The constant of the galvanometer and the method of calculating
the specific resistance have already been fully considered. It
should be noted here, however, that as the resistance of the insula-
tion apparently, or really, increases with the time of electrification,
one should always give, in stating the value of the resistance or
the resistivity of the insulation, the time of electrification used
in obtaining the value given. It is also found that the resistivity
is greatly affected by changes of temperature and care should
therefore be used to state the temperature which corresponds to
the value given. Thus, in stating the resistivity of gutta percha,
we should say: resistivity = 4.5 X 10 14 at temperature 20 C.
ART. 904] INSULATION RESISTANCE OF CABLES 197
198 MEASURING ELECTRICAL RESISTANCE [ART. 905
and was obtained by deflection method with electrification of
905. Factory Testing Set for Insulation Measurements.
Wherever insulation measurements are to be made frequently, as
in factories where cables are manufactured, it is necessary, or at
least very desirable, to provide a full equipment designed expressly
for the purpose. Such an equipment, called a " factory cable-testing
set " made by The Leeds and Northrup Company of Philadel-
phia, Pa., has been upon the market for several years. This set
is illustrated in Fig. 905a. Fig. 905b gives a top view of the set
which shows the connections.
The galvanometer, and lamp and scale are not shown in the
illustration. The chief features of this set are: All the instru-
ments, except the galvanometer, and lamp and scale, are mounted
upon a hard rubber plate to give high insulation between earth and
the instruments. The wire connections are carried from the tops
of hard rubber posts, petticoat-insulated. Except where the wires
rest upon the posts, they are air-insulated. The switches and
keys are arranged for convenient manipulation.
The instruments of the outfit consist of a D' Arson val galva-
nometer of high resistance and about 1200 megohms sensibility;
a lamp and scale, the latter 1 meter long; a standard resistance of
0.1 megohm mounted in a cylindrical case; an Ayrton universal
shunt which gives multiplying powers of 1, 10, 100, 1000, 10,000
and infinity. There is a highly insulated double-pole, double-
throw switch, and a key which has a lock-down device by which
it may be permanently closed. The methods of using this set
for insulation measurements do not differ in principle from those
already described, while detailed instructions for its use will be
furnished by the makers of the set.
RESISTANCE AS DETERMINED WITH ALTERNATING
1000. Remarks upon Resistance when Determined with Alter-
nating Current. We call attention to the sense in which the
term resistance is used in this work. It is a quantity which, in
general, may be determined by direct currents and is denned as
the fall of potential in any portion of a circuit, divided by the
direct current which flows in that portion of the circuit. In
metallic conductors, Ohm's law is obeyed, when regard is had to
the temperature of the conductor.
When, however, the current in a circuit is not steady, or alter-
nates, there will be a fall of potential in any portion of the circuit
which is the product of a certain quantity R ac and that compo-
nent of the current which is in phase with the electromotive force,
or more properly the fall of potential. The quantity R ac is often
a different quantity than that which is denned as ohmic resistance.
It represents, in addition to a true ohmic resistance, anything
which causes energy losses of whatever character occuring in the
portion of the circuit considered. These energy losses may be
due to hysteresis of iron in the circuit, to an electric absorption of
dielectrics, to power wasted by currents and electrostatic poten-
tials (resulting in currents external to the circuit) induced in
neighboring circuits, to electric radiation and to other causes not
to be classed with ohmic resistance. In one sense this quantity
Rac is a resistance, but it varies greatly with changes in fre-
quency, current density, etc., and is not to be considered as a
constant quantity or as a true ohmic resistance. From a broad
viewpoint, a work of this character, which is intended to
give a full treatment of the methods of measuring resistance,
should include, also, all useful methods of measuring alternat-
ing-current resistance. The subject is, however, so extensive
that we should transgress our purposes and limitations if we
considered them here in full. We shall, however, describe one
200 MEASURING ELECTRICAL RESISTANCE [ART. 1001
method,* devised by the author, which will, in general, enable an
alternating-current resistance to be determined with sufficient
precision to meet commercial requirements. We proceed to a
description of the method.
