Question 8.6: Consider again the two-phase, permanent-magnet stepping moto......
Consider again the two-phase, permanent-magnet stepping motor of Example 8.5. Calculate the rotor position which will result if the phase currents are controlled to be sinusoidal functions of a reference angle θ_{ref} in the form
i_1=I_0 \cos θ_{ref}\\i_2=I_0 \sin θ_{ref}
Substitution of the current expressions into the torque expression of Example 8.5 gives
T_{mech} = T_0 (i_1 \cos θ_m + i_2 \sin θ_m) = T_0I_0 (\cos θ_{ref} \cos θ_m + \sin θ_{ref} sin θ_m)
Use of the trigonometric identity \cos (α – β) = \cos α \cos β + \sin α \sin β gives
T_{mech} = T_0 I_0 \cos (θ_{ref} – θ_m)
From this expression and using the analysis of Example 8.5, we see that the rotor equilibrium position will be equal to the reference angle, i.e., θ_m = θ_{ref}. In a practical implementation, a digital controller is likely to be used to increment θ_{ref} in finite steps, which will result in finite steps in the position of the stepping-motor.