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THE TRANSMISSION OF ALTERNATING-CURRENT POWER WITH
SMALL EDDY-CURRENT LOSSES*
By A. H. M. ARNOLD, D.Eng., Associate Member.
[From the National Physical Laboratory.]
(Paper first received 2$th July, and in final form 23rd November, 1936.)
SUMMARY
A method is outlined of designing single-phase conductors
with small eddy-current losses. Experimental results are
given verifying the theory and the accuracy of the formula
for calculating the eddy-current losses.
(1)
(2)
(3)
(4)
(5)
For each value of a there is one shape of equi-inductance
line only.
It is of interest to note that when two conductors,
shaped in accordance with equi-inductance lines, are
oppositely electrified, the surface stress is uniform over
the whole surface of each conductor.
TABLE OF CONTENTS
Introduction.
Equi-inductance Lines for Single-phase Systems.
Effect of Finite Thickness of Conductors.
Experimental Work.
Design of Conductors for Single-phase Systems.
(1) INTRODUCTION
The problem considered in this paper is the transmission of alternating-current power along a single-phase
system of two conductors with the minimum of eddycurrent losses due to the non-uniform distribution of
current across the section of the conductors. The two
conductors are assumed straight, parallel, and of uniform and equal sections, and, further, the dielectric
current is assumed to be negligible, so that the lines of
current flow are normal to the section. The problem is
then purely two-dimensional, since the shape, size, and
disposition of the sections are the only variables in space.
The conductor current is assumed to vary sinusoidally
with respect to time.
It is well known that the non-uniform distribution of
current across the section of a conductor results from the
unequal inductances of the various filaments of the conductor. If it were possible to shape the section so that
all filaments had equal inductance there would be no
eddy-current loss. Such a section does not exist. If,
however, the section be represented by a line, and it is
assumed that the dimension of the section normal to the
line, the thickness, is infinitely small, then it is possible
to find two lines, representing the sections of the " go
and return " conductors of a single-phase system, which
satisfy the requirement of equal inductance of all their
parts. Such lines will be termed equi-inductance lines.
The ratio of the shortest distance between the lines to
the arc-lengthf of each line is defined by the symbol a.
• The Papers Committee invite written communications, for consideration
with a view to publication, on papers published in the Journal without being
read at a meeting. Communications (except those from abroad) should reach
the Secretary of The Institution not later than one month after publication of
the paper to which they relate.
•f The term " arc-length " is used for the length of the equi-inductance line
ni order to distinguish this dimension from the length of the conductor.
A
Y
Fig. 1.—Equi-inductance lines.
Arc-length of each line = s \ f l .
Minimum separation between lines •» LM j ' LM/s
(2) EQUI-INDUCTANCE LINES FOR SINGLEPHASE SYSTEMS
Fig. 1 shows two equi-inductance lines for a singlephase system. Conditions of symmetry require that
these lines shall be symmetrical about the axes XX
and YY.
[ 395]
396
ARNOLD: THE TRANSMISSION
OF ALTERNATING-
Fig. 2.—Family of equi-inductance lines.
The lines are all of equal arc-length and are drawn in their correct positions relative to the axes of symmetry XX and YY.
The inductance of a small element of the line at the
point P is equal to the sum of the mutual inductances of
all the elements of both the lines with the small element
at P, or, in symbols,
that the losses in a conductor, shaped approximately to
an equi-inductance line, may be calculated within a few
per cent from a formula for the eddy-current losses in an
B
Inductance at P = - ilog
s
(1)
where A and B move over the whole arc-length (s) of
their respective lines. If the integral in equation (1)
has the same numerical value for any position of P on
the line, then the line will be an " equi-inductance " line.
The author has been unable to find a solution of
equation (1) in terms of known functions, but the required equi-inductance lines have been obtained by a
process of approximate integration and successive
approximations. The family of lines over a range of a
from 0-025 to 1*5 is shown in Fig. 2. Only one halfline is shown, as the other half is symmetrical. It may
be seen that when a is very small, corresponding to a
small spacing, the lines approximate to two parallel
straight lines. When a is large, corresponding to a large
spacing, the lines approximate to circles, finally reaching
the circular shape when a = oo.
(3) EFFECT OF FINITE THICKNESS OF
CONDUCTORS
Conductor sections must have finite area, and therefore
can only approximate to an equi-inductance line which
has no area. The approximation may be made as close
as is desired by making the thickness of the section very
small, and the dimension normal to the thickness a close
approximation to the shape of an equi-inductance line.
The inductances of the elementary filaments of the conductor will then be very nearly equal and the eddycurrent losses will be very small.
