III, for the manner of making the connections and the meaning
of the symbols used.
ART. 207] VOLTMETER AND AMMETER METHODS
C XTX Reads D
R
31
( vwwvwvw
B 3
y
^wwwvww
B 2
c;
=> wwwwvv
AWVWVWVW
w
II
II-
A/WWWW * tB 2
Reads C?
VWWWW\/ i
III
FIG. 207.
Cij C 2 , etc., are currents and R is the known resistance of the
deflection instrument used. D, di, and d 2 are deflections each of
which is assumed in all cases to be proportional to the E.M.F.
applied at the terminals of the instrument. Measurements of this
kind would ordinarily be made with a Weston voltmeter, reading
to 150 volts, and 110 volts on the supply mains, or with a milli-
voltmeter, having a known resistance of from 200 to 500 ohms in
32 MEASURING ELECTRICAL RESISTANCE [ART. 207
series with it, and current supplied from one or two cells of storage
battery.
If direct current is supplied to the system, a direct-current
deflection instrument must be used, as a Weston voltmeter, a
millivoltmeter with a series resistance, or a galvanometer of known
resistance.
If the current is alternating, then an alternating-current Weston
voltmeter may be used, or other alternating-current deflection
instrument of the electrodynamometer-type with a scale marked
to give indications which are directly proportional to the E.M.F.
at the instrument terminals or to the current thru it.
The resistances x\ and Xz are determined by knowing R, the
resistance of the deflection instrument, and by taking three instru-
ment readings:
1st. Determine the deflection when the instrument is connected
to B l and Bz (Fig. 207, I). Call this D,
2nd. Determine the deflection when the instrument is connected
to Bi and B 3 (Fig. 207, II). Call this di.
3rd. Determine the deflection when the instrument is connected
to B 3 and Bz (Fig. 207, III). Call this d*.
If in case 1 the deflection goes off the scale or in cases 2 and 3
the readings are but a very small fraction of the total scale, the
method is not applicable, without modification, in the former case,
and in the latter case the resistances Xi and Xz are too high to be
satisfactorily measured by this method. Having taken the above
three readings it will be shown that
R (D di dz) , t x
Xi = - j (1)
dz
and
R (D di dz) , \
Xz = -j - (A)
As the resistance y does not enter into Eqs. (1) and (2) it may
have any value without influencing the result.
It is to be noted that, as x\, Xz, and y are in series and form a
closed loop, this method determines the resistances of portions of
a closed circuit without cutting it.
We further note from Eqs. (1) and (2) that
ART. 2071 VOLTMETER AND AMMETER METHODS 33
and that if Xi = infinity, the deflection d 2 = 0, and hence,
This is the ordinary expression used in measuring a resistance with
a voltmeter, and has the same form as Eq. 8, par. 201.
Equations (1) and (2) above are obtained as follows: Let k be
the constant of the deflection instrument, such that the E.M.F.'s
at the instrument terminals are proportional to the deflections.
Then,
Ci- ^ , (5)
/I#i
*^ 2 ' /?
c = *TST - ^
#1 +: r~H
^ / _ X\ _ KCli
Cl := ^~^ Cl ~ IT
or. d= ^^ (7)
kdt
R
Cr _ "2 f
2 D i ^2 ~B~
C 2 = . (8)
(9)
Hence we have the two relations,
kD kdi(R+xi)
Rx\
A;Z) /bc?2(jR+^2) /ir , x
and 5- = ^ ' (10)
The factor k cancels from equations (9) and (10) and x l and 2 are
found, by simple algebraic transformations, to have the values
given above in Eq. (1) and Eq. (2). If any one of the equations
(1), (2) and (4) are solved for R, it is seen that we have a method,
when one resistance is known, of determining the internal resist-
ance of a deflection instrument. The method is often useful for
finding the resistance of a galvanometer when no other instrument
is available. As the E.M.F. of a single cell of battery will throw
34
MEASURING ELECTRICAL RESISTANCE [ART. 208
a galvanometer off its scale, it is necessary to use a feeble E.M.F.
which may be obtained by opposing two nearly equal cells, or by
taking the drop between two near points on a low-resistance wire.
The above method is illustrated by the following measurements :
The deflection instrument used was a millivoltmeter with a resist-
ance in series with it, so that [Eqs. (1) and (2)] R = 102.3 ohms.
