has the resistance 10 n ohms. Thus, with ten coils arranged as
in the figure and having the above values, all values in steps of
n (I) ohms from to 10 n ohms may be obtained. Thus, referring
to the figure, if the plugs 0, 1, 2, 3, . . . 10 are all in, the resist-
ance from a to b is ohm, if 1, 2, 3, . . . 10 are all in, the resist-
ance is 1 ohm. If plugs 0, 1, 2 are removed, the resistance is
3 ohms, etc., that is, the resistance obtained between a and b will
always be that which is stamped opposite the lasb plug toward
the left, which is not removed. Since the current passes thru all
the plugs not removed, in parallel, the contact resistance is greatly
reduced and this contact resistance decreases as the resistance
value plugged decreases.
This method is an excellent arrangement when it is required
to obtain ten regularly ascending values of very low resistance,
as, for example, 0.001, 0.002, 0.003 . . . 0.01 ohm. Greater pre-
cision of adjustment can be gotten and maintained by this parallel
arrangement of coils, of relatively high resistances, than from coils
arranged in series.
If the entire rheostat is based upon this principle and has several
decades it would only be required to have the values run from
to 9 n ohms in the decades, except the one of lowest denomina-
tion. For obtaining the succession of values from to 9 n ohms
the coils so joined in parallel are n (2, 6, 12, 20, 30, 42, 56,
72, 9) ohms. This decade plan is deserving of more attention
than it has received, for excellent results may be obtained for
accurately varying resistance in steps as low as 0.001 ohm.
The above described selections and dispositions of resistance
units for Wheatstone-bridge rheostats comprehend those known
which are of interest. We shall describe now the methods in use
for arranging the resistances in the ratio arms of a bridge.
508. Arrangements of Resistances for the Ratio Arms of
Wheatstone Bridges. The most simple disposition of ratio
coils, and one which is found in most inexpensive Wheatstone-
bridge boxes, made for laboratories and, the use of students, is
shown in Fig. 508a. The required ratio, as 10 to 100, is obtained
by withdrawing a plug from each of the arms A and B. The con-
tact resistance of the remaining plugs in each arm enters into the
ratio and there is no provision made for reversing the two arms.
In Fig. 508b is shown how connections and plugs may be dis-
posed so that the ratio arms may be reversed.
This is the classical arrangement which is used in the so-called
" English Post Office " bridge. By changing two plugs from
holes 1, 2, to holes 3, 4, the positive pole of the battery is joined
to the end of A and the negative pole to the end of B or vice
MEASURING ELECTRICAL RESISTANCE [ART. 508
versa. By so reversing the ratio arms and twice balancing the
bridge and then taking the mean value of X to be the true value,
any errors of adjustment in the ratio coils is eliminated when
unity ratio is used. However, it is always desirable so to choose
the values in the ratio arms that a large portion of the rheostat
will be required for getting a balance, and it is only when the
ratio is unity that this can be done with the ratio coils used both
direct and reversed. This requirement limits the usefulness of
reversible ratio arms to those cases where unity ratio may be
used. Thus, suppose the capacity of the rheostat is 10,000 ohms,
and may be varied in steps of 1 ohm. To set the bridge to an
accuracy of 0.1 of 1 per cent would require that at least 1000 ohms
of the rheostat be brought into service. If a resistance of 900
ohms is to be measured, a ratio of 10 to 1, with the ratio arms
direct, would utilize 9000 ohms of the rheostat, but when the ratio
arms are reversed only 90 ohms of the rheostat .could be utilized,
which would necessitate a setting that might be less accurate than
0.5 of 1 per cent. In such a case one would either have to aban-
don the advantage of reversible ratio arms and set the ratio 10
to 1, or use unity ratio and reverse the ratio arms, in which case
the rheostat setting would be 900 ohms, which, being variable
in steps of 1 ohm, would permit settings to 1 part 'in 900. The
advantage, then, of reversible ratio arms is chiefly confined to the
measurement of resistances of the same order of magnitude but
smaller than the total resistance of the rheostat. When the
instrument maker can be trusted to accurately adjust the ratio
coils in a Wheatstone bridge the reversible feature is scarcely
worth its extra cost as applied to the " Post Office " type of
509. Schone's Arrangement of Ratio Arms. This very
superior disposition of ratio coils was described by 0. Schone,
in " Zeitschrift fur Instrumentenkunde," May,
1898. It is now extensively used in America and
by its superiority deserves to supersede all other
arrangements of resistances for reversible ratio arms.
