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 TemperatureThe resistivity of copper versus temperature can be calculated using the following formula:

)()()( 00 TTTT CuCuCu

where is a reference temperature and is a constant give by

0T Cu

Cin/ 10668.2 o9 Cu

or:Cm/ 10775.6 o11

Cu

At , we have C20o0 T

Cin/ 10679.0)( o60 TCu

or:Cm/ 10724.1)( o8

0 TCu

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.

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…

A. Integral S/P

B. Fractional S/PS/P takes a fractional number.

Integral S/P may cause extensive cogging or detent torque since all pole faces will line up with slot openings at the same time.

S is multiples of mP.

Cogging torque: torque from the interactions between rotor poles and stator teeth. Use slot skew or fractional S/P can reduce it.

Number of Turns per Coil

pkgaepkgaerated NfNfV ,,,ˆ44.4ˆ2

,,

2 g pkg pk

B DlP

1.1/ˆawa NkN sdpw kkkk

where

DlBqkfCV

Npkgwe

ratedc

,

,

221.1

Na is the number of series turns per phase of armature windingC is the number of parallel circuits of armature winding

CPqNN ca /

To consider leakage flux

Number of Conductor Positions per Slot on Stator

aNC cs 2 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.

,

,

1.12

rateds

e w g pk

aV CC

f qk B Dl

Maximum and Average Flux Density Average flux per pole:

pkgaeaepkg

avg

d,

0 ,,

2sin

0

pkg ,

Average flux density per pole:

DlP

PDlB pkgavg

avg 2,,

,

2/

or:P

DlB avgpkg 2

,2

,

P

DlB pkgpkg

,,

2Since

pkgpkgavg BBB ,,2, 4.04

Specific magnetic loading

Typically, take TB avg 6.0, or TB pkg 5.1, in design.

Machine Size (1)

rms current per unit length of the armature circumference, ,(2 )( / ) 2a A rated a A rated

a

m N C I C mN IK

D D

Na is the number of series turns per phase of armature windingC is the number of parallel circuits of armature winding

, 2a

A rateda

DKImN

Specific electric loading:

Machine Size (2)

Apparent power , ,rated rated A ratedS mV I

, 2a

A rateda

DKImN

pkgaerated NfV ,,

ˆ2 PDlB pkg

pkg,

,

2

a

apkaerated mN

DKP

DlBNfmS

22ˆ2

mpkgawm

rated BKknlD

S 60120

2)(

2

,

2

2

proportional to power density

awa NkN ˆ

PlDKkBfS awpkgerated

2

,22

120Pnf m

e

2 where , pkg

awm

BKk

Defined as: magnetic shear stress

Machine Size (3)

Machine of Volume 260

,2

2

pkgam

rated

w BKnS

klD

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

coolingon depends

12

1 , 200

2

amerated KfCC

PSlD

fe = 60 Hzcommon steel

cooled)-(liquid/MVA in 375

cooled)-(hydrogen/MVA in 700

cooled)-(air/MVA in 1400

30

30

30

C

C

C

Length/Diameter Ratio (1)The length/diameter ratio of a machine is defined as the ratio of the length and the stator bore diameter:

DlrlD

Discussions:

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 lengthand the pole pitch:

P

lr

asp wherePD

P

We can find the relationship between aspect ratio and length/diameter ratio:

PrP

Dllr lD

P

asp

Stator Core DiameterStator Core diameter (outer diameter) 0D

0

0

Two-pole: 2.1Four-pole: 1.7

D DD D

Stator Slot Design

sDS

Use 65 V/mil ground insulation Use 0.375-in slot wedge Use 0.125-in coil separator and top stick

sss bdb 73

0.4 0.6s s sb

sss bt

Stator Conductor Size

, /A rateds

a

I CJ

aA

where Aa is stator (armature) conductor cross section area and can be determined from the above formula together with:

2

2

2

Air-cooled: 2500 A /in

Hydrogen-cooled: 4000 A /in

Water-cooled: 7000 A /in

s rms

s rms

s rms

J

J

J

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 diameterdwc: covered wire diameterAwb: bare conductor cross section areaAwc: covered conductor cross section area

American Wire Gauge (AWG) 8.251463 0.8905257 G

wbd

log8.251463

log 0.8905257

wbd

G

Or

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

An empirical formula for effective air gap size:

The actual air gap size can be further tuned using an electromagneticsimulation software.

0, ,

ˆ4 1.5 2aa pk A rated

eff

NB Ig P

From

0,

,

ˆ4 1.5 2aeff A rated

a pk

Ng IB P

eff cg k g

/eff cg g k

Part 3 – Round Rotor Designfor Generator with Field

Winding

Number of Poles

For round rotor machine with field winding, take P = 2 or 4.

