it would come if the wire were uniform.
ART. 405] THE WHEATSTONE-BRIDGE NETWORK 73
It is also evident that within an error of the second order which
need not be considered,
5r m = jf-R. (3)
The results expressed in Eq. (2) should be plotted in a curve as
follows:
Divide the axis of ordinates into distances li, 1%, 1 3 , . . . l n ,
corresponding to the various positions found upon the wire where
a balance was obtained. At each of these points raise ordinates
+10
+ 5
- 5
-10
-15
100 200 300 400 500 600
Millimeters
FIG. 405b.
700
900 1000
equal to 8li, 5? 2 , 5Z 3 , etc., to 5Z( n _i). The values of dl m may have
different signs and some of the ordinates may extend below the
axis. Thru the ends of the ordinates draw a smooth curve.
From this curve values which lie between the determined values
of dl m may be taken.
Fig. 405b shows such a curve drawn from actual observations
made upon a manganin wire 1 meter long. This wire had been
scraped in its middle portion for about one third of its length in
order to exaggerate its inequality.
We can now find an exact expression for the value of any un-
known resistance when measured with a slide-wire bridge in which
the calibrated wire is used.
74
MEASURING ELECTRICAL RESISTANCE [ART. 405
In measuring the resistance X with the slide-wire bridge in the
manner indicated in Fig. 405c, we have
W
X = m r (^
~ R-W m r '
where R is the total resistance of the wire and r the known resist-
ance.
P R-W m
FIG. 405c.
FromEq. (1)
Hence
X =
mR -f ndr
r.
nR (mR + ndr m )
Placing in Eq. (5) the value of dr m given in Eq. (3) we have
m - -
X =
r.
(5)
(6)
Since n and L occur as a ratio we can express them in any units
we choose provided the same units are used for each. Take
L = 1000, corresponding to 1000 millimeters for a slide wire
1 meter long, and take n equal 1000. Then
8l m
X =
m
1000 - (m + dl m )
r.
(7)
To use Eq. (7) proceed as follows:
Obtain a balance of the bridge by sliding the contact p along the
wire. Note the distance in millimeters (or in divisions equal to
TuW f the length of the wire, if ( this is not 1 meter long) from
the left-hand end of the wire to the point of balance. This dis-
tance will be m. From the curve find the value of dl m which
corresponds to m scale divisions (to millimeters in the case of a
ART. 406] THE WHEATSTONE-BRIDGE NETWORK
75
wire 1 meter long). Add this value to the reading m, and call the
result a. Then
a
X
1000 - a
r.
(8)
The quantity
may be taken from the table (Appendix
1000 - a
I, 1) as in other cases. If a calibration of a slide wire is made
carefully in this way and used, a slide-wire bridge may become
an accurate device for measuring resistance.
A trial was made of the application of this correction. The
wire used was the same one for which the calibration curve is
shown plotted in Fig. 405b. The known resistance r was given
different values and a resistance was measured using the con-
nections shown in Fig. 405c. The values obtained for X were
calculated by Eq. (7) and are exhibited below.
r in
m in
x,
x.
Per cent
ohms
millimeters
MM
m + 6Z w
measured
true value
error
25
790 8
+ 8.95
799.75
99.84
100
-0 16
100
507 1
- 7.00
500.10
100.04
100
+ 04
200
348.2
- 14.80
333.40
100.03
100
+ 03
300
261 5
-11.70
249.80
99.89
100
-0 11
The failure to get higher precision resulted from the crudeness
of the apparatus. This consisted of a wire stretched upon a
wooden meter stick with no provision for the sliding contact other
than the blade of a knife held in the hand.
406. The " Kelvin- Varley Slides." This is a device whereby
the slide wire of a slide-wire bridge may be replaced with sets of
resistance coils. The latter are so disposed that without an
excessive number of coils the ratio
may be obtained
1000 - a
and a be varied in steps as small as desired. This device con-
nected in a Wheatstone-bridge network is represented in Fig. 406.
In the arrangement shown there are in row 1, 11 coils, each of
resistance r. Spanning always two coils are the two contacts
PI, pi, which may be moved together over the studs or blocks to
which the coil terminals are joined. In row 2 there are also
11 .coils, each of resistance - , and two traveling contacts p%, p 2)
o
76
MEASURING ELECTRICAL RESISTANCE [ART. 406
which always span two coils of this row. In row 3 there are
10 coils, each of resistance ^= and one traveling contact p.
&o
Now, it is evident, if the resistance from contact pi to pi of
all below these contacts is equal to 2 r, that two coils of row 1
are always shunted with a resistance equal to these two coils.
