which has been termed " the figure of merit/' may be reached
with a fair degree of satisfaction. A clear physical conception
of this feature and what it means will be a useful preliminary
ART. 1504]
DEFLECTION INSTRUMENTS
351
to any estimation of the worth of any current-measuring in-
strument.
Let (Fig. 1504) G a and G b represent, diagramatically, two sys-
tems of any type of moving-coil galvanometer. Let these systems
be held by suspensions S a and S b attached at points pi, p 2 and
pi and p2.
FIG. 1504.
Suppose the coil-winding of each is on a metal frame of such
cross-section and conductivity that, when the system rotates in
the magnetic field, its return to zero from a deflection is just
aperiodic in virtue of the currents induced in the frame. To have
this condition always fulfilled, one may conceive the conductivity
of the frame to vary whenever the moment of inertia of the sys-
tem, the torsion of the suspension, and the strength of the mag-
netic field are varied. This premised, let such a current pass
thru each system that it will be rotated thru a standard angle
6, which may be made always the same by varying the strength
of the current. Evidently the current which will be required to
produce this deflection will depend upon many factors, chief
among which are the strength and uniformity of the magnetic
field, the torsional force of the suspension, the length and number
of turns in the coil, and the degree of freedom from magnetic
impurities in the system.
The system, starting from rest, will require a certain time T
after the current is applied, to complete a certain fractional part
of its aperiodic deflection. To be definite, assume that the de-
flection is practically completed, when it has reached within 0.05
of 1 per cent of its final deflection. This time T will be, for all
352 MEASURING ELECTRICAL RESISTANCE [ART. 1504
purposes of practical computation, the same as the time of a
complete oscillation of the same system undamped. If 7 is the
moment of inertia of any galvanometer system, then the square
of its time of deflection, as above denned, or of a complete oscilla-
tion if undamped, is proportional to this moment of inertia or
Now so arrange matters that the same current thru each of the
two systems G a and Gb will produce in each the standard angular
deflection b. Then the sensibility S m of each system is the same
and this sensibility will be inversely proportional to the current
required to produce the deflection. If, however, the system G a
has a moment of inertia I a , and the system Gb a greater moment of
inertia /&, then G a will reach its standard deflection in a shorter
time than G b . The ratio will maintain
By hypothesis, the sensibilities, that is, the currents required
thru each to produce the standard deflection, are equal. Since,
however, G a deflects in a shorter time than Gb its suspension might
be weakened until its time of deflection equals that of Gb. But
with a weaker suspension it will take less current to produce
the standard deflection. Hence, with equal times to make the
standard deflection, the system G a is more sensitive than the
system (7&.
Conversely, to make Gb deflect the same amount in the same
time as G a , its suspension must be stiffened, and with a stiffer
suspension it will require more current than G a for the standard
deflection. In this respect it is a less sensitive galvanometer
than G a .
As it is always possible to vary within wide limits the torsional
force of a galvanometer suspension, a galvanometer which is
quick but not sensitive can be made more sensitive at the expense
of quickness by changing its suspension, and a galvanometer
which is slow but sensitive can be made quicker at the expense of
sensibility. If we call S m the sensibility of a particular galva-
nometer and T its period, then the product S m ^ cannot be in-
creased by changes of the above character. We shall call this
ART. 1504] DEFLECTION INSTRUMENTS 353
product proportional to the useful sensibility of any particular
galvanometer and write
fff?- a)
By T we must understand the time of a complete oscillation,
if the galvanometer is undamped, or the time it takes to reach
its final deflection within 0.05 of 1 per cent, if it is magnetically
damped to be just aperiodic.
For practical purposes of comparison of galvanometers, the
time may be considered the same for the instrument in either of
these conditions.
