Design of Synchronous
Generator with Round Rotor
Part 1 – General Considerations
Synchronous Generator System
Rotor Peripheral Speed
The maximum allowable peripheral speed of the rotor is a central consideration in
machine design. With present-day steel alloys, rotor peripheral speeds of 50,000
ft/min (or about 250 m/s) represent the design limit.
1 ft/min = 0.0051 m/s
Resistivity of Copper vs Temperature
The resistivity of copper versus temperature can be calculated
using the following formula:
 Cu (T )   Cu (T0 )   Cu (T  T0 )
where T0 is a reference temperature and
give by
 Cu is a constant
 Cu  2.668  10 9   in/ o C
or:
 Cu  6.775  10 11   m/ o C
At T0  20 o C , we have
 Cu (T0 )  0.679  10 6   in/ o C
or:
 Cu (T0 )  1.724  10 8   m/ o C
Part 2 – Armature Design
Number of Armature Slots
For a m-phase synchronous machine, the number of armature slots (S)
must be multiples of m. This will guarantee all the phases are balanced.
A. Integral S/P
S is multiples of mP.
Example: For a 2 pole, 3 phase machine, the number of slots
can be 6,12,18,24,30,36… For a 4 pole, 3 phase machine, the
number of slots can be 12, 24,36,48,60…
Integral S/P may cause extensive cogging or detent torque since
all pole faces will line up with slot openings at the same time.
Cogging torque: torque from the interactions between rotor poles
and stator teeth. Use slot skew or fractional S/P can reduce it.
B. Fractional S/P
S/P takes a fractional number.
Number of Turns per Coil
V , rated 
where
2 f e Nˆ a  g , pk  4 .44 f e Nˆ a  g , pk
g , pk 
2Bg , pk Dl
P
Nˆ a  kw Na / 1.1 kw  k p kd ks
Na  PqNc / C
Na is the number of series turns per phase of armature winding
C is the number of parallel circuits of armature winding
Nc 
1 .1V , rated C
2 2 f e qk w B g , pk Dl
To consider leakage flux
Number of Conductor Positions per Slot on Stator
Cs  2N ca
for double layer winding.
In the above expression, Cs includes hollow conductor positions
for cooling (about 25%) and additional 15% - 25% of both height
and width tolerance of conductors (for insulation, slot liner, etc.)
in factor a. a can take about 1.6 – 2.
Cs 
1.1aV , rated C
2 f e qk w B g , pk Dl
Maximum and Average Flux Density
Average flux per pole:
 g , av



0
 g , pk sin  ae d  ae


 g , pk
2
 g , pk

0
Average flux density per pole:
B g , av 
 g , av
 Dl / P

 DlB g , av
2  g , pk P
 2 Dl
2
or:
 g , pk 
2P
Specific magnetic loading
g , pk 
Since
B g , av 
4

2
B g , pk
2Bg , pk Dl
P
 0 .4 B g , pk
Typically, take B g , av  0 . 6 T or B g , pk  1 . 5T in design.

Machine Size (1)
Specific electric loading:
rms current per unit length of the armature circumference
m (2 N a C )( I A,rated / C ) 2 mN a I A,rated

Ka 
D
D
Na is the number of series turns per phase of armature winding
C is the number of parallel circuits of armature winding
 I A , rated 
 DK a
2 mN a
Machine Size (2)
Apparent power
S rated  mV , rated I A , rated
2 f e Nˆ a  g , pk
V , rated 
 S rated
 g , pk 
2Bg , pk Dl
P
2 B pk Dl  DK
ˆ
 m 2 f e N a
P
2 mN
 S rated 
a
S rated
2
2 2


k w K a Bg , pk 
m
2
( D l ) nm
120
60
proportional to power density
2mN a
Nˆ a  k w N a
a
D 2l
2  f e B g , pk k w K a
P
2
I A,rated 
 DK a
nm P
fe 
120
where  m  k w K a
Bg , pk
2
Defined as: magnetic shear stress
Machine Size (3)
 60 2  S rated
 D l   2 
  k w  nm K a Bg , pk
2
 Volume of Machine
Discussions:
 The more advanced cooling technology (larger Ka), the
smaller the volume.
 The larger the rated apparent power Srated, the larger the
volume.
 The faster the machine speed nm , the smaller the volume.
 The larger the gap magnetic field Bg,pk (through using
advanced materials with larger magnetic saturation, etc.),
the smaller the volume.
Generator Size - Experience
1
1
D 2l
depends on cooling
 C0 , C0  2

2 f e m K a
S rated P
C0  1400 in 3 /MVA (air - cooled)
C0  700
in 3 /MVA (hydrogen- cooled)
C0  375
in 3 /MVA (liquid - cooled)
fe = 60 Hz
common steel
Length/Diameter Ratio (1)
The length/diameter ratio of a machine is defined as the ratio of the
length and the stator bore diameter:
rlD
Discussions:
l

