Alfred Still.

Principles of electrical design; d. c. and a. c. generators online

. (page 18 of 30)
Online LibraryAlfred StillPrinciples of electrical design; d. c. and a. c. generators → online text (page 18 of 30)
Font size
QR-code for this ebook


below the teeth will therefore be

0.28 X 9 X ^[(17.5) 2 - (9.5) 2 ] == 428 Ib.

Item (57) : Weight of Iron in Teeth.

0.28 X 1 X 0.521 X 9 X 57 = 75 Ib.

Items (58) to (60): Iron Losses. Refer Art. 31. The watts
per pound are read off the curve of Fig. 34 on page 102; thus,
for the armature core we have,

Bf 15,000 X 20
1,000 1,000

whence watts per pound = 3; and total watts = 3 X 428 =
1,284.

Similarly, for the teeth (item (59)) we get a loss with full-
load flux of 390 watts. The total iron loss of 1,674 watts, being
2.23 per cent, of the output, is rather higher than the average
as given in the table on page 104; but if the temperature rise
is not excessive, it will not be necessary to reduce the flux
densities.

Item (61): Total Loss to be Radiated from Armature Core.
Refer Art. 34, page 109. The copper loss to be added to the
total iron loss is the amount of item (50).

Items (62 to (64): Cooling Surfaces. Refer Art. 34, page 107.

Cooling surface item (62) = w X 19.5 X 11 = 675 sq. in.
Cooling surface item (63) = w X 9.5 X 11 = 329 sq. in.

Cooling surface item (64) = ^ [(19.5) 2 - (9.5) 2 ] X 4 = 1,820
sq. in.



212 PRINCIPLES OF ELECTRICAL DESIGN

Item (65): Temperature Rise. Refer Art. 34, page 107. The
radiating coefficient for the cylindrical surfaces is, by formula
(54),

i 50Q I Q 070
For the outside surface w c = - - 0.0457

lUUjUUU

v ^ - -j 1,500 + 1,500

For the inside surface w c = - ' = 0.03

1UU,UUU

In the case of the ducts, it should be noted that the radial
depth of the armature stampings is large because of the wide
polar pitch, and instead of taking the velocity of air through
the vent ducts as one-tenth of the outside peripheral velocity,

we shall assume a lower value, making v d = r^- Thus, the cool-

1.4

ing coefficient for the ducts and ends will be, by formula (56),



The procedure is exactly as followed in the example on page 111,
and the temperature rise is found to be 37C., which is within
the specified limit of 40.

Items (66) and (67): Equivalent Air Gap. Refer Art. 36,
page 117.

The permeance of the air gap over one slot pitch at the center
of the pole face where the actual clearance is 5 = Y in. may be
written,



= 98
and the equivalent air gap, as given by formula (58), will be



6.45 X 11 X 1.076 ._

-98~ - = 0-78 cm.

= 0.307 in.

Item (68): Drawing of Equivalent Flux Lines. Refer Art. 41,
page 129.

Before completing the drawing of the pole shoe as shown in Fig.
78, it is well to estimate the cross-section of the pole core, as



PROCEDURE IN DESIGN OF D.C. GENERATOR 213

this will be helpful in deciding upon the most suitable shape of
the pole shoe. The full-load flux per pole (item (51)) is 6,970,000
maxwells; and if we assume a leakage coefficient of 1.2 (Art.
56, page 187), the flux to be carried by the pole core is 8,370,000
maxwells, approximately. The density in the core may be as
high as 16,000 gausses, and the cross-section is therefore
8,370,000



16,000 X 6.45
cross-section,



= 81 sq. in. If the pole core is made of circular



Diameter = -\|81 X = 10.15
= (say) 10 in.

