*4.1.1. Static end effects*

This is the phenomenon which refers to the generation of alternating magnetic field in addition to the magnetic traveling field component. The process of generation of alternating magnetic field at different instances is shown in Fig. 13.

**Instant t1 = 0:** The 3-phase currents are of the values shown by phasor diagram in Fig. 13.b and 13.c with maximum in phase A. The current distribution in the primary winding relevant to these values is shown in Fig. 13.a and its first space harmonic is represented by curve J in Fig. 13.b. The distribution of magneto-motive force Fm in the air-gap corresponding to this linear current density has the cosine form with the maximum value at both edges of the primary part shown in Fig. 13.b. This mmf generates the magnetic flux which consists of two components: alternating flux Φa shown in Fig. 13.a and traveling flux component. The distribution of these two components Ba and Bt as well as the resultant flux density (Ba + Bt) are shown in Fig. 13.c.

**Instant t2 = ¼ T (where T is sine wave period):** The 3-phase currents are of value shown by phasor diagram in Fig. 13.d with zero in phase A. These currents make the distribution of the first harmonic of mmf Fm as shown in Fig. 13.d. Since there is no mmf at primary edges, the alternating flux component Φa does not occur and the traveling flux Bt is the only available component.

**Instant t3 = ½ T:** After a half period, the currents are of values shown by phasor diagram in Fig. 13.e with the maximum negative value in phase A. The relevant mmf distribution reveals its maximum negative value at primary edges which generates the magnetic flux Φa represented by its flux density Ba (see Fig. 13.e) which adds to the traveling flux component Bt.

two categories as follows:

**4.1. End effects** 

*4.1.1. Static end effects* 

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

The edge effects phenomena caused by finite length of both armatures can be classified into

One obvious difference between LIM and conventional rotary machines is the fact that in LIM the magnetic traveling field occurs at one end and disappears at another. This generates the phenomena called end effects. End effects can be categorized into two smaller groups

This is the phenomenon which refers to the generation of alternating magnetic field in addition to the magnetic traveling field component. The process of generation of alternating

**Instant t1 = 0:** The 3-phase currents are of the values shown by phasor diagram in Fig. 13.b and 13.c with maximum in phase A. The current distribution in the primary winding relevant to these values is shown in Fig. 13.a and its first space harmonic is represented by curve J in Fig. 13.b. The distribution of magneto-motive force Fm in the air-gap corresponding to this linear current density has the cosine form with the maximum value at both edges of the primary part shown in Fig. 13.b. This mmf generates the magnetic flux which consists of two components: alternating flux Φa shown in Fig. 13.a and traveling flux component. The distribution of these two components Ba and Bt as well as the resultant flux

**Instant t2 = ¼ T (where T is sine wave period):** The 3-phase currents are of value shown by phasor diagram in Fig. 13.d with zero in phase A. These currents make the distribution of the first harmonic of mmf Fm as shown in Fig. 13.d. Since there is no mmf at primary edges, the alternating flux component Φa does not occur and the traveling flux Bt is the only

**Instant t3 = ½ T:** After a half period, the currents are of values shown by phasor diagram in Fig. 13.e with the maximum negative value in phase A. The relevant mmf distribution reveals its maximum negative value at primary edges which generates the magnetic flux Φa represented by its flux density Ba (see Fig. 13.e) which adds to the traveling flux

linear motor end effects (Mosebach et all, 1977, Poloujadoff et all, 1980).

 End effects: which occurs in the tubular part of the motor. Transverse edge effects: which exists in the rotary part.

called: static end effects and dynamic end effects.

magnetic field at different instances is shown in Fig. 13.

density (Ba + Bt) are shown in Fig. 13.c.

available component.

component Bt.

**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 component. The first component Ba does not exist in motors with infinity long primary part, which is the case in conventional rotary machines.

