**2.3 Hybrid actuator**

Hybrid stepping motors generally consist of a toothed mobile fitted with permanent magnets. The **Figure 4** shows the structure of a hybrid motor, [9].

**Figure 2.** *Switched reluctance actuator with longitudinal flux configuration.*

**Figure 3.** *Switched reluctance actuator with modular structure.*

#### **Figure 4.**

*Hybrid linear stepper actuator.*


#### **Table 1.**

*Comparison of different linear actuators.*

This type of motor has both the advantages of the permanent magnet motor, which has a high torque, and those of the switched reluctance motor, which makes it possible to obtain a large number of steps per cycle. However, the iron losses are relatively large and therefore penalize this structure.

The movement of hybrid motors results from the superposition of the force developed by the reluctant effect of the teeth and the force created by the magnet.

The contribution of the amplitudes and the geometric periods of these forces makes it possible to achieve very diverse static characteristics. In fact, the magnet placed in the hybrid structure ensures a certain distribution of the field lines. The supply of the coils produces a switching phenomenon of the field lines more or less important depending on the intensity of the supply current by acting on the orientation of the fields it is possible to control the variation of the resulting force.

The **Table 1** gives a comparative study of the different configurations of the linear actuator studied above.

In what follows, our study will focus on switched reluctance actuators.

#### **3. Operating principal of switched reluctance actuator**

The force develop by a switched reluctance actuator is explained using the elementary principle of electromechanical energy conversion in a solenoid, as shown in **Figure 5**.

The switched reluctance machine belongs to the family of electromagnetic converters with single excitation.

Mechanical energy is produced by the displacement of a ferromagnetic material, placed in a magnetic field in order to maximize the flux in the circuit [10–13].

Generally, this type of actuator have only one degree of motion corresponding either to a translation or to a rotation around an axis, as shown in the figure.

**63**

*Incremental Linear Switched Reluctance Actuator DOI: http://dx.doi.org/10.5772/intechopen.96584*

relation to the stator is indicated by the angle

Where

**Figure 5.**

of the winding.

φ and λ

*Elementary circuit of a stepper motor.*

the moving part moves.

notch. Thus, we define:

governing its operation.

**4.1 The electrical model**

the derivative of flux linkages

The hatched parts represent the guiding of the moving part of which the position is identified by the distance x in the **Figure 5**. The position of the rotor in

> λ φ

totalized induction flux depending on the position and the supply current.

λ λ

**4. Analytic modeling of linear switched reluctance**

λ(*I,x*).

*j jj*

*U RI*

For a given position of the moving part, the magnetic circuit is the seat of a

To establish the equations governing the operation of the electromagnetic linear actuator, we consider the variations of the energy stored in the magnetic field when

The axis of a moving tooth is identified by its distance x with the axis of a fixed

Investigating the operational behavior of the switched reluctance actuator requires a mathematical model based on the electrical and mechanical equations

An elementary equivalent circuit for the switched reluctance actuator can be obtained by neglecting the mutual inductance between the phases. Assuming that each phase of the motor consists of a coil with resistance R and inductance *L(I, x*), the applied voltage U to a phase is equal to the sum of the resistive voltage drop and

θ

in the **Figure 5**.

= *n* (1)

= (*I x*, ) (2)

λ λ <sup>=</sup> Magnetic energy : − = <sup>∫</sup> (4)

denote respectively the flow through a turn and the total flow

*mag I cte W Id*λ <sup>=</sup> Magnetic energy : = ∫ (3)

*co cte co W dI*

*<sup>j</sup>*( )

= + , (5)

*d xI*

*dt* λ

*Incremental Linear Switched Reluctance Actuator DOI: http://dx.doi.org/10.5772/intechopen.96584*

**Figure 5.** *Elementary circuit of a stepper motor.*

*Computational Optimization Techniques and Applications*

**Type of actuator Switched reluctance** 

*Comparison of different linear actuators.*

linear actuator studied above.

converters with single excitation.

shown in **Figure 5**.

**actuator**

relatively large and therefore penalize this structure.

