Since dc motors of various types are used extensively in control systems, it is essential for analytical purposes that we establish a mathematical model for the dc motor. Let us consider the case of a separately excited dc motor with constant field excitation. The schematic representation of the model of a dc motor is shown in Figure. We will investigate how the speed of the motor responds to changes in the voltage applied to the armature terminals. The linear analysis involves electrical transients in the armature circuit and the dynamics of the mechanical load driven by the motor. At a constant motor field current If, the electromagnetic torque and the generated emf are given by

T_e = K_mi_a

e_a = K_mω_m

where K_m = kI_f is a constant, which is also the ratio e_a/ω_m. In terms of the magnetization curve, e_a is the generated emf corresponding to the field current If at the speed ωm. Let us now try to find the transfer function that relates m(s) to V_t(s).