Then if either r or 7*1 be varied until the galvanometer shows no
154 MEASURING ELECTRICAL RESISTANCE [ART. 801
Suppose the standard R is 10 5 ohms, and r is plugged at its ex-
treme value 10 4 ohms and n is plugged at 1 ohm, then
x = -r- 10 5 = 10 9 ohms, or 1000 megohms
is the greatest resistance which can be measured with this dis-
position of resistances. If R is a megohm then ten times this
resistance may be measured, provided the galvanometer is suffi-
Suppose the galvanometer will deflect one scale division with
10~ 9 ampere, or one tenth of a scale division with 10~ 10 ampere,
and we wish to measure x to within 1 part in 1000, when x = 10 9
ohms and R = 10 5 ohms. We have to inquire what E.M.F. E
must be applied to the bridge at the points a and 6. The
approximate value of this E.M.F. is easily found as follows:
First, suppose the bridge is balanced when the unknown resist-
ance has the value x, then very approximately, the current i
which will flow thru x is, i =
The fall of potential over x with the bridge balanced is
E = ix, (2)
and, if x receives an increment dx this fall of potential is
E + dE = ix + i dx. (3)
Hence dE = i dx, or
dE = -dx. (4)
Now dE is the E.M.F. effective to send a current 5i thru the
galvanometer and this is, very approximately, 5i = . Hence
dE = x di which value of dE put in Eq. (4) gives
Since x = 10 9 ohms, di = 10~ 10 ampere and dx = 10 6 ohms, for
an accuracy of 1 part in 1000, we obtain
E = 10 18 X jp = 100 volts.
One hundred volts then is the necessary E.M.F. for measuring
1000 megohms, by the bridge method, to an accuracy of 0.1 of 1
per cent with a galvanometer which will give one division deflec-
tion with 10~ 9 ampere.
ART. 802] THE MEASUREMENT OF HIGH RESISTANCE 155
The chief precaution to observe in applying this method is to
make sure that the galvanometer and the 0.1 megohm are per-
fectly insulated from earth. This is easily accomplished by
setting the apparatus upon plates of glass or hard rubber and
running the connections thru the air, supporting the wires on
glass or hard rubber.
It is well also to include in the battery circuit a resistance of
not less than 1000 ohms to protect the galvanometer from injury,
should the resistance being measured break down.
802. Use of a Capacity in Connection with a Wheatstone
Bridge for High-Resistance Measurements. This method,
which is taken from " Measurement of Electrical Resistance,"
by W. A. Price, is added for the sake of completeness. The
author thinks, however, there would not be much occasion for its
employment in view of the better methods which are available.
The diagram, Fig. 802, is almost self-explanatory.
Here a condenser C is charged to the difference of potential
of the points a, 6 by moving the levers of the key so lever q is
insulated and lever p makes connection with point 1, and then
the condenser is discharged thru the galvanometer Ga by moving
the levers so p is insulated and q makes connection with point 2.
The resistance r or r x is adjusted until, when the condenser is
connected for discharge, there is no deflection. When this adjust-
MEASURING ELECTRICAL RESISTANCE [ART. 803
ment is effected, a and 6 must be at the same potential and the
ordinary Wheatstone-bridge formula holds, giving
X = -R.
The advantage which the method is supposed to possess is one of
greater sensitiveness, which results from the fact that C has
time to become fully charged by the slow leak of current thru X,
and this charge is then able to expend its energy suddenly upon
the galvanometer, producing a deflection far greater than would
be obtained by the same degree of unbalance of the bridge and
with the galvanometer joined directly to the points a and b. In
using the method the condenser C should be chosen of as great
capacity as possible and the galvanometer should be of very high
803. Major Cardew's Electrometer Method of Measuring a
High Resistance. In this method, proposed by Major Cardew,
(Fig. 803) the standard R which must be made variable and the
high resistance X to be measured are joined in series and their free
ends a and b are connected to the quadrants of an electrometer.
The vane is joined to the point of junction c of the two resist-
ances. The resistance R is varied until the electrometer shows
no deflection. Then X = R. In using this method it would not
be necessary to produce an exact balance. The deflection of the
electrometer is noted when R is too small, and calling R the re-
sistance and d the deflection we then increase R by an amount 5R
and again note the deflection d' which should be in the opposite
ART. 806] THE MEASUREMENT OF HIGH RESISTANCE 157
direction. If for small deflections we assume the deflections to
be proportional to the potential applied to the vane, we have
804. The Measurement of High Resistances, Unassociated
with an Appreciable Capacity ; Deflection Methods. The
measurement of the specific resistances of insulating materials,
the insulation resistance of electrical apparatus, etc., is not a
measurement which usually demands high precision. The resis-
tance of insulating materials is subject to considerable fluctuation
from temperature changes and other causes and hence the less
precise, but more convenient and sensitive deflection methods
are to be preferred to the null methods which are so superior in
the case of medium and low resistances.