**Permanent magnet** 

**Hybrid actuator**

**actuator**

Resolution High resolution Medium Medium Thrust Force Weak force High force High force Operating frequency high frequency low frequency high frequency

This type of motor has both the advantages of the permanent magnet motor, which has a high torque, and those of the switched reluctance motor, which makes it possible to obtain a large number of steps per cycle. However, the iron losses are

The movement of hybrid motors results from the superposition of the force developed by the reluctant effect of the teeth and the force created by the magnet. The contribution of the amplitudes and the geometric periods of these forces makes it possible to achieve very diverse static characteristics. In fact, the magnet placed in the hybrid structure ensures a certain distribution of the field lines. The supply of the coils produces a switching phenomenon of the field lines more or less important depending on the intensity of the supply current by acting on the orientation of the fields it is possible to control the variation of the resulting force. The **Table 1** gives a comparative study of the different configurations of the

In what follows, our study will focus on switched reluctance actuators.

The force develop by a switched reluctance actuator is explained using the elementary principle of electromechanical energy conversion in a solenoid, as

The switched reluctance machine belongs to the family of electromagnetic

placed in a magnetic field in order to maximize the flux in the circuit [10–13].

either to a translation or to a rotation around an axis, as shown in the figure.

Mechanical energy is produced by the displacement of a ferromagnetic material,

Generally, this type of actuator have only one degree of motion corresponding

**3. Operating principal of switched reluctance actuator**

**62**

**Table 1.**

**Figure 4.**

*Hybrid linear stepper actuator.*

The hatched parts represent the guiding of the moving part of which the position is identified by the distance x in the **Figure 5**. The position of the rotor in relation to the stator is indicated by the angle θin the **Figure 5**.

Where φ and λ denote respectively the flow through a turn and the total flow of the winding.

$$
\mathcal{A} = n\phi \tag{1}
$$

For a given position of the moving part, the magnetic circuit is the seat of a totalized induction flux depending on the position and the supply current.

$$
\mathcal{A} = \mathcal{A}\left(I, \mathfrak{x}\right) \tag{2}
$$

To establish the equations governing the operation of the electromagnetic linear actuator, we consider the variations of the energy stored in the magnetic field when the moving part moves.

The axis of a moving tooth is identified by its distance x with the axis of a fixed notch. Thus, we define:

$$\mathbf{Magnetic energy} : \mathcal{W}\_{\text{mag}} = \left[Id\lambda\right]\_{\text{l-cte}} \tag{3}$$

$$\mathbf{Magnetic } co-\text{energy}: \mathcal{W}\_{co} = \left[\mathcal{\lambda} dI\right]\_{\mathbb{A}-cte} \tag{4}$$

#### **4. Analytic modeling of linear switched reluctance**

Investigating the operational behavior of the switched reluctance actuator requires a mathematical model based on the electrical and mechanical equations governing its operation.

#### **4.1 The electrical model**

An elementary equivalent circuit for the switched reluctance actuator can be obtained by neglecting the mutual inductance between the phases. Assuming that each phase of the motor consists of a coil with resistance R and inductance *L(I, x*), the applied voltage U to a phase is equal to the sum of the resistive voltage drop and the derivative of flux linkages λ(*I,x*).

$$\mathbf{U}\_{\rangle} = \mathbf{R}\_{\rangle} \mathbf{I}\_{\rangle} + \frac{d\boldsymbol{\lambda}\_{\rangle}(\mathbf{x}, \mathbf{I})}{dt} \tag{5}$$

j represents the index of the phase with j = 1, 2,3, indicates in order the phases A, B and C.

