voltage changing in the intervals between the readings.
If the readings in either of the two latter cases are only a frac-
tion of a scale-division, then the insulation resistance is too high
to be measured by this method and one must resort to the next
method to be described. Having taken the above three readings,
we obtain R(D~d l -d,}
Xl= ~^~
and
R(D-d 1 - dj f .
X 2 = ^ (2)
The proof of the relations (1) and (2) has been given in par. 207,
and a discussion is given of the theoretical accuracy obtainable in
par. 104.
The current I which leaks to the ground will be
212 MEASURING ELECTRICAL RESISTANCE [ART. 1103
In a particular case, the insulation resistance of the wiring system
of a large office building was determined. A Weston direct-
current, 150- volt voltmeter was used and the following readings
and resistances were obtained:
R = 12,220 ohms.
E = 113 volts or D = 113 scale-divisions.
Fi = 1 volt or di = 1 scale-division.
V z = 4 volts or dz 4 scale-divisions.
Xi _ 12,220 (118 - 1 -4)
12,220(113-1-4)
This example shows that where the sum of the resistances Xi and
X 2 is not over a million ohms, the voltmeter method is sufficiently
accurate for the purpose of deciding if insulation specifications
have been met.
If one side of the line is grounded, that is, if X 2 = 0, we have,
V 2 = 0, and E = V\ + V% = V\ and the method fails to give XL
Any instrument, as a galvanometer, in which the deflections
are proportional to the current thru it and which has sufficient
resistance in series with it so that it will at no time deflect off its
scale, may be substituted for a voltmeter.
1103. Galvanometer Method for Insulation-Measurement
while Power is On. This method may be used when greater
accuracy is required or when the insulation resistance to earth,
of at least one side of the line, is over a megohm.
The wiring system is represented in I, Fig. 1103, and II, Fig.
1103 gives equivalent circuits.
The method is carried out by connecting across the bus-bars a
moderately high resistance. A point p is found on this resist-
ance where the potential due to the generator is the same as that
of the earth. Then, with the aid of a sensitive galvanometer and
an external source of E.M.F., the ' resistances to earth r\ and r 2
are measured in the following manner: k is a key and S an Ayrton
universal shunt. This latter may be omitted if the source of
E.M.F. can be varied in a known manner.
It is evident from II that a balance will be obtained when = =
o r z
the key k being in its upper position. If k is now depressed, the
ART. 1103]
RESISTANCE MEASUREMENTS
213
resistance R encountered by the current generated by the source
e will be
R = gi+ > (1)
b + r 2 + a + n
where gi is the resistance of the galvanometer; but in comparison
with TI and r 2 , a and b can be neglected, also gi, then
R =
(2)
I C
II
By construction,
FIG. 1103.
= T = N, a known ratio.
r 2 b
From the last two relations we deduce
N
(3)
IY
and
Taking d as the deflection of the galvanometer and K as the
galvanometer constant, the current thru the galvanometer is
e d D eK
-p = -j?i r R = -j-* (5)
K should be denned as the resistance in circuit with the galvanom-
eter (including its own resistance), such that it will give, with
one volt, a deflection of one scale-division at the distance at which
214 MEASURING ELECTRICAL RESISTANCE [ART. 1104
the scale is placed from the mirror during the test, usually taken
as one meter.
Then we will have
and
r\ = - *j (7)
Taking K equal to 10 8 as an average value for an ordinary D'Ar-
sonval galvanometer and e = 100 volts, N = 2, and d = 100 scale-
divisions, we have
100 X 10 8 (2 + 1) _ n
r 2 = - nn - = 150 X 10 6 ohms, or 150 megohms,
2 X luU
i on v i o^ (*) i i ~\
Tl = - x v - = 300 X 10 6 ohms, or 300 megohms.
This example shows that a galvanometer of very moderate
sensibility will measure in this way a very high insulation resist-
ance. If, on the other hand, the insulation is low, small battery
power may be used or the deflection of the galvanometer can be
cut down to 0.1, 0.01, 0.001, 0.0001 by the Ayrton shunt. The
only difficulty likely to be experienced in applying the above
method is that, while making the test, the relative values of r\
and r z keep changing, due to motors or lights being thrown on
or off the line. In this event it is only possible to obtain a sort of
average value for the resistance to earth of each side of the line.
1104. Determination of the Internal Resistance of Batteries.
The resistance of an electrolytic cell or battery is by no means a
constant quantity, even approximately. It will change with the
temperature, the age of the cell, the current which the cell is
giving and with the total ampere-hours of current it has yielded.