A laminated pole shoe of the shape shown in Fig. 78 will be
suitable. The stampings would be riveted together and at-




FIG. 78. Graphical construction for calculating permeance of flux paths.

tached by screws to the face of the cylindrical pole core. The
distance aO, measured on the armature surface, is one-half
of item (21), or 5.5 in.; while al is one-half of the pole pitch r
(item (20)), and measures 7.67 in. The tip of the pole is shaped
so that the air gap increases from the point / outward, being }/%
in. greater at the ends of the pole arc than at the center. The
extreme point of the pole shoe is rounded off* with a %Q-m.
radius. The reference points on the armature surface have been
chosen for convenience at intervals of 10, except in the neighbor-
hood of the pole tip, where the selected points are only 5 apart.



214



PRINCIPLES OF ELECTRICAL DESIGN



The construction of the flux lines is as explained in Art. 41,
and the measurements taken off the drawing have the following
values :



Points on armature


Length of equivalent flux
line = I cm.


Permeance per square centi-
meter = I/I


a, b, c, d, e, f


0.78


1.28


9


0.93


1.075


h


1.17


0.855


i


2.03


0.492


3


4.13


0.242


k


7.12


0.141


I


13.1


0.076


m


20.1


0.05




FIG. 79. Curves of air gap permeance and open circuit flux distribution-

Item (69): Permeance Curve. Refer Art. 39. The curve
marked P in Fig. 79 has been plotted from the above figures.
It shows the variation of air-gap permeance between pole and
armature at all points from the center of pole face to a point,
m, 10 beyond the geometric neutral.

Item (70): Actual Tooth Densities. Refer Art. 37, page 119.
In order to make use of formula (62), giving the relation between



PROCEDURE IN DESIGN OF D.C. GENERATOR 215



B g and B t at high densities, it will be found convenient to prepare
a table similar to the one below.



Bt


TT




d + M S










M(rf + 5)


tig


21,000


450


46.7


0.217


10,360


22,000


670


33.0


0.224


10,950


23,000


950


24.2


0.233


11,600


24,000


1,360


17.6


0.245


12,300


25,000


2,000


12.5


0.264


13,100



Values of the actual tooth density, B tj from 21,000 to, say,
25,000 gausses are assumed, and the corresponding values of H











































































^^r**"


+^


**


*"


24000




























^


^^






























- <"


*a


^x-


^^^














&3

II 9QQOO


















^


^




















01














]


^


X 1






















3
O 29000












/


/


























Jj










/


/




























3

C
<u 21000








/


/






























Q *.iuw
X






/


/


































20000












































/




































IQOOfl




/













































































100 200 300 400 500 600 700 800 900 1000 1200 1400 1600 1800 2000
Magnetizing Force (Gilberts per Centimeter) = H

FIG. 80. B-H Curve for armature stampings high values of
magnetization.

and M are then found by referring to the B-H curve, Fig. 80,
which is similar to Fig. 4 except that the quantities concerned
are expressed as B and H because this is more convenient for
obtaining /z.

The quantity which is a function of ju> in formula (62), can
now be determined, and the corresponding values of B readily
calculated.

In this example, we shall consider the tooth density in. the



216



PRINCIPLES OF ELECTRICAL DESIGN



root, or narrowest part, of the tooth; and t in the formula is
therefore taken as 0.466 in. (item 34). The results of this
calculation are shown graphically in the upper dotted curve
of Fig. 81.



Tooth Density, JBj , ( Gausses )




























/
























>


//


























//


























/


/
























/


A
























/




F


jrmi


la((


2)














I




/
























I


/














API


arer


Formula (63) L
t Tooth Density!


1 /


























1


[/






















































I/


























1




























f




























f


























/




























/


























i


/






















































1

























2000300040005000600070008000 10000 12000 14000
Air Gap Density, B g, ( Gausses )

FIG. 81. Curve giving relation between air gap and tooth densities.

