on 2, or E out of circuit.
a to b
EI made zero.
P =5.4 ohms.
E =0.75X original E.
Ei = 2x original EI.
P =3 ohms.
Polarity of E reversed.
1 1 ii. Alternating-current Methods of Measuring the Re-
sistance of a Battery. While several other direct-current
methods of measuring the internal resistance of a battery are
described in the older treatises, they will not be mentioned here,
as they are all very much inferior to alternating-current methods
and more troublesome. Of the alternating-current methods of
value, two are bridge methods and one an electrodynamometer
method, in which the value of the resistance is given as equal to
a known metallic resistance.
1 1 12. Bridge Method. Telephone Detector. This method
is similar to the method of Kohlrausch, described in connection
with Fig. 1120c, for measuring the resistance of an electrolyte (see
also par. 1124). The arrangement provided is intended for the
measurement of the internal resistance of a battery when this is
yielding a known current. As the resistance of any battery,
and especially -one which polarizes readily, varies with the current
which it gives, it is practically useless to obtain its internal
resistance if the current to which this resistance corresponds is
The arrangement shown in the diagram will enable the internal
resistance to be obtained and the current which the battery gives
to be measured or calculated.
The source of the measuring current, which is rapidly alternat-
ing, is a small induction coil I which is operated by one or two
cells of battery B. The bridge is an ordinary slide-wire bridge
which will be found very suitable for the measurement. The
detector to indicate when the bridge is balanced is a telephone.
It is convenient if this is provided with a head band.
Connections are made as indicated in Fig. 1112. There is
shown in the diagram in dotted line a condenser C in the main
circuit and a condenser Ci in the telephone circuit. The object in
using these (which may be cheap paper condensers of one micro-
farad capacity) is to confine the direct current produced by the
cell being measured, to the circuit c S% Si around the bridge.
If this is done it becomes very easy to calculate the current which
the cell is giving, if its E.M.F. has been determined by means of
a voltmeter. The condensers will not interfere with the passage
of alternating current sufficient for the measurement. If the con-
densers are omitted the current from the cell may still be deter-
mined by inserting a low-reading ammeter or millimeter at q.
The bridge is balanced for a minimum sound or no sound in
the telephone by moving the contact p upon the slide wire Si, S*.
As the scale of a slide-wire or meter bridge is usually divided into
1000 divisions, we have for the internal resistance of the cell, the
where a is the reading from the end of the scale nearest the cell
228 MEASURING ELECTRICAL RESISTANCE [ART. 1112
and R is the fixed resistance in the bridge. This latter should be
quite non-inductive and preferably of about the same magnitude
as the resistance of the cell. For values of - , see Appen-
dix I, 1. The current from the cell (when the condensers are
used) is now simply
where I is the resistance of the bridge wire.
It may be required to determine the value of X when the cell
is in circuit with a lower resistance than the arms of the bridge.
The cell may be made to yield a larger current by joining to its
terminals a known resistance r (shown in the diagram in dotted
line). If the bridge is now balanced we have
r+X~ 1000 -a
V R r(1 fA\
X = -
The current may be measured with an ammeter or milliammeter
inserted at q, or, if E is measured with a voltmeter and the con-
densers are used, it may be calculated from the formula
For the information given by this measurement to have the highest
value, the temperature of the cell, the current which it gives, and
its corresponding resistance should all be recorded. The author
made a careful test of this method which developed points of
A meter bridge with a slide wire of 14 ohms resistance was
selected for the test. The alternating current was supplied by a
very small induction coil about 7 cms long and 2.5 cms in diam-
eter. It was first run by one and later by two small cells of storage
battery. Its vibrator gave a high-pitched note but not loud.
The condensers used were a 1 microfarad and a 0.5 microfarad mica
condenser, the latter being in the current circuit and the former in
the telephone circuit. The telephone was supplied with an ivory
plug attached to a cord which could be used with advantage to
stop up one ear to keep out the sound of the coil.