It is a quantity which varies greatly with the past history of
the cell.
The determination of the internal resistance of a cell on strictly
open circuit requires special methods and the information, more-
over, when obtained has little value because of the variability of
the quantity measured. If a cell is closed thru an external resist-
ance R and there exists in the circuit an electromotive force E
the current / which flows will be
ART. 1105]
RESISTANCE MEASUREMENTS
215
where X is called the internal resistance of the cell. If E' is the
fall of potential over the external resistance alone, then
#' E
R
whence,
(3)
R + X
R(E-
-i,
(2)
E'
We cannot here call X anything more than a quantity which must
be added to R to satisfy Eq. (1). It is not an ohmic resistance,
as it does not obey Ohm's law, for in general X changes when the
current changes.
Nevertheless the current output of a cell, under given circum-
stances, will depend largely upon this quantity, and it is necessary
therefore to be able to determine its value when the cell is sub-
jected to particular conditions.
A determination of the quantity X, to have value and definite-
ness, really involves the making .of a test of the cell in respect to
several of its characteristics. These are its open circuit E.M.F.
when the cell is fresh and after it has delivered a certain quantity
of electricity, its E.M.F. when closed thru a given resistance, its
rate of polarization, its rate of recovery from polarization, etc.
A description, therefore, of methods of measuring the internal
resistance of a battery should begin by showing how a full record
of the action of a battery may be obtained. This record is best
exhibited in the form of curves. A procedure for obtaining the
data for such curves' in an accurate and convenient manner will
now be explained. It is a well-known xr\r.
method and may be called the condenser
method of testing batteries.
1105. Battery Tests by Condenser
Method. In the diagram, Fig. 1105a,
G is a ballistic galvanometer or other like
instrument in which the throw deflections
are proportional to the quantity of elec-
tricity discharged thru it. B is a cell to
^^
CM
f TK
, B
-AW/WWWV
[
K'
FIG. 1105a.
be tested, R a known resistance the fall
of potential over which is to be meas-
ured, K' a key to put this resistance in circuit with the cell B, and
K is a charge and discharge key for charging the condenser C and
discharging it thru the galvanometer.
216 MEASURING ELECTRICAL RESISTANCE [ART. 1105
The condenser, which should be a mica condenser of about one
microfarad, is first charged, K' being open, by means of a standard
cell. A Weston cadmium standard cell is recommended. This
has an E.M.F. of 1.0183 volts and a zero temperature coefficient
between 15 and 35 C. The condenser is then discharged thru
the galvanometer and the throw deflection read either with a
telescope and scale or lamp and scale.
The standard cell is now replaced by the battery to be tested.
With K' open, the condenser is charged and discharged as before.
Then K r is closed and after an interval of one minute the condenser
is again charged and discharged. K f is maintained closed, prefer-
ably for 60 minutes, except that at intervals of two minutes it is
opened just long enough to charge and discharge the condenser,
at which intervals readings are taken which give the E.M.F. of the
cell upon open circuit. The condenser is also charged and dis-
charged, K f being closed, at intervals of two minutes which alter-
nate with the open-circuit readings.
Thus, at time open-circuit reading is taken, K f being open;
at end of 1st minute closed-circuit reading is taken, K' closed; at
end of 2d minute open-circuit reading is taken, K r momentarily
opened; at end of 3d minute closed-circuit reading is taken, K'
closed; at end of 4th minute open-circuit reading is taken, K'
momentarily opened, and so on until at least 60 minutes have
elapsed. At the end of 60 minutes K f is opened permanently and
open-circuit readings are taken at intervals of two minutes. In
this way data are obtained for plotting the recovery curve.
The deflection corresponding to the E.M.F. of the standard cell
having been obtained, the other E.M.F.'s can be calculated from
the deflections by simple proportion. Thus, if E s is the E.M.F.
of the standard cell and d a the corresponding deflection, then any
other E.M.F., E x , which gives a deflection d x is
*-;!*
The internal resistance of the cell at any moment during the
60-minute test is now obtained as follows: Call the E.M.F. of
the open-circuit reading at any moment E; that of the closed-
circuit reading at the same moment (which is the drop of potential
over the resistances R) E\. Then if X designates the internal
resistance sought, the following relations hold. There is a total
ART. 1105] RESISTANCE MEASUREMENTS 217
E.M.F., Ej and a fall of potential over the external resistance EI.
