the two rods are at the same temperature.
With a D' Arson val galvanometer of such a sensibility that
10~ 8 ampere will produce a deflection of one scale-division with
the scale at a meter distance, as the detecting instrument, the ratio
coils RI and R z may assume resistances considerably higher than
100 ohms, so that the contact resistances of the sliders p\ and p 2
can be entirely neglected.
This method gives results in low-resistance measurements
similar to those obtained with a Kelvin double bridge to be later
described. But in this method a balance must be obtained twice,
instead of only once as with the Kelvin double bridge.
The method is very conveniently applied when the conductor
3-4 is a 5-ohm meter bridge, the wire of the meter bridge resting
upon a meter scale marked off in millimeters. The resistance of
this wire per centimeter of length must be accurately known. If
the conductor 1-2 is of low resistance, then RI would be made
smaller than R 2 , so that a length I on 1-2 of, say, 50 cms. would
correspond in resistance with a length of, say, 75 cms. on the
bridge wire.
This method may likewise be applied to the measurement of
low resistances between potential points. In this case pi would
first be set on one potential point, and a balance obtained with
p z at b, then on the other potential point and a balance be again
r>
obtained with p 2 at 61, the ratio ^ being so chosen that L would
KZ
be a considerable proportion of the length of the bridge wire.
Then if x is the resistance sought between potential points, and
if p is the resistance of the bridge wire 3-4 per centimeter of length,
ART. 403] THE WHEATSTONE-BRIDGE NETWORK 61
we have as above,
* = |L P , (2)
which gives x in ohms, when L is in centimeters.
403. The Carey-Foster Method. The Carey-Foster method
of using a slide-wire bridge, while being a very elegant precision
method, is not as much employed in this country as formerly.
It, nevertheless, deserves a somewhat extended consideration.
Tho the method was originally devised for the measurement of
very low resistances, it is even better adapted to the accurate
comparison of medium or even high resistances. The success
with which the Carey-Foster method may be applied to precision
work in comparing resistances depends a good deal upon how well
the apparatus which is needed for carrying out the method is de-
signed and made. We can only give here the theory of the method
referring the reader to trade publications for a description of the
mechanical features of the Carey-Foster bridges which are upon
the market.
The unique feature of the Carey-Foster modification of the
Wheatstone slide-wire bridge consists in interchanging the stand-
ard resistance, which is in one arm of the bridge, and the resistance
under comparison which is in the adjacent arm. A reading is
taken of the position of the contact on the slide wire necessary
for a balance before and after the resistances are interchanged.
By this procedure all resistances other than those being measured,
as well as all constant thermal E.M.F.'s in the bridge circuits, are
eliminated.
The connections for the Carey-Foster bridge method are shown
in Fig. 403a. S is a standard resistance coil and Si is a coil of
approximately the same value, which is to be compared with S.
Ri and R 2 are two fixed resistance coils of nearly equal value.
di and a 2 are the number of units of length of the portions of the
bridge wire to the left and right respectively of the galvanometer
contact p. If p is the resistance of one unit of length of the
bridge wire, then pai and pa 2 represent the resistance of the bridge
wire to the left and to the right of the galvanometer contact.
n and n\ may be taken to represent all resistances of unknown
value in the two bridge arms. As above stated, the Carey-Foster
method provides for eliminating these unknown resistances by
taking two readings of the position of the galvanometer contact
62
MEASURING ELECTRICAL RESISTANCE [ART. 403
p on the bridge wire, one reading when the coils S and Si have
the position shown in Fig. 403a and one when these coils are inter-
changed. The bridge is mechanically so constructed that this
interchange of the standard coil and the coil under comparison
can be quickly and easily made.
The dotted line a, Fig. 403a, represents a low-resistance shunt
to the bridge wire and its leads, which may or may not be used.
When used it has the effect of magnifying the distances that the
contact p must be moved along the bridge wire to obtain a balance
in the two positions which the coils S and Si are made to assume.
The general theory of the method is the same whether this shunt
is or is not used.
When S and Si have the positions shown in Fig. 403a, and the
contact p is so chosen that no current flows in the galvanometer
circuit,
Ri = S + n + pai (}
7~) ~ o [ j ^ \ /
where p is the resistance per unit length of the bridge wire.