The method was first tried, using for the R and the X resistances
two 10-ohm non-inductive manganin coils. It was found under
these circumstances that the sound in the telephone was sufficiently
loud, and that a point could be found upon the slide wire which
gave complete silence. The loudness of the sound was scarcely
affected by cutting- out of circuit the two condensers. The setting
could be made to within 0.5 of a millimeter. The bridge balanced
at a = 501, showing that, as the coils were equal, the wire was
practically of equal resistance either side of its middle point.
A new Columbia dry cell was now tested, first without a shunt
and later with a shunt of 1 ohm. The resistance R was made
0.123 ohm. It was found now that it was impossible to obtain
silence in the telephone and that it was difficult to set the sliding
contact closer than 2 or 3, and sometimes 7 or 8 millimeters. The
continuance of the sound in the telephone was attributed to the
electrostatic capacity of the cell, and this was shown to be the case
by putting an equal number of cells, joined in series, in the two
arms of the bridge when a balance giving complete silence could be
obtained, as in the case of metallic resistances. It should be
remembered that to accurately balance a bridge with alternating
current it is necessary that the " time constant " of its adjacent
arms shall be the same. The smaller the cell and the higher its
resistance the more accurately and easily can it be measured by
this method. A set of six dry cells, some very old, were joined in
series and with polarities mutually opposed, and it was found easy
to balance the bridge accurately because by the arrangement in
series the resistance was increased more than the capacity. Thus,
when one has several cells it is easier to measure the resistance
of a number joined in series than to measure the resistance of one.
Some of the results obtained are recorded below:
Columbia dry cell (standard size)
Initial E.M.F. E = 1.47 volts
# = 0.123 ohm (1)
372. 3 Mean
0.0729 ohm, calculated resistance
#=0.223 ohm (2)
237. 2 Mean
0.0693 ohm, calculated resistance
230 MEASURING ELECTRICAL RESISTANCE [ART. 1113
Mean resistance obtained by (1) and (2) is X = 0.0711. De-
parture from mean is 2.5 per cent. The current flowing was
0.1 ampere. Same cell as above shunted with 1 ohm, or r = 1.
Settings, a. R = 0.123 ohm.
375.6 mean. This gives -^^ = 0.0740 ohm,
or X = 0.0799 ohm. The current was 1.4 amperes.
An old Mesco dry cell was also measured. Its resistance was
found to be about 5 ohms, but this resistance rose during the
From these measurements it is to be concluded,
1st. That with cells of moderate size a close setting for a balance
with complete silence in the telephone is impossible.
2d. That the resistance of a dry cell is an extremely variable
3d. That the method is well adapted to metallic resistances,
small cells of high resistance or to a number of cells in series,
but is not accurate to more than from 3 to 5 per cent for low
resistance, single cells.
1113. Bridge Method; Electrodynamometer Detector. This
method may be applied in precisely the same way as the method
for measuring the resistance of an electrolyte described in con-
nection with Fig. 1120d, except that a condenser of considerable
capacity should be in circuit with the fixed coil of the electro-
dynamometer. No current from the battery can then flow thru
both the fixed and movable coil of the electrodynamometer and
so influence its deflection. If this instrument is of the suspended
coil type (as designed by the late Prof. Henry A. Rowland and
described in par. 1001) it will have ample sensibility when used
in this way. Thus on a circuit of 60 cycles and 100 volts the
current thru the fixed coil of the dynamometer will be very
i = ZirNVC = 6.28 X 60 X 100 X C = 3768 C amperes,
where C is the capacity in farads in the circuit. If C is 2.5 X 10" 6
the current will be 0.0942 ampere, which is sufficient. The direct
current from the cell is determined most simply by measuring it
with an ammeter or milliammeter in circuit with it.
1114. Electrodynamometer Substitution Method (Author's
Method) . This method has a much wider range of usefulness
than for the particular measurement here described. Its applica-
tion to the measurement of the effective resistance of a circuit
containing iron when carrying alternating current has been already
described in Chapter X, and therefore its application to the deter-
mination of the internal resistance of a battery may be indicated
The electrodynamometer should be of the Rowland type. The
cell under test and accessory apparatus are connected as in I, II
and III, Fig. 1114.
Referring to diagram I it will be noted that the three-point
double-throw switch S when in position 1, shown in full line,
makes the connections indicated more simply in diagram II, and
when in position 2, shown in dotted line, makes the connections
indicated more simply in diagram III.