In describing these methods we shall reserve for separate
paragraphs the methods of measuring the insulation of cables and
condensers, as the presence of an appreciable capacity must con-
siderably modify the procedure.
805. The Galvanometer and Accessory Apparatus for High-
Resistance Measurement. The instrument most used and
best adapted to very high-resistance measurements by a deflec-
tion method is the galvanometer. In connection with the galva-
nometer a standard resistance is required. This standard may
be either a megohm or a one-tenth megohm. Because of the
expense of the former the latter is now almost universally employed.
To increase the range of measurement the galvanometer is gen-
erally used with a high-resistance shunt which is made variable.
This shunt serves the same purpose in high-resistance measure-
ments as the variable ratio arms of a Wheatstone bridge in the
measurement of medium resistances. Special types of highly
insulated keys and insulating posts and plates complete the
accessories required. We proceed to give the theory and uses
of galvanometer shunts.
806. Galvanometer Shunts. There are two types of gal-
vanometer shunts, the ordinary and the universal or Ayrton shunt.
In the use of the ordinary shunt the resistance of the galvanom-
eter must be known. In the diagram, Fig. 806a, let S be the
resistance of the shunt and g the resistance of the galvanometer.
The main current C will divide thru the shunt and galvanometer
158 MEASURING ELECTRICAL RESISTANCE [ART. 806
in the inverse ratio of their resistances. If C 8 is the current in
the shunt and C g the current in the galvanometer,
? and C 4- C C
C- , d-IlU. O0 T U 8 U,
C ' = j^S C (D
is the current thru the galvanometer,
is the current thru the shunt.
To obtain the value of the main current from the current thru
the galvanometer, we have from Eq. (1)
The quantity M = ^^ is called the multiplying power of the
shunt, namely, it is the quantity by which the galvanometer
current must be multiplied to obtain the main current.
If we wish to make the current in the galvanometer - of the
main current, that is, to reduce the sensibility of the galvanometer
to TTJ: we must make S = ,, .,
M M I
For putting this value of S in Eq. (1) we have
The introduction of the shunt, however, changes the resistance
of the circuit. After the galvanometer is shunted its resistance
If it is necessary to keep the resistance of the circuit constant,
ART. 806] THE MEASUREMENT OF HIGH RESISTANCE 159
when a shunt is added to the galvanometer, there must be intro-
duced into the circuit a resistance which is
It is customary to make shunt boxes so that M may be given
such values as 1, 10, 100, 1000, and 10,000. We should then have
Mi = 1 S = oo
M 3 = 100
Some shunt boxes are provided also with means for adding the
proper resistance in series with the circuit to maintain the resist-
ance of the circuit constant when differ-
ent values are given to the shunt. One
arrangement used is shown in Fig. 806b.
Here the shunts are the coils Si, S 2 ,
S$ and the compensating resistances the
coils 0i, 02, 03, and 4 . A plug inserted
at d puts the circuit directly to the gal-
vanometer without a shunt. A plug in-
serted at b and &', for example, shunts
the galvanometer with the shunt $2 and
puts into the circuit the compensating
resistances 0i + 2 , and similarly for
plugs inserted at a, a' or c, c' '. A plug
at e short circuits the galvanometer and puts into the circuit the
compensating resistance 0i + 2 + 3 + 04, which sum equals the
resistance of the galvanometer alone.
This type of shunt should be wound with wire of the same tem-
perature coefficient as the wire with which the galvanometer is
wound. Practically all galvanometers are wound with copper
MEASURING ELECTRICAL RESISTANCE [ART. 807
wire and this changes in resistance about 4 per cent for every 10 C.
change in temperature. Unless the coils in the shunt have the
same temperature coefficient and are maintained at the same tem-
perature as the coil in the galvanometer (a matter hard to realize
in practice) unallowable errors may result from the employment
of this type of shunt. Furthermore every shunt must be adapted
to the particular galvanometer with which it is to be used.