D
For fixed mechanical speed, the machine power rating depends on D2l.
•As the l/D increases, the rotor diameter decreases and thus the
moment of inertia decreases. Also the rotor peripheral speed decreases.
•As the l/D increases, the machine length increases and the rotor is prone to
exhibit critical frequencies at lower speeds that can result in shaft flexure to
the point that the rotor strikes the stator bore.
• If the l/D is too large, it is difficult to cool.
• If the l/D is too small, the leakage inductance of end turns can severely
affects machine performance.
Length/Diameter Ratio (2)
Some people use aspect ratio, which is defined as the ratio of the length
and the pole pitch:
rasp 
l
where
P
P 
D
P
We can find the relationship between aspect ratio and length/diameter
ratio:
rasp 
l
P

l P
P
 rlD
D 

Stator Core Diameter
Stator Core diameter (outer diameter) D0
Two-pole: D0  2.1D
Four-pole: D0  1.7 D
Stator Slot Design
s 
D
S
0.4 s  bs  0.6 s
3bs  d s  7bs
t s   s  bs
 Use 65 V/mil ground insulation
 Use 0.375-in slot wedge
 Use 0.125-in coil separator and top stick
Stator Conductor Size
I A,rated / C
Js 
aAa
where Aa is stator (armature) conductor cross section
area and can be determined from the above formula
together with:
Air-cooled: J s  2500 A rms /in 2
Hydrogen-cooled: J s  4000 A rms /in 2
Water-cooled: J s  7000 A rms /in 2
1 A/in2 = 0.00155 A/mm2
J. J. Cathey, Electric Machines: Analysis and Design Applying MatLab, pp. 482,
McGraw Hill, 2001.
Round Wire Structure
This figure shows the wire structure, including the bare conductor,
an insulation layer and an optional bonding layer.
dwb: bare conductor diameter
dwc: covered wire diameter
Awb: bare conductor cross section area
Awc: covered conductor cross section area
American Wire Gauge (AWG)
d wb  8.251463   0.8905257 
G
Or
 d wb 
log 

8.251463 

G
log  0.8905257 
G: wire gauge (typically an integer)
dwb: bare wire diameter in mm.
Relative Wire Resistance versus Wire Gauge
 Increasing wire gauge by 1, increases copper loss by 26%.
 Decreasing wire gauge by 1, decreases copper loss to about 79% of
the previous gauge.
Current Capacity versus Wire Gauge
The maximum allowable current density varies roughly between 1 Arms/mm2 to
10 Arms/mm2. In confined volumes, the lower limit 1Arms/mm2 may be too
high. Similarly, with active cooling, the upper limit 10 Arms/mm2 may be too
conservative.
Round Wire with Film Insulation (1)
Round Wire with Film Insulation (2)
Square Wire with Film Insulation
Air Gap Size
From
Ba , pk
4 0 Nˆ a

1.5 2 I A,rated
 g eff P
g eff  k c g
An empirical formula for effective air gap size:
g eff
4 0 Nˆ a

1.5 2 I A,rated
 Ba , pk P
g  g eff / k c
The actual air gap size can be further tuned using an electromagnetic
simulation software.
Part 3 – Round Rotor Design
for Generator with Field
Winding
Number of Poles
For round rotor machine with field winding,
take P = 2 or 4.
Phasor Diagram
EA
jX S I A

E A  V  jX s I A

V

IA
Pick up torque angle  (T full load =Tmaxsin ) and power factor pf  cos  .
From X s I A cos   E A sin 
 E A  K B X s I A , K B  cos  / sin 
Example: If  =30 o , pf=0.85 lagging, K B  1.7.
Steady
E A  K B X s I A  B Steady

K
B
f , pk
B a , pk
V  E A cos   X s I A sin   ( K B cos   sin  ) X s I A

Ba , pk
B g , pk

X sIA
1

V
K B cos   sin 
Rotor Slot Selection (1)
total number of slots on rotor N r  2nr P
nr is integer
N
slots on rotor per pole half : n r  r
2P
D r
rotor pole pitch :  r 
P
pole width : 0.2 r  W f  0.3 r
Angular slot pitch (in elec. radian) :
Dr  PW f
1
P Dr  PW f
2 

2n r P
( Dr / 2) 2
2 Dr n r
 2 Dr
Arc length between two adjacentslots : t 
P
rotor slot width : 0.4t   b f  0.5t  rotor toot h width : t f  t   b f
Number of Conductors in Rotor (1)
The length of the ith field coil:L fi  2(l  W f  i
 2 Dr
P
)
Assume the number of conductors (Cf ) are the same in the each slot
Total length of the field winding:
Note: use single layer concentric winding on rotor.
nr
 2 Dr
i 1
P
LF  C f P  2(l  W f  i
)
Number of Conductors in Rotor (2)
LF  C f X f
nr
 2 Dr
i 1
P
where X f  P  2(l  W f  i
0.7VF max  I F ,rated
)  2nr P(l  W f )   2 Dr nr (nr  1)
 LF
 J f C f X f
Af
where Af is the cross section area of the field conductor, J f is the allowable
current density, and  is the resistivity of copper at working temperature.
0.7VF max
Cf 
J f X f
The calculated results will be rounded to an integer. Jf depends
on rotor cooling. See next slide
Rotor Cooling
1 A/in2 = 0.00155 A/mm2
Rotor Rated Field Current and Slot Size
Empirical design requires maximum magnetic field from field
winding is about KB times maximum magnetic field from armature winding:
Steady
B Steady

