Electromagnetic Design of Aircraft Synchronous Generator
with High Power-Density
Thomas Wu and Tony Camarano
University of Central Florida, Orlando, FL 32816
Jon Zumberge and Mitch Wolff
Air Force Research Lab, Wright Patterson, OH 45431
Eric S. Lin
ANSYS Corp., Pittsburg, PA 15219
and
Hao Huang and Xiaochuan Jia
General Electric – Aviation Systems LLC, Vandalia, OH 45377
This paper discusses the methodology for the electromagnetic design of an aircraft
synchronous generator with high power-density. A new method is proposed to more
accurately model the air-gap of a salient pole rotor through expanding the inverse of an
effective air-gap function. The corresponding magnetic fields from the rotor and stator
windings, as well as the expressions of back EMF, are derived using the air-gap model. The
stator inner diameter and length are designed by considering a proper cooling scheme and
maximum peripheral-speed of the rotor. This allows for design of the stator winding and slot
geometry, including the derivation of a formula for the stator core thickness. The air-gap
and salient pole shoe face can be designed using the desired specifications for power factor
and torque angle. The rotor windings and geometry are subsequently designed. Following
the above procedure, a 200 KVA high power-density synchronous generator with 12 krpm
rotational velocity is obtained. Finally, the design is verified and finely tuned using ANSYS
RMxprt, Maxwell FEM software, and SimuLink.
Nomenclature
αCu
αg
Af
Bf,pk Ba,pk
Bg,pk
C
C0
Cf
Dr D
gav
Ia,rated
IF,rated
Jf Ja
Ka
kB
kv
kw
l
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
coefficient of copper
air-gap coefficient
cross-sectional area
field and phase peak magnetic flux density
air-gap magnetic flux density
number of parallel circuits in a phase winding
cooling coefficient
number of turns for a salient pole
rotor and stator bore diameter
average air-gap
phase winding current
field winding current
field and phase winding current density
cooling technology coefficient
magnetic flux density coefficient
field voltage margin
winding factor
stator length
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lf
Lmd Lmq
Lmf
Lsf
lturn
m
Nf Na
nm
θd
P
pemb
ρCu
Rf Rs
rlD
S
Srated
T0
VFmax
vr
Wt
A
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
total mean length of field winding
direct and quadrature magnetizing inductance
field magnetizing inductance
field-armature mutual inductance
average length of each field winding turn
number of phases
field and phase effective number of series turns
revolutions per minute
direct axis angle
number of poles
pole embrace
resistivity of copper
field and armature resistance
length-diameter ratio
number of slots
rated apparent power
reference temperature
maximum field voltage
maximum allowable peripheral-speed
pole shoe width
I. Introduction
N aircraft electrical system is responsible for the generation, control, and distribution of electrical power
within the aircraft. A typical system uses 115 VAC (400 Hz), 270VDC, and 28VDC1-4. Most contemporary
high power-density aircraft generators are designed to provide between 30 to 250 kW and operate at angular
speeds from 7200 to 27000 rpm. The typical topology of an aircraft generator is shown in Fig. 1. The three-phase
synchronous generator includes an outer stator with the windings distributed according to phase and an inner rotor
with compact DC windings. The field windings receive excitation from a synchronous brushless exciter with threephase windings on the rotor and concentrated windings on the stator. This is used in conjunction with a PM
brushless exciter. The synchronous generator, synchronous exciter, and PM exciter share the same rotor shaft.
The number of stator slots can range from 24 to 108, depending on the desired slots per pole per phase. In
general, a larger number of slots per pole per phase combined with a double layer lap winding structure will reduce
the effects of higher-order harmonics in magnetic flux density and air-gap MMF. A typical range for salient rotor
poles is from 2 to 12. The design analyzed in this paper is a 30 slot, 10 pole machine with a rated apparent power of
200 kVA operating at 12 krpm.
Figure 1. Aircraft generator topology.
The design methodology for the synchronous generator is discussed in Section 2. This includes general design
considerations, detailed descriptions of the armature and salient rotor winding and geometry design, and analytical
estimation for equivalent model inductances and resistances. Section 3 will explain the process of generating a
design solution from theory as well as implementation using RMxprt and Maxwell FEM simulation tools.
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Simulation results and post processing is discussed in Section 4, followed by a conclusion of the overall design
process in Section 5.
II. Design Methodology
A. General Considerations
One of the primary design parameters for machine design is the maximum allowable peripheral-speed of the
rotor. Modern steel-alloys have a rotor peripheral-speed design limit of about 50,000 ft/min (about 250 m/s). The
maximum rotor diameter Dr can be estimated using
D
r m ax

