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METHODS OF
MEASURING ELECTRICAL RESISTANCE
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METHODS OF MEASURING
ELECTRICAL RESISTANCE
BY
EDWIN F. NORTHRUP, PH.D.
Mem. A. I. E. E., Am. Electrochem. Soc., Inventors Guild,
Am. Phys. Soc., Franklin Institute, Fellow A. A. A. S.
Palmer Physical Laboratory, Princeton University
McGRAW-HILL BOOK COMPANY
239 WEST 39TH STREET, NEW YORK
6 BOUVERIE STREET, LONDON, E.G.
1912
Engineering
Library
COPYRIGHT, 1912,
BY THE
McGRAW-HILL BOOK COMPANY
Stanbopc ipress
F. H.GILSON COMPANY
BOSTON, U.S.A.
PREFACE.
THIS treatise contains a compilation of many methods of
measuring electrical resistance, most of which are fully described.
Some of the methods are new and are described here for the first
time. Several are illustrated with records of sample measure-
ments. While it is not claimed that the work is exhaustive, the
author has selected for presentation all methods which in his
judgment are useful, for commercial tests and measurements, for
purposes of instruction in educational institutions and for appli-
cation in technical and research laboratories. Rules for the
estimation of errors are briefly considered in the first chapter.
One chapter is devoted to methods of measuring temperature by
means of resistance measuring apparatus, and in another chapter
methods are considered for locating faults upon telephone and
other land lines. While few descriptions of specific types of in-
struments are given, two chapters are devoted to a consideration
of the broad principles which should apply when designing, select-
ing and using apparatus intended for .the measurement of elec-
trical resistance. An appendix contains data and information
useful in connection with the subjects treated. Methods employed
for the absolute determination of the ohm are not considered
because few persons have occasion to make this determination.
In the examples recorded to illustrate specific methods, it may
at times appear to some that the precision obtained is unsatis-
factory. The measurements recorded, however, are real and not
hypothetical cases, and they were made under such working con-
ditions as ordinarily obtain. They are thought, therefore, to be
more instructive than specially selected cases where the measure-
ments have been made with unusual skill and care resulting in
exceptionally high precision.
The author has felt justified in writing upon methods of
measuring electrical resistance, because for over twenty years he
has been engaged in electrical measurement, and for over seven
years he was connected with The Leeds and Northrup Company
257915
VI PREFACE
of Philadelphia, Pa. (but with whom he now has no association),
which manufactures electrical resistance measuring apparatus.
He hopes by recording the experience acquired he may benefit
those who are interested in similar lines.
Doubtless this book is not free from errors and defects. Any
reader noting such will confer a favor upon the author by pointing
them out.
The author acknowledges his indebtedness to Mr. J. W. Wright
of the Bell Telephone Company of Pennsylvania, for his careful
reading of the chapter, " Elementary Principles of Fault Location,"
and for the valuable suggestions which he offered for its improve-
ment. He adds his acknowledgment to The Leeds and Northrup
Company for the loan of electrotypes. He further takes this
opportunity to express gratitude to his wife, Margaret Stewart
Northrup, for her encouragement to proceed with the work and
for her unremitting assistance in the preparation of the manu-
script.
EDWIN F. NORTHRUP.
PALMER PHYSICAL LABORATORY,
PRINCETON UNIVERSITY.
December, 1912.
CONTENTS.
PREFACE.
CHAPTER I. EXTENT, CHARACTER AND PRECISION OF
ELECTRICAL MEASUREMENT. THEORY OF ERRORS.
OHMIC RESISTANCE.
ART. PAGE
100. Electrical Measurement 1
101. Some General Principles 2
102. Comments on Accuracy and Method 4
103. Elements of the Theory of Errors 7
104. Application of the Theory of Errors 12
105. Comments on Ohmic Resistance 16
CHAPTER II. RESISTANCE MEASURED WITH DEFLECTION
INSTRUMENTS; VOLTMETER AND AMMETER METHODS.
