the variations in c are one half as great as the variations in X.
302. Illustration of the Practical Advantages of Differential
Circuits. To see the advantages of the above properties of
differential circuits let the diagram, Fig. 301b, be reconstructed
as given in Fig. 302.
46
MEASURING ELECTRICAL RESISTANCE [ART. 302
In Fig. 302, 1, 2, 3 are three binding posts of the differential
apparatus and 4, 5 are the terminals of a resistance X which is
supposed to be placed at a distance, which may be very great,
from the differential instrument. If the resistance of the lead wire
A is made equal to the resistance of the lead wire B, Eq. (16) ( 301)
still holds. For, giving A a resistance and B an equal resistance
is not different from increasing by an equal amount the resistance
of each of the windings g and g, and as the resistance of these
windings does not appear in either Eq. (15) or Eq. (16) ( 301) the
J
o
o
k
ir
3 jl
A 4 X
FIG. 302.
equal resistances A and B do not enter these equations. In other
words the resistance X, and changes in the resistance X, may be
determined by knowing the constant resistances R, I, S and the
resistance c which is varied for obtaining a balance; the resistances
g and g } A, B, and C not appearing.
The chief industrial application of the above principles is in
the measurement of temperature with electrical-resistance ther-
mometers which may be placed at a distance from the reading
instrument. The resistances of the lead wires which only need
to be three in number can vary (provided the A and B leads are
equal in resistance) without affecting the accuracy with which the
resistance of the thermometer is determined, and hence the tem-
perature which is a function of its resistance.
Again, the method may be employed for the determination of
any unknown resistance X. If the contact p is set at the middle
of slide resistance fk and the resistance R is varied until a balance
ART. 302] NULL METHODS 47
is obtained, we have, without regard to the resistance of the lead
wires, X = R.
If R is made equal to X at a particular temperature, then
small variations in X, due to temperature changes, can be accu-
rately determined by moving p over the slide resistance fk till the
galvanometer is balanced. The variations in X are then given
by the relation
2 SI
sx = s+l s - . .. (
If fk is a slide wire of uniform resistance, the variations in c
can be read directly on a millimeter scale and thus the curve giving
the relation between the resistance and the temperature of a wire
can be very accurately determined. As S can be given any value,
let it equal infinity, then dX = 2 be. This shows that when the
method is used for determining temperature coefficients, the highest
value of c (which cannot exceed that of I) will not be greater than
one half the total change which takes place in X with the greatest
variation in temperature employed. For example, suppose it is
required to determine the temperature coefficient of a copper coil
which has a resistance of 10 ohms at C. in the range to 100 C.
At 100 C the resistance of the copper coil would be about 14.2 ohms.
Then its increase in resistance is 4.2 ohms and the resistance of the
slide wire fk could not be less than 2.1 ohms. The resistance R
would be so chosen that the galvanometer would be balanced
when X was at C. and p is set at /, or when c = 0. Let it next
be required to determine the temperature coefficient of a 1-ohm
coil in the same range. The increase in resistance of this coil in
the 100 C. range would only be 0.42 ohm and the distance to
move on the slide wire would only correspond to 0.21 ohm or 0.1
of its length. If, however, the slide wire is shunted with a resist-
ance of a proper value (and R is always so adjusted that with the
1-ohm coil at C. we have c = f or a balance), we can again have
c = I when the 1-ohm coil is at 100 C., and thus determine the
temperature coefficient of 1 ohm with the same percentage pre-
cision as 10 ohms.
To find the proper value to give S we assume an approximate
increase in the resistance of the 1-ohm coil when the temperature
increases 100 C. We then solve Eq. (1) for S and find
48
MEASURING ELECTRICAL RESISTANCE [ART. 303
Since 5X is to be 0.42 ohm, and I has previously been made 2.1
ohms, and 8c is to be practically the whole length of the slide
wire, or 2.1 ohms, we derive
It should be noted in the above methods that the only movable
contact is in the battery circuit, and hence variations in its resist-
ance in no wise affect the readings.
Analysis shows that, for obtaining greatest sensibility, each wind-
ing of a differential instrument should have a resistance which is
approximately equal to the resistance external to that winding.
Or, approximately, we should have
V I /'OX
9 = x + 2 ' (3)
303. Differential Galvanometer Used with Shunts. This
method is perhaps better applied with a differential galvanometer
FIG. 303.
of the moving-magnet type. Some of these galvanometers are
made with a bell-shaped magnet hanging in a hole in a copper
sphere to damp the oscillations of the system. The current
coils are two in number and can be individually moved along a
horizontal bar so as to approach or recede from the system.
