**2.1. X-Y motors**

X-Y motors, also called planar motors, are the machines which are able to translate on a plane, moving in the direction defined by two space co-ordinates. They may be usefully employed for precision positioning in various manufacturing systems such as drawing devices or drive at switch point of guided road/e.g. railway. The representative of X-Y motors is shown schematically in Fig. 1. Primary winding consist of two sets of three phase windings placed perpendicularly to one another. Therefore, magnetic traveling fields produced by each winding are moving perpendicularly to one another as well. Secondary part can be made of non-magnetic conducting sheet (aluminum, copper) backed by an iron plate. The motor with a rotor rectangular grid-cage winding is another version that can be considered.

Induction Motors with Rotor Helical Motion 249

forces: linear (axially oriented) and rotary, which are the products of two magnetic fields and currents induced in the rotor. By controlling the supply voltages of two armatures independently, the motor can either rotate or move axially or can perform a helical motion.

**Figure 2.** Scheme of twin armature rotary-linear induction motor (Mendrela et al., 2003).

The last class of multi-DoMF motors has spherical structure. The rotor is able to turn around axis, which can change its position during the operation. Presently, such actuators are mainly proposed for pointing of micro-cameras and laser beams, in robotic, artificial vision, alignment and sensing applications (Bolognesi et al., 2004). In larger sizes, they may be also used as active wrist joints for robotic arms. Fig. 3 shows one of the designs in which the rotor driven by two magnetic fields generated by two armatures moving into two directions perpendicular to one another. This design is a counterpart of twin-armature rotary-linear

**Figure 3.** Construction scheme of twin-armature induction motor with spherical rotor (Mendrela et al., 2003).

**2.3. Spherical motors** 

motor.

**Figure 1.** Construction scheme of X-Y induction motor (Mendrela et al., 2003).

The forces produced by each of traveling fields can be independently controlled contributing to the control of both magnitude and direction of the resultant force. This in turn controls the motion direction of the X-Y motor.

## **2.2. Rotary-linear motors**

Mechanical devices with multiple degrees of freedom are widely utilized in industrial machinery such as boring machines, grinders, threading, screwing, mounting, etc. Among these machines those which evolve linear and rotary motion, independently or simultaneously, are of great interest. These motors, which are able effectively generate torque and axial force in a suitably controllable way, are capable of producing pure rotary motion, pure linear motion or helical motion and constitute one of the most interesting topologies of multi-degree-of freedom machines (Bolognesi et al., 2004). Some examples of such actuators have already been the subject of studies or patents (Mendrela et al., 2003, Giancarlo & Tellini, 2003, Anorad, 2001). A typical rotary-linear motor with twin-armature is shown in Fig. 2. A stator consists of two armatures; one generates a rotating magnetic field, another traveling magnetic field. A solid rotor, common for the two armatures is applied. The rotor consists of an iron cylinder covered with a thin copper layer. The rotor cage winding that looks like grid placed on cylindrical surface is another version that can be applied. The direction of the rotor motion depends on two forces: linear (axially oriented) and rotary, which are the products of two magnetic fields and currents induced in the rotor. By controlling the supply voltages of two armatures independently, the motor can either rotate or move axially or can perform a helical motion.

**Figure 2.** Scheme of twin armature rotary-linear induction motor (Mendrela et al., 2003).

### **2.3. Spherical motors**

248 Induction Motors – Modelling and Control

X-Y motors, also called planar motors, are the machines which are able to translate on a plane, moving in the direction defined by two space co-ordinates. They may be usefully employed for precision positioning in various manufacturing systems such as drawing devices or drive at switch point of guided road/e.g. railway. The representative of X-Y motors is shown schematically in Fig. 1. Primary winding consist of two sets of three phase windings placed perpendicularly to one another. Therefore, magnetic traveling fields produced by each winding are moving perpendicularly to one another as well. Secondary part can be made of non-magnetic conducting sheet (aluminum, copper) backed by an iron plate. The motor with

a rotor rectangular grid-cage winding is another version that can be considered.

**Figure 1.** Construction scheme of X-Y induction motor (Mendrela et al., 2003).

turn controls the motion direction of the X-Y motor.