The other quantities in the table below have the same meanings
as in Eqs. (1), (2) and in Fig. 207. The source of current was a
single cell of storage battery and the resistances measured were
manganin resistance coils. For the sake of generality a resistance y
(Fig. 207) of 200 ohms was used to shunt the resistances Xi and x*.
The results obtained are given in the table below.
D
di
d,
X
true value
X
found
Per cent
error
100.5
31 5
63.0
Zi = 10
z 2 = 20
9.74
19.48
-2.6
-2.6
100.8
72.2
14.4
zi = 100
x,= 20
100.88
20.12
+0.88
+0.6
It should be noted that in the first case the difference
D (di + d 2 ) = 6 divisions, only, and in the second case it
equals 14.2 divisions and that the precision is, as should be ex-
pected, better in the latter case.
The theoretical precision which can be obtained by this method
is fully discussed in par. 104.
208. Limitations of Voltmeter Methods. It remains to
point out the limitations of voltmeter methods when applied to
the measurement of resistances of over a megohm.
-in,
FIG. 208
Let m and mi be two direct-current mains between which the
potential is e.
Let V be the voltmeter of internal resistance R and let re be a
resistance (over a megohm) which is to be measured. With the
ART. 209] VOLTMETER AND AMMETER METHODS 35
connections made as in Fig. 208, first put the switch s on a and
read the deflection of the voltmeter, which call d 2 , then on 6 and
read the deflection, which call d\. By the theory and formula
given in par. 201, Eq. (8), we have
Ordinary conditions would be met by assuming that the mains
furnish 110 volts, that the voltmeter has one division to the volt
and reads 150 volts at the limit of its scale, and that it has a re-
sistance of 100 ohms to the volt or a total of 15,000 ohms. We
shall then have d 2 = 110 and R = 15,000.
Solving Eq. (1) for d\, we find
Rd 2
" x + R
If we assume that x 10 6 ohms, or one megohm, we find d\ =
1.625 + divisions.
The expression for calculating the maximum percentage error
is,
[See Eq. (4), 104] where - is the fractional part of the larger
deflection which can be read. Suppose that the readings are
taken with great care so that they are accurate to within 0.05 of
a division, then
_
p 20X110" 2200'
and we find by Eq. (2) that
p 110 110 + 1. 625 v ,
Ep ~ 2200 X 1.625 X 110 - 1.625 *
This is a fairly representative case and it shows that one megohm
is about the largest resistance which can be measured with a
150-volt voltmeter, with even very moderate precision. If the
voltmeter is a 300-volt instrument and this potential is available
the accuracy would of course be correspondingly increased.
209. Resistance Measured with a Voltmeter and an Ammeter.
Connect the ammeter in series with the resistance and the volt-
36
MEASURING ELECTRICAL RESISTANCE [ART. 209
meter in shunt, as indicated in Fig. 209. Calling R the resistance
of the voltmeter, x the resistance to be measured, / the current
Voltmeter
FIG. 209.
measured by the ammeter, and V the potential at the terminals
of the voltmeter, we have, by Ohm's law,
Rx V
x =
(1)
It appears from Eq. (1) that if R is very large as compared with
y
V the term ~ can be neglected.
K
As the above method is often used to measure the resistance of
an incandescent lamp while carrying full load current, the follow-
y
ing example is given to show the importance of the term -^ in
tc
this case: Let the current / be measured with an ammeter,
which has a full scale value of 1 ampere, and let the voltage drop
over the lamp be read with a 150-volt voltmeter which has 100
ohms resistance to the volt or a total of 15,000 ohms. If the line
current is 0.5 ampere, then the true resistance of the lamp, if the
voltmeter reads 110 volts, is,
110
x =
0.5-
110
= 223.1 ohms.
15,000
If the last term in the denominator is neglected we obtain for the
value of the resistance,
x' = ^r = 220 ohms.
This result is 1.39 per cent lower than the true value, which is
223.1 ohms.
ART. 210] VOLTMETER AND AMMETER METHODS
37
The practical way to measure the resistance of a 110- volt in-
candescent lamp by this method, using Weston instruments, would
be to divide the volts read by the current and add 1.4 per cent
to the result.
A trial was made of this method using the same tungsten lamp
and voltmeter employed in the test to illustrate the method
shown in Fig. 203. The resistance of a carbon 60- volt lamp joined
in series with the tungsten lamp was determined at the same
time. The resistance R of the Weston voltmeter was 15,660
ohms. The ammeter was a Weston instrument reading 1 ampere
fora full scale. The results were calculated by Eq. (1), par. 209,
and are exhibited in the table below:
/
V
X
Remarks
Tungsten Lamp.