v (in len g th /s )
v (in len g th /m in )
r
r

1 .2 f 
1 .2 n (in rev /m in ) 
m
m
(1)
where vr is the maximum allowable peripheral-speed and nm is the rpm of the machine5. It is important to note that
Eq. (1) is an approximation and is used to provide simplified design guidance when choosing an appropriate
diameter for the rotor.
The resistivity of copper windings will vary with temperature. Machine working temperature varies depending
on application and should be taken into account. The resistivity of copper versus temperature can be calculated using
Cu (T )  Cu (T0 )   Cu (T  T0 )
(2)
where T0 is a reference temperature of 20°C, αCu is equal to 2.668e-9 Ω in/°C, and ρCu is equal to 0.679e-6 Ω in/°C at
20°C.
B. Stator Design
The number of armature slots per pole may either be integral or fractional. A m-phase synchronous machine will
have S slots that are multiples of mP, where P is the number of machine poles. However, integral S/P may lead to
excessive cogging torque because all pole faces will align with slot mouths simultaneously. A fractional S/P value is
generally used in order to reduce cogging torque. Although S/P is fractional, the number of slots should still be a
multiple of the number of phases.
A relationship between machine size and other machine parameters has been derived using rated phase voltage
and current. It can be shown that
 60 2  S rated
D 2l   2 
(3)
  k w  nm K a Bg , pk
D and l are the stator bore diameter and stator length, respectively. The winding factor kw is for the primary machine
harmonic and is derived using air-gap MMF analysis. The volume of the machine is proportional to Eq. (3) and the
following discussions are generally accurate. A larger Ka, which is a parameter for quality of cooling technology,
will allow for a smaller machine. The faster the machine speed nm, the smaller the volume. A larger gap magnetic
flux density Bg,pk can be obtained by using advanced materials with larger magnetic saturation; this will also
decrease volume. However, a larger rated apparent power Srated will increase the volume of the machine.
A similar approach can be seen in the relationship between machine size, apparent power, and number of poles.
The constant relating these parameters is
D 2l
(4)
C 
0
C0 
Srated P
1
1

2 2 f e m K a
(5)
C0 is dependent on the cooling technology and should be small in order to reduce machine volume5. The value of C0
for the synchronous generator design studied in this paper is 92 in3/MVA, which is for spray cooling. This number
was tuned through previous experience and knowledge of the aircraft synchronous generator cooling technology.
The length-diameter ratio of a machine is defined as the ratio of the length and the stator bore diameter, meaning
rlD 
l
D
(6)
The machine power rating depends on D2l for a fixed mechanical speed. As rlD increases, the rotor diameter
decreases, causing the moment-of-inertia to decrease. In this case, the rotor peripheral-speed will also decrease. As
rlD increases, the machine length increases and the rotor is prone to exhibit critical frequencies at lower speeds. This
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can result in shaft flexure, causing the rotor to strike the stator. If rlD is too large, the machine is difficult to cool.
However, if rlD is too small, the leakage inductance of end-turns can severely affect machine performance.
Armature conductor cross-sectional area is also dependent on machine cooling. It can be written as
Aa 
I a , rated / C
(7)
Ja
where Ia,rated is the rated current for one phase winding and C is the number of parallel circuits in the phase winding.
The current density values given machine cooling in Table 1 can be used for Ja6.
Table 1. Current density values dependent
on cooling technology.
Cooling Type
Enclosed Machine
Air Surface Cooling
Air Duct Cooling
Liquid Cooling
Spray Cooling
Ja (A/in2)
3000 ~ 3500
5000 ~ 6000
9000 ~ 10000
15000 ~ 20000
≥ 20000
The armature slot geometry and corresponding dimensions are shown in Fig 2. In general, the following ranges
yield a satisfactory design of the armature slot:
s 
D
S
,
0.4 s  bs  0.6 s ,
3bs  d s  7bs ,
ts   s  bs
The defined length dc can be shown, for a good design, to be
dc 
D
1 .6 P
(8)
Figure 2. Stator slot geometry.
C. Rotor Design
The number of field conductors is an important design consideration. Figure 3 defines the parameters used in the
design of pole geometry and windings. If the average length of each turn of the field winding is assumed to be
l turn  2l  Wt
(9)
then the total mean length of the field winding is approximately
l f  PC f l turn
(10)
The number of turns Cf are assumed to be the same for each salient pole. Assuming that VFmax is the maximum
voltage of the field winding, is can be shown that
 l
kV VF max  I F , rated Cu f  J f  Cu PC f lturn
Af
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(11)
where Af is the cross-sectional area of the field conductor, kv (0.7-1) provides a certain design margin, and Jf is the
allowable current density and depends on cooling. Therefore,
(12)
kV VF max
Cf 
J f  Cu Plturn
The calculated results will be rounded to an integer. Referring to Figure 4, the following approximations for the
respective geometry will provide a satisfactory design of the salient pole:
pemb

