g e (θ) = μ 0 H (θ) = 0
,
g e (θ)
shown in Fig. 5. The field windings is wound around
3 stator poles, so the period of the MMF is π 2 and the
The maximum phase inductance Lmax can be calculated as:
Lmax =
3
where μ 0 is the permittivity of the air, l e is the effective
length of the machine, N p is the turn number of the phase
windings.
When the rotor pole is in line with the stator pole, the
minimum phase inductance Lmin can be calculated as the
summation of the stator leakage inductance Lsmin and the
⎧ 60 N f if
⎪ π θ,
⎪
⎪N i ,
⎪ ff
⎪
π
⎪ 60 N f if
(θ − ),
F (θ) = ⎨−
π
5
⎪
⎪
⎪− N f if ,
⎪
2π
⎪ 60 N f if
⎪⎩ π (θ − 5 ),
π
60
π
11π
<θ≤
60
60
11π
13π
<θ≤
.
60
60
13π
23π
<θ≤
60
60
23π
2π
<θ≤
60
5
0<θ≤
(20)
rotor leakage inductance Lrmin .
Lmin = Lsmin + Lrmin .
(14)
For the machine in this paper:
24
8
π
36
0
∫
μ 0 Dle N p2
dθ ,
(15)
π
μ 0 Dl e N p2
24
36
Lsmin =
dθ .
8 0
2 g e (θ)
(16)
Lsmin =
2 g e (θ)
∫
The inductance between field winding and phase A
Lfa has the same trend with La . Because the field coils are
also wounded around the stator poles, the linear inductance
between field winding and phase A Lfa′ .
′ =
Lfa
Nf
La′ ,
Na
Fig. 4 − The pole overlap angle.
(17)
where N f and N a are the turn number of each field coil
and phase coil.
Since the field winding reluctance is a constant, the
linear inductance of the field winding can be given as
πD0
μ 0 nα s
l e N f2
(18)
24
,
Lf′ =
2g
where n is the average number of the stator poles that have
flux.
3.3. RADIAL AIR GAP FLUX DENSITY MODEL
Without the consideration of the saturation, the radial air
gap flux density model can be described as:
Fig. 5 − The magnetomotive force of the field winding.
4. SIMULATION AND EXPERIMENTAL
VERIFICATION OF THE ANALYTICAL MODEL
To verify the analytical models been proposed in this
paper, a 2D-FEA model of the 24-slot 18-pole DSEG was
built to analyze its static and transient magnetic field, as
shown in Fig. 6.
4
Analysis of a novel doubly salient brushless generator
Fig. 6a − The flux of the generator iron core.
11
The self inductance of the field winding is calculated
according to the analytical model in (18), and the result is
compared with 2D-FEA method in Fig. 7(b). The analytical
result has a slightly higher error, because the winding coil
at the position of 0 rad in Fig. 1(b) can not have the average
flux. But it confirms that the analytical model in (19) is
correct.
And the total reluctance and self inductance of the field
winding is almost constant, and the new machine will not
suffer from serious cogging torque ripples and voltage ripples.
If we establish a coordinate system as shown in Fig. 6(a),
the radial magnetic flux density of typical points can be
calculated with (10), (19) and (20), if we set the field
winding current as 4 A. These points have distinct cyclical
characteristics, and the radial flux density curve can be
calculated with curve fitting method, as shown in Fig. 8 (a)
and (b), when the flux of phase C and phase A have the
maximum value.
As the machine has 24 stator poles and 18 rotor poles,
the period of the magnetic flux density is π , and every four
poles have the same flux density waveform with the former
four poles. Compared with 2D-FEA results, most of the
analytical results are higher than the 2D-FEA results,
because the iron core MMF is ignored. Because the positive
magnetic flux should be equal to the negative magnetic flux
within a certain range, the flux density of the poles with
small overlap angle is much higher than the one with large
overlap angle.
Fig. 6b − The photograph of the machine.
Table 1 gives key parameters of the 24-slot 18-pole fourphase DSEG.
Table 1
Key parameters of the 24-slot 18-pole four-phase DSEG
Item
Stator pole number
Stator pole arc
Stator out diameter [mm]
Stator inner diameter [mm]
Value
24
0.667
210
150.6
Winding turns of each phase
72
Axial length [mm]
70
Item
Value
Rotor pole
18
Rotor pole arc
0.5
Rotor out diameter (mm) 150
Airgap (mm)
0.3
Winding turns of each
140
field coil
In Fig. 7(a), the 2D-FEA results and the analytical results
of the inductances between phase windings and field
windings are investigated. It shows that both of them have
the same trend. The analytical results are larger than the
2D-FEA results, because all of the phase flux and field flux
are passing through the iron core. Because the MMF of the
iron core is nonlinear, it is very hard to describe it with
mathematical models. So only the 2D-FEA method should
be applied to get an accurate result. Compared with the
traditional method to model the inductance of the reluctance
machine by fitting multi-linear after the 2D-FEA [11], the
inductance analytical models proposed in this paper are
time-saving and can reveal the nature characteristics.
