Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis

*Sergey Gandzha and Dmitry Gandzha*

## **Abstract**

An analysis of electric machines with axial magnetic flux is given. First, the effect of commutation on the electromagnetic moment and electromagnetic power is analyzed. Two types of discrete switching are considered. The analysis is performed for an arbitrary number of phases. The first type of switching involves disabling one phase for the duration of switching. The second type of switching involves the operation of all phases in the switching interval. The influence of the pole arc and the number of phases on the electromagnetic moment and electromagnetic power is investigated. The conclusion is made about the advantage of the second type of switching. It is recommended to increase the number of phases. Next, the classification of the main structures of the axial machine is carried out. Four main versions are defined. For each variant, the equation of the electromagnetic moment and electromagnetic power is derived. This takes into account the type of commutation. The efficiency of the selected structures is analyzed. The comparative analysis is tabulated for choosing the best option. The table is convenient for engineering practice. This chapter forms the basis for computer-aided design of this class of machines.

**Keywords:** brushless electric machine, axial gap electric machine, discrete commutation, multiphase electric machine, electromagnetic moment, electromagnetic power, permanent magnet, cylindrical magnet, segment magnet, diamagnetic anchor

#### **1. Introduction**

Low-and medium-power electric drives based on brushless electric machines are widely used both in industrial applications and in special-purpose products (space, medicine, robotics). Traditionally, brushless electric machines with a radial magnetic flux are used for this purpose. This is due to the good specific energy indicators of these electric machines, well-established technology of their production [1–7].

Recently, brushless electric machines with axial magnetic flux (BMAMF) have been increasingly used for these purposes. These electric machines are actively developing, and we can talk about the formation of a new class of brushless electric drives that are competitive with traditional brushless electric drives. There is a process of transition from the design of individual products to the development of an industrial range of electric machines of this type. International and domestic practice confirms this trend [8–12].

The following reasons can explain the active introduction of electric machines of this class into production:


microelectronics that can implement control algorithms of almost any complexity. However, it is necessary to recognize the great complexity of implementing this type

In contrast to analog switching, in discrete switching, electronic keys on the inter-switching interval have only two States: on and off. Discrete switching by the nature of the current is divided into one-half-period switching, when the current in the phase sections flows in only one direction, and two-half-period switching, when

Single-half-period switching is simple enough to implement, since it requires only one key per switched phase. However, the armature phase is connected to the source only in the zone of polarizing which creates a positive electromagnetic moment. After that, the phase is in the disconnected state without generating electromagnetic torque. To maintain the required torque, it is necessary to increase the linear load on the connected phases, which leads to increased losses and reduced efficiency. Due to the worst energy performance, Single-half-period switching is

Types of two-half-period switching are distinguished by the time the phase is connected in the inter-switching interval. A distinction should be made between

In the future, we will consider 180-degree switching and 180-(180/m)-degree

At present, there is a steady trend towards an increase in the number of phases

• indicators that characterize the quality of output parameters are improved, in

Increased reliability is a determining factor when the number of phases increases. According to the method of connecting multiphase windings to each other, three

particular, the pulsation of the electromagnetic moment is reduced;

options are possible: in the "star", in the "ring", with independent connection

If the number of phases is more than three, when connected to a "star", the current flows only through the phases that have the highest and lowest potential on

The connection of the winding to the ring is rarely used due to the increased current through the keys. When connecting in a "star" and "ring", it is difficult to

Galvanically isolated phases require a large number of power keys (four keys per phase), but they provide the greatest reliability in case of open and short circuit

of switching. For this reason, it is not considered in the work.

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis*

*The classification of the main types of switching of BMAMF.*

*DOI: http://dx.doi.org/10.5772/intechopen.95945*

180-degree switching and 180-(180/m) degree switching.

of the anchor winding. This is due to the following factors:

• improved energy performance, in particular efficiency;

• increased reliability in case of failures of one or more phases.

the switched keys. This reduces the power of the multiphase BMAMF.

the current flows in both directions.

also excluded from further analysis.

(galvanically isolated phases).

ensure high reliability in case of phase failures.

switching.

**9**

**Figure 1.**

It should be noted that, despite the urgent need for practical implementation, theoretical research on the analysis and synthesis of electric machines of this class is episodic and scattered. As a rule, developers analyze one design for a special drive. The results of these studies are quite difficult to transform into another constructive type. The influence of the electronic switch on the engine characteristics has not been fully studied.

Recently, there has been a tendency to increase the number of phases to improve reliability [13]. At the same time, the switching theory for multiphase BMAMF execution is not sufficiently covered in scientific publications and requires further improvement and development.

There is no unified theory for calculating electric machines of this class that would link electromagnetic power with electromagnetic loads and basic dimensions, taking into account design features.

Thus, the existing contradiction between the practical need for implementation and the insufficiently developed theory of analysis and synthesis is the main source of further development of electric machines of this class, which determines the relevance of scientific research in this area.

### **2. Analysis of the effect of commutation on the electromagnetic moment for any number of phases of the anchor winding for BMAMF**

The winding of the brushless electric machine is connected to the power source via a commutator. Its principle of operation is as follows: when the rotor is turned, those sections of the armature winding are connected to the source, which are most profitable to pass current through from the point of view of creating an electromagnetic moment.

Switching the valve machine should be understood as connecting and disconnecting the phases of the armature winding with electronic keys to the power source. The main characteristics of the electric machine depend on the choice of switching method: moment, power consumption and useful power, efficiency, current.

The classification of the main types of switching of brushless machines is shown in **Figure 1**. Analog switching implements vector control of the brushless machine. Vector control is a control method that generates harmonic phase currents and controls the magnetic flux of the rotor. Currently, vector control systems are well developed in theory and implemented in practice. They have a wide range of applications, due to the development of power electronics, which allows you to create reliable and relatively cheap converters, as well as the development of high-speed

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis DOI: http://dx.doi.org/10.5772/intechopen.95945*

#### **Figure 1.**

The following reasons can explain the active introduction of electric machines of

• at present, the industrial production of powerful magnets with high values of residual induction and coercive force has been intensively developed, which allowed to concentrate the energy of the magnetic field in small volumes and

• modern development of computing tools and special software allows you to optimize the geometry of BMAMF for efficient use of the volume occupied by them. At the same time, optimally designed BMAMF under conditions of limited size can have better specific weight size and energy indicators

• modern technologies allow to create BMAMF more economical to manufacture

It should be noted that, despite the urgent need for practical implementation, theoretical research on the analysis and synthesis of electric machines of this class is episodic and scattered. As a rule, developers analyze one design for a special drive. The results of these studies are quite difficult to transform into another constructive type. The influence of the electronic switch on the engine characteristics has not been fully studied. Recently, there has been a tendency to increase the number of phases to improve

reliability [13]. At the same time, the switching theory for multiphase BMAMF execution is not sufficiently covered in scientific publications and requires further

**2. Analysis of the effect of commutation on the electromagnetic**

profitable to pass current through from the point of view of creating an

Switching the valve machine should be understood as connecting and disconnecting the phases of the armature winding with electronic keys to the power source. The main characteristics of the electric machine depend on the choice of switching method: moment, power consumption and useful power, efficiency, current. The classification of the main types of switching of brushless machines is shown in **Figure 1**. Analog switching implements vector control of the brushless machine. Vector control is a control method that generates harmonic phase currents and controls the magnetic flux of the rotor. Currently, vector control systems are well developed in theory and implemented in practice. They have a wide range of applications, due to the development of power electronics, which allows you to create reliable and relatively cheap converters, as well as the development of high-speed

There is no unified theory for calculating electric machines of this class that would link electromagnetic power with electromagnetic loads and basic dimen-

Thus, the existing contradiction between the practical need for implementation and the insufficiently developed theory of analysis and synthesis is the main source of further development of electric machines of this class, which determines the

**moment for any number of phases of the anchor winding for BMAMF**

The winding of the brushless electric machine is connected to the power source via a commutator. Its principle of operation is as follows: when the rotor is turned, those sections of the armature winding are connected to the source, which are most

this class into production:

reduce the size of electric machines;

*Emerging Electric Machines - Advances, Perspectives and Applications*

compared to radial electric machines;

and reliable in operation.

improvement and development.

electromagnetic moment.

**8**

sions, taking into account design features.

relevance of scientific research in this area.

*The classification of the main types of switching of BMAMF.*

microelectronics that can implement control algorithms of almost any complexity. However, it is necessary to recognize the great complexity of implementing this type of switching. For this reason, it is not considered in the work.

In contrast to analog switching, in discrete switching, electronic keys on the inter-switching interval have only two States: on and off. Discrete switching by the nature of the current is divided into one-half-period switching, when the current in the phase sections flows in only one direction, and two-half-period switching, when the current flows in both directions.

Single-half-period switching is simple enough to implement, since it requires only one key per switched phase. However, the armature phase is connected to the source only in the zone of polarizing which creates a positive electromagnetic moment. After that, the phase is in the disconnected state without generating electromagnetic torque. To maintain the required torque, it is necessary to increase the linear load on the connected phases, which leads to increased losses and reduced efficiency. Due to the worst energy performance, Single-half-period switching is also excluded from further analysis.

Types of two-half-period switching are distinguished by the time the phase is connected in the inter-switching interval. A distinction should be made between 180-degree switching and 180-(180/m) degree switching.

In the future, we will consider 180-degree switching and 180-(180/m)-degree switching.

At present, there is a steady trend towards an increase in the number of phases of the anchor winding. This is due to the following factors:


Increased reliability is a determining factor when the number of phases increases. According to the method of connecting multiphase windings to each other, three options are possible: in the "star", in the "ring", with independent connection (galvanically isolated phases).

If the number of phases is more than three, when connected to a "star", the current flows only through the phases that have the highest and lowest potential on the switched keys. This reduces the power of the multiphase BMAMF.

The connection of the winding to the ring is rarely used due to the increased current through the keys. When connecting in a "star" and "ring", it is difficult to ensure high reliability in case of phase failures.

Galvanically isolated phases require a large number of power keys (four keys per phase), but they provide the greatest reliability in case of open and short circuit

**Figure 2.** *Representation of air gap induction and linear load.*

failures. Taking into account the development trends of power electronics for the production of hybrid assemblies, we choose to focus on the study of this type of connection of multiphase BMAMF windings.

Thus, two-half-period (180-(180/m)-degree switching and two-half-period 180 degree switching with galvanically isolated phases are selected for further analysis.

For the selected switching types, we will carry out the further analysis.

#### **2.1 Analysis of 180-(180/m) - degree commutation for BMAMF with any phases**

To analyze the commutation, we determine the interaction of amperes-turns with the magnetic field of permanent magnets at different positions of the armature and inductor relative to each other. In this case, a trapezoidal one with an equivalent amplitude replaces the actual distribution of induction in the air gap [13].

By the pole overlap coefficient, we mean the ratio:

$$a = \frac{b\_m}{\tau},\tag{1}$$

*di* – elementary section current; *dr* – length of an elementary section.

*DOI: http://dx.doi.org/10.5772/intechopen.95945*

the pole division in electrical degrees;

*The diagram for the (180–180/m)-degree commutation.*

within the pole division.

**Figure 3.**

**Figure 4.**

**11**

*Model for axial gap machine.*

For an elementary moment, we can write the equation

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis*

where *r* – radius of the elementary section location.

where *αel* – angular coordinate in electrical degrees.

Imagine the induction as the product of a base value equal to the amplitude of the induction in the air gap and the relative function of the change in the induction

By analogy with induction, we express the linear load function on the average radius of the disk of a generalized axial machine depending on the pole division

where *Asr* - the amplitude of the linear load on the average radius of the disk. *Ai αel* ð Þ , *x* – relative function of the linear load change for the i-th phase within

*x –* offset of the beginning of the first phase relative to the neutral. Given that there is a relationship between geometric and electrical degrees

*dM* ¼ *rdF* ¼ *Brdidr*, (3)

*B* ¼ *BδB*ð Þ *αel* , (4)

*A x*ð Þ¼ *AsrAi αel* ð Þ , *x* , (5)

where *bm*- width of the pole;

*τ* - distance between of the neutrals.

The linear load of the phase conductors is represented as rectangles, equal in width to the phase zone, and equal in amplitude to the average linear load of the phase (**Figure. 2**). The analysis is carried out for relative values, taking the base value of the induction amplitude and the average linear load on the average diameter of the axial machine. This representation of an electric machine for analysis should be recognized as traditional.

**Figure 3** shows a diagram of the positions of the amperes of the turns in the inter-switching interval for the two pole divisions and the moments of connection and disconnection of the corresponding phases.

We derive the equation of the electromagnetic moment for a generalized axial machine, which is a disk with a distributed current layer that is permeated by a magnetic flux (**Figure 4**).

The equation of an elementary electromagnetic force acting on an infinitesimal section of a generalized axial machine can be written in the following form based on Ampere's law.

$$dF = \text{Biddr} \tag{2}$$

where *B* – elementary section induction;

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis DOI: http://dx.doi.org/10.5772/intechopen.95945*

*di* – elementary section current; *dr* – length of an elementary section. For an elementary moment, we can write the equation

$$d\mathcal{M} = r dF = B rid dr,\tag{3}$$

where *r* – radius of the elementary section location.

Imagine the induction as the product of a base value equal to the amplitude of the induction in the air gap and the relative function of the change in the induction within the pole division.

$$B = B\_{\delta} B(a\_{\mathrm{el}}),\tag{4}$$

where *αel* – angular coordinate in electrical degrees.

By analogy with induction, we express the linear load function on the average radius of the disk of a generalized axial machine depending on the pole division

$$A(\mathbf{x}) = A\_{sr} A\_i(\alpha\_{el}, \mathbf{x}),\tag{5}$$

where *Asr* - the amplitude of the linear load on the average radius of the disk. *Ai αel* ð Þ , *x* – relative function of the linear load change for the i-th phase within the pole division in electrical degrees;

*x –* offset of the beginning of the first phase relative to the neutral.