Thus the resistance from pi to p\, when the contacts bear upon
the studs, will be r. There will be thus in row 1 ten equal
resistances, each of value r. Similarly, if the resistance from
contact pz to p z of all below these contacts is , then there will
5
be ten resistances in row 2, each equal to - Lastly, in order that
o
the total resistance of the 10 coils of row 3 shall equal , each
o
of the 10 coils must have a resistance ~r- With this arrangement
the total resistance from c to d will be 10 r ohms. The value
of the reading a will now be given by the positions occupied by
the left-hand contacts. With the positions shown in the figure
/i
becomes
the reading is a = 375 and the ratio
375
1000 - 375
1000 - a
0.6000 (see table, Appendix I, 1).
The same principle of subdivision may be indefinitely extended
to read in steps as small as desired. The one represented in the
ART 406] THE WHEATSTONE-BRIDGE NETWORK 77
diagram reads to 1 part in 1000. The Kelvin- Varley slides is a
device equivalent to a long slide wire. It may be given any
accuracy of adjustment desired. The cost of the mechanical
construction required and the many coils which must be adjusted
have limited the use of this otherwise excellent arrangement.
CHAPTER V.
WHEATSTONE-BRIDGE METHODS. VARIABLE RHEO-
STAT. ARRANGEMENTS OF RESISTANCES.
PER CENT BRIDGE. SUGGESTIONS
FOR USING BRIDGE.
500. Wheatstone-bridge Methods with Variable Rheostat. -
To carry out these methods, arrangements are provided for
setting the ratio arms to a chosen fixed ratio, and for varying the
resistance in the rheostat arm until a balance is obtained.
To change the setting of the ratio and to change the resistance
in the rheostat involves, by all ordinary methods, contact resist-
ances in at least two arms of the bridge. This is a possible
source of error which will be better understood from a consider-
ation of Fig. 500.
In this figure let a and b represent the two ratio arms, R the
rheostat and X the resistance to be determined. To change the
ratio a to 6 (in bridges of ordinary construction) it is necessary
to change the value of at least one of these resistances. This
may be accomplished by moving a point of contact, as p&, to
different positions along the arm 2 to 3. But there will be some
contact resistance, however good the construction, at the point p^
78
ART. 501] WHEATSTONE-BRIDGE METHODS 79
Call this contact resistance 8b. Likewise to change the value
of the rheostat arm 1 to 4 will require with any type of construc-
tion, at least one movable contact, as p R , which will have some
contact resistance which we may call 5R.
The equation of balance of the bridge, then, becomes
a R + dR
5b
If we expand the second member of Eq. (1) and neglect the prod-
uct 5b dR we find
(2)
The last term of Eq. (2) represents a necessary error in estimating
the value of X, which is due solely to contact resistances in two
arms of the bridge. In the special type of construction described
in par. 510, the contact resistance 66 is done away with by
shifting the galvanometer terminal to different positions along a
or b, these being soldered together at joint 2. The error which
still remains is the second term of the right-hand member of the
relation
X = -R + -dR. (3)
a a
This source of error should be reduced to a minimum by a
proper mechanical construction of the rheostat. To accomplish
this and at the same time make a rheostat which may be con-
veniently varied in small steps thru a wide range has given oppor-
tunity for a wide variety in design. The resistance coils or units
may be arranged in numerous ways and the mechanical construc-
tion for putting various resistance values in circuit may be greatly
varied. Both brass blocks, to be connected by plugs, and brushes
which can be moved over studs arranged in the arc of a circle,
called a dial, are extensively used. We shall now describe the
various methods of arranging the coils, but will omit a description
of mechanical constructions.
501. Arrangements of Resistances in Wheatstone-bridge
Rheostats. This subject was discussed by the author in an
article published in the Electrical Review, June and July, 1903,
80 MEASURING ELECTRICAL RESISTANCE [ART. 501
and much of what follows under this heading and the next is
take from that publication:
The fundamental purpose of a Wheatstone-bridge rheostat,
whether employing plugs and blocks or dials and sliding contacts,
is to provide means of obtaining the largest possible number of
values from the fewest possible number of accurately adjusted
resistance units and to do this without introducing into the circuit
objectionable contact resistances.
Where the number of coils is made greater than the least number
required theoretically, it is done to give some convenience of
working or to increase the ease or simplicity with which the values
are added up.
