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American Institute of Aeronautics and Astronautics

2

l f = total mean length of field winding

Lmd Lmq = direct and quadrature magnetizing inductance

Lmf = field magnetizing inductance

L sf = field-armature mutual inductance

l turn = average length of each field winding turnm = number of phases

N f N

a = field and phase effective number of series turns

nm = revolutions per minuteθ d = direct axis angle

P = number of poles

pemb = pole embraceρCu = resistivity of copper

R f R s = field and armature resistance

r lD = length-diameter ratio

S = number of slots

S rated = rated apparent powerT 0 = reference temperature

V Fmax = maximum field voltage

vr = maximum allowable peripheral-speedW t = pole shoe width

I. Introduction N aircraft electrical system is responsible for the generation, control, and distribution of electrical powerwithin 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 rotorwith compact DC windings. The field windings receive excitation from a synchronous brushless exciter with three-

phase 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. Ingeneral, 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 of200 kVA operating at 12 krpm.

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 adesign solution from theory as well as implementation using RMxprt and Maxwell FEM simulation tools.

A

Figure 1. Aircraft generator topology.




3

Simulation results and post processing is discussed in Section 4, followed by a conclusion of the overall design

process in Section 5.

II. Design MethodologyA. 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). Themaximum rotor diameter Dr can be estimated using

(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

(2)

where T 0 is a reference temperature of 20°C, αCu is equal to 2.668e-9 in/°C, and ρCu is equal to 0.679e-6 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 toexcessive 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

(3)

D and l are the stator bore diameter and stator length, respectively. The winding factor k w 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 thefollowing discussions are generally accurate. A larger K a, 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 B g,pk can be obtained by using advanced materials with larger magnetic saturation; this will alsodecrease volume. However, a larger rated apparent power S rated 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(4)

(5)

C 0 is dependent on the cooling technology and should be small in order to reduce machine volume5. The value of C 0

for the synchronous generator design studied in this paper is 92 in3

/MVA, which is for spray cooling. This numberwas 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

(6)

The machine power rating depends on D2l for a fixed mechanical speed. As r lD increases, the rotor diameter

decreases, causing the moment-of-inertia to decrease. In this case, the rotor peripheral-speed will also decrease. As

r lD increases, the machine length increases and the rotor is prone to exhibit critical frequencies at lower speeds. This

0 0( ) ( ) ( )Cu Cu CuT T T T

2

2

,

60 2rated

w m a g pk

S D l

k n K B

0 2

1 1

2 e m a

C K

2

0

rated

D l C

S P

lD

l r

D

(in len gth /s) (in leng th /m in )

m ax 1.2 1.2 (in rev/m in)

v vr r D

r f nm m




4

can result in shaft flexure, causing the rotor to strike the stator. If r lD is too large, the machine is difficult to cool.

However, if r lD 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

(7)

where I a,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 J a

6.

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:

The defined length d c can be shown, for a good design, to be

(8)

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

(9)

then the total mean length of the field winding is approximately

(10)

The number of turns C f are assumed to be the same for each salient pole. Assuming that V Fmax is the maximum

voltage of the field winding, is can be shown that

, /a rated

a

a

I C A

J

, 0.4 0.6 , 3 7 , s s s s s s s s s s

Db b d b t b

S

P

Dd c

6.1

t turn W l l 2

turn f f l PC l

turn f Cu f

f

f Cu

rated F F V l PC J A

l I V k

,max

Table 1. Current density values dependent

on cooling technology.

Cooling Type J a (A/in2)

Enclosed Machine 3000 ~ 3500

Air Surface Cooling 5000 ~ 6000

Air Duct Cooling 9000 ~ 10000

Liquid Cooling 15000 ~ 20000

Spray Cooling ≥ 20000

Figure 2. Stator slot geometry.




5

(11)

where A f is the cross-sectional area of the field conductor, k v (0.7-1) provides a certain design margin, and J f is the

allowable current density and depends on cooling. Therefore,

(12)

The calculated results will be rounded to an integer. Referring to Figure 4, the following approximations for therespective geometry will provide a satisfactory design of the salient pole:

The pole embrace pemb for the design analyzed in this paper is 0.7.

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

| E A|>|VΦ|.

The relationship between the peak values of the field and phase magnetic flux densities is notated

(13)

and it is assumed that

(14)

It can be shown using Fig. 4 that(15)

(16)

turnCu f

F V f

Pl J

V k C

max

pk a

pk f

B

B

Bk

,

,

s B A V k E

sincos Bk

tan)tan(q

d

qq

d d

X

X

I X

I X

Figure 3. Rotor salient pole geometry and diameters.

Figure 4. Phasor diagram of the dq currents and voltages.

embmax

( 2 )sin( ), ( 2 ) 0.45 ~ 0.65 , (0.6 ~ 0.7)

(0.3 ~ 0.5) , ( ) / 2, (0.2 ~ 0.3)

t p r t ra r

sh r tp t p r ra t tp

pW D g W D H D D

P P

D D H H H D D H H




6

Typically a power factor is specified, which then defines Φ. Using Eq. (15) and Eq. (16) and the assumption

(17)

δ and ζ can be obtained. An estimation of the effective air-gap across the salient pole has been derived as follows:

(18)

The gap coefficient is approximately

(19)

An approximation of the rated field current using field and phase magnetic flux densities can be written as

(20)

The angle of the phase voltage is assumed to be zero in Eq. (20). The sign of Φi depends on whether the load isleading or lagging. N a and N f 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

(21)Through a similar derivation for (20), an estimation of the average air-gap is found:

(22)

A function of the gap versus θ d 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 areshown below. These parameters are used for effective modeling of the machine for high-level system simulation.

