Design
cu ations or A
of the Field Form C, enerators Fundamental This constant (denoted by Wieseman3
DAVID GINSBERG MEMBER AIEE
as Al) is the ratio of the peak fundamental flux density of the no-load field form to the peak actual flux density of the field form. Values of Cl are given on Figure 3 in terms of the pole embrace and the ratio of maximum air gap to minimum air gap. The ratio of air gap to pole pitch
Synopsis: This paper presents simplified formulas and curves for rapid and accurate evaluation of steady-state and transient performance of a-c generators. The calculations provide for the determination of no-load, zero-power-factor and full-load field current, short-circuit ratio, leakage reactance, synchronous reactance, transient reactance,
the papers to the machines in question is that they apply predominantly to synchronous generators having open-slot gp p choosgnrtrghvnepnsodp ti stators and air gaps with a length of more has been ignored and, in this respect than 3/4 per cent of the pole pitch. It these curves differ from those of Wiesealso was found that the conventional man.3 The values were derived by igmethods of calculating the magnetomo- noring the flux in the interpolar spaces,
cuit and short-circuit time constants, and
tive force required by the steel were quite inaccurate. For generators having small air gaps and highly saturated iron the calculated and test values could differ from each other by more than 100 per cent. Such inaccuracies obscure the opportunities for design improvement which can be realized by better machine proportions, by full utilization of new materials, and by taking advantage of recent developments in automatic regu lators. It is the purpose of this paper to extend and simplify the results of the published literature on the subject and to formulate a systematic design procedure which can be applied with accuracy to the machines in question.
d-c time proposed load core apply to
constant. A simple formula is for rough approximation of noloss. These formulas and curves very small machines with semienclosed slots, very short air gaps and high flux densities in the iron, as well as to the larger machines covered by previous papers on this subject.
THE expanding
use of electric equipI ment by the armed forces has created a demand for lightweight generators having exceptionally good wave form and transient performance. The greatest demand is for generators from 0.15 kw to 150 kw. It was found that many designers of a-c generators within this range
assuming that the flux density at any pointiunder the pole shoeis inversely proportional to the air gap length at that point, and applying the Fourier integral to obtain the magnitude of the fundamental. Aithough the flux in the inter-
he spal. lth tg ough polar space is large enough to have a slight effect on some design constants, it has a negligible effect on C1 because it is far removed from the pole axis and its main contribution is to the harmonic content of the flux wave. Although this assumption may not be justified for machines with very large ratios of air gap to pole pitch having very small values of pole embrace, it is justified by tests and calculations of all machines within the writer's experience. A comparison of a
of sizes completely ignored transient
A-C Generator Design Sheets
Paper 50-201, recommended by the AIEE Rotating Machinery Committee and approved by the AIEE' Technical Program Committee for presentation at the AIEE Middle Eastern District Meeting, Baltimore, Md., October 3-5, i950. Manuscript submitted September i6, i949; made available for printing August 7, i950.
This procedure follows Kilgore'si methods and terminology, where applicable It is incorporated in a 2-page design sheet Figures 1 and 2 in which an actual generator design is worked out as an example. The calculation of all but seven quantities can be completed from information directly available on the design sheets and without reference to any other curves or formulas. The seven quantities which require supplementary data for their determination are C1, Cp, steel magnetization curves, core-loss facslo cost tors, sltcntnt K1, and chording leakage factor Kx. The factor Z on the bottom of Figure 2 represents the effect of
machines is listed in Table I. Full test data is not available for all machines, since the tests were made by various agencies and were not, in general, intended for use in design studies. Further experimental verification has been obtained by the use of exploring coils to make oscillograms of the no-load field forms of a number of generators. These field forms were then analyzed to compare the actual values of Ci (and C,) with the calculated values. The elimination of the ratio of air gap to pole pitch as a parameter makes it possible to express the values of Ci by means of a single family of curves and to apply these curves to machines whose
the pole embrace on the tooth-tip and zig-zag leakages and the derivation was based on considerations presented in a 1918 paper by Doherty and Shirley.2
dimensions are outside the range of Wieseman's3 curves. These curves, like Wieseman's, were calculated for pole faces which form a continuous arc of a
characteristics because of the absence of criteria for these characteristics and because they did not have the time to formulate a systematic design procedure from the various papers on the subject published over a period of more than 30 years. These papers agree with each other in principle, but differ from each other in terminology, in simplifying assumptions and in calculating methods. The difficulty of applying the results of
DAVID GINSBERG is with the United States Army
Engineer Researcb and Devrelopment Laboratories, Fort Belvoir, Va. The author wishes to express his appreciation to Leo J. Misenheimer and R. Powers for their assistance in preparing this paper.
1274
calculated and test results for various
Ginsberg-Design Calculations for A-C Generators
AIBE TRANSACTIONS
Figure 1. A-c generator design sheet number I ELECTRICAL EN(WNEERING BRANCHAB~E.E.DEFARTMENT ENGINEER RESEARCH8a DEVELOPMENT LABORATORIES ,FT. BELVOIR,VA.
4OOeg-6 A.C. GENERATOR DESIGN SHEET Cont. No. SHEET I OF 2. 60 KWL PF2! VOLTS .2PHASES.L FREQ. RPM~2 INS..A MFG EM CO AMPSI-L oamete r lbi
ID
S%01
-!P 3~~ i ~ hi de
Stator Slots
Damper Winding
d
Field 2 Poles Series
Armature
No. slots ___________27 (18 used) 1 Ciro. Y Connection _______ 4 Series turns/coil ____ Slot throw _______1- 11
dr
gmsin
gmnax 0.05
822
ge
sin90(pitch)0. 918
bh
IC9 p sin 90 (pitch~~~
sin
P
b p h
(sn
90o(.29
Conductor cu ~~~~8
W
Total series
Amps/sq.inch
Azspe/sq. inch _________ Damper
~~nd ring:
-s(4.4g ge T5 (4.4g
+ +
0.75b1)
0.75b1)b 1
0.053 0.614 7.56
Teeth ___ Core
+ b5) =or( 7Ts(5.5g 5)
5.5g
2.O5
Spider
1.4
(
0O M
hh
-1.40
30,000o
40.0- 64,300~ 4.96 75,400
75,300 51,800
10.4 15.1
+ b
slot
or
0.10
NI/inch
effective series CC 7T+ SinCC 77 4 sin( U.T)
_
_
w
ampere-conductors
.