200. Assumptions 20
201. Voltmeter Method. Circuit Includes a Known Resistance.
Method 1 20
202. Voltmeter Method. Circuit Includes an Unknown Resistance. 22
203. Voltmeter Method. Circuit Includes a Known Resistance.
Method II 23
204. Comparing Potential Drops with a Deflection Instrument;
Special Case 24
205. Voltmeter Method Using a Shunt 26
206. Deflection Method. Resistance Measured by Substitution 27
207. Voltmeter Method. Circuit Forms Loop of Three Unknown Re-
sistances, Two of which are to be Determined 30
208. Limitations of Voltmeter Methods 34
209. Resistance Measured with a Voltmeter and an Ammeter 35
210. Remarks Upon the Methods of Chapter II 37
211. Ohmmeters and Meggers 38
CHAPTER III. NULL METHODS. RESISTANCE MEASURED
BY DIFFERENTIAL INSTRUMENTS.
300. Remarks on Null Methods 40
301. Properties of Differential Circuits 41
302. Illustration of the Practical Advantages of Differential Circuits . . . 45
303. Differential Galvanometer Used with Shunts 48
304. The Differential Telephone 50
vii
Vlii CONTENTS
CHAPTER IV. THE WHEATSTONE-BRIDGE NETWORK.
SLIDE-WIRE-BRIDGE METHODS.
ART. PAGE
400. Network of the Wheatstone Bridge 51
401. Uses of the Slide-wire Bridge 54
402. Comparison of Resistances by Modified Slide- wire Bridge 58
403. The Carey-Foster Method 61
404. Galvanometer Resistance. Measured, Using the " Second Prop-
erty" of the Bridge 69
405. Calibration of Bridge Wire 70
406. The " Kelvin- Varley Slides" 75
CHAPTER V. WHEATSTONE-BRIDGE METHODS. VARIABLE
RHEOSTAT. ARRANGEMENTS OF RESISTANCES.
PER CENT BRIDGE. SUGGESTIONS
FOR USING BRIDGE.
500. Wheatstone-bridge Methods with Variable Rheostat 78
501. Arrangements of Resistances in Wheatstone-bridge Rheostats. . . 79
502. Rheostat Coils; Classical Arrangements 81
503. Northrup's Four-coil Arrangement 82
504. Five-coil Combinations 86
505. Decade System of Feussner 87
506. Decade System of Irving Smith 87
507. Multiple Arrangements 87
508. Arrangements of Resistances for the Ratio Arms of Wheatstone
Bridges 89
509. Schone's Arrangement of Ratio Arms 91
510. Nonreversible Ratio Arms Adjustable without Contact Resistances. 92
511. Wheatstone Bridge Arranged for Reading in Per Cent 93
512. Remarks upon the Use of the Wheatstone Bridge 94
CHAPTER VI. THE MEASUREMENT OF Low RESISTANCE.
600. Introductory Statement 100
601. Low Resistance Measured with an Ammeter and a Millivolt-
meter 101
602. To Measure the Resistance of Sections of a Closed Circuit; General
Method 102
603. To Measure the Resistance Between Two Points on a Bus-bar . . 106
604. Measurement of the Current in a Bus-bar 108
605. Measurement of the Resistance of Underground Mains 110
606. Comparison of Low Resistances by the Modified Slide-wire
Bridge 114
607. Comparison of Low Resistances by the Carey-Foster-bridge
Method 114
608. Comparison of Low Resistances with a Potentiometer 115
609. The Kelvin Double Bridge; A Network of Nine Conductors 115
610. Theory of the Kelvin Double Bridge 117
CONTENTS ix
ART. PAGE
611. Sensibility Which Can be Obtained With the Kelvin Double
Bridge , 120
612. Methods of Applying the"Kelvin Double Bridge Principle 123
613. Plan of Procedure for Making and Recording a Measurement. . . . 126
614. Sample of a Low-resistance Measurement; Resistivity of Mag-
nesium 127
CHAPTER VII. THE DETERMINATION OF ELECTRICAL
CONDUCTIVITY.