232 MEASURING ELECTRICAL RESISTANCE [ART. 1114
The introduction of the condenser C in the main circuit is to
prevent any direct current from the battery passing thru the
fixed coil of the dynamometer. The alternating-current mains
of 110 volts may be used as the source of E.M.F.
Now it is evident by inspecting diagrams II and III that the
resistance of the cell Ba is equal to r, provided the point p has been
adjusted upon the resistance r until the deflection of the electro-
dynamometer is the same when, with the switch S, the connec-
tions are changed from II to III and vice versa. This equivalance
of the resistance of the cell Ba and the resistance r will hold
accurately provided the resistance p' is made equal to the total
resistance p of the hanging coil circuit. The further assumption
must also be made that the electrostatic capacity of the battery
is small. A considerable capacity reactance in the battery would
necessitate a small correction of the same general character as
that discussed in par. 1002. It is thought, however, that the
magnitude of this correction would in general be so small that it
could be entirely disregarded. When the measurement can be
made upon a number of cells of the same size and kind at the same
time the effect of reactance can be reduced to any desired extent
by joining a number of cells in series, for the resistance measured
will increase directly with the number of cells joined in series while
the capacity will decrease.
A trial of this method was made by the author. It gave excel-
lent results and a brief description of the test follows :
Four Daniell cells were made up in glass jars with porous cups.
These were joined in series and not opposing. The internal resist-
ance was measured, using the method and connections shown in Fig.
1114. The electrodynamometer was of the Rowland type. Both
its fixed coils and hanging coil system had a carrying capacity of
0.1 ampere. The source of current was 120 volts A.C. and the
frequency, at the time of the test, was 59.9 cycles per second.
The condenser C was a mica condenser of 1.75 microfarads.
The current flowing in the main circuit was 0.077 ampere. The
dynamometer deflection was 218 divisions. To make this de-
flection the same with the connections first as in III, and then as
in II, it was necessary to make r = 7.64 ohms; hence, the resist-
ance of the four cells in series was 7.64 ohms, making the aver-
age resistance of each cell 1.91 ohms. This method gave good
results without any difficulty arising and the sensibility was
ART. 1116] RESISTANCE MEASUREMENTS 233
found to be ample. At the time of the test the resistance thru
which the battery could flow was 300 ohms, this being the value
given to p. The method gave in another trial with these same
cells so connected that their E.M.F. 's opposed, under which cir-
cumstances the cells yielded practically no current, the value 1.81
ohms as the mean resistance for each cell.
1115. Galvanometer Deflection Methods for Obtaining the
Resistance of a Battery. Though the following methods are
well known they are not much used, for the primary battery has
assumed a subordinate position as a source of electric current.
However, for the sake of completeness in the treatment of this sub-
ject we shall describe them briefly.
Many types of modern cells, especially storage-battery cells,
have an extremely low internal resistance, and in any of the
methods for measuring this resistance it is very advantageous,
when one has a number of similar cells, to join as many of them as
possible in series opposing their E.M.F.'s. In this way the resist-
ances of the cells are added in series, the electrostatic capacity is
reduced and the resultant E.M.F. is small, which permits of
smaller resistances in the circuits being used. Even when there
is an even number of cells, the resultant E.M.F. is usually suf-
ficient to furnish enough current for the measurement if an
ordinary D' Arson val galvanometer is the measuring instrument.
It must be remembered that the internal resistance measured
includes the connecting wires to the cells and the contact re-
sistance under the binding posts. These resistances must be
taken into account and allowed for whenever great accuracy is
1116. Diminished Deflection Method. The battery, of
which the resistance X is to be deter-
mined (Fig. 1116), is joined in series
with a resistance and a galvanometer.
Ordinarily the galvanometer must be
shunted with a low-resistance shunt,
but where a low-sensibility galva-
nometer is used, as a tangent gal-
vanometer, and the battery has a
c o m p a r a t i v e 1 y high resistance, the FlQ
shunt may be omitted. Calling g the
resistance of the galvanometer, and s the resistance of its shunt,
234 MEASURING ELECTRICAL RESISTANCE [ART. 1116
E the E.M.F. of the battery, and 7*1 the resistance used, the value
of the current which flows is
t! - ~ - = Kd,. (1)
Here d\ is the deflection of the galvanometer and K is a constant.