The "apparent" tooth density at the bottom of the tooth is,
by formula (63),

1.076 X ir



which enables us to plot the lower dotted curve of Fig. 81. The
actual density in the iron of the teeth is almost exactly expressed,
for the low values, by formula (63); while at very high densities,
the actual tooth density approaches more and more nearly the
values calculated by formula (62) without ever quite reaching
them. It is therefore possible to draw a curve, such as the full



PROCEDURE IN DESIGN OF D.C. GENERATOR 217

line in Fig. 81, which very closely expresses the true relation
between the tooth density and the average density over the slot
pitch, for the entire range of values from zero to the highest
attainable.

Item (71): Saturation Curves for Air Gap, Teeth, and Slots.
Refer Art. 38, page 121, and Art. 42, page 132. We are now in
a position to plot curves similar to- those of Fig. 49, page 133;
and, in order to obtain a proper value for the ampere-turns neces-
sary to overcome the reluctance of the teeth, the correction for
the taper of teeth should be applied. The results of the calcu-
lations for the teeth are shown in tabular form; the meaning of
the different columns of figures being as follows:

First column: Assumed values of air-gap density B g , including
the highest value likely to be attained.

Second column: The corresponding values of the density B t
at the bottom of the tooth (read off the full-line curve of Fig. 81).

Third column: The magnetizing force H, calculated, when
necessary, by applying SIMPSON'S rule (Formula 64), as explained
in Art. 38.

Fourth column: The ampere-turns required to overcome the
reluctance of the teeth, being

Hd e X 2.54

0.47T

where d e is the " equivalent" length of tooth; its numerical value,
in this example, being (1 -f 0.25) - 0.307 = 0.943 in.



B .


B t


H


(SI)t


12,000


24,400


790


1,500


10,000


22,100


347


642


8,000


19,300


110


210


6,000


15,800


20


38



As an example of the method of calculation, consider the value
B g = 10,000; the corresponding value of tooth density, as read
off Fig. 81, is B t = 22,100. This is the actual density at the
root of the tooth. Referring to items (32) and (34), it is seen
that, over a distance of 1 in., the width of tooth changes by the
amount 0.576 - 0.466 = 0.11 in. The width of tooth at the
distance d e from the bottom of tooth (see Fig. 38) is therefore



218 PRINCIPLES OF ELECTRICAL DESIGN
0.466 + (0.11 X 0.943) = 0.57; and the density at this point is
B w = 22,100 X = 18,100



At the halfway section, B m = - 5Li = 2 0,100

The corresponding values of H, as read off the B-H curves, Fig. 80
and Fig. 2, are:

At bottom, H n = 700

At middle, H m = 310

At top, H w = 144.
By formula (64), we have,

700 2 X 310 , 144
Average H - -g- H - - ^ - + -^- = 347

which is appreciably higher than the value of H at the section
halfway between the two extremes. This difference will, how-
ever, hardly be noticeable on low values of tooth density; and
indeed the somewhat tedious work involved in the above calcu-
lation is quite unnecessary with small. values of tooth density,
because the ampere-turns required to overcome tooth reluctance
are then, in any case, but a small percentage of the air-gap
ampere-turns. The values of H, in the above table, for B g =
8,000 and B g = 6,000, are those corresponding to the average
values of the tooth density (B m ).

Having plotted in Fig. 82 the curve for the teeth only, the
straight line for the air-gap proper can now be drawn for the
points under the center of the pole face where the equivalent
air gap is d e = 0.307 in. Obviously, since HI = 0.47r>S/, and
B has the same value as H in air, we may write,



= 0.62B,

This gives us the line marked A in Fig. 82. The addition to this
curve of the ampere-turns for the teeth and slots, results in the
curve marked a, 6, c, d, e, f, which may be used for all points
under the pole where the permeance has the same value as at a.
The curves for the other points on the armature surface may now
be drawn as explained in Art. 42.