700. Standards of Conductivity; Their Relation. Useful Formulae. . 132
701. The Measurement of Conductivity 140
702. The Hoopes' Bridge for Conductivity Determinations; Described 140
703. The Hoopes' Bridge; Operations Required for Using 143
704. Precautions to Observe in Using Hoopes' Bridge 144
705. Other Methods of Measuring Conductivity 145
706. Equipment for Conductivity Determination; The Standard Re-
sistance Variable 145
707. Method of Using Variable Resistance Standard for Conductivity
Determinations 147
708. Method of Calculating Conductivity from Resistance Data 148
709. Conductivity Determinations with Fixed Resistance Standard
and Variable Ratios 150
CHAPTER VIII. THE MEASUREMENT OF HIGH RESISTANCE.
800. High Resistance Specified and Described 152
801. Wheatstone-bridge Method of Measuring a Resistance from 10 to
1000 Megohms 153
802. Use of a Capacity in Connection with a Wheatstone Bridge for
High-resistance Measurements 155
803. Major Cardew's Electrometer Method of Measuring a High Re-
sistance 156
804. The Measurement of High Resistances, Unassociated with an
Appreciable Capacity; Deflection Methods. . 157
805. The Galvanometer and Accessory Apparatus for High-resistance
Measurement 157
806. Galvanometer Shunts 157
807. The Ayrton or Universal Shunt '. 160
808. Galvanometer Constant, Obtained by Using an Ayrton Shunt. . . . 166
809. Insulation Measurements with a Galvanometer and an Ayrton
Shunt 167
810. Measurement of High Resistances by Leakage Methods 170
811. Theory of Leakage of Condensers 170
812. High Resistance Measured by Leakage; Method 1 171
813. High Resistance Measured by Leakage; Method II 175
814. Insulation Resistance of a Celluloid Condenser Obtained by the
Method of Leakage 176
815. High Resistance Measured by Leakage; Method III 180
X CONTENTS
CHAPTER IX. INSULATION RESISTANCE OF CABLES.
ART. PAGE
900. Introductory Note 185
901. Formula for Calculating the Insulation Resistance of a Cable 185
902. Theorem upon the General Relation Between Capacity and
Resistance 186
903. Application of Theorem to the Measurement of a High Resistance
by Leakage 189
904. Insulation Resistance of a Long Cable by Deflection Methods .... 191
905. Factory Testing Set for Insulation Measurements 198
CHAPTER X. RESISTANCE AS DETERMINED WITH ALTERNAT-
ING CURRENT.
1000. Remarks upon Resistance when Determined with Alternating
Current ., . . . 199
1001. To Measure an Alternating-current Resistance; Apparatus
Required 200
1002. Description of Circuits and Theory of Method 201
1003. Directions for Using, and Test of Method 207
CHAPTER XI. RESISTANCE MEASUREMENTS WHEN THE
RESISTANCE INCLUDES AN ELECTROMOTIVE FORCE.
1100. Material Included Under this Title 210
1101. Measurement of Insulation of an Electric Wiring System while the
Power is On .' . . . 210
1102. Voltmeter Method for Insulation Measurement while the Power
is On.... 210
1103. Galvanometer Method for Insulation Measurement while Power
is On 212
1104. Determination of the Internal Resistance of Batteries 214
1105. Battery Tests by Condenser Method 215
1106. Mance's Method of Measuring the Internal Resistance of a
Battery . . . .' 218
1107. Voltmeter and Ammeter Methods of Measuring the Internal
Resistance of a Battery 220
1108. A Word on Polarities 221
1109. "Voltmeter and Ammeter Method; Principle of Polarities Illus-
trated 222
1110. Total Resistance of a Network between Two Points when the
Branches of the Network Contain Unknown E.M.F's 224
1111. Alternating-current Methods of Measuring the Resistance of a
Battery 226
1112. Bridge Method; Telephone Detector 226
1113. Bridge Method; Electrodynamometer Detector. . 230
1114. Electrodynamometer Substitution Method (Author's Method). . 231
1115. Galvanometer Deflection Methods for Obtaining the Resistance of
a Battery. 233
CONTENTS XI
AKT.
