Unipolar Stepper Motor

Stepper motor with center taps on two-phase windings

Library:
Simscape / Electrical / Electromechanical / Reluctance & Stepper

Description

The Unipolar Stepper Motor block represents a stepper motor that has center taps on the two phase windings. The winding currents and mechanical output are defined by the following equations:

$e_{A +} = - K_{m} ω \sin (N_{r} θ)$

$e_{A -} = K_{m} ω \sin (N_{r} θ)$

$e_{B +} = K_{m} ω \cos (N_{r} θ)$

$e_{B -} = - K_{m} ω \cos (N_{r} θ)$

$\frac{d i_{A +}}{d t} = (v_{A +} - R i_{A +} - e_{A +}) / L$

$\frac{d i_{A -}}{d t} = (v_{A -} - R i_{A -} - e_{A -}) / L$

$\frac{d i_{B +}}{d t} = (v_{B +} - R i_{B +} - e_{B +}) / L$

$\frac{d i_{B -}}{d t} = (v_{B -} - R i_{B -} - e_{B -}) / L$

$J \frac{d ω}{d t} + B ω = T_{e}$

$T_{e} = - K_{m} (i_{A +} - i_{A -} - \frac{e_{A +} - e_{A -}}{R_{m}}) \sin (N_{r} θ) + K_{m} (i_{B +} - i_{B -} - \frac{e_{B +} - e_{B -}}{R_{m}}) \cos (N_{r} θ) - T_{d} \sin (4 N_{r} θ)$

$\frac{d θ}{d t} = ω$

where:

e_A+ is the back emf induced across the A+ to A0 half-winding.
e_A- is the back emf induced across the A- to A0 half-winding.
e_B+ is the back emf induced across the B+ to B0 half-winding.
e_B- is the back emf induced across the B- to B0 half-winding.
i_A+ is the current flowing from the A+ port to the A0 center tap port.
i_A- is the current flowing from the A- port to the A0 center tap port.
i_B+ is the current flowing from the B+ port to the B0 center tap port.
i_B- is the current flowing from the B- port to the B0 center tap port.
v_A+ is the voltage at the A+ port relative to the A0 center tap port.
v_A- is the voltage at the A- port relative to the A0 center tap port.
v_B+ is the voltage at the B+ port relative to the B0 center tap port.
v_B- is the voltage at the B- port relative to the B0 center tap port.
K_m is the motor torque constant.
N_r is the number of teeth on each of the two rotor poles. The Full step size parameter is (π/2)/N_r.
R is the half-winding resistance. For example, it is the resistance between A+ and A0 ports.
L is the half-winding inductance. For example, it is the inductance between A+ and A0 ports.
R_m is the magnetizing resistance.
B is the rotational damping.
J is the inertia.
ω is the rotor speed.
Θ is the rotor angle.
T_d is the detent torque amplitude.
T_e is the electrical torque.

If the initial rotor is zero or some multiple of (π/2)/N_r, the rotor is aligned with the A-phase winding. If a positive current flows from the A+ port to the A0 center tap port, then the stepper acts to stay aligned with the A-phase. Equivalently, a positive current flowing from the A0 center tap port to the A- port also acts on the rotor to stay aligned with the A-phase.

The Unipolar Stepper Motor block produces a positive torque acting from the mechanical C to R ports for either of the following sequences. Both sequences assume the rotor initial angle is zero or some multiple of (π/2)/N_r.

Sequence	Center taps connected to ground	Center taps connected to positive supply
1	Positive current from A+ to A0	Positive current from A0 to A-
2	Positive current from B+ to B0	Positive current from B0 to B-
3	Positive current from A- to A0	Positive current from A0 to A-
4	Positive current from B- to B0	Positive current from B0 to B-

Averaged Mode

If you set the Simulation mode parameter to Averaged, both for a Unipolar Stepper Motor block and for the Unipolar 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 non-slipping 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 +10V across the A+ and A- terminals is interpreted as a step rate demand of ten steps per second. See the Unipolar Stepper Motor Driver reference page for more information on how to connect the Unipolar Stepper Motor Driver 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 will not generally be 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), 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.

If you simulate or predict slip, MathWorks recommends that you do some validation runs comparing Stepping and Averaged modes before using the averaged model representation for simulation studies.

Thermal Ports

The block has five optional thermal ports, one for each of the four half-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 exposes the Temperature Dependence and Thermal Port parameters. These parameters 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. For more information on using thermal ports in actuator blocks, see Simulating Thermal Effects in Rotational and Translational Actuators.

Assumptions and Limitations

The 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.
All four half-windings are assumed to be identical, and therefore have the same resistance temperature coefficient, alpha, and the same thermal mass.
To use Averaged mode, the Unipolar Stepper Motor block must be directly connected to a Unipolar 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.

Ports

Conserving

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`A+` — Top A-phase electrical connection
electrical

Electrical conserving port associated with the top A-phase electrical connection.

`A0` — A-phase center tap connection
electrical

Electrical conserving port associated with the A-phase center tap connection.

`A-` — Lower A-phase electrical connection
electrical

Electrical conserving port associated with the lower A-phase electrical connection.

`B+` — Top B-phase electrical connection
electrical

Electrical conserving port associated with the top B-phase electrical connection.

`B0` — B-phase center tap connection
electrical

Electrical conserving port associated with the B-phase center tap connection.

`B-` — Lower B-phase electrical connection
electrical

Electrical conserving port associated with the lower B-phase electrical connection.

`C` — Case
mechanical

Mechanical rotational conserving port associated with the case.

`R` — Rotor
mechanical

Mechanical rotational conserving port associated with the rotor.

`HA+` — Thermal port for winding between A+ and A0
thermal

Thermal port for winding between A+ and A0. For more information, see Thermal Ports.

`HA-` — Thermal port for winding between A- and A0
thermal

Thermal port for winding between A- and A0. For more information, see Thermal Ports.

`HB+` — Thermal port for winding between B+ and B0
thermal

Thermal port for winding between B+ and B0. For more information, see Thermal Ports.

`HB-` — Thermal port for winding between B- and B0
thermal

Thermal port for winding between B- and B. For more information, see Thermal Ports.

`HR` — Rotor thermal port
thermal

Thermal port for rotor. For more information, see Thermal Ports.

Parameters

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Electrical Torque

`Simulation mode` — Simulation mode
`Stepping` (default) | `Averaged`

Select Stepping or Averaged. Use Averaged only if the block is connected directly to a Unipolar Stepper Motor Driver block also running in Averaged mode.