0.479
65.8
138.5
The resistance of the tungsten
0.317
32.7
103.8
lamp increases 1.05 ohms per
volt in range 33 to 66 volts.
Carbon Lamp.
0.620
65.7
106.6
The resistance of the carbon
0.270
32.8
122.3
lamp decreases 0.477 ohm per
volt in range 33 to 66 volts.
Calculating the resistance of the tungsten lamp at 33.9 volts
from the resistance found at 32.8 volts on the assumption that the
resistance increases 1.05 ohms per volt, we find this resistance to
be 105.06 ohms. By the method of par. 203, we found the resist-
ance at this voltage to be 104.8 ohms. The two measurements
are thus seen to be in agreement within 0.25 of 1 per cent.
210. Remarks Upon the Methods of Chapter II. The
methods given above for measuring resistance with deflection
instruments of convenient type are the ones which are most often
used and are best suited to practice. They may be modified and
extended in a variety of ways, but the fundamental principles
involved would not be altered. The methods as given are
typical and will cover almost every case that may arise. They
are sufficient to illustrate the principles involved, and, if these are
understood, modifications may easily be made to meet any special
requirements. In considering the theoretical precision of the
deflection methods given above, reference should be made to the
elementary discussion of the theory of errors in Chapter I.
38 MEASURING ELECTRICAL RESISTANCE [ART. 211
2ii. Ohmmeters and Meggers. These instruments are
devices for reading resistances directly. Ohmmeters are of two
general types: those which read a resistance by the deflection of
a pointer over a scale, this deflection being proportional to the
resistance, but independent, within wide limits of the value of the
testing current employed; and those which operate on the prin-
ciple of a slide-wire bridge, which is balanced, the balanced con-
dition being indicated by a galvanometer or a telephone. The
scale under the slide wire is laid off in ohms so that the resistances
are read directly in ohms.
The first type is constructed by the maker so as to operate
directly by simply connecting the resistance to be read to two bind-
ing posts and reading the deflection of a pointer. A description
of them involves describing not methods of measurement, but in-
struments, and they will not be further considered here.
The second type should be included under a description of
" balance " methods of measuring resistances.
" Megger" is a trade name given to a type of deflection ohm-
meter, which is very fully described in trade publications. It is
claimed that they give all results which can be obtained with volt-
meters in the measurement of resistances and with greater facility
and precision. This claim is probably justified. If such an in-
strument is available it should be used in preference to a voltmeter
for many kinds of resistance measurements.
The " Megger" was designed and patented by Mr. Sidney
Evershed of London, for the rapid determination of high resist-
ances. Following is a very brief description of this instrument
and some of the claims made for it.
It is a direct-reading ohmmeter with a direct-current hand-
driven dynamo mounted in the same case. The instrument is
arranged for easy portability and rough usage. The scale is
graduated in ohms so that no calculations are required. The
hand-driven dynamo delivers current at 100, 250, or 1000 volts
(according to the range and sensibility of the particular type used)
and this makes the instrument independent of outside current
supply. The instrument is similar in principle to a differential
galvanometer so devised that moderate change of dynamo voltage
does not vary the scale reading. The entire instrument weighs
about 20 Ibs. It is claimed to have a range from 0.01 to 2000
megohms. Two scales of high-range meggers are shown full size
ART. 2111 VOLTMETER AND AMMETER METHODS
39
in Fig. 211. From an inspection of these scales one may estimate
the possible precision attainable on the assumption that the instru-
ment itself is accurately calibrated.
FIG. 211.
The instrument is sold in America by James G. Biddle, Phila-
delphia, Pa., and is fully described in his catalogue No. 740.
This being an instrument rather than a method of measurement,
the author must refer the reader to the above-mentioned pamphlet
for a more detailed description.
CHAPTER III.
NULL METHODS. RESISTANCE MEASURED BY
DIFFERENTIAL INSTRUMENTS.