The theoretical combinations of resistance coils .which are
possible are given by the following relations: If there are n coils
these can be joined in various combinations, in series, or in parallel,
or in parallel and series or again in mixed arrangements. The total
number of combinations that can be formed, using the coils singly
and by joining them in series combinations, is
T = 2 n - 1. (1)
The same total number of combinations can be formed if the coils
are taken singly and joined in all parallel combinations^ but since
the coils used singly are the same for both arrangements, we
have as the total number of combinations possible for n coils used
singly and joined in series and in parallel combinations
N = 2 n -l-n+2 n -l or N = 2 n+1 - (n + 2). (2>
We will not, in general, by combining coils in all these ways, obtain
as many different values of resistance as there are combinations
of coils, for many of the combinations of coils give the same re-
sistance values even tho the resistance of each coil be different.
Thus, the much-used set of coils, of 1, 2, 3, 4 ohms, respectively,
can be joined in series in7 7 = 2 4 1 = 15 different ways, but
in 5 cases the same values of resistance are repeated, giving but
10 different values of resistance for this series of coils. The total
number of different values of resistance which can be obtained
from a given number of coils becomes in any particular case a
complex problem, if the number of coils be large.
If coils are arranged in all possible ways, we can get from 2 coils
4 arrangements, from 3 coils 17 arrangements and from 4 coils
106 arrangements. Thus it is often possible, when only a few
ART. 502] WHEATSTONE-BRIDGE METHODS 81
standards of resistance are at hand, to obtain, nevertheless, a
large variety of values.
502. Rheostat Coils; Classical Arrangements. To arrange
resistances in combinations so a series of values can be quickly
and easily added up has led to the employment of at least five
methods, tho only three of these are now in common use.
Siemens' Plan. The first plan is due to Siemens and consists
in joining a series of coils between blocks, the coils having the
values 1, 2, 4, 8, 16, 32, etc.
Resistance is thrown in circuit by removing plugs from between
the blocks. This plan employs the smallest possible number of
coils for attaining a given range of values, but, as the summing
up of the values is not an easy matter, this method is now
obsolete.
The 1, 2, 3, 4 Plan. By this plan all values from 1 to 10 ohms,
in steps of 1 ohm, are obtained by the coils 1, 2, 3, 4; all values
from 10 to 100 ohms, in steps of 10 ohms, by the coils 10, 20,
30, 40; all values from 100 to 1000 ohms in steps of 100 ohms, by
the coils 100, 200, 300, 400. By this arrangement of coils there
must be as many plugs as coils, the required values being obtained
by withdrawing plugs. As many plugs will be withdrawn as there
are coils in the circuit.
The 1, 2, 2 , 5 Plan. This arrangement of coils is used in pre-
cisely the same way as the one above. Most resistance boxes in
which plugs are removed to throw resistance in the circuit are
based on one or other of these two plans.
The 1,1,3,5 Plan. 1, 1, 3, 5; 10, 10, 30, 50; etc., will give ten
consecutive values in each unit's place, like the two above. This
arrangement, so far as the author knows, has not been used in any
resistance boxes placed upon the market.
There are no other four numbers which add up to ten which
will give ten consecutive values.
The Decade Plan. In this arrangement, as ordinarily applied,
there are 9 or 10 one-ohm coils for the units' place, 9 or 10 ten-ohm
coils for the tens' place, 9 or 10 one-hundred-ohm coils for the
hundreds' place, and so on. Each series of coils of the same value
is designated a decade. The connections as usually made are as
shown in Fig. 502.
It is apparent from this diagram that any value in any one
decade can be obtained by inserting between a bar and a block
82
MEASURING ELECTRICAL RESISTANCE [ART. 503
one, and only one, plug. It also appears that if several decades
are in series any value up to the limit of the set can be read off
directly from the position of the plugs, without any addition
whatever.
10 10 10 10 10 10 .10
It is evident that in the first four plans the values of resistance
required are obtained by withdrawing plugs which must be laid
aside, with liability of being lost, and one must make sure that
all the remaining plugs are well seated so as not to introduce
unknown contact resistances. Furthermore, as many plugs must
be employed as there are resistance units and to obtain any given
value may require the manipulation of a large number of plugs.
In the decade plan, upon the other hand, there is but one plug
used to a decade and this is always in service and hence not readily
mislaid. The use of only one plug to the decade makes it easy
to ascertain that this is tightly fitted in its place. When the value
is finally obtained, by manipulating only the one plug to the decade,
this value is readily read off without any mental summing up of
values. Again the decade plan alone permits of obtaining a suc-
cession of values by means of sliding contacts or dial switches, a
method which is becoming deservedly more appreciated.