Phasor Diagram

,From cos sin cos / sins A A A B s A BX I E E K X I K

AE

AIV

S AjX I

cos sin ( cos sin )A s A B s AV E X I K X I

,

,

1cos sin

a pk s A

g pk B

B X IB V K

A s AjX E V I

maxPick up torque angle ( = sin ) and power factor cos .full loadT T pf

, ,Steady Steady

A B s A f pk B a pkE K X I B K B

oExample: If =30 , pf=0.85 lagging, 1.7.BK

Rotor Slot Selection (1)

integer is 2rotor on slots ofnumber total

r

rr

nPnN

PNn r

r 2 :half poleper rotor on slots

rr

fr

rr

fr

nDPWDP

DPnPWD

22)2/(1

2

:radian) elec.(in pitch slot Angular

2

PDr

r

:pitch polerotor

rfr W 3.00.2 : widthpole

PD

t r2 :slotsadjacent obetween twlength Arc

ff b tt :h widthrotor toot t.bt. f 5040 :slot widthrotor

Number of Conductors in Rotor (1)

The length of the ith field coil: )(2 2

PDiWlL r

ffi

Assume the number of conductors (Cf ) are the same in the each slot

Total length of the field winding:

rn

i

rffF P

DiWlPCL

1

2 )(2

Note: use single layer concentric winding on rotor.

Number of Conductors in Rotor (2)

)1()(2)(2 where 21

2

rrrfr

n

i

rff

ffF

nnDWlPnPD

iWlPX

XCLr

max ,0.7 FF F rated f f f

f

LV I J C XA

ff

Ff XJ

VC

max7.0

The calculated results will be rounded to an integer. Jf depends on rotor cooling. See next slide

where is the cross section area of the field conductor, is the allowable

current density, and is the resistivity of copper at working temperature. f fA J

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 Steadyf pk B a pkB K B

0, ,

ˆ4 1.5 2 Steady aa pk A rated

eff

NB Ig P

0, ,

4 wf fSteadyf pk f rated

eff

k NB I

g P

, ,,

ˆ ˆ1.5 2 2.12B a A rated B a A ratedf rated

wf f wf f

K N I K N II

k N k N

where Nf is total number of series turns in field winding: rff nPCN

kwf is rotor winding factor given is next page.

rotor slot cross section area Af :,f rated

ff

IA

J

Round Rotor Winding Factor

1

1

cos[(2 1) / 2]r

r

n

r

wf n

Nk

N

This is the case when sr is even.

2r rs n

1 2If rnN N N

1

cos[(2 1) / 2]rn

r

wfr

kn

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 laggingMaximum allowable rotor peripheral speed 50,000 ft/minfor 20% overspeedDirectly cooled stator (water)Directly cooled rotor (hydrogen)

In the design, initially picked up48 stator slots, 20 rotor slots5/6 stator coil pitchVfmax = 600V Not skewed

Details in sgDesign.m

Part 4 – Round Rotor Designfor Surface Mount Permanent

Magnet Generator

Magnetic Circuit Analysis

For a multi-pole surface mount rotor

md

Poles

aD

D

0g m mgH H d 0 0g m mgB H d

g

0

m m m

m g

d B Ag H A

m m g gB A B A

Working Point for Permanent Magnetics (1)

B

0H0 cH

rB

)( cc

r HHHBB HHH

HBBH c

c

r )(

max( )To get (BH) 0 ,

2 2r c

m mBH B HB HH

0 mRH

mRB

Maximum Energy Point

Working Point for Permanent Magnetics (2)

Load Line:

0

0

gmm m

m

c m

AdB Hg A

P H

1 1.2rm

Pc is called permeance coefficient

//

g g gm mc

m m m g m

A A gdPg A A d

RR

PP

Define: (1 )m m r m m cB B H H

0.5 0.8m Typically pick up:

0

rrm

c

BH

0 1m m

c rmm m

BPH

Airgap Magnetic Field from PM Rotor

/2 /2

/2 /2

2 cos( ) ( ) cos( )2

sin4 2

PM PM

PM PMrh m ae ae m ae ae

PM

m

B B h d B h d

hB

h

, 1,3,5...

cos( ) cos( )2

g rotor Rhh

Rh rh de rh d

B B

PB B h B h

sin2PM

phk h

pitch factor for the hth harmonic

, g rotorBmB

mB2

2P M

PM

2PM

22PM

de2

P M

2de dP

2PM

PM embrace: PM

electrical angle

Phasor Diagram

,From cos sin cos / sins A A A B s A BX I E E K X I K

AE

AIV

S AjX I

cos sin ( cos sin )A s A B s AV E X I K X I

,

,

1cos sin

a pk s A

g pk B

B X IB V K

A s AjX E V I

maxPick up torque angle ( = sin ) and power factor cos .full loadT T pf

, ,Steady Steady

A B s A f pk B a pkE K X I B K B

oExample: If =30 , pf=0.85 lagging, 1.7.BK

Air Gap Size and PM Thickness

Initial total effective air gap size:

0, ,

ˆ4 1.5 2ˆ

aa pk A rated

total

NB Ig P

ˆ ' ' /ctotal total total m rmg k g g g d

0,

,

ˆ4ˆ 1.5 2atotal A rated

a pk

Ng IB P

From:

From: mc

d Pg

(1 ) ' '

m total

m m total rm

g gd g

( )g mA A

1m

c rmm

P

Carter’s coefficient

totalg

sdst

s

0sb 0sd 1sd

Effective Air Gap ˆ 'total c totalg k g

where the Carter’s coefficient

2

0 0 0

0

2 'atan ln 12 ' 2 '

sc

s s total ss

total s total

kb b g b

g b g

20

05 '

sc

ss

total s

kb

g b

approximately