1116. Diminished Deflection Method .............................. 233
1117. Kelvin's Method .......................................... 235
1118. Siemens' Method .......................................... 236
1119. Resistance of Electrolytes .................................. 238
1120. The Method of Kohlrausch for Measuring the Resistivity of an
Electrolyte .......................... .... 240
1121. Determination of Relative Resistivities of Electrolytes ......... 244
1122. Hering's Liquid Potentiometer Method for Determining Elec-
trolytic Resistances .................................... 247
1123. The Substitution Method ................................... 248
1124. Resistance of " Grounds" (Bell Telephone Method) ............ 248
CHAPTER XII. ELEMENTARY PRINCIPLES OF FAULT LOCATION.
1200. Fault Location 251
1201. Faults Occurring on Land Lines 252
1202. Problems in Fault Location 253
1203. Relation of Principles to Practice in Fault Location 253
1204. Location of a Ground upon a Single Line with Only an Earth
Return 253
1205. Loop Methods for Locating Grounds or Crosses 258
1206. Notes on the Varley Test 265
1207. Modified Loop Methods to Meet Special Conditions 267
1208. Where the Faulty Wire is of Known Length and there is Only
One Good Wire of Unknown Length and Resistance 271
1209. One Good Wire of Unknown Length and Two Faulty Wires
Equal in Length and Resistance 276
1210. Methods of Applying Corrections in Loop Tests 278
1211. Location of Grounds on High-tension Cables 281
1212. Location of Faults upon Low-tension Power Cables 283
1213. Method of Locating Grounds upon Heavy, Short, Underground
Cables 284
1214. The Location of Opens 284
1215. Location of Inductive Crosses 288
1216. Comments on Practice and Accuracy in Fault Location 290
1217. A Word on Fault-locating Apparatus 293
CHAPTER XIII. MEASUREMENT OF TEMPERATURE BY THE
MEASUREMENT OF RESISTANCE.
1300. Remarks on Temperature and Thermometry 296
1301. Electrical-resistance Thermometry 297
1302. Construction of Resistance Thermometers 302
1303. Methods of Reading Resistance Thermometers 308
1304. Slide-wire Bridge Method 308
1305. Differential-galvanometer Method 312
1306. Resistance-thermometer Bridge with Two Traveling Contacts. . . 312
1307. Use of Dial Bridges for Temperature Measurements 314
xii CONTENTS
ART. PAGE
1308. Kelvin Double-bridge Method of Reading Temperature 315
1309. Direct-deflection Method of Reading Temperatures 317
1310. Deflection Methods; Using Constant Currents 319
1311. The Measurement of Extremely High Temperatures 322
CHAPTER XIV. INSTRUMENTS USED FOR MEASURING RESISTANCE.
SOME GENERAL PRINCIPLES CONSIDERED.
1400. Proposed Treatment of Subject .............................. 324
1401. Conformity in the Parts of an Outfit ......................... 324
1402. Sensibility and Accuracy ................................... 325
1403. Resistance Standards ....................................... 326
1404. Resistance Boxes and Wheatstone Bridges; General Remarks. . . . 331
1405. Watt Capacity of Resistance Units ................. .......... 332
1406. Construction of Resistance Spools ........................... 333
1407. The Precision of Coils in Resistance Sets ..................... 334
1408. Some Features of Outside Construction ....................... 335
CHAPTER XV. DEFLECTION INSTRUMENTS AND GALVANOMETERS.
1500. Distinction Between Indicators and Deflection Instruments ..... 338
1501. Pointer Type, Flat-coil Galvanometers ....................... 340
1502. Sensitive Galvanometers for Refined Measurements of Resistance
and Insulation Testing ................................. 346
1503. The Equation of Motion of a Galvanometer System ............ 346
1504. Comparison of Galvanometers ............................... 349
1505. Description of One Type of High-sensibility Galvanometer ..... 361
APPENDIX.