Items (72) to (76) : Flux Distribution on Open Circuit. Refer
Arts. 40, 41, and 42. From the point R on the permeance curve



PROCEDURE IN DESIGN OF D.C. GENERATOR 219

(Fig. 79) draw a straight line to the point on the datum line
immediately below the pole tip, in order to obtain the curve A
of flux distribution on open circuit, all as explained on page
131 (Art. 41). Measure the area of this flux curve, and draw



11000



10000




1000 2000 3000 4000 5000 6000 7000 8000

Ampere -Turns per Pole

FIG. 82. Saturation curves for air-gap, teeth, and slots.



the dotted rectangle of equal area. The height of this rectangle
represents the average air-gap density over the pole pitch, and

i u a / x 6,430,000

its numerical value is B g (average) = ^ ^ x 15 34 X 11 =

5,910 gausses.

Using this length as a scale for measuring other ordinates of the



220



PRINCIPLES OF ELECTRICAL DESIGN



flux curve, the value of B g at all points on the armature periphery
can be determined.

The ampere-turns between pole and armature, due to field
excitation on open circuit, have at every point on the armature



surface the value SI =



B f .



0.47T X permeance per square centimeter



9000



M.M.F. Curves

Ampere Turns per Pole

Field M.M.F. (Open Circuit)

\ Field M.M.F. (Full Load)




I kj ihg f e d c b a, b c d e f ghijk I m

Reference Points

FIG. 83. Curves showing distribution of m.m.f. between pole and

armature.

The actual figures are given in the following table.



Point on armature surface


P, q . cm.


B g


SI


o, 6, c, d, e, /,


1.28


7,840


4,860


9


1.075


6,580


4,860


h


0.855


5,220


4,850


i


0.492


2,910


4,700


J


0.242


I7305v


4,275


k


0.141


581


3,270


I


0.076









From these figures the dotted m.m.f. curve of Fig. 83 has been
plotted. Observe now that the m.m.f. represented by 4,860
ampere-turns per pole is not sufficient to overcome the reluctance
of the teeth, and in order to obtain the required total flux on



PROCEDURE IN DESIGN OF D.C. GENERATOR 221



open circuit, the m.m.f. between pole face and armature core,
(i.e., bottom of slots) must exceed this value. Referring again
to Fig. 82, it will be seen that, for a density of 7,840 gausses under
the pole face, the m.m.f. to overcome tooth reluctance amounts
to a little over 200 ampere-turns. The ordinates of the dotted
m.m.f. curve of Fig. 83 may therefore be increased throughout
in the proper proportion, the maximum addition being 200
ampere-turns. This corrected curve may now be used to plot
with the aid of the magnetization curves of Fig. 82 the actual
distribution of flux over the armature surface, when the effect



12000
11000
10000











































































^


^




































^


^


^


^


'\


\












soon




















^


^


^


^


B






\\


















A/






/"


?


-X


^










N


\\










6000
5000








/




/


/




^














^


\ u
















1


/




,/


^


















\
















/


/


/






















I


















/


/
























\










90OT1






/


//


























l\
















//


/


























H











1000
2000
3000
4000
5000




^


y/






























^


^>






']


AH


siti


m








Direction of Travel of Conductors




^


s












































V












Flux Curves
( Distribution of Flux Density
over Armature Surface )


















\


























x















































I kjihg f e d c b a b c d e f ghijk I m

Reference Points

FIG. 84. Curves of flux distribution over armature surface.

of tooth saturation is taken into account. This has been done
in Fig. 84, where the curve marked A is similar to the flux curve
of Fig. 79 except in so far as its shape may be modified by tooth
saturation.

The procedure above described for obtaining the actual flux
distribution curve is logical and correct; but for practical pur-
poses it is usually permissible to assume that the curve A of
Fig. 79 shows the actual flux distribution, the slight modifica-
tion brought about by variable degrees of tooth saturation being
neglected. It is then a simple matter to plot the required open
circuit field m.m.f. curve directly by taking from Fig. 82 the



222 PRINCIPLES OF ELECTRICAL DESIGN

values of SI corresponding to each known value of the air-gap
density, B g .

