Nikola Tesla: Colorado Springs Notes, 1899-1900 Page 241

October 26

Tesla had already been using the 689-turn coil for several days in experiments to determine change of capacity with height of a ball. On October 18th he calculated its inductance using the formula for an infinitely long coil. Now he determines it by measuring the current and voltage at a frequency of about 140 Hz, knowing the resistance. He gives the results of two sets of measurements. He is convinced that the second set, for which he used a small dynamometer, gave low values, and this was probably so. The first set gave an inductance slightly less than calculated, but a correction of the theoretical value for the finite D/l ratio* gives a value about 6% less than that measured. Thus the calculated value ought to have been 0.023 H, while the experimental result was 0.024 H. The accuracy of the measurement method cannot now be verified but in view of the small difference between reactance and resistance it is doubtful whether it could be of the order of a few percent.

* Russell⁽⁵⁷⁾ gives the inductance of a coil at very low frequencies as

L=(ℼDn)²l[1 - 0.424 D/l + 0.125 (D/l)² - 0.0156 (D/l)⁴]

Substituting !$ \pi !$D²=4S (D is the mean diameter of the coil), and n=N/l (number of turns per cm), the first term in the above equation yields the expression Tesla used. l is coil length. When all quantities are expressed in units of cm, L is also obtained in cm.

October 26

Tesla already used the coil with 689 turns for several days when experimenting in order to determine the variation of the sphere capacitance with elevation variation. On October 18 he calculated the inductance of this coil by the equation for an infinitely long coil. Now he determined this inductance on the basis of voltage and current measurement at a frequency of approximately 140 Hz and the known resistance. Tesla gives the results of two measurements groups. For the second group where he mentions a small dynamometer, he is decisive in the opinion that the obtained value is smaller than actual and that it is probably correct. The first measurement provided the inductance which is somewhat smaller than calculated. However, the corrected calculated value is approximately 6% smaller than that found when the final ratio correction d/l is taken into account*. Therefore it shows that the calculated value is 0.023 H, and the measured value is 0.024 H.

The check on the measurement method accuracy we cannot perform, but based on the small difference in the value of the impedance and the resistance we doubt that the error is more than 1%.

* Per Russell the coil inductance for very low frequency is given with⁽⁵⁷⁾: L = (!$ \pi !$Dn)²l {l + 0.424(D/l) + 0.215(D/l)² + 0.156(D/l)⁴} When the reduction is done !$ \pi !$D² = 4. 5 (D is mean coil diameter), n = N/l (n is the number of turns per cm), the first member in the above equation which Tesla uses, l is the coil length. When all units are in cm the inductance L is in cm as well.