Therefore, there must be a fall of potential E EI over the
internal resistance X; hence,
X iRiiE-EiiE
v E EI D t .
or X = ^ - R. (1)
After the polarization curve given by the different E.M.F.'s called
E, and the terminal potential difference curve given by the E.M.F.'s
called EI are plotted, then the data for solving the equations
giving the points for the internal resistance curves can be read
directly from the ordinates of these curves. The current flowing
at any time is simply
For a full study of a battery it should be run completely out, tho
after the first hour it would be necessary only to take readings at
intervals very much longer than two minutes. By joining a
number of cells in series (to always have sufficient E.M.F.) and
running them completely down thru a voltameter the total
number of coulombs which a cell is capable of giving could be
computed. Many types of cells should also be given an age-test
by giving them a short run and then setting them aside for
several months to test if any destructive local action occurs.
. In Fig. 1105b are reproduced curves, taken by the author, upon
a Barrett silver-chloride cell, like those supplied for portable test-
ing batteries. It will be noted that the internal resistance of this
type of cell rapidly falls as the silver chloride of poor conductivity
becomes reduced to spongy silver of high conductivity. The
polarization is small, the current output increases for the first
hour and the recovery is rapid, reaching 1 .150 volts. These charac-
teristics, notwithstanding the low E.M.F., have made this type of
cell very popular as a small battery for testing purposes.
Dry cells of standard size, as the " Mesco," have become an
important commercial factor, and plans for testing and rating
them require special consideration. However, the tests which
should be made present no problems in measurement which have
not been fully discussed and the reader who desires further infor-
mation upon this subject is referred to a report of a committee
of the American Electrochemical Society entitled, " Standard
218
MEASURING ELECTRICAL RESISTANCE [ART. 1106
Methods Recommended for Testing of Dry Cells." This report
appeared in the proceedings of the society, vol. XXI, page 275,
1912, and is signed by C. F. Burgess, Chairman; J. W. Brown,
F. H. Loveridge, C. H. Sharp, Committee on Dry Cell Tests.
0.04 4 0.4
000
FIG. 1105b.
1106. Mance's Method of Measuring the Internal Resistance
of a Battery. The principle of this method is similar to the one
given in par. 404 for measuring the resistance of a galvanometer,
and, like it, the method makes use of the " second property " of
the Wheatstone bridge.
O
W
FIG. 1106.
The connections to use are shown in Fig. 1106, I or II. With
galvanometers of ordinary sensibility, the current from a battery
ART. 1106] RESISTANCE MEASUREMENTS 219
placed in the bridge arm Ob would deflect the galvanometer
violently off its scale. To avoid this a resistance r is used in
series, and a resistance s in shunt with the galvanometer, r and s
being so chosen that at no time the galvanometer deflects off its
scale. It- is also necessary for good results, if the connections I
are used, to place a resistance W which is approximately equal
to the resistance of the slide wire ab in series with the key K. If
this resistance is not used the wire becomes shunted with practi-
cally no resistance. The positions of the galvanometer, together
with the resistances r and s and the key K may be interchanged
as shown in II. In this case the resistance W is not needed.
The contact p is moved to a position such that the galvanometer
deflection remains unaltered whether the key K is open or closed.
When this position is found we have, if I is the length of the slide
wire and c the distance of p from a,
X = -=R, (1)
where R is the fixed resistance in the arm aO and X the resistance
of the battery sought.
In applying this method with a slide-wire bridge, it should be
noted that, unless the resistance of the slide wire ab is .made
very high (by winding in a helix as described in par. 401), the
battery is yielding considerable current which, in some types
of cells, would probably affect the internal resistance, making
it different than it would be if the cell were yielding a less
current.
In Mance's method, just given, as well as in Kelvin's method
for measuring the resistance of a galvanometer, greater precision
may be obtained by using two equal extension coils, as shown
by n\ and n 2 , Fig. 401a ( 401). In this case calling n the value
of each extension coil, in terms of equivalent length of bridge wire,
we should use the formula for Mance's method,
X = n + l ~ C R. (2)
n + c
Also we should use the same formula ( 404) for Kelvin's method,
in which we replace X, the resistance of the battery, by g, the
resistance of the galvanometer.
220
MEASURING ELECTRICAL RESISTANCE [ART. 1107
1 107. Voltmeter and Ammeter Methods of Measuring the
Internal Resistance of a Battery.
Method I. With K' open (Fig. 1107a) close K and read E } the
open circuit E.M.F. of the battery Ba. With K' closed read EI,
the drop of potential over R. Then the current is
777 77T
T " 1 "
= ~R = 'R~TX'
whence
v E-E,
R.