The value of the resistance is now changed from r\ to r 2 and the
current which then flows is
t, = - Kd*. (2)
If the galvanometer is a tangent galvanometer, then we must
write, instead of Kdi and Kdz, ii = K tan 0i and i z = K tan 2 ,
where 0i and 2 are angular deflections.
Eqs. (1) and (2) make the assumption (never strictly true) that
the E.M.F. of the cell remains unchanged when the current is
changed from i\ to it. From the two relations above, we easily derive
ridi - r z d 2 gs
A = j - j - (6)
d} di g + s
If r 2 is chosen so that d 2 = -^ , Eq. (3) becomes
It should be recalled that TI is the resistance that gives the
deflection di, and r 2 is the larger resistance which halves it.
It sometimes happens that a calibrated galvanometer is used to
read the E.M.F. of a thermocouple. Now the resistance of a
thermocouple will change with its depth of immersion in the hot
place, and with the length of the lead wires used. The above
method could be conveniently employed to determine the resist-
ance of the thermocouple circuit from binding post to binding
post of the galvanometer. In this case the galvanometer would
have no shunt and its resistance g would be known. Also the
resistance r\ would be zero and thus, if a resistance r 2 is inserted
quickly in the thermocouple circuit before the temperature of the
hot junction has time to change we would find, by Eq. (3),
X^ - rt-g. (5)
ART. 1117] RESISTANCE MEASUREMENTS
Or if r 2 is so chosen as to halve the deflection d\,
X - r, - g. (6)
The above measurement is made under the best conditions when
r\ H -. is less than X.
1117. Kelvin's Method. The ob-
ject of this modification of the re-
duced deflection method is to maintain
the deflection of the galvanometer
unchanged and then it makes no dif-
ference what the law of the deflection
of the galvanometer may be. With
the circuits arranged as in Fig. 1117
we have, for the current thru the gal-
X (s + g + r) + s (g + r)
The value of the shunt is now changed to si and r is changed to ri,
so the same current as before goes thru the galvanometer. Then,
If in the second case the shunt Si is made infinity then Eq. (3)
X Oi + g + rO + si (g +
From Eqs. (1) and (2)
,, _ ssi (TI r)
/ ' x ' -i + g)
The method, like the former, assumes that the E.M.F. of the
battery remains unaltered when the current which it delivers is
For this method to be applied practically, the galvanometer
must be very insensitive or shunted, or two cells of nearly equal
E.M.F. must be joined in series with their polarities opposed.
If the galvanometer is shunted, then in place of g we must use the
shunted resistance of the galvanometer.
236 MEASURING ELECTRICAL RESISTANCE [ART. 1118
1118. Siemens' Method. The arrangement of circuits for
applying this method is shown diagrammatically in Fig. 1118.
Here the circuit a c b is a slide wire of uniform resistance upon
which a contact c may be moved. Ba is the battery of which the
internal resistance is to be determined and Ga is a galvanometer
which has its sensibility reduced by any means which does not
include a series resistance. It may be shunted, in which case the
value assigned to its resistance must be that of its shunted resist-
Let X = the internal resistance of Ba, to be found,
g = the resistance of the galvanometer (or, if shunted, its
p = resistance from point c to point 6,
q = resistance from point c to point a,
R = resistance from point c to point o,
E = E.M.F. of battery,
/ = current from battery, and
i = current thru galvanometer circuit.
For brevity, write Q = q + X and P = p + g.
Then Q + P = K, a constant.