**2.2. Rotary-linear motors** 

The forces produced by each of traveling fields can be independently controlled contributing to the control of both magnitude and direction of the resultant force. This in

Mechanical devices with multiple degrees of freedom are widely utilized in industrial machinery such as boring machines, grinders, threading, screwing, mounting, etc. Among these machines those which evolve linear and rotary motion, independently or simultaneously, are of great interest. These motors, which are able effectively generate torque and axial force in a suitably controllable way, are capable of producing pure rotary motion, pure linear motion or helical motion and constitute one of the most interesting topologies of multi-degree-of freedom machines (Bolognesi et al., 2004). Some examples of such actuators have already been the subject of studies or patents (Mendrela et al., 2003, Giancarlo & Tellini, 2003, Anorad, 2001). A typical rotary-linear motor with twin-armature is shown in Fig. 2. A stator consists of two armatures; one generates a rotating magnetic field, another traveling magnetic field. A solid rotor, common for the two armatures is applied. The rotor consists of an iron cylinder covered with a thin copper layer. The rotor cage winding that looks like grid placed on cylindrical surface is another version that can be applied. The direction of the rotor motion depends on two

**2.1. X-Y motors** 

The last class of multi-DoMF motors has spherical structure. The rotor is able to turn around axis, which can change its position during the operation. Presently, such actuators are mainly proposed for pointing of micro-cameras and laser beams, in robotic, artificial vision, alignment and sensing applications (Bolognesi et al., 2004). In larger sizes, they may be also used as active wrist joints for robotic arms. Fig. 3 shows one of the designs in which the rotor driven by two magnetic fields generated by two armatures moving into two directions perpendicular to one another. This design is a counterpart of twin-armature rotary-linear motor.

**Figure 3.** Construction scheme of twin-armature induction motor with spherical rotor (Mendrela et al., 2003).

$$B = B\_m \exp\left[j\left(\alpha t - \frac{\pi}{\tau\_o}\theta - \frac{\pi}{\tau\_z}z\right)\right] \tag{1}$$

$$\frac{F\_{\theta}}{F\_{\text{Z}}} = \text{ct} \, g \, a \; , \qquad F\_{\theta} = F \cos a \,\tag{2}$$

$$B(\mathbf{t}, \theta\_1, \mathbf{z}\_1) = B\_m \exp\left[j\left(\omega \mathbf{t} - \frac{\pi}{\mathbf{r}\_\theta} \theta\_1 - \frac{\pi}{\mathbf{r}\_\mathbf{z}} \mathbf{z}\_1\right)\right] = \nu ar \tag{4}$$

$$
\omega t - \frac{\pi}{\mathfrak{r}\_{\theta}} \theta\_1(t) - \frac{\pi}{\mathfrak{r}\_{\mathfrak{r}}} \mathbf{z}\_1(t) = \psi(t) \tag{5}
$$

$$
\omega - \frac{\pi}{\mathfrak{r}\_{\theta}} \omega\_{\theta} - \frac{\pi}{\mathfrak{r}\_{\mathfrak{z}}} u\_{\mathfrak{z}} = \omega\_{\mathfrak{z}} \tag{6}
$$

$$
\omega\_{1\theta} = 2\pi\_{\theta} f \quad , \qquad u\_{1\mathbf{z}} = 2\pi\_{\mathbf{z}} f \tag{7}
$$

$$
\omega - \frac{\omega}{\omega\_{1\theta}} \omega\_{\theta} - \frac{\omega}{u\_{1\varepsilon}} u\_{\varepsilon} = \omega\_{2} \tag{8}
$$

$$
\omega\_2 = \omega \text{s} \tag{9}
$$

$$\mathbf{s}\_{\theta \mathbf{z}} = \mathbf{1} - \frac{\omega\_{\theta}}{\omega\_{1\theta}} - \frac{u\_{\mathbf{z}}}{u\_{1\mathbf{z}}} \tag{10}$$

$$\mathbf{s}\_{\theta} = \mathbf{1} - \frac{\omega\_{\theta}}{\omega\_{1\theta}} \tag{11}$$

$$P\_m = mR\_2' \left. \frac{1 - s\_{\theta x}}{s\_{\theta x}} I\_2'^2 \right. \tag{12}$$

$$P\_m = m! I\_2'^2 \frac{\kappa\_2'}{s\_{\theta x}} \left(\frac{\omega\_\theta}{\omega\_{1\theta}} + \frac{u\_x}{u\_{1x}}\right) \tag{13}$$

$$P\_m = \,^p P\_{m\theta} + P\_{mz} \tag{14}$$

$$P\_{m\theta} + P\_{mz} = \ \ m R\_2' \ \frac{1 - s\_{\theta}}{s\_{\theta x}} l\_2'^2 + \ \ m R\_2' \ \frac{1 - s\_x}{s\_{\theta x}} l\_2'^2 \tag{15}$$

**Figure 9.** Equivalent circuit of rotor of rotary-linear induction motor with mechanical resistance split into two components.

