Induction-Coil Potentials Page 574

Calculation of the Secondary Potential.

The formulæ used in the calculation of the frequencies, amplitudes, and damping factors of the two oscillations are collected here:—

\[ 8\pi^2(n_1^2,\, n_2^2) = \frac{1}{1-k^2}\left[\frac{1}{L_1 C_1} + \frac{1}{L_2 C_2} \pm \sqrt{\left(\frac{1}{L_1 C_1} - \frac{1}{L_2 C_2}\right)^2 + 4k^2 \cdot \frac{1}{L_1 C_1} \cdot \frac{1}{L_2 C_2}}\right], \]

\[ A_1 = \frac{2\pi L_{21} i_0 n_1 n_2^2}{(n_2^2 - n_1^2)\cos\delta_1}, \]

\[ A_2 = \frac{2\pi L_{21} i_0 n_1^2 n_2}{(n_2^2 - n_1^2)\cos\delta_2}, \]

\[ k_1 = 4\pi^2 n_1^2\left(\frac{\theta_1 + \theta_2}{2} - \beta\right), \]

\[ k_2 = 4\pi^2 n_2^2\left(\frac{\theta_1 + \theta_2}{2} + \beta\right), \]

where

\[ \theta_1 = \frac{1}{2} R_1 C_1 = \frac{1}{2} \frac{R_1}{L_1} \cdot L_1 C_1, \]

\[ \theta_2 = \frac{1}{2} R_2 C_2 = \frac{1}{2} \frac{R_2}{L_2} \cdot L_2 C_2, \]

\[ \beta = -\frac{2\pi^2 n_1^2 n_2^2}{n_2^2 - n_1^2}(\theta_1 - \theta_2)(L_1 C_1 - L_2 C_2), \]

and \(i_0\) is the current interrupted in the primary.

These expressions are, with a change of notation, the same as those used in the former papers.

In the case of the phase-angles \(\delta_1\), \(\delta_2\), however, new formulæ are required owing to the fact that in the former papers no account was taken of the difference between the effective resistances and the steady-current values. This affects the initial conditions of the problem, which now become at \(t = 0\),

\[ -C_1 \frac{dV_1}{dt} = i_0, \quad V_1 = R_0 i_0^*, \quad \frac{dV_2}{dt} = 0, \quad V_2 = 0, \]

where \(R_0\) is the resistance of the primary for steady currents.

^* If the condenser is connected across the interruptor in the usual way the corresponding initial condition is \(V_1 = 0\), and the solution for this case may be obtained by putting \(\theta_0 = 0\) in the final equation given below.