The flux linkage depends on the current and the translator position. Then the flux expression becomes:

$$\mathbf{U}\_{\rangle} = \mathbf{R}\_{\rangle}\mathbf{I}\_{\rangle} + \frac{\partial \lambda}{\partial \mathbf{I}} \frac{d\mathbf{I}\_{\rangle}}{dt} + \mathbf{I}\_{\rangle} \frac{\partial \lambda\_{\rangle}}{\partial \mathbf{x}} \frac{d\mathbf{x}}{dt} \tag{6}$$

By mean the magnetization curve **Figure 6**, the magnetic field energy can be determined for a fixed translator position as function of current and linkage flux.

$$\mathcal{W}\_{\rm co} = \bigwedge\_{\rm o} \lambda dI \tag{7}$$

The force developed by the switched reluctance linear actuator is proportional to the change in mechanical energy as a function of mechanical displacement. It can be given by:

$$F(I, \mathfrak{x}) = \frac{\partial W\_{\alpha}}{\partial \mathfrak{x}} \tag{8}$$

The total instantaneous electromagnetic force Ft is the sum of the q individual phase forces.

$$F\_{\varepsilon} \left( I \,\bigg|\mathcal{X}\right) = \sum\_{j=1}^{q} F \left( I \,\bigg|\mathcal{X}\right) \tag{9}$$

For a linear flux model of switched reluctance actuator it is λ ( *x LxI* ) = ( ) . Thus, neglecting magnetic saturation gives.

$$\mathcal{W}\_{co} = \bigwedge\_{0}^{l} dI = \frac{1}{2} L(\infty) I^2 \tag{10}$$

Hence,

$$F(I,\infty) = \frac{1}{2}I^2 \frac{dL}{dt} \tag{11}$$

**65**

*Incremental Linear Switched Reluctance Actuator DOI: http://dx.doi.org/10.5772/intechopen.96584*

> δ.

maximum when the teeth are in aligned position.

*x*

2

**5. Specificity of the control of linear actuator**

synchronized with the real position of the moving part.

of a phase.

follows:

**4.2 The mechanical model**

mechanical step of the actuator.

positioned opposite the latter.

(see Eq. (14)) [5].

dental pole pitch

The thrust force is then proportional to the derivative of the inductance with respect to the displacement of the mobile x and to the square of the supply current

The inductance *L (I, x*) is a periodic function of x, with a period equal to the

The inductance is minimal when the teeth are in unaligned position, and it is

The inductance depends on the moving position. The partial derivative of the

( ) *Lj <sup>x</sup> L j*

<sup>∂</sup> = − − − <sup>∂</sup> <sup>1</sup> 22 2 sin <sup>1</sup>

By replacing Eq. (12) in Eq. (11), the electromagnetic force developed by each

*F xI IL x j j j* ( ) ( ) ππ

12 2 2 , sin <sup>1</sup>

The mechanical movement of the actuator is described by the equation deduced from the fundamental principle of dynamics characterizing a linear movement,

> *vs l* ( ) ( ) *d x dx m D fs g v F F xI dt dt* + + +=

The special feature of the switched reluctance linear actuator is to ensure a continuous incremental translational movement. In other words, each supply pulse must correspond to a constant elementary displacement, this is correspond to the

Then, to ensure continuous movement, it is necessary on the one hand to have several phases, on the other hand, the successive supply of the phases must be

A determined number of pulses causes a corresponding number of steps by the actuator. In addition, the succession of determined pulses generated by a control circuit at a well determined frequency makes it possible to impose a continuous movement of the moving part at constant speed. At each pulse of the control, induce that the poles of the supplied phase closest to the stator poles, and they are

Like all electric motors Linear, actuators can be driven in open loop or closed loop for applications, which require high precision and high positioning quality.

The linear actuator is essentially an electric actuator, which requires an electronic power converter to change the operating frequency and the magnitude of the applied voltage. The main characteristics of electronic converters used for linear actuators generally require full operation of two quadrants (half H-bridge), a high switching voltage is necessary for the rapid establishment and extinction of the current.

 δ

<sup>=</sup> − −

δ

 δ  π

> π

2 3 (13)

<sup>2</sup> in , (14)

<sup>3</sup> (12)

ππ

phase of the separately supplied switched reluctance actuator is expressed as

2 1

inductance with respect to the moving position can be expressed by:

δ

**Figure 6.** *Magnetization curve of the studied actuator.*

*Incremental Linear Switched Reluctance Actuator DOI: http://dx.doi.org/10.5772/intechopen.96584*

*Computational Optimization Techniques and Applications*

B and C.

be given by:

phase forces.