Summarizing, the resultant magnetic flux density distribution is a combination of the traveling wave component B�(t) and alternating magnetic field B�(t) denoted by:

$$B(t, \mathbf{z}) = B\_t \cos\left(\omega t - \frac{\pi}{\tau\_p}\mathbf{z} + \delta\_m\right) + B\_a \sin\left(\omega t - \frac{\pi}{\tau\_p} + \delta\_a\right) \tag{23}$$

Induction Motors with Rotor Helical Motion 261

currents damp the magnetic field in the air-gap at entry in order to keep zero flux linkage for the secondary circuit. At the exit edge the secondary eddy currents tries to sustain the magnetic flux linkage outside the primary zone the same as it was before the exit. This leads to damping magnetic flux at the entry edge and to appearance of magnetic flux tail beyond the exit edge (Fig. 16.a). The distribution of the primary current J1 is uniform over the entire region. The envelop of the eddy currents induced in the secondary J2 shown in Fig. 16.a is

The eddy currents at the entry and exit edges attenuate due to the fact that the magnetic energy linked with these currents dissipate in the secondary resistance. Thus, the lower is the secondary resistance the more intensive is damping at entry and the longer is the tail

The currents are induced in the secondary over the entire primary length due to slip of the secondary with respect to the travelling component of primary magnetic flux. These currents superimpose the currents that are due to the entry and exit edges. The resultant eddy current envelop is shown schematically in Fig. 16.b. The flux density distribution in the air-gap and current density in the primary windings are also shown in Fig. 16.b. As it is illustrated the primary current density is uniformly distributed along the primary length only if the coils of each phase are connected in series and the symmetry of 3-phase currents is not affected by the end effects. The magnetic flux density distribution has the same shape and changes in a same pattern in both stages, but due to the rotor current reaction, the second stage has a lower magnetic flux density (B). However, primary current density is higher at the second

**Stage II:** rotor (secondary part) moves with a speed less than the synchronous speed

stage if the primary winding is supplied in these two cases with the same voltage.

<sup>u</sup> <sup>1</sup> <sup>B</sup> <sup>t</sup>

In general, end effect phenomena leads to non-uniform distribution of:

u 1

**Figure 15.** End effect explanation: (Bt) - travelling component of magnetic flux density in the air-gap

Primary

Secondary

2 3

Exit

relevant to the magnetic flux density distribution in the air-gap.

beyond the exit.

Entry

 magnetic field in the air-gap, current in the secondary, driving force density,

Thus, this contributes to: lower driving force, higher power losses,

power loss density in the secondary.

(u1) speed of traveling magnetic field (u) speed of the rotor.

where τp is pole pitch and δ� is phase angle.

**Figure 14.** The envelope of the resultant magnetic flux density in the air-gap of four pole linear motor due to presence of the alternating magnetic field (τp - pole pitch).

When only the travelling wave exists, the envelop of flux density distribution in the air gap is uniform over the entire length of the primary core but the second term deforms the air gap field distribution to the shape shown in Fig. 14. The alternating flux contributes to the rising of additional power losses in the secondary and to producing of braking force when one part of LIM motor is moving with respect to the other one (Mosebach et all, 1977, Poloujadoff, et all, 1980, Amiri, et all, 2011). This component occurs no matter what is the value of the speed of the secondary part (Poloujadoff, 1980). The envelope of the resultant magnetic flux density for the four-pole motor is no longer uniform as shown in Fig. 14.

### *4.1.2. Dynamic end effects*

The dynamic end effects are the entry and exit effects that occur when the secondary moves with respect to the primary part. This phenomenon will be explained in two stages:

### **Stage I:** secondary part moves with synchronous speed

There are no currents induced in the rotor (within the primary part range) due to traveling magnetic field component (since the secondary moves synchronously with travelling field). However, the observer standing on the secondary (see Fig. 15) feels relatively high change of magnetic flux when enters the primary part region and when leaves this region at exit edge. This change contributes to rising of the eddy currents at both the entry and exit edges. These currents damp the magnetic field in the air-gap at entry in order to keep zero flux linkage for the secondary circuit. At the exit edge the secondary eddy currents tries to sustain the magnetic flux linkage outside the primary zone the same as it was before the exit. This leads to damping magnetic flux at the entry edge and to appearance of magnetic flux tail beyond the exit edge (Fig. 16.a). The distribution of the primary current J1 is uniform over the entire region. The envelop of the eddy currents induced in the secondary J2 shown in Fig. 16.a is relevant to the magnetic flux density distribution in the air-gap.

The eddy currents at the entry and exit edges attenuate due to the fact that the magnetic energy linked with these currents dissipate in the secondary resistance. Thus, the lower is the secondary resistance the more intensive is damping at entry and the longer is the tail beyond the exit.