S and Si are now interchanged and a balance is again obtained
by sliding p to a new position on the bridge wire. Calling the
two lengths of bridge wire, which are now to the left and to the
right of the slider a/ and a 2 ',
Ri _ Si + n -f pai /2\
7") f | j / \ /
Changing the forms of Eqs. (2) and (3) by adding unity to each
side, we obtain
Ri + R 2 S + Si + p (di + a 2 ) + n + ni
pa 2
(3)
ART. 403] THE WHEATSTONE-BRIDGE NETWORK 63
Ri + R 2 S + Si + P (ai + a*') + n + ni ,
and "" '
Changing the position of the galvanometer contact p does not
alter the total resistance of the bridge wire, hence,
P (ai + a 2 ) = P (a/ + a 2 ')- (5)
The numerators of the second members of Eqs. (4) and (5) are,
therefore, equal, and, as the second member of each of the two
equations is equal to the same quantity, the denominators of the
two equations are equal.
Thus, we have
Si + pa z + m = S + pat + ni,
or
-Si = S - P (a, - a/), (6)
or, as paz paz' = pai pai = p (a/ ai),
Si=>S-p (a/ - ai). (7)
Eqs. (6) and (7) show that the difference in the resistance of the
standard resistance S and the resistance Si under comparison is
equal to the resistance of a certain length of bridge wire.
If, once for all, the resistance p per unit length of the bridge
wire be determined, then the difference in the resistance of any
coil under comparison and a standard coil is given in ohms with
great accuracy.
There are several methods known for determining the value of p.
A simple but inferior method is to balance two standard coils the
difference of whose resistances S and Si is accurately known, then
S Si /0 s
P = ^ / (8)
&2 #2
On account of differences of temperature and temperature
changes in the coils S and Si it is not simple to accurately deter-
mine the difference in their resistances. We have, therefore, found
the following method of determining the value of p, by means of
four readings, to be very accurate and satisfactory :
It is necessary for the determination to use a coil Si the value
of which does not need to be known, but which is sufficiently
near the resistance of coil S to make it possible to obtain a balance
with p (Fig. 403b) not far from the center of the bridge wire. An
ordinary resistance spool or resistance box will answer very well.
The resistance may be varied by shunting or by any rheostat
64
MEASURING ELECTRICAL RESISTANCE [ART. 403
method until a value is reached by trial which will balance the
standard S with p near the center of the bridge wire.
C is another coil or box of resistance coils with which S may be
shunted by closing the contact K. This shunt coil should have
about 100 times the resistance of S and its resistance need, not be
known to a high degree of accu-
racy.
The four readings are taken as
follows:
Reading 1 is taken with K
open and S and Si in the posi-
tions shown in Fig. 403b. Call
this reading a\. Reading 2 is
taken with S, together with coil
C, and Si interchanged and K
open. Call this reading a 2 . S t
together with coil C, and Si are
now interchanged again so that they have their original positions.
K is now closed, shunting S with (L Call the resistance of S when
shunted Si. Reading 3 is then taken, which we will call a/.
Lastly S together with coil C is interchanged with Si, K is closed,
and reading 4 is taken, which we call a* .
The value of p is then deduced as follows:
Before S was shunted
K
FIG. 403b.
After shunting S
Si = S p (a 2 ai).
F
Si = Si f -p (aj - a/)
Eliminating Si from Eqs. (9) and (10), we derive
Si f -S
p =
0,1
Since Si' =
CS
C + S
, the value of p may also be expressed
p =
(9)
(10)
(ID
(12)
When low-resistance coils are being compared the distance that
the galvanometer contact p, Fig. 403a, moves over the bridge wire
in obtaining a balance with S and Si in the two positions is often
very small.
ART. 403] THE WHEATSTONE-BRIDGE NETWORK 65
In order, therefore, that the same slide wire may serve for com-
paring coils of both high and low resistance it has been found
advantageous to be able to shunt the bridge wire with a low-resist-
ance shunt. This shunt is represented in dotted line in Fig. 403a
and called v.
The lower the resistance of this shunt the greater is the range
of motion of p over the bridge wire. If the bridge wire is cali-
brated (that is the value of p determined) after being shunted, no
error is introduced by such shunting, provided the resistances of
the leads n and n\ to the bridge wire do not alter after the calibra-
tion is made. This possible error is avoided by making these leads
of heavy copper cable.
The bridge may be provided with three bridge wires of different
resistances, any one of which may be used at will. Two shunts
may be provided for the wire of lowest resistance, and thus there
is altogether the equivalent of five different bridge wires. This
arrangement adapts the apparatus to making direct comparisons
with a wide range of resistances.