The average ordinate of the curve A in Fig. 84 as obtained
by dividing the area under the curve by the length of the base
is found to check within 1 per cent, of the required amount
(average B g 5,910 gausses). Had there been an appreciable
difference between the calculated and measured areas of the
flux curve, it would have been necessary to correct the m.m.f
curve of Fig. 83, and re-plot the flux curve A of Fig. 84.

Items (78) and (79): Flux Distribution under Load. Refer
Art. 43, page 137. The curve of armature m.m.f., of which the
maximum value is 3,565 ampere-turns (item (17)), may now be
drawn in Fig. 83. The point k has been selected for the brush
position because the (positive) field m.m.f. has here about the
same value as the (negative) armature m.m.f. From the re-
sultant m.m.f. curve, the flux curve B of Fig. 84 is plotted, the
area of which measured between brush and brush is found
to be 99.5 sq. cm. while curve A measures 107 sq. cm. The
average air-gap density is therefore less than before current
was taken from the armature, and the loss of flux is due partly
to distortion (tooth saturation) and partly to the demagnetizing
ampere-turns (brush shift).

Items (80) to (83): Corrected Full-load Flux Distribution.
Refer Art. 43. The e.m.f. to be developed at full load is 238.3
volts, obtained by adding the numerical values of items (45),
(46), and (48), to the full-load terminal voltage. The final flux

1 0fi 3 ^ 238
curve C should therefore have an area of -



sq. cm.; the number 106.3 being item (74). In order to estimate
the probable increase in field excitation to obtain this increase
of flux, we may follow the method outlined on page 138. The
ampere-turns necessary to bring up the flux from the reduced
value under curve B to the original value under curve A } are
calculated by assuming that the air-gap density under the pole

99 5
has changed from 7,800 gausses to 7,800 X ^y = 7,260 gausses;

and the ampere-turns necessary to increase the developed volts
from 220 to^ 238.3 are calculated by assuming that the air-gap
density under the pole must be raised from 7,800 to 7,800 X

238 3

= 8,450 gausses. The total additional excitation is indi-



cated by the distance SS' in Fig. 82, its value being about 900



PROCEDURE IN DESIGN OF D.C. GENERATOR 223

ampere-turns. This must be added to the open-circuit m.m.f.
curve of Fig. 83, all the ordinates of which must be increased in

. 5,050 + 900 _, .

the ratio - , n , n . The new resultant m.m.f. curve ob-

o,uoU

tained by adding this full-load field m.m.f. to the armature
m.m.f. may now be used to plot the final full-load flux curve
C of Fig. 84.

Items (84) and (85) : Diameter of Commutator. Refer Art. 53,
page 181.

A diameter of commutator not exceeding three-quarters of
the armature diameter will be suitable. Let us try a diameter

13 5

D c = 13.5 in., making v c = 3,070 X - ;r^ = 2,130ft. per minute.

iy.o

This dimension is subject to correction if the thickness of the
individual bar does not work out satisfactorily.

Items (86) to (88): Number of Commutator Bars. Refer Art.
27, page 93. On a 220- volt machine, the potential difference
between adjacent commutator segments might be anything
between 2.5 and 10 volts (page 94). If we provide the same
number of commutator bars as there are slots on the armature,
the average voltage between the segments at full load would

230 X 4

be about ?= = 8.63, which is within the limits obtained in
o/

practice. At the same time commutation will be very much
improved by having a smaller number of turns between the
tappings, and since the number of inductors in each slot is
not divisible by 4, we shall have to provide 57 X 3 = 171
commutator segments.

Items (93) to (99): Dimensions of Brushes. Unless a very
soft quality of carbon is used, the current density over brush-
contact surface does not usually exceed 40 amp. per square
inch. Taking 35 as a suitable value, the contact surface of all

S26
brushes of the same sign will be -^~- = 9.32 sq. in., or 4.66 sq.

in. per brush set. If the brush covers three bars, the width
will be W = 0.247 X 3 = 0.741 in. Let us make this dimen-
sion % in. The total length of brushes per set, measured in
a direction parallel to the axis of the machine, will then be
l c 4.66^0.75 = 6.23, or (say) 6 in., which can be made up of
six brushes each 1 in. by % in.