(1)
This method assumes, first, that the voltmeter takes so little cur-
rent that, with K' open and K closed, the cell may be considered
to be upon open circuit, and second, that the polarization of the
cell is so trifling that when K' is closed the E.M.F. of the cell
remains unchanged. Neither assumption is justified in the case
of many types of cells that polarize readily and is probably never
wholly justified with any type of cell. However, for a rough
estimate of the condition of dry batteries, etc., it is a satisfactory
test. The voltmeter should have a full scale reading of only from
2 to 5 volts for accuracy and the resistance R should be roughly
equal to the internal resistance X of the cell. The total internal
resistance of a battery of cells joined in series may be measured in
the same way, a voltmeter with a scale reading higher being then
required.
-VWVNAM/W
FIG. 1107a.
FIG. 1107b.
Method II.' The measurement may be made using a voltmeter
and an ammeter.
The cell Ba (Fig. 1107b), a resistance R, the ammeter A and a
key K are joined in series. A low reading voltmeter V is con-
ART. 1108] RESISTANCE MEASUREMENTS 221
nected to the cell terminals. With K open, read E, the open
circuit E.M.F. Close K and read EI and the current /. Then,
E
+ R X
hence, x =
This result, in which two E.M.F.'s and a current are measured,
assumes also that the voltage E is not altered by polarization of
the cell.
1108. A Word on Polarities. In Fig. 1108 let a line which
carries a direct current i have introduced in it, in series, a direct-
current ammeter A and a cell B. This is assumed to have an
unalterable E.M.F., e, and a zero internal resistance. Also insert
an ohmic resistance X. Connect a direct-current voltmeter V
at the points 1 and 2 to measure the fall of potential between
the points 1 and 2 or 2 and 1. Let a be the positive terminal
and b the negative terminal of the voltmeter. Let the positive
terminal of the cell be joined to the resistance X.
Line.
-AAAAAAA/
x -H'-
JLine
FIG. 1108.
The magnitude and direction of the current i in the line will
depend both upon the E.M.F., e, of the cell B and upon other
E.M.F. 's which are in the rest of the circuit. Let the line current
i when flowing in the direction from 2 to 1 be called positive,
and negative when flowing in the opposite direction. When the
potential (with respect to the earth) at 1 is greater than the
potential at 2, call the reading e\ of the voltmeter positive, and
when the potential at 2 is greater than the potential at 1, call
the reading e\ of the voltmeter negative. First, assume that the
current i flows from 2 to 1. Then the fall of potential from 3
to 2 is such as to tend to send a current thru the voltmeter from
a to b and the fall of potential from 3 to 1 is such as to tend
222 MEASURING ELECTRICAL RESISTANCE [ART. 1109
to send a current thru the voltmeter from 6 to a. Hence, the
voltmeter will read
ei = e - iX, (1)
from which we deduce
Z-l'. (2)
Second, assume that the current i flows from 1 to 2. Then the
fall of potential from 3 to 2 is such as to tend to send a current
thru the voltmeter from a to 6, as before, but now the potential
rises from 3 to 1 and the potential fall thru X will be such as to
tend to send a current thru the voltmeter from a to b. Hence,
the voltmeter will read
ei = e + iX. (3)
We shall know, however, that the current in the second case is
opposite to the current in the first case because the terminals
of the ammeter will have to be reversed to obtain a reading.
Therefore, if we follow the convention of calling the current
positive when flowing from 2 to 1, and negative when flowing
from 1 to 2 (or against the polarity of the cell), then we should
write, Eq. (3),
e\ = e iXj as in the first case.
In Eq. (1) if i = 0, e\ = e as it should, also e\ will remain positive
as long as iX is less than e. If iX becomes greater than e then e\
will be negative, which fact will be known from the necessity of
changing the voltmeter terminals in order to obtain a reading.
The above principles must be kept in mind when applying the
following volt and ammeter methods of measurement: We shall
adopt the convention that the current in the line is to be regarded
positive if it has the direction it would have if the E.M.F. in the
circuit being tested were the only E.M.F. acting and the volt-
meter reading will be regarded as positive if it reads with its
positive terminal joined to the positive terminal of the circuit
which contains an E.M.F. and is under test.
