| Paul Krause | Oleg Wasynczuk | Steven Pekarek | Timothy O'Connell |
The equivalent circuit of a dc machine with a stator field winding is shown below. We are assuming that commutation is ideal and the back voltage is a function of rotor speed. This is a good approximation when the rotor has many coils.
The voltage equations are $$\begin{split} \left[\begin{array}{c} v_f \\ v_a \end{array}\right] = \left[\begin{array}{cc} r_f+pL_{FF} & 0\\ \omega_rL_{AF} & r_a+pL_{AA} \end{array}\right] \left[\begin{array}{c} i_f\\ i_a\end{array}\right] \end{split}~~~~~~~~~~\text{(1)}$$ where the $\omega_rL_{AF}i_f$ term is the back voltage due to the commutator picking the maximum positive portion of the full-wave rectified voltage induced in the armature (rotor) windings. This term is generally written as $\omega_r k_v$ for a permanent-magnet field whereupon $$v_a = r_ai_a+pL_{AA}i_a+\omega_rk_v~~~~~~~~~~\text{(2)}$$ where $r_a$ and $L_{AA}$ are the resistance and self inductance of the armature windings, $k_v$ is the voltage constant and $\omega_r$ the rotor speed. The torque equation is $$T_e = k_vi_a.~~~~~~~~~~\text{(3)}$$ The relationship between torque and rotor speed is $$T_e = Jp\omega_r + B_m\omega_r+T_L.~~~~~~~~~~\text{(4)}$$ The steady-state equations for a permanent-magnet dc machine are $$V_a = r_aI_a+\omega_rk_v~~~~~~~~~~\text{(5)}$$ $$T_e = k_vI_a.~~~~~~~~~~\text{(6)}$$ A plot of the steady-state torque versus rotor speed $\omega_r$ is given below.
The DC Drive and Torque Control module follows. You can advance to it by clicking the corresponding link at either the top or bottom of this page. You can also go back to the Introduction using the corresponding links at those locations.