4p =
0
armature reaction dm
lphClnKd -
p
LFJ P ~~~~~~~~~ __
.
h4-Cp ____________, 34xi
3
Trb
t
0. 095' A-
P 9~~~~~~~~~~~~~~~~~~-Tl84.9
hr D
.58
11.00
(No.
=
1
ge
2NI
950 3.8 9.8
2.3 74.1
24 1.1 WL 24 1.1 WL
9.4 3.65
19.4 5.1
slots)
(0.93) =40.0
D-Dst slot(0: 93)
=
4.96
P=Ib(-3
p l9p(.3
1.00 1.
0.58
TEKTH
COR LENGTH
POLE
Slobhsr(D)D
132
1.
SPIDER
(d50+d5
0.761
0.614 g
KTx1n3 =2573x103
(RM
0.420rCj
b3
b ______ne
29 0.72 WL 0.72 WL
0.05
[o.817o0.22 + 0.075)1 L0.24-0.01 81
80,400
Pcuns
lb.
566 4.3 114
15 4
57,400
WAT
2iNI
10800 7 15
83,500
1.4
0.053
NI/inch
33,500 71,000
2.06r
7.56
_____
1.49 2.06
6.9 42
Watts ____
10.3
86.5
31 5.6 '21
_
WINP p
-0.883
0.4S5
P0OIOE
~~~~~~~~~~~~1 8.248 1Acore
29-
__
-0.588 V(Phases)--
0.131
4r h1, d3 or ds
b2C.42
604 -_
~ET ClEd "-
0.63
d1
area
=
/-
F 9
h2 d2
.
iso ~~~~~~~
STE ~~~~~~GaLW Grade
B
Area 84.9
______
Subtotal____
Pole
b s or b 1
1e 1.75 1e 2 Ih 3.875 1 4.375 hb 0.26
0.6E3
71~
Lgth.
Gap )Fg)
-3.875
bb
Nc. of Bars
Area 0.3
OD m
T
1
2.00
Useful
Data
Edin:Thck.0.125
F
~~~~~2.5 2. 06
.87a 10. 87
tf.
Coil length Resist./terms 0 750 C __
CM
54
r
1.45 D510t HF ht 0.50 Conductor insul. 18.3 3L.4 .0 h tf .1 Dme9-2 2 15.6 0.147 2 l~ ~ ~ ~ ~_____ ~ ~ i 0.614 bta____Aspider 0.41 C1 1644 144 4088 ~1396 Soskw1.0C 406 Slot_skw_13_30p
insul.HF
Toaeiscond.,
h f2
*17
#17
Cmductor
A.
0.053_
(n )0hf,
BSd
Total
7.02 6. 92 0.05
I h = n ep
875
FIELD HEATIln I-i-
1133
2)(1
+
b )
Radiating Sfc.
ih It) T
+ (t
f1
L
h2
2hf
1=18.3)ht I l- _
MrZT (2hf1I tf 1 + tf2f2) 2 .120~~~~~~~~~~~~~ L
.1.
single circle whose radius is located on a radial line through the center of the pole embrace. The simplification of the curves for Cl make it easy to tell by inspection how changes of pole embrace and pole face shape will affect certain design constants. Work now is being done to utilize these simplified curves for C1 and the simplified curves for Cp discussed in the following into anaytialmetodwmg into an analytical method of predetermining machine dimensions from the specified performance requirements.
Symbols Used in Figures 1 and 2 A = effective series ampere conductors A,= total area of air gap Ap=area of pole body A= total area of teeth
K0 =slot reactance factor
Lf = total field circuit I= length of stator coreinductance (including vents)
conductors per inch extension of diamond portion of AA=ampere 1ei=axial B' = flux density at the voltage behind the stator winding
potier reactance B,7= air gap density (air gap density at the pole center half over tooth pitch) =tooth width averaged btmway down the teeth bbar = width of rectangular damper bar bh = width of pole head bp =width of pole bs =width of stator slot Pl= width of stator tooth at tip Pole Constant Cp Cm-= "demagnetizing factor" ratio of equivalent field ampere turns to maximum sine wave armature ampere Curves for the determination of Cp turns (ratio of field to armature amare included in Figure 3, which also contains the curvesforC,iscussepere turns for same fundamental flux) = pole constant" ratio of average to CP maximum viously. The curves for the portion of the field form flux under the pole, were derived analyC,,= winding constant tically using the simplifying assumptions C1 = "fundamental of field form" the ratio of the maximum the previously discussed in with actual maximum fundamental value of fieldtoform previously connection with dixscussed in connection the C1 curves except, of course, that a (figured with no saturation) direct integration of flux density over the Connection = number of parallel paths, type of circuit (for example, 2-circuit Y) pole embrace was substituted for the bore (outer diameter of punchFourier integral. In order to take ac- D =frame ings) count of the flux in the interpolar d -stator bore a correction factor for Cp also is plotted in d,= rotor diameter Figure 3. The correction factor was ob- d,, =outside spider diameter tained by plotting Wieseman's3 values d,j = inside spider diameter terminal voltage of Ko against the ratio of air gap to pole Et=rated rated phase voltage pitch and averaging the results for vanl- F=total no-load ampere turns ous ratios of maximum gap to minimum Fdm = demagnetizing ampere turns per pole FFL = field ampere turns at full load gap. The values of CP, which represents the ratio of the average flux density to F,,=air gap ampere turns per pole at rated the actual peak flux density of the field F1' = airvoltage gap ampere turns at (1+XI) Xrated form, can not be obtained directly from voltage Wieseman's paper. They can be derived Fo= no-load ampere turns per pole from Wieseman's curves for K, by means Fo' = total no-load field ampere turns at (1+XI) Xrated voltage of the following formula FSC=field ampere-turns for symmetrical steady-state short circuit at rated Cp = 0.636C1Ko armature current From Figure 3, it is seen that the af- f-=rated frequency = single air gap fect ratiofairgaptpg of the g effective air gap fect of the ratio of ar gap to pole ptch, though finite, is not large and, for ma... =single air gap at pole tip chines with relatively small air gaps, can gmin = single air gap at center of pole face be disregarded. hbar = depth of rectangular damper bar The simplicity of the single family of hhh = depth of pole head depth of pole body curesfor for design desi for Cpfclitaofthes