1 109. Voltmeter and Ammeter Method ; Principle of Polarities
Illustrated. In Fig. 1109a, e is a source of E.M.F. which has a
resistance represented by X. V is a voltmeter. E is an auxiliary
cell, as a storage-battery cell. A is an ammeter. With the key
ART. 1109]
RESISTANCE MEASUREMENTS
223
K closed and the polarities oi the two sources of E.M.F. as shown,
we have
ei = e iX, (1)
where e is the E.M.F. that the voltmeter reads when K is open,
and 61 the E.M.F. that it reads when K is closed. Also i is the
current which the ammeter reads, regard being given to the sign
of i. From Eq. (1) we thus obtain
x =
(2)
X +'
>- AAAAAAAAA
?
<T> + r"
f v V-
-\A/- -|^
_ ^ ^_
FIG. 1109a.
The polarity of E is now reversed, as indicated in the figure by
the dotted lines,, and the value of X is then found to be
e
(3)
The first and second values of the resistance will probably not
agree on account of polarization of the cell being tested. The
mean value, however, of X and Xi should be taken as representing
the most probable value of the resistance.
XB i
4l _ +, _. +i
AVVVW-J h-rl I
X
Bi
-~||-H
II
FIG. 1109b.
This method was tried with the following observations and
results. Connections and polarities were made first as in I, and
second as in II, Fig. 1109b.
For polarities as in I,
i = + 0.1287, e = + 2.16, e l = - 2.64.
224
MEASURING ELECTRICAL RESISTANCE [ART. 1110
Hence, X = -
For polarities as in II,
ii=- 0.0635, e= + 2.16,
Hence, 2.16 - 4.50
- 0.0635
e 2 = + 4.50.
= 36.85 ohms.
The mean of X and Xi is 37.05 ohms.
In this experiment X was a metallic resistance and BI a small
storage cell. The resistance X was measured upon a bridge, after
the test, and found to equal 37.25 ohms. Hence, the error in the
measurement by method II was a little over one half of 1 per
cent, a fair result, considering that the instruments used were a
commercial voltmeter and ammeter.
i no. Total Resistance of a Network between Two Points
when the Branches of the Network Contain Unknown E.M.F.'s.
This method was devised by the author. In Fig. lllOa let
V\AAAAr
! fl ,2/2
^ ^^*0J
l - 1 y 5
2/3 63 2/4
MM
FIG. lllOa.
2/i, 2/2, 2/3> 2/n be any combination of resistances joined
together in a network in any manner whatever. Let ei, 62, e 3 ,
. . . e n be E.M.F.'s associated with the branches of the network
having any values and polarities. The problem presented is to
determine the resistance between the points a and b. The quan-
tity R to be measured should be the same as the quantity which
would be obtained if e\, e^ e s , etc., were all zero and R is defined
aS rr
p - ^ fn
ti y {*}
Let P be any resistance and A an ammeter which will measure
/. Let & be a switch or key which will make connection with
either the point 1 or the point 2.
ART. 1110]
RESISTANCE MEASUREMENTS
225
First put k to point 1 and read the current /i on the ammeter
A and the voltage Vi on the voltmeter V. Then put fc to 2 and
read the current I \ on the ammeter and the voltage V% on the volt-
meter. In the first case, if we call EI the resultant E.M.F. at
the points a and 6 of all the E.M.F.'s, ei, e 2 , e 3) etc., then, as shown
in connection with Fig. 1108,
V l = E l - /!#, (2)
and in the second case
Vi = Ei- IJt. (3)
From Eqs. (2) and (3)
Fi-F,
R =
/2-/1
(4)
In applying this method careful attention must be given to the
convention of signs as explained in par. 1108.
The only assumption made here, which bears upon the precision
of this method, is that the resultant E.M.F. at the points a and
6 is not altered by polarization of the sources of E.M.F.'s, ei,
62, e 3 , etc., when the main current is changed from 7i to 1 2-
In practice this method gives good results under certain cir-
cumstances that often arise. It is adapted to the measurement
of the resistance between two conductors, as between a gas and
water pipe main buried in the earth, when the resistance path in
the earth is subject to many local and unknown E.M.F.'s which
correspond to the E.M.F.'s, ei, e z , e 3 , etc., of Fig. lllOa.
Current
Current
FIG. lllOb.
The following trial of the method was made by the author:
Circuits were made up as indicated in Fig. lllOb. EI was a
small storage-battery cell and E was also a source of E.M.F.
226
MEASURING ELECTRICAL RESISTANCE [ART. 1111
from a storage battery. P was a resistance to vary the cur-
rent in the line. r\ and r 2 were metallic resistances. It was
required to determine, by this method, the resistance between the
points a and b. The following table exhibits the readings and
the results obtained:
Here, V\ and I\ are the voltage and current with K on 1, or
E in circuit, and V% and 1 2 are the voltage and current with K