Hence,

flux expression becomes:

j represents the index of the phase with j = 1, 2,3, indicates in order the phases A,

*j j*

λ

∂ ∂ (6)

<sup>∂</sup> <sup>=</sup> <sup>∂</sup> , (8)

, , (9)

λ

<sup>2</sup> (10)

*dt* <sup>=</sup> <sup>1</sup> <sup>2</sup> , <sup>2</sup> (11)

( *x LxI* ) = ( ) . Thus,

(7)

*I dt x dt*

The flux linkage depends on the current and the translator position. Then the

By mean the magnetization curve **Figure 6**, the magnetic field energy can be determined for a fixed translator position as function of current and linkage flux. *I W dI co* = λ∫ 0

The force developed by the switched reluctance linear actuator is proportional to the change in mechanical energy as a function of mechanical displacement. It can

( ) *Wco F Ix*

The total instantaneous electromagnetic force Ft is the sum of the q individual

( ) ( ) *<sup>q</sup>*

*j F Ix F Ix* = =∑1

( ) *<sup>I</sup> W dI L x I co* = = λ

( ) *dL F Ix I*

0

∫ <sup>2</sup>

1

*t*

For a linear flux model of switched reluctance actuator it is

neglecting magnetic saturation gives.

*x*

*j jj j dI dx U RI I*

=+ +

∂λ∂

**64**

**Figure 6.**

*Magnetization curve of the studied actuator.*

The thrust force is then proportional to the derivative of the inductance with respect to the displacement of the mobile x and to the square of the supply current of a phase.

The inductance *L (I, x*) is a periodic function of x, with a period equal to the dental pole pitch δ.

The inductance is minimal when the teeth are in unaligned position, and it is maximum when the teeth are in aligned position.

The inductance depends on the moving position. The partial derivative of the inductance with respect to the moving position can be expressed by:

$$\frac{\partial L\_j}{\partial \mathbf{x}} = -\frac{2\pi}{\delta} L\_1 \sin \left[ \frac{2\pi\mathbf{x}}{\delta} - (j - 1)\frac{2\pi}{3} \right] \tag{12}$$

By replacing Eq. (12) in Eq. (11), the electromagnetic force developed by each phase of the separately supplied switched reluctance actuator is expressed as follows:

$$F\_{\slash}\left(\infty, I\right) = \frac{1}{2} I\_{\slash}^2 L\_1 \frac{2\pi}{\delta} \sin\left(\frac{2\pi}{\delta} \varkappa - (j - 1)\frac{2\pi}{3}\right) \tag{13}$$

#### **4.2 The mechanical model**

The mechanical movement of the actuator is described by the equation deduced from the fundamental principle of dynamics characterizing a linear movement, (see Eq. (14)) [5].

$$m\frac{d^2\mathbf{x}}{dt^2} + D\_v\frac{d\mathbf{x}}{dt} + f\_i\text{sign}\left(\boldsymbol{\nu}\right) + F\_l = F\left(\mathbf{x}, I\right) \tag{14}$$

#### **5. Specificity of the control of linear actuator**

The special feature of the switched reluctance linear actuator is to ensure a continuous incremental translational movement. In other words, each supply pulse must correspond to a constant elementary displacement, this is correspond to the mechanical step of the actuator.

Then, to ensure continuous movement, it is necessary on the one hand to have several phases, on the other hand, the successive supply of the phases must be synchronized with the real position of the moving part.

A determined number of pulses causes a corresponding number of steps by the actuator. In addition, the succession of determined pulses generated by a control circuit at a well determined frequency makes it possible to impose a continuous movement of the moving part at constant speed. At each pulse of the control, induce that the poles of the supplied phase closest to the stator poles, and they are positioned opposite the latter.

Like all electric motors Linear, actuators can be driven in open loop or closed loop for applications, which require high precision and high positioning quality.