These and other obvious advantages of the decade plan are in
part offset by the necessity, if the decades are arranged as in
Fig. 502, of using a larger number of resistance units than is
required by the previous plans.
503. Northrup's Four-coil Arrangement. An arrangement of
coils has been devised by the author which secures to the decade
ART. 503]
WHEATSTONE-BRIDGE METHODS
83
plan the further advantage of requiring no more coils than the
1, 2, 3, 4 or the 1, 2, 2, 5 plans. This arrangement may be ex-
plained as follows: Let the terminals of the 1 ohm and 2 ohm coils,
and the points of union of the other coils be numbered (1), (2),
(3), (4). (5), as shown in Fig. 503a. The current enters at point
(1) and leaves the coils at point (5), traversing 1, 3', 3, 2 = 9
ohms in all. If this series is multiplied by any factor n, then
n (1 + 3' + 3 + 2) = n 9 ohms. It will be seen that if the
1 (1)
AAAVVWWW +
(2)'
AA/WWWWV
AAAAA/WWW
(3)
(4)
A/WWWWW -
(5)
FIG. 503a.
points (1) and (5) are connected, all the coils are short-circuited,
and the current will traverse zero resistance. If the points (2)
and (5) are connected, the 3', 3 and 2 ohm coils will be short-
circuited and the current will traverse 1 ohm. By extending the
process so that we connect two, and only two, points at a time,
it is possible to obtain the regular succession of values n (0, 1, 2,
3, 4, 5, 6, 7, 8, 9), the last value being obtained when no points
are connected. The following table shows the points which must
be connected to obtain each of the above values and the coils
which will be in circuit for giving each value:
84
MEASURING ELKCTRICAL RESISTANCE [ART. 503
Value
Points connected
Coils used
(5-1)
1
(2-5)
1
2
(4-1)
2
3
(2-4)
1, 2
4
(3-5)
1, 3'
5
d-3)
3, 2
6
(2-3)
1, 3, 2
7
(5-4)
1, 3, 3'
8
d-2)
3', 3, 2
9
(0)
1, 3', 3, 2
Fig. 503b shows the method of connecting these points two at
a time, with the use of a single plug.
The circles in the diagram represent two rows of ten brass
blocks each. To the first two blocks at the top of the rows, the
points 5 and 1 are connected, to the second two the points 2 and
5 are connected, and so on, no points being connected at the last
pair of blocks. It is evident that if a plug be inserted between
the blocks 1 and 5, the points 1 and 5 are connected, giving the
value 0; if between the blocks 2 and 5, the points 2 and 5 are con-
nected, giving the value 1, and so on. The value 9 is obtained
when the plug is disposed of by being inserted in the last pair of
blocks which have no connections.
The only combinations of four coils which will give the dec-
ade in the above manner are the coils n (1, 3, 2', 2), n (1, 1', 4, 3)
and n (1, 3, 3', 2) where n may have any value.
The above method of obtaining the decade with only four coils,
as well as the ordinary decade arrangement, can be applied to a
dial or sliding brush construction. To accomplish this, 20 studs,
or ten pairs, are arranged in a circle. To make the required con-
nections, the pairs of studs at opposite ends of a diameter of the
circle must be successively joined together. This can be done
by rotating, with a handle at the center of the circle, a single
connecting bar or brush which will join successively pairs of studs
to give the values 0,1, 2, 3, 4, 5, 6, 7, 8, 9. The 20 studs are fas-
tened to the top surface of a hard rubber plate and there extends
down from each stud thru the plate a shaft slightly longer than
a resistance-unit or spool. On four of these shafts (beneath the
rubber plate) four resistance spools are mounted the resistance
values of these spools for a units' dial being I, 2, 3, 3' ohms; for a
ART. 503]
WHEATSTONE-BRIDGE METHODS
85
tens' dial 10, 20, 30, 30' ohms; for a hundreds' dial 100, 200, 300,
300' ohms, etc. The ends of the wire of the spools are joined,
one to the end of the shaft on which a spool is mounted, and
one to the end of an adjacent shaft. The ends of other shafts
are cross-connected in the manner shown in Fig. 503c. This
gives a view of the connections as seen from underneath the
A
1, 10, 100 or
1000 Ohms
2, 20, 200 or
2000 Ohms
3, 30, 300 or
3000 Ohms
3, 30, 300 or
3000 Ohms
FIG. 503c.
rubber plate. The circles in full line represent ends of resistance
spools. The circles in dotted line represent the brass studs
upon the upper side of the rubber plate. The larger circle in
dotted line represents a handle at the center of the circle of studs
located above the upper side of the plate. The bar or brush
shown in dotted line connects one pair of studs after another as
the handle is turned. The resistance value between the points
A and B with the bar in the position shown in Fig. 503c is 5 ohms.
