Electromechanical systems are intricate configurations that effectively combine electrical and mechanical elements to achieve a desired outcome. Central to many of these systems is the DC motor, a device that converts electrical energy into mechanical motion, enabling various applications ranging from simple fans to complex robotic mechanisms.

A key component of the DC motor is the armature, a rotating circuit positioned within a magnetic field. As an electric current passes through the armature, it encounters a force due to the interaction with the magnetic field, producing torque. This torque initiates the rotation of the rotor, thus converting electrical energy into mechanical motion. The voltage induced in the armature is directly proportional to its speed, a phenomenon known as back electromotive force (EMF).

To analyze the behavior of a DC motor, we apply electrical principles to the armature circuit. By employing a loop equation and transforming it via the Laplace method, we can elucidate the relationship between the armature current (i_a), the applied armature voltage (V_a), and the back EMF (E_b). The equation is given by:

Where R_a represents the armature resistance and E_b represents the back EMF.

In the s domain, the torque (T) produced by the motor is directly proportional to the armature current, described by:

​Here, k_t is the torque constant. This torque can also be written in terms of the inertia (J) of the rotor:

By expressing the torque in terms of the angular position (θ) of the motor shaft and simplifying it, one can derive the transfer function. Assuming the armature inductance to be negligible compared to the armature resistance, the simplified transfer function of the DC motor becomes:

This transfer function provides a comprehensive understanding of the motor's dynamic response, linking the electrical input to the mechanical output and facilitating the design and control of electromechanical systems.