), W p  ( Dr  2 H t )  0.45 ~ 0.65  , Dra  (0.6 ~ 0.7) Dr
P
P
Dsh  (0.3 ~ 0.5) Dr ,
H tp  H t  H p  ( Dr  Dra ) / 2,
H t  (0.2 ~ 0.3) H tp
Wt  ( D  2 g max ) sin(
The pole embrace pemb for the design analyzed in this paper is 0.7.
Figure 3. Rotor salient pole geometry and diameters.
The phase diagram in Fig. 4 shows relationships between dq currents, field and phase winding flux linkage, and
dq reactance. The resistance of the phase windings in neglected in Fig. 4. The power factor of the load is lagging,
and
the
machine
is
over excited,
meaning
|EA|>|VΦ|.
Figure 4. Phasor diagram of the dq currents and voltages.
The relationship between the peak values of the field and phase magnetic flux densities is notated
kB 
B f , pk
Ba , pk
(13)
E A  k BVs
(14)
cos   k B sin 
(15)
X d Id
X
 tan(   )  d tan
X qIq
Xq
(16)
and it is assumed that
It can be shown using Fig. 4 that
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Typically a power factor is specified, which then defines Φ. Using Eq. (15) and Eq. (16) and the assumption
X q  (0.6 ~ 0.8) X d
(17)
δ and ζ can be obtained. An estimation of the effective air-gap across the salient pole has been derived as follows:
g 'eff ( d ) 
g av
P
1   g cos(2  d )
2
(18)
The gap coefficient is approximately
g  2
1  ( Lmq / Lmd )
Lmq / Lmd  0.4 ~ 0.8
1  ( Lmq / Lmd )
(19)
An approximation of the rated field current using field and phase magnetic flux densities can be written as
I F ,rated 
k B 1.5 2 Nˆ a 1  ( g / 2)   g cos 2( i   r ) I a ,rated
Nˆ (1  ( / 2))
f
(20)
g
The angle of the phase voltage is assumed to be zero in Eq. (20). The sign of Φi depends on whether the load is
leading or lagging. Na and Nf are the effective number of series turns in the field and phase windings, respectively.
From Eq. (20), an estimation of the field conductor cross-sectional area can be determined using
Af  I F ,rated / J f
(21)
Through a similar derivation for (20), an estimation of the average air-gap is found:
g av  (6 /  )( Nˆ a / P )(  0 / Bg , pk ) 1  ( g / 2)   g cos 2(i   r ) 2 I a ,rated  k B cos   sin  
(22)
A function of the gap versus θd is obtained by substituting Eq. (19) and Eq. (22) into Eq. (18).
D. Resistance/Inductance Estimation
Analytical estimations for armature and field resistances, magnetizing inductances, and mutual couplings are
shown below. These parameters are used for effective modeling of the machine for high-level system simulation.
The armature winding resistance is estimated as
(23)
R  ( N / C )(  l ) / A
s
a
Cu turn
a
where Na is the number of series turns per phase, C is the number of parallel circuits, and lturn is the estimated length
of a winding turn. The field winding resistance is estimated as
(24)
R  ( l ) / A
f
Cu f
f
The inductances can be estimated using the following relationships, derived using the dq frame:
LA 
80 Dl  Nˆ a

 g av  P
LB 
g
2



2
LA

3
3
Lmd  ( LA  LB )  LA (1  g )
2
2
2

3
3
Lmq  ( LA  LB )  LA (1  g )
2
2
2
Lmf
8 Dl  Nˆ f
 0

 g av  P
(25)
(26)
(27)
(28)
2

g
 (1  )

2

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(29)
Lsf 
III.
80 Dl  Nˆ a Nˆ f

 g av  P 2

g
 (1  )