These disadvantages are overcome in the universal or Ayrton
shunt, which is the kind now almost universally in use. The
theory and use of the Ayrton shunt is as follows :
807. The Ayrton or Universal Shunt. Fig. 807a shows the
disposition of the circuits employed, a, b, c, d, e, are resist-
ance coils of manganin or other low-temperature-coefficient wire.
These are joined in series and the galvanometer terminals are
permanently connected to the terminals of the series. One ter-
minal of the main circuit is permanently joined at one end, as at
* a 2 b 3 _ c 4 _d 5 e
the point 6, and an arrangement is provided by which the other
terminal p may be moved to any of the points 1, 2, 3, 4, 5, 6.
R represents the total resistance and E the E.M.F. included in the
main circuit. The resistance of the galvanometer alone is g.
It was shown, Eq. (3), par. 806, that the multiplying power of
any shunt is M = ^^ , where S is the total resistance of the
shunt and g is the resistance of the galvanometer. We can now
construct for the different resistances in the galvanometer circuit
and for the different resistances which shunt the galvanometer,
when p is moved from the point 1 to 2 to 3, etc., the following
ART. 807] THE MEASUREMENT OP HIGH RESISTANCE 161
p on point
Res. in gal. circuit
Value of shunt, S
Multiplying power of shunt
M ^+a+6+c+d+e '
M 6 = Infinity.
It now appears from the 4th column of this table that the
relative value of the multiplying power of the shunt for any two
positions of the contact p is independent of the resistance of the
galvanometer. Thus, calling a+b+c+d+e=r
It follows that, if the resistances are chosen so e = 0.0001 r,
d + e = 0.001 r, c + d + e = 0.01 r and 6 + c + d + e = 0.1 r,
the sensibility possessed by the galvanometer (when shunted with
the resistance r) will become 0.1, 0.01, 0.001, or 0.0001 as great
according as the contact p rests on point 2, .3, 4, or 5. This
result is obtained theoretically with a galvanometer of any resis-
tance and with any value given to the total resistance r. The
question then arises: what considerations govern the value which
should be given to r? It will be observed, if the resistance r
is made very high as compared with the resistance of the galva-
nometer, that, with the contact on point 3 or 4, the galvanometer
has thrown in series with it a very considerable resistance which
will reduce greatly the current C in the main circuit (Fig. 807a)
unless the resistance R in this main circuit is also very high. On
the other hand, if the resistance r is made very small, as com-
pared with g, the galvanometer being permanently shunted with
a low resistance has its intrinsic sensibility much reduced. Also,
162 MEASURING ELECTRICAL RESISTANCE [ART. 807
if this is a D' Arson val galvanometer it will be overdamped
when r is small, and the coil will move sluggishly. Experience
and practice show that r should be chosen approximately ten times
the average resistance of the galvanometers which are to be used
with the shunt.
In considering the principle of the Ayrton shunt it should be
carefully noted, that, while the shunt reduces the sensibility of
any galvanometer in a definite way it does not in general reduce
the current thru the galvanometer in the same definite way. Thus,
if we call C' the main current when p (Fig. 807a) is on point 1,
the galvanometer current will be
r C f
Cf _ ' ni _
a i ^ TI /r '
g + r Mi
If the contact is now moved to some other point as 3, and we call
the main current which is then flowing C'" the galvanometer
current will be
, c + d + e
g + r ~ Ms '
The ratio of the galvanometer currents in these two cases is
C r ' r C' M 3 C'
Only when the external resistance R is very large, so that the
effective resistance of the entire circuit remains practically con-
stant for the different positions of the shunt contact, will the
current C"' be the same as the current C'. In this case only will
the ratio of the galvanometer currents for any two positions of
the shunt contact be in the inverse ratio of the multiplying powers
of the shunt for these two positions.
If by any device the main current C is maintained exactly
constant, then the shunt will exactly cut down the galvanometer
current in the same way it cuts down the sensibility. In measur-
ing insulation resistances, R is usually very large and the Ayrton
shunt may be used not only as a device to reduce galvanometer
sensibility but also to reduce in like manner the galvanometer
current by known and fixed amounts. It is in this latter way and
for this purpose that the Ayrton shunt is chiefly used and it is
therefore necessary to investigate the magnitude of the errors
introduced, under different conditions of use, when it is assumed
ART. 807] THE MEASUREMENT OF HIGH RESISTANCE 163
that the galvanometer current is cut down in the same proportion
as the galvanometer sensibility.