I. TABLE.
(1) Values of .................................... 363
II. MATHEMATICAL QUANTITIES AND RELATIONS.
(1) Functions of IT and e .................... ............... 365
(2) English-Metric and Metric-English Conversions .......... 365
(3) Formulae for the Conversion of Temperature Scales ....... 366
(4) Formulae for Temperature Coefficients ................... 366
(5) Relations Between Resistance and Conductivity .......... 367
(6) Conversions from Practical to Electrostatic, to Electro-
magnetic Units. (C.G.S. System) .................... 371
(7) Approximation Formulae. Certain other Expressions ...... 372
III. WIRE DATA AND FORMULA.
(1) Wire Table ........................................... 374
(2) The Ohm ............................................ 375
(3) Resistance of Wire Wound in a Channel ................. 375
(4) Certain Formulae for Wire ........ 376
CONTENTS Xlll
ABT. PAQE
IV. PHYSICAL DATA.
(1) Resistivity of Mercury . 376
(2) Resistivities at 20 C. Densities and Melting Points of the
Solid Elements 377
(3) Data on a Few Alloys 379
(4) Standard Solutions for Calibrating Purposes 379
(5) Resistivities of Insulators 38
INDEX.. 381
METHODS OF MEASURING
ELECTRICAL RESISTANCE
CHAPTER I.
EXTENT, CHARACTER, AND PRECISION OF ELEC-
TRICAL MEASUREMENT. THEORY OF
ERRORS. OHMIC RESISTANCE.
100. Electrical Measurement. The quantities to measure
and the methods of making measurements are more numerous in
electrical science than in any other, and when there is added the
special tests required in connection with industrial practice, the
extent of the subject is far too great for a full and adequate treat-
ment in a single volume. The author has chosen, therefore, to
present but one phase of the general subject methods of meas-
uring electrical resistance. We shall do well, however, to first
consider briefly what is usually comprehended under the subject
of electrical measurement, the nature of the problems involved
and some of the fundamental principles which pertain to all kinds
of electrical measurement. This will give a clearer understand-
ing of the relation which methods of measuring resistance bear
to electrical measurement in general.
In addition to several more or less fundamental quantities,
industrial practice requires the determination or measurement of
other quantities, such as: reactance; impedance; frequency of a
periodically varying current or E.M.F. ; phase differences; power
factor; location of short circuits in coils; ratios of transformers;
location of faults in linear conductors. Very many special deter-
minations are also made, such as the calibration of instruments,
the testing of conductors, conductor-insulation and all kinds of
electric power machinery, etc. Special methods of measurement
have been devised in many cases to meet these various requirements.
All the electrical quantities may be steady or constant in value,
or they may vary in a determinate manner. In the latter case
1
2 ; '
ELECTRICAL RESISTANCE [ART. 101
the methods of measurement and the instruments employed are
usually quite different from those in the former case. Electrical
measurements are, therefore, ordinarily considered under direct-
current measurements and alternating-current measurements. Cur-
rents and E.M.F.'s which vary rapidly in an entirely indeter-
minate manner are now studied and measured with the aid of the
oscillograph.
The quantities which principally require consideration in
ordinary industrial electrical measurements are the following:
Quantity
Symbol
Dimensions
(El' mag.
System)
Practical units
Quantity of electricity
Potential difference
Q
v
L\U\
rfM'T- 2
Coulomb.
Volt.
Electromotive force
Electric current
Ohmic resistance
Resistivity
E
I
R
p
L\M\T~*
L>M*T~ l
LT-i
L 2 T~ l
Volt.
Ampere.
Ohm.
Ohm-centimeter.
Conductance
G
L~ 1 T
Mho.
Conductivity . .
cr
L~ 2 T
Mho per centimeter
Capacity . .
c
L- 1 T Z
Microfarad.
Inductance . . .
L
L
Henry.
Specific inductive capacity. .
Strength of magnetic pole ...