Experimental results given later in the paper show
Fig. 3.—Rectangular sections for close spacings.
Equi-inductance lines shown dotted.
isolated tubular conductor, provided that an appropriate
modification is made in the definition of one of the terms.
Such a formula has already been published by the
CURRENT
POWER
WITH
SMALL
author* for the case of a tubular conductor remote from
all other conductors, and is reproduced here for convenient reference. It is:—
R
= 1 + a(z)(l - tf)
(2)
in which R' = alternating-current resistance of conductor, R = direct-current resistance of conductor,
jS = 2tfd, t = thickness of conductor (cm.), = | (outside
diameter—inside diameter), d = outside diameter of conductor (cm.), a(z) is a function of z, z = 8ir2tzfa, f = frequency (cycles per sec), and a = conductivity of conductor (c.g.s. units).
As the two conductors of a single-phase system of
tubular conductors are brought nearer together from an
EDDY-CURRENT
LOSSES
397
Since the section is shaped only approximately to an
equi-inductance line, equation (2) can only be used, provided the value of R'/R is fairly small. It has been
found experimentally that the equation is correct within
a few per cent, provided z is not greater than 10 or
R'lR is not greater than 2. For this range of z the
following simple approximate formula for a[z) may be
used:—
7z2
()
(3)
At a frequency of 50 cycles per sec, z does not exceed
10 if the thickness is less than 2 • 2 cm.
Equation (3) gives values for a(z) having errors of less
than 0 • 1 per cent, provided z is less than 10. Equation (2)
may now be written in the form
B
( N\
\
\\
\
/
/
.
v
^/
1
1
1
1
1
\
I
\
\
\
1
1
1
1
1
(
//
/
\
\
V
D
Fig. 4.—Channel sections for moderate spacings.
Equi-inductance lines shown dotted.
infinite distance apart, the eddy-current losses increase on
account of proximity effect. If, however, at the same
time, the shapes of the sections are altered so that they
always approximate to the equi-inductance lines corresponding to the spacing between conductors, then no
proximity effect will be introduced, and equation (2) may
be used to calculate the eddy-current losses for all spacings
between conductors. A new definition of jS is, however,
required.
For the tube,
j8 = 2f/(Outside diameter of tube)
= 27rt/(Outside circumference of tube)
For section shaped approximately
inductance lines,
to the equi-
j8 = 27rf/(Arc-length of section measured on side nearest
to return conductor)
* Journal I.E.E., 1936, vol. 78, p. 582, equation (3).
nation (9) is not required in this work.
equatio
The more precise
R
(315 + 3z2 - 0-002Z4)
(4)
For values of z above .10, equation (2) should be used,
and the value of a(z) should be obtained from Table 1 in
the previous paper;* but the error of the equation may
be rather large.
A conductor with a section shaped approximately to
an equi-inductance line has minimum loss for a given
thickness, but the loss increases slowly as the shape
departs from the true shape. A certain amount of
deformation will not appreciably affect the validity of
equation (4), and at the same time may enable a simpler
section to be obtained. Experience alone is the best
guide to the amount of deformation permissible, but as
a rough rule it may be taken that the average distance
between the centre line of the section and the equiinductance line should not exceed the thickness of the
* hoc. dt.
ARNOLD: THE TRANSMISSION
398
conductor, and the maximum distance should not exceed
twice the thickness of the conductor. With this rule in
mind, and provided j8, the ratio of 2TT times the thickness
of the conductor to the length of the section, is not less
than about 0* 16, it will be found that one of three simple
sections can be used for all spacings between conductors.
When the spacing is small, a lying between 0 and 0 • 3, a
OF ALTERNATING-
spacing between the conductors from the correct spacing
for theoretical minimum loss is shown. At low frequencies the eddy-current losses are increased in accordance with theory. At high frequencies, minimum
loss occurs at a spacing somewhat greater than the
theoretically correct spacing. The effect is most pronounced in the case of the rectangular conductors. At
Table 1
(1)
(2)
(3)
(4)
Rectangular section, close spacings
Channel section, moderate spacings
Low inductance
Little rigidity for resisting shortcircuit forces
Large surface area for dissipating heat
Heat dissipation may be reduced
on account of proximity of
return conductor
Moderate inductance
Great rigidity for resisting shortcircuit forces
Large surface area for dissipating heat
Heat dissipation unaffected by
return conductor
rectangular section is a sufficiently close approximation
to an equi-inductance section. Fig. 3 shows an example,
with the equi-inductance line drawn as a dotted line.