C, facilitates itsuse its use for culrves hi =depth of copper in slot (including inanalysis and calculation. Comparison of sulation) calculations and test data indicates that h2 = depth of slot to the copper the curves have a high degree of accuracy Ip or Iph=rated phase current Kd= distribution factor for the types of machines considered. factor = sine 90 (pitch) Kf = chord KT= total flux, kilolines, in the machines Steel Magnetizing Curves the value of which would exist if density were uniform and equal to magneizingforcefor te airgap i K - the maximum The __nelzn frefrtea gps K-lot chording, leakage factor calculated by dividing 3.19 (the permeability of air) into the peak actual flux density of the air gap. This peak core is plotted against flux density on actual flux density is the value obtained Figure 4. These curves differ radically by following the procedure of the de- from the published d-c curves for the same sign sheet and equals the ratio 4r/A. steel because the latter do not take into The magnetizing force for the teeth and account the unequal distribution of flux an
spacesinsX POadvai teeEpl=
=
1276
Kid = damper slot reactance factor
le2= straight extension of stator and winding (average) Ih = length of pole head including any magnetic coil supports = length of pole Ip It = depth of stator slot MLT=mean length of turn
m =number of phases Np = series turns per pole on the field
nb = number of damper bars per pole ne =total effective conductors P = number of poles q =slots per phase per pole RF = total field circuit resistance ra = armature resistance per phase
St = total number of stator slots (including
empty) So = number of slots per phase of stator slots used Si = number = short-circuit SCR ratio Ta = d-c time constant Td' = short-circuit transient time constant Tdo'= open circuit transient time constant W =total series conductors X= treamuelakgracne true armature leakage reactance Xad= direct axis armature magnetizing re-
actance
X = quadrature-axis armature magnetizing reactance Xd = = direct-axis synchronous reactance Xdu' unsaturated transient reactance
XF' =field leakage reactance
X1=armature leakage reactance Xq = quadrature-axis synchronous reactance
y =number of slots spanned by the coil Z =zig-zag leakage correction factor
a= ratio of pole arc to pole pitch
O=minimum
number of poles having an integral number of slots
0= slot skew in electrical degrees 1= pole leakage flux 4p= total flux per pole 4r = total flux in the machine the value of which would exist if density were uniform and equal to the maximum r = pole pitch measured at the stator bore -i damper bar pitch (average if not uniform) -T = pole pitch on rotor diameter Ts =stator slot pitch =
Xa =armature magnetizing permeance XB= phase belt permeance
Xb= damper bar winding permeance XD0 = quadrature-axis damper winding permeance
XDd = direct-axis subtransient winding per meance
Xe = end connection permeance
XF= field leakage permeance XFe =-field and leakage permeance XFs=field side leakage permeance permeance XsX,t==slot pole tip damper winding densities that exist in the teeth and core of an a-c generator. The usual textbook method of calculating ampere-tulrns is to calculate the maximum flux density in a part, determine from the published d-c
Ginsberg-Design Calculations for A-C Generators
AIEE TRANSACTIONS
Figure 2. A-c generator design sheet number 2
EN0WER RESEARtHaDVELOPIENTLAORATORIES,FT.BELVOIR,VA.
ELECTRICAL ENGNEERING BRANCH,1V18EEDEFARTMENT
PF0-80 VOLTS120 PHASES.
KW 5
|A
_
\1Td
d
|
Ep
For open slots
MP x 10i
Xl
LEAKAGE REACTANCE
+
(
mq
Ep +
YoeKd
=
+1) /
I
20
phases, X3
0
- number of phases. q = slots per phase per pole.
r2 hl
]24Xa 4
aa
1100] 4
Xe-4
e1Y 2
3
bt2
ON1.41+2.651+1.81=14.90
d
Xad = I
(CmC1)
Xq
Cql A= 2.7 x 0.356 -= O.906
=
2.27
le.9g
Aa = a
± = Pge
Cql
40+ 1
426
.
sinc r =
REACTANCES
0.091
XF1
C1
Cp
+
Unsaturated
XL = X,
trans.
Saturated trans.
xFS + AFe
10
0-055rj +
(dr
bh
2hh - 0.4hf )-bp
-
Negative
=2.0
4(lh 1) + 2hf -
+
0.5 b3
Xfd
=
0.0183
(For direct-axis use dimensions of end bar near
X2
sequence.
0.o006 x 3.04
-
0.0183
.38 r bh 6Pt=
Ab
-
2
T
[cos
Ab+i+XH 2 ___+_)X1 1b' s,L
-
]
3.04mKKd
4. 3
[
[
x 0~'0532f seh (nb-1) I1= 6.38 Vr5.490541
V-03
6.38 (Kld + 0-5) =
Kld = damper
45'
X,Bo =|
w/o dampers
A
°xo
=
~
16.7
0.0063
AD
=0 b[r rTja ir=0.75 oC
1950, VOLUME 69
L
l
x 4.1 =
=
X1)
0.136
=
Not
applicable
2C(h1+2h3)
2
4
+
< C ~~~~~~~~~~~~~~~Lf
-
-1?
X6q
Tdo do
trans.:
=
S. C. transient
0.0246
=
+
a
+
1.35
_
Rf
8222 xW-388(4.02) 52-1
F)=
2id Tdo' = :T = 0.10 Xd 0.126
Armature (or DC): T =
2
rura2lTfr5
*
3.7q7x.073
= .005
0.02463
05+_
A | x-
XDq
KX = aK 4mq
-29 q
2
(Use dimensions of center bars ) = g X01 =
=
Open-circ.
slot reactance factor
X5q
Kxo
+
0 139
TIME CONSTANTS (Seconds)
-I~.O@S
QUADRATURE-AXIS DAMPER LEAKAGE
0.132
(O.7 Xa) _
Kp
+ B +
)
2
+ 0.625 + 0.5 = .4S3 _
0.18
XO
=
w/dampers
|
1
=
(X5
1/2
=
pole )
=
X\u
0.88
=
0.11440.906=1.12 X= 0.114t0.091t0.205
= X+
Zero-sequence
DIRECT-AXIS DAMPER LEAKAGE
X=d X X
+
Jw/dampers. Xj X Dd = direct-axiS lw/o dampers. X1 = X, = = Subtransient wl dampers. Xj = Xl + quadrat.axisl w/o dampers. Xj = Xq = -
P 10) P
r
3(1.48 o6
4.25
XI
0.114+2.27*2.1
=
+ X aq
Subtransient
6
_F = XFS = 4.25L3thh t L rr
+ Xad
Xq = X1
Quadrature-axis syn.
-
X,
Xd =
Direct-axis synch.