K
B
f , pk
B a , pk
4 0 Nˆ a
Steady
1.5 2 I A,rated
Ba , pk 
 g eff P
4 0  kwf N f 
Steady
B f , pk 
I
 geff  P  f ,rated
I f ,rated
K B  1.5  2 Nˆ a I A,rated 2.12 K B Nˆ a I A,rated


kwf N f
kwf N f
where Nf is total number of series turns in field winding:
kwf is rotor winding factor given is next page.
rotor slot cross section area Af :
Af 
I f ,rated
Jf
N f  PC f nr
Round Rotor Winding Factor
This is the case when sr is even.
sr  2nr
nr
kwf 
N cos[(2  1)


1
r
nr
N


1
If N1  N 2   N nr
nr
kwf 
cos[(2  1)


1
nr
r
/ 2]
/ 2]
Comprehensive Design Example
Design a 3 phase turboalternator with the following specifications:
500 MVA Y connected 24 kV (terminal voltage) 60 Hz
3600 rpm 2 pole 0.85 pf lagging
Maximum allowable rotor peripheral speed 50,000 ft/min
for 20% overspeed
Directly cooled stator (water)
Directly cooled rotor (hydrogen)
In the design, initially picked up
48 stator slots, 20 rotor slots
5/6 stator coil pitch
Vfmax = 600V
Not skewed
Details in sgDesign.m
Part 4 – Round Rotor Design
for Surface Mount Permanent
Magnet Generator
Magnetic Circuit Analysis
For a multi-pole surface mount rotor
dm
Da
D
g
Poles
gH g  H m d m  0
 gBg  0 H m d m  0
dm
Bm Am


g
0 H m Ag
Bm Am  Bg Ag
Working Point for Permanent Magnetics (1)
Maximum Energy Point
B
Br
BmR
0 Hc
0 HmR
Br
B
(H  H c )
Hc
To get (BH) max
0 H
Br
 BH 
(H  H c )H
Hc
 ( BH )
Br
Hc

 0  Bm 
,Hm  
H
2
2
Working Point for Permanent Magnetics (2)
 rm 
Br
0 H c
1   rm  1.2
Load Line:
Bm  
d m Ag
 0 H m 
g Am
  Pc   0 H m 
Define: Bm   m Br
H m  (1   m ) H c
Typically pick up:  m  0.5  0.8
Pc is called permeance coefficient
Ag / g
d m Ag
R m Pg
Pc 



g Am
Am / d m R g Pm
 Pc  
Bm
m

 rm
0H m
1 m
Airgap Magnetic Field from PM Rotor
PM embrace:
B g , rotor
  PM 
  PM
Bm

 Bm
 PM
 PM
2
2
B g , rotor 

h 1,3,5...
2


electrical angle
PM
2
2 
PM
2
2
 de
 de 
B Rh
B Rh  B rh cos( h de )  B rh cos( h
PM

P
d )
2
   PM /2
2   PM /2
B
h

d


cos(
)
(  Bm ) cos( h ae ) d  ae 
m
ae
ae






/2



/2
PM
2  PM
  
sin  h PM 
4
2 

Bm


h
 Brh 
pitch factor for the
hth harmonic
  
k ph  sin  h PM 
 2 
P
d
2
Phasor Diagram
EA
jX S I A

E A  V  jX s I A

V

IA
Pick up torque angle  (T full load =Tmaxsin ) and power factor pf  cos  .
From X s I A cos   E A sin 
 E A  K B X s I A , K B  cos  / sin 
Example: If  =30 o , pf=0.85 lagging, K B  1.7.
Steady
E A  K B X s I A  B Steady

K
B
f , pk
B a , pk
V  E A cos   X s I A sin   ( K B cos   sin  ) X s I A

Ba , pk
B g , pk

X sIA
1

V
K B cos   sin 
Air Gap Size and PM Thickness
From:
Ba , pk
4 0 Nˆ a
1.5 2 I A,rated

 gˆ total P
Initial total effective air gap size:
gˆ total
From:
gˆ total  k c g 'total
4 0 Nˆ a
1.5 2 I A,rated

 Ba , pk P
g 'total  g  d m / rm
dm
 Pc
g
Pc 
Carter’s coefficient
 g  (1   m ) g 'total
d m   m g 'total rm
( Ag  Am )
m
 rm
1 m
Effective Air Gap
gˆ total  kc g 'total
where the Carter’s coefficient
kc 
s

2 b s 0 
bs 0
g ' total

atan
ln
s 
1 

2 g ' total
 
bs 0


2
 b s 0   

 
2
'
g
total   


approximately
kc 
s
bs20
s 
5 g ' total  bs 0
ts
g total
bs0
s
ds0ds1
ds