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

256 Induction Motors – Modelling and Control

F

2003).

following form:

F (u)

u1z uz

(20)

(21)

u θ

Fθ

Fz

F (u) = lz

F

Flz sin \*

F (u) = <sup>l</sup>θ

Flθ cos \*

u1θ

u

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

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

> (,) *l z dF dF u u du du*

 0 *lz z dF du*

0, *<sup>l</sup> dF du* 

**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

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

A

uθ

F (u) <sup>l</sup>

uz

K

B

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, but play significant role in linear motors.

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 effects are obtained.

The end effect has been also included in the analysis of rotary-linear motors in the literature (Mendrela et al., 2003, Krebs et all, 2008, Amiri et all, 2011). This inclusion was done by applying Fourier's harmonic method when solving the Maxwell's equations that describe motor mathematically (Mendrela et al., 2003). This approach was also applied to study the linear motor end effects (Mosebach et all, 1977, Poloujadoff et all, 1980).

Induction Motors with Rotor Helical Motion 259

IC

x

z

I B

IA

IC IB

IC

IB

I A

z

z

z

z

IA

F<sup>m</sup>

Bt + Ba

Bt + Ba

**Figure 13.** The process of generation of alternating magnetic field at different instances in 4-pole tubular motor: (Φa) – alternating component of magnetic flux, (Ba) – alternating component of magnetic flux density in the air-gap, (Bt) - travelling component of magnetic flux density in the air-gap, (J) – linear

current density of the primary part, (Fm) – magneto-motive force of primary part in the air-gap.

Analysing the above phenomenon in time, one may find that magnetic flux density has two components: Ba which does not move in space but changes periodically in time called alternating component and Bt which changes in time and space is called traveling

J

Secondary part

t1 = 0

(a)

(b)

(c)

(d)

(e)

t1 = 0

B

Bt

Fm

Bt

B

t2= 1/4 T

t3= ½ T

Ba

B

Ba

Bt


Primary part a

The edge effects phenomena caused by finite length of both armatures can be classified into two categories as follows:

