# Mathematical Model of the PM Stepper Motor

## Model description:

The equations describing the stepped motor in the attached image are given as follows:

\begin{align*} \dot{x}_1 &= -K_1x_1 + K_2x_3\sin{(K_5x_4)} + u_1 \\ \dot{x}_2 &= -K_1x_2 + K_2x_3\sin{(K_5x_4)} + u_2 \\ \dot{x}_3 &= -K_3x_1\sin{(K_5x_4)} + K_3x_2\cos{(K_5x_4)} - K_4x_3 - K_6\sin{(4K_5x_4)} - \tau_L/J \\ \dot{x}_4 &= x_3, \end{align*}

where $K_1 = R/L$, $K_2 = K_m/L$, $K_3=K_m/J$, $K_4 = B/J$, $K_5 = N_r$, $K_6 = K_D/J$, $K_5 = N_r$, $K_6 = K_D/J$, $u_1=v_a/L$.

 $i_a, i_b$, and $v_a, v_b$ currents and voltages in phase $A$ and $B$, respectively. $L$ and $R$ self-inductance and resistance of each phase winding $K_m$ motor torque constant $N_r$ number of rotor teeth $J$ rotor inertia $B$ vicious friction constant $\omega$ rotor speed $\theta$ motor position $\tau_L$ Load torque

The term $K_D\sin{(4N_r\theta)}$ represents the detent torque due to the permanent rotor magnet interacting with the magnetic materia of the stator poles. $K_D$ is typically 5% to 10% of the value of $K_mi_0$, where $i_0$ is the rated current.

The state variables $x_1, x_2, x_3$ and $x_4$ are assigned by $x^{\mathrm T}=[i_a, i_b, i_c, i_d]^{\mathrm T}.$

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## Publication details:

 Title Position Control of a PM Stepper Motor by Exact Linearization Publication Type Journal Article Authors Zribi, M., and Chiasson J.