### **3.3. Electromechanical characteristics**

Unlike conventional rotary motors with the curvy characteristics of electromechanical quantities versus slip, electromechanical quantities in rotary-linear motor cannot be interpreted in one dimensional shape and should be plotted in a surface profile as a function of either slip ��� or speed components ���������. The circumferential speed u� is expressed as follows:

$$
\mu\_{\theta} = R\_r \, . \,\omega\_{\theta} \tag{16}
$$

Induction Motors with Rotor Helical Motion 255

Fθ

Fz

F

u θ

� and F��

� *,* forces

FLθ

In order to determine the operating point of the machine set, let the rotor be loaded by two machines acting independently on linear (axial) and rotational directions with the load force

FLz

uz

Fl θ

**Figure 11.** Load characteristics for IM-2DoMF, F��: load force in axial direction, F�� : load force in rotary

The equilibrium of the machine set takes place when the resultant load force is equal in its absolute value and opposite to the force developed by the motor. The direction of the electromagnetic force F of the motor is constant and does not depend on the load. Thus, at steady state operation both load forces F�� and F�� acts against motor force components

*l z*

To draw both load characteristics on a common graph, the real load forces F�� and F�� acting

The transformed load characteristics drawn as a function of u� and u�, as shown in Fig. 12, are the surfaces intersecting one another along the line segment KA����. This line segment that forms the F�(u) characteristic is a set of points where the following equation is fulfilled:

> ' ' *l lz F F*

*lz*

*F*

sin *lz*

*F*

(17)

(18)

(19)

F� and F� in the same direction if the following relation between them takes place:

*z lz F F ctg F F*

 

separately on rotational and linear directions should be transformed into F��

cos *l*

*F*

, '

acting on the direction of the motor force *F*. These equivalent forces are:

'

*l*

*F*

characteristics shown in Fig. 11.

FLz

direction.

where R� is the rotor radius.

As an example, the force-slip characteristic of a typical rotary-linear motor is plotted in Fig. 10.

**Figure 10.** Fig. 10. Electromechanical characteristic of the induction motor with a rotating-travelling field (Mendrela et al., 2003).

In order to determine the operating point of the machine set, let the rotor be loaded by two machines acting independently on linear (axial) and rotational directions with the load force characteristics shown in Fig. 11.

254 Induction Motors – Modelling and Control

into two components.

as follows:

10.

where R� is the rotor radius.

field (Mendrela et al., 2003).

E0

**3.3. Electromechanical characteristics** 

X's2 R'2

**Figure 9.** Equivalent circuit of rotor of rotary-linear induction motor with mechanical resistance split

Unlike conventional rotary motors with the curvy characteristics of electromechanical quantities versus slip, electromechanical quantities in rotary-linear motor cannot be interpreted in one dimensional shape and should be plotted in a surface profile as a function of either slip ��� or speed components ���������. The circumferential speed u� is expressed

As an example, the force-slip characteristic of a typical rotary-linear motor is plotted in Fig.

**Figure 10.** Fig. 10. Electromechanical characteristic of the induction motor with a rotating-travelling

R'2 1 - s s z

P <sup>m</sup>

P mz

�� ������ �� (16)

R'2 1 - s<sup>z</sup> s z

**Figure 11.** Load characteristics for IM-2DoMF, F��: load force in axial direction, F�� : load force in rotary direction.

The equilibrium of the machine set takes place when the resultant load force is equal in its absolute value and opposite to the force developed by the motor. The direction of the electromagnetic force F of the motor is constant and does not depend on the load. Thus, at steady state operation both load forces F�� and F�� acts against motor force components F� and F� in the same direction if the following relation between them takes place:

$$\frac{F\_{\theta}}{F\_{z}} = \frac{F\_{l\theta}}{F\_{lz}} = c \text{tg}\,\alpha = \frac{\tau\_z}{\tau\_{\theta}}\tag{17}$$

To draw both load characteristics on a common graph, the real load forces F�� and F�� acting separately on rotational and linear directions should be transformed into F�� � and F�� � *,* forces acting on the direction of the motor force *F*. These equivalent forces are:

$$F\_{l\theta} = \frac{F\_{l\theta}}{\cos \alpha} \quad , \quad F\_{lz} = \frac{F\_{lz}}{\sin \alpha} \tag{18}$$