While the sensibility S m is inversely proportional to the current
needed for the standard deflection, this current, if everything else
remains the same, will be less as the number of turns in the coil
is increased. Of two galvanometers which are to be used on the
same constant-current circuit, and which are alike in all features
except in respect to number of turns, that one which has the more
turns will be the more sensitive. We can call, therefore, the
sensibility of a galvanometer for use on a constant-current cir-
cuit a quantity which is proportional to its number of turns n,
and inversely proportional to the current i required to produce a
standard deflection, or
&,?. (2)
i
Hence, its useful sensibility is
To increase the number of turns we may proceed in either or both
of two ways; the size of the insulated wire may be diminished and
the same winding space be filled, or the wire may be kept the same
size and the dimensions of the cross-section of the winding chan-
nel may be increased. By the first method the moment of inertia
of the system remains nearly the same. It would remain exactly
the same, if in altering the size of the wire no alteration were made
in the density of the coil by changing the ratio of insulation to
wire, thru a change of wire size. By the second method, the
moment of inertia, and hence T 2 , will be changed unless the length
of the turns are also diminished in a proper proportion. We have
seen that U, the quantity which we have called the useful sen-
354 MEASURING ELECTRICAL RESISTANCE [ART. 1504
sibility of a galvanometer, cannot be changed by changing the
torsional force of its suspension, but it may be changed by
changing the coil winding. Thus, in changing n, if the moment
of inertia only is changed T 2 will be changed and U will be changed
because the period changes, but if n is changed in such a manner
as not to change the period then U will change again because S mj
the sensibility, changes. If n is changed in such a way as to vary
both S m and T 2 , U will still change unless the exceptional condi-
Tl
tion is met, that n so changes that 7 remains constant. We con-
clude, by the above line of reasoning, that useful sensibility is a
constant property of a particular galvanometer with a particular
winding, but a quantity which usually varies when the coil windings
are changed. But if we divide the useful sensibility by the num-
ber of turns in the coil and write
we obtain the new quantity F which has been designated the
" figure of merit " of a galvanometer.
The figure of merit of a galvanometer is a kind of " specific
quantity " which attaches to every galvanometer. As the use-
ful sensibility of a galvanometer cannot be improved by changing
the torsional force of its suspension so also we cannot increase the
figure of merit of a galvanometer by changing either its suspension
or the turns which fill a winding space of fixed volume. A galva-
nometer with a certain " figure of merit " is potentially, so to
speak, capable of having a certain chosen period with a certain
accompanying sensibility and number of turns, or a certain chosen
sensibility with a certain accompanying period and number of turns,
but to increase the figure of merit changes must be made in the
field strength or in the proportioning of the galvanometer parts.
As " figure of merit " attaches as a specific property to every
galvanometer, it serves in a useful way to compare the intrinsic
worths of various types of instruments. If, however, we wish to
compare the figures of merit of different galvanometers, we cannot
do so practically by using the expression above in its present
form, because there is no easy way of counting the number of turns
in the coils after the galvanometers are built. It is necessary,
therefore, to find how the resistance of the coil is related to its
ART. 1504] DEFLECTION INSTRUMENTS 355
number of turns, for this is a quantity easily measured. To do
this write
, .
where Zi = length of mean turn,
W = diameter of wire and
p = specific resistance of wire.
If S = cross-section of channel and
d = double thickness of wire insulation, then
S TrS
n =
(W + d) 2 irW 2 +2wWd'
(6)
the term -K d 2 being neglected, as being very small.
From Eq (6) we derive irW 2 = , and putting this
value of vW 2 in Eq. (5) we obtain
Solving this quadratic [see appendix II, 7, Eq. (16)] and using the
positive sign before the radical, we obtain
n a VRS + R 2 W 2 d 2 - RW d. (8)
When this value of n replaces n in Eq. (4) we have the result-
ing expression, not involving n, for the figure of merit of a
galvanometer,
p _- _ 5 _ /q\
T 2 [(RS + R 2 W 2 d 2 )* - RW d] '
Except in cases where galvanometers wound with coarse wire
are compared with galvanometers wound with very fine wire, the
thickness d of the insulation may be neglected. We may con-
sider S constant and we have, when we do this,
-^ acJ- (10)
T*VR
S
^=. is the usual expression for the figure of merit of any galva-
T 2 VR
nometer. In comparing galvanometers by it, the supposition is
made that thickness of insulation is neglected and that the gal-
vanometers compared are wound with wire of the same specific
resistance. The specification which we shall adopt to define S m
is as follows:
356 MEASURING ELECTRICAL RESISTANCE [ART. 1504
With the scale at 1000 scale divisions from the mirror, of a
mirror galvanometer, the sensibility S m is the number of megohms
which must be in the galvanometer circuit so that with an E.M.F.
of one volt in the circuit, there will result a deflection of one scale
division. With this understood, we can write
A galvanometer would have, then, a unit figure of merit, if its
time of a complete oscillation is one second (or which is practically
the same thing, if its time of a periodic return to zero within 0.05
of 1 per cent of its previous deflection is one second), and the re-
sistance of its winding is one ohm, and if, with one megohm in
series and one volt in circuit, its deflection, on a scale 1000 scale
divisions from its mirror, is one division.
