Stepper motor suitable for whole-, half- and micro-stepping representation

**Library:**Simscape / Electrical / Electromechanical / Reluctance & Stepper

The Stepper Motor block represents a stepper motor. It uses the input pulse trains, A and B, to control the mechanical output according to the following equations:

$${e}_{A}=-{K}_{m}\omega \mathrm{sin}({N}_{r}\theta )$$

$${e}_{B}={K}_{m}\omega \mathrm{cos}({N}_{r}\theta )$$

$$\frac{d{i}_{A}}{dt}=\left({v}_{A}-R{i}_{A}-{e}_{A}\right)/L$$

$$\frac{d{i}_{B}}{dt}=\left({v}_{B}-R{i}_{B}-{e}_{B}\right)/L$$

$$J\frac{d\omega}{dt}+B\omega ={T}_{e}$$

$${T}_{e}=-{K}_{m}\left({i}_{A}-\frac{{e}_{A}}{{R}_{m}}\right)\mathrm{sin}\left({N}_{r}\theta \right)+{K}_{m}\left({i}_{B}-\frac{{e}_{B}}{{R}_{m}}\right)\mathrm{cos}\left({N}_{r}\theta \right)-{T}_{d}\mathrm{sin}\left(4{N}_{r}\theta \right)$$

$$\frac{d\theta}{dt}=\omega $$

where:

*e*and_{A}*e*are the back electromotive forces (emfs) induced in the A and B phase windings, respectively._{B}*i*and_{A}*i*are the A and B phase winding currents._{B}*v*and_{A}*v*are the A and B phase winding voltages._{B}*K*is the motor torque constant._{m}*N*is the number of teeth on each of the two rotor poles. The_{r}**Full step size**parameter is (π/2)/*N*_{r}.*R*is the winding resistance.*L*is the winding inductance.*R*is the magnetizing resistance._{m}*B*is the rotational damping.*J*is the inertia.*ω*is the rotor speed.*Θ*is the rotor angle.*T*is the detent torque amplitude._{d}*T*is the electrical torque._{e}

If the initial rotor is zero or some multiple of
(π/2)/*N*_{r}, the rotor is aligned with the
phase winding of pulse A. This happens when there is a positive current flowing from the
**A+** to the **A-** ports and there is no current
flowing from the **B+** to the **B-** ports.

Use the Stepper Motor Driver block to create the pulse trains for the Stepper Motor block.

The Stepper Motor block produces a positive torque
acting from the mechanical **C** to **R** ports when
the phase of pulse A leads the phase of pulse B.

If you set the **Simulation mode** parameter to
`Averaged`

, both for a Stepper
Motor block and for the Stepper Motor
Driver block that controls it, then the individual steps are not
simulated. This can be a good way to speed up simulation. In Averaged mode, under
nonslipping conditions, the motor and driver are represented by a second-order
linear system that tracks the specified step rate. The demanded step rate is
determined directly from voltage across **A+** and
**A-**. So, for example, a voltage of +10 V across the
**A+** and **A-** terminals is interpreted as
a step rate demand of 10 steps per second. See the Stepper Motor Driver block reference page for more information
on how to connect the driver block to your step angle controller.

Averaged mode includes a slip estimator to predict whether the stepper motor would
have slipped if running in Stepping simulation mode. Slip is predicted if the motor
torque exceeds the **Vector of maximum torque values** parameter
value for longer than one step period, the step period being determined from the
current step rate demand. Upon detecting slip, the simulation will proceed or stop
with an error, according to the **Action on slipping** parameter
value. If you choose the action that lets the simulation continue, note that
simulation results may be incorrect. When slipping occurs, the torque generated by
the motor is not generally the maximum available torque; the maximum torque is only
achieved if the stepper controller detects slip and adjusts the step rate command
accordingly.

The dynamics of the equivalent second-order system are determined from the values
that you specify for the **Approximate total load inertia** and
**Maximum step rate command** parameters. It is important that
you set as accurate values as possible for these parameters, so that the step rate
command is tracked, and the block does not generate false slipping warnings or
errors.