When the same or equal currents are sent in opposite directions
thru the two coils, one or both of these can be moved into a
position where the galvanometer is perfectly differential. If this
adjustment is made and it be assumed that the resistance of its
two coils are alike and known, and if they can be relied upon to
remain constant, we can, by the following well-known method,
measure any resistance X. Let the circuits be arranged as in
Fig. 303.
ART. 303] NULL METHODS 49
Here g\ and g% are the two windings or coils supposed to be
alike in resistance, so gi = g 2 = g, the resistance of each. Each of
these coils is shunted with resistances Si and S 2 respectively.
R is a fixed resistance and X a resistance to be measured.
Denote by / the current in the battery circuit and let i% denote
the current in the right-hand coil and ii the current in the left-
hand coil. To find an expression for the value of X, when the
galvanometer is balanced, we make use of the following law of the
division of currents:
Let a circuit divide into two branches of resistances x and y,
and let the resistance x carry a current ii and the resistance ?/, a
current i' 2 and let ii + iz = I be the total current; then
77 '7*
ii = 7 / and i' 2 = - /;
x+y x+y
that is, the current in either branch is equal to the resistance of
the other branch divided by the total resistance around the circuit
times the total current.
Applying this principle to the circuits shown in Fig. 303, we
have
Y [ s *9*
j _ &+02 &
Y _L P -4- l i J_
A- i ft ~t~ TY i " ~r
oi-T0i
and
^2^2
T~
Q Q _|_
01 + 91 &2 + ^2
If the galvanometer is so adjusted that g\ = gi = g, and we
alter Si, S 2 , and R until there is no deflection, we get from Eqs. (1)
and (2)
(3)
^* t
The method expressed by equation (3) can only give accurate
results on the assumption that the resistance g is either the same
at different temperatures, or is known at the temperature at which
the measurement is made. To realize the first assumption the
50 MEASURING ELECTRICAL RESISTANCE [ART. 304
windings must be made of some low-temperature coefficient
wire, as manganin, which has a high resistance, and to realize
the second assumption requires the taking of accurate temperature
readings of the coils. Moreover, it would in this case be impossible
to make the shunts Si and S 2 bear any fixed relation to- the re-
sistance of the windings. From these considerations the method
is seen to be inferior to the Wheatstone-bridge methods of measur-
ing resistance which are in vogue. But, with galvanometer
windings of manganin and shunts of suitable values, the precision
and the range of resistance-measurement possible can be made
equal to that of the ordinary post-office type of Wheatstone
bridge. For example, suppose
Si = jy and S 2 = infinity,
then, by Eq. (3), X = 100 R, and if R can be varied in steps of
1 ohm from to 10,000 ohms we can measure a resistance X =
1,000,000 ohms. Or, suppose
^ 2 = QQ anc * $1 = infinity?
then, by Eq. (3), X= 0.01 R, and if R = 1 ohm, X = 0.01 ohm.
The above method is thus seen to require, for precision and
range, a specially constructed galvanometer and shunts which
have values of at least ^ and gV of the resistance of a coil winding
and a 10,000-ohm rheostat variable in steps of 1 ohm. It has,
therefore, little to recommend it in competition with the bridge
methods to be described, and is given here chiefly to complete
the treatment of the differentially wound instrument as used for
resistance-measurements .
304. The Differential Telephone. The magnet of a tele-
phone may be wound with a differential winding. This instru-
ment can then be used to indicate when there is an equality
between two currents which are alternating or unsteady. The
two wires of the differential winding should be twisted together
and wound bifilar upon the magnet. The theory of this instru-
ment is complicated, and the circumstances under which it can
be used to advantage are limited, therefore we shall not give
further discussion to it here.
CHAPTER IV.
THE WHEATSTONE-BRIDGE NETWORK.
BRIDGE METHODS.
SLIDE-WIRE
400. Network of the Wheatstone Bridge. The Wheatstone
bridge is a network of six conductors and should be distinguished
from the Kelvin double bridge (to be discussed later) which is a
network of nine conductors. While the Wheatstone net or bridge
may assume many forms, the essential electrical properties are
the same in all.
This network of six conductors may be represented in three
ways which are equivalent. They are presented in I, II, and III,
Fig. 400a.