As this unit has received no name, we shall call it, for con-
venience, a D' Arson.
We can say, also in accord with the above definition of sensi-
bility, that the sensibility is unity when one microampere pro-
duces the standard deflection. Hence, if i m = the microamperes
in the galvanometer circuit,
If E m microvolts are applied at the terminals of the galvanom-
eter of resistance R, we have
E m R
lm =: ~R r = E~ '
Putting this value of S m in Eq. (11) gives
'*-$' (12)
The relation (12) defines the figure of merit of a galvanometer
in terms of its resistance, period, and the number of microvolts
applied at its terminals to produce the standard deflection.
As an example of the use of relation (11) suppose we have a
galvanometer with a complete period of 5 seconds, a coil resistance
of 400 ohms, and which deflects one scale division with one volt
acting thru 500 megohms, then its figure o'f merit is
500
F= - 7= = 1 D'Arson.
5 2 A/400
ART. 1504] DEFLECTION INSTRUMENTS 357
If this galvanometer were given a longer period, by using a
weaker suspension, its sensibility would be larger, but F would
not be altered by this change. It is evident that the coil might
be wound, using the same size wire, to a smaller resistance, but
if this were done the mass of the coil would be less and hence T 2
would be smaller. Both of these changes would contribute to a
greater figure of merit. On the other hand, a smaller resistance
would mean a smaller number of turns which would reduce the
sensibility and hence diminish the figure of merit. Thus it is
always open to the designer to so choose the winding and pro-
portion the coil and to so arrange the strength of the magnetic
field and other factors that F shall be large. The success with
which he does this determines in considerable measure the per-
fection of his design.
It must not be forgotten, however, that the number of D'Arsons
possessed by a galvanometer is not necessarily a final measure of
its fitness for actual service. It may possess faults of many kinds
which more than offset a large figure of merit. Chief among such
is zero shift and magnetic impurities in the system, which two, in
fact, generally go together. There are also other common defects,
as small coil clearance, inaccessibility of the parts, a poor optical
system, an unproportional scale, a provoking tendency of the
system to respond to small tremors, and a host of other minor
defects which the user soon observes and condemns.
If it were not necessary to load a galvanometer system with a
mirror or a pointer for the purpose of reading the deflections, it
would be possible by proper designing and by a great diminution
in the size of the moving parts to realize an instrument which
would possess an enormous figure of merit as compared with an
ordinary moving-coil galvanometer. This has in fact been done
in the case of the Einthoven String Galvanometer, which has
about 3000 times the number of D' Arsons of a good moving-coil
galvanometer using a mirror and scale. We are led thus to the
general consideration of what are the possibilities of obtaining a
great figure of merit for galvanometers of the deflection type.
In every galvanometer of this type we may consider the moment
of inertia of its moving system as made up of two parts : One part
is the moment of inertia which is contributed by the mirror, the
pointer, or whatever device may be attached to the system which
is required for reading the deflections of the instrument. We
358 MEASURING ELECTRICAL RESISTANCE [ART. 1504
may make this reading device small but we cannot dispense with
it altogether and preserve the instrument as a galvanometer of a
type to which the name is ordinarily applied. Indeed, there are
many practical considerations which soon put a limitation upon a
continual diminution of "mass of these parts. The other part is
the moment of inertia which belongs to the moving wire or mag-
nets, which constitute the effective working element of the system.