Referring to Fig. 807a, the current thru the galvanometer is
c * = ji c > a)
where M is the multiplying power of the shunt (which takes
values MI, M 2 , etc., according as p is on point 1, 2, etc.).
Call R a the shunted value of the galvanometer resistance and
R the resistance in the main circuit. Then if E is the E.M.F.
of the source,
c - E
R + R,
and c " E <
Let the contact be upon a point p such that M = M p and
R s = R 8 f , then the galvanometer current will be
Now move the contact to a point q such that M = M q and
R, = R 8 "', then the galvanometer current will be
r " = 1 u\
M q R+R a "
By taking the ratio of Eq. (4) to Eq. (3) we find
C " M p R+R 8 '
C ' M q R + R a '
-p I -p r
Eq. (5) shows that, in so far as p , *, differs from unity the
rt + H a
ratio of the galvanometer currents for any two positions of the
shunt differs from the inverse ratio of the multiplying powers of
the shunt for the two positions.
1 -1- Ra '
R+R 8 f ~R
R + R."
we note that when R is very large the fraction is practically unity,
that is, the current in the main circuit is practically constant,
while the current in the galvanometer is changed in the same ratio
but inversely as the multiplying power of the shunt is changed.
MEASURING ELECTRICAL RESISTANCE [ART. 807
The following typical case will serve to show the magnitude of
the errors which actually result from assuming that the current
in the galvanometer is altered in the ratio - - Reference is here
made to Fig. 807b.
By means of the handle H the contact can be moved to positions
1, 0.1, 0.01, 0.001, 0.0001, 0, Inf. In the case selected the shunt
is intended for use with galvanometers having from 100 to 500
ohms resistance. We shall take
g = 350 ohms,
r = 3000 ohms,
and first assume that the external resistance is
R = 10 5 ohms.
= r = 3000 ohms and M 1 =
b + c + d + e = 0.1 r = 300 ohms and M 2 = 10 Mi,
c +d+e= 0.01 r =
d + e = 0.001 r=
e = 0.0001 r =
30 ohms and M 3 = 100 Mi,
3 ohms and M 4 = 1000 MI,
0.3 ohm and M 5 = 10,000 MI.
ART. 807] THE MEASUREMENT OF HIGH RESISTANCE 165
The shunted galvanometer resistance then takes the following
gr 350 X 3000
R, 1 = - = - ^^ - = 313 ohms.
g + r 3350
p In _
g + r
(350 + 2700) X 300 QnQ
- oorrt - = oUo.U,
g + r
(350 + 2700 + 270) X 30
_ OQ 7Q
IV _ (g+a+b+c)(d+e) _ (350 +2700 +270 +27) X 3 _ QQ7
p v _
a + b + c+d)e _ (350 + 2700 + 270 + 27 + 2.7) X 0.3
g + r 3350
If we now put these numerical values in expressions of the form
given in Eq. (5), we obtain the following values for the current
in the galvanometer with the shunt in positions 1, 2, 3, 4, 5:
C n C ? v 1Q5 + 313
" 10 X 105+303
r m C,i 105+313
100 A 105+30
r TV C ff i 105+313
* 1000 X 1Q5+3
Cff ~ 10000 X 1Q5+0.3
In this case it is seen that the last terms differ very little from
unity. Thus the largest departure is for the position 5, where
IP 4- 313
10 5 + 0.3
With an external resistance as great as 100,000 ohms and a shunt
of total resistance 3000 ohms it is legitimate to assume, for most
work, that the shunt cuts down the galvanometer current in the
same way as it cuts down the galvanometer sensibility, that is,
inversely as the multiplying power of the shunt. If, on the other
hand, R is made as low as 1000 ohms the error for position 5 of
the shunt would be as much as 31.3 per cent.
166 MEASURING ELECTRICAL RESISTANCE [ART. 808
We can now find from the galvanometer deflection, with the
shunt set in any position, the value of the main current C as
follows: By the principle embodied in Eq. (1), par. 807, the cur-
rent thru the galvanometer is equal to the main current divided
by the multiplying power of the shunt, or, in general,
. , O r
L ~M =
808. Galvanometer Constant, Obtained by Using an Ayrton
Shunt. To determine the constant of the galvanometer using
an Ayrton shunt to which the galvanometer is permanently
attached (as in Fig. 807b) we may proceed as follows:
Let C' = MiCg' be the main current with the shunt in position 1 ,
C" = M^Cg" be the main current with the shunt in position 2,
C'" = M z Cg f be the main current with the shunt in position
3, with similar expressions for the other shunt positions.
