A paper to be presented at the 29th Annual Con* vention of the American Institute of Electrical Engineers, Boston, Mass., June 25, 1912.

Copyright, 1912. By Α. I. Ε. E. (Subject to final revision for the Transactions.)

CHARACTERISTICS AND APPLICATIONS OF VIBRA- TION GALVANOMETERS

BY FRANK WENNER

In the vibration galvanorneter we have a type of synchronous motor which is distinctly different from all the ordinary types of dynamo electric machines. Further it does not have any of the characteristics of any of the ordinary galvanometers and except for the fact that it is used in the detection or measure­ ment of small currents and voltages, it should not be called a galvanometer. In galvanometers, except when used in the measurement of transient currents or quantity of electricity, the moving system is displaced until we have an equality of static couples acting on it. In the vibration galvanometer the equilibrium condition is an equality between integral values of the product of the current and generated voltage and the mechanical power dissipated in various ways as in ordinary electric motors when operated without a load. It therefore behaves more like an electric motor than like a galvanometer. Further, since it is used only with alternating currents and op­ erates in synchronism with the current it must necessarily have some of the characteristics of a synchronous motor. As a motor the efficiency of conversion was found in a particular case to be as high as 97J per cent while the power required to maintain an easily discernable amplitude of vibration was of the order of 10-11 watts. One of the large turbo-generators would therefore furnish the power necessary to operate a thou­ sand-million-billion such machines. While the vibration galvanometer will probably never be used in the ordinary way for driving other machines yet it is of con­ siderable interest from a theoretical standpoint and possesses 1073 1074 WENNER: VIBRATION GALVANOMETERS [June 25 certain characteristics which make it a most valuable instrument for use in a large number of alternating current measurements. A number of different forms of galvanometers have been and are being used. One form is very similar to an oscillograph except that in general the period is longer and the oil for damping is omitted. Another form is very similar to the D'Arsonval galvanometers of the marine type except that the coil is narrower and the suspensions tighter. In all cases provision is made for changing the period of the moving system. This is usually done by changing the length or tension of the suspensions. Where a large range in frequency is desired the suspensions are often made bifilar. Passing an alternating current through the coil causes it to vibrate back and forth past its equilibrium position, hence the name vibration galvanometer. The amplitude of the vibration, which is a measure of the current, is ordinarily determined by observing the broadening of a line image as seen in a small mirror attached to the moving system. As regularly used in the detection or measurement of current or voltage the natural or free frequency of the moving system is made to correspond with the frequency of the alternating current to be detected. As a result the amplitude of the vibration is much larger than it would be under almost any other condition. The sensitivity is large only to currents of the frequency to which the moving system of the galvanometer is tuned. It is therefore possible to use currents of almost any wave form, even in those null methods in which an exact balance can be obtained only with a current free from all higher harmonics. In all such measurements we may make our observations and calculations just as if we were using current having a sine wave form since the galvanometer responds only very feebly to the 3rd, 5th and higher harmonic components. It is this characteristic combined with its extremely high sensitivity at low frequencies which makes the vibration galvanometer a most valuable in­ strument in various kinds of alternating current measurements. Before we can make such progress in the design of an instru­ ment or machine it is necessary that we know definitely the re­ lation between its various constants. In some cases this knowl­ edge is necessary before we can even use a well designed and constructed instrument or machine to its best advantage. We shall therefore show the relation which exists between the ampli­ tude of the vibration and the impressed voltage in terms of the 1912] WENNER: VIBRATION GALVANOMETERS 1075 intrinsic constants of the instrument and of the electric circuit in which it may be used. Since here we have a mechanical and an electrical oscillating system so connected that they must necessarily operate at the same frequency we shall make free use of the analogy existing between such systems. Passing a current through the winding produces a mechanical torque tending to displace the moving system of the galvanometer from its equilibrium position. The torque is proportional to the current and we shall let G be the proportionality factor or the displacement constant. In addition we shall use the follow­ ing notation, in which vectors are designated by bold faced type and make use of the following well known relations:

IN THE ELECTRICAL SYSTEM IN THE MECHANICAL SYSTEM L = inductance κ = inertia constant r = electrical resistance D = damping constant or mechanical resistance C = capacity U = restoring constant anal- agous to 1/C ef = impressed voltage Gi = displacing torque i = current s = rate of displacement or angular velocity q = charge = / i δ t Φ = displacement = f sτ t