1001. To Measure an Alternating-Current Resistance; Appa-
ratus Required. For carrying out this method in a precise
manner the apparatus required is a frequency meter to measure
the frequency of the current used (which must be known, as the
quantity being measured will vary with frequency), an alternating-
current ammeter to give roughly the value of the current (for the
alternating-current resistance will also, in general, depend upon the
value of the current) , a three-point double-throw switch for quickly
changing connections, resistances, and an electrodynamometer.
This last piece of apparatus should have sufficient capacity in its
current coils to carry the full current without heating. Its hang-
ing or potential coils should be two in number and so arranged as
to form a system which is perfectly astatic in respect to the earth's
field. The constant of the instrument will then be the same for
direct and alternating currents. All good electrodynamometers
are constructed in this way. Either the Rowland deflection type
or Siemens' type constructed to be astatic, may be used. The
method to be described was tested, using a Rowland deflection-
type electrodynamometer. This instrument is illustrated in
Fig. 1001, and is in part described as follows in The Leeds and
Northrup Company 1911 Catalogue, No. 74:
" The electrodynamometer proper consists of two pairs of
fixed coils and a swinging coil. One pair of fixed coils on the out-
side of the case is adapted to carrying currents as great as 50
amperes; one pair on the inside of the case is suitable for currents
of 0.1 ampere. The swinging coil is adapted to currents as great
as 0.1 ampere. The position of the fixed coils is such that the
swinging coil turns in a field of force, which is nearly uniform
for the angle thru which the coil moves. The dynamometer has
attached to it a scale and telescope which are placed at a fixed
distance from the mirror attached to the swinging coil. The
scale and telescope rest on an arm which can be swung up out
of the way when the instrument is not in use. The scale is not
* This method was first described by the author in a paper presented before
the American Institute of Electrical Engineers at the Boston Meeting, June 24-
ART. 1002] RESISTANCE WITH ALTERNATING CURRENT 201
provided with a lateral adjustment. The coil is brought to zero by
turning a micrometer screw which rotates the suspension tube.
Thus, when the instrument is adjusted to read zero with no current
flowing, the coil is always in the same position in the field. This
requirement must be filled in order that the constants of the instru-
ment, after being once determined, shall not alter. The hanging-
coil system has two coils connected to form an astatic combination
so that the instrument will not be affected in its deflection by the
earth's field when used with direct currents."
1002. Description of Circuits and Theory of Method. In
I and II, Fig. 1002, GG are the fixed coils and hh the hanging,
astatic system of the electrodynamometer. The hanging system
has an ohmic resistance a and there is joined in series with this a
non-inductive resistance p'. Let p + a = p, the entire resist-
ance of the hanging-coil system. In the instrument, illustrated
in Fig. 1001, the resistance a is about 18 ohms. It has a minute
inductance, which is approximately 0.00045 henry. When p' is
moderately large and non-inductive, we may consider, without
sensible error, that the alternating current thru the hanging
system is in phase with its E.M.F., even when the frequency is
high. We shall so consider it in all that follows.
MEASURING ELECTRICAL RESISTANCE [ART. 1002
A represents a coil which contains iron. It is assumed that
this coil has a certain ohmic resistance Rdc as measured by direct
current, and a different resistance R as measured by an alternat-
ing current of a given value, wave form, and frequency. It is
this latter resistance (not the impedance or inductance of A)
which the method will enable us to determine. The resistance r
is any resistance capable of carrying the full current. It may be
a coil inductively wound but it must not contain iron or have such a
section and resistivity that its resistance on alternating current
will be different from its resistance on direct current due to hyster-
esis, skin effect, or other cause. By a sliding contact p, means
must be provided for tapping upon this resistance at any point
along its length as the diagram illustrates, p is a non-inductive
resistance, which is equal to a + p', the resistance of the hang-
ing-coil circuit. As will later be shown the connections can in-
stantly be changed from the arrangement shown in I to that
shown in II and vice versa.