Given that there is a relationship between geometric and electrical degrees

#### **Figure 3.**

failures. Taking into account the development trends of power electronics for the production of hybrid assemblies, we choose to focus on the study of this type of

Thus, two-half-period (180-(180/m)-degree switching and two-half-period 180 degree switching with galvanically isolated phases are selected for further analysis. For the selected switching types, we will carry out the further analysis.

**2.1 Analysis of 180-(180/m) - degree commutation for BMAMF with any phases**

To analyze the commutation, we determine the interaction of amperes-turns with the magnetic field of permanent magnets at different positions of the armature and inductor relative to each other. In this case, a trapezoidal one with an equivalent

*<sup>α</sup>* <sup>¼</sup> *bm*

The linear load of the phase conductors is represented as rectangles, equal in width to the phase zone, and equal in amplitude to the average linear load of the phase (**Figure. 2**). The analysis is carried out for relative values, taking the base value of the induction amplitude and the average linear load on the average diameter of the axial machine. This representation of an electric machine for analysis

**Figure 3** shows a diagram of the positions of the amperes of the turns in the inter-switching interval for the two pole divisions and the moments of connection

We derive the equation of the electromagnetic moment for a generalized axial machine, which is a disk with a distributed current layer that is permeated by a

The equation of an elementary electromagnetic force acting on an infinitesimal section of a generalized axial machine can be written in the following form based on

*<sup>τ</sup>* , (1)

*dF* ¼ Bdidr (2)

amplitude replaces the actual distribution of induction in the air gap [13].

By the pole overlap coefficient, we mean the ratio:

where *bm*- width of the pole; *τ* - distance between of the neutrals.

should be recognized as traditional.

magnetic flux (**Figure 4**).

Ampere's law.

**10**

and disconnection of the corresponding phases.

where *B* – elementary section induction;

connection of multiphase BMAMF windings.

*Emerging Electric Machines - Advances, Perspectives and Applications*

*Representation of air gap induction and linear load.*

**Figure 2.**

*The diagram for the (180–180/m)-degree commutation.*

**Figure 4.** *Model for axial gap machine.*

$$a = 2pa\_{el},\tag{6}$$

*a*1*n*1ð Þ¼ *i*, *x*

*b*1*n*1ð Þ¼ *i*, *x*

following formula

1 *<sup>π</sup>el* <sup>ð</sup> �*πel*þ*<sup>i</sup>*

1 *<sup>π</sup>el* <sup>ð</sup> �*πel*þ*<sup>i</sup>*

torques (one phase is switched of)

0

*DOI: http://dx.doi.org/10.5772/intechopen.95945*

BB@

0

BB@

*πel <sup>m</sup>* þ*x*

*πel <sup>m</sup>* þ*x*

*<sup>m</sup>* þ*x*

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis*

*<sup>m</sup>* þ*x*

*M*<sup>∗</sup> <sup>180</sup>�<sup>180</sup> *m*

*sr*ð Þ <sup>180</sup>�180*=<sup>m</sup>* ¼

where *k*1– number of terms in the harmonic series; *i* – the number of the phase; m – number of phases.

ð Þ �1 sin ð Þ *n*1*αel dαel* þ

ð Þ �1 cosð Þ *n*1*αel dαel* þ

The total torque of the anchor winding will be created as a sum of the phases

ð Þ¼ *<sup>x</sup>* <sup>X</sup>*<sup>m</sup>*�<sup>1</sup>

The medium torque for this type of commutation may be described by the

Ð *πel m* <sup>0</sup> *M*<sup>∗</sup>

It is possible to see from diagram 3 that different wires conductors create different torque because they take place in different magnet field conditions. The

*M*<sup>∗</sup>

ð Þ¼ *α*, *m*

2

**2.2 Analysis of 180-degree commutation for BMAMF with any phases**

The diagram of the 180-degree commutation is presented in **Figure 6**.

Let us name this factor as the efficiency factor of the anchor for (180–180/m) degree commutation, This factor depends from pole factor and number of the phases. The curves for (180–180/m)-degree commutation (Eq. 19)) are presented in **Figure 5**. The curves analysis shows that it is necessary to increase pole factor and

*AsrBδD*<sup>2</sup>

The curves analysis shows that it is necessary to increase pole factor and number

We will use the same harmonic analysis method for 180-degree commutation.

We may use the same formulas (Eq. (9), Eq. (12), Eq. (14)) for induction in the

*i*¼1

*M*<sup>∗</sup>

<sup>180</sup>�180*=<sup>m</sup>*ð Þ *x dx πel m*

> *sr*ð Þ 180�180*=m πel*

*srLringKef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

ð *i πel <sup>m</sup>* þ*x*

sin ð Þ *n*1*αel dαel*

cosð Þ *n*1*αel dαel*

*fi* ð Þ *x :* (17)

*:* (18)

*:* (19)

*:* (20)

1

CCA ;

1

CCA,

(16)

(15)

ð Þ *<sup>i</sup>*�<sup>1</sup> *<sup>π</sup>el <sup>m</sup>* þ*x*

> ð *i πel <sup>m</sup>* þ*x*

ð Þ *<sup>i</sup>*�<sup>1</sup> *<sup>π</sup>el <sup>m</sup>* þ*x*

�*πel*þð Þ *<sup>i</sup>*�<sup>1</sup> *<sup>π</sup>el*

�*πel*þð Þ *<sup>i</sup>*�<sup>1</sup> *<sup>π</sup>el*

*M*<sup>∗</sup>

factor below can estimate the efficiency of anchor wire:

*Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

number of the phases for increasing the electromagnetic torque. The absolute moment may be defined by the following formula

*Msr*ð Þ <sup>180</sup>�180*=<sup>m</sup>* <sup>¼</sup> *<sup>π</sup>*

of the phases for increasing the electromagnetic torque.

air gap, current load, and torque for the one phase.

**13**

where 2*p* – the number of poles of a generalized axial machine.

The expression for the current of an elementary section can be written as follows

$$di(\mathbf{x}) = \frac{2\pi r\_{sr} A\_{sr} A\_i(a\_{el}, \mathbf{x})}{2\pi r} r 2p da\_{el} = 2pr\_{sr} A\_{sr} A\_i(a\_{el}, \mathbf{x}) da\_{el},\tag{7}$$

Substituting (Eq.4) and (Eq.4) in (Eq.3), we obtain an expression for the electromagnetic moment of the elementary section that creates the i-th phase

$$d\mathcal{M}\_{\hat{f}}(\mathbf{x}) = 2pr\_{sr}A\_{sr}B\_{\delta}B(a\_{\rm el})A\_{i}(a\_{\rm el},\mathbf{x})rdrda\_{\rm el}.\tag{8}$$

To determine the moment of the i-th phase, we take the integral over the disk surface of a generalized axial machine

$$\begin{split} M\_{\hat{\rm f}}(\mathbf{x}) &= \int\_{r\_{\rm in}}^{r\_{\rm out}} \int\_{0}^{2\pi} d\mathbf{M}\_{\hat{\rm f}}(\mathbf{x}) = 2pr\_{sr}A\_{sr}B\_{\delta} \int\_{r\_{\rm in}}^{r\_{\rm out}} \int\_{0}^{2\pi} B(a\_{\rm el})A\_{i}(a\_{\rm el})r dr da\_{\rm el} \\ &= \frac{p}{2}A\_{sr}B\_{\delta}D\_{gr}^{2}L\_{ring} \int\_{0}^{\pi\_{d}} B(a\_{\rm el})A\_{i}(a\_{\rm el}) da\_{\rm el},\end{split} \tag{9}$$

where *Dsr* – average ring diameter of a generalized axial machine, *Lring* – ring thickness of the generalized axial machine (**Figure 4**). For the basic moment value, we take the expression

$$M\_b = \frac{p}{2} A\_{sr} B\_\delta D\_{sr}^2 L\_{ring}.\tag{10}$$

Then the dependence of the relative moment on the displacement of the armature relative to the inductor for the i-th phase will have the form.

$$M\_{fi}^{\*} = \int\_{0}^{\pi\_{el}} B(a\_{el}) A\_{i}(a\_{el}, \varkappa) da\_{el} \tag{11}$$

Decompose the function of induction and linear load into a harmonic series. We take into account the symmetry of the curves relative to the coordinate axes.

Relative value of induction in the air gap

$$B(a\_{el}) = \sum\_{n=1}^{k} a\_n \sin \left(n a\_{el} \right);\tag{12}$$

$$a\_{\mathfrak{n}} = \frac{2}{\pi\_{d}} \begin{pmatrix} \frac{\mathfrak{s}\_{d}(1-a)}{2} & \frac{\mathfrak{s}\_{d}(1-a)}{2} + \mathfrak{a}\mathfrak{s} & \frac{\mathfrak{s}\_{d}(1-a)}{2} + \mathfrak{a}\mathfrak{s} \\\\ \int\_{0}^{\frac{\mathfrak{s}\_{d}(1-a)}{2}} \mathfrak{m}\_{d}(1-a) da\_{d} + \int\_{\frac{\mathfrak{s}\_{d}(1-a)}{2}}^{\frac{\mathfrak{s}\_{d}(1-a)}{2} + \mathfrak{a}\mathfrak{s}} \sin(n\mathfrak{a}\_{d}) da\_{d} + \int\_{0}^{\frac{\mathfrak{s}\_{d}(1-a)}{2}} \frac{2(1-a)}{\mathfrak{s}\_{d}(1-a)} \sin(n\mathfrak{a}\_{d}) da\_{d} \end{pmatrix} \tag{13}$$

where *k* – number of terms in the harmonic series. Relative value of the linear load of the i-th phase:

$$A\_i(a\_{el}, \boldsymbol{\chi}) = \sum\_{n1}^{k1} a \mathbf{1}\_{n1}(i, \boldsymbol{\chi}) \sin \left(n \mathbf{1} a\_{el} \right) + b \mathbf{1}\_{n1}(i, \boldsymbol{\chi}) \cos \left(n \mathbf{1} a\_{el} \right) \tag{14}$$

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis DOI: http://dx.doi.org/10.5772/intechopen.95945*

$$a\mathbf{1}\_{n1}(i,\mathbf{x}) = \frac{1}{\pi\_{el}} \left( \begin{array}{c} -\pi\_{d} + i\frac{\pi\_{d}}{\pi} + \mathbf{x} \\\\ \int\_{-\pi\_{d} + (i-1)\frac{\pi\_{d}}{\pi} + \mathbf{x}} (-1)\sin\left(n\mathbf{1}a\_{d}\right)da\_{d} + \int\_{(i-1)\frac{\pi\_{d}}{\pi} + \mathbf{x}} \sin\left(n\mathbf{1}a\_{d}\right)da\_{d} \end{array} \right); \tag{15}$$

$$b\mathbf{1}\_{n1}(i,\mathbf{x}) = \frac{1}{\pi\_{el}} \left( \begin{array}{c} -\pi\_{d} + i\frac{\pi\_{d}}{\pi} + \mathbf{x} \\\\ -\pi\_{d} + (i-1)\frac{\pi\_{d}}{\pi} + \mathbf{x} \end{array} \begin{array}{c} \frac{\pi\_{d}}{\pi\_{m}} \cdot \mathbf{x} \\\\ \cos\left(n\mathbf{1}a\_{d}\right)da\_{d} \end{array} \right), \tag{16}$$

where *k*1– number of terms in the harmonic series;

*i* – the number of the phase; m – number of phases.

The total torque of the anchor winding will be created as a sum of the phases torques (one phase is switched of)

$$\mathcal{M}\_{180-\frac{10}{m}}^\*(\boldsymbol{\omega}) = \sum\_{i=1}^{m-1} \mathcal{M}\_{fi}^\*(\boldsymbol{\omega}).\tag{17}$$

The medium torque for this type of commutation may be described by the following formula

$$M\_{sr(180-180/m)}^{\*} = \frac{\int\_0^{\frac{\pi\_d}{m}} M\_{180-180/m}^{\*}(\infty)d\infty}{\frac{\pi\_d}{m}}.\tag{18}$$

It is possible to see from diagram 3 that different wires conductors create different torque because they take place in different magnet field conditions. The factor below can estimate the efficiency of anchor wire:

$$K\_{\epsilon f\left(180-\frac{180}{m}\right)}(a,m) = \frac{M\_{sr(180-180/m)}^{\*}}{\pi\_{el}}.\tag{19}$$

Let us name this factor as the efficiency factor of the anchor for (180–180/m) degree commutation, This factor depends from pole factor and number of the phases. The curves for (180–180/m)-degree commutation (Eq. 19)) are presented in **Figure 5**. The curves analysis shows that it is necessary to increase pole factor and number of the phases for increasing the electromagnetic torque.

The absolute moment may be defined by the following formula

$$M\_{sr(180-180/m)} = \frac{\pi}{2} A\_{sr} B\_{\delta} D\_{sr}^2 L\_{ring} K\_{\epsilon f \left(180-\frac{180}{m}\right)}.\tag{20}$$

The curves analysis shows that it is necessary to increase pole factor and number of the phases for increasing the electromagnetic torque.

#### **2.2 Analysis of 180-degree commutation for BMAMF with any phases**

We will use the same harmonic analysis method for 180-degree commutation. The diagram of the 180-degree commutation is presented in **Figure 6**.

We may use the same formulas (Eq. (9), Eq. (12), Eq. (14)) for induction in the air gap, current load, and torque for the one phase.

*α* ¼ 2*pαel*, (6)

*r*2*pdαel* ¼ 2*prsrAsrAi αel* ð Þ , *x dαel*, (7)

*B*ð Þ *αel Ai*ð Þ *αel rdrdαel*

*srLring :* (10)

*B*ð Þ *αel Ai αel* ð Þ , *x dαel* (11)

*an* sin ð Þ *nαel* ; (12)

2 1ð Þ � *α*

*<sup>π</sup>el*ð Þ <sup>1</sup> � *<sup>α</sup>* sin ð Þ *<sup>n</sup>αel <sup>d</sup>αel*,

(13)

ð *πel*ð Þ 1�*α* <sup>2</sup> þ*πα*

0

*B*ð Þ *αel Ai*ð Þ *αel dαel*, (9)

*dMfi*ð Þ¼ *x* 2*prsrAsrBδB*ð Þ *αel Ai αel* ð Þ , *x rdrdαel:* (8)

where 2*p* – the number of poles of a generalized axial machine.