300. Remarks on Null Methods. When some quantity of
known value is varied in a known way until it is made equal to,
or to some known multiple of, a quantity being measured, and
when the attainment of this condition is indicated by some de-
tector then showing no deflection, the measurement is said to be
made by a "null" or "balance" method. The most familiar
example of the above is weighing with a chemical balance. Here
a mass is the unknown quantity. Known masses are placed in
the pan until the balance, which is here the detecting instrument,
shows no deflection. Other examples are: The measurement of
resistance with a Wheatstone bridge, where known resistances are
varied until they are made a known multiple of an unknown
resistance. The zero deflection of a galvanometer indicates in
this case when the known resistances have been perfectly adjusted;
the measurement of potential differences with a potentiometer,
where a known potential difference is varied in a known manner
until it equals an unknown potential difference under measure-
ment. The zero deflection of a galvanometer, the detecting in-
strument, shows in this case also when the equality is attained.
Null or balance methods of measurement are generally more
sensitive and accurate than deflection methods: 1st, because it
is unnecessary, as in deflection methods, to know the constant of
the detector; 2d, because it is unnecessary to know the law of the
deflection of an indicating instrument, or to rely upon the accuracy
or the constancy of the calibration of a scale; 3d, because quanti-
ties of the same kind are simply matched against, or compared
with, one another; 4th, because it is easier to detect a small
departure from the zero of an indicating instrument than it is to
read the exact extent of a deflection.
For reasons of the above character, weighing, bridge methods
of measuring resistance, and potentiometer methods of measuring
potential differences are the most accurate used.
40
ART. 301] NULL METHODS 41
Balance or null methods of measurement have, however, the
disadvantages of requiring more apparatus, of consuming more
time, and of requiring more manual manipulation than do deflec-
tion methods. If the quantity under measurement is subject to
fluctuations, which it is desirable to note, then strictly null methods
are less suitable than deflection methods. In case the quantity
being measured is fluctuating, it is generally impossible to perform
the manual manipulations with sufficient rapidity to maintain the
balanced condition and, even if this can be done, to record the
settings.
There are methods, which we shall discuss later, which measure
the quantity by approximately matching it with a known quantity,
deflections being used to determine the small differences. These
balance-deflection methods are of great value in industrial elec-
trical measurements, as they combine precision and speed.
The strictly null methods are chiefly valuable in connection
with standardization measurements. Those which are useful in
the measurement of medium resistances will receive full attention.
The methods to be discussed now are those which make use of a
differentially wound galvanometer and those which employ some
form of the network of resistances generally known as the Wheat-
stone bridge.
The differential methods have certain advantages in some im-
portant commercial applications and therefore deserve a careful
consideration.
301. Properties of Differential Circuits. Almost any instru-
ment which deflects with the passage of an electric current thru
it may be differentially wound; namely, so wound with a double
winding that a current thru one winding is exactly neutralized, in
its action to produce a deflection, by an equal current thru the
other winding. A differential winding may be applied to milli-
ammeters, voltmeters, galvanometers of the moving-magnet or
of the moving-coil type, and to telephones. When applied to
direct-current instruments the circuits have certain interesting
and useful properties, to a consideration of which we now pro-
ceed. The properties to be discussed are quite independent of
the type of the differential instrument. This may be a moving-
magnet or a moving-coil galvanometer, but for definiteness let
us fasten our attention upon a differentially wound D' Arson val
galvanometer.
42
MEASURING ELECTRICAL RESISTANCE [ART. 301
One assumption must always be made, and a second assumption
generally. The first is that equal currents, but opposite in their
electromagnetic action, in the two windings shall be without influ-
ence upon the movable system. The second is that the resistances
between binding posts of the two windings shall be the same. A
third assumption is sometimes required for certain classes of
refined work. It is, that the resistances oT the two windings
shall remain constant under temperature changes in the room
and with varying quantities of current in the windings. This
last assumption can only be met, as a rule, with difficulty or at
the sacrifice of sensibility, as when coils are wound with wire of
high resistance to secure a negligible temperature coefficient.
II
FIG. 301a.
To test if the instrument is truly differential join the two wind-
ings in series as shown at I, Fig. 301a. Then, if the number of
turns in the windings gi and g% are equal and wound in opposite
directions, the same current traversing these windings should
produce no turning moment on the system. It is found in prac-
tice, however, that howsoever carefully the coil is wound to be
made differential, some tendency will remain for the system to
turn. Suppose the winding g z slightly predominates. If a very
high resistance S is shunted around this winding, it may be
adjusted until the system is entirely without tendency to rotate,
whatever value the current 7 has; provided this current is not
so large as to greatly heat the windings.
