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MEASUREMENT OF INDUCTANCE BY ANDERSON'S METHOD, USING ALTERNATING CURRENTS AND A VIBRATION GALVANOMETER.
By EDWARD B. Rosa and FREDERICK W. GROVER.
1. HISTORY OF THE METHOD.
Several modifications of Maxwell's method of comparing an inductance with a capacity have been proposed in order to obviate the double adjustment of resistances necessary in that method. Maxwell showed
that if (1) the bridge is balanced for steady currents and at the same time (2) the resistances are so chosen that there is no deflection of the galvanometer when the battery current is suddenly closed or broken, then L=CRQ=CPS
where L is the inductance in the arm A D, the resistance of which is
a Electricity and Magnetism, $ 778.
Q, C is the value of the capacity in parallel with R, and P, R, and S are noninductive resistances.
In order to satisfy both of these conditions two of the arms of the bridge must be varied simultaneously, so that the balance for steady currents may be maintained while the balance for transient currents is sought. This is generally a tedious process, although by means of a small variable inductance in Q, in addition to the inductance to be measured, and a multiple valued condenser the process might be considerably accelerated.
In 1891 Professor Anderson proposed“ an important modification of Maxwell's method, which consisted in joining the condenser to a point E, separated from C by a variable resistance r. The bridge being balanced for steady currents by varying any one of the four arms of the bridge, the balance for transient currents is then made by
varying r, which does not disturb the balance of the bridge for steady currents. This change, which rendered the two adjustments independent, removed at once a most serious difficulty and made the method thoroughly practicable.
Anderson's demonstration for the case of transient currents gives for the value of the inductance (changing the letters to correspond to fig. 2) L=C [r (Q+8)+PS]
(2) If r=0, L=CPS, as in Maxwell's method.
In the use of Anderson's method r may be small, so that CPS is the principal part of the expression for the inductance, or it may be larger, and the first term, Cr (Q+S), represents the larger part of L. Thus a considerable range of values of inductance may be measured
a Phil. Mag., 31, p. 329, 1891.
without changing the arms of the bridge or the capacity of the condenser.
Stroud and Oates a have proposed another modification of Maxwell's method, which they have used with much success in measuring inductances. Instead of employing an interrupted current from a battery, as Anderson had done, they used an alternating current and an alternating-current galvanometer, the latter being essentially a d'Arsonval galvanometer, with the field magnet laminated and strongly excited by an alternating current from the generator. The galvanometer was thus made very sensitive, and to increase the sensitiveness still further the resistance r was placed outside the bridge, as shown in Fig. 3. It will be seen that this arrangement differs from Maxwell's only in separating the point B from the terminal of the condenser by the
auxiliary adjustable resistance r, which in Anderson's method is in the galvanometer circuit between C and D. As the resistance r is sometimes several hundred ohms, it reduces the sensibility when in the galvanometer circuit, whereas in the arrangement of Fig. 3 the electromotive force can be increased if r is large, and so keep the same current in the bridge as when r is small, and thus maintain the sensibility.
The expression for the inductance L in Stroud's method (changing the letters to correspond with Fig. 3) is L=C[r (Q+P)+PS],
(3) which closely resembles the formula for Anderson's method, but dif. fers in having Q+Pin the first term instead of Q+S.
a Phil. Mag., 6, p. 707, 1903.
Professsor Fleming hus pointed out that Stroud's arrangement may be regarded as conjugate to Anderson's, the galvanometer and source of current being interchanged, Fig. 4. In this case the formula is exactly the same as for Anderson's method. If, however, Fig. 4 be rearranged so as to agree with Fig. 3, it will be found that the arms P and S are interchanged, and consequently that these letters must be interchanged in the formula for L. This changes equation (2) into equation (3).
Fleming and Clinton have employed Anderson's method for the measurement of small inductances, using a battery and a rotating commutator and galvanometer," and later Fleming employed an interrupted current, produced by a vibrating armature, and a telephone.
During the past two years we have employed Anderson's method for the measurement of both large and small inductances, using (1) a battery as a source of current and a d'Arsonval galvanometer, with a rotating commutator to interrupt and reverse simultaneously the current and galvanometer terminals; or (2), what has proved more satisfactory, an alternating current and a vibration galvanometer, the latter being tuned to the frequency of the current furnished by the generator.
2. ADVANTAGES OF THE METHOD.
We have found the method rapid and convenient in practice and the vibration galvanometer sufficiently sensitive to permit very accurate settings. As compared with other methods of accurately measuring inductance, it possesses striking advantages, some of which will here be specifically mentioned.
a Phil. Mag., 5, p. 493; 1903.
• Phil. Mag., 7, p. 586; 1907.
(a) All methods of measuring inductances without the use of a condenser (or other known inductance) require an accurate knowledge of the frequency of the alternating current employed. It is not difficult to determine accurately the mean frequency of an alternating current, even when the generator is inaccessible, as a counter may be employed to record on a chronograph the number of revolutions in a given time; moreover, the speed of the generator may be maintained sufficiently constant to enable good settings to be made. But to hold the speed steady enough to make settings of a high order of accuracy is difficult and requires an assistant to control the speed. With Anderson's method, even with a tuned galvanometer, slight changes of frequency are not detrimental, and hence the labor of taking the observations is greatly reduced.
(6) The inductance is determined in terms of a capacity, in addition to several resistances, which are also required in other methods of measuring inductance. A capacity can be measured by Maxwell's bridge method, using a commutator, with very great exactness, provided care is taken in choosing the resistances of the arms of the bridge, and also provided the temperature of the condenser is taken and a temperature correction subsequently applied whenever necessary. The capacity of a condenser is not the same for slow charges as for rapid charges, and hence, if Anderson's method is used for transient currents, the capacity employed in the formula should correspond to the conditions of the experiment. As the successive makes and breaks of the current are likely to be irregular, the result would be that the effective capacity would vary slightly in successive trials, even with the best mica condensers. On the other hand, using an interrupted or alternating current of constant frequency, the capacity is uniform and definite, and if it is measured at the same frequency there is no uncertainty as to its value. In our experiments we employ an eight-pole generator, giving four complete cycles in each revolution. To this generator is joined the commutator which is employed in charging and discharging the condenser when measuring its capacity, the commutator having four segments, and hence charging and discharging the condenser four times in each revolution. Thus the frequency of charge and discharge of the condenser may be made exactly the same in use as when its capacity is measured. The change of capacity of a condenser with the frequency is very slight, but in measurements of the highest accuracy it is well to eliminate the slight uncertainty due to change of frequency.
a Bulletin of the Bureau of Standards, No. 2, 1905.