Ε' (D ef G I (2)

I ê*1 (3) i S s (4) a Q ë*1 (5) Φ φ ηiPt (6)

1 r + i(pL-l/Cp) (7)

c GJi s = (R D + i(Kp-U/p) W or if we let X represent the electrical reactance and M represent the mechanical reactance

0) 1076 WENNER: VIBRATION GALVANOMETERS [June 25

s = <10>

G I Φ = ip(D + iM) (12)

Here £7' is the total voltage available for producing current or the sum of the impressed voltage Ε and generated voltage

Et. The voltage generated by the relative motion of the magnet and winding is proportional to the amplitude and in quadrature with the vibration. It is also proportional to the frequency and proportional to the displacement constant and it is easily shown that

Eg = - ip G Ö (13) therefore j E-~ipG$

1 = r + iX (14)

This value of / substituted in equation (12) gives GE Φ = p [(-rM-D X) + i{Dr-MX + G2)] (15) or GE Φ = ρ V(-rM-'DX)* + (Dr-MX+ G2)2 (16) or G Ε €»<*"+<•>

p VJ-r M -D Χ)2 + (Dr-M X + G2)2 (17> where -(Dr-MX + G2) tan a = Mr + DX : (18)

Equation (16) gives the relation between the amplitude of the vibration, the impressed voltage, the displacement constant, the frequency, and the other constants of the electric and me­ chanical systems expressed as resistances and reactances. Where the instrument is to be used in precise measurements we usually desire as high a sensitivity as can conveniently be obtained, i.e. we wish as large an amplitude of the vibrations for a given small impressed voltage as we can conveniently get. An inspection of equation (16) does not at once suggest the 1912] WENNER: VIBRATION GALVANOMETERS 1077 relations which should be made to exist between the various quantities D, r, M, X, and G to give the best sensitivity. Since power is absorbed or converted into heat proportional both to D and to r the most natural beginning is to make both small. The mechanical resistance or damping constant D is a definite constant of the galvanometer and as we shall see later one of the most important points to be looked to both in the design and the construction is to make this constant as small as possible. The electrical resistance r is the total resistance of the circuit in which the galvanometer is used. As the resistance of a part of this circuit is usually fixed from other considerations the particular value of the resistance of the galvanometer is of little importance providing it is less than say \ the total resistance of the circuit. As vibration galvanometers are usually constructed provision is made for varying the free period of the moving system so that the mechanical reactance can be adjusted to zero for any frequency in the range over which it is expected that the gal­ vanometer will be used. The electrical reactance is usually very small in comparison with the electrical resistance. It can however, be varied by placing capacity or inductance in series with the winding of the galvanometer. If the galvanometer is so constructed that the displacement constant G can be varied; and all instruments intended for use in circuits of different resistances should, if a high sensitivity is desired, be so constructed; we can then adjust each or all of the remaining quantities, the electrical reactance X, the me­ chanical reactance Af, and the displacement constant G. If M, is adjusted to zero and if ζ is small or adjusted to zero it will be seen that the sensitivity is a maximum when

G2 = Dr Under these conditions we have

Φ = 27Wr (19) or since we observe the total amplitude of the vibration and measure the root mean square value of the voltage

Ρ V 2 Dr • (20)

as has been shown by the author.1 1. Bulletin, Bureau of Standards, Vol. VI, p. 376; reprint 134; 1909. 1078 WENNER: VIBRA TION G A L VA NOME TERS [June 25

Under these conditions the generated voltage is half as large as and in direct opposition to the impressed voltage. This we know to be the conditions under which the mechanical power developed is a maximum. This, then is a condition which gives the maximum sensitivity with the particular values of r and D. Since to get the maximum sensitivity it is only necessary to make such adjustments as will bring the generated voltage in direct opposition to and make it half as large as the impressed voltage it should be possible to bring about this condition by changing any two of the three constants X, M, and G. If these constants are changed one after the other in small steps or con­ tinuously it can be shown by differentiation that the sensitivity becomes a maximum for changes

G2M of X, when X = D2 + M2 (21)

G2X of M, when M = r2 + X2 (22)

and of G, when G =

Since the last of these equations is the product of the other two it follows that if any two of the constants are adjusted so that they simultaneously have their best value the third constant also has its best value or the sensitivity is the maximum attain­ able. When this adjustment, which is a double one, and in some cases will have to be made by successive approximations, is carried out we have

Χ M T=D (24) or the electrical time constant equal to the mechanical time constant and μΤΎ> = Ό <25) or the electromagnetic damping equal to the mechanical damping. When we^have the first of these relations the generated voltage is in direct opposition to the impressed voltage. When we have 1912] WENNER: VIBRATION GALVANOMETERS 1Τ79 the second the generated voltage is half as large as the impressed voltage. When we have both relations.