For rectangular sections jS may be taken as 2TT times the
thickness divided by the sum of one long side and two
short sides, i.e. 27r//(Length ABCD in Fig. 3). For
moderate spacings, a lying between 0-2 and 0*7, a
channel section, as shown in Fig. 4, may be used. For
channel sections /3 may be taken as 2TT times the thickness divided by the sum of the three longest sides, i.e.
27rf/(Length ABCD in Fig. 4). For large spacings, with a
greater than 0-6, the tubular conductor may be used.
Table 1 shows the advantages and disadvantages of each
section.
Large inductance
Great rigidity for resisting shortcircuit forces
The inner surface will not be able
to dissipate heat quickly
Heat dissipation unaffected by
return conductor
high frequencies the current is concentrated mostly on
the surface of the conductors, and the effect can be explained in this manner: When the conductors are very
close together the current is concentrated only on the
surface of the conductor nearest to the return conductor
and the two end surfaces. As the separation is increased,
21
2-0
1-9
1-8
(4) EXPERIMENTAL WORK
1-7
Experimental measurements of eddy-current losses
were made on three conductor sections of copper. The
conductors employed were each 20 ft. long. Two pairs
of conductors had rectangular sections with a ratio of
depth to thickness of 16 to 1 and 8 to 1 respectively.
The third pair of conductors had a channel section.
The experimental method employed was the same as
that used for measuring the eddy-current losses in solid
and tubular conductors, and has already been described.*
The conductors of rectangular cross-section may be considered to approximate to the equi-inductance lines for
zero spacing between conductors, and minimum loss may
therefore be expected when the conductors are close
together. The dimensions of the channel section are
shown in Fig. 6, and this section approximates closely to
an equi-inductance line if the separation between conductors is 4 • 3 cm. The results obtained when the conductors were tested at these spacings are shown in
Table 2, and it may be seen that the greatest discrepancy between the experimental results and the calculations by equation (4) is 3 per cent. The range of
frequency employed to obtain different values of a(z) was
from 25 cycles per sec. to 600 cycles per sec.
In the curves, Figs. 5 and 6, the effect of altering the
1-6
• Journal I.E.E., 1935, vol. 77, p. 55; and 1936, vol. 78, p. 588.
Tubular section, large spacings
Z-ll-2
1-4
1-3
1-2
11
1-0
2 3 4 5 6 7 8 9
10
Spacing between condttctors;cm
Fig. 5.—Values of R'/R for rectangular-section conductor
with varying spacing between conductors.
Spacing for theoretical minimum loss is zero.
Experimental points shown by crosses.
Conductor-section dimensions 10-17 cm. x 0-637 cm.
it may be seen from the equi-inductance lines that some
of the current will tend to flow on the surface of the
conductor farthest from the return conductor near to the
ends. Thus the effective section will be slightly increased, with a resultant fall in eddy-current losses. A
similar explanation applies to the channel sections,
although the effect is less pronounced.
CURRENT POWER
WITH SMALL EDDY-CURRENT
In designing a system of conductors, therefore, the
actual separation should be made greater than that indicated by the equi-inductance lines, if the calculated value
of R'/R is high.
LOSSES
399
By a few trial calculations it is found that the minimum
value of R' occurs when z is greater than 4.
A very simple equation, accurate to 2 decimal figures,
may be used for a(z) when z lies between 4 and 9. This
equation is:—
26
a(z) = l(z
- 2)
(6)
2-5
Inserting this value into equation (5), we have
2-4
2-3
s
-sii
2-2
2-1
Table 2
2-0
•Z-10-9
-2-55-
1-9
a(z)
A = Theoretical
value of R'lR,
calc. from
B = Experimental value
of R'jR
B/A
1-8
17
Z-7-5
1-6
1-5
1-4
1-3
\
5TZ-3-9
1-2
1-1
1-0
0
1 2 3 4 5 6 T '
Spacing1 between conductors, cm
Fig. 6.—Values of R'/R for channel-section conductors with
varying spacing between conductors.
Spacing for theoretical minimum loss = 4-3 cm.
Experimental points shown by crosses.
(5) DESIGN OF CONDUCTORS FOR SINGLEPHASE SYSTEMS
Theory
If all the dimensions of the conductors are fixed except
the thickness, and the thickness is gradually increased
from zero, the alternating-current resistance will fall,
reach a minimum, then rise, and will finally oscillate with
ever-decreasing amplitude of oscillation about a fixed
value. It is clear that it is useless to add further copper
after the first minimum resistance has been reached, since
none of the succeeding minima is as low as the first, and
the thickness of conductor which gives this first minimum
resistance should never be exceeded in any design. A
lesser thickness may, of course, often be employed
advantageously.
