Having now obtained the constants of Table XI, the theoretical distribution of angular velocities around each of the circular graphs of Figures 7 to 11 were calculated by equation (39), and these theoretical values are designated by numerals on the inside of the circular graphs. The values of R'-R and of X-X corresponding to these theoretical values of w were then plotted as the circles on the rectangular graphs of motional resistance and motional reactance in Figures 4, 5, and 6. It is seen that the agreement of the computed and observed points in these Figures 4 to 6, while not exact throughout the entire range, is yet sufficiently good to show that the theory is essentially correct. Another significant point in the theory is the interpretation we have given to the depression angle 23 of the circular graphs. We interpreted 3 to be the angle by which the magnetic flux lags behind the magnetizing current in the telephone receivers. To test this point, this angle of lag of magnetic flux behind magnetizing current was independently measured with the experimental bipolar receiver. This receiver had a separate secondary, or exploring, coil wound around the ends of its poles, near the diaphragm. The e. m. f. generated in this exploring coil is in phase with the time rate of change of flux; and the phase of this e. m. f. was compared with the phase of the alternating current in the exciting coils in two ways (1) by a three-voltmeter method and (2) by an alternating current potentio meter. In the three-voltmeter method, a known resistance was put in series with the exciting coils, and one end of the exploring coil was connected to the point between the exciting coil and the known resistance. With the frequency and the e. m. f. about the telephone kept the same as that used in the bridge measurements i. e., the e. m. f. of 1 volt, and the frequency near the resonant frequency of the diaphram voltages were measured about the known resistance 20 ohms), about the exploring secondary, and about the two in series. These voltages, being small, were measured by a crystal rectifier in series with a galvanometer- the galvanometer and rectifier having been calibrated immediately before and after the experiment by an a. c. potentiometer operating at the frequency employed. The reading of voltage were very consistent, and were as follows in a typical case: $ G. W. Perse: Phys. Review, 25. p. 31, 1907; ibid., 28. p. 153, 1909. 66 about both with secondary reversed = 0.161 Substitution of the first three of these values in the formula for an obtuse-angle oblique triangle gives 79°, as the angle by which the secondary voltage leads the primary current. This is the angle by which the time derivative of the magnetic flux leads the primary current. The flux itself lags its time derivative by 90°, and therefore lags the primary current by 90°-79° 11°. = Again, a substitution of the first, second, and fourth value of above table in the formula for an acute-angle oblique triangle gives for the flux lag angle the value 11.5°. This angle of lag of flux behind the magnetizing current was found to be nearly independent of the frequency. To illustrate this, and as a further confirmation of the result obtained by the three-voltmeter method with the crystal rectifier and galvanometer as voltmeter, a second measurement was made by an entirely different method; namely, by the use of a Drysdale alternating-current potentiometer, with a 60-cycle current, and with a vibration galvanometer as indicating instrument. The method employed in this experiment consisted in first measuring the magnitude and phase of the primary current, and then the magnitude and phase of the voltage in the secondary winding. The difference between these two phases, subtracted from 90°, gives the required angle of magnetic flux lag. Balance was in each case indicated by getting a zero deflection of the vibration galvanometer. This method gave 12.5° as the angle by which the flux in the telephone lags behind the magnetising current. The three values obtained by direct measurement for the flux lag, which should be the angle ẞ according to the theory above proposed, are 11°, 11.5°, and 12.5°; whereas half the depression angle, for this telephone, which, according to the theory, should also be the angle ẞ, is 13.2°. The agreement is not as good as might be desired for a perfect confirmation of the proposed theory; but in view of the difficulty of measuring small angles of lag in circuits containing voltages of the order of 0.1 volt, and in view of the fact that the experimental telephone receiver constructed for this purpose had to be complicated by auxiliary secondary windings and also unfortunately had a diaphragm mounted in such a way as to have a very large temperature coefficient of vibration period, which rendered difficult an accurate determination of the points of the circular graph, the writers believe that the departure of a degree or two in the value of ẞ, as obtained by direct measurement from its value as obtained by the circular graphs, is not unsatisfactory. VII. SUMMARY OF RESULTS. 1. The resistance and inductance of several telephone receivers were measured over a wide range of frequencies with their diaphragms both free and damped. 2. The damped resistance is approximately a quadratic function of the angular velocity of impressed e. m. f. (see equations (1) and (2)). 3. Although the damped resistance and the damped inductance both change with the frequency of e. m. f., their product is approximately constant, independent of the frequency, over a considerable portion of the range of audible frequencies (see eq. (3) and (4) and Table VII). 4. The damped reactance of one form of standard bipolar Bell receiver is approximately equal to its damped resistance, over a considerable range of frequency; so that the current lags the e. m. f. by 45° (see Figure 2). 5. The free resistance and reactance of telephone receivers go through marked changes with changes in frequency of constant e. m. f. in the neighborhood of the natural frequency of their diaphragms (cf. Figures 4 6). 6. The motional resistance and motional reactance (by which is meant excess of free resistance of reactance over damped resistance or reactance) conform accurately to certain simple laws as follow: I. The motional reactance plotted as ordinates against the motional resistance as abscissas, as the frequency of constant impressed e. m. f. is changed from zero to infinity, gives a circular locus, with various interesting characteristics. (cf. Figures 7 12). II. The rectangular plots of motional reactance and motional resistance against angular velocity of constant impressed e. m. f. give curves somewhat analogous to the curves of index of refraction and absorption of light in an optical medium in the neighborhood of an absorption band (cf. Figures 4-6). 7. The power taken by a telephone receiver when sounding at 0.3 volt applied voltage may exceed by 68% the power taken from the same e. m. f. when the diaphragm is damped (Figures 4-7 and Tables VIII-X). 8. A theoretical explanation of the phenomena is given, and computations are submitted in comparison of experiment and theory (Headings V and VI). 9. The vibration constants of the diaphragms of the several receivers are deduced and collected (Table XI). HARVARD UNIVERSITY, CAMBRIDGE, MASS. |