This part of the total moment of inertia can be modified at will
by the designer with the object of making the figure of merit of
the galvanometer as large as possible. The question then arises,
has the figure of merit a maximum value which the most skilful
designing cannot exceed? If there were no " dead parts " attached
to the system for reading deflections, then, theoretically, a gal-
vanometer could be given, by proper designing, an indefinitely
great figure of merit. This realization is obtained practically in
the Einthoven String Galvanometer. But as long as galvanom-
eters continue to be instruments the deflections of which are read
with mirrors or pointers, there will be a theoretical maximum
figure of merit which cannot be exceeded. The author has
shown* that when we have chosen the moment of inertia of the
winding of a rectangular coil, equal to the moment of inertia of
the " dead parts," mirror or pointer, the resistance of the coil re-
maining always the same, we have designed the proportions of the
coil such, that the galvanometer, in this respect, has the greatest
figure of merit which it is possible to give it.
Every consideration shows that in starting out to design a
galvanometer of any type which is to have a large number of
D'Arsons, one should begin by carefully considering the selection
of the mirror, pointer, or other contrivance essential to reading
the deflections, so that this contrivance may have the least possible
moment of inertia; for it is the mass of these " dead parts " which
ultimately sets a limit to the maximum figure of merit obtainable.
The dispensing of " reading parts " in the Einthoven String
Galvanometer is the essential reason for its enormous number of
D'Arsons. Similar considerations hold for oscillographs, watt-
meters, pointer voltmeters and ammeters and other deflection
instruments in which a high figure of merit is desired.
* See pages 255-256 Jour, of the Franklin Institute, October, 1910,
"The Comparison of Galvanometers and a New Type of Flat-coil Galvanom-
eter."
ART. 1504] DEFLECTION INSTRUMENTS 359
In the above discussion attention has been confined chiefly
to galvanometers intended for use on constant current or nearly
constant-current circuits. This is the condition which applies
when galvanometers are used for measuring insulation resistance
by direct deflection methods. It also applies, tho to a less extent,
when galvanometers are used, with considerable resistance external
to themselves, in Wheatstone and Kelvin double-bridge measure-
ments of resistance. When, however, galvanometers are de-
signed for use with thermocouples and for reading millivolt or
microvolt drops over low resistances, many points of design, as
the galvanometer resistance, damping, etc., should receive atten-
tion; but our limits will not permit a discussion of this phase of
the subject. When dealing with galvanometers for use on con-
stant potential circuits, it is preferable to use, as the expression
for the figure of merit, the relation (12) given above. In this
expression when E m = I microvolt, T = 1 second, and R = 1 ohm,
the figure of merit is unity, and then we may call the unit, a
microvolt D' Arson.
We have collected in a table the essential characteristics of
sixteen different well known types of galvanometers, for which
see table at end of paragraph.
Number 16 is a galvanometer of the D'Arsonval type made
by The Leeds and Northrup Company, which has the very
high figure of merit of 2.80 D'Arsons. Its megohm sensibility,
however, is but 121.8. Theoretically, a very fine suspension
could be used until its sensibility reaches that of No. 13 which is
1750 megohms. Practically, however, this would not be feasible,
because the suspension would be finer than any wire on the
market except Wollaston wire. But were such a fine suspension
used, the more serious difficulty would arise that the coil would
then be influenced to a relatively great degree by traces of mag-
netic matter in the coil. This would produce a large zero shift
on reversed deflections. Thus, for high sensibility work, as
in cable testing, where the longer period is not too serious a dis-
advantage, galvanometer No. 13 would be a much better instru-
ment to use, altho its figure of merit is but 11.4 per cent of
that of No. 16.
In respect to Table I the following remarks may be added.
For the Einthoven String Galvanometer (No. 1) the resistance of
the string was reduced to its copper equivalent, and a magnifica-
360
MEASURING ELECTRICAL RESISTANCE
[ART. 1504
tion of 100 was taken as the equivalent of a scale at 1000 scale
divisions from a mirror.
The data for No. 2 was taken from Siemens and Halske's re-
print No. 30, and the data for No. 4 was taken from an article by
Dr. H. Sack in the same reprint. The data for No. 5 is from
The Cambridge Scientific Instrument Company's catalogue. The
data for No. 16 is based on the average of five instruments designed
by the author, and made by The Leeds and Northrup Company
in August, 1910.
The Weston Voltmeter (No. 7) shows a figure of merit of 0.115
D'Arson. In considering the meaning of this low figure, it must
be remembered that a light system has to carry a pointer which
must be heavy enough to be perfectly rigid. The same system,
fitted with a mirror, would show a much higher figure of merit.