According to this arrangement all the ratio coils
have one of their terminals joined to a common
bar connector which corresponds to the block
marked C of Fig. 508a. The other terminal of
each coil is joined to a separate block. The scheme
is given in Fig. 509.
The bar A on one side of these blocks is joined
to the rheostat R, and the bar B, on the other side,
to an X post.
In the ordinary use of this arrangement two plugs
only are used. One plug is inserted between the
bar A and one of the blocks 1, 1', 10, 10', etc., of
the central row, and the other plug is inserted be-
tween the bar B and any one of the blocks of the
central row, except the one which the other plug
joins to bar A.
The construction generally embodies two ratio coils of each
value. Referring to Fig. 509, if one wishes to obtain a unity
ratio, as 1000 to 1000', one plug would be inserted between the
block 1000 and the bar A , and the other plug between the block
1000' and the bar B. This disposition of the plugs joins the end
of the 1000 ohm coil to the rheostat and the end of the 1000' ohm
coil to the X post. If, now, one plug is inserted between the
1000' block and bar A, and another plug between the 1000 block
and bar B, the ratio arms become reversed; that is, the 1000' ohm
coil is joined to the rheostat, and the 1000 ohm coil to the X
When uneven ratios are used the same ratio can be obtained by
four different combinations. If we wish to obtain the ratio 1 to
10, we can plug between A and 1 and B and 10 and get 1 to 10,
or between A and 1' and B and 10 and get 1' to 10, or between
MEASURING ELECTRICAL RESISTANCE [Aux. 510
A and 1 and B and 10' and get 1 to 10', or between A and 1'
and B and 10' and get I' to 10'.
To obtain the reciprocal set of ratios, like the above, we would
plug A and 10, B and 1, and get 10 to 1; A and 10', B and 1,
and get 10' to 1; A and 10, B and 1', and get 10 to 1'; A and 10',
B and 1', and get 10' to 1'.
By using more than two plugs and connecting certain of the
coils in parallel combinations, a large number of other ratios may
be obtained. For example, we can plug between A and 100 and
A and 100', and between B and 1000 and get the ratio 50 to 1000,
or we can plug between A and 1000 and A and 1000' and between
B and 100 and get the ratio 500 to 100.
With this arrangement of ratio coils it is seen that errors due
to plug contacts become practically nil, because only two plug
contacts enter the circuit, while with even ratios it is only the
difference in the resistance of the two plug contacts which affects
510. Nonreversible Ratio Arms Adjustable without Contact
Resistances. A very excellent arrangement of variable, but
nonreversible ratio arms, which involves no contact resistances
in the ratio arms, is shown in Fig. 510. The ratio values may be
varied by moving a brush contact, which is joined to the battery,
over studs as indicated in the figure.
To calculate the odd values required for ratio coils the solution
must be found for equations of the form,
ART. 511] WHEATSTONE-BRIDGE METHODS 93
a + b _!_
a + b + c
b + c + d _ 1A
In the case selected we have five equations and six unknowns,
hence some one of the quantities, as a, must be assumed as known.
If we choose a = 1 ohm, then the solution of the above equations
a = l, 6 = 8.1818, c = 41.3182, d = 41.3182, e = 8.1818, / = !.
We note that a = /, b = e, and c = d, and therefore there are but
two odd values of resistance to adjust to give the five different
ratio settings, 100, 10, 1, 0.1, 0.01.
The method may be indefinitely extended and is an excellent
arrangement to use with bridges in which the rheostat values are
varied by means of dials and sliding-brush contacts, for then all
resistance changes in the box can be effected with dials and no
plugs are required. By this arrangement the ratio arms, being
free from contact resistances, give ratios just as accurately as the
coils are adjusted.
511. Wheatstone Bridge Arranged for Reading in Per Cent.
It frequently happens that the problem of very rapidly measuring
a large number of resistance units of even values as 1, 10, 100
ohms, etc., is presented. Instrument makers have this to do in
checking up the precision of the coils in resistance boxes. In such
cases it is of little interest to know the absolute number of ohms
by which any coil is in error, the important question being what is
the per cent accuracy of any coil.
