Direct current motors are widely used in a plethora of applications, ranging from industrial to modern electric (and intelligent) vehicle applications. Most recent operation methods of these motors involve drives that are designed based on an adequate knowledge of the motor dynamics and circulating currents. However, in spite of its simplicity, accurate discrete-time models are not always attainable when utilising the Euler method. Moreover, these inaccuracies may not be reduced when estimating the currents and rotor speed in sensorless direct current motors. In this paper, we analyse three discretisation methods, namely the Euler, second-order Taylor method and second-order Runge-Kutta method, applied to three common types of direct current motor: separately excited, series, and shunt. We also analyse the performance of two of the most simple Bayesian filtering methods, namely the Kalman filter and the extended Kalman filter. For the comparison of the models and the state estimation techniques, we performed several Monte Carlo simulations. Our simulations show that, in general, the Taylor and Runge-Kutta methods exhibit similar behaviours, whilst the Euler method results in less accurate models.