**Stage II:** rotor (secondary part) moves with a speed less than the synchronous speed

The currents are induced in the secondary over the entire primary length due to slip of the secondary with respect to the travelling component of primary magnetic flux. These currents superimpose the currents that are due to the entry and exit edges. The resultant eddy current envelop is shown schematically in Fig. 16.b. The flux density distribution in the air-gap and current density in the primary windings are also shown in Fig. 16.b. As it is illustrated the primary current density is uniformly distributed along the primary length only if the coils of each phase are connected in series and the symmetry of 3-phase currents is not affected by the end effects. The magnetic flux density distribution has the same shape and changes in a same pattern in both stages, but due to the rotor current reaction, the second stage has a lower magnetic flux density (B). However, primary current density is higher at the second stage if the primary winding is supplied in these two cases with the same voltage.

**Figure 15.** End effect explanation: (Bt) - travelling component of magnetic flux density in the air-gap (u1) speed of traveling magnetic field (u) speed of the rotor.

In general, end effect phenomena leads to non-uniform distribution of:


260 Induction Motors – Modelling and Control

which is the case in conventional rotary machines.

where τp is pole pitch and δ� is phase angle.

B

*4.1.2. Dynamic end effects* 

�(�� �) � �� ��� ��� � �

due to presence of the alternating magnetic field (τp - pole pitch).

**Stage I:** secondary part moves with synchronous speed

component. The first component Ba does not exist in motors with infinity long primary part,

Summarizing, the resultant magnetic flux density distribution is a combination of the

� � ���� �� ��� ��� � �

z τ<sup>p</sup> <sup>3</sup> τ <sup>p</sup>

**Figure 14.** The envelope of the resultant magnetic flux density in the air-gap of four pole linear motor

When only the travelling wave exists, the envelop of flux density distribution in the air gap is uniform over the entire length of the primary core but the second term deforms the air gap field distribution to the shape shown in Fig. 14. The alternating flux contributes to the rising of additional power losses in the secondary and to producing of braking force when one part of LIM motor is moving with respect to the other one (Mosebach et all, 1977, Poloujadoff, et all, 1980, Amiri, et all, 2011). This component occurs no matter what is the value of the speed of the secondary part (Poloujadoff, 1980). The envelope of the resultant magnetic flux density for the four-pole motor is no longer uniform as shown in Fig. 14.

The dynamic end effects are the entry and exit effects that occur when the secondary moves

There are no currents induced in the rotor (within the primary part range) due to traveling magnetic field component (since the secondary moves synchronously with travelling field). However, the observer standing on the secondary (see Fig. 15) feels relatively high change of magnetic flux when enters the primary part region and when leaves this region at exit edge. This change contributes to rising of the eddy currents at both the entry and exit edges. These

with respect to the primary part. This phenomenon will be explained in two stages:

��

� ��� (23)

traveling wave component B�(t) and alternating magnetic field B�(t) denoted by:

��

power loss density in the secondary.

Thus, this contributes to:


Due to dynamic end effects, the resultant magnetic flux density in the air-gap can be expressed as a summation of three flux density components as follows (Greppe, et all, 2008):

Induction Motors with Rotor Helical Motion 263

half-wave length is τ��. The B� wave is caused by the core discontinuity at the entry end and the B� wave is caused by the core discontinuity at the exit end, hence, both are called end effect waves. Both waves have an angular frequency ω, which is the same as that of power supply. They have the same half wave-length, which is different from half-wave length (equal to pole pitch) of the primary winding. The traveling speed of the end waves is given by �� = 2 f τ�� and is the same as the secondary speed if high speed motors is studied. However, in low speed motors, the speed of the end waves can be much higher than that of secondary (Yamamura, 1972). The length of penetration of entry end wave α� depends on motor parameters such as gap length and secondary surface resistivity. The impact of these parameters on α� are quiet different at high speed motors and low speed motors. As a result, α� is much longer at high speed motors with respect to low speed motors. Also, in the high-speed motors, half wave length τ�� is almost linearly proportional to the speed of secondary and is independent from gap length and secondary surface resistivity while it is dependent to such parameters at low speed motors (Yamamura, 1972). Therefore, the speed of the end waves is equal to the secondary speed at high speed motors regardless the value of parameters such as supply frequency, gap length and surface resistivity, while in low speed motors, end wave's speed depends on such parameters and may reach to even higher than synchronous speed at low slip region. The super-synchronous speed of the end-effect wave at motor speed lower and

close to synchronous speed occurs only in low speed motors (Yamamura, 1972).