φ- E 2pVDr (26)

as given above and has been shown recently by Butterworth.2 If we express the amplitude of the vibration in terms of the broadening of the line image 1 meter from a mirror attached to the moving system, the voltage in microvolts, the resistance in ohms and the mechanical constants in c.g.s. units then

Ε φ = 2.0 f VDr (27)

when / is the frequency or if ν is the sensitivity the ratio of the number representing the amplitude of the vibration to the num­ ber representing the voltage

v- 20 v - rvw (28)

Since usually it is not practical to place a variable inductance in series with the galvanometer and by adjusting it and the mechanical reactance to bring about the best condition, and since few of the galvanometers now in use are provided with a means for adjusting- G we are often obliged to be content with the best condition we can obtain by the adjustment of the free frequency only. If then this adjustment is made and the elec­ trical reactance is small in comparison with the electrical resist­ ance then equation (16) takes the form

G Ε Φ = pW+~&) (29)

and these are the relations under the more usual conditions of use. To the person who is considering using a vibration galva­ nometer the question as to what sensitivity can readily be ob- 2. Proceedings Phys. Soc, London, Vol. XXIV, p. 77, 1912. 1080 WENNER: VIBRATION GALVANOMETERS [June 25 tained is of much more importance than the effect of various constants upon the sensitivity. In this connection we may state that three years ago the author determined the constants of three vibration galvanometers andiound the voltage sensitivity in millimeters per microvolt at 100 cycles as follows:

0.0014 0.0061 and 0.0075.

Recently Mr. Silsbee of the Bureau of Standards made, ac­ cording to the suggestions of the author, a new coil for one of the older galvanometers of the D'Arson val type in fact one of the three just mentioned. With the new coil the voltage sensitivity at 25 cycles is 0.47. The instrument was designed for use in a bridge having a resistance of about one ohm. It has since been found that it will be necessary to increase the resistance in the bridge arrangement so that the total resistance of the galvanometer circuit will be about four ohms. Under these conditions the sensitivity will be 0.28, while if the magnet is strengthened just a little so as to give the best conditions the sensitivity will be 0.40. With the new coil the galvanometer differs in so many ways from the others that the relative merits cannot be represented by figures giving the ratio of the voltage sensitivities which range from 60 to 300. It cannot be used at a frequency of 100 neither can the others be used at a fre­ quency of 25. It was designed to operate at the lower frequency and to be used in connection with a low resistance bridge. When so used its sensitivity is about the same as that of a good direct current D'Arsonval galvanometer designed for and used in connection with a bridge of the same resistance. We have gone rather fully into the matter of the sensitivity and the adjustments necessary to get the maximum sensitivity attainable with any particular galvanometer when used in any particular circuit. We have done this because the matter is one of considerable importance to many of those working with such instruments and a matter which seems not to be very well understood. A further consideration of equation (16) will bring out other characteristics of the vibration galvanometer. In cases where the electrical reactance is small in comparison with the resistance so that it need not be considered we have 1912] WENNER: VIBRA TION G ALVA NO METERS 1081

2 or since M = p Κ — U/p and U = p0 K, where p0 is the free frequency of the moving system,

r V K2 (po2 - p2)2 + p2 (Dr + G2)2/r2 W

If Po = Ρ

Po (Dr + G2) ΟΓ p (Dr + G2) (29)

If po is very different from p, Κ (po2 — p2) is large in com­ parison with p (Dr + G2) so we may write

G Ε ^

Φ = —. ^ / . ď ΓΟ\ or Κ = 2 2 ± r 2C (V ~ ί> ) ±rX (^>ο - Ρ2) (30) or we have

Vo ( r ro /r \ 2 π Kr

= ~V~ ±(/0-///0) Df + Q2

If the galvanometer has an inertia constant of 0.02, a damping constant of 0.01 and a displacement constant of 10,000 (none of which are exceptional values) then