The bridge, as mechanically designed, often consists of two
separate units.
One unit, which we may designate " the coil holder," consists
of a hard-rubber base upon which are mounted massive copper
bars for holding and connecting the ratio coils and the resistance
standards, also the commutating device for interchanging the
standard and the coil under comparison.
The other unit, which may be called the " bridge," consists of a
hard- wood base upon which are fastened the three slide wires and
three scales. A sliding contact maker is also a part of the
" bridge."
The two units are joined together electrically by low-resistance
cables.
The Carey-Foster method, as above described, is especially useful :
(1) For comparing with fundamental standards of resistance,
coils of approximately the same resistance. For this purpose the
bridge is adapted to comparing coils with standards of resistance.
(2) Coils may be compared with the standards when they differ
in value from the standards very considerably, provided one is
supplied with a resistance set of only very moderate accuracy.
If the resistance coil has a value higher than the standard, it is
shunted with the coils of a resistance box until the value of the
66 MEASURING ELECTRICAL RESISTANCE [ART. 403
shunted combination is sufficiently near that of the standard to
enable an exact balance to be obtained with one of the bridge
wires. If the coil under comparison has a lower value than the
standard, then the standard is shunted. The exact method of
procedure will be explained later.
(3) A valuable application of the method is the obtaining of
temperature coefficients of resistance coils or specimens of wire
of any kind, and exceedingly accurate results can be obtained.
(4) The method is very useful in adjusting a number of resist-
ance spools to the same value. For this a bridge wire is used, of
the same kind and size as the wire of the resistance spools. A
single reading will then give at once without any calculation the
exact length of wire that must be cut off to make the resistance of
the spool equal to that of the standard.
(5) A Carey-Foster bridge, together with a few reliable standard
resistances, enables one to check up the coils of a Wheatstone
bridge or standard resistance box with great accuracy.
(6) Very low resistances, such as the contact resistance of plugs
in a resistance box or the resistance of lead wires, are conveniently
measured with the Carey-Foster bridge. For such cases a thick
metal bar of known resistance may occupy the place of a standard
resistance coil.
When one wishes to compare with a standard resistance another
resistance differing from it considerably, we may do so by shunt-
ing the standard resistance, if it has the higher value or the
resistance to be compared, if it has the higher value. The pro-
cedure is then the same as in other cases.
Suppose first that the resistance under comparison is greater
than the standard. Then if we shunt this resistance with a known
resistance a- we have
where D = a 2
or we obtain Si = a ^ . (13 )
a o ~r pU
If, second, the resistance is less than the standard, the standard
may be shunted with a resistance a and we have
-81 = Wr- - PD. (14)
ART. 403] THE WHEATSTONE-BRIDGE NETWORK
67
The value of a does not need to be known with great accuracy
if its value is considerably greater than the resistance which it
shunts. Thus if a = 100 S, and is in error one-tenth of one per
cent, the calculated value of S will be in error only about one-
hundredth of one per cent.
It may happen that we wish to compare a coil of, say, nominally
10,000 ohms resistance with a standard coil of that resistance. In
general the coil to be compared will differ so much from the standard
that a balance cannot be obtained with the slider on the bridge
wire. In this case a variable resistance may be inserted in series
with the coil of lower resistance and adjusted until a balance is
made possible with the slider on the bridge wire. This added
resistance, if not known, can then be measured and its value added
or subtracted as may be required from the coil being compared.
Again it may be required to measure a resistance which is lower
than the total length of one of the two bridge wires. Such a case
would be where we wish to obtain the value of the lead wires
connecting the mercury cups of the bridge to a coil in a resistance
box. This case is .easily met by connecting together one set of
mercury cups by a copper bar of negligible resistance, or a bar of
resistance metal of negligible temperature coefficient and known
resistance.
Such a bar, then, takes the place of a standard coil, serving, in fact,
as a standard of zero or very low resistance. If R Q is the resistance
of this short-circuiting rod, and R m the low resistance being meas-
ured we have the same relation as that expressed by Eq. (6), namely,
R m = RQ p (a-2, 0,2).
Following are a few specimen readings which will serve to
illustrate a satisfactory method of procedure. Some readings are
also given below which were taken in calibrating one of the bridge
wires. The numerical calculation of the final result is also given.
a i
A]
r-i
Temp,
st'd.