In the three representations like letters designate the same
conductors and the same points of junction. Following the
lettering of the diagrams, it will be seen, from a well-known prop-
erty of the bridge, that, if the resistances 6, c, x, y have the
relation
yb = ex (1)
there will be no current in a if there is an E.M.F, in z, and, con-
versely, there will be no current in z if there is an E.M.F. in a.
Under these circumstances the two conductors a and z are said
to be conjugate conductors. Other pairs of conductors can become
51
52
MEASURING ELECTRICAL RESISTANCE [ART. 400
conjugates. By referring to diagram I we can write, from the
symmetry of the diagram, the following relations:
If yb = ex, then z and a are conjugates.
If za = by, then x and c are conjugates.
If xc = az, then y and 6 are conjugates.
The above relations belong to what we shall term the " first prop-
erty " of the Wheatstone bridge.
We shall presently show that if any two conductors are con-
jugates, in whatever part of the network an E.M.F. be placed, the
current which flows through one of the conjugates is independent
of the resistance of the other conjugate. For example, if in
diagram III a and z are conjugates and an E.M.F. is placed in
branch c, the current which will flow in z will be the same what-
ever may be the resistance of its conjugate a. This principle
is made use of later in determining the internal resistance of a
battery and the resistance of a galvanometer. We shall term this
the " second property " of the Wheatstone bridge. The above-
mentioned " first " and " second " properties of the Wheatstone
bridge may be deduced as follows:
A
FIQ. 400b.
If, in Fig. 400b, we place an E.M.F. in the branch BC we shall
have the distribution of currents shown in the diagram, where
i e is the current in c,
i a is the current in a,
i x is the current in x,
i e i y is the current in y,
i a i x is the current in 6,
and i c i y + i x is the current in z.
As before (Fig. 400a), x, y, z, a, b, c designate ohmic resistances.
If it is true that x and c are conjugates, then the E.M.F. in c will
ART. 400] THE WHEATSTONE-BRIDGE NETWORK 53
produce no current in x, and hence we shall have i x = 0. Assume,
then, the current i x is zero and determine the relation of the re-
sistances necessary to produce this result. With i x = 0, the points
A and will be at the same potential, and we shall have by Ohm's
law the fall in potential from B to A = ai a and the fall in potential
from B to = y (i c i v ) and these will equal each other, or
aia = y (i c ~ i v ). (2)
Likewise (remembering that we have assumed i x = 0) we shall
have
bi a = z (i c - i y ). (3)
Hence, taking the ratio of Eq. (2) to Eq. (3), we obtain
a y
1 = -, r za = by,
which is the relation necessary to make x and c conjugates. The
above proves the " first property " of the bridge.
To prove the " second property " assume that an E.M.F. exists
in some branch of the bridge, as AC (Fig. 400b). Then a cur-
rent will flow in the branch BC as well as in the branch AO.
Assume now a counter E.M.F. to be introduced into the branch
BC. This E.M.F. cannot produce any current in the conjugate
AO. In particular choose this counter E.M.F. such as to just
reduce the current in BC to zero. But when an E.M.F. is intro-
duced into BC, which reduces the current in this branch to zero,
we have done the equivalent of opening the circuit BC. Hence
it follows that the current in AO resulting from an E.M.F. in
AC is unaffected by any change in the resistance of BC.
The mathematical relations which, hold for the Wheatstone
bridge when this is unbalanced, namely, when no two conductors
are conjugates, are complicated and require lengthy calculations
to derive. These relations are given in such standard works as
Kempe's " Handbook of Electrical Testing." They are rarely
made use of in the practical employment of the Wheatstone bridge,
and we do not consider that it would be advisable to give a general
treatment of these relations here. We shall, therefore, proceed
at once to. a consideration of those features which are useful to
know and understand in applying the Wheatstone bridge to such
electrical measurements as arise in practice.
Represent the bridge as in Fig. 400c. Here KI and K 2 are the
two conductors which are to become conjugates when the bridge
54
MEASURING ELECTRICAL RESISTANCE [ART. 401
is balanced. If a, 6, c, d represent the resistances of their re-
a c
spective arms, this condition is filled when T = j If an y three of
these four resistances are known the other is given by this relation.
In conformity with custom we shall
call the two adjacent arms which
do not contain the unknown resist-
ance the " ratio arms " and the
arm adjacent to the unknown re-
sistance the " rheostat " of the
bridge. It is evident that the bridge
may be balanced either by varying
the ratio arms or by maintaining
the latter fixed and varying the
rheostat. When the former plan is
followed, the bridge usually takes
the form of a so-called slide-wire or slide-resistance bridge and
when the latter plan is followed the bridge is usually a so-called
plug-type or dial-type bridge.