*Emerging Electric Machines - Advances, Perspectives and Applications*

*dMfi*ð Þ¼ *x* 2*prsrAsrB<sup>δ</sup>*

where *Dsr* – average ring diameter of a generalized axial machine, *Lring* – ring thickness of the generalized axial machine (**Figure 4**).

> *Mb* <sup>¼</sup> *<sup>p</sup>* 2

> > ð*<sup>π</sup>el* 0

take into account the symmetry of the curves relative to the coordinate axes.

X *k*

*n*¼1

ð *πel*ð Þ 1�*α* <sup>2</sup> þ*πα*

> *πel*ð Þ 1�*α* 2

ture relative to the inductor for the i-th phase will have the form.

*B*ð Þ¼ *αel*

ð*<sup>π</sup>el* 0

*di x*ð Þ¼ <sup>2</sup>*πrsrAsrAi <sup>α</sup>el* ð Þ , *<sup>x</sup>* 2*πr*

surface of a generalized axial machine

ð*rout rin*

¼ *p* 2 ð<sup>2</sup>*<sup>π</sup>* 0

*AsrBδD*<sup>2</sup>

*srLring*

For the basic moment value, we take the expression

*M*<sup>∗</sup> *fi* ¼

Relative value of induction in the air gap

*<sup>π</sup>el*ð Þ <sup>1</sup> � *<sup>α</sup>* sin ð Þ *<sup>n</sup>αel <sup>d</sup>αel* <sup>þ</sup>

*k*1

*n*1

where *k* – number of terms in the harmonic series. Relative value of the linear load of the i-th phase:

*Mfi*ð Þ¼ *x*

*an* <sup>¼</sup> <sup>2</sup> *πel*

**12**

ð *πel*ð Þ 1�*α* 2

0

BB@

2*α*

*Ai <sup>α</sup>el* ð Þ¼ , *<sup>x</sup>* <sup>X</sup>

0

The expression for the current of an elementary section can be written as follows

Substituting (Eq.4) and (Eq.4) in (Eq.3), we obtain an expression for the electromagnetic moment of the elementary section that creates the i-th phase

To determine the moment of the i-th phase, we take the integral over the disk

*AsrBδD*<sup>2</sup>

Then the dependence of the relative moment on the displacement of the arma-

Decompose the function of induction and linear load into a harmonic series. We

sin ð Þ *nαel dαel* þ

*a*1*<sup>n</sup>*1ð Þ *i*, *x* sin ð Þþ *n*1*αel b*1*<sup>n</sup>*1ð Þ *i*, *x* cosð ÞÞ *n*1*αel* ; (14)

ð*rout rin*

ð<sup>2</sup>*<sup>π</sup>* 0

#### **Figure 5.**

*The dependences of the efficiency factor of the anchor wire for (180–180/m)-degree commutation from pole factor and number of phases.*

#### **Figure 6.**

*The diagram for the 180-degree commutation.*

For the total relative torque, we may write the following formula (all phases are switched on).

$$M\_{180}^\*(\mathbf{x}) = \sum\_{i=1}^m M\_{fi}^\*(\mathbf{x}). \tag{21}$$

We can create the same conclusion for this type of the commutation. It is necessary to increase the pole factor and number of the phases for increasing the

*The dependences of the efficiency factor of the anchor for 180-degree commutation from pole factor and number*

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis*

*DOI: http://dx.doi.org/10.5772/intechopen.95945*

**2.3 The compare of the (180–180/m)-degree commutation and 180- degree**

We will use the same sizes and the same current load for both types of

<sup>¼</sup> *Kef* <sup>180</sup> *Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

Let us compere both commutations for pole factor 0.8. It is typical pole factor

The curve shows that 180-degree commutation has the advantage but for big numbers of phases this advantage is decreasing. It is possible to do this compere for

We show the advantage of the 180-degree commutation with using the electro-

The studies have shown that to increase the efficiency of the system it is necessary to increase the number of phases and use the180-degree commutation but it is necessary to say that the cost of electronic control system will be increasing with increasing the number of phases. Therefore, the researcher has to take a mind this

It should be noted that as the number of phases increase, the difference in efficiency of these type of commutation decreases too. Since the (180–180/m) degree commutation is simpler to implement, so it can be chosen for practice.

*:* (25)

*Msr*<sup>180</sup> *Msr*ð Þ <sup>180</sup>�180*=<sup>m</sup>*

for brushless machine. The curve of this analysis is shown in the **Figure 8**.

another type of the pole factors and we will have the same conclusion.

Let us compere the efficiency of this type commutation. The electromagnetic

electromagnetic torque.

commutation.

**Figure 7.**

*of phases.*

magnetic torque.

**15**

torque will be the criteria for this analysis.

**commutation for BMAMF with any phases**

The ratio of the total electromagnetic torques is

information when he will create the real equipment.

The study can define the following results:

The medium torque

$$\mathcal{M}\_{sr180}^{\*} = \frac{\int\_{-\frac{\pi\_d}{2m}}^{\frac{\pi\_d}{2m}} \mathcal{M}\_{180}^{\*}(\infty) d\infty}{\frac{\pi\_d}{m}}.\tag{22}$$

The efficiency factor of the anchor for 180-degree commutation

$$K\_{\mathfrak{gl}180}(\mathfrak{a},m) = \frac{M\_{sr180}^\*}{\pi\_{el}}.\tag{23}$$

The curves for this factor (Eq.(23)) are presented in the **Figure 7**. The absolute moment may be defined by the following formula

$$M\_{sr180} = \frac{\pi}{2} A\_{sr} B\_{\delta} D\_{sr}^2 L\_{ring} K\_{\epsilon f 180}.\tag{24}$$

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis DOI: http://dx.doi.org/10.5772/intechopen.95945*

#### **Figure 7.**

For the total relative torque, we may write the following formula (all phases are

*The dependences of the efficiency factor of the anchor wire for (180–180/m)-degree commutation from pole*

*Emerging Electric Machines - Advances, Perspectives and Applications*

*i*¼1

*M*<sup>∗</sup>

*M*<sup>∗</sup> *sr*180 *πel*

*fi* ð Þ *x :* (21)

*:* (22)

*:* (23)

*srLringKef* <sup>180</sup>*:* (24)

<sup>180</sup>ð Þ¼ *<sup>x</sup>* <sup>X</sup>*<sup>m</sup>*

Ð *πel* 2*m* �*πel* 2*m M*<sup>∗</sup> <sup>180</sup>ð Þ *x dx πel m*

*M*<sup>∗</sup>

*M*<sup>∗</sup> *sr*<sup>180</sup> ¼

The efficiency factor of the anchor for 180-degree commutation

*Kef* <sup>180</sup>ð Þ¼ *α*, *m*

The curves for this factor (Eq.(23)) are presented in the **Figure 7**. The absolute moment may be defined by the following formula

2

*AsrBδD*<sup>2</sup>

*Msr*<sup>180</sup> <sup>¼</sup> *<sup>π</sup>*

switched on).

**Figure 6.**

**14**

**Figure 5.**

*factor and number of phases.*

The medium torque

*The diagram for the 180-degree commutation.*

*The dependences of the efficiency factor of the anchor for 180-degree commutation from pole factor and number of phases.*

We can create the same conclusion for this type of the commutation. It is necessary to increase the pole factor and number of the phases for increasing the electromagnetic torque.

Let us compere the efficiency of this type commutation. The electromagnetic torque will be the criteria for this analysis.

#### **2.3 The compare of the (180–180/m)-degree commutation and 180- degree commutation for BMAMF with any phases**

We will use the same sizes and the same current load for both types of commutation.

The ratio of the total electromagnetic torques is

$$\frac{M\_{sr180}}{M\_{sr(180-180/m)}} = \frac{K\_{ef180}}{K\_{ef\left(180-\frac{180}{m}\right)}}.\tag{25}$$

Let us compere both commutations for pole factor 0.8. It is typical pole factor for brushless machine. The curve of this analysis is shown in the **Figure 8**.

The curve shows that 180-degree commutation has the advantage but for big numbers of phases this advantage is decreasing. It is possible to do this compere for another type of the pole factors and we will have the same conclusion.

We show the advantage of the 180-degree commutation with using the electromagnetic torque.

The studies have shown that to increase the efficiency of the system it is necessary to increase the number of phases and use the180-degree commutation but it is necessary to say that the cost of electronic control system will be increasing with increasing the number of phases. Therefore, the researcher has to take a mind this information when he will create the real equipment.

It should be noted that as the number of phases increase, the difference in efficiency of these type of commutation decreases too. Since the (180–180/m) degree commutation is simpler to implement, so it can be chosen for practice.

The study can define the following results:

**Figure 9.**

**Figure 10.**

**Figure 11.**

**Figure 12.**

**17**

*BMAMF with cylindrical magnets and a smooth anchor with ring coils.*

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis*

*DOI: http://dx.doi.org/10.5772/intechopen.95945*

*BMAMF with segment magnets and a smooth anchor with trapezoidal.*

*BMAMF with segment magnets and a smooth armature with toroidal coils.*

*Classification of BMAMF.*

#### **Figure 8.**

*The compere of the (180–180/m)-degree commutation with the 180-degree commutation for pole factor 0.8.*


These conclusions determine the trend of increasing efficiency, but in practice, it is necessary to calculate the price for choosing the best type of commutation.

#### **3. Analysis of electromagnetic power of various designs of the BMAMF**

BMAMF have a large number of designs. Different designs in the same dimensions develop different torque and power. To compare the effectiveness of various designs, we will classify them.

#### **3.1 Classification of various BMAMF designs**

The shape of active elements that create an electromagnetic moment can classify a large number of design modifications of BMAMF. The shape of permanent magnets may be cylindrical, prismatic and segmental magnets. BMAMF phase coils can have ring, trapezoidal, wave, and toroid shapes. The combination of various permanent magnet shapes with armature winding shapes creates a variety of BMAMF designs [14, 15]. The classification of BMAMF structures is shown in **Figure 9**.

The main design versions of BMAMF active parts with various forms of magnets and coils are shown below. On their basis, it is possible to build various design modifications. Let us call these models basic. **Figure 10** shows a BMAMF with cylindrical magnets and ring coils. **Figure 11** shows a BMAMF with segment magnets and trapezoidal coils. **Figure 12** shows a BMAMF with segment magnets and toroid coils.

The development of computational models for the above structures has its own peculiarities. Let us output the values of the electromagnetic moment and electromagnetic power for the basic versions shown in **Figure 10**–**13**.

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis DOI: http://dx.doi.org/10.5772/intechopen.95945*

**Figure 9.** *Classification of BMAMF.*

1.The electromagnetic torque will be increasing for both type commutations

*The compere of the (180–180/m)-degree commutation with the 180-degree commutation for pole factor 0.8.*

2.180-degree commutation has the advantage with different phases and different pole factors if compare it with the (180–180/m)-degree commutation.

These conclusions determine the trend of increasing efficiency, but in practice, it is necessary to calculate the price for choosing the best type of commutation.

**3. Analysis of electromagnetic power of various designs of the BMAMF**

BMAMF have a large number of designs. Different designs in the same dimensions develop different torque and power. To compare the effectiveness of various

The shape of active elements that create an electromagnetic moment can classify a large number of design modifications of BMAMF. The shape of permanent magnets may be cylindrical, prismatic and segmental magnets. BMAMF phase coils can have ring, trapezoidal, wave, and toroid shapes. The combination of various permanent magnet shapes with armature winding shapes creates a variety of BMAMF designs [14, 15]. The classification of BMAMF structures is shown in **Figure 9**. The main design versions of BMAMF active parts with various forms of magnets

and coils are shown below. On their basis, it is possible to build various design modifications. Let us call these models basic. **Figure 10** shows a BMAMF with cylindrical magnets and ring coils. **Figure 11** shows a BMAMF with segment magnets and trapezoidal coils. **Figure 12** shows a BMAMF with segment magnets and toroid coils. The development of computational models for the above structures has its own peculiarities. Let us output the values of the electromagnetic moment and electro-

magnetic power for the basic versions shown in **Figure 10**–**13**.

with increasing the number of phases.

*Emerging Electric Machines - Advances, Perspectives and Applications*

**3.1 Classification of various BMAMF designs**

designs, we will classify them.

**Figure 8.**

**16**

**Figure 10.** *BMAMF with cylindrical magnets and a smooth anchor with ring coils.*

**Figure 11.** *BMAMF with segment magnets and a smooth anchor with trapezoidal.*

**Figure 12.** *BMAMF with segment magnets and a smooth armature with toroidal coils.*

**Figure 13.** *BMAMF with segment magnets and toothed anchor.*

### **3.2 Electromagnetic torque and electromagnetic power for BMAMF with cylindrical magnets and smooth armature with ring coils**

Let us define the electromagnetic moment of the phase in the position at which it has the maximum value. This is the position at which the axis of the ring coil coincides with the geometric neutral. A sketch of the magnetic system and the armature winding is shown in **Figure 14**. To facilitate reference to the dependencies given below, we denote this design as model 1.

The real value of the magnetic induction in the gap is replaced by its average value, assuming that it does not change within the pole division.

The ratio between the maximum induction and the average induction is determined by the formula

$$B\_{sr} = B\_{\delta} \frac{\mathbb{S}\_{pole}}{\mathbb{S}\_{\tau}},\tag{26}$$

*Rsr* – average radius of the magnetic system ring;

2 ð*π*

0

*Mmaxfmod*<sup>1</sup> ¼ *p*

where *Ws* – number of turns in the coil section; *drsr*– the average diameter of the coil section ring.

where *Dsr* – average ring diameter of the magnetic system;

*dMα*1*<sup>j</sup>* þ *dMα*2*<sup>j</sup>*

*Sketch of the magnetic system and armature windings BMAMF with ring armature windings.*

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis*

*DOI: http://dx.doi.org/10.5772/intechopen.95945*

X *Ws*

*j*¼1

Let us express the average ring diameter of the coil section in terms of the ring

From the angle of rotation, the moment of the phase changes according to the

where *γel* – turn of the armature relative to the inductor in electrical degrees. Let us determine the maximum electromagnetic moment of the machine for

� � <sup>¼</sup> *iaBsrDsrdvj*, (30)

*Mfmod*<sup>1</sup> *γel* ð Þ¼ *Mmaxfmod*<sup>1</sup> cos *γel* ð Þ, (32)

*Mj* ¼ *iaBsrDsrdrsrpWs*, (31)

*Mj* ¼

*rvj* – radius of the j-th turn. Moment of the j-th turn

**Figure 14.**

*dvj* – diameter of the j-th turn. The maximum moment of phase

thickness of the magnetic system.

various switching options.