Temp,
coil, a/
Temp,
coil, <Z(
Mean
temp.
Res., st'd.
Res. of coil
at observed
temp.
512.0
510.5
+ 1.5
23.8
19 '.4
19.6
19.5
100.0068
100.0062
471.0
554.0
- 83.0
23.8
26.5
26.5
26.5
100.0068
100.0403
453.0
586.0
-133.0
24.6
30.4
30.2
30.3
100.0070
100.0607
Here column (1) gives the readings in millimeters of the length of
scale to the left of the contact when an end A of the commutator
68
MEASURING ELECTRICAL RESISTANCE [A RT . 403
is towards the ratio coils. Column (2) gives the scale readings
after reversing the commutator. Column (3) gives the difference
between the two sets of readings, regard being had to the sign.
Column (4) is the observed temperature in degs. C. of the standard.
Column (5) is the observed temperature of the coil at the time of
taking reading a/. Column (6) is the temperature of the same
at the time of taking reading ai. Column (7) is the mean of the
two temperatures. Column (8) is the resistance of the standard
corresponding to the observed temperatures in column (4), these
resistances being calculated using the certified value and tempera-
ture coefficient of the standard, or more conveniently read from a
curve plotted to show the relations between resistance and tempera-
ture of the standard. Column (9) gives the calculated resistances
at the different mean temperatures of column (7) of the coil under
comparison. When a sufficient number of readings are taken for
different temperatures a very accurate curve showing the changes
in resistance due to temperature changes may be plotted.
With the particular bridge wire used in obtaining the above
results p = 0.000404 ohm per millimeter of wire, and thus, if S
be the resistance of the standard given in column (8) for particular
temperatures, the results in column (9) are obtained from the
relation Si = S - 0.000404 (a/ - ai). See Eq. (7), par. 403. The
actual readings and the resistances used for obtaining the above
value of p are as follows:
10-ohm standard, No. 1592, called Si. 10 ohms shunted with
1000 ohms, called Si'.
Si' = j p = 9.90300,
10.00206 1000
S - Si = 9.90300 - 10.00206 = - 0.09906.
l
<Z 2
V
a 2 '
Temp,
standard
Res., st'd.
Res. shunted.
526.5
526.5
526.1
526.1
526.3
497.0
497.5
497.5
497.5
497.4
649.0
648.5
648.5
648.5
648.4
374.0
374.5
374.5
374.5
374.4
30.6
a
te
i (
10.00206
u
11
10.000
tl
1C
((
mean readings.
p =
Si'
- 0.09906
- ai f - a* 374.4 + 526.3 - 648.4 - 497.4
ART. 404] THE WHEATSTONE-BRIDGE NETWORK
69
or
P =
- 0.09906
- 0.000404.
- 245.1
404. Galvanometer Resistance. Measured, Using the " Second
Property " of the Bridge. This method, called Kelvin's method,
offers a good example of the employment of the " second property "
of the bridge. The method would be employed where only one
galvanometer, the one whose resistance is to be determined, is
available. The method is equally applicable for determining the
resistance of a millivoltmeter.
Make the connections as shown in Fig. 404.
r i r 2
-AMM/ 1 ' WVWWWWWWVN
Ba
FIG. 404.
The resistance R should be chosen about equal to the resistance
g of the galvanometer. The E.M.F. of the battery Ba will gen-
erally be too great to be applied directly to the bridge, as the
galvanometer will deflect off its scale. The E.M.F. may be re-
duced to - - of its value in the manner indicated in Fig. 404.
It should be so adjusted that the deflection of the galvanometer
(or other deflection instrument having a resistance g to be deter-
mined) remains upon the scale at all times. K is a key. It will
be found that the deflection of the galvanometer is varied when
the sliding contact p is moved along the slide wire ab and the
deflection will also vary when the key is closed, except for one
position of the slider p. The measurement is made by finding
this position of p where the deflection of the galvanometer remains
unaltered when the key K is open or closed. When this position
70 MEASURING ELECTRICAL RESISTANCE [ART. 405
is found the battery circuit and the key circuit are conjugates,
and then by the " second property " of the Wheatstone bridge
( 400) we have
R c I c n
jTtt or 9= R > (1)
when I is the length of the bridge wire and c is the distance from
end a. This method, due to Lord Kelvin, gives very good results
when correctly applied. The source of E.M.F. applied to the bridge
and the key may be interchanged when the same results follow.