To meet the above requirements of a Wheatstone bridge, the
author devised the method and connections given below in Fig. 511.
Bridges were constructed, embodying the connections of Fig. 511
and placed in continuous service, which would give by a direct
reading the per cent value of any coil being measured in terms
of the standard employed, The readings of the last of the four
MEASURING ELECTRICAL RESISTANCE [ART. 512
dials used were in steps of 0.001 of 1 per cent, and the range of
the bridge was from 95 per cent to 106 per cent of the standard.
As a method may be applied for eliminating the lead resistances,
the per cent bridge may be used advantageously for resistances from
1 ohm up, with errors not exceeding 0.001 of 1 per cent.
The diagram, Fig. 511, is practically self-explanatory.
1- 500^- Spool
To use this method, one first connects the lead wires together,
which go to the coil to be measured. All the bridge dials are set
so as to read an even 100 per cent. Then the lead wires which
go to the standard resistance are also joined together at the ends
which connect to the standard, and they are varied in length until
the galvanometer shows a balance. This means that the lead
resistances to the X coil are equal to the lead resistances to the
standard coil, and as the X coil is never very greatly different in
value from the standard, the lead resistances eliminate.
The bridge requires for its practical use a complete set of re-
sistance standards, against which to match the resistance coils to
This form of the Wheatstone bridge is of great value to the
instrument maker who has many coils to measure with both
rapidity and precision.
512. Remarks upon the Use of the Wheatstone Bridge. In
arranging to use a Wheatstone bridge with accuracy, speed, and
convenience, one should select with care a suitable galvanometer
or other detector for indicating when the bridge is balanced. In
addition to its' greater sensibility a galvanometer is more ad-
vantageous than detectors of the telephone type, in that the deflec-
ART. 512] WHEATST ONE-BRIDGE METHODS 95
tions are to the right or to the left, according as the rheostat adjust-
ment is higher or lower than the setting required for a balance;
whereas with the telephone the sound increases equally and with-
out distinction for a departure from the setting of the rheostat in
either direction from that which gives a true balance. Further-
more, when a telephone is used, the current through the bridge arms
must be made variable to cause a sound in the telephone, and
correct values of resistance will only be obtained by meeting the
condition, not always possible of fulfillment, that the four arms
of the bridge are without appreciable capacity or inductance.
These considerations practically necessitate, for the general use
of the Wheatstone bridge, the employment of a galvanometer as
the instrument to show when the bridge is balanced.
The variety and the types of bridges and the methods of their
employment are so great that no general rules can be laid down
as to what kind of galvanometer will best serve the purpose. How-
ever, a few general considerations may be mentioned. Except for
special requirements, a mirror galvanometer of the D'Arsonval
type, having a resistance of from 100 to 500 ohms, will be found
convenient, and will have ample sensibility if it will show a deflec-
tion of one division on a scale 1000 divisions from the mirror with a
current of 0.005 microampere. The galvanometer should be just
aperiodic to save time in waiting for the deflections to return to
zero and increased satisfaction will be found in working in pro-
portion as the period of the galvanometer is made shorter. A
galvanometer of the D'Arsonval type having a period of three
seconds, a resistance of 200 ohms and a sensibility of 200 megohms,
can easily be constructed and will admirably meet nearly all the
requirements of Wheatstone-bridge work of high precision. It
should be recalled that a galvanometer sensibility is expressed in
megohms when, with the scale at 1000 scale divisions from the
mirror, the sensibility S m is the number of megohms which must
be in the galvanometer circuit, so that, with an E.M.F. of 1 volt
in the circuit; there will result a deflection of one scale division.
One should distinguish sensibility from " figure of merit " which
is defined by the equation
* See par. 1504, Eq. (11), also article by Edwin F. Northrup in the Journal
of the Franklin Institute, Oct., 1910, entitled "The Comparison of Galvanom-
eters and a New Type of Flat-coil Galvanometer."
96 MEASURING ELECTRICAL RESISTANCE [ART. 512
where T is the undamped complete period of the galvanometer
and R the resistance of its coil.