2

(30)
Design and Simulation Results
The analytical design theory described above can be used to generate the specifications of geometry, windings,
source excitation, and effective resistance and inductance modeling for a high power-density aircraft synchronous
generator. The design analyzed here is a 200 kVA, 12 krpm, 3-phase machine. The number of poles and slots chosen
are 10 and 30, respectively. The operating temperature is set at 250°C, with a defined maximum field voltage of 50
V. From these parameters, an analytical design can be produced. The geometry and machine specifications are put
into ANSYS RMxprt modeling software to create an initial simulation design. The generator specifications are
shown in Table 2. Tables 3 and 4 contain the generator stator and rotor details, respectively. Figures 5, 6, and 7
show generator, stator slot, and rotor pole geometry, respectively. Table 5 shows the exciter specifications, followed
by exciter rotor and stator details in Tables 6 and 7, respectively. Figures 8, 9, and 10 show exciter, rotor slot, and
stator pole geometry, respectively.
Table 2. Designed generator parameters.
nm
fe
Srated
VΦ
IΦ
Twork
VFmax
IFmax
Sexciter
Ja
Jf
kB
kv
pf
Φ
δ
Rs
Rf
Lmd
Lmq
Lmf
Lsf
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
S
Nc
D
L
D0
kw
bs0
bs
ds0a
ds0b
ds1
gmin
Aslot
Aa
Acond
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
12000
1000
200
115.4339
577.5309
250
50
154.93
8.1546
20000
20000
1.5
0.7
0.95
-18.1949
33.0649
0.007788
0.22138
8.7912e-5
3.956e-5
0.00669
0.000626
rpm
Hz
kVA
VRMS
ARMS
°C
V
A
kVA
A/in2
A/in2
°
°
Ω
Ω
H
H
H
H
mechanical speed
electrical frequency
apparent power
phase voltage
phase current
work temperature
max field voltage
max field current
exciter apparent power
armature current density
field current density
flux density coefficient
field design margin
power factor
power angle
torque angle
armature winding resistance
field winding resistance
d magnetizing inductance
q magnetizing inductance
field magnetizing inductance
armature-field mutual inductance
Figure 5. Generator geometry.
Table 3. Designed generator stator.
30
1
6.3872
4.471
8.0882
0.82699
0.05
0.26755
0.025
0.025
0.40132
0.02079
0.10737
0.02888
0.05775
in
in
in
in
in
in
in
in
in
in2
in2
in2
slots
turns per coil
stator bore diameter
stator length
stator core diameter
winding factor
stator slot mouth width
stator slot width
stator slot mouth depth
stator slot shoulder depth
stator slot depth
minimum air-gap
area of stator slot
bare area of each coil
total area of bare coils per slot
Figure 6. Generator stator slot geometry.
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Table 4. Designed generator rotor.
=
=
=
=
=
=
=
=
=
=
=
=
=
=
P
pemb
Cf
Dr
Dra
Dsh
vr
Wt
Ht
Wp
Hp
AslotR
Af
AcondR
10
0.7
10
6.3456
4.2516
2.5383
19935
1.3643
0.3141
0.8083
0.7329
0.2100
0.00775
0.07747
in
in
in
ft/min
in
in
in
in
in2
in2
in2
poles
pole embrace
turns per rotor pole
rotor core diameter
rotor diameter at pole bottom
shaft diameter
rotor peripheral speed
pole shoe width
pole shoe depth
pole leg width
pole leg length
area of half rotor slot
bare area of field conductor
total area of bare conductors
Table 5. Designed exciter parameters.
nm
fe
Srated
VΦ
IΦ
Twork
VFmax
IFmax
SPMexciter
Ja
Jf
kB
kv
pf
Φ
δ
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
12000
600
8.5
21.3767
132.5433
250
5
301.1639
1.5851
20000
20000
1.7
0.7
0.95
-18.1949
28.2937
rpm
Hz
kVA
VRMS
ARMS
°C
V
A
kVA
A/in2
A/in2
°
°
mechanical speed
electrical frequency
apparent power
phase voltage
phase current
work temperature
max field voltage
max field current
PM generator apparent power
armature current density
field current density
flux density coefficient
field design margin
power factor
power angle
torque angle
Figure 7. Generator rotor pole geometry.
Figure 8. Exciter geometry.
Table 6. Designed exciter rotor (armature).
S
Nc
D
L
vr
Dsh
kw
ba0
ba
da0a
da
gmin
Aslot
Aa
Acond
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
21
3
2.5008
0.75024
7856
1.2812
0.96299
0.05
0.14965
0.025
0.2993
0.0242
0.044789
0.006627
0.039763
in
in
ft/min
in
in
in
in
in
in
in2
in2
in2
slots
turns per coil
Rotor diameter
rotor length
rotor peripheral speed
shaft diameter
winding factor
armature slot mouth width
armature slot width
armature slot mouth depth
armature slot depth
minimum air-gap
area of armature slot
bare area of each coil
total area of bare coils per slot
Figure 9. Exciter rotor slot geometry.
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Table 7. Designed exciter stator (field).
P
pemb
Cf
Ds
Dsa
D0
Wt
Ht
Wp
Hp
AslotF
Af
AcondF
=
=
=
=
=
=
=
=
=
=
=
=
=
6
0.75
5
2.5492
3.4414
4.2253
1.0272
0.1115
0.6532
0.3346
0.2231
0.01506
0.07529
in
in
in
in
in
in
in
in2
in2
in2
poles
pole embrace