401. Uses of the Slide-wire Bridge. The slide-wire bridge
(often referred to as the " meter bridge " because a slide wire a
meter long stretched over a meter scale is used) was one of the
earliest if not the earliest form of a Wheatstone bridge.
FIG. 400c.
7H
n
R
n
Ga rvwv-j
r
i
h
c
1 n_ ^
1 l-
2
C
A
7 P
J
7> ,
fci
t ~
FIG. 401a.
j
This form of the Wheatstone bridge is diagrammatically shown
in Fig. 401a. Here R is a known resistance, which may be given
various convenient values, x is the resistance to be measured and n\
and ri2 are two resistances which may be inserted at the ends of the
slide wire.
To use this bridge, a resistance R is chosen which, preferably,
is as nearly as possible equal to the resistance x to be measured.
ART. 401] THE WHEATSTONE-BRIDGE NETWORK 55
The resistances n\ and n% should be chosen low enough so that the
bridge may be balanced by sliding the sliding contact p to some
point on the bridge wire. The battery and galvanometer may
be located as shown in the diagram but the position of these may
be reversed. If I is the length of the bridge wire and c the distance
from A to p in millimeters, and if n\ and n z are resistances deter-
mined in equivalent millimeters of length of bridge wire we have for
a balance
x. ". n 2 + / c'
* = ??tl=- C fl. '(1)
ni + c
In practice the two resistances n\ and n 2 would be made equal
and may then be called rc, and I would be 1000 mm.
Then
g = n + 1000-e
n -f c
The value which may be given to the resistances n and n will
depend upon how nearly it is possible to choose the resistance R
to be like the resistance x. Since the greatest value c can have
is 1000, and a balance still be obtained with the slider upon the
wire, we assume c to have this value and solve Eq. (2) 'for n and
thus find,
1000 x ,_.
n = /T=V < 3 >
as the maximum value n may have. Suppose it is possible to
always choose R so that the value of the resistance x which we
D
have to measure, will never be less than ^ . Then we could make
Namely, n could equal the resistance of 1000 mm of bridge wire.
If the resistance R is obtained from a small plug resistance box,
this is almost always practicable. Using, then, this value of n,
Eq. (2) becomes
_ 2000 - c
'1000 + c
56 . MEASURING ELECTRICAL RESISTANCE [ART. 401
The effect of using resistances n and n each of which are equal in
resistance to 1000 mm of bridge wire, is, virtually, to make this
wire three times as long. Furthermore, by care in construction, the
resistances n and n may be made to include the unavoidable resist-
ances of the joints and copper straps between the points e\ and
0i, and e<i and g% (Fig. 401a). When, as in the more ordinary
use of the slide-wire bridge, the resistances n and n are assumed
equal to 0, and the formula
(5)
is used, errors are sure to result from a neglect of the small and un-
known resistances which exist between the galvanometer termi-
nals 0i, 02 and the ends of the wire e\, e^ where the scale begins and
ends. The use of the resistances n and n limits the range of the
bridge (in the case of Eq. (4), from x = 2 R to x = 0.5 R), but gives
greater precision than can be obtained without them. When
these resistances are not used and the connecting resistances g\ to
ei and 02 to 6 2 are kept very small by a good construction of the
bridge, the range of measurement is, theoretically, from zero to
infinity, tho in practice the precision for either extreme is very
low. If the method expressed in Eq. (5) is used, it is always
desirable to choose R as nearly equal to x as possible so as to bring
the balance point near to the center of the wire. It should be
noted that Eq. (5) can be written,
x = (reciprocal of scale reading X 1000 1) R.
Thus by using a table of reciprocals, or a slide rule, the values of
x, when a number of measurements are to be made, may be cal-
culated simply and rapidly. For R one would choose values, as
1, 10, 100, or 1000, etc.
In the above methods employment is made of a slide wire of
high resistance alloy one meter in length. It is now possible to
obtain alloys of practically zero temperature coefficient and
having 60 times the resistivity of copper. If the slide wire is
No. 24 B. & S. gauge its resistance per meter might be made
about 5 ohms.
It is very desirable to be able to produce a uniform slide resist-
ance of 100 or more ohms. This is accomplished by certain
well-known makers, by winding on a long mandril of about 3 mm
diameter an insulated resistance wire and then withdrawing the
ART. 401] THE WHEATSTONE-BRIDGE NETWORK
57
mandril. This leaves a long helix of insulated wire with the turns
close together. This helix is laid in a groove in a strip of hard
M.F:
Std.0ap
or f
EetS
XCap
or
>td.