0.091
2.27 x 9.94 2
3
o.as8
a
XF =Xad [l
e
7
p2 0e8
5 ''
Xaq)
Xa - 2.57 x 0.883 x 1.0
32 2.75)
4 (1.0 (21e2 + le 1) -3.875=
330095
(Xad,
EFFECTIVE FIELD LEAKAGE
A\Dd
0.006
m
(negligible)
MAGNETIZING REACTANCE
Xa
S1ge
1.41
slot skew in electrical degrees
NOTE: For 3
AF
52
2 2 _- + - + 0.2 +b 1 Tse 3bs Kp Ed bs
0.8e
= chording leakage factor
K1 = slot reactance factor = =
d
Kl
p2KXd 2S1
Slots
=
2
2122
.W8.96= 0,114 Ai
Total slots/pole
Co AMPS
'ewx3.875x132 x0.829
52 1
4.44 CBd)
SHEET 2 OF 2.
MFG EM
INS A
*d
For semi-enclosed
0
I
)
Xe
X1
Kx
)
/3 x slots spaTmed
5 J
(xA
x
RPM!6-q
FREQ. 60
I
|flneKd
lp
CBg
v2
W44 OO9ena- 468
Cont. No.
A.C. GENERATOR DESIGN SHEET
_
=
l. l)j]
11x 2.68
DT
= 4.1
0. 252 x 1. 432 = 0.
318
_
_
_
_
_
||B
Ginsberg-Design Calculations for A -C Generators
1277
Table I Generator Number
1
3
2
5
4
7
6
8
Phase .............. and 3 .......... 3........... 3 1.......... 1 and 3 ...... 3..... 3.......... 3........... Kw ................ 5.......... 5........... 25......... ......... 60.......... 60 ........... 100 .100 Power factor ......... 0.80 .......... 0.80 ..... 0.80 .......... 0.90 ......... 0.80 .......... 0.80 ........... 0.80 .0.60 Volts ............... 120 .......... 208/120 ..... 208/120 .......... 208/120 ......... 208/416 .......... 450 ........... 450 .450 60 ..... 400.......... 60.......... 60......... 60........... Frequency .......... 60 .......... 60.60
Rpm............... 3,600 d................. 7.02 g.................. 0.05
Ffl
. SCR Xd . Xd'
1,860
3,600
6.5 0.08
3,420 ..........
.....
9.0.......... 0.02 ..........
..... .....
Test Calc Test 503 ... 1,000 ...1,000
Calc 503
Fg..
.......... .......... ..........
...
Calc 308
...
1,200 .........
17.0 ......... 0.175 .........
1,800 .......... 1,200
...........
1,200
13.0 .......... 17.0 ........... 17.0 0.110 .......... 0.099 ........... 0.09
........... ........... ...........
1,200
20.37 0.115
Test Test Calc Calc Calc Test Calc Test test Calc Test 310 ... .2,340 ... 2,360 ... 2,600 ... 2,600 ... 1,800...1,860 ... 1,940 ... 1,900...1,800... 1,850
3,500 ... 3,480 ... 6,850 ... 6,950* ..4,400 ... 4,420 ... 4,650...4,600 0.504 ... 0.505 ... 0.848 ... 0.850 ... 1.84 ... 1.90 .... 2.04... 2.16 ... 0.83... 0.83 ... 1.46 ... 1.42 ... 1.37 ... 1.32... 1.10 ... 1.11 2.38 ... 2.38 ... 1.29 ... 1.29 ... 0.56 ... 0.52 ... 0.518... 0.48 ... 1.45... 1.44 ... 0.96 ... 0.98 ... 1.07... 1.02 ... 1.09 ... 1.10 0.18 0.19... 0.17 ... 0.085 ... 0.080 .... 0.10 ... 0.184 ... 0.19 ... 0.135...0.132...0.127 ... 0.133...0.152 ... 0.164 0 .115 .. 0.07 ... 0.05 .0.115.. 0.113... 0.09... 0.10 Xd0..132 ... 1.35 ... 1.32 ... 1.30 .. 0.078...0.083 ...1,... l.fi ... 1.40 0.012... 0.02 .0.2... 0.18.. 0.167.. 0.172 Td .0.10 ... 0.11 ... 0.18 ...
.
...
...
Tdo'
* Test value at 211 volts, 0.792 power factor.
steel curves, the corresponding magnetizing force in ampere-turns per inch, and to multiply this magnetizing force by the length of the magnetic path in inches. In applying this method to such a part as the core, where the flux density varies nearly sinusoidally along the length of the magnetic path, the actual magnetomotive force is the sum of the products of the magnetizing forces of each increment of the path by the length of the corresponding increment. The magnetizing force for each increment can be obtained from the d-c steel curves for the flux density prevailing in that increment.
The curves of Figure 4 were obtained for different values of maximum flux density by an averaging process. The flux densities were assumed to vary sinusoidally in accordance with the formula B=Bma,, sin X, where X was increased from zero to 90 degrees in equal 2
C _
4O
-;,
£DlJ>2D3X jL,b
9!min/$'OLE -
_
-
_ _
-
ulot;
PITCH
_0 _
7
a li 0 71 .20
ID5SJ;' ;1,_i30 i
Chording Leakage Factor, K,
Core Loss
The chording leakage factor is given in Figure 1 of Kilgore's1 article, Figure 2 of Alger's7 article, and in Figure 143, page 219 of Kuhlmann's design text.8
/B 2(f 1.4
Core loss in watts =m5o(WL)/k6
where W=weight of part (core or teeth) B =maximum flux density in part (kilolines per square inch) f = fundamental frequency in part (cycles per second) (WL) =watts per pound guaranteed by the manufacturer at 10,000 gauss, 60 cycles per second, for the grade and
thicknessofth estee laminationsused
For a more accurate calculation, Spooner's method4,56 can be used.
._ ttA/t .l250
XC
/ /7 2 4
-6E>-4X7 _4
:0275
Slot Factor K,
/Imalent. _17T0 he slot factor K1
is the permeance of
sl30 th ot leakage path and the formulas are
given in numerous text books and technical papers. Using the nomenclature of CL1a0the design sheet, the value can beob& 2 _ tamned for a rectangular semienclosed slot
0.
7N
XA
-7,
2f> 3 g W F ,;
B.F I0
by the formula
-a X
7
_
+ 8
m ' Yb
Figure 3. Peak fundamental and average of no-load flux wave
1278
sometimes led to ridiculous results.