ART. 301]
NULL METHODS
43
Next join the windings in parallel, as in II, Fig. 301a, and vary
a small resistance n in series with gi, or a resistance r 2 in series
with / 2 , until ii = i 2 , when the system will again give no deflection.
With these two adjustments it is possible to make the system differ-
ential and to give the two windings exactly the same resistance
namely, the total resistance from post 1 to post 2 will equal the
total resistance from post 3 to post 4. The above adjustments
should be made by the instrument maker and are explained here
only to show their possibility.
If the windings g\ and g 2 are of copper and differ very much in
resistance, then changes in temperature will destroy the relation
between the resistance of # 2 and the shunt S and the adjust-
ment to differentially will not be permanent. With proper
original construction, however, such a defect should not exist. .
It will now be assumed in what follows that the instrument is
truly differential and that the two windings from binding post to
binding post have the same resistance. It cannot be assumed
that the resistances of the windings will remain constant when
these are of copper, but if they change with temperature they will
change alike; because the two windings are made by winding a
double wire and are therefore
intimately associated, and hence
always at the same temperature.
We proceed, first, to prove
an interesting and useful prop-
erty of the circuits of a differen-
tial galvanometer when these
are made up in the manner in-
dicated in Fig. 301b.*
Let 1, 2, and 3, 4 be the
binding posts of the differential
instrument (made strictly dif-
ferential by means of adjust-
ments as described above).
Is
A/WW-
FIG. 301b.
Let g and g be its two windings so disposed that equal currents
i and i flowing in the direction indicated by arrows, produce no
deflection.
Let fk be a uniform slide-wire resistance along which a sliding
* Described by author in the Trans, of the Electrochemical Society, May,
909, Vol. XV, p. 340.
44 MEASURING ELECTRICAL RESISTANCE [ART. 301
contact t may be moved, and let this slide wire be shunted by a
resistance S.
Let R be any fixed resistance from to / and X any resistance
from to A; which may be varied.
Now, it is evident that if R and X are nearly equal there will be
some position for t on the slide wire fk such that the currents i
and i thru the two windings will be equal and the system give no
deflection.
If I is the length of fk (and is proportional to its resistance) , we
wish to determine how the distance c of t from / varies when
X varies. If c varies uniformly with variations in X we have a
means of measuring small changes in X with high precision.
The signification of all the symbols used will be easily under-
stood by a reference to Fig. 301b, without further explanation.
By Kirchoff's laws,
i i c i a = are the currents which enter and leave /, (1)
i + i a id = are the currents which enter and leave k, (2)
iX + i d d = E is the E.M.F. in circuit OktQ, (3)
iR + i c c = E is the E.M.F. in circuit OftO, (4)
iS -f idd i c c = is the E.M.F. in circuit fSkf. (5)
Subtracting (4) from (3) gives
i (X-R)+ i d d - i c c = 0. (6)
From (1) i e = i - i 8 . (7)
From (2) i d = i + i a . (8)
Putting the values of i c and i d in (6) gives
i (X - R) + di + di s - d + ci a = 0, (9)
or i s (c + d) + i (X -R) - i (c-d)= 0. (10)
Putting the values of i c and i d in (5) gives
i 8 S + di + di s ci + ci a = 0,
. i(c-d)
Also from (10) we have
.^tfr-cO-^-B). (12)
Equating the second members of (11) and (12) we derive
R(c + d + S)+S(c-d)
ART. 302] NULL METHODS 45
In (13) replace d'by its value I c and we obtain
v _ 2cS , R(l + S)-lS . .
~S + l^ S + l
The last term of (14) is a constant.
Call this K and we have, finally,
K. (15)
Differentiating (15), we find
dX = 2S
8c ~ S + l'
OTdc = ^ l dX. (16)
Eq. (16) shows that c varies proportionally to X and only
S + l
g as rapidly as X.
^ o
This result means that in the case of a uniform slide resistance,
the slide resistance may be given any convenient value greater
than the minimum allowed. It may then be shunted to give the
particular resistance range desired with any chosen length of
slide resistance.
By minimum resistance allowed, we are to understand a resist-
ance which is large enough to permit a balance to be obtained with
the sliding contact upon the slide resistance fk when X changes
from its least to its greatest value.
For example, if when X has a minimum value the contact t is
at point / for a balance, it must be possible to obtain a balance
with the contact t at, or to the left of, the point k when X has its
greatest value.
Putting Eq. (16) in the form
we see that when S is infinity, that is, when the shunt is absent,