The armature winding resistance is estimated as

(23)

where N a is the number of series turns per phase, C is the number of parallel circuits, and l turn is the estimated length

of a winding turn. The field winding resistance is estimated as

(24)

The inductances can be estimated using the following relationships, derived using the dq frame:

(25)

(26)

(27)

(28)

(29)

d q X X )8.0~6.0(

)2

2cos(1)('

d g

avd eff P

g g

1 ( / )2

1 ( / )

mq md

g

mq md

L L

L L

/ 0.4 ~ 0.8mq md L L

,

,

ˆ1.5 2 1 ( / 2) cos 2( )

ˆ (1 ( / 2))

B a g g i r a rated

F rated

f g

k N I I

N

, / f F rated f I J

0 , ,ˆ(6 / )( / )( / ) 1 ( / 2) cos 2( ) 2 cos sinav a g pk g g i r a rated B g N P B I k

( / )( ) / s a Cu turn a R N C l A

( ) / f Cu f f R l A

2

0ˆ8 a

A

av

N Dl L

g P

3 3

( ) (1 )2 2 2

g

md A B A L L L L

2

0ˆ8

(1 )2

f g

mf

av

N Dl L

g P

3 3( ) (1 )

2 2 2

g

mq A B A L L L L

2

g

B A L L




7

(30)

III. Design and Simulation ResultsThe 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 synchronousgenerator. 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 putinto 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, andstator pole geometry, respectively.

Table 2. Designed generator parameters.nm = 12000 rpm mechanical speed

f e = 1000 Hz electrical frequencyS rated = 200 kVA apparent power

V Φ = 115.4339 VRMS phase voltage

I Φ = 577.5309 ARMS phase current

T work = 250 °C work temperature

V Fmax = 50 V max field voltage

I Fmax = 154.93 A max field current

S exciter = 8.1546 kVA exciter apparent power

J a = 20000 A/in2 armature current density

J = 20000 A/in2 field current density

k B = 1.5 flux density coefficient

k v = 0.7 field design margin

pf = 0.95 power factor

Φ = -18.1949 ° power angle

δ = 33.0649 ° torque angle

R s = 0.007788 Ω armature winding resistance

R f = 0.22138 Ω field winding resistance

Lmd = 8.7912e-5 H d magnetizing inductance

Lmq = 3.956e-5 H q magnetizing inductance

Lmf = 0.00669 H field magnetizing inductance

L sf = 0.000626 H armature-field mutual inductance

Table 3. Designed generator stator.

S = 30 slots

N c = 1 turns per coil

D = 6.3872 in stator bore diameter

L = 4.471 in stator length

D0 = 8.0882 in stator core diameter

k w = 0.82699 winding factor

b s0 = 0.05 in stator slot mouth width

b s = 0.26755 in stator slot width

d s0a = 0.025 in stator slot mouth depth

d s0b = 0.025 in stator slot shoulder depth

d s1 = 0.40132 in stator slot depth

g min = 0.02079 in minimum air-gap

A slot = 0.10737 in2area of stator slot

Aa = 0.02888 in2 bare area of each coil

Acond = 0.05775 in2total area of bare coils per slot

Figure 5. Generator geometry.

Figure 6. Generator stator slot geometry.

0

2

ˆ ˆ8(1 )

2

a f g

sf

av

N N Dl L

g P






9

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 tothe software specifications. The design has some freedom in specifications of the stator slot mouth. Any change ofthis 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.

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 willvary 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 3-

phase voltage waveform for the Maxwell models.

Figure 11. Maxwell 2D FEM model. Figure 12. Maxwell 3D FEM model.

Table 8. Analytical compared with

numerical results. Units are Henry.

Inductance Analytical Numerical

Lmd 8.7912e-5 8.8128e-5

Lmq 3.956e-5 4.2603e-5

Lmf 0.00669 0.00622

L sf 0.000626 0.0005412

Figure 10. Exciter stator pole geometry.

Table 7. Designed exciter stator (field). P = 6 poles

pemb = 0.75 pole embrace

C = 5 turns per rotor pole

D s = 2.5492 in Stator bore diameter

D sa = 3.4414 in Stator diameter at pole bottom

D0 = 4.2253 in Stator core diameter

W t = 1.0272 in pole shoe width

H t = 0.1115 in pole shoe depth

W p = 0.6532 in pole leg width

H p = 0.3346 in pole leg length

A slotF = 0.2231 in2 area of half field slot

A = 0.01506 in2 bare area of field conductor

cond = 0.07529 in2 total area of bare conductors




10

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 synchronousgenerator, exciter, controller, and rectifiers. The main generator output voltage and current versus time is shown in

Fig. 16, respectively.

Figure 13. Numerical results for phase voltages.

Induced phase voltages for an open circuit 200kVA, 12krpm 3-phase synchronous generator. Field current

excitation: 60 VDC.

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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IV. ConclusionAn analytical design methodology has been developed which creates a relatively accurate design for high power-

density 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

1J. F. Gieras, Advancements in Electric Machines, Springer, 2008, Chap. 4.2 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, 2003.5J. J. Cathey, Electric Machines: Analysis and Design Applying MatLab, McGraw Hill, 2001, pp. 477.6T. 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.

Figure 16. Output voltage and current.

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Ansys Max Well