If the galvanometer current is C g ' = K\ d\ for the shunt in
position 1, where K\ is a constant and d\ the galvanometer deflec-
tion, we have
C' = M&tdi
where K is another constant. Similarly
C" = M 2 #i d z = 10 M&i d 2 = 10
C'" = 100Kd 3 ,etc.
where JR.', R 8 ", R s ' n ', etc., are the shunted galvanometer resistances
with the contact on positions 1, 2, 3, etc. We therefore obtain
R + R a f==Kdl '
or K = -T-,
JK -f- ri a Cti
or K = 0.1 R _^ Rf , j-, or K = 0.01 _^ , J-etc.
Or, in general,
Where N refers to the numbers 1, 0.1, 0.01, etc., stamped upon the
ART. 809] THE MEASUREMENT OF HIGH RESISTANCE 167
shunt for the shunt position used, R a is the shunted galvanometer
resistance for that position and d is the galvanometer deflection.
In determining the constant K it is customary to take R so large,
usually 10 5 ohms, that R 8 is negligible in comparison. For insula-
tion testing this approximation may be permitted and we have,
with sufficient precision for many purposes,
809. Insulation Measurements with a Galvanometer and an
Ayrton Shunt. When the insulation resistance to be measured
has a comparatively small capacity and dielectric absorption the
procedure is very simple and is carried out in practice as follows:
A galvanometer (usually a D'Arsonval instrument), a standard
one-hundred-thousand-ohm box, an Ayrton shunt, and the resist-
ance to be measured are joined as in Fig. 809.
In a convenient type of construction, the battery key 6 is
combined with the Ayrton shunt for compactness and also for
convenience in manipulation. The handle a controls the key b
168 MEASURING ELECTRICAL RESISTANCE [ART. 809
and is mounted so that it projects up thru the handle which
operates the shunt. By depressing a, the contact b is closed.
This is arranged so that it can be locked in the closed position.
If the shunt and the D'Arsonval galvanometer are properly
related the latter will be just aperiodic.
The shunt may be held in one hand while its handle and key is
manipulated with the other hand. Both the shunt and one-
hundred-thousand-ohm box should be highly insulated and are,
for this purpose, often constructed entirely of hard rubber.
To make a measurement of insulation resistance, of a short
length of cable for example, the procedure would be as follows: A
battery of 50 or 100 dry cells is used. These should be first joined
directly to the one-hundred-thousand-ohm box instead of to the
cable as indicated in the figure. The constant of the galvanometer
may now be obtained. This is not the same constant as that given
by Eq. (2) in par. 808, which is the true galvanometer constant.
It is an arbitrary constant defined by the relation D = -^ .
Here G is the constant sought, N the shunt setting 0.1, 0.01, 0.001,
or 0.0001. The 0.1 is the one hundred thousand ohms expressed
in megohms, and D is the galvanometer deflection which is
obtained with the particular battery used for the measurement.
Thus we have
G= IQN' (1)
In obtaining this constant, the shunt is first set at zero. The
battery circuit is then closed by depressing the battery key b
and the shunt is moved, first to N = 0.0001, and the deflection
no-ted. If this is less than 25 small scale-divisions, the shunt is
moved to N = 0.001, and, if still less than 25 scale-divisions, to
N = 0.01. The final deflection, which should not be less than
25 divisions, is noted and called D. The shunt setting also being
noted, the constant is given by expression (1) above.
The constant G having been thus determined the insulation
resistance may now be measured by the following procedure:
First connect the battery to the cable or resistance to be measured,
as shown in Fig. 809. Reset the shunt to position zero. The one-
hundred-thousand-ohm box, or 0.1 megohm, may be left in circuit
or it may be short-circuited. In the former case the resistance
measured will include this and will be 0.1 megohm too large.
ART. 809] THE MEASUREMENT OF HIGH RESISTANCE 169
Close the battery key b. If the cable has any considerable capac-
ity and dielectric absorption, sufficient time must be allowed for
the cable to become fully charged. No definite time for this can
be specified and this matter will receive further treatment later
on. But assuming electrification is complete, move the shunt
successively to positions 0.0001, 0.001, 0.01, etc., until the deflec-
tion obtained is as large as possible and yet remains upon the
galvanometer scale. Note this deflection calling it d, and also
the shunt position used and call it Ni. Then in the same way that