A rough estimate of the no-load iron losses for teeth and core can be obtained from the following equation
- 1/t/M2D5 ^//gg
/ /, // X W
.2
The permeance of a semicircular slot bottom is equal to 0.14, so that for a semienclosed rectangular slot with semicircular bottom di 2d, d3
O
4 . 7X g ?B BR X e 7Y
5-degree increments. For each increment, the value of magnetizing current was determined from the published d-c magnetization curves for the steel and the sum was divided by the number of terms. The use o-these curves has made accurate prediction of saturation curves possible where the conventional method
_b1 b1+b2
Discussion The design method outlined takes into account only the performance characteristics of the machine but does not attempt to evaluate losses or heat dissipation factors accurately. This was done partly because of the fact that no method is known for the accurate predetermination of temperature rises and partly because for the generators under consideration, the design will usually be dictated by performance rather than by temperature rise. This condition will become increasingly true as the use of high-temperature insulation becomes more prevFor the present, some indication of reasonable temperature rises is afforded by using such criteria as d21, current densities, ampere-conductors per inch of periphery, and cooling air velocities. For the machines in question, the values of d21, can be assumed to be pro-
portional to (kw/rpm) 0.64
In basing the design on transient per-
3b2
. . The permeance of a circularte slottowith a opeing f nomalsizeat is
K--d+0.625
K1=-+ +-+0.14 b, bl+b2 3b2
tablished. An important criterion was ~~~~~~~~~~the ability of the generator to start and aclrt
oos
opeetsaln
of motors and opening of contactors dur-
ing the accelerating period, it is necessary
blthat the minimum rms voltage of the
Ginsberg-Design Calculations for A-C Generators
AJEE TRANSACTIONS
generator, as determined by oscillograph, should not drop to less than 70 per cent of rated voltage. It also is necessary that during the accelerating period, after the automatic voltage regulator action has stabilized, the generator voltage should be at least 80 per cent of rated voltage. These values are based on dropout voltage of 65 per cent for magnetic contactors and a pull-out torque of 200 per cent for induction motors. At 70 per cent voltage the pull-out torque is 128 per cent, leaving some margin for acceleration when driving a load having constant torque characteristics. The following equation is used for determining the minimum transient rms voltage in per unit
design sheets to calculate total leakage _ _ - - - _i- |; |l flux. The results obtained so far do not - __ 14 - - _ appear to require a separate calculation 7_ for total leakage flux. _ 2 In calculating full-load and zeroi10 a0 00 4i O 6 M 80 9 SK 1100 2 D power-factor ampere-turns the factor (1+XF'), was used to make some allow- x, z_ ance for the increase in ampere-turns of |l i, the field pole because of the increase of from the componentthe bElli leakage flux resulting of field required to neutralize the arma- Z90,_20 30) 40 50 60) 7O e 90 0m1MlO 13 140y 2 ture demagnetizing action.
Appendix 1. Derivation of Equations for C
ZLZ L
tIT'dl
(zL+x,d,zL+xd)et/TP_ where' where
ZL = impedance of starting motor in per unit Xd=direct axis synchronous reactance Xd'=transient reactance of generator in per unit
t=response time for regulator (0.1 second for most mechanical regulators) Tdl'=generator time constant for loadZL T&'=generator open-circuit time constant
23,4
a2o idr
J
1
ing
density
a
a[ L
2
For the air gap shown in Figure 5 4gn r/2cos d 4gn = X Cl g ir cos E f ai2
Jo
air/2
sin (
C1 1.27 sin2
Tr
\\
\
\ \
\
/
xE, \\\\\ \ I R \\ r ,aT \\\\ X
/ 9
\ )*LH/ \s/
1950, VOLUME 69
__
gx
Apedx1.Deiaino Derivation of Appendix Equation for Cp Cp is the average value of flux density
divided by the maximum value of flux
Figure 5. Length of air gap as a function of pole shape and angular displacement from
pole center
gaR-Itcosir2- HzsinI
gop gxaMaximum gap
point
grmR-H-r A =R-r-gn
\\RRadiusatony of stotor bore gx a
\
tan'-/g-1 1
2ir).
g agap
2 -J
sin2
cos 4 de 1+ -1j \sin240 (sin2 \gn 2/ \
Pole Face
9 2 / gnal
/-
X
Pole
\
-, IX
tan--1sin1
=12a7s/2 air
1g o
1
\sin 2/
For a uniform air gap
C1 =1.27f °
1 1
a2
Stator Bore
\ ~~d /\\
1 /
fa1/2cosfdct g Jo
wr/2cos g ad
a Minimum
//
31
AMPER-TURNS
C1 1.27
1r
9
12 3 14
56 7 8910 112 PER INCH
Assuming the effect of flux in the interpolar spaces to be negligible
4g Tsin
sions on the flux in the interpolar spaces. This would make it possible to calculate field harmonics accurately without resorting to laborious flux plotting for each new generator design. The leakage factor XFo , used in Figures 1 and 2 is calculated in accordance with the formulas of Kilgore.' It is correct only for the detennination of transient reactance since it evaluates the linkages of the leakage flux with the field winding rather than the total leakage flux. However, the value XF' has been used in the
3c_|
4. Magnetizing curves for steel, showFigure the effect of sinusoidal distribution of flux
B2cos XIX-X3.19
ZL+Xd
Further work is necessary in order to
'
-6W
4 Nl C= B1= - gn NI 7r m m 3.19 cos d V,2/2 0k d_____2_s I
Tdl'=TdoIZL+Xd generalize the effects of machine dimen-
7C
The field form at no load has no even harmonics, it is symmetrical with respect to the origin of co-ordinates, and the fundamental component can therefore be expressed by: (Refer to Figure 5)
Emin = ZL + ZL±Xd
.|l.
8
= Radius of pole face ~~~~~~~~~~r
siWfg cosrlk /risie|rC l- Hsfl4 R-H
*CI-Sir H=~~~~~H*Offset of center of polel HCOs4H Si ~~~~~~~~facearcfrom center g9 CR-Hr)+±C).(*r)Siri of stator circle I 2r _ cu Pole embrace |g
s~~~
gn+2r+H.r)sin4 r)irL
pohsition I~9x gn+§@H+H ng9utlar PO~~~~* zp respect to centerl 2r I.'+I r)= . X2a~~~~~~~~with pole 2rsn ~~~~~~~of
Ginsberg-Design Calcukations for A-C Generators
of
12'79
density. The average value of flux density is the total flux under the pole divided by the area under the pole pitch. Assuming that the flux density at any point is inversely proportional to the air gap at that point, and neglecting the flux in the interpolar spaces: (See Figure 5) B
max_3.19 NI gn
f/w/BdA B=2 -
B av
lRrJ o
pole area
BdA=iX pagesv493-513. pageE Trnscios3vlue471ebuay198 1Rr 2/
=-
ra7r/2 3.19 NI 1Rd4 g
ar/2 do
2
Bav = -X3.19 7r
=
-
,J0
g
air/2
Bap
Bmaa.