A trial was made of the above method in measuring the resist-
ance of a Weston millivoltmeter. The connections were made
as in Fig. 404. A slide-wire meter bridge was used, the resistance
of the slide wire being 0.1397 ohm per centimeter (a much higher
resistance than is usual in this type of bridge).
Ba was a storage cell. R was selected 10 ohms. The value
found for the resistance corresponding to the 200-millivolt scale
was
I- c n 100 - 49.1 ,
g = R = 2771 10 = 10.366 + ohms.
c ~ty . i
A second trial was made in which R was again chosen 10 ohms,
but use was made of extension coils each of 100 ohms. These
were inserted at the ends of the bridge wire, as shown in Fig. 401a
above. Remembering that the resistance of the bridge wire had
been found to be 0.1397 ohm per centimeter the result obtained
becomes
p 100 + (100- 36) X 0.1397, n inQ79
R - 10Q + 36XQ.1397 = 10 ' 372 hms '
Here l\ = the total length of the bridge wire in centimeters mul-
tiplied by the resistance of the wire per centimeter, and Ci = the
length in centimeters from left end of wire of sliding contact mul-
tiplied by its resistance per centimeter. This last result differs
from the first by about 0.06 per cent which shows that both
methods are fairly precise.
The method is too insensitive, as applied above, for measuring
the high resistance of a voltmeter.
1 405. Calibration of Bridge Wire. When a slide-wire bridge
is used and high precision is required in resistance measurements
it is necessary to calibrate the slide wire for uniformity of resist-
ance. A manganin wire, if carefully selected and handled, will
ART. 405] THE WHEATSTONE-BRIDGE NETWORK
71
be fairly uniform in resistance. Nevertheless this fact should be
determined, and, if it is found not to be uniform, corrections should
be applied in as simple a manner as possible. We may make the
calibration and obtain an expression which will give the values of
the corrections to apply as follows:
In Fig. 405a, p, p, etc., represent ten or more resistance coils
which are exactly alike in resistance. They may have, conven-
If
m 5 6
H[^ VWWW
FIG. 405a.
iently, a resistance of about 100 ohms each. The absolute resist-
ance of these coils does not need to be known and it is easy to
adjust a number of such coils to be equal in resistance with the
aid of a Wheatstone bridge of very simple construction. One of
the terminal wires of each spool may be shortened or lengthened
by trial until each coil matches some one taken as a standard.
If the spools are wound with manganin wire the difficulty of
maintaining the temperature sufficiently constant while the adjust-
ments are being made is not great.
The terminals of these coils, n in number and joined in series,
are connected by means of heavy leads cp and dq to the terminals
of the bridge wire to be calibrated. It is assumed that this wire
can be divided into divisions of known length by means of a scale
near to, or lying underneath, the wire. A cell of battery, with
some resistance P in series, is joined to the ends of the bridge
wire at the points p and q (not at c and d). A galvanometer has
one terminal connected to a traveling contact and' the other
terminal t is arranged so that connection may be made at the
various points between the coils, as 1, 2, 3, etc.
72 MEASURING ELECTRICAL RESISTANCE [ART. 405
Then, if the contact t is at point 1 a balance of the galvanometer
will be obtained when the sliding contact is at a distance li from
the end of the wire, or, in general, when the contact t is at any
point m' between the coils, a balance will be obtained when the
sliding contact is at a distance l m from the end p of the wire. As
the coils p, p, etc., are all of equal resistance it is evident that the
wire becomes in this way divided up into n lengths of equal re-
sistance. If the wire is not uniform in resistance these n lengths
will not be equal.
Let R the total resistance of the wire from p to q and
L = the total length of the wire,
then R will be of the total resistance.
n n
If we call TFi the resistance of - of the length of the wire or, in
general, W m the resistance of of the length of the wire, we shall
71
have
W m =R + 8r m , or Wm =^R-8r m ,
according as W m > or < R. Choosing the first case
8r m =W m -R. (1)
Here dr m is the small difference in resistance between of the
n
length of the wire and of the total resistance of the wire.
Again, let l m = the distance from end p at which a balance is
obtained when the terminal t is between the mth coil and the
(m + l)th coil. Then
n n
Choosing the first case
X7 7 ^ T ffy\
Oi m = l m Li. (Z)
Here 8l m expresses the difference in the lengths (at points 1, 2,
3, etc. . . . n) where a balance comes on the actual wire and where