The Coupled Circuit by the Method of Generalized Angular Velocities

The generalized impedance z of an oscillating circuit may be formed from the generalized angular velocity of oscillation, n, by analogy with the alternating-current circuit. Equating this generalized impedance to zero, and solving for n, gives the free generalized angular velocities of oscillation. The real and imaginary portions of these free generalized angular velocities are used to find respectively the damping factors and frequencies of oscillation of the circuit. From z may be found the threshold impedance of the circuit n(dz/dn). Dividing the initial voltage of the circuit by this threshold impedance gives the initial amplitudes of current oscillation. The use of these two rules determines the complete expression for the oscillating current in any oscillating-current network. The method applies to the simple inductively coupled circuit. A complete exact solution for the case of primary condenser discharge may be readily obtained. The method is of particular service in numerical problems. An approximate method of solving the biquadratic obtained when coupled circuits are considered is given in an appendix to the paper.