2
B
f
nJ
g
.
/ar/2
2
rJo
1 +gr __: 1
\gn
1
/sin2 air
sin2
2
The integral is of the form 1
2J
d(240)
a2 /(a2 ll+- )-2 cos 2 4
air 0.636 sin 2 C Xsm CP x
.
xxT
'*I-1+sin 2-2 n
gxaFr - sin i2 -1+ gn
tan-
L
air(
ainr
\
tan i'j
sin 2
J
Cp=0.636 tan-Xar 2 tan-
I
n
aCosi2ar
1
2
/ gx/gn___
air 2L! Cos2
R eferences 1. CALCULATION OF SYNCHRONOUS3 MACHIINE CONSTANTS, L. A. Krilgore. AIEE Transactions, volume 50, December 1931, pages 1201-14. 2. RE:ACTANCE OF SYNCH:RONOUS MACHINE:S AND B. Shirley.
AITSAFFPLlCAsTaION, R.ol. Doe°hrty, 0.2
AIE rnscios,vlue37 ar 2 91,pae
3. GRAPHICAL DETERMINATION OF MAGNETIC FIELDS, R. W. Wieseman. AIEE Transactions, volume 46, February 1927, pages 141-54.
1280
saturation.
Local saturation around slots
carrying high currents should be checked.
Depression of the originally sinusoidal air gap flux due to tooth saturation occurs, but
can be approximated.
required to drive the flux through the yoke
11. THE REACTANCES OF SYNCHRONOUS MAB. L. Robertson. AIEE CHINES, R. H.volume Park, 47,
Transactions,
February 1928, pages
Discussion Discussion
do d4
methods to calculate the saturation curves are insufficient. In minimum size machines, the actual physical flux, that is, air gap flux plus local leakage flux, must be calculated closely to permit the use of high magnetic densities without resulting over-
N. Y., second edition, 1946.
12. DETERMINATION OF SYNCHRONOUS MACHINE CONSTANTS BY TEST, S. H. Wright. AIEE Transactions, volume 40, December 1931, pages 1331-1.
2=-27_gn agn=a
rederive and modify his formulas, for
example, leakage flux and reactance formulas, as his design proceeds. Mr. Ginsberg states that the customary
9. MAPPING MAGNETIC AND ELECTROSTATIC FIELDS, A. D. Moore. Electric Journal (East Pittsburgh, Pa.), volume 23, July 1926, pages 355-62. 10. THE INTERPOLAR FIELDS OF SATURATED MAGNETIC CIRCUITS, Th. Lehmann. AIEE
514-36.
7r
-2
8. DESIGN OF ELECTRICAL APPARATUS, John H. Kuhlman. John Wiley and Sons, New York,
Journal, volume 46, December 1927, pages 1411-14.
dok
For a uniform gap
C
4. TOOTH PULSATION IN ROTATING MACHINES, T. Spooner. AIEE Transactions, volume 43, ' . SURFACE LoS T Spooner I. F Ennard MRON AIEE Transactions, volume 43, February 1924, pages 262-81. 6. NO-LOAD INDUCTION MOTOR CORE LOSsES, T. Spooner, C. W. Kincaid. AIEE Transactions, volume 48, April 1929, pages 645-55. 7.ANCETHE CALCULATION OF THE ARMATURE REACT OF SYNCHRONOUS MACHINES, P. L. Alger.
Paul W. Franklin (Bendix Aviation CorN. J.): poration, Teterboro, oN po ratiTer Jo) Mr. Ginsberg is to be congratulated for his publication, which regrettably is one of the few papers dealing with actual design calculations. Whereas there is no lack in the number of theoretical papers, actual design methods are rarely published and thus the average designer cannot always benefit from the advances of theory. Mr. Ginsberg developed his paper for the design of light-weight minimum-size machinery for the Armed Forces. It is of interest to note that for aircraft this type of equipment was required and produced years ago. The specifications as to weight1 er g.Teseiiain egtto and performance are st difficult bulk, attain. This led to the development of a special line of rotating equipment, the performance of which indicates that sooner or later the principles and procedure employed will be adopted by manufacturers, wherever light-weight minimum size equipment is essential. The design of minimum size equipment was made possible by the use of Silicone varnish, inorganic insulation, high strength materials, and to a certain extent by the development of practical power selenium rectifiers. The use of Silicone varnish permitted a design which was not limited by temperature. High strength materials permitted speeds of 24,000 rpm and more for comparatively large ratings and selenium rectifiers permit the use of transductors and may, to a certain extent, replace the commutator on d-c generators. The theory, used for these designs is of course the same as for commercial machines, except that it is refined and specialized to a very high degree. In order to get the most out of a certain amount of material, a good designer cannot use commercial proportions, density data and empirical curves, but must be prepared to
The ampere turns
are usually much less than obtained by commercial formulas and may be obtained either as Mr. Ginsberg indicates, or by using the results of investigations published.
In the same way, the pole-to-pole
leakage formulas and for commercial design mustabe flux disregar eachaseshol must bedlsregarded and each case should be calculated on its own merits. Direct and quadrature axis coefficients become a function of the saturation in the vicinity
of the air gap and a field weakening occurs,
due to flux distortion under load, similar to the uncompensated d-c machine. All these points of consideration complicate the design work to a great extent, but on the other hand permit the design of
a light-weight machine where each dimen-
sion is reduced to the minimum and where each Component carries Just as much as it can and not more. The result of this is that today 400-cycle machines can be built, with magnetic densities determined
buily
by
sagnetic denot detemped
entirely by saturation and notby temperature, and that often with a 100 per cent been
increased
two
four
times that
cercial thout detriment timent too commercial mcn machines without
a reasonable lifetime. It appears to the writer, that the designer of such equipment must be well versed in theory and rather unprejudiced by commercial design methods. He should dis card empirical curves and semiempirical formulas and take nothing for granted. Under theseasaconditions sheet ser neatrorda f design design serves only as a neat record for final design data, but by no means as a method. R. C. Powers (Electric Machinery Manufacturing Company, Minneapolis, Minn.): Although Mr. Ginsberg's paper presents no new concepts in the design of a-c generators, it does present a simplified procedure for such machinery. The design designing sheet itself is laid out in such a way that a complete picture of a machine can be had with little more than a glance and the performance can be determined quite readily. In the introduction of this paper, the author has stressed the importance of establishing criteria for the design and performance of relatively small a-c generators. In developing the steel magnetization curves (Figure 4), the flux densities were assumed to vrary sinusoidally. In small machines of the salient-pole type, however, where the ratios of air gap length to pole pitch are relatively small and where the flux densities are relatively high, the flux wave departs greatly from a sine wave. Test oscillograms have proved such waves
to
Ginsberg-Design Calculations for A-C Generators
approach
a squared condition.
