# Stepper Motor

Implement stepper motor model

**Libraries:**

Simscape /
Electrical /
Specialized Power Systems /
Electrical Machines

## Description

The Stepper Motor (STM) block implements a generic model that represents two most popular families of stepper motors:

Variable-reluctance stepper motors

Permanent-magnet or hybrid stepper motors

The Stepper Motor model consists of electrical and mechanical sections. The electrical section is represented by an equivalent circuit, configuration of which depends on the motor type. The equivalent circuits assume that the magnetic circuit is linear (no saturation) and the mutual inductance between phases is negligible. The mechanical section is represented by a state-space model based on inertia moment and viscous friction coefficient.

This figure shows the equivalent circuit for one phase in a variable-reluctance stepper motor.

In this model, *R _{a}* and

*L*(

_{a}*θ*), respectively, represent the resistance and the inductance of phase A winding. The winding inductance varies as a function of the rotor position:

L(_{a}θ) =
L_{0} +
L_{1}cos(N),_{r}θ | (1) |

where,

*L*_{0}is the average inductance.*L*_{1}is the maximum inductance variation.*N*is the rotor teeth number._{r}

At the reference position (*θ* = 0), the rotor tooth is fully aligned
with the A-axis pole to achieve the maximum A-phase winding inductance.

The total electromagnetic torque produced by the motor is the sum of the torques produced by the motor phases:

$${T}_{e}={\displaystyle \sum _{x=1}^{m}0.5{i}_{x}^{2}\frac{d{L}_{x}}{d\theta}},$$

where,

*m*is the number of phases.*i*is the winding current in phase_{x}*x*.*L*is the inductance function of phase_{x}*x*winding.

This figure shows the equivalent circuit for one phase in a permanent-magnet (PM) or hybrid stepper motor.

In this model, *R _{a}* and

*L*, respectively, represent the resistance and inductance of A-phase winding. Due to the large value of the air gap introduced by the magnets, the winding inductance of the PM or hybrid stepper motor can be considered to be independent of the rotor position. The voltage source

_{a}*e*(

_{a}*θ*) represents the motor back electromotive force (EMF), which is a sinusoidal function of the rotor position:

$${e}_{a}(\theta )=-p{\psi}_{m}\mathrm{sin}(p\theta )\frac{d\theta}{dt},$$

where,

*p*is the number of pole pairs. The number of pole pairs*p*is given by*p*=*Nr*/2.*ψ*is the motor maximum magnetic flux._{m}

Note that at the reference position (*θ* = 0), the north pole on the
rotor is fully aligned with the A-axis pole to achieve zero value of the A-phase back
EMF.

The electromagnetic torque produced by a two-phase PM or hybrid stepper motor is equal to the sum of the torque resulting from the interaction of the phase currents and magnetic fluxes created by the magnets and the detent torque, which results from the saliency of the rotor:

T =
–_{e}pψsin(_{m}i_{a}pθ)
–
pψsin(_{m}i_{b}pθ
– π/2) –
T_{dm}sin(mNrθ). | (2) |

where,

*m*is the number of phase (*m*=2) of the motor.*Nr*the number of teeth on the rotor (*Nr*= 2**p*).

### How to Get Stepper Motor Parameters

The parameters used in the stepper model are usually obtained from the manufacturer data sheets. In case the parameters are not available, you can determine them from experimental measurements.

**Variable-Reluctance Stepper Motor Parameters**

The parameters provided by manufacturer data sheets are usually: number of phases,
holding torque, step angle, voltage per phase, current per phase, winding resistance,
*R _{a}*, maximum inductance,

*L*, average inductance,

_{max}*L*, and rotor inertia,

_{0}*J*.

**Permanent-Magnet/Hybrid Stepper Motor Parameters**

The parameters provided by manufacturer data sheets are usually:

number of phases

holding torque

step angle

voltage per phase

current per phase

winding resistance,

*R*_{a}winding inductance,

*L*_{a}rotor inertia,

*J*

The maximum detent torque, *T _{dm}*, is not
always specified. This parameter can be assumed to be equal to 1-10% of the maximum
holding torque.

The maximum flux linkage, *ψ _{m}*, is not always
specified. This parameter can be obtained experimentally by driving the motor to a
constant speed,

*N*, in rpm, and by measuring the maximum open-circuit winding voltage,

*E*, in V.

_{m}The parameter ψ_{m} is then computed by the following
relation:

ψ =
(30/_{m}pπ)(E/_{m}N), | (3) |

where *p* is the number of pole pairs given by *p* =360 / (2*m*·*step*). Here *m* = phase number, *step* =
step angle in degrees.

## Examples

## Ports

### Input

### Output

### Conserving

## Parameters

## References

[1] T. Kenjo, A. Sugawara,
*Stepping Motors and Their Microprocessor Controls*, 2nd Edition,
Oxford University Press, Oxford, 2003.

[2] P. Acarnley, *Stepping
Motors - A guide to theory and practice*, 4th Edition, The Institution of
Electrical Engineers, London, 2002.

## Extended Capabilities

## Version History

**Introduced in R2008a**