For (180–180/m)-degree commutation

law of cosine

**19**

where *Spole*– surface of the pole,

*Sτ*– area of the pole division/.

Let us choose an arbitrary j turn with an anchor current ia. On this turn, we select an elementary conductor with length dl on the left and right sides. These elementary conductors will be affected by elementary forces, like conductors in a magnetic field, which will be directed to the center of the anchor coil. These forces can be decomposed into components on the X-axis and on the y-axis. The forces on the Y-axis will compensate for each other as equal in magnitude and opposite in direction.

The electromagnetic moment will only be created by forces directed along the xaxis. Elementary moment of the j-th turn:

$$dM\_j = dM\_{a1\natural} + dM\_{a2\natural},\tag{27}$$

where

$$d\mathcal{M}\_{a\mathbf{l}\mathbf{j}} = dF\_{a\mathbf{l}\mathbf{x}\mathbf{j}} \left( R\_{\mathbf{r}} + r\_{\mathbf{v}\mathbf{j}} \sin a\_1 \right) = i\_a B\_{\mathbf{r}} r\_{\mathbf{v}\mathbf{j}} \cos a\_1 \left( R\_{\mathbf{r}} + r\_{\mathbf{v}\mathbf{j}} \sin a\_1 \right) da\_1. \tag{28}$$

$$d\mathbf{M}\_{a2\circ} = dF\_{a2\circ\mathbf{j}} \left( R\_{\boldsymbol{\pi}} + r\_{\boldsymbol{v}\boldsymbol{\jmath}} \sin a\_1 \right) = i\_a B\_{\boldsymbol{\pi}} r\_{\boldsymbol{v}\boldsymbol{\jmath}} \cos a\_2 \left( R\_{\boldsymbol{\pi}} + r\_{\boldsymbol{v}\boldsymbol{\jmath}} \sin a\_2 \right) da\_2. \tag{29}$$

where *dF<sup>α</sup>*1*xj*, *dF<sup>α</sup>*2*xj* – elementary forces acting on the left and right halves of the coil;

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis DOI: http://dx.doi.org/10.5772/intechopen.95945*

*Rsr* – average radius of the magnetic system ring; *rvj* – radius of the j-th turn. Moment of the j-th turn

$$\mathbf{M}\_{j} = \int\_{0}^{2\pi} \left( d\mathbf{M}\_{a\mathbf{j}\mathbf{j}} + d\mathbf{M}\_{a\mathbf{2}\mathbf{j}} \right) = \mathbf{i}\_{a} B\_{sr} D\_{sr} d\_{v\mathbf{j}},\tag{30}$$

where *Dsr* – average ring diameter of the magnetic system; *dvj* – diameter of the j-th turn. The maximum moment of phase

$$\mathbf{M}\_{\text{maxf}} \mathbf{m}\_1 = p \sum\_{j=1}^{W\_s} \mathbf{M}\_j = \mathbf{i}\_d \mathbf{B}\_{sr} \mathbf{D}\_{sr} p \mathbf{W}\_s,\tag{31}$$

where *Ws* – number of turns in the coil section;

*drsr*– the average diameter of the coil section ring.

Let us express the average ring diameter of the coil section in terms of the ring thickness of the magnetic system.

From the angle of rotation, the moment of the phase changes according to the law of cosine

$$M\_{fmod1}(\chi\_{\rm el}) = M\_{maxfmod1} \cos \left(\chi\_{\rm el}\right),\tag{32}$$

where *γel* – turn of the armature relative to the inductor in electrical degrees. Let us determine the maximum electromagnetic moment of the machine for various switching options.

For (180–180/m)-degree commutation

**3.2 Electromagnetic torque and electromagnetic power for BMAMF with**

has the maximum value. This is the position at which the axis of the ring coil coincides with the geometric neutral. A sketch of the magnetic system and the armature winding is shown in **Figure 14**. To facilitate reference to the dependencies

Let us define the electromagnetic moment of the phase in the position at which it

The real value of the magnetic induction in the gap is replaced by its average

*Spole Sτ*

, (26)

The ratio between the maximum induction and the average induction is

*Bsr* ¼ *B<sup>δ</sup>*

Let us choose an arbitrary j turn with an anchor current ia. On this turn, we select an elementary conductor with length dl on the left and right sides. These elementary conductors will be affected by elementary forces, like conductors in a magnetic field, which will be directed to the center of the anchor coil. These forces can be decomposed into components on the X-axis and on the y-axis. The forces on the Y-axis will compensate for each other as equal in magnitude and opposite in

The electromagnetic moment will only be created by forces directed along the x-

<sup>¼</sup> *iaBsrrvj* cos *<sup>α</sup>*<sup>1</sup> *Rsr* <sup>þ</sup> *rvj* sin *<sup>α</sup>*<sup>1</sup>

<sup>¼</sup> *iaBsrrvj* cos *<sup>α</sup>*<sup>2</sup> *Rsr* <sup>þ</sup> *rvj* sin *<sup>α</sup>*<sup>2</sup>

where *dF<sup>α</sup>*1*xj*, *dF<sup>α</sup>*2*xj* – elementary forces acting on the left and right halves

*dMj* ¼ *dM<sup>α</sup>*1*<sup>j</sup>* þ *dM<sup>α</sup>*2*<sup>j</sup>*, (27)

*dα*1*:* (28)

*dα*2*:* (29)

**cylindrical magnets and smooth armature with ring coils**

*Emerging Electric Machines - Advances, Perspectives and Applications*

value, assuming that it does not change within the pole division.

given below, we denote this design as model 1.

*BMAMF with segment magnets and toothed anchor.*

determined by the formula

direction.

**Figure 13.**

where

of the coil;

**18**

where *Spole*– surface of the pole, *Sτ*– area of the pole division/.

axis. Elementary moment of the j-th turn:

*dM<sup>α</sup>*1*<sup>j</sup>* ¼ *dF<sup>α</sup>*1*xj Rsr* þ *rvj* sin *α*<sup>1</sup>

*dM<sup>α</sup>*2*<sup>j</sup>* ¼ *dF<sup>α</sup>*2*xj Rsr* þ *rvj* sin *α*<sup>1</sup>

$$\begin{split} M\_{\max\bmod 1}(180 - \frac{180}{m}) &= M\_{fmd1} \sum\_{i=1}^{m-1} \cos \left( -\frac{\pi}{2} + \frac{\pi}{m} + \frac{\pi}{m}(i-1) \right) \\ &= \frac{1}{2} i\_{4} B\_{sr} D\_{sr} L\_{ring} pW\_{s} \sum\_{i=1}^{m-1} \cos \left( -\frac{\pi}{2} + \frac{\pi}{m} + \frac{\pi}{m}(i-1) \right) \\ &= \frac{\pi}{2} A\_{sr} B\_{sr} D\_{sr}^{2} L\_{ring} K\_{mod1}(180 - \frac{100}{m}), \end{split} \tag{33}$$

where *Kmod*1 180�<sup>180</sup> ð Þ *<sup>m</sup>* – the coefficient of efficiency of model 1 for (180–180/m) degree commutation

$$K\_{mod1\left(180-\frac{180}{m}\right)} = \frac{\sum\_{i=1}^{m-1} \cos\left(-\frac{\pi}{2} + \frac{\pi}{m} + \frac{\pi}{m}(i-1)\right)}{2m}.\tag{34}$$

The graphical dependence of this coefficient on the number of phases for (180–180/m) - degree commutation (Eq. (34)) is shown in **Figure 15**.

For 180-degree commutation

$$\begin{split} M\_{\max\ mod(1|180)} &= M\_{\text{find}1} \sum\_{i=1}^{m} \cos\left(-\frac{\pi}{2} + \frac{\pi}{2m} + \frac{\pi}{m}(i-1)\right) \\ &= \frac{1}{2} i\_{a} B\_{sr} D\_{sr} L\_{\text{ring}} pW\_{s} \sum\_{i=1}^{m} \cos\left(-\frac{\pi}{2} + \frac{\pi}{2m} + \frac{\pi}{m}(i-1)\right) \\ &= \frac{\pi}{2} A\_{sr} B\_{sr} D\_{sr}^{2} L\_{\text{ring}} \mathbf{K}\_{\text{mod1}(180)}. \end{split} \tag{35}$$

where *K*mod1 180 ð Þ – efficiency coefficient of model 1 on the number of phases for 180- degree commutation

$$K\_{mod1(180)} = \frac{\sum\_{i=1}^{m} \cos\left(-\frac{\pi}{2} + \frac{\pi}{2m} + \frac{\pi}{m}(i-1)\right)}{2m}.\tag{36}$$

quantitative evaluation, we introduce the commutation comparison coefficient as the ratio of electromagnetic powers (180–180/m) - degree commutation and

*Dependence of the efficiency coefficient of model 1 on the number of phases for (180–180/m) - degree commutation.*

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis*

*DOI: http://dx.doi.org/10.5772/intechopen.95945*

*Dependence of the efficiency coefficient of model 1 on the number of phases for 180 - degree commutation.*

¼

The graphical dependence of this coefficient on the number of phases is shown

The dependence analysis shows that for model 1, 180-degree commutation has an advantage with a small number of phases. As the number of phases increases,

A sketch of the magnetic system and the BAMF armature winding of this design

**3.3 Electromagnetic moment and electromagnetic power for BMAMF with**

(41) .

*Kmod*1 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*Kmod*1 180 ð Þ*Kef* <sup>180</sup>

*Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

*Pel mod*1 180�<sup>180</sup> ð Þ *<sup>m</sup> Pel mod*1 180 ð Þ

180-degree commutation:

this advantage decreases.

in **Figure 17**.

**21**

**Figure 16.**

**Figure 15.**

*Kcomp*\_*com*\_*mod*<sup>1</sup> ¼

**segment magnets and trapezoidal coils**

is shown in **Figure 18**. Let us refer to this design as model 2.

The graphical dependence of this coefficient on the number of phases for 180 degree switching (Eq.(36)) is shown in **Figure 16**.

Physical meaning of the efficiency coefficients of the model is to determine the share that the phases invest in creating the maximum moment.

Let us determine the average electromagnetic moment and electromagnetic power for the model1 BMAMF for various switching options, taking into account the efficiency coefficients derived above.

For (180–180/m) - degree commutation:

$$M\_{sr\,mol1\left(180-\frac{180}{m}\right)} = \frac{\pi}{2} A\_{sr} B\_{sr} D\_{sr}^2 L\_{ring} K\_{mod1\left(180-\frac{180}{m}\right)} K\_{ef\left(180-\frac{180}{m}\right)};\tag{37}$$

$$P\_{el\ mol1\left(180-\frac{180}{m}\right)} = \frac{\pi}{2} A\_{sr} B\_{sr} \alpha D\_{sr}^2 L\_{ring} K\_{mod1\left(180-\frac{180}{m}\right)} K\_{ef\left(180-\frac{180}{m}\right)},\tag{38}$$

where *ω* – rotation speed in rad/s. For 180 - degree commutation:

$$M\_{sr\,mod1(180)} = \frac{\pi}{2} A\_{sr} B\_{sr} D\_{gr}^2 L\_{ring} K\_{mod1(180)} K\_{ef180} \,\text{;}\tag{39}$$

$$P\_{el\ mol1(180)} = \frac{\pi}{2} A\_{sr} B\_{sr} \alpha D\_{gr}^2 L\_{ring} K\_{mod1(180)} K\_{eff180} \,. \tag{40}$$

It is of theoretical interest to choose the most efficient type of switching for model 1 with the same electromagnetic loads and in the same dimensions. For

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis DOI: http://dx.doi.org/10.5772/intechopen.95945*

quantitative evaluation, we introduce the commutation comparison coefficient as the ratio of electromagnetic powers (180–180/m) - degree commutation and 180-degree commutation:

$$K\_{comp\\_com\\_mod1} = \frac{P\_{el\\_mod1\left(180-\frac{100}{m}\right)}}{P\_{el\\_mod1\left(180\right)}} = \frac{K\_{mod1\left(180-\frac{100}{m}\right)}K\_{ef\left(180-\frac{100}{m}\right)}}{K\_{mod1\left(180\right)}K\_{ef\left(180\right)}}\tag{41}$$

The graphical dependence of this coefficient on the number of phases is shown in **Figure 17**.

The dependence analysis shows that for model 1, 180-degree commutation has an advantage with a small number of phases. As the number of phases increases, this advantage decreases.

#### **3.3 Electromagnetic moment and electromagnetic power for BMAMF with segment magnets and trapezoidal coils**

A sketch of the magnetic system and the BAMF armature winding of this design is shown in **Figure 18**. Let us refer to this design as model 2.

*Mmax mod*1 180�<sup>180</sup> ð Þ *<sup>m</sup>*

where *Kmod*1 180�<sup>180</sup> ð Þ *<sup>m</sup>*

degree commutation

¼ *Mfmod*<sup>1</sup>

¼ 1 2

¼ *π* 2

*Kmod*1 180�<sup>180</sup> ð Þ *<sup>m</sup>*

For 180-degree commutation

*Mmax mod*1 180 ð Þ ¼ *Mfmod*<sup>1</sup>

180- degree commutation

¼ 1 2

¼ *π* 2 X*m*�1 *i*¼1

*Emerging Electric Machines - Advances, Perspectives and Applications*

*iaBsrDsrLringpWs*

*AsrBsrD*<sup>2</sup>

¼

X*m i*¼1

*iaBsrDsrLringpWs*

P*<sup>m</sup>*

share that the phases invest in creating the maximum moment.

