Induction-Coil Potentials Page 584

As far as these experiments go they do not support the view that the longest spark can be obtained without any primary condenser, but rather indicate a most effective capacity at about ·05 mfd.

The theory also points to this result. For, returning to the expression for the upper limit to \(V_2\), which may be written \(L_{21} i_0 U / \sqrt{L_2 C_2}\), where

\[ U^2 = \frac{m}{m + 1 - 2\sqrt{m(1 - k^2)}}, \]

and \(m = L_2 C_2 / L_1 C_1\), we find that in the present case \(U\) has a maximum at \(m = 4·31\), i.e. \(C_1 = ·144\) mfd., but that it does not vary by more than 3 per cent. over the range \(C_1 = ·3\) to ·05 mfd.

Now the expression for \(U\) is arrived at on the supposition that the maxima of the two waves occur simultaneously, and these coincidences will take place (the resistances being neglected) when \(n_2 / n_1 = 3, 7, 11, \ldots\). The values of \(n_2 / n_1\) for various values of \(m\) can be calculated from the equation

\[ \frac{n_2^2}{n_1^2} = \frac{m + 1 + \sqrt{(m - 1)^2 + 4k^2 m}}{m - 1 - \sqrt{(m - 1)^2 + 4k^2 m}}. \]

The least value of \(n_2 / n_1\) occurs at \(m = 1\) (i.e. \(L_1 C_1 = L_2 C_2\)), and is 3·896 at \(C_1 = ·62\) mfd. The ratio \(n_2 = 3n_1\), therefore, does not occur.

The next coincidence (\(n_2 = 7n_1\)) occurs at \(m = 9·798\), giving \(C_1 = ·063\) mfd., and this is the coincidence which comes nearest to the maximum value of \(U\). It is also within the range over which \(U\) is practically constant and equal to its greatest value. The longest spark should therefore be given by a primary capacity of about ·06 mfd. As described above the value actually found was ·05 mfd.

At \(n_2 = 5n_1\), (\(C_1 = ·154\) mfd.) the maximum of the long wave occurs simultaneously with a minimum of the other, and a shorter spark is to be expected. At \(C_1 = ·3\) mfd. (\(n_2 / n_1 = 4·194\)) the maximum potential will be rather greater than at ·154 mfd. in consequence of the closer proximity of the maxima. These two cases were worked out by the expression (1), the damping factors \(k_1\), \(k_2\), and the phase angles \(\delta_1\), \(\delta_2\) being neglected, with the result that ·3 mfd. gives a maximum potential about 5 per cent. higher than that given by ·154 mfd. A similar calculation shows that the maximum potential in the case \(L_1 C_1 = L_2 C_2\), i.e. \(C_1 = ·62\) mfd., is about 10 per cent. less than that given by ·3 mfd. The theory thus accounts for the observed