turns per rotor pole
Stator bore diameter
Stator diameter at pole bottom
Stator core diameter
pole shoe width
pole shoe depth
pole leg width
pole leg length
area of half field slot
bare area of field conductor
total area of bare conductors
Figure 10. Exciter stator pole geometry.
RMxprt is used to verify the dq inductances, armature and field resistances, and rated apparent power of the
generator. Some fine tuning for conductor cross-sectional area is often necessary to match the analytical results to
the software specifications. The design has some freedom in specifications of the stator slot mouth. Any change of
this geometry will affect the dq inductances. The inductances from the analytical result are used as a reference to
iterate on the geometry until a satisfactory implementation is obtained. Once the design has been verified, it is
transferred directly into ANSYS Maxwell 2D and 3D FEM software. Figures 11 and 12 show the 2D and 3D FEM
models, respectively. The 2D simulation uses only the portion of the model with unique winding structure and
assumes the rest of the machine has symmetry.
Figure 11. Maxwell 2D FEM model.
Figure 12. Maxwell 3D FEM model.
In order to verify the analytical design, FEM analysis and a flux linkage method for calculating self and mutual
inductance are used. The flux linkages can be found using a sweep of the field and phase currents in the Maxwell
software. Since the machine operates within the magnetic saturation region, the flux linkages and inductances will
vary with current. A numerical function for the dq inductances versus current is built and compared with the
analytical derivations. The analytical values should be approximately equal to the peak inductances versus current.
Table 8 shows the numerical results compared with the analytical derivations. Figure 13 shows the open circuit 3phase voltage waveform for the Maxwell models.
Table 8. Analytical compared with
numerical results. Units are Henry.
Inductance Analytical Numerical
Lmd
Lmq
Lmf
Lsf
8.7912e-5
3.956e-5
0.00669
0.000626
8.8128e-5
4.2603e-5
0.00622
0.0005412
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Figure 13. Numerical results for phase voltages.
Induced phase voltages for an open circuit 200kVA, 12
krpm 3-phase synchronous generator. Field current
excitation: 60 VDC.
SimuLink is employed to perform high-level simulations of the generator when interfaced with control and load7.
Figure 14 shows the upper-level block diagram of the generator-exciter synchronous machine attached to a 200 kW
load. The detailed SimuLink model for the machine is shown in Fig. 15. This model includes the synchronous
generator, exciter, controller, and rectifiers. The main generator output voltage and current versus time is shown in
Fig. 16, respectively.
Figure 14. High-level SimuLink model.
Figure 15. 3-Phase synchronous generator-exciter model. Main blocks from left to right: controller, exciter,
exciter rectifier, main generator, main rectifier.
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Figure 16. Output voltage and current.
IV.
Conclusion
An analytical design methodology has been developed which creates a relatively accurate design for high powerdensity aircraft synchronous generators. Important design parameters such as rated apparent power, mechanical
speed, machine poles, and stator slots can be specified for a 3-phase generator. The equations and design process
described in this paper produce a guideline for the machine geometry, winding parameters, and source excitation.
FEM simulation and post processing verify the design method using a numerical flux linkage method to confirm
machine inductances. The design methodology can be used to develop new and innovative high power-density
synchronous generators. The performance of these designs can be simulated, verified, and optimized to fabricate
high-quality, reliable machines.
References
1
J. F. Gieras, Advancements in Electric Machines, Springer, 2008, Chap. 4.
P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of Electric Machinery and Drive Systems, 2nd Edition, Wiley,
2002.
3
C.M. Ong, Dynamic Simulation of Electric Machinery, Prentice Hall, 1998.
4
A.E. Fitzgerald, C. Kingsley, Jr., and S. D. Umans, Electric Machinery, 6th Edition, page 270, McGraw-Hill, 2003.
5
J. J. Cathey, Electric Machines: Analysis and Design Applying MatLab, McGraw Hill, 2001, pp. 477.
6
T. A. Lipo, Introduction to AC Machine Design, Wisconsin Power Electronics Research Center, University of Wisconsin,
2007, pp. 356-358.
7
Jie Chen, Thomas Wu, Jay Vaidya and “Nonlinear Electrical Simulation of High-Power Synchronous Generator System,”
2006 SAE Power Systems Conference.
2
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