¼ *π* 2

*AsrBsrωD*<sup>2</sup>

2

2

*AsrBsrD*<sup>2</sup>

*AsrBsrωD*<sup>2</sup>

It is of theoretical interest to choose the most efficient type of switching for model 1 with the same electromagnetic loads and in the same dimensions. For

¼ *π* 2

*Msr mod*1 180 ð Þ <sup>¼</sup> *<sup>π</sup>*

*Pel mod*1 180 ð Þ <sup>¼</sup> *<sup>π</sup>*

*AsrBsrD*<sup>2</sup>

*AsrBsrD*<sup>2</sup>

*Kmod*1 180 ð Þ ¼

degree switching (Eq.(36)) is shown in **Figure 16**.

the efficiency coefficients derived above. For (180–180/m) - degree commutation:

*Msr mod*1 180�<sup>180</sup> ð Þ *<sup>m</sup>*

where *ω* – rotation speed in rad/s. For 180 - degree commutation:

*Pel mod*1 180�<sup>180</sup> ð Þ *<sup>m</sup>*

**20**

(180–180/m) - degree commutation (Eq. (34)) is shown in **Figure 15**.

cos � *<sup>π</sup>*

P*m*�<sup>1</sup>

cos � *<sup>π</sup>*

2 þ *π m* þ *π*

X*m*�1 *i*¼1

*srLringKmod*1 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*<sup>i</sup>*¼<sup>1</sup> cos � *<sup>π</sup>*

<sup>2</sup> <sup>þ</sup> *<sup>π</sup>* 2*m* þ *π*

X*m i*¼1

where *K*mod1 180 ð Þ – efficiency coefficient of model 1 on the number of phases for

The graphical dependence of this coefficient on the number of phases for 180-

Physical meaning of the efficiency coefficients of the model is to determine the

Let us determine the average electromagnetic moment and electromagnetic power for the model1 BMAMF for various switching options, taking into account

*<sup>i</sup>*¼<sup>1</sup> cos � *<sup>π</sup>*

The graphical dependence of this coefficient on the number of phases for

� �

cos � *<sup>π</sup>*

– the coefficient of efficiency of model 1 for (180–180/m)-

*<sup>m</sup>* ð Þ *<sup>i</sup>* � <sup>1</sup>

<sup>2</sup> <sup>þ</sup> *<sup>π</sup>* 2*m* þ *π*

*srLringKmod*1 180 ð Þ*:* (35)

� �

<sup>2</sup> <sup>þ</sup> *<sup>π</sup> <sup>m</sup>* <sup>þ</sup> *<sup>π</sup> <sup>m</sup>* ð Þ *<sup>i</sup>* � <sup>1</sup> � �

� �

cos � *<sup>π</sup>*

<sup>2</sup> <sup>þ</sup> *<sup>π</sup>* <sup>2</sup>*<sup>m</sup>* <sup>þ</sup> *<sup>π</sup> <sup>m</sup>* ð Þ *<sup>i</sup>* � <sup>1</sup> � �

*srLringKmod*1 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*srLringKmod*1 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*<sup>m</sup>* ð Þ *<sup>i</sup>* � <sup>1</sup>

2 þ *π m* þ *π*

� �

*<sup>m</sup>* ð Þ *<sup>i</sup>* � <sup>1</sup>

, (33)

<sup>2</sup>*<sup>m</sup> :* (34)

*<sup>m</sup>* ð Þ *<sup>i</sup>* � <sup>1</sup>

<sup>2</sup>*<sup>m</sup> :* (36)

*Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

*Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

*srLringKmod*1 180 ð Þ*Kef* 180; (39)

*srLringKmod*1 180 ð Þ*Kef* <sup>180</sup>*:* (40)

; (37)

, (38)

Electromagnetic moment of an arbitrary i-th phase

*DOI: http://dx.doi.org/10.5772/intechopen.95945*

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis*

*M*ð Þ *α*,*r fimod*<sup>2</sup> ¼ *M*ð Þ *α*,*r WS*

linear relationship.

*Kmod*2 180�<sup>180</sup> ð Þ *<sup>m</sup>*

magnetic system;

*<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* <sup>∗</sup>

neutral by *<sup>x</sup>* <sup>¼</sup> *<sup>π</sup>*

*<sup>M</sup>*max *mod*2 180�<sup>180</sup> ð Þ *<sup>m</sup>*

**23**

coefficient of pole overlap;

defined by the following expression

ð Þ¼ *α*, *m*

winding at the real coefficient of pole overlap.

<sup>2</sup>*<sup>m</sup>* (see **Figure 19**)

*<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* <sup>∗</sup>

(180–180/m) - degree commutation

<sup>¼</sup> <sup>X</sup>*<sup>m</sup>*�<sup>1</sup> *i*¼1

¼ 2*iaB<sup>δ</sup>*

*r*2 *out* � *<sup>r</sup>*<sup>2</sup> *in* *<sup>p</sup>* <sup>¼</sup> <sup>2</sup>*iaB*ð Þ *<sup>α</sup> <sup>r</sup>*<sup>2</sup>

The maximum moment of all phases in the intercommutation interval will depend on the pole overlap coefficient of the magnetic system, determined by equation (Eq. (1)), the number of phases, and the type of switching. Let us determine the influence of these factors on model 2. By analogy with model 1, we introduce the efficiency coefficient of model 2, which is the ratio of the maximum moment of the armature winding at the real pole overlap coefficient to the maximum moment at the theoretical pole overlap coefficient equal to 1.0. As a rule, the distance between the side faces of permanent magnets is the same on the inner and outer diameters. In this case, the pole overlap coefficient changes linearly when moving from the inner diameter to the outer one. Therefore, analytical expressions can be derived for the average diameter of a ring with magnets and further use this

For (180–180/m)-degree commutation, the efficiency coefficient of the model is

where *αsr* – real coefficient of pole overlap on the average ring diameter of the

*<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup> max* – maximum moment of the armature winding at the real

*<sup>M</sup>*ð Þ *<sup>α</sup>sr* <sup>¼</sup> 1, *<sup>m</sup>* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup> max* – the maximum moment of the armature winding with

The relative value of the maximum moment of the armature winding can be determined from the expression (Eq. (17)) for the offset of the armature from the

<sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

The physical meaning of the efficiency coefficient of model 2 is similar to the

Taking into account the introduced efficiency coefficient of the model, we can

<sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup> max* – relative value of the maximum moment of the armature

*π* 2*m* � � <sup>¼</sup> <sup>X</sup>*<sup>m</sup>*�<sup>1</sup>

*i*¼1

¼ *π* 2

*AsrBδD*<sup>2</sup>

*M*<sup>∗</sup> *fi π* 2*m*

� �*:* (47)

*srLringKmod*2 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*:*

(48)

¼

*<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* <sup>∗</sup>

*π*

<sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup> max*

*<sup>m</sup>* ð Þ *<sup>m</sup>* � <sup>1</sup> , (46)

*<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup> max <sup>M</sup>*ð Þ *<sup>α</sup>sr* <sup>¼</sup> 1, *<sup>m</sup>* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup> max*

a pole overlap coefficient equal to 1, which is only theoretically possible;

<sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup> max* <sup>¼</sup> *<sup>M</sup>*<sup>∗</sup>

coefficient for model 1. The graphical dependence of the coefficient for (180–180/m) - degree switching (Eq.(46)) is shown in **Figure 20**.

write the following expression of the maximum electromagnetic moment for

*<sup>M</sup>*ð Þ *<sup>α</sup>*, *<sup>m</sup> fi mod*2*Kmod*2 180�<sup>180</sup> ð Þ *<sup>m</sup>*

<sup>2</sup> *WSpKmod*2 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*out* � *<sup>r</sup>*<sup>2</sup> *in*

<sup>2</sup> *WSp:* (45)

**Figure 17.** *Dependence of the commutations comparison coefficient for model 1.*

**Figure 18.**

*Sketch of the magnetic system and armature winding BMAMF with segment magnets and trapezoidal armature windings.*

Let us select elementary conductors of length dr on an arbitrary j-turn of the winding and define elementary moments for them. They will be a function of the angular position and radius of the elementary section.

$$dM(a,r)\_{\mathfrak{l}\mathfrak{j}} = dM(a,r)\_{\mathfrak{j}\mathfrak{j}} = i\_a B(a) r dr,\tag{42}$$

where *r –* radius, where the elementary conductor is located. Electromagnetic moment j-th of the turn

$$\mathcal{M}(a,r)\_j = \int\_{r\_{\rm in}}^{r\_{\rm out}} d\mathcal{M}(a,r)\_{4j} + \int\_{r\_{\rm in}}^{r\_{\rm out}} d\mathcal{M}(a,r)\_{2j} = 2i\_a \mathcal{B}(a) \frac{r\_{\rm out}^2 - r\_{\rm in}^2}{2}. \tag{43}$$

Electromagnetic moment of the phase section

$$\mathcal{M}(a,r)\_{W\_{\mathcal{S}}} = \sum\_{j=1}^{W\_{\mathcal{S}}} d\mathcal{M}(a,r)\_{j} = 2i\_{d}\mathcal{B}(a)\frac{r\_{out}^{2} - r\_{in}^{2}}{2}W\_{\mathcal{S}}.\tag{44}$$

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis DOI: http://dx.doi.org/10.5772/intechopen.95945*

Electromagnetic moment of an arbitrary i-th phase

$$\mathcal{M}(a,r)\_{\text{fixed2}} = \mathcal{M}(a,r)\_{W\_{\text{S}}} p = 2i\_a B(a) \frac{r\_{out}^2 - r\_{in}^2}{2} W\_{\text{S}} p. \tag{45}$$

The maximum moment of all phases in the intercommutation interval will depend on the pole overlap coefficient of the magnetic system, determined by equation (Eq. (1)), the number of phases, and the type of switching. Let us determine the influence of these factors on model 2. By analogy with model 1, we introduce the efficiency coefficient of model 2, which is the ratio of the maximum moment of the armature winding at the real pole overlap coefficient to the maximum moment at the theoretical pole overlap coefficient equal to 1.0. As a rule, the distance between the side faces of permanent magnets is the same on the inner and outer diameters. In this case, the pole overlap coefficient changes linearly when moving from the inner diameter to the outer one. Therefore, analytical expressions can be derived for the average diameter of a ring with magnets and further use this linear relationship.

For (180–180/m)-degree commutation, the efficiency coefficient of the model is defined by the following expression

$$K\_{mod2\left(180-\frac{100}{m}\right)}(a,m) = \frac{M(a\_r,m)\_{\left(180-\frac{100}{m}\right)\max}}{M(a\_r=1,m)\_{\left(180-\frac{100}{m}\right)\max}} = \frac{M(a\_r,m)^{\frac{\ast}{\ast}}\_{\left(180-\frac{100}{m}\right)\max}},\tag{46}$$

where *αsr* – real coefficient of pole overlap on the average ring diameter of the magnetic system;

*<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup> max* – maximum moment of the armature winding at the real coefficient of pole overlap;

*<sup>M</sup>*ð Þ *<sup>α</sup>sr* <sup>¼</sup> 1, *<sup>m</sup>* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup> max* – the maximum moment of the armature winding with a pole overlap coefficient equal to 1, which is only theoretically possible;

*<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* <sup>∗</sup> <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup> max* – relative value of the maximum moment of the armature winding at the real coefficient of pole overlap.

The relative value of the maximum moment of the armature winding can be determined from the expression (Eq. (17)) for the offset of the armature from the neutral by *<sup>x</sup>* <sup>¼</sup> *<sup>π</sup>* <sup>2</sup>*<sup>m</sup>* (see **Figure 19**)

$$\left(M(a\_{\mathcal{V}},m)\right)^{\*}\_{\left(180-\frac{100}{m}\right)\max} = M^{\*}\_{\left(180-\frac{100}{m}\right)}\left(\frac{\pi}{2m}\right) = \sum\_{i=1}^{m-1} M^{\*}\_{\hat{\mathcal{V}}}\left(\frac{\pi}{2m}\right). \tag{47}$$

The physical meaning of the efficiency coefficient of model 2 is similar to the coefficient for model 1. The graphical dependence of the coefficient for (180–180/m) - degree switching (Eq.(46)) is shown in **Figure 20**.

Taking into account the introduced efficiency coefficient of the model, we can write the following expression of the maximum electromagnetic moment for (180–180/m) - degree commutation

$$\begin{split} M\_{\text{max}\,\text{mod}2\left(180-\frac{100}{m}\right)} &= \sum\_{i=1}^{m-1} M(a,m)\_{\text{f}\,\text{mod}2} K\_{\text{mod}2\left(180-\frac{100}{m}\right)} \\ &= 2i\_a B\_\delta \frac{r\_{\text{out}}^2 - r\_{\text{in}}^2}{2} W\_\delta p K\_{\text{mod}2\left(180-\frac{100}{m}\right)} = \frac{\pi}{2} A\_{\text{sr}} B\_\delta D\_{\text{sr}}^2 L\_{\text{ring}} K\_{\text{mod}2\left(180-\frac{100}{m}\right)}. \end{split} \tag{48}$$

Let us select elementary conductors of length dr on an arbitrary j-turn of the winding and define elementary moments for them. They will be a function of the

*Sketch of the magnetic system and armature winding BMAMF with segment magnets and trapezoidal armature*

*dM*ð Þ *α*,*r* <sup>1</sup>*<sup>j</sup>* ¼ *dM*ð Þ *α*,*r* <sup>2</sup>*<sup>j</sup>* ¼ *iaB*ð Þ *α rdr*, (42)

*out* � *<sup>r</sup>*<sup>2</sup> *in*

*out* � *<sup>r</sup>*<sup>2</sup> *in* <sup>2</sup> *:* (43)

<sup>2</sup> *WS:* (44)

*dM*ð Þ *<sup>α</sup>*,*<sup>r</sup>* <sup>2</sup>*<sup>j</sup>* <sup>¼</sup> <sup>2</sup>*iaB*ð Þ *<sup>α</sup> <sup>r</sup>*<sup>2</sup>

*dM*ð Þ *<sup>α</sup>*,*<sup>r</sup> <sup>j</sup>* <sup>¼</sup> <sup>2</sup>*iaB*ð Þ *<sup>α</sup> <sup>r</sup>*<sup>2</sup>

angular position and radius of the elementary section.