For most uses of the slide-wire Wheatstone bridge and other
types in which the coils are adjusted to an accuracy of not better
than 0.05 of 1 per cent, it is unnecessary to use a reflecting type
of galvanometer, with either telescope and scale or lamp and
scale. A small pointer galvanometer, of 100 or 200 ohms
resistance, having a sensibility such that with 1 volt and
250,000 ohms in circuit the pointer will deflect 1 millimeter on
its scale, will be found amply sensitive and very convenient to
The relative positions of the battery and the galvanometer in
the Wheatstone-bridge circuits should be chosen to meet the
condition that the terminals of the galvanometer shall connect
such junction points of the four arms of the bridge as will make
as nearly as possible the resistance external to the galvanometer
equal to the resistance of the galvanometer itself. For example,
if the resistance X is 10 ohms, and the ratio arms are made 100
ohms to 1 ohm, the terminals of the galvanometer, if this has a
resistance of 100 ohms or more, should be connected, one to the
junction point of the 100 with the 1000 ohms coil of the rheostat,
and the other to the junction point of the 1 with the 10 ohms coil.
Maxwell gives, in his " Electricity and Magnetism," Vol. I, par.
348, the following rule: "Of the two resistances, that of the bat-
tery and that of the galvanometer, connect the greater resistance so
as to join the two greatest to the two least of the four resistances."
In modern practice, one generally uses a battery of 4 or 6 volts, and
then reduces the current in the bridge circuit to a suitable value
by the use of a resistance in series with the battery, and the posi-
tion occupied by the galvanometer should be chosen with refer-
ence to its own resistance only, as compared with the resistances
of the bridge arms. The object to be obtained is that the circuit
external to the galvanometer should be as nearly as possible that
of the galvanometer itself, without regard to the battery resistance.
Hence the rule at the beginning of the paragraph. The object for
this choice of position of the galvanometer is to give the arrange-
ment the maximum sensibility, but, with the current-carrying
capacity of the manganin coils now in use, of practically zero
temperature coefficient, and with the high sensibility of easily
obtainable galvanometers, the sensibility is generally adequate
ART. 512] WHEATSTONE-BRIDGE METHODS 97
however the position of the galvanometer is chosen, and the im-
portance of fulfilling the above conditions is slight.
The safe watt capacity of the coils of a Wheatstone bridge will
vary from one-quarter to four watts per coil according to its con-
struction. If this watt capacity of a coil is greatly exceeded the
coil may be heated to a point where the resistance is permanently
changed, even though the insulation is not charred. It should
always be remembered that the watt load put on any coil in a
bridge is equal to the square of the potential applied at its ter-
minals divided by its resistance, and that, as a rule, this quantity
should never exceed 1 watt. Unless one should forget and make
connections which would bring an excessive voltage at the ter-
minals of a coil, it is always well in order to avoid this danger to
keep an external resistance in circuit with the battery. This will
limit, for any connections of the bridge, the flow of current to a
Even though the rheostat of a bridge is incapable of being varied
by very small steps, one can measure resistances with exactness
by making use of the deflections of the galvanometer after a
balance has been obtained within an adjustment of the smallest
step of the rheostat. The procedure is as follows: The current
furnished by the battery being assumed constant and the deflec-
tions of the galvanometer proportional to the current through it,
one takes note of the permanent deflection of the galvanometer
when the resistance of the rheostat required for a balance is set
too small, in a final adjustment, by the smallest step in the rheo-
stat. Call R this resistance and d the deflection. Then increase
the resistance of the rheostat by one of its smallest steps, say one
ohm, and observe the deflection then obtained which will be in
the opposite direction to the one previously obtained. Call this
deflection d f . The true value of X will then be given by the
where a and b are the resistances in the ratio arms, and s the
value in ohms of the smallest step in the rheostat.
When the resistance to be measured is wholly unknown one
should proceed, in seeking a balance, in a systematic manner. To
avoid violent deflections of the galvanometer this may be tempo-
rarily shunted with a low resistance, which shunt is removed when
98 MEASURING ELECTRICAL RESISTANCE [ART. 512
a balance is nearly obtained. It is well to start with unity ratio
and with zero resistance in the rheostat. A quick tap of the key
will cause a moderate deflection of the shunted galvanometer in
one direction. 1000 ohms may now be put in the rheostat and
the key be again tapped. A deflection in the opposite direction
will now indicate that the resistance lies between and 1000 ohms.