"Item (100) : Length of Commutator. The spaces between brushes
(which will depend upon the type of brush holder) and the



224 PRINCIPLES OF ELECTRICAL DESIGN

clearances at the ends, including an allowance for " staggering"
the brushes, will probably require a minimum axial length of
commutator surface of 7J^ in.

Item (101): Flux Cut by End Connections. Refer Art. 48.
Assuming for the constant in formula (72) on page 159, the
average value k = 2.4, we have:

$ e = 0.4 \/2 X 2.4 X 3 X 83.4 X ^ X 2.75 [ (log. y) - 1 ]
= 31,300 maxwells.

Item (102) : Slot Flux. Refer Art. 49. The equivalent slot
flux, by formula (80), page 164, is

1.6 X TT X 1 X 3 X 83.4 X 11 X 2.54



6X05



maxwells -



Items (103) and (104) : Average Flux Density in Commutating
Zone. Refer Art. 49. By formula (81) page 164:

3>c = (2 X 11,730) + 31,300 = 54,760 maxwells. The
average density is, therefore, by formula (83) :

54,760
B < = 1.083 X 11 X 6.46 = 712 gaUSSeS '

Items (105) and (106) : Flux Densities at Beginning and End of
Commutation. The value of item (104) is the density of the
magnetic field at the middle of the zone of commutation. After
the current in the short-circuited coil has passed through zero
value (a condition attained only when the coil as a whole is
moving in a neutral field), the field should increase in strength
until, at the end of commutation, it is of such a value as to
develop I C R volts in the short-circuited coil. The resistance, R,
of the coil of one turn is 0.00132 ohm (item (42)), and the e.m.f.
to be developed in the coil at the beginning and end of commu-
tation is therefore 0.00132 X 83.4 = 0.11 volt, or 0.055 volt in
each coil-side. The flux to be cut by each coil-side to develop
this voltage is:

Maxwells per centimeter __ volts X 10 8 _
of armature periphery " rate of cutting, in centimeters per

second

= 0.055 X 10 X 3 )070 X ?2 X 2.54
= 3,530



PROCEDURE IN DESIGN OF D.C. GENERATOR 225



and this corresponds to a density of



3 530
^ 2 54 = 126 gausses.



The ideal flux density at beginning of commutation is therefore
712 126 = 586 gausses, and at the end of commutation, 712+
126 = 838 gausses.

Item (107) : Sparking Limits of Flux Density in Zone of Com-
mutation. Refer Art. 52. The time of commutation, in seconds,



5000 -



4000 -



3000



2000



1000 -




1000 -



I k j i h g

FIG. 85. Flux distribution in zone of commutation.

is approximately t c = ^r = o 130 v 12 = 0-00176 sec.; and, by
formula (88) page 177;

$ = % X 0.00176 X 10 8
= 132,000 maxwells.

The variation in flux density which is permissible when carbon
brushes of medium hardness are used, will therefore be

132,000
1.083 X 11 X 6.45 ==1^20 gausses.

A portion of the full-load flux curve C of Fig. 84 has been re-
drawn in Fig. 85 to a larger scale, and the flux density required

15



226 PRINCIPLES OF ELECTRICAL DESIGN

to obtain ideal commutation, together with the permissible
variation from this value, are indicated on the same diagram.
It will be seen that there should be no difficulty in obtaining
sparkless commutation at full load with the brushes in the selected
position (over the point k) ; but the brush might be moved with
advantage 4 or 5 nearer to the leading pole tip; and, al-
though the final flux curve C would be slightly modified, it
would not depart materially from the line drawn in Fig. 85.
The angular degrees referred to are so-called " electrical" de-
grees, because the pole pitch has been divided into 180 parts; the



Online LibraryAlfred StillPrinciples of electrical design; d. c. and a. c. generators → online text (page 18 of 30)