We wish first to find the general expression for the power
which the wattmeter measures when the connections are those
shown in I, Fig. 1002. Call / the current in the fixed coils of the
dynamometer, i the current in the hanging-coil circuit, < the phase
angle between the currents i and /. Then the deflection of the
where K is the instrumental constant of the dynamometer. This
constant in the case of a deflection instrument of the Rowland type
will change slightly with the magnitude of the deflection. There
is also an inductive action of the current in the fixed coil which
tends to induce a current in the hanging-coil circuit when the
plane of the movable system is not vertical to the plane of the
fixed coils. This inductive action may vary in a complicated
ART. 1002] RESISTANCE WITH ALTERNATING CURRENT 203
way, so Eq. (1) cannot be taken as strictly true. If, however, the
system is deflected by means of the torsion head, when there is
no current thru the instrument, so that when current is introduced
the system is brought back to the position where its plane is
vertical to the plane of the fixed coils, then the inductive action is
null and the relation given by Eq. (1) may be considered to hold
very exactly. In the use of the dynamometer which follows, the
system should be deflected by means of the torsion head, when
there is no current flowing, to such an extent that, on introducing
current, the instrument reads roughly at the zero of the scale.
With this precaution observed the theoretical relations will" be
found to hold very exactly.
If we call V the impressed E.M.F. between the points a and
6, I, then the current thru the hanging-coil circuit will be
As stated above the current i will be approximately in phase with
the E.M.F., V, because the inductance of the hanging coils is
By Eqs. (1) and (2) we have
But IV cos <f> is the entire power W t . This power is the sum of
two parts, W the power consumed in A between the points a and
b and W the power consumed in the hanging-coil circuit. The
value of this latter is
W = (4)
Thus we have
D=*W t , (5)
D= ~p~( W ' p)'
From Eq. (6)
and from Eq. (5)
r,-a ( 8 )
204 MEASURING ELECTRICAL RESISTANCE [ART. 1002
is generally a small quantity, p is known very precisely and
V can be obtained with a voltmeter, hence Eq. (7) enables the
true power spent in A to be accurately obtained. It is Eq. (8),
however, which we wish to use in measuring the alternate-current
resistance of A.
With the connections as shown in I, the torsion head is turned,
so that with the current (steady as possible) which is flowing the
deflection reads near the zero of the scale. The total power then
being registered is given by Eq. (8).
The connections are now quickly changed to those shown in
II. The main current will not be altered by this change in con-
nections, for the resistance p is simply made to change places
with an equal resistance. The total power which is registered,
however, will now be
W t ' = ^ (9)
when D' is the deflection which the dynamometer now gives. The
contact p is moved along the resistance r until the deflection
D' is made equal to the deflection D, then W t r will be equal to W t .
Since the main current I is the same for the connections I and II
we have Wf = PR , = w ^ (10)
or R' = r'. (11)
Here the quantity R' is not the alternating-current resistance of
the coil A but it is the alternating-current resistance of this coil
when shunted with the non-inductive resistance p. Similarly r'
is the alternating-current resistance of r when shunted with the
non-inductive resistance p.
We can write
R > = J?K K > and K = - *,
P + R P + r
The alternating-current resistance of two parallel circuits when
one or both of the branches contain reactance is not given by the
same expression, as applies when the branch circuits are without
reactance, hence the ordinary expression for branch circuits
without reactance, namely P . n * must be multiplied by some
p + K
factor K f , the value of which we now have to determine; also the
ART. 1002] RESISTANCE WITH ALTERNATING CURRENT 205
It is shown in " Alternating Currents " by Bedell and Crehore,
pages 238 to 241, how the alternating-current resistance, or, as
they call it, the equivalent resistance of any number of parallel
circuits having self-induction and carrying alternating current,
may be expressed. It is there shown that, in general,
~A 2 + V
^ _ Ri | Rz
/?, Xl \ X2
in which expressions RI, R%, etc., are ohmic resistances and xi,
x 2 , etc., are reactances of the several branches.
We can now find expressions which will give the values of K'
and k. Here we have
and Bu =
We cannot, because of the necessity of brevity, give here the
purely algebraic processes required for obtaining the final expres-
sion and so we shall present only the final results, which are as
If/ = 1 j __ P X *
Call the fractional expressions a and a\ respectively, then
K' = 1+ a and k = I + ai.
This gives, because of Eq. 11,
R = r
It will be shown that, in general, when a sensitive electrodyna-
mometer is used, a and a\ are very small quantities which in most
cases can be neglected.
