The magnetic field within electrical machines causes an interaction between the electrical and mechanical dynamics of the system. In the simplest cases, when the rotor mean position is central in the stator, the interaction manifests itself mainly as a negative stiffness between the rotor and the stator. When the rotor mean position is offset relative to the stator, then components of force arise whose frequency in the stationary frame is twice the electrical frequency of the supply. For induction machines in particular, both the electrical system and the mechanical system may be quite complex dynamically in the sense that over the range of frequencies of interest, it is necessary to consider a number of degrees of freedom in both the electrical part of the model and the mechanical part. This work sets out a structured and formal approach to the preparation of such models. Each different combination of voltage and slip is examined separately. In each case, the first step is to compute the steady-state reference solution for machine currents as a function of time. Then, the electro-magnetic behaviour of the electrical machine is linearised around that reference solution. The result is a linear time-dependent model for the electromagnetic behaviour which is then easily coupled with a linear model for the mechanical dynamics. The mechanical dynamics are usually stationary. Floquet methods can then be applied to determine whether the system is stable and the response of the system to mechanical or electrical perturbations can be computed quickly. The analysis method is applied to a particular three-phase induction machine which has parallel paths integrated into its winding structure in the sense that each of the phases is split into a "Wheatstone-bridge" arrangement following. Currents passing diametrically through a phase in the vertical direction account for the main torque-producing components of stator field. Currents passing diametrically through the phase in the horizontal direction account for transverse forces. The parallel paths can be switched to open-circuit or closed-circuit without affecting the torque-producing function of the machine and all of the stator conductors contribute to torque-production.For a number of combinations of voltage and slip, the machine is stable irrespective of whether the parallel paths are open-circuit or not but the effective damping of the machine for synchronous vibration is shown to be much higher with the parallel paths in closed-circuit.