Among other advantages of this method of arranging four
resistance units to give the decade in dial form may be men-
tioned the following: The traveling bar or brush contact can be
rotated continuously in either direction, as there are no electric
connections made to it. Hence, one can pass directly from the
value to 9 by turning back one stud. The construction is very
economical and there are only four coils in place of nine to put
and keep in accurate adjustment. It is easy to construct the
traveling bar of a number of thin copper leaves, which make a
right angle turn at each end, so as to give an end bearing of many
copper leaves upon the faces of the brass studs, thus securing a
certain and low-resistance contact with the studs. The disad-
vantages are slight, one being that 20 instead of 10 studs, as in
86 MEASURING ELECTRICAL RESISTANCE [ ART. 504
the ordinary arrangement of coils, are required. Also, at those
times where the contact, in passing from one stud to the next,
joins both together, the resistance value is not that of the stud
touched which reads the lower value, but is some odd value of
resistance. Hence, in using the dials the battery or galvanometer
key should be open while the dial is being turned. This objec-
tion has no weight in Wheatstone-bridge work, but it has some
weight in certain classes of work where it is desirable to follow
rapidly changing resistances with the battery and galvanometer
keys closed.
On the whole this type of dial construction, used in an ordinary
Wheatstone bridge, is economical, accurate, and highly satis-
factory in service.
504. Five-coil Combinations. The author pointed out * that
by using five coils the decade, to 9 inclusive, can be obtained in
a manner similar to that used for obtaining the decade with four
coils, as described above, from the values,
n(l, 1,2,2,3),
n (1,2, 2,2,2),
and n(l, 1, 1, 1,5).
Also that eleven values, namely to 10 inclusive, may be obtained
from the following arrangements of five coils:
n (1,2, 3, 2,2),
* (1,5,1, 1,2),
n(l, 3, 1,3, 2),
n(l, 1, 1,3,4),
n(l, 1, 1,4, 3),
n(l, 1,4, 1,3),
n (1,3, 1,3,2),
n(l,2, 1,4,2).
It was further pointed out that by traveling a single contact,
in the manner used with the four-coil arrangement, the general
method may be indefinitely extended. Any number of successive
values may be obtained with the use of many less coils than the
values obtainable. Thus from the seven coils,
n(l, 1,3,1,3,3,2),
we can get fifteen consecutive values inclusive of 0. It was also
* Electrical Review, July 18, 1903, Vol. 43, page 75.
ART. 507]
WHEATSTONE-BRIDGE METHODS
87
pointed out that in the above methods a sliding contact or a dial
switch may be used in place of a plug for making the connections.
505. Decade System of Feussner. In Fig. 505 is shown a
disposition (credited to M. Feussner) of blocks and resistance
units, whereby the values 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 may be obtained
by using five coils and only one
traveling plug. The resistance
units used are n (1, 1, 1, 1, 5)
where n has values 1, 10, 100,
1000, etc. An examination of the
figure makes the system easy to
understand. The four-coil decade
being still better, this method is
not likely to be much used in this
country.
506. Decade System of Irving
Smith. In Fig. 506 is shown a
disposition of 5 coils consisting of
n (1, 2, 2, 2, 2) where n has
values 1, 10, 100, 1000, etc.,
whereby the decade may be ob-
tained by traveling a single plug.
The system is easily understood
from an examination of the figure.
FIG. 506.
Mr. Smith has used this system with a dial construction to which
it is well adapted.
507. Multiple Arrangements. If it be required to obtain a
very low contact resistance of the plugs, to enable the last decade
88 MEASURING ELECTRICAL RESISTANCE [ART. 507
row to be varied with some precision in steps of 0.01 ohm or even
0.001 ohm, then a multiple arrangement of coils may be employed
with advantage. The principle whereby regularly increasing
values of resistance are obtained by joining coils in multiple is
given in Fig. 507.
FIG. 507.
It may easily be shown that when ten coils have values n (2, 6,
12, 20, 30, 42, 56, 72, 90, 10) ohms, where n has any value, their
resistance when all are joined in multiple is n ohms. Also the
nine coils n (6, 12, 20, 30, 42, 56, 72, 90, 10) have the resistance
2 n ohms when joined in multiple, the eight coils n (12, 20, 30, 42,
56, 72, 90, 10) the resistance 3 n ohms, the seven coils n (20, 30,
42, 56, 72, 90, 10) the resistance 4 n ohms, etc., to n (10) which