If you run the motor in Averaged mode with the optional thermal ports exposed (see
Thermal Ports and Effects), then heat is added to the thermal ports, assuming that
the windings are always powered even when the step rate command is zero. The block
makes adjustments for half stepping and for reduced torque (and winding currents) at
higher speeds. For these adjustments to be correct, the **Vector of maximum
torque** parameter values must be correct. For half stepping, at zero
speed the heat generated by the block is the average of that generated when stopped
at a half step and at a full step.

To validate Averaged mode model configurations where you predict slip to occur, compare results with the same simulation performed in stepping mode.

The block has three optional thermal ports, one for each of the two windings
and one for the rotor. These ports are hidden by default. To expose the thermal
ports, right-click the block in your model, and then from the context menu
select **Simscape** > **Block
choices** > **Show thermal port**.
This action displays the thermal ports on the block icon, and adds the
**Temperature Dependence** and **Thermal
Port** tabs to the block dialog box. These tabs are described
further on this reference page.

Use the thermal ports to simulate the effects of copper resistance and iron losses that convert electrical power to heat. If you expose these ports, winding resistance is assumed linearly dependent on temperature, and is given by:

R = R
(1 + α (_{0}T – T
))_{0} | (1) |

where:

*R*is the resistance at temperature*T*.*R*is the resistance at the measurement (or reference) temperature_{0}*T*. Specify the reference temperature using the_{0}**Measurement temperature**parameter.α is the resistance temperature coefficient, which you specify with the

**Resistance temperature coefficients, [alpha_A alpha_B]**parameter. A typical value for copper is 0.00393/K.

The block calculates temperature of each of the windings and the rotor by

$$M\frac{dT}{dt}=Q$$

where

*M*is the thermal mass. Specify this value for the windings using the**Winding thermal masses, [M_A M_B]**parameter, and for the rotor using the**Rotor thermal mass**parameter.*T*is the temperature. Specify the initial values for the windings using the**Winding initial temperatures, [T_A T_B]**parameter, and for the rotor using the**Rotor initial temperature**parameter.*Q*is the heat flow, which is calculated from the iron losses of the windings:$$\begin{array}{l}{Q}_{A}={i}_{a}{}^{2}{R}_{A}\left(1-{\rho}_{m}/100\right)\\ {Q}_{B}={i}_{B}{}^{2}{R}_{B}\left(1-{\rho}_{m}/100\right)\\ {Q}_{R}={Q}_{A}\left({\rho}_{m}/100\right)+{Q}_{B}\left({\rho}_{m}/100\right)\end{array}$$

where

*ρ*is the percentage of magnetizing resistance associated with the rotor. Specify this percentage using the_{m}**Percentage of magnetizing resistance associated with the rotor**parameter.

The model is based on the following assumptions:

This model neglects magnetic saturation effects and any magnetic coupling between phases.

When you select the

**Start simulation from steady state**check box in the Simscape™Solver Configuration block, this block will not initialize an**Initial rotor angle**value between –π and π.To use Averaged mode, the Stepper Motor block must be directly connected to a Stepper Motor Driver block also running in Averaged mode.

The Averaged mode is an approximation, and exact step tracking compared to the Stepping mode should not be expected.

Slip detection in Averaged mode is approximate, and depends on a good estimate for load inertia and maximum step rate. Incorrect values may result in false slip detection.

When simulating slip in Averaged mode, it is assumed that the stepper motor controller adjusts the step rate command so as to achieve maximum possible torque.

[1] M. Bodson, J. N. Chiasson, R. T. Novotnak and R. B. Rekowski. “High-Performance Nonlinear Feedback Control of a Permanent Magnet Stepper Motor.” IEEE Transactions on Control Systems Technology, Vol. 1, No. 1, March 1993.

[2] P. P. Acarnley. *Stepping Motors: A Guide to
Modern Theory and Practice*. New York: Peregrinus, 1982.

[3] S.E. Lyshevski. *Electromechanical Systems,
Electric Machines, and Applied Mechatronics*. CRC, 1999.