By inspection of the diagram it will be seen that if c is moved to
6 the current thru the galvanometer will be greater than if c is at
some intermediate point between a and b. Also if c is moved to
a the current thru the galvanometer will again be greater than if
c is at some intermediate point between a and b. Hence, gener-
ally, there is some point c between a and 6 where the current thru
the galvanometer is a minimum. It is this point, which, when
ART. 1118] RESISTANCE MEASUREMENTS 237
found by trial, will give the value of the resistance sought. With
the contact at any point c on the slide wire, we have
p \ % >
Q = K - P,
" PK - P 2 + KR
It is required to move c until i is a minimum or until - = ii is a
Thus we have
and this expression is a maximum when
K = 2P, (5)
or when Q + P = 2 P, or Q = P
That is, when
2 + ^ = P + 0, or X = g+p-q. (6)
Thus the value of X is found by moving the contact c upon the
slide wire until the deflection of the galvanometer is reduced to a
minimum. As all the resistances are known we can calculate
the current which the battery is yielding, provided we know its
Thus, Eq. (1) becomes
/ = E (p + g + R) ^
7/J (T, 4- n -4- 7?^
or / =7
Analysis shows (see Kempe, " Handbook of Electrical Testing,"
p. 161) that the measurement is made under the best condi-
tions when p + q is not less than the greater of the two quan-
tities R + X and R + g. Also R should be less than the greater
MEASURING ELECTRICAL RESISTANCE [ART. 1119
of the two quantities X and g, and the galvanometer resistance
should preferably not exceed X.
1119. Resistance of Electrolytes. The resistance of electro-
lytes, as sulphuric acid, salt solutions, etc., could be measured
with a Wheatstone bridge in the usual way if it were not for the
fact that, as soon as a measuring current passes thru the electro-
lyte the electrodes polarize and an E.M.F. is developed which
opposes the E.M.F. which sends current thru the electrolyte.
To understand this clearly consider the diagram, Fig. 1119.
Let C be an electrolytic cell in one arm of a Wheatstone bridge.
Let a, b, c, d be resistances, and i and ii be currents. If the
resistances a, b and d are always so chosen that the bridge is
balanced, we shall have the potential drop V a from 1 to 2 equal
the potential drop Vb from 1 to 4. Also the potential drop V c
from 2 to 3 will equal the potential drop Vd from 4 to 3. Or
ai = bii, and V c = di\, whence
Now the potential V c will be equal to the current i times the
resistance c of the cell, less the opposing E.M.F., E, of polariza-
tion of the cell,
or V c = ic - E. (2)
ART. 1119] RESISTANCE MEASUREMENTS 239
It may be assumed that for a very small current i which has
flowed for a short time t the E.M.F. of polarization is propor-
tional to the quantity of electricity that has passed thru the cell,
or, what is the same thing,
E = Kit. (3)
V c = ic - Kit. (4)
Putting this value of V c in Eq. (1) we obtain
'c = ~-\-Kt. (5)
This last relation shows that the resistance c of an electrolyte,
which would be measured by a balanced Wheatstone bridge, will
seem to increase with the time that the current i is kept flowing
thru the electrolyte, and that it will always be higher than the
true resistance of the electrolyte. For this reason the use of the
Wheatstone bridge with direct current is not suited to the measure-
ment of the resistance of an electrolyte. If, however, an alter-
nating current be substituted for a direct current and a detector,
responsive to alternating current, be substituted for the galvanom-
eter, the principle of the Wheatstone bridge may be used with
convenience and accuracy. This is because the E.M.F. of polari-
zation, produced by the current in one direction and which would
lead to a balancing of the bridge giving too high a value of the
resistance, will, upon the reversal of the current, either be neutral-
ized or, if not neutralized, will lead to a balancing of the bridge
giving too low a value of the resistance. Thus the setting actu-
ally obtained for a balance is the same, whether polarization is
neutralized or not, as would be required were there no polarization.
Thus, to merely measure the resistance of an electrolyte, with-
out attempting to determine its specific resistance, it is only
necessary to place the electrolyte in a vessel provided with two
electrodes of thin gold or platinum and connect this vessel into
one arm of a Wheatstone bridge. The other arms are resistances
which are highly non-inductive. The bridge is balanced for alter-
nating current. A telephone is a suitable detector and a small in-
duction coil with a secondary winding furnishes from its secondary
a very suitable source of alternating current. The alternating cur-
rent obtained from a small induction coil is filled with harmonics
and gives a clearer and sharper sound in the telephone, conse-
240 MEASURING ELECTRICAL RESISTANCE [ART. 1120
quently a more accurate balance, than an alternating current from