The transformed load characteristics drawn as a function of u� and u�, as shown in Fig. 12, are the surfaces intersecting one another along the line segment KA����. This line segment that forms the F�(u) characteristic is a set of points where the following equation is fulfilled:

$$
\vec{F\_{l\theta}} = \vec{F\_{lz}}\tag{19}
$$

**Figure 12.** Determination of the operating point A of the machine set with IM-2DoMF (Mendrela et al., 2003).

The load characteristic F�(u) intersects with the motor characteristic at points A and B, where the equilibrium of the whole machine set takes place. To check if the two points are stable the steady state stability criterion can be used, which is applied to rotary motors in the following form:

$$\frac{dF}{du} < \frac{dF\_l(\mu\_{\partial'}, \mu\_z)}{du} \tag{20}$$

Induction Motors with Rotor Helical Motion 257

(22)

machines and the traveling field of linear motors are in the mathematical description special cases of the rotating–traveling field. If the wave length remains steady, pole pitches along both axis (τ� and τ�) will vary by changing the motion direction of field waves. For example: if the wave front (see Fig. 4) turns to θ axis, then τz = ∞. This makes the formula (1) changes

which is the flux density function for rotary motor. The α = 0 and according to Eqn (2) the force F = Fθ what is the case for rotary motors. On the other hand, turning the wave completely toward z axis leads to infinity pole pitch value along θ axis (τ� = ∞). By inserting this into Eqn (1) and (10) the description of both field and slip expressed by two

In other word, the mathematical model of the rotary-linear motor is a general form of conventional, one dimensional motors and can be reduced at any time to the model either of

The twin-armature rotary-linear induction motor, which is the object of this chapter consists of two armatures what makes this machine a combination of two motors: rotary and tubular linear, whose rotor are coupled together. This implies that the phenomena that take place in each set of one-degree of mechanical freedom motors also occur in the twin armature rotarylinear motor in perhaps more complex form due to the complex motion of the rotor. One of these phenomena is called end effects and occurs due to finite length of the stator at rotor axial motion. This phenomenon is not present in conventional rotating induction machines,

These effects are the object of study of many papers (Yamamura, 1972, Greppe et all, 2008, Faiz & Jafari, 2000, Turowski, 1982, Gierczak & Mendrela, 1985, Mosebach et all, 1977, Poloujadoff et all, 1980). In the literature, end effects are taken into account in various ways. In the circuit theory a particular parameter can be separated from the rest of equivalent circuit elements, and it represents the only phenomena that are caused by finite length of primary part of linear motor. This approach has been done in (Pai et all, 1988, Gieras et all, 1987, Hirasa et all, 1980, Duncan & Eng, 1983, Mirsalim et all, 2002). Kwon et al, solved a linear motor (LIM) with the help of the FEM, and they suggested a thrust correction coefficient to model the end effects (Kwon et all., 1999). Fujii and Harada in (Fujii & Harada, 2000) modeled a rotating magnet at the entering end of the LIM as a compensator and reported that this reduced end effect and thrust was the same as a LIM having no end effects. They used FEM in their calculations. Another application of FEM in analysing LIMs is reported by (Kim & Kwon, 2006). A d-q axis equivalent model for dynamic simulation purposes is obtained by using nonlinear transient finite element analysis and dynamic end

exp *<sup>m</sup> BB j t*

space coordinates turns to the description of such quantities in linear motors.

**4. Edge effects in rotary-linear induction motors** 

but play significant role in linear motors.

to:

rotary or linear motors.

effects are obtained.

$$\frac{dF\_{l\theta}}{du\_{\theta}} > 0, \quad \frac{dF\_{lz}}{du\_z} > 0 \tag{21}$$

Applying this criterion, point A in Fig. 12 is stable and point B is unstable.

### **3.4. Conversion of mathematical model of IM-2DoMF into one of IM-1DoMF**

The mathematical model of IM-2DoMF presented in previous subsections is more general than the one for linear or rotating machines. The rotating magnetic field wave of rotating machines and the traveling field of linear motors are in the mathematical description special cases of the rotating–traveling field. If the wave length remains steady, pole pitches along both axis (τ� and τ�) will vary by changing the motion direction of field waves. For example: if the wave front (see Fig. 4) turns to θ axis, then τz = ∞. This makes the formula (1) changes to:

$$B = B\_m \exp\left[j\left(\alpha t - \frac{\pi}{\tau\_\vartheta} \theta\right)\right] \tag{22}$$

which is the flux density function for rotary motor. The α = 0 and according to Eqn (2) the force F = Fθ what is the case for rotary motors. On the other hand, turning the wave completely toward z axis leads to infinity pole pitch value along θ axis (τ� = ∞). By inserting this into Eqn (1) and (10) the description of both field and slip expressed by two space coordinates turns to the description of such quantities in linear motors.

In other word, the mathematical model of the rotary-linear motor is a general form of conventional, one dimensional motors and can be reduced at any time to the model either of rotary or linear motors.