Magnetic induction
Magnetizing force
Magnetic permeability
Electric energy. ...
k
m
B
H
M
w
number
L 3 M 1 T- 1
L-$M*T-i
L~*M*T-i
number
L 2 MT~ 2
No name.
No name.
Gauss.
Gilbert per Cm.
No name.
Joule.
Electric power
Length .
p
L
L 2 MT~*
L
Watt.
Centimeter
Mass.
M
M
Gram
Time
T
T
Second.
101. Some General Principles. Measurement always in-
volves the process of finding, by direct or by indirect means, how
many times some quantity, which we choose to call the unit, is
contained in some other quantity of the same kind, the magni-
tude of which we wish to determine. In the actual process of
measurement, the quantity, which is selected as the unit, must
be something more than a mere abstraction or definition. It
must be represented by a concrete thing. Thus the unit of
length, in the metric system, is defined as the one ten-millionth
of the earth's quadrant and is called the meter, but the real unit
of length is an actual distance between two marks upon a par-
ART. 101] EXTENT OF ELECTRICAL MEASUREMENT 3
ticular bar of metal. The entire scientific world has agreed to
call the distance between these two marks the real unit of length,
which is called a meter. There are numberless, more or less
accurate copies of this standard of length, and whenever an actual
length measurement is made, the procedure consists essentially
in finding how many times the length of one of these copies of
the standard meter is contained in the length being measured.
The concrete thing which embodies and represents the unit of
definition is generally and properly called a standard. The stand-
ard may represent the unit exactly, as in the case of the meter,
or it may be a known multiple or fraction of the unit repre-
sented. Thus the column of pure mercury of uniform cross-sec-
tion which is 106.3 centimeters long and has a mass of 14.4521
grams, or a cross-section of 1 mm 2 is, when at a temperature of
C., an exact concrete realization of the unit of resistance,
called the ohm. A standard cadmium cell, on the other hand,
which bears a certified value of 1.0183 volts is a concrete reali-
zation which represents a known multiple of the unit of electro-
motive force.
It is desirable and convenient that a standard should represent
the unit exactly or some simple even multiple of it, but the diffi-
culties in the way of accomplishing this are often great, as in the
case of the standard of E.M.F. just cited. For exact work it is
customary to construct standards to equal the unit or its even
multiple as nearly as possible, and furnish certificates giving the
exact value in terms of the unit represented. Thus a standard
one-microfarad condenser is seldom constructed to equal one
microfarad closer than one-tenth or one-quarter of one per cent,
tho condensers can be compared with one another much closer
than this. The value stamped upon the standard is called the
nominal value, and the certificate which goes with it states the
amount, generally, or at least preferably, in percentage by which
it is greater or less than its nominal value.
When a measurement is made, it is determined by some
selected method of procedure how many times the magnitude of
the standard is contained in the quantity measured. If this quan-
tity is smaller than the standard, then, of course, it will be found
to contain the standard a fractional number of times.
The measurement may be in error from two causes ; either the
standard may be larger or smaller than it is certified, in which
4 MEASURING ELECTRICAL RESISTANCE [ART. 102
case the quantity being measured will be called smaller or largei
than it really is, or the process of finding how many times the
quantity measured contains the magnitude of the standard may
be incorrectly applied. The responsibility of the first error rests
with the standard and of the second error with the user of the
standard.
In industrial measurements, the one engaged in making the
measurements, that is, in finding how many times the magnitude
of the standard he employs is contained in the quantity he is
measuring, seldom undertakes a test of the precision of his stand-
ards. He relies upon others for this, and herein rests the vast
importance and responsibility of a Standards' Bureau like the one
we have in Washington. It can be shown that, in a final analysis,
the accuracy of most of the measurements made in America rests
upon the facilities, the intelligence, and conscientious care of this
Bureau.
102. Comments on Accuracy and Method. The precision
with which any measurement is made may vary all the way from
a rough estimation to the high refinement attained by a prolonged
investigation where the total error may be reduced to a few parts