206 MEASURING ELECTRICAL RESISTANCE [ART. 1002
We have the following cases:
1st. a and ai are negligible. Then
R = r> (12)
2nd. a and on are not negligible but are very nearly equal.
R = r.
In these two cases the alternating-current resistance sought
may be taken as numerically equal to the resistance r.
3d. on = 0, but a is not negligible. In this case
R = r - . - (13)
4th. a and on are not negligible and are unequal, but p is very
large. Then again we can take R = r.
Consideration of a single example under case 3d will suffice
to show the magnitude of the error which may be introduced by
omitting the correction. The example chosen is from an actual
measurement. With the electrodynamometer available, only
0.1 ampere could be passed through the fixed coil and hence, the
potential drop over the coil A and over the resistance r being small,
the resistance p had necessarily to be taken very small to give the
requisite sensibility. If the dynamometer coils could have carried
several amperes (as is ordinarily the case), p would have been
much larger and the error would be much less. In the example
ai = and
-[(R + P )*
300 (2 X 3.14 X 60 X 0.036) 2
" [(11 X 300) 2 + (2 X 3.14 X 60 X 0.036) 2 ] 11 '
or a 0.052, nearly.
R = r - l ' = 0.948 r.
Thus, if we had called R = r, the error would have been about
5.2 per cent, R being assumed too large. This conclusion was
checked experimentally. Without changing the ohmic resistance
of the coil A its inductance, which was capable of variation, was
ART. 1003] RESISTANCE WITH ALTERNATING CURRENT 207
varied from 0.003 to 0.036 henry, and in the first case, using the
uncorrected formula, R = 10.94 ohms, and in the second case,
using the same formula, R = 11.62 ohms, or 6 per cent too large,
which is in fairly close agreement with the calculated result of
5.2 per cent.
If the fixed coils of the dynamometer had been made to carry
10 amperes instead of 0.1 ampere, p could have been 100 times as
large, in which case the correction factor would reduce to about
0.05 of 1 per cent.
The above adjustments having been made, direct current can
be made to replace the alternating current and in the same way
we find the direct-current resistance of A. It will be
Rdc = ri. (14)
is the ratio of the alternating-current to the direct-current resist-
ance of the circuit A. This ratio may take a value of 2 or more.
It should be clearly understood just what is meant by the
quantity R, which this method measures. It is a quantity which,
expressed in ohms and multiplied by the square root of the mean
square value of the alternating current thru the circuit, expressed
in amperes, will give the square root of the mean square value of
that component of the impressed E.M.F. expressed in volts, which
is in phase with the current. Or it is the quantity which, when
multiplied by the mean square value of the current, will give the
power in watts which is being dissipated in the circuit. In draw-
ing the triangle of E.M.F.'s of an inductive circuit one sometimes
represents the component of the E.M.F. which is in phase with
the current by the product of the current and the direct-current
resistance Rd C . This procedure may lead to considerable error
in circuits in which there are other losses than the I 2 Rd C losses.
In such circuits the alternating-current resistance R should always
1003. Directions for Using, and Test of Method. For mak-
ing the above measurement the apparatus is assembled and con-
nected as shown in Fig. 1003.
D is the electrodynamometer with its hanging system h. The
heavy fixed coils or the light wire fixed coils are used according to
MEASURING ELECTRICAL RESISTANCE [ART. 1003
the magnitude of the current which is selected for the measure-
ment. In the Rowland instrument the light wire fixed coils will
carry 0.1 ampere and the heavy wire coils will carry 50 amperes.
C is the alternating-current ammeter and F the frequency meter.
P is a rheostat to control the main current, s is a switch to shift
from direct current to alternating current and vice versa. S is
the three-point double-throw switch which in position 1 makes
the connections shown in I, Fig. 1002 and which in position 2
makes the connections shown in II, Fig. 1002. r is best obtained
from a slide-wire rheostat of considerable current capacity. It does
not need to be non-inductive, but must contain no iron. If its
reactance is just equal to that of the coil being measured a = ai
and R = r exactly.