*Dependence of the commutations comparison coefficient for model 1.*

*Emerging Electric Machines - Advances, Perspectives and Applications*

*dM*ð Þ *α*,*r* <sup>1</sup>*<sup>j</sup>* þ

*WS*

*j*¼1

Electromagnetic moment of the phase section

*<sup>M</sup>*ð Þ *<sup>α</sup>*,*<sup>r</sup> WS* <sup>¼</sup> <sup>X</sup>

Electromagnetic moment j-th of the turn

*r* ð*out*

*rin*

*M*ð Þ *α*,*r <sup>j</sup>* ¼

**Figure 17.**

**Figure 18.**

*windings.*

**22**

where *r –* radius, where the elementary conductor is located.

*r* ð*out*

*rin*

Average electromagnetic moment and electromagnetic power for (180–180/m) degree commutation

$$\begin{split} M\_{\text{smod2}\left(180-\frac{100}{m}\right)} &= M\_{\text{max}\,\text{mod2}\left(180-\frac{100}{m}\right)} K\_{\mathcal{H}\left(180-\frac{100}{m}\right)} \\ &= \frac{\pi}{2} A\_{\text{sr}} B\_{\delta} D\_{\text{sr}}^2 L\_{\text{ring}} K\_{\text{mod2}\left(180-\frac{100}{m}\right)} K\_{\mathcal{H}\left(180-\frac{100}{m}\right)}; \end{split} \tag{49}$$

*<sup>M</sup>*ð Þ *<sup>α</sup>sr* <sup>¼</sup> 1, *<sup>m</sup>* ð Þ <sup>180</sup> *max* – maximum moment of the armature winding with a pole

Efficiency coefficient of model 2 for 180 - degree switching at different the coefficient of pole overlap on the average diameter of the ring of the magnetic system (Eq. (51)) is shown at **Figure 22**. We can see that this coefficient does not

The relative value of the maximum moment of the armature winding can be determined from the expression (Eq.(21)) for the zero offset of the armature from

ð Þ <sup>180</sup> *max* – relative value of the maximum moment of the armature

ð Þ <sup>180</sup> *max* ð Þ¼ <sup>0</sup> <sup>X</sup>*<sup>m</sup>*

*i*¼1

*M*<sup>∗</sup>

*r*2 *out* � *<sup>r</sup>*<sup>2</sup> *in*

*fi* ð Þ 0 *:* (52)

<sup>2</sup> *WSpKmod*2 180 ð Þ

*:* (56)

*srLringKmod*2 180 ð Þ*Kef*ð Þ <sup>180</sup> ; (54)

*srLringKmod*2 180 ð Þ*Kef*ð Þ <sup>180</sup> *:* (55)

*Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

(53)

overlap coefficient equal to 1, which is only theoretically possible;

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis*

ð Þ <sup>180</sup> *max* <sup>¼</sup> *<sup>M</sup>*<sup>∗</sup>

Maximum electromagnetic torque for 180-degree commutation

*srLringKmod*2 180 ð Þ*:*

*<sup>M</sup>*ð Þ *<sup>α</sup>:<sup>r</sup> fimax mod*2 180 ð Þ*Kmod*2 180 ð Þ <sup>¼</sup> <sup>2</sup>*iaB<sup>δ</sup>*

The average electromagnetic torque and electromagnetic power for 180-degree

2

By analogy with the previous analysis, for quantitative evaluation, we introduce the switching comparison coefficient, as the ratio of electromagnetic powers (180–

¼

The graphical dependence of this coefficient on the number of phases and the

It should be noted that such a comparative analysis makes sense only for the same electromagnetic loads: induction in the air gap and linear load on the average

These dependencies are of great practical importance. Their analysis shows that for the same electromagnetic loads, magnetic systems with a high value of the pole overlap coefficient have an advantage for any number of phases. Graphic dependences of the switching comparison coefficient are below 1.0. However, for magnetic systems with a pole overlap coefficient of 0.7–0.5, which is very typical for practice, the advantage is (180–180/m) - degree commutation for the number of

Given that (180–180/m) - degree commutation has a simpler and cheaper technical implementation, this theoretical conclusion is of great practical importance.

*AsrBδωD*<sup>2</sup>

*AsrBδD*<sup>2</sup>

*Kmod*2 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*Kmod*2 180 ð Þ*Kef* <sup>180</sup>

winding at the real coefficient of pole overlap.

*DOI: http://dx.doi.org/10.5772/intechopen.95945*

*<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* <sup>∗</sup>

*i*¼1

*AsrBδD*<sup>2</sup>

*<sup>M</sup>*sr*mod*2 180 ð Þ <sup>¼</sup> *<sup>M</sup>*max *mod*2 180 ð Þ*Kef*ð Þ <sup>180</sup> <sup>¼</sup> *<sup>π</sup>*

*<sup>P</sup>*el*mod*2 180 ð Þ <sup>¼</sup> *<sup>π</sup>*

2

*Pel mod*2 180�<sup>180</sup> ð Þ *<sup>m</sup> Pel mod*2 180 ð Þ

180/m) - degree commutation and 180-degree commutation:

coefficient of the pole arc (Eq.(56)) is shown in **Figure 23**.

¼ *π* 2

*Kcomp*\_*com*\_*mod*<sup>2</sup> ¼

phases starting from 3 and higher.

**25**

diameter of the disk of the magnetic system.

depend on the number of phases.

the neutral (see **Figure 21**)

*<sup>M</sup>*max *mod*2 180 ð Þ <sup>¼</sup> <sup>X</sup>*<sup>m</sup>*

commutation

*<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* <sup>∗</sup>

$$P\_{\text{elmod2}\left(180-\frac{100}{m}\right)} = \frac{\pi}{2} A\_{sr} B\_{\delta} \alpha D\_{sr}^2 L\_{ring} K\_{mod2\left(180-\frac{100}{m}\right)} K\_{\epsilon\left(180-\frac{100}{m}\right)}.\tag{50}$$

By analogy, we derive the equations for 180-degree commutation.

For 180-degree switching, the efficiency coefficient of the model is defined by the following expression

$$K\_{mad2(180)}(a,m) = \frac{M(a\_r,m)\_{(180)\,\max}}{M(a\_r=1,m)\_{(180)\,\max}} = \frac{M(a\_r,m)^\*\_{(180)\,\max}}{\pi},\tag{51}$$

where *<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* ð Þ <sup>180</sup> *max* – maximum moment of the armature winding at the real coefficient of pole overlap;

**Figure 19.**

*Position of multiphase armature winding sections at the maximum electromagnetic moment for (180–180/m) degree commutation.*

#### **Figure 20.**

*Efficiency coefficient of model 2 for (180–180/m) - degree switching at different values of the number of phases and the coefficient of pole overlap on the average diameter of the ring of the magnetic system.*

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis DOI: http://dx.doi.org/10.5772/intechopen.95945*

*<sup>M</sup>*ð Þ *<sup>α</sup>sr* <sup>¼</sup> 1, *<sup>m</sup>* ð Þ <sup>180</sup> *max* – maximum moment of the armature winding with a pole overlap coefficient equal to 1, which is only theoretically possible;

*<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* <sup>∗</sup> ð Þ <sup>180</sup> *max* – relative value of the maximum moment of the armature winding at the real coefficient of pole overlap.

Efficiency coefficient of model 2 for 180 - degree switching at different the coefficient of pole overlap on the average diameter of the ring of the magnetic system (Eq. (51)) is shown at **Figure 22**. We can see that this coefficient does not depend on the number of phases.

The relative value of the maximum moment of the armature winding can be determined from the expression (Eq.(21)) for the zero offset of the armature from the neutral (see **Figure 21**)

$$\left(M(a\_{\rm tr},m)\right)^{\*}\_{\left(180\right)\max} = M^{\*}\_{\left(180\right)\max}\left(\mathbf{0}\right) = \sum\_{i=1}^{m} M^{\*}\_{\hat{f}i}\left(\mathbf{0}\right).\tag{52}$$

Maximum electromagnetic torque for 180-degree commutation

$$\begin{split} M\_{\text{max}\,\text{mod}2(180)} &= \sum\_{i=1}^{m} M(a.r)\_{\text{fmax}\,\,\text{mod}2(180)} K\_{\text{mod}2(180)} = 2i\_a B\_\delta \frac{r\_{\text{out}}^2 - r\_{\text{in}}^2}{2} W\_S p K\_{\text{mod}2(180)}, \\ &= \frac{\pi}{2} A\_{sr} B\_\delta D\_{sr}^2 L\_{\text{ring}} K\_{\text{mod}2(180)}. \end{split} \tag{53}$$

The average electromagnetic torque and electromagnetic power for 180-degree commutation

$$M\_{\text{srmod2}(180)} = M\_{\text{max}\,\text{mod2}(180)} K\_{\text{ef}(180)} = \frac{\pi}{2} A\_{\text{sr}} B\_{\delta} D\_{gr}^2 L\_{\text{ring}} K\_{\text{mod2}(180)} K\_{\text{ef}(180)};\tag{54}$$

$$P\_{\text{elmod2}(180)} = \frac{\pi}{2} A\_{sr} B\_{\delta} a o D\_{sr}^2 L\_{ring} K\_{mod2(180)} K\_{ef(180)}.\tag{55}$$

By analogy with the previous analysis, for quantitative evaluation, we introduce the switching comparison coefficient, as the ratio of electromagnetic powers (180– 180/m) - degree commutation and 180-degree commutation:

$$K\_{comp\\_com\\_mod2} = \frac{P\_{el\\_mod2\left(180-\frac{180}{n}\right)}}{P\_{el\\_mod2\left(180\right)}} = \frac{K\_{mod2\left(180-\frac{180}{n}\right)}K\_{ef\left(180-\frac{180}{n}\right)}}{K\_{mod2\left(180\right)}K\_{ef\left(180\right)}}.\tag{56}$$

The graphical dependence of this coefficient on the number of phases and the coefficient of the pole arc (Eq.(56)) is shown in **Figure 23**.

It should be noted that such a comparative analysis makes sense only for the same electromagnetic loads: induction in the air gap and linear load on the average diameter of the disk of the magnetic system.

These dependencies are of great practical importance. Their analysis shows that for the same electromagnetic loads, magnetic systems with a high value of the pole overlap coefficient have an advantage for any number of phases. Graphic dependences of the switching comparison coefficient are below 1.0. However, for magnetic systems with a pole overlap coefficient of 0.7–0.5, which is very typical for practice, the advantage is (180–180/m) - degree commutation for the number of phases starting from 3 and higher.

Given that (180–180/m) - degree commutation has a simpler and cheaper technical implementation, this theoretical conclusion is of great practical importance.

Average electromagnetic moment and electromagnetic power for (180–180/m) -

*srLringKmod*2 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*srLringKmod*2 180�<sup>180</sup> ð Þ *<sup>m</sup>*

For 180-degree switching, the efficiency coefficient of the model is defined by

where *<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* ð Þ <sup>180</sup> *max* – maximum moment of the armature winding at the real

*Position of multiphase armature winding sections at the maximum electromagnetic moment for (180–180/m)-*

*Efficiency coefficient of model 2 for (180–180/m) - degree switching at different values of the number of phases*

*and the coefficient of pole overlap on the average diameter of the ring of the magnetic system.*

*Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

*Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

*Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

ð Þ 180 *max*

*<sup>π</sup>* , (51)

<sup>¼</sup> *<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* <sup>∗</sup>

; (49)

*:* (50)

<sup>¼</sup> *<sup>M</sup>*max *mod*2 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*AsrBδD*<sup>2</sup>

*AsrBδωD*<sup>2</sup>

By analogy, we derive the equations for 180-degree commutation.

*<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* ð Þ <sup>180</sup> *max <sup>M</sup>*ð Þ *<sup>α</sup>sr* <sup>¼</sup> 1, *<sup>m</sup>* ð Þ <sup>180</sup> *max*

¼ *π* 2

*Emerging Electric Machines - Advances, Perspectives and Applications*

¼ *π* 2

degree commutation

*<sup>M</sup>*sr*mod*2 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*<sup>P</sup>*el*mod*2 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*Kmod*2 180 ð Þð Þ¼ *α*, *m*

the following expression

coefficient of pole overlap;

**Figure 19.**

**Figure 20.**

**24**

*degree commutation.*

**Figure 21.**

*Position of multiphase armature winding sections at the maximum electromagnetic moment for 180-degree commutation.*

Electromagnetic moment j-th of the turn

**Figure 24.**

*windings.*

*M*ð Þ *α*,*r <sup>j</sup>* ¼

Electromagnetic moment of the phase section

*<sup>M</sup>*ð Þ *<sup>α</sup>*,*<sup>r</sup> WS* <sup>¼</sup> <sup>X</sup>

*M*ð Þ *α*,*r fimod*<sup>3</sup> ¼ *M*ð Þ *α*,*r WS*

We will consider this in the following equations.

is defined by the following expression

ð Þ¼ *α*, *m*

*Kmod*3 180�<sup>180</sup> ð Þ *<sup>m</sup>*

**27**

*r* ð*out*

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis*

*DOI: http://dx.doi.org/10.5772/intechopen.95945*

*rin*

*WS*

*j*¼1

*dM*ð Þ *<sup>α</sup>*,*<sup>r</sup> <sup>j</sup>* <sup>¼</sup> <sup>2</sup>*iaB*ð Þ *<sup>α</sup> <sup>r</sup>*<sup>2</sup>

*Sketch of the magnetic system and armature windings BMAMF with segment magnets toroidal armature*

*dM*ð Þ *<sup>α</sup>*,*<sup>r</sup> <sup>j</sup>* <sup>¼</sup> <sup>2</sup>*iaB*ð Þ *<sup>α</sup> <sup>r</sup>*<sup>2</sup>

*<sup>p</sup>* <sup>¼</sup> <sup>2</sup>*iaB*ð Þ *<sup>α</sup> <sup>r</sup>*<sup>2</sup>

Maximum value of the electromagnetic moment of an arbitrary i-th phase

By analogy with model 2, we introduce the efficiency coefficient of model 3, which is the ratio of the maximum moment of the armature winding at the real pole overlap coefficient to the maximum moment at the theoretical pole overlap coefficient equal to 1.0. It should be noted that the electromagnetic moment for model 3 is 2 times higher than the electromagnetic moment of model 1. This can be seen from the comparison of equation (Eq. (47)) and equation (Eq. (61)). From a physical point of view, this is because with the same external and internal diameters of the magnetic system, the electromagnetic moment in model 3 is created from 2 sides.