Greppe, et all, 2008).

number of pole-pairs (Mendrela, 2004).

**4.2. Transverse edge effects** 

The entry-end-effect wave decays relatively slower than the exit-end-effect wave and unlike exit-end-effect wave, is present along the entire longitudinal length of the air-gap and degrades the performance of the high speed motor. The exit-end-effect wave attenuates much faster due to the lack of primary core beyond the exit edge. Therefore, the influence of the exit field component B� on motor performance is less than that of the entry component B�, and it may be disregarded for many applications Gieras et all, 1987, Hirasa et all, 1980,

For the motors with the number of magnetic pole pairs greater than 2 if the synchronous speed is below 10 m/s the end effects can be ignored. For the motors with higher synchronous speeds the influence of end effects can be seen even for the motors with higher

The transverse edge effect is generally described as the effect of finite width of the flat linear motor and is the result of x component of eddy current flowing in the solid plate secondary (Fig. 17.b). Since, there are no designated paths for the currents, as it is in cage rotors of rotary motors, the currents within the primary area are flowing in a circular mode (Fig. 17.b). These currents generate their own magnetic field, whose distribution is shown schematically in Fig. 17.a. This magnetic field shown schematically as Br in Fig. 18 subtracts from the magnetic field Bs generated by the primary part winding. The resultant field has non-uniform distribution in transverse direction (x axis) (Fig. 19). This

**Figure 16.** Distribution of primary current (J1), secondary current (J2) and magnetic flux density in the air-gap (B): (a) u =u1 (b) u < us.

$$B\left(t, z\right) = B\_t \cos\left(\alpha t - \frac{\pi}{\tau\_p} z + \left.\delta\_m\right| + \left.B\_1 e^{-z/a\_1} \cos\left(\alpha t - \frac{\pi}{\tau\_{pe}} z + \left.\delta\_1\right| + \left.B\_2 e^{+z/a\_2} \cos\left(\alpha t + \frac{\pi}{\tau\_{pe}} z + \left.\delta\_2\right|\right)\right|\right)\right) \tag{24}$$

All the three terms of this equation have the same frequency and are steady with respect to time t. The first term is the traveling wave moving forward at synchronous speed. The second term is an attenuating traveling wave generated at the entry end, which travels in the positive direction of z and whose attenuation constant is 1 α⁄ � and its half-wave length is τ��. The third term of Eqn (24) is an attenuating traveling wave generated at the exit end, which travels in the negative direction and whose attenuation constant is 1 α⁄ � and half-wave length is τ��. The B� wave is caused by the core discontinuity at the entry end and the B� wave is caused by the core discontinuity at the exit end, hence, both are called end effect waves. Both waves have an angular frequency ω, which is the same as that of power supply. They have the same half wave-length, which is different from half-wave length (equal to pole pitch) of the primary winding. The traveling speed of the end waves is given by �� = 2 f τ�� and is the same as the secondary speed if high speed motors is studied. However, in low speed motors, the speed of the end waves can be much higher than that of secondary (Yamamura, 1972). The length of penetration of entry end wave α� depends on motor parameters such as gap length and secondary surface resistivity. The impact of these parameters on α� are quiet different at high speed motors and low speed motors. As a result, α� is much longer at high speed motors with respect to low speed motors. Also, in the high-speed motors, half wave length τ�� is almost linearly proportional to the speed of secondary and is independent from gap length and secondary surface resistivity while it is dependent to such parameters at low speed motors (Yamamura, 1972). Therefore, the speed of the end waves is equal to the secondary speed at high speed motors regardless the value of parameters such as supply frequency, gap length and surface resistivity, while in low speed motors, end wave's speed depends on such parameters and may reach to even higher than synchronous speed at low slip region. The super-synchronous speed of the end-effect wave at motor speed lower and close to synchronous speed occurs only in low speed motors (Yamamura, 1972).

The entry-end-effect wave decays relatively slower than the exit-end-effect wave and unlike exit-end-effect wave, is present along the entire longitudinal length of the air-gap and degrades the performance of the high speed motor. The exit-end-effect wave attenuates much faster due to the lack of primary core beyond the exit edge. Therefore, the influence of the exit field component B� on motor performance is less than that of the entry component B�, and it may be disregarded for many applications Gieras et all, 1987, Hirasa et all, 1980, Greppe, et all, 2008).