In
AIBE TRANSACTIONS
i+o.6(Xag+Xad)+XagXad _ iF VI + 1.2 Xag+ Xag2
L
indicated by Figure
]
Fg _
/
/Z
OPEN-CIRCUIT "-0.8 P.F.SAT.
/SAT.
/es
o
4
of this paper but
higher effective permeabilities than those indicated by manufacturers' d-c magnetiza-
RPF SAT.
Figure 1 determinin the the constantsChC, and and CtCv determining from these oscillograms, however, they, constants
tion curves at corresponding inductions. Stator flux density-field intensity curves for grades of steel other than 29 gauge, 0.72 watt-loss can be interpolated from this curve 'and I the proper manufacturers' d-c magnetization curve. As expressed in this paper, there is a present demand for small light-weight a-c generators. The trend is toward higher flux densities, and, in order to maintain optimum electric and magnetic balances, the current densities become correspondingly higher. Recent developments in high temperature materials and high permeability electric steels have made such machines more practical from an economy standpoint. In high temperature a-c generators having good transient performance, armature resistance becomes increasingly effective; it may become as much as 250 per cent and more of the armature leakage reactance. In calculating the short-circuit ratio, reactances, and load excitations of a machine (refer to Figures 1 and 2), therefore, armature resistance cannot be ignored. The author's expression for full load excitation at 0.8 power factor (Figure 1) should be modified slightly:
fo too,s oap suchoawaygtha, i3 results(magnel tingl ves) wallye srmeas whnen m,cil, andrthe flu About thf armeotai e fr ilogas FFLng 1a+ 0.6(Xag+Xad) +XaoXad 1 prepared for this anindeped independent ths paper prveptigar ior paper,nan 1 L. on wbasibing conduted for invoes purpose of obtainig relb sato mag+XF'IFv+Fo'[1XXF'I netization curves. The fl.ux wave was
ine fro3 thos curvesdiffered included in this paper (Figure o). Tests indicate that these differences occur and a sinearewave der-Of mine from Figure 3 and mined flux is assumed in obtaining the flux density-field intensity curves, the end substantiresults (magnetizing for forces) a arere substantla
o
et
_
t e
analyzed by the same method as Mr. Ginsberg describes in this paper with the exception that an oscillogram of an actual air gap flux wave was used for analysis instead of a pure sine wave of flux. The flux densityfield intensity curves resulting from this investigation were effectively identical with those of Figure 4 of the paper. However, these curves proved to be too optimistic when applied to analyses of machines already in existence and tested; that is, the curves indicated excessively high effective iron permeabilities. Therefore, a new approach was used to determine reliable magnetization curves. The total magnetomotive forces at various inductions were determined by test for several machines using 29 gauge, 0.72 wattloss steel. The air gap magnetomotive forces were calculated, and the pole and spider magnetomotive forces determined from manufacturer's d-c magnetization curves at these inductions. The tooth and core flux densities were calculated at the corresponding inductions. Tooth and core magnetization curves were then plotted in such a way as to provide stator magnetomotive forces so that the total calculated machine magnetomotive forces
met the test values of total magnetomotive force at all inductions. These flux densityfield intensity curves for all machines effectively coincided with one another to result in a single curve for 29 gauge, 0.72 watt-loss steel. Test results of additional machnmes designed in accordance with the design sheets (Figures 1 and 2) of this paper and by the use of this curve met the calculated performance estimates very closely in all cases. This curve indicates lower effective permeabilities than those
1950, VOLUME 69
V
This expression is evident from test results, and from Figure 1 of this discussion.
E. R. Martin (University of Michigan, Ann Arbor, Mich.): The importance of transient factors on a small single-phase generator would appear to be relatively unimportant, and it is unfortunate that a 3phase design was not used to illustrate the calculations. The design itself is not according to usual practice. The stator has 27 slots which makes the winding unsymmetrical, A lap winding is used while a concentric winding would give better wave shape and use about 14 per cent less copper. The rotor is evidently of special construction, since the pole dimensions leave no room for a shaft, so the revolving field must be supported between nonmagnetic rings with stub shafts attached, which is expensive construction and not very satisfactory. In this case there is no spider, contrary to
calculations shown. The core density is higher than the tooth density while the reverse is generally used. The teeth are tapered, being wider near the air gap which makes it appear that a slot die was used which was made for another machine. The test value of core loss was not given in Table I, and that is one item that is difficult to calculate. A good 3-phase design would have made a much better foundation for the calcula-
discourages the use of empirical formulas.
The author heartily agrees with this philosophy and hopes that he will find the opportunity to describe these methods more fully. It would be interesting to compare the calculation of one of Mr. Franklin's generators by means of the methods presented in this paper with a calculation by Mr. Franklin's method. The need for a better knowledge of flux densities in the useful and leakage paths is recognized. In a recent conversation with Mr. Wieseman of the General Electric Company, the author learned that Mr. Wieseman has developed a machine for making 2-dimensional and 3-dimensional flux plots. Mr. Franklin seems to have overlooked one important application of the proposed design method. To the extent that the paper has adhered closely to formulas rigorously derived from fundamental principles, the design sheets are much more useful than a mere record for final design data. The orderly array of the formulas and curves provide a clear indication of the effect of the various design parameters on generator performance and facilitate the selection of these parameters in accordance with actual limitations rather than average
practice. Mr. Powers makes a number of
significant
points which call for specific answers. As he correctly points out, the stator magnetization curves were based on a sinusoidal distribution linear of flux distribution density, although discontinuous (for aa per-
fectly flat wave) would Wh-ilenearly correct for many the generators. of thsagmn a theoretcalize dity of thes argument has been realized, itof fu pointed out that the flux distribution S relat equation fierc betnat andvsine wae dflu ferences between actuda mainly to the unt where saturatep onf magnei in magnet flux
be more
thoeia vaidt
t q othe difference motive forces are negligibl small. In the sataed portos g g y saturated portions of the magnetic cir-
cuit, the sine wave approximation is quite
bore thisut. thexrmahinve borne thns outa The experimental cu vesfield f coefficietszreferedutoeb Mr. Powersfwas madeat thewrer suggestio in the course of a Government development the cn elet Machine
Manufacturing Company. At that time magnetization curves had been calculated for electrical grade steel only and it is not surprising that the ampere-tur values were optimistic compared to Mr. Powers' test data made with 0.72 wattstator
loss steel. A theoretical stator magnetization curve for 0.72 watt-loss steel has since been calculated and is compared in the following table with Mr. Powers' curve embodied in a report to the Engineer Research and Development Laboratories.
tions.