For (180–180/m)-degree communication the efficiency coefficient of the model

¼

<sup>2</sup>*<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* <sup>∗</sup>

*π*

*<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup> max <sup>M</sup>*ð Þ *<sup>α</sup>sr* <sup>¼</sup> 1, *<sup>m</sup>* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup> max* *out* � *<sup>r</sup>*<sup>2</sup> *in*

*out* � *<sup>r</sup>*<sup>2</sup> *in* 2

*out* � *<sup>r</sup>*<sup>2</sup> *in*

<sup>2</sup> *:* (58)

*WS:* (59)

<sup>2</sup> *WS*2*p:* (60)

<sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup> max*

*<sup>m</sup>* ð Þ *<sup>m</sup>* � <sup>1</sup> *:* (61)

#### **Figure 22.**

*The efficiency coefficient of model 2 for 180-degree commutation at different values of the pole overlap coefficient on the average diameter of the ring of the magnetic system.*

**Figure 23.**

*Dependence of the comparison coefficient (180–180/m) - degree commutation and 180-degree commutation for model 2.*

#### **3.4 Electromagnetic moment and electromagnetic power for BMAMF with segment magnets and toroidal coils**

The analysis will be carried out by analogy with the previous models. A sketch of the magnetic system and armature winding BMAMF of this design is shown in **Figure 24**. Let us designate this design as model 3.

Let us select an elementary conductor of length dr on an arbitrary j-th turn of the winding and determine the elementary moment for it.

$$d\mathcal{M}(a,r)\_j = i\_a B(a) r dr. \tag{57}$$

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis DOI: http://dx.doi.org/10.5772/intechopen.95945*

#### **Figure 24.**

*Sketch of the magnetic system and armature windings BMAMF with segment magnets toroidal armature windings.*

Electromagnetic moment j-th of the turn

$$\mathcal{M}(a,r)\_j = \int\_{r\_{in}}^{r\_{out}} d\mathcal{M}(a,r)\_j = 2i\_a \mathcal{B}(a) \frac{r\_{out}^2 - r\_{in}^2}{2}.\tag{58}$$

Electromagnetic moment of the phase section

$$\mathcal{M}(a,r)\_{W\_S} = \sum\_{j=1}^{W\_S} d\mathcal{M}(a,r)\_j = 2i\_a \mathcal{B}(a) \frac{r\_{out}^2 - r\_{in}^2}{2} W\_S. \tag{59}$$

Maximum value of the electromagnetic moment of an arbitrary i-th phase

$$M(a,r)\_{
\text{final3}} = M(a,r)\_{W\_{\text{S}}}p = 2i\_a B(a) \frac{r\_{out}^2 - r\_{in}^2}{2} W\_{\text{S}} 2p. \tag{60}$$

By analogy with model 2, we introduce the efficiency coefficient of model 3, which is the ratio of the maximum moment of the armature winding at the real pole overlap coefficient to the maximum moment at the theoretical pole overlap coefficient equal to 1.0. It should be noted that the electromagnetic moment for model 3 is 2 times higher than the electromagnetic moment of model 1. This can be seen from the comparison of equation (Eq. (47)) and equation (Eq. (61)). From a physical point of view, this is because with the same external and internal diameters of the magnetic system, the electromagnetic moment in model 3 is created from 2 sides. We will consider this in the following equations.

For (180–180/m)-degree communication the efficiency coefficient of the model is defined by the following expression

$$K\_{mod3\left(180-\frac{100}{m}\right)}(a,m) = \frac{M(a\_r,m)\_{\left(180-\frac{100}{m}\right)\max}}{M(a\_r=1,m)\_{\left(180-\frac{100}{m}\right)\max}} = \frac{2M(a\_r,m)^{\*}\_{\left(180-\frac{100}{m}\right)\max}}{\frac{\pi}{m}(m-1)}.\tag{61}$$

**3.4 Electromagnetic moment and electromagnetic power for BMAMF with**

*Dependence of the comparison coefficient (180–180/m) - degree commutation and 180-degree commutation for*

*The efficiency coefficient of model 2 for 180-degree commutation at different values of the pole overlap*

*Position of multiphase armature winding sections at the maximum electromagnetic moment for 180-degree*

*Emerging Electric Machines - Advances, Perspectives and Applications*

the magnetic system and armature winding BMAMF of this design is shown in

The analysis will be carried out by analogy with the previous models. A sketch of

Let us select an elementary conductor of length dr on an arbitrary j-th turn of the

*dM*ð Þ *α*,*r <sup>j</sup>* ¼ *iaB*ð Þ *α rdr:* (57)

**segment magnets and toroidal coils**

**Figure 22.**

**Figure 23.**

*model 2.*

**26**

**Figure 21.**

*commutation.*

**Figure 24**. Let us designate this design as model 3.

*coefficient on the average diameter of the ring of the magnetic system.*

winding and determine the elementary moment for it.

The relative value of the maximum moment of the armature winding, which is created on one side of the working air gap, can be determined from the expression (Eq. (17)), for the displacement of the armature from the neutral poles by an amount *<sup>x</sup>* <sup>¼</sup> *<sup>π</sup>* <sup>2</sup>*<sup>m</sup>* .

The graphical dependence of the coefficient for (180–180/m) - degree commutation is shown in **Figure 25**.

Maximum electromagnetic torque for (180–180/m)-degree commutation

$$M\_{\max\,\text{mod3}\left(180-\frac{100}{m}\right)} = \sum\_{i=1}^{m-1} M(a, m)\_{\text{fi\\_mod3}} K\_{\text{mod3}\left(180-\frac{100}{m}\right)} = $$

$$= 2i\_a B\_\delta \frac{r\_{out}^2 - r\_{in}^2}{2} W\_S p K\_{\text{mod3}\left(180-\frac{100}{m}\right)} = \frac{\pi}{2} A\_{sr} B\_\delta D\_{sr}^2 L\_{ring} K\_{mod3\left(180-\frac{100}{m}\right)}.\tag{62}$$

Average electromagnetic moment and electromagnetic power for (180–180/m) degree commutation, based on the above

$$\begin{split} M\_{\text{smod3}\left(180-\frac{100}{m}\right)} &= M\_{\text{max}\,mod3\left(180-\frac{100}{m}\right)} K\_{\text{eff}\left(180-\frac{100}{m}\right)} \\ &= \frac{\pi}{2} A\_{\text{sr}} B\_{\delta} D\_{\text{sr}}^2 L\_{\text{ring}} K\_{\text{mod3}\left(180-\frac{100}{m}\right)} K\_{\text{eff}\left(180-\frac{100}{m}\right)} \end{split} \tag{63}$$

*<sup>M</sup>*max *mod*3 180 ð Þ <sup>¼</sup> <sup>X</sup>*<sup>m</sup>*

commutation

**29**

**Figure 26.**

*i*¼1

*AsrBδD*<sup>2</sup>

*<sup>M</sup>*sr*mod*3 180 ð Þ <sup>¼</sup> *<sup>M</sup>*max *mod*3 180 ð Þ*Kef*ð Þ <sup>180</sup> <sup>¼</sup> *<sup>π</sup>*

*<sup>P</sup>*el*mod*3 180 ð Þ <sup>¼</sup> *<sup>π</sup>*

¼ *π* 2

*the average ring diameter of the magnetic system.*

*DOI: http://dx.doi.org/10.5772/intechopen.95945*

*Kcomp*\_*com*\_*mod*<sup>3</sup> ¼

*<sup>M</sup>*ð Þ *<sup>α</sup>:<sup>r</sup> fimax mod*3 180 ð Þ*Kmod*3 180 ð Þ <sup>¼</sup> <sup>2</sup>*iaB<sup>δ</sup>*

*Model 3 efficiency coefficient for 180-degree commutation with different values of the pole overlap coefficient on*

The average electromagnetic torque and electromagnetic power for 180-degree

2

By analogy with the previous analysis, for quantitative evaluation, we introduce the switching comparison coefficient for model 3 as the ratio of electromagnetic powers (180–180/m) - degree communication and 180-degree commutation:

¼

Since the analytical dependences of the efficiency coefficient for model 2 and model 3 are multiples of 2, the graphical dependence of this coefficient on the number of phases for different values of the pole overlap coefficient completely coincides with the curves shown in **Figure 23** for model 3, we can draw conclusions similar to those for model 2 regarding the advantages of commutations types when

**3.5 Electromagnetic moment and electromagnetic power for BMAMF with**

The armature winding for this design can be made by analogy with a radial design (wave or loop) or toroidal. It should be noted that the windings for these three options would differ only in the shape of the frontal parts, which will only affect the calculation of active and inductive resistances. The active zone with copper (the groove-tooth zone) will be identical for all variants. Consequently, the electromagnetic processes of mutual conversion of electromagnetic and mechanical energy will also be identical. This allows you to combine all design types with a

*AsrBδωD*<sup>2</sup>

*AsrBδD*<sup>2</sup>

*Kmod*3 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*Kmod*3 180 ð Þ*Kef* <sup>180</sup>

*srLringKmod*3 180 ð Þ*:*

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis*

2

*Pel mod*3 180�<sup>180</sup> ð Þ *<sup>m</sup> Pel mod*3 180 ð Þ

changing the number of phases and the pole overlap coefficient.

**segment magnets and toothed anchor**

*r*2 *out* � *<sup>r</sup>*<sup>2</sup> *in* 2

*srLringKmod*3 180 ð Þ*Kef*ð Þ <sup>180</sup> ; (67)

*srLringKmod*3 180 ð Þ*Kef*ð Þ <sup>180</sup> *:* (68)

*Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

*WSpKmod*3 180 ð Þ

*:* (69)

(66)

$$P\_{\text{ebmod3}\left(180-\frac{100}{m}\right)} = \frac{\pi}{2} A\_{sr} B\_{\delta} \alpha D\_{sr}^2 L\_{ring} K\_{mod3\left(180-\frac{100}{m}\right)} K\_{\varepsilon\left(180-\frac{100}{m}\right)}.\tag{64}$$

By analogy, we derive the equations for 180-degree switching.

For 180-degree commutation the efficiency coefficient of the model is defined by the following expression

$$K\_{mad3(180)}(a,m) = \frac{M(a\_{gr},m)\_{(180)\,\max}}{M(a\_{gr} = 1,m)\_{(180)\,\max}} = \frac{2M(a\_{gr},m)^\*\_{(180)\,\max}}{\pi}.\tag{65}$$

The relative value of the maximum moment of the armature winding can be determined from the expression (1.16) for the zero offset of the armature from the neutral (see **Figure 21**).

By analogy with model 2, the efficiency coefficient of model 3 for this type of switching will not depend on the number of phases and will be determined only by the value of the overlap coefficient on the average ring diameter of the magnetic system. The graphical dependence of this coefficient is shown in **Figure 26**.

Maximum electromagnetic torque for 180-degree commutation

**Figure 25.**

*The efficiency coefficient of model 3 for (180–180/m) - degree commutation at different values of the number of phases and the coefficient of pole overlap on the average diameter of the ring of the magnetic system.*

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis DOI: http://dx.doi.org/10.5772/intechopen.95945*

#### **Figure 26.**

The relative value of the maximum moment of the armature winding, which is created on one side of the working air gap, can be determined from the expression (Eq. (17)), for the displacement of the armature from the neutral poles by an

The graphical dependence of the coefficient for (180–180/m) - degree commu-

¼ *π* 2

Average electromagnetic moment and electromagnetic power for (180–180/m) -

<sup>¼</sup> *<sup>M</sup>*max *mod*3 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*AsrBδD*<sup>2</sup>

*<sup>M</sup>*ð Þ *<sup>α</sup>*, *<sup>m</sup> fi mod*3*Kmod*3 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

<sup>¼</sup> <sup>2</sup>*<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* <sup>∗</sup>

*AsrBδD*<sup>2</sup>

*srLringKmod*3 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*srLringKmod*3 180�<sup>180</sup> ð Þ *<sup>m</sup>*

For 180-degree commutation the efficiency coefficient of the model is defined

The relative value of the maximum moment of the armature winding can be determined from the expression (1.16) for the zero offset of the armature from the

By analogy with model 2, the efficiency coefficient of model 3 for this type of switching will not depend on the number of phases and will be determined only by the value of the overlap coefficient on the average ring diameter of the magnetic system. The graphical dependence of this coefficient is shown in **Figure 26**. Maximum electromagnetic torque for 180-degree commutation

*The efficiency coefficient of model 3 for (180–180/m) - degree commutation at different values of the number of*

*phases and the coefficient of pole overlap on the average diameter of the ring of the magnetic system.*

¼

*:* (62)

(63)

*:* (64)

*srLringKmod*3 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

ð Þ 180 *max*

*<sup>π</sup> :* (65)

*Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

Maximum electromagnetic torque for (180–180/m)-degree commutation

<sup>¼</sup> <sup>X</sup>*m*�<sup>1</sup> *i*¼1

*WSpKmod*3 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*Emerging Electric Machines - Advances, Perspectives and Applications*

¼ *π* 2

*AsrBδωD*<sup>2</sup>

By analogy, we derive the equations for 180-degree switching.