The linear actuator is essentially an electric actuator, which requires an electronic power converter to change the operating frequency and the magnitude of the applied voltage. The main characteristics of electronic converters used for linear actuators generally require full operation of two quadrants (half H-bridge), a high switching voltage is necessary for the rapid establishment and extinction of the current.

Generally the linear incremental actuators are controlled by a static converter in most applications, the force generated by each phase of the actuator is proportional to the square of the phase current (see Eq. (13)). The circuit and the control strategy are directly related to the performance and characteristics of the actuator. Several topologies are presented with a reduced number of power switches, faster excitation, faster demagnetization, high efficiency and high power through continuous research [14]. Conventionally, there has always been a trade-off between obtaining some advantages and losing others with each topology.

Assuming that the edge effects is neglect, then the variation of inductance is linear as shown in **Figure 7**. The characteristic of the inductance is periodic and the periodicity of the inductance is equal to 2 / π*q* , *q* number of phases.

The physical meaning of the different regions in **Figure 7** is as follows.

In the [*x1*-*x2*] zone, the inductance begins to increase as the mobile moves. When the poles of the moving part meet the stator poles, the inductance reaches its maximum value. In this region, the actuator operates in an increasing inductance regime where the slope of the inductance is positive where a positive force is developed by the actuator.

The area in the gap [*x2*-*x3*] the teeth of the mobile and the stator are completely aligned. In this interval, the inductance is constant and in this case the actuator cannot generate any force even if the phase supply is kept constant.

#### **5.1 Different control methods**

Incremental actuator supplies are generally classified into five modes [14]:

**Mode 1**: only one phase is supplied by the nominal current *I*n, in this case that the mechanical step of the actuator is defined, the phase supply sequences is shown in **Figure 8**.

**Mode 2:** two successive phases are supplied at the same time by the current. Indeed, the force is greater by a factor 2 than the first mode.

**Mode 3:** the alternating combination of the two previous modes allows operation in half-step, In this control mode, the phases of the actuator are supplied in order in accordance with the cyclogram shown in **Figure 9**.

**Mode 4**: this mode, commonly called "Ministepping" consists in multiplying the intermediate positions by supplying each phase with fractions of the nominal current, this corresponds to the extension of operation in mode 4.

**67**

*Incremental Linear Switched Reluctance Actuator DOI: http://dx.doi.org/10.5772/intechopen.96584*

**5.2 Drive circuit for switched reluctance actuator**

actuator converters (**Figure 10**) [14].

through both switches.

**Figure 8.** *Full step command.*

**Figure 9.**

*Half steps command.*

inverter components.

reluctance actuators because of its ability to operate efficiently.

Several application using linear switched Reluctance actuator using an adjustable speed drive. Asymmetric half bridge converter is very popular for switched

The power switch used in converter is a transistor. However, in industrial applications, the other types of power switches are used, mostly Thyristor, power IGBTs, or even MOSFETs. A dc voltage source is necessary for supplying the power converter and motor phases. The dc source may be from batteries or mostly a rectified ac supply with a filter to provide a dc input voltage source to the switched reluctance

First, consider that the phase *1* is supplied when the upper and lower transistors *T1* and *T4* are switched on. Then, *+V*DC voltage applied to the phase winding. Therefore, a current is established and increases in the windings of the phase

When the poles of the supplied phase reach the aligned position with the poles of the stator, in this case the switches are turned off. Phase current then slowly decreases by freewheeling through one transistor and one diode. When both transistors are off, the phase winding will supplied by voltage -VDC. Indeed, the phase current then quickly decreases through both diodes. By appropriately coordinating the above three switching states, phase current of the switched reluctance actuator can be controlled. The major advantages of the asymmetric bridge converter are the independent control of each motor phase and the relatively low voltage rating of the

By supplying the first phase maintains the translator in a stable equilibrium position, if the supply current of the first phase is cut off and if the second phase is supplied, then a positive force is developed by the second phase which moves the translator to the second equilibrium position. Conversely, if the excitation is

**Figure 7.** *Inductance and force curves as function of position.*

*Incremental Linear Switched Reluctance Actuator DOI: http://dx.doi.org/10.5772/intechopen.96584*

**Figure 8.** *Full step command.*

*Computational Optimization Techniques and Applications*

some advantages and losing others with each topology.

periodicity of the inductance is equal to 2 /

developed by the actuator.