After the settings for p have been found, the connections are
broken at (3) and (4) and the direct-current resistance value of r
is measured with a Wheatstone bridge or by any other convenient
The resistances p and p' may be obtained best from plug or dial
decade resistance boxes. These may be high, 10,000 ohms or
so, depending entirely upon the current used, the magnitude of
the resistance being measured and upon the sensibility of the
The torsion head may be turned so that the no-current-deflection
is between 100 and 200 divisions of the scale. By then adjusting
P the deflection with current on may be made to come near the
zero of the scale.
ART. 1003] RESISTANCE WITH ALTERNATING CURRENT 209
It will be found, if A consists of an ironless variable standard
of inductance, that the variable standard may be set to any
inductance value without greatly altering the deflection. The
change in the deflection will be less as p is made larger.
This method will be found useful in measuring the alternating-
current resistance of steel-cored copper or aluminum cables which
differ considerably from their direct-current resistance.
The following test was made of this method.
This test was made in a manner to show how large a correction
would be required when p is chosen only 300 ohms and L is varied
between 0.003 and 0.036 henry.
The resistance measured was that of a Hartmann and Braun
variable standard of inductance, which should, of course, show
the same value on direct and on alternating current and at what-
ever value its inductance is set.
(a) L = 0.003 henry.
With alternating current in circuit,
D t = 244 (no current),
D = D' = (current flowing),
R = r = 10.94 ohms,
7 = 0.08 ampere,
N = 60.2 cycles,
p = 300 ohms.
(b) L' = 0.036 henry.
With alternating current in circuit,
D t = 253 (no current),
D = D' = (current flowing),
R = r = 11.62 ohms,
/ = 0.08 ampere,
N = 60.2 cycles,
p = 300 ohms.
With direct current in circuit,
D t = 244 (no current),
D = D' (current flowing),
R dc = r = 10.94 ohms,
Idc = 0.08 ampere,
p = 300 ohms.
RESISTANCE MEASUREMENTS WHEN THE RESIST-
ANCE INCLUDES AN ELECTROMOTIVE FORCE.
1 100. Material Included Under this Title. We pass now to
a consideration of the methods employed for the measurement of
resistance, when an electromotive force is included in the portion
of the circuit the resistance of which is to be measured.
The methods to be considered are those employed for measuring
the insulation resistance of lighting or power circuits while the
power is on, the internal resistance of batteries, the resistance of
the earth between electrodes where the disturbing effects of earth
currents must be considered, and the resistance of electrolytes
subject to an E.M.F. of polarization. Resistance measurements
of the above character require special consideration at some
noi. Measurement of Insulation of an Electric Wiring Sys-
tem while the Power is On. It is not infrequently required
to measure the insulation between the gas pipes and each bus-bar
of an interior wiring system, as that of a city office building, at a
time and under circumstances when it is impracticable to shut off
Two methods are given for making this measurement.*
1102. Voltmeter Method for Insulation Measurement while
the Power is On. Let Fig. 1102 represent any wiring, system
in which Xi and X 2 represent the insulation resistances between
the bus-bars BI and B 2) and the earth (the gas or water mains
being considered to have the potential of the earth).
The diagrams, I, II, III, of Fig. 207, show circuits equivalent
to the circuits represented by Fig. 1102. In these diagrams y
represents the unknown resistance of all the lamps, motors, etc.,
which are connected across the line.
If the bus-bars are supplied with direct current a Weston
* These methods were first described by the author, in the Electrical World
and Engineer, May 21, 1904, vol. xliii, pages 966-7.
ART. 1102] RESISTANCE MEASUREMENTS 211
direct-current voltmeter should be used. If the current is alter-
nating then an alternating-current voltmeter of the electro-
dynamometer type will be required. The resistances Xi and X z
are determined by knowing R, the resistance of the voltmeter,
and by taking three voltmeter readings di, d z and D which cor-
respond to three voltages Fi, Vz, and E indicated in Fig. 1102.
1st. Obtain the deflection D which corresponds to the voltage
E between the bus-bars.
2d. Connect the voltmeter between the bus-bar BI and the gas
or water pipe and obtain the deflection di which corresponds to
the voltage V\.
3d. Connect the voltmeter between the bus-bar B 2 and the
gas or water pipe and obtain the deflection d 2 which corresponds
to the voltage F 2 . The last two readings should follow the first
as quickly as possible so that there will be less chance of the line