500 ohms should now be plugged in the rheostat and, if the deflec-
tion is like the first one when the key is tapped, the resistance is
known to lie between 500 and 1000 ohms.
By proceeding in this manner the resistance is narrowed down,
with only a few trials, very close to its actual value. One should
now choose the value of the ratio so that in obtaining a final
balance the largest possible portion of the rheostat is brought into
service. The final balancing is made with the shunt removed
from the galvanometer and the procedure to be followed is pre-
cisely that adopted for weighing with a delicate balance. With
a galvanometer in which the coil is visible the preliminary balanc-
ing is usually effected by directly observing the movements of the
coil. In the final adjustments only is it necessary to observe the
movements of the coil by looking thru the telescope, or by ob-
serving the spot of light on the scale.
No definite limitations can be laid down for the useful resist-
ance range of a Wheatstone bridge, as this depends upon the
range of its rheostat and upon the number and precision of the
coil-values provided in its ratio arms. Ordinarily, Wheatstone
bridges should be considered adaptable for the fairly accurate
measurement of resistances which lie between 1 and 1,000,000
ohms, though this range is often exceeded in both directions with
The precision of measurements possible with a Wheatstone
bridge depends upon a variety of circumstances, such as, the
value of the resistance being measured, the accuracy of the coils
in the bridge, the possibility of reversing the ratio arms to elimi-
nate their error, and the care with which contact resistances are
allowed for, or guarded against. In routine work, for resistances
of the same order of magnitude as the total resistance of the
bridge rheostat, a precision of 0.04 of 1 per cent may be considered
fairly good, tho the author owns a bridge which can be relied
upon to measure resistances in the range from 10 to 10,000 ohms
to an accuracy better than 0.02 of 1 per cent.
ART. 512] WHEATSTONE-BRIDGE METHODS 99
Since the advent of manganin coils with their practically zero
temperature coefficients, little regard need be given to the interior
temperature of the bridge. The temperature of the resistance
being measured, however, unless this is also of a zero temper-
ature coefficient material, must be carefully observed.
THE MEASUREMENT OF LOW RESISTANCE.
600. Introductory Statement. When one is about to make
an electrical measurement, it is often not possible to choose the
best method because the apparatus for this is not available.
For this reason it is desirable to be acquainted with alternative
methods, and this consideration leads us to describe several
methods for measuring low resistances, tho the one known as the
" Kelvin-double-bridge " method is preeminently the most accu-
rate and elegant, and, when apparatus suitable for its application
is to be had, should be chosen in preference to any other.
While there is no sharp distinction between medium and low
resistance we may, for convenience, consider any resistance which
is less than one ohm as low. Ordinary methods, applicable to
medium resistances, fail to give precision with low resistances
either on account of contact resistances which are likely to enter
the circuit which contains the resistance being measured or be-
cause a low resistance is often a short conductor, and errors in the
determination of the exact length measured are apt to enter.
Both these sources of error are avoided by providing the re-
sistance measured and the standard with which it is compared
with potential points. The resistance which is determined is the
resistance which lies between two potential points, when the lines
of current flow thru the low resistance have a particular distribu-
tion. It should be remarked, that, if the resistances of several
conductors are given, each provided with fixed potential points
to which connections may be made, these conductors cannot be
joined in series or in parallel combinations to obtain a known
resultant resistance. For this reason standards of resistance,
provided with fixed potential points, are unsuited for obtaining
other values by series or parallel combinations.
For the measurement then of low resistance of widely varying
range, one needs to be provided with a series of low-resistance
ART. 601] THE MEASUREMENT OF LOW RESISTANCE 101
precision standards. A set of precision standards which would
be quite complete would consist of 1, 0.1, 0.01, 0.001, 0.0001
ohm, the resistance values being in every case between fixed
potential points. The standards would each have, therefore,
two-current and two-potential terminals. They are usually
mounted in metal cylindrical boxes which can be filled with kero-
sene or paraffin oil to permit of an accurate determination of their