With the thickness as the only variable, the directcurrent resistance of a conductor is inversely proportional
to the thickness, and therefore inversely proportional to
the square root of z. From equation (4), therefore, we
may write,
(1) Rectangular section 10-17 cm. x 0-637 cm., j8 = 0-35,
separation between conductors 0 • 03 cm.
0-99 9
1-003
0-469
0-005
1-004
0-99 8
1-014
0-938
0-020
1-016
0-99 5
058
1-875
0-076
1-063
0-98 6
237
3-99
0-308
1-254
0-98 3
649
7-48
0-822
1-678
l-00 0
094
11-21
1-326
2-095
(2) Rectangular section 10-17 cm. x 1 • 274 cm., j3'= 0-63,
separation between conductors 0 • 02 cm.
1-051
l-00 0
1•864 0-075
1•051
1-184
0-99 7
3 •73
0-273
1•187
7 •46
1•561
1-570
0-819
1-00(3) Channel section 6- 35 cm. x 2 -55 cm. X 0-635 cm.
{outside dimensions), j8 = 0*35, separation
between conductors 4*39 cm.
0-456
0-912
1-824
3-88
7-29
10-91
0-005
0-018
0-072
0-292
0-794
1-290
1-004
1-015
1-059
1-241
1-655
2-065
1-004
1-021
1-079
1-280
1-664
2-028
l-000
i-oo6
l-019
1-03J
1'OOB
0-982
Thus
dR' _
dz
The minimum value of R' is obtained when dR'/dz = 0,
i.e. when
im)
«
Table 3 shows the values of z, a(z), and R'lR, for the
minimum value of R'.
The permissible thickness of conductor is therefore
h r
i
dependent on the value of /?. An example will now be
Rf = —r\l + o(z)[l — •£/?]/. where h is a constant . (5) given to show the method of design suggested.
400
ARNOLD: THE TRANSMISSION
OF ALTERNATING-CURRENT POWER
Example: To Design a Single-phase System of Adopting a value of a of 0-6, Fig. 7 shows a suitable
Conductors for a Frequency of 50 cycles per channel section, with the equi-inductance line for a = 0 • 6
sec. with an Effective Conductor Section of fitted to it.
The necessary separation between the conductors is
10 sq. in.
First approximation.
Table 3
Assume R'/R = 1 • 5. Then copper section required
Values
of
z,
a(z),
and
R'jR, for minimum value ofR'
= 15 sq. in. Assuming a temperature-rise of 30 deg. C.
from an ambient temperature of 20° C,
p
z
a{z)
R'/R
0
0-1
0-2
0-3
0-4
0-5
0-6
0-7
0-8
4-67
5-10
5-41
5-84
6-34
6-89
7-52
8-26
9-11
0-40
0-47
0-51
0-58
0-65
0-73
0-83
0-94
1-07
1-40
1-42
1-46
1-49
1-52
1-55
1-58
1-61
1-64
Conductivity of copper = cr = 0-00052 c.g.s. units
Thickness of section = t = — . / ( — )
2T7\ \2/cr/
For z = 5, t = 1 • 56 cm.; for 2 = 9 , t = 2 • 09 cm.
Assume t = \^ in. = 1-75 cm. Then mean arc-length of
section = 22 in., and
11
22
= 0-20
14' I
-ley;
Fig. 7
Second approximation.
For jS = 0 • 20, minimum R' occurs when z = 5-4, i.e.
when t = 1-62 cm. Take t = £ in. = 1 • 59 cm. Then
mean arc-length of section = 24 in.
The designer now has to choose his section. A
rectangular section, 24 in. x f in., would be very awkward to handle. The tubular conductor would be most
compact, but the necessary separation between conductors to minimize proximity losses would be large, and
the heat dissipation from the inner surface poor. The
channel section offers the most advantages in this case,
and it could be made in three or more pieces of rectangular
section, if desired.
The value of a has next to be selected. If a is taken
too small, the channel section will be little more compact
than the rectangular section, while if a is taken too large
it is difficult to fit the section to the equi-inductance line.
seen from Fig. 7 to be 16|in. The actual effective
section obtained with these conductors will now be worked
out.
Copper section = (14 X §) + (4f x f X 2) sq. in.
= 14-2 sq. in.
277 X
2
I
z = 8TT X 1-5872 X 50 X 0-00052 = 5-17
a(z) = 0-48
R'IR= 1-44
Effective copper section = 14-2/1-44 = 9-9 sq. in.
If a smaller spacing between conductors is required,
then it will be necessary to shorten the two short sides of
the channel, and lengthen the long side.
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