Flux density (kilolines). .......... 105..911i5. 120. i25 Mmf by Powers' test data.36......3 64. 118. 166. .240 Mmf from caclte .uv.2....6.i8.1620
David Ginsberg: Mr. Franklin's discussion emphasizes the importance of returning to fundamental principles of design and
As Mr. Powers points out, his data were obtained for machines with an almost rectangular flux wave and for a vrery small pole embrace, yet the agreement between
Ginsberg-Design Calculations for A-C Generators
1281
test and calculated curves is well within the limit of experimental error. The alternative implied by Mr. Powers' discussion is to have separate magnetization curves for every conceivable pole-shoe and air gap configuration. The writer believes that this would be a step backward. Mr. Powers is correct in stating that the voltage drop due to armature resistance cannot be ignored in concentrated machines operated at high temperatures. This factor has been taken into account in the formula for full-load field excitation by basing it on the stator flux required to generate the arithmetic sum of the terminal voltage plus the leakage reactance drop instead of the vector sum of these two quantities. Assuming the stator resistance to be of the same order of magnitude as the leakage reactance, this takes the resistance into account and simplifies the calculation. Mr. Powers implies that his proposed equation corrects a theoretical mathematical error in the writer's formula. It appears that his proposed formula contains typographical errors. For example, the subscript "ag" should be "aq" and the factor "1 + F"' should apparently be a leakage drop factor such as " 1 + Xi." Taking these obvious corrections for granted, Mr. Powers' formula would be theoretically more correct than that in the paper if he had not overlooked a very important physical phenomenon. This phenomenon involves the effect of armature reaction on the pole leakage flux. At a no-load per-unit voltage of 1 + X, the pole body flux density is the corresponding value of B' in the design sheet. However, at full load the pole leakage increases in
proportion to the following quotient: stator plus gap plus armature reaction
ampere-turns divided by stator plus gap
ampere-turns. This increases the polebody flux density and magnetomotive force. It is a very large factor where the pole bodies are highly saturated. By rigorous mathematical standards the writer's formula for full-load 0.80 power factor is incorrect but it is relatively simple and contains compensating errors which take account of the increased pole leakage under load. Its accuracy may be estimated by reference to Table I of the paper. Since the paper was written, the writer has applied his method to the design of a number of 50-cycle, 60cycle, and 400-cycle machines which have since been built and tested. The accuracy of the full-load and no-load field currents have been within 4 per cent of the predicted values even though the designs covered a wide range of ratings and extreme differences in magnetic saturation. It also
1282
has been found by mathematical analysis that the expression for field excitation can be still further simplified with negligible loss of accuracy. The simplified formula is:
FFL = [0-90F(1 +Xd) + F'- F0' ] [1 XF' I
. . . Mr. Martin questions the importance of transient factors on small single-phase generators and raises several questions concernig the construction of the machine used as an illustrative example of the design method. Transient response is very important for small generators since they are frequently called upon to start and accelerate motors whose locked-rotor current may be more than twice the rated current of the generator. The need for good transient response has been borne out by military experience. It also should be noted that some of the transient characteristics are used for the accurate calculation of steady-state values. This is particularly true of single-phase machines in which negative-sequence impedance determines steady-state characteristics. The 27-slot winding is not unsymmetrical but is a typical fractional slot winding so chosen as to permit the elimination of multiples of the third harmonic. The rotor of this machine actually does have a shaft and the term "spider," used in the calculations, refers to that portion of the magnetic circuit of the rotor which is located between the poles. The fact that the pole density is greater than the tooth density and that "the reverse is generally used" is of minor importance compared with the attainment of design goals. Mr. Franklin's discussion has clearly explained the fallacy of blind adherence to custom. Slot dies are normally used for a large number of machines and are not usually allocated to any specific machine. The slot die used on the generator calculated in the design sheets is one of the standard dies used by a prominent manufacturer and was selected as being the most suitable available die. The actual open-circuit core loss on this generator was 105 watts at rated volts. A small single-phase generator was deliberately chosen for illustrative purposes to show that the proposed design method applies even to these machines which have been ignored by the literature. Several oral discussions also were made at the meeting, and the author takes this opportunity to answer them. Mr. W. R. Hough of the Reliance Electric and Manufacturing Company expressed an interest in d-c generator design cal-
culations similar to those for a-c generators presented in this paper. Such d-c generator design calculations have been incorporated by the writer into a design sheet which has been used with some success at the Engineer Research and Development Laboratories for about one year. Publication of the d-c generator design calculations has been withheld until the method has been substantiated by sufficient experience with actual machines. The suggestion of Sterling Beckwith of the Allis-Chalmers Manufacturing Company to express the magnetomotive forces of the magnetic circuit in per-unit values is a novel one and appears to present an opportunity for broadening the value of design data on specific machines. The approval of this paper by Mr. L. A. Kilgore of the Westinghouse Electric Corporation is particularly gratifying since his paper of 19311 furnished much of the background for the present paper. His comments concerning the need for a synthetic design method are encouraging. As stated in my paper, considerable work has already been done toward this goal. The initial equations are set up to solve for generator dimensions in terms of desired values of generator weight, electrical output, transient reactance, and square inches per watt of heat dissipating surface. A number of simplified analytical expressions have been derived to make this approach practical. For example, the constant C1 can be calculated from the formula: g
C, =1.27 -) n
-0.35
Sin
(9OXpole embrace)
Certain attitudes toward accurate equations expressed in the paper and in the foregoing discussions may seem contradictory but they are actually consistent. In striving toward a synthetic design method which could be used for development, no attempt was made to simplify the equations by assuming factors based on customary design procedure or by ignoring factors which are negligible only in conventional machines. A high degree of accuracy even for unconventional designs is necessary if safety factors are to be reduced to a bare minimum. On the other hand, every effort was made to eliminate complicated expressions whose only justification was their mathematical rigor. REFERENCE 1. CALCULATION OF SYNCHRONOUS MACHINE AND TIME CONSTANTS CONSTANTS-REACTANCES AFFECTING TRANSIENT CHARACTERISTICS, L. A. Kilgore. AIEE Transactions, volume 50, December 1931, pages 1201-14.
Ginsberg-Design Calcukations for A-C Generators
AJEE TRANSACTIONS