(a)
(b)
Fig. 7 − a) The inductances between phase windings and field windings;
b) self inductance of the field winding.
Liwei Shi, Bo Zhou
12
5
5. CONCLUSION
(a)
(b)
Fig. 8 − The magnetic flux density in middle of airgap of the new DSEG:
a) rotor pole align with phase A; b) rotor pole align with phase C.
Overall, the results with the analytical model have the
same trend with the 2D-FEA model, and the analytical
model proposed in this paper can be used to analyze the
static characteristics of the WFDSM.
A prototype machine of 12/9-pole four-phase DSEG has
been developed with the same parameters shown in Table I,
as shown in Fig. 6 (b).
Figure 9 shows the 2D-FEA and the experiment voltage
waveform of the four-phase DSEG at the field current of
5 A, 3000 r/m. The experimental waveform is similar to a
square wave, and it agrees well with the simulated ones,
which verifies the reasonable structure and good output
performances of the novel four-phase WFDSM.
(a)
(b)
Fig. 9 − Voltage waveforms of the new WFDSG:
a) 2D-FEA, b) experiment.
1
The four-phase DSEG has a broad application prospects
in the aerospace, automotive industry, which needs high
reliability for the whole system. This paper presents the
inductance model and a nonlinear radial air gap flux
density model of the new DSEG.
It can be concluded from the models that the magnetic
circuit model is very helpful to find the most critical
structural parameters of the machine. The pole arc
coefficients and the pole numbers determine the inductance
waveform directly. The phase inductance will decrease if
we increase the pole number with the same airgap diameter,
because the magnetic flux leakage will increase. The FEA
and experiments of the WFDSG show that the proposed
models can calculate the radial air gap flux density and the
winding inductances.
The analytical model shows the key parameters which
affect the electromagnetic characteristics of the machine.
Therefore, this paper founds the design basis of the
multiphase DSEG and other reluctance machines.
ACKNOWLEDGEMENTS
This work was supported and funded by the National
Natural Science Foundation of China (51477075) and the
Provincial Natural Science Foundation (ZR2014JL035).
Received on May 15, 2015
REFERENCES
1. Z. Chen, H. Wang, Y. Yan, A doubly salient starter-generator with
two-section twisted-rotor structure for potential future aerospace
application, IEEE Trans. Ind. Electron., 59, 9, pp. 3588–3595, 2012.
2. Z. Zhang, Y. Tao, Y. Yan, Investigation of a new topology of hybrid
excitation doubly salient brushless DC generator, IEEE Trans. Ind.
Electron, 59, 6, pp. 2550–2556, 2012.
3. A. M. El-Refaie, Fault-tolerant permanent magnet machines: A review,
IET Electr. Power Appl., 5, 1, pp. 59–74, 2011.
4. Y.Wang, Z. Deng, A multi-tooth fault-tolerant flux-switching permanentmagnet machine with twisted-rotor, IEEE Trans. Magn., 48, 10,
pp. 2674–2684, 2012.
5. M. Cheng, W. Hua, J. Zhang, W. Zhao, Overview of stator-permanent
magnet brushless machines , IEEE Trans. Ind. Electron., 58 , 11 ,
pp. 5087–5101, 2011.
6. W. Zhao, M. Cheng, X. Zhu, W. Hua, X. Kong, Analysis of fault
tolerant performance of a doubly salient permanent magnet motor
drive using transient cosimulation method, IEEE Trans. Ind. Electron.,
55, 4, pp. 1739–1748, 2008.
7. Z. Zhu, Y. Pang, D. Howe, et al., Analysis of electromagnetic
performance of flux-switching per-manent-magnet machines by
nonlinear adaptive lumped parameter magnetic circuit model, IEEE
Trans. Magnetics, 41, 11, pp. 4277–4287, 2005.
8. L. Shi, B. Zhou, Comparative Study of a fault-tolerant multiphase
wound-field doubly salient machine for electrical actuators, Energies,
8, 4, pp. 3640–3660, 2012.
9. B. Gaussens, E. Hoang, O. dela Barrière, et al., Analytical Approach
for Air-Gap Modeling of Field-Excited Flux-Switching Machine: NoLoad Operation, IEEE Trans. on Magn. , 48, 9, pp. 2505–2517, 2012.
10. B. D. Vărăticeanu, P. Minciunescu, Modeling and analysis of dualsided coreless linear syschronous motor, Rev. Roum. Sci. Techn. –
Électrotechn. et Énerg., 60, 1, pp. 17–27, 2015.
11. N.-M. Modreanu, M.-I. Andrei, Mihaela Morega, T. Tudorache, Brushless
DC micro-motor with surface mounted permanent magnets, Rev.
Roum. Sci. Techn. – Électrotechn. et Énerg., 59, 3, pp. 237–247, 2014.