Induction-Coil Potentials Page 585

principal maximum spark-length at about \(C_1 = ·05\) mfd., and for the second (smaller) maximum at about \(C_1 = ·3\) mfd.

Other cases may be obtained by varying the self-inductance of the primary circuit. If, for example, the four sections of the primary coil are connected in parallel \(L_1\) is reduced to approximately one sixteenth, \(L_{21}\) and \(L_{12}\) to one quarter, of their former values, and \(k^2\) is unaltered. Calculating as above, we find that the maximum value of \(U\) is given by \(m = 4·31\), i.e. \(C_1 = 2·304\) mfd. The nearest coincidence (\(n_2 = 7n_1\)) occurs at \(m = 9·798\), or \(C_1 = 1·013\) mfd., and the longest spark should be given by a capacity of about this value. By experiment the longest sparks were found to occur at \(C_1 = 1·025\) mfd., the variation being however very slight over the range 1·0 to 1·05 mfd.

A case of special interest is obtained by diminishing the coefficient of coupling so as to bring in the ratio \(n_2 / n_1 = 3\). The least possible value of \(n_2 / n_1\) is \(\sqrt{(1 + k)(1 - k)}\), consequently in order that \(n_2\) should be equal to \(3n_1\), \(k\) must not be greater than ·8. In this case it is also possible to make the maximum value of \(U\), and the ratio \(n_2 / n_1 = 3\), occur at the same value of the primary capacity. The necessary value of \(k^2\) is found by eliminating \(m\) between

\[ m = \frac{1}{1 - k^2} \]

and

\[ \frac{m + 1 + \sqrt{(m - 1)^2 + 4k^2 m}}{m - 1 - \sqrt{(m - 1)^2 + 4k^2 m}} = 9. \]

This leads to the equation

\[ 91k^4 - 164k^2 + 64 = 0 \]

for \(k^2\), giving \(k^2 = ·571\), the other root being greater than unity.

Consequently with \(k^2 = ·571\) (\(k = ·756\)), and \(L_1 C_1 = ·429 L_2 C_2\), an exceptionally high secondary potential should be given by an induction-coil.

The most effective way of diminishing \(k^2\) is to introduce external self-inductance into the primary circuit, since this does not involve any diminution of \(L_{21}\). In the present case (primary sections in series) the value of \(L_1\) required to give \(k^2 = ·571\) is ·241 henry, and the primary capacity (\(C_1 = ·429 L_2 C_2 / L_1\)) is ·198 mfd. On trying the experiment it was found that by introducing the necessary external self-inductance the maximum spark-length for a given primary current was considerably increased. Thus at