**5.1 Different control methods**

in **Figure 8**.

Generally the linear incremental actuators are controlled by a static converter in most applications, the force generated by each phase of the actuator is proportional to the square of the phase current (see Eq. (13)). The circuit and the control strategy are directly related to the performance and characteristics of the actuator. Several topologies are presented with a reduced number of power switches, faster excitation, faster demagnetization, high efficiency and high power through continuous research [14]. Conventionally, there has always been a trade-off between obtaining

Assuming that the edge effects is neglect, then the variation of inductance is linear as shown in **Figure 7**. The characteristic of the inductance is periodic and the

π

The area in the gap [*x2*-*x3*] the teeth of the mobile and the stator are completely aligned. In this interval, the inductance is constant and in this case the actuator

Incremental actuator supplies are generally classified into five modes [14]: **Mode 1**: only one phase is supplied by the nominal current *I*n, in this case that the mechanical step of the actuator is defined, the phase supply sequences is shown

**Mode 2:** two successive phases are supplied at the same time by the current.

**Mode 3:** the alternating combination of the two previous modes allows operation in half-step, In this control mode, the phases of the actuator are supplied in

**Mode 4**: this mode, commonly called "Ministepping" consists in multiplying the intermediate positions by supplying each phase with fractions of the nominal

The physical meaning of the different regions in **Figure 7** is as follows. In the [*x1*-*x2*] zone, the inductance begins to increase as the mobile moves. When the poles of the moving part meet the stator poles, the inductance reaches its maximum value. In this region, the actuator operates in an increasing inductance regime where the slope of the inductance is positive where a positive force is

cannot generate any force even if the phase supply is kept constant.

Indeed, the force is greater by a factor 2 than the first mode.

order in accordance with the cyclogram shown in **Figure 9**.

current, this corresponds to the extension of operation in mode 4.

*q* , *q* number of phases.

**66**

**Figure 7.**

*Inductance and force curves as function of position.*

**Figure 9.** *Half steps command.*

#### **5.2 Drive circuit for switched reluctance actuator**

Several application using linear switched Reluctance actuator using an adjustable speed drive. Asymmetric half bridge converter is very popular for switched reluctance actuators because of its ability to operate efficiently.

The power switch used in converter is a transistor. However, in industrial applications, the other types of power switches are used, mostly Thyristor, power IGBTs, or even MOSFETs. A dc voltage source is necessary for supplying the power converter and motor phases. The dc source may be from batteries or mostly a rectified ac supply with a filter to provide a dc input voltage source to the switched reluctance actuator converters (**Figure 10**) [14].

First, consider that the phase *1* is supplied when the upper and lower transistors *T1* and *T4* are switched on. Then, *+V*DC voltage applied to the phase winding. Therefore, a current is established and increases in the windings of the phase through both switches.

When the poles of the supplied phase reach the aligned position with the poles of the stator, in this case the switches are turned off. Phase current then slowly decreases by freewheeling through one transistor and one diode. When both transistors are off, the phase winding will supplied by voltage -VDC. Indeed, the phase current then quickly decreases through both diodes. By appropriately coordinating the above three switching states, phase current of the switched reluctance actuator can be controlled. The major advantages of the asymmetric bridge converter are the independent control of each motor phase and the relatively low voltage rating of the inverter components.

By supplying the first phase maintains the translator in a stable equilibrium position, if the supply current of the first phase is cut off and if the second phase is supplied, then a positive force is developed by the second phase which moves the translator to the second equilibrium position. Conversely, if the excitation is

changed from phase *1* to phase *3* the force developed by phase 3 is negative, moving the translator in the negative direction to the phase *3* equilibrium position.

The **Figure 11** presents the characteristics of the forces developed by a threephase structure supplied separately. By applying a resistant force *F*l when phase 1