The design of this instrument, as every one knows, is most scien-
tifically worked out, and the fact that its figure of merit is low
simply emphasizes the fact that one must exercise great caution
against estimating the real worth of an instrument by this feature
alone.
No.
Type of instrument
Method of reading
Critical
resistance
for damp-
Instru-
ment
resist-
ance
Megohm
sensi-
bility
Com-
plete
period
Figure of
merit in
D' Arsons
F- Sm
ing
R
Sm
T
T*VR
*1
Einthoven string
Microscope, 100 fold
Aperiodic
287
5.2
0.01
3000.00
magnification
*2
Dubois Rubens, iron-
clad moving mag-
net
Mirror
Aperiodic
290
12200.0
6.0
20.00
3
Siemens & Halske,
high sensibility ....
Mirror
120
200
2500.0
12.0
1.24
*4
Siemens & Halske,
high sensibility... .
Mirror
290
1220
6
2 00
*5
Ayrton-Mather
Mirror
Undamped
20
52 6
3.5
0.96
6
R. W. Paul, single
pivot millivolt-
meter
Pointer
25
50
21.4
3.0
0.33
7
Weston, voltmeter. . .
Pointer
Aperiodic
75
0.25
0.5
0.115
8
Weston, portable galv
Pointer
3000
280
13.8
1.5
0.37
9
L. &N., typeP
Mirror
140
124
85.0
8.5
0.10
10
L. & N., marine type
Mirror
Aperiodic
1660
21.4
2.0
0.131
11
L. & N., No. 2300
standard four coil. .
Pointer
300
300
15.54
2.5
0.14
12
13
L. & N., typeH
L. & N., No. 2280
Mirror
Aperiodic
544
295.0
7.0
0.26
wide coil
Mirror
Aperiodic
1420
1750.0
12.0
0.32
14
L. & N., steel magnet
Pointer
500
230
8.8
1.3
0.344
15
L. & N., special small
four coil
Mirror
500
85
6.4
1.0
0.695
16
L.&N.,No.2280nar-
row coil
Mirror
200
180
121.8
1.8
2.80
Data not obtained by author.
ART. 15051
DEFLECTION INSTRUMENTS
361
FIG. 1505.
1505. Description of One Type of High-sensibility Galvanom-
eter. As it is not our purpose to discuss the structural details
of galvanometers as made by the various instrument makers we
can do no more in this respect than briefly describe, and give
an illustration of, a single type of high-sensibility galvanometer
362 MEASURING ELECTRICAL RESISTANCE [ART. 1505
(galvanometer 2280; Nos. 13 and 16 in the above table). The
instrument is shown in Fig. 1505. It is furnished with a coil of
medium width when intended for use in insulation testing and
where high sensibility is more important than a quick working
period. It may also be furnished with a very narrow coil, when
it becomes best adapted to Wheatstone-bridge work, low resist-
ance measurements by the Kelvin double-bridge principle and
general laboratory work in connection with potentiometers, ther-
mocouples, etc.
Its more prominent features may be summarized as follows :
The tube which contains the coil system makes one unit and the
magnet another, so different tubes with coils of different charac-
teristics may be fitted to the same magnet.
The coil system is always exposed to full view.
The deflections are very closely proportional to the current
passed thru the coil.
The damping of the system may be varied thru wide limits by
removable copper rectangles which fit upon the coil.
The suspended system may be locked by a clamping device
when the instrument is to be carried about.
The instrument as a whole is very highly insulated from ground
by hard rubber, petticoat-insulated leveling screws.
The figure of merit is high, the narrow coil type reaching 2.8
D'Arsons, while the wider coil type of 1500 ohms resistance has a
megohm sensibility of 1700 megohms with a period of 12 seconds.
The suspension is easily replaced if broken, an event which is
made unlikely by means of a protecting spring at the top of the
suspension tube.
APPENDIX
I. TABLE.
a
(1) Values of
1000 -a
U
lits
100
10
1
2
3
4
5
6
7
8
9
.00
0000
1001
2004
3010
4016
5025
6036
7049
8064
9082
1
.0
1010
1112
1214
1317
1420
1523
1626
1730
1833
1937
2
.0
2011
2145
2250
2354
2459
2564
2670
2775
2881
2987
3
.0
3093
3199