2wKr 6.3, if r = 100 ohms (1011 c.g.s. units) Dr + G2 or 12.6 if r is large

Then if the fundamental frequency of the voltage is 60 and the instrument is tuned to this frequency a substitution of 60 for/o and of 180 and 300 for / gives the ratio of the sensitivity of the instrument to the fundamental as compared with its sensitivity to the 3rd and 5th harmonic as follows:

With 100 ohms With high resistance

Si/S3 = 3000 or = 6000

St/Si = 9000 or = 18,000

If the instrument is tuned to the 3rd or 5th harmonic a sub­

stitution of 180 and 300 for /0 and 60 for / gives the ratio of 1082 WENNER: VIBRATION GALVANOMETERS [June 25

the sensitivity to the harmonics as compared with its sensitivity to the fundamental as follows:

With 100 ohms With high resistance Sz/Si = 1000 or = 2000 St/Si = 1650 or = 3300

It will thus be seen that the galvanometer when tuned to the funamental is (if the resistance of the circuit is 100 ohms or more) at least 3000 times as sensitive to the fundamental as to any of the harmonics. If then the voltage used in testing has a third and higher harmonics amounting to not more than 3 per cent, the accuracy to which a balance may be established cannot be limited by the presence of unbalanced harmonic com­ ponents of the voltage, unless the accuracy sought is better than 1 in 100,000. It is this characteristic of not responding to the harmonic components of the current passing through it together with its high sensitivity to the fundamental component both of the voltage and current which has led to the use of the vibration galvanometer in various alternating current measure­ ments. It will also be seen that where a galvanometer having the constants just considered is tuned to a frequency of three or five times the fundamental and the resistance is high the instru­ ment is 2000 or more times as sensitive to the harmonics as to the fundamental. The instrument can therefore be used to read the harmonic components of the voltage directly from the amplitude of the vibration and it should be possible if the har­ monics are small to obtain their values to better than 0.1 per cent of the fundamental. In this method of determining the higher harmonic components of the electromotive force use is made of the mechanical resonating system of the galvanometer instead of an electrical resonating system. If the frequency of the impressed voltage is varied from below that for which the amplitude of the vibration is a maximum to above this value the angle between the generated and impressed voltage changes by nearly π radians or 180 4eg. The change in angle with frequency is largest at the frequency which gives the maximum amplitude. In some cases we have a noticeable change in the phase angle when the frequency changes by only a few hundredths per cent. Considerably below the particular frequency the current lags behind the impressed voltage by 1912] WENNER: VIBRA TION G AL VA NO METERS 1083 nearly 7τ/4 radians or 45 deg. and considerably above this fre­ quency it leads by about the same amoufit. If then the stator windings of. a very small two-phase induction motor were connected to the same voltage supply, one through a resistance and the other throuh a large vibration galvanometer we would have, in general, an elliptical rotating field at the stator and the direction of rotation would depend upon whether the frequency of the impressed voltage were above or below that for which the amplitude of the vibration of the galvanometer would be a maximum. It is not at all improbable that such an arrangement might be used to accomplish some particular end in connection with the generation or transmission of electrical power-, such for example as the indication or regulation of the frequency. In engineering work the vibration galvanometer is being used but little. When its characteristics become more widely known and when it becomes known that it is not a delicate instrument various uses will no doubt be found for it. As a laboratory instrument we may mention that it is being used in the Bureau of Standards in connection with: The Anderson bridge for the comparison of self inductance with- capacity and resistance. One or more bridge methods for the comparison of self with mutual inductance. Bridge method for the comparison of the capacities and phase angles of condensers. Bridge methods for comparing the resistances and time con­ stants of wire resistance standards. The Thomson bridge method in the comparison of time con­ stants and resistances, to alternating currents, of standards of low resistance. In the determination of the ratio transformation and the phase angle between primary and secondary voltages and currents of potential and current transformers. It has also been used in various other precision measure­ ments and the indications are that it will soon be used in still others. In the National Physical Laboratory in England the vibra­ tion galvanometer is also being much used. Of the various ap­ plications there we may mention: The absolute measurement of resistances by a two-phase alternating current method. 1084 WENNER: VIBRATION GALVANOMETERS [June 25

The testing of transformer steel using a null method for de­ termining the total losses. A modified Carey Foster method for comparing self and mutual inductance. In most laboratories, however, the vibration galvanometer has met with less favor than it deserves.