Phone
or f
Spiral of Insulated Wire
Insulation removed where
Contact Bears
FIG. 40 Ib.
wood or hard rubber, or in a groove in the periphery of a disk of
hard rubber and cemented in place with thick shellac. When the
58 MEASURING ELECTRICAL RESISTANCE [ART. 402
shellac is dry and hard, the insulation is removed along one side
of the helix parallel with its axis. A sliding contact is arranged
to slide over the bared portion of the helix so as to make contact
at any point along its length. In this manner a substantial slide
resistance may be made which can be given a resistance of as
much as 1000 ohms to the meter. By rubbing one side of the helix
with emery cloth such a slide resistance can be made very uni-
form in resistance from one end to the other.
When such a slide-wire resistance is used, the bridge wire is
usually made circular in form and the contact is moved around
the periphery of the circular disk by means of a handle at its
center. The disk supporting the slide wire is placed underneath
a rubber plate, and a circular scale and a pointer which moves
with the contact are placed above the rubber plate. The con-
struction is illustrated in Fig. 401b.
If the scale above the rubber plate is properly laid off, in the
manner indicated in the figure, it is only necessary to multiply
the scale reading by the value of the fixed resistance R which,
being chosen 1, 10, 100, or 1000 ohms, makes the instrument
direct reading. Hence it becomes a slide-resistance ohmmeter.
When mounted with a small portable pointer galvanometer it
becomes a portable ohmmeter and is a convenient laboratory tool.
The instrument as regularly made is capable of giving measure-
ments of an accuracy of 0.25 of 1 per cent when a standard is chosen
which will bring the reading not far from the center of the scale.
In the classes of slide-wire bridge measurements as described
above, sufficient sensibility may be obtained from a pointer gal-
vanometer of from 30 to 100 ohms resistance of the types made
by Paul, of London; The Leeds and Northrup Company, of
Philadelphia, Pa. ; or by Edward Weston.
The type of slide-wire bridge measurements, described in con-
nection with Fig. 40 la, with a portable pointer galvanometer used
as a detecting instrument is very suitable for instruction in the
use of the Wheatstone-bridge principle and is to be recommended
for college laboratories.
402. Comparison of Resistances by Modified Slide- wire
Bridge. This method, which is applicable to low-resistance
conductors, is explained as follows (Fig. 402) : Let 1-2 be a linear
conductor of uniform resistance, the resistance per unit length
of which we wish to determine, and let 3-4 be a linear conductor
ART. 402] THE WHEATSTONE-BRIDGE NETWORK
59
of uniform resistance, the resistance per unit length of which is
given. Ri and R 2 are ratio coils.
-1 1
J a a L
"^ U '-' ^ b l
1 f
|l4
i
Ba K
FIG. 402
The two conductors are joined together, no special regard being
taken to make this a low-resistance contact. The battery is
joined to the ends 1 and 4 with a key K in circuit. With con-
tact pi set at a, move contact p 2 to some point of balance b.
Next move contact pi to i which is some accurately measured
distance I from a. Now move contact p 2 to some point 61 where
a balance is again obtained and accurately measure or read off
on a scale beneath the conductor the distance L of 61 from b.
When a balance is first obtained the relation holds:
R i _ resistance a to
R z resistance 6 to 0*
When the balance is next obtained the relation holds :
Ri resistance a\ to
Hence,
atoO
btoO
resistance 61 to
a to
bitoO'
It is proved in algebra that, if
or
6 toO
en to O bi to
Now
and
hence,
x w , , x y x
- . then - =
y z w z w
(a to 0) (ai to 0) = (a to ai)
(6toO) - (fcitoO) = (b to 61);
a to
a to
b to 61 ~ 6 toO
60 MEASURING ELECTRICAL RESISTANCE [ART. 402
or, *~ r W- (1)
Itz
Eq. (1) states that the resistance z of a length / of the rod 1-2 is
r>
-^ times the resistance r of a length L of the rod 3-4. If this last
KZ
is known the other is given, and the method enables one to compare
in a very simple way and with inexpensive apparatus the resistance
of a given length of one conductor, as a rod of aluminum, with that
of another conductor, as a rod of copper.
If the conductor used as a standard has the same temperature
coefficient as the conductor which is being compared with it, no
regard has to be taken of temperature other than to be sure that