For the motors with the number of magnetic pole pairs greater than 2 if the synchronous speed is below 10 m/s the end effects can be ignored. For the motors with higher synchronous speeds the influence of end effects can be seen even for the motors with higher number of pole-pairs (Mendrela, 2004).

## **4.2. Transverse edge effects**

262 Induction Motors – Modelling and Control

Due to dynamic end effects, the resultant magnetic flux density in the air-gap can be expressed as a summation of three flux density components as follows (Greppe, et all, 2008):

**Figure 16.** Distribution of primary current (J1), secondary current (J2) and magnetic flux density in the

<sup>1</sup> 1 2 <sup>2</sup> , cos cos cos *z z*

All the three terms of this equation have the same frequency and are steady with respect to time t. The first term is the traveling wave moving forward at synchronous speed. The second term is an attenuating traveling wave generated at the entry end, which travels in the positive direction of z and whose attenuation constant is 1 α⁄ � and its half-wave length is τ��. The third term of Eqn (24) is an attenuating traveling wave generated at the exit end, which travels in the negative direction and whose attenuation constant is 1 α⁄ � and

*B t z B t z Be t z Be t z*

*p pe pe*

 

 

(24)

1 2 / /

 lower motor efficiency, lower power factor.

air-gap (B): (a) u =u1 (b) u < us.

*t m*

 The transverse edge effect is generally described as the effect of finite width of the flat linear motor and is the result of x component of eddy current flowing in the solid plate secondary (Fig. 17.b). Since, there are no designated paths for the currents, as it is in cage rotors of rotary motors, the currents within the primary area are flowing in a circular mode (Fig. 17.b). These currents generate their own magnetic field, whose distribution is shown schematically in Fig. 17.a. This magnetic field shown schematically as Br in Fig. 18 subtracts from the magnetic field Bs generated by the primary part winding. The resultant field has non-uniform distribution in transverse direction (x axis) (Fig. 19). This non-uniform distribution of the magnetic field and circular pattern of the secondary currents contribute to the increase of power losses, decrease of motor efficiency and reduction of maximum electromagnetic force (Boldea & Nasar, 2001).

Induction Motors with Rotor Helical Motion 265

**Figure 19.** The resultant magnetic flux density distribution in the air-gap at different secondary

in the air-gap is damped, but at the exit edge it increases.

linear speed greater than zero (u > 0).

As the rotary-linear motor is concerned, the transverse edge effect occurs for rotary armature. This effect has here more complex form due to the additional axial motion of the rotor. The above transverse edge effects superimpose on entry and exit effects whose nature is the same as discussed earlier for linear part of the rotary-linear motor. This motion makes the flux density distribution distorted as shown in Fig. 20. At the entry edge the flux density

**Figure 20.** Resultant magnetic flux density in the air-gap of rotary part of the IM-2DoMF motor with

slips.

**Figure 17.** Transverse edge effect explanation: (a) The resultant magnetic flux distribution, (b) eddy current induced in the secondary.

**Figure 18.** The distribution of magnetic flux density Bs produced by the primary current and Br by the secondary currents.

current induced in the secondary.

secondary currents.

non-uniform distribution of the magnetic field and circular pattern of the secondary currents contribute to the increase of power losses, decrease of motor efficiency and

**Figure 17.** Transverse edge effect explanation: (a) The resultant magnetic flux distribution, (b) eddy

**Figure 18.** The distribution of magnetic flux density Bs produced by the primary current and Br by the

reduction of maximum electromagnetic force (Boldea & Nasar, 2001).

**Figure 19.** The resultant magnetic flux density distribution in the air-gap at different secondary slips.

As the rotary-linear motor is concerned, the transverse edge effect occurs for rotary armature. This effect has here more complex form due to the additional axial motion of the rotor. The above transverse edge effects superimpose on entry and exit effects whose nature is the same as discussed earlier for linear part of the rotary-linear motor. This motion makes the flux density distribution distorted as shown in Fig. 20. At the entry edge the flux density in the air-gap is damped, but at the exit edge it increases.

**Figure 20.** Resultant magnetic flux density in the air-gap of rotary part of the IM-2DoMF motor with linear speed greater than zero (u > 0).