*<sup>M</sup> <sup>α</sup>sr* ð Þ , *<sup>m</sup>* ð Þ <sup>180</sup> *max <sup>M</sup>*ð Þ *<sup>α</sup>sr* <sup>¼</sup> 1, *<sup>m</sup>* ð Þ <sup>180</sup> *max*

¼ *π* 2

amount *<sup>x</sup>* <sup>¼</sup> *<sup>π</sup>*

¼ 2*iaB<sup>δ</sup>*

<sup>2</sup>*<sup>m</sup>* .

tation is shown in **Figure 25**.

*r*2 *out* � *<sup>r</sup>*<sup>2</sup> *in* 2

*<sup>M</sup>*max *mod*3 180�<sup>180</sup> ð Þ *<sup>m</sup>*

degree commutation, based on the above

*<sup>M</sup>*sr*mod*3 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*<sup>P</sup>*el*mod*3 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*Kmod*3 180 ð Þð Þ¼ *α*, *m*

by the following expression

neutral (see **Figure 21**).

**Figure 25.**

**28**

*Model 3 efficiency coefficient for 180-degree commutation with different values of the pole overlap coefficient on the average ring diameter of the magnetic system.*

$$\begin{split} M\_{\text{max}\,\text{mod}\,3(180)} &= \sum\_{i=1}^{m} M(a.r)\_{\text{fmax}\,\,\text{mod}\,3(180)} K\_{\text{mod}\,3(180)} = 2i\_a B\_\delta \frac{r\_{\text{out}}^2 - r\_{\text{in}}^2}{2} W\_\text{S} p K\_{\text{mod}\,3(180)}, \\ &= \frac{\pi}{2} A\_{sr} B\_\delta D\_{sr}^2 L\_{\text{ring}} K\_{\text{mod}\,3(180)}. \end{split} \tag{66}$$

The average electromagnetic torque and electromagnetic power for 180-degree commutation

$$M\_{\text{surmod}\mathfrak{z}(180)} = M\_{\text{max}\,\text{mod}\mathfrak{z}(180)} K\_{\mathfrak{z}f(180)} = \frac{\pi}{2} A\_{\text{gr}} B\_{\delta} D\_{\text{gr}}^2 L\_{\text{ring}} K\_{\text{mod}\mathfrak{z}(180)} K\_{\mathfrak{z}f(180)};\tag{67}$$

$$P\_{\text{elmod3}(180)} = \frac{\pi}{2} A\_{sr} B\_{\delta} a \nu D\_{gr}^2 L\_{ring} K\_{mod3(180)} K\_{ef(180)}.\tag{68}$$

By analogy with the previous analysis, for quantitative evaluation, we introduce the switching comparison coefficient for model 3 as the ratio of electromagnetic powers (180–180/m) - degree communication and 180-degree commutation:

$$K\_{comp\\_com\\_mod3} = \frac{P\_{el\\_mod3\left(180-\frac{140}{m}\right)}}{P\_{el\\_mod3\left(180\right)}} = \frac{K\_{mod3\left(180-\frac{140}{m}\right)}K\_{ef\left(180-\frac{140}{m}\right)}}{K\_{mod3\left(180\right)}K\_{ef\left(180\right)}}\,\tag{69}$$

Since the analytical dependences of the efficiency coefficient for model 2 and model 3 are multiples of 2, the graphical dependence of this coefficient on the number of phases for different values of the pole overlap coefficient completely coincides with the curves shown in **Figure 23** for model 3, we can draw conclusions similar to those for model 2 regarding the advantages of commutations types when changing the number of phases and the pole overlap coefficient.

#### **3.5 Electromagnetic moment and electromagnetic power for BMAMF with segment magnets and toothed anchor**

The armature winding for this design can be made by analogy with a radial design (wave or loop) or toroidal. It should be noted that the windings for these three options would differ only in the shape of the frontal parts, which will only affect the calculation of active and inductive resistances. The active zone with copper (the groove-tooth zone) will be identical for all variants. Consequently, the electromagnetic processes of mutual conversion of electromagnetic and mechanical energy will also be identical. This allows you to combine all design types with a

toothed anchor into one basic model. We denote it as model 4.to analyze the tooth structure, we apply a well-known technique: we place all the ampere-conductors in a uniform layer on the armature surface in the working air gap with an equivalent linear current load. The value of the induction in the gap for this model will be considered equivalent to the real induction. If this assumption is accepted, all analytical expressions, including the electromagnetic moment, electromagnetic power, and model efficiency coefficients, will be similar to the expressions for model 3.

**Table 1** shows a comparison of models for the variant: pole arc coefficient 0.8,

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis*

**Table 2** shows a comparison of models for the variant: pole arc coefficient 0.8,

**Tables 1** and **2** show that model 4 is the most efficient in terms of the electro-

These tables are convenient to use in practice for choosing the design and type of

Model 1 1 0.577 0.288 0.113

Model 2 1.733 1 0.5 0.195

Model 3 3.466 2 1 0.391

Model 4 8.87 5.12 2.56 1

Model 1 1 0.667 0.333¤ 0.13

Model 1 1.499 1 0.5 0.195

Model 1 3.0 2 1 0.391

Model 1 7.68 5.12 2.56 1

*Comparison of the efficiency of models based on the developed electromagnetic moment for 120-degree switching.*

**The model to compare with Model 1 Model 2 Model 3 Model 4**

**The model to compare with Model 1 Model 1 Model 1 Model 1**

number of phases 3,120-degree commutation.

*DOI: http://dx.doi.org/10.5772/intechopen.95945*

number of phases 3,180-degree commutation.

magnetic moment and electromagnetic power.

switching depending on the project.

**Comparison of 3-phase models for 120-degree commutation and the pole**

**Comparison of 3-phase models for 180-degree commutation and the pole**

*A similar comparison for 180-degree commutation.*

**arc coefficient 0.8**

**Model for comparison**

**Table 2.**

**31**

**arc coefficient 0.8**

**Model for comparison**

**Table 1.**

The average electromagnetic torque and electromagnetic power for (180–180/ m)-degree commutation on the basis of the above

$$\begin{split} M\_{\text{smod}4} \left( {}\_{180-\frac{100}{m}} \right) &= M\_{\text{max}\,\text{mod}4} \left( {}\_{180-\frac{100}{m}} \right) K\_{\text{eff}\left( {}\_{180-\frac{100}{m}} \right)} \\ &= \frac{\pi}{2} A\_{\text{sr}} B\_{\delta} D\_{\text{sr}}^2 L\_{\text{ring}} K\_{\text{mod}4\left( {}\_{180-\frac{100}{m}} \right)} K\_{\text{eff}\left( {}\_{180-\frac{100}{m}} \right)}, \end{split} \tag{70}$$

$$P\_{\text{elmod}4\left(180-\frac{180}{m}\right)} = \frac{\pi}{2} A\_{sr} B\_{\delta} a D\_{sr}^2 L\_{ring} K\_{mod4\left(180-\frac{180}{m}\right)} K\_{\varepsilon\left(180-\frac{180}{m}\right)}.\tag{71}$$

where *Kmod*4 180�<sup>180</sup> ð Þ *<sup>m</sup>* – the efficiency coefficient of the model, determined by **Figure 25**.

The average electromagnetic torque and electromagnetic power for 180-degree commutation

$$M\_{\text{surmod4}(180)} = M\_{\text{max}\,\text{mod4}(180)} K\_{\text{ef}(180)} = \frac{\pi}{2} A\_{\text{gr}} B\_{\text{ef}} D\_{\text{ring}}^2 K\_{\text{mod4}(180)} K\_{\text{ef}(180)};\tag{72}$$

$$P\_{\rm elmol4(180)} = \frac{\pi}{2} A\_{sr} B\_{\delta} \alpha D\_{gr}^2 L\_{ring} K\_{mod4(180)} K\_{ef(180)}.\tag{73}$$

where *Kmod*4 180 ð Þ – the efficiency coefficient of the model, determined by **Figure 26**.

It should be noted that, despite the analogy with model 3, the value of the average electromagnetic power and electromagnetic moment for model 4 would be approximately 4–6 times higher due to higher values of electromagnetic loads (induction in the gap and linear current load on the average armature diameter).

#### **3.6 Comparative analysis of structures at (180–180/m) - degree commutation and 180-degree commutation**

This analysis allows us to make a qualitative assessment of the effectiveness of models in terms of the development of the electromagnetic moment in the same volumes. Prong models are more efficient than models with a smooth anchor due to the large values of electromagnetic loads. For a BMAMF with a smooth anchor, due to different values of the model coefficient, we can conclude: model 2 is more efficient than model 1 and model 3 is more efficient than model 2 and, accordingly, model 1. A quantitative analysis of this efficiency is of Practical interest.

We perform this analysis using the following method: for a fixed number of phases, we determine the ratio of electromagnetic powers for different models and different switching options. For a BMAMF with a smooth armature, the value of the electromagnetic loads will be considered the same. The air gap induction value for the toothed armature is approximately 1.6 times higher than the gap induction value for the model 3 smooth armature (80% of the residual permanent magnet induction for the toothed armature and 50% for the smooth armature). Approximately the same ratio can be assumed for linear loads. These relations are confirmed by practical tests. Therefore, for a comparative analysis for model 4, you can enter an increasing coefficient of 2.56 compared to model 3. The Results are summarized in the table.

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis DOI: http://dx.doi.org/10.5772/intechopen.95945*

**Table 1** shows a comparison of models for the variant: pole arc coefficient 0.8, number of phases 3,120-degree commutation.

**Table 2** shows a comparison of models for the variant: pole arc coefficient 0.8, number of phases 3,180-degree commutation.

**Tables 1** and **2** show that model 4 is the most efficient in terms of the electromagnetic moment and electromagnetic power.

These tables are convenient to use in practice for choosing the design and type of switching depending on the project.


#### **Table 1.**

toothed anchor into one basic model. We denote it as model 4.to analyze the tooth structure, we apply a well-known technique: we place all the ampere-conductors in a uniform layer on the armature surface in the working air gap with an equivalent linear current load. The value of the induction in the gap for this model will be considered equivalent to the real induction. If this assumption is accepted, all analytical expressions, including the electromagnetic moment, electromagnetic power, and model efficiency coefficients, will be similar to the expressions for

The average electromagnetic torque and electromagnetic power for (180–180/

*Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

*srLringKmod*4 180�<sup>180</sup> ð Þ *<sup>m</sup>*

– the efficiency coefficient of the model, determined by **Figure 25**.

*Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

*Kef* <sup>180</sup>�<sup>180</sup> ð Þ *<sup>m</sup>*

*srLringKmod*4 180 ð Þ*Kef*ð Þ <sup>180</sup> ; (72)

*srLringKmod*4 180 ð Þ*Kef*ð Þ <sup>180</sup> *:* (73)

; (70)

*:* (71)

*srLringKmod*4 180�<sup>180</sup> ð Þ *<sup>m</sup>*

The average electromagnetic torque and electromagnetic power for 180-degree

2

*AsrBδωD*<sup>2</sup>

where *Kmod*4 180 ð Þ – the efficiency coefficient of the model, determined by

It should be noted that, despite the analogy with model 3, the value of the average electromagnetic power and electromagnetic moment for model 4 would be approximately 4–6 times higher due to higher values of electromagnetic loads (induction in the gap and linear current load on the average armature diameter).

**3.6 Comparative analysis of structures at (180–180/m) - degree commutation**

model 1. A quantitative analysis of this efficiency is of Practical interest.

This analysis allows us to make a qualitative assessment of the effectiveness of models in terms of the development of the electromagnetic moment in the same volumes. Prong models are more efficient than models with a smooth anchor due to the large values of electromagnetic loads. For a BMAMF with a smooth anchor, due to different values of the model coefficient, we can conclude: model 2 is more efficient than model 1 and model 3 is more efficient than model 2 and, accordingly,

We perform this analysis using the following method: for a fixed number of phases, we determine the ratio of electromagnetic powers for different models and different switching options. For a BMAMF with a smooth armature, the value of the electromagnetic loads will be considered the same. The air gap induction value for the toothed armature is approximately 1.6 times higher than the gap induction value for the model 3 smooth armature (80% of the residual permanent magnet induction for the toothed armature and 50% for the smooth armature). Approximately the same ratio can be assumed for linear loads. These relations are confirmed by practical tests. Therefore, for a comparative analysis for model 4, you can enter an increasing coefficient of 2.56 compared to model 3. The Results are summarized in the table.

*AsrBδD*<sup>2</sup>

<sup>¼</sup> *<sup>M</sup>*max *mod*4 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*AsrBδωD*<sup>2</sup>

*AsrBδD*<sup>2</sup>

2

m)-degree commutation on the basis of the above

¼ *π* 2

> ¼ *π* 2

*Emerging Electric Machines - Advances, Perspectives and Applications*

*<sup>M</sup>*sr*mod*4 180�<sup>180</sup> ð Þ *<sup>m</sup>*

where *Kmod*4 180�<sup>180</sup> ð Þ *<sup>m</sup>*

commutation

**Figure 26**.

**30**

*<sup>P</sup>*el*mod*4 180�<sup>180</sup> ð Þ *<sup>m</sup>*

*<sup>M</sup>*sr*mod*4 180 ð Þ <sup>¼</sup> *<sup>M</sup>*max *mod*4 180 ð Þ*Kef*ð Þ <sup>180</sup> <sup>¼</sup> *<sup>π</sup>*

**and 180-degree commutation**

*<sup>P</sup>*el*mod*4 180 ð Þ <sup>¼</sup> *<sup>π</sup>*

model 3.

*Comparison of the efficiency of models based on the developed electromagnetic moment for 120-degree switching.*


#### **Table 2.**

*A similar comparison for 180-degree commutation.*

## **4. Conclusion**

BMAMP is a new class of electric machines. Their use is expanding for electric drives for General industrial applications and special applications in medicine, space and robotics. The theory of their analysis is not fully developed. These studies expand the possibilities of this analysis. The following main conclusions can be drawn from the presented research.

**References**

[1] Gandzha, S.A Application of the Ansys Electronics Desktop Software Package for Analysis of Claw-Pole Synchronous Motor / S.A. Gandzha, B.I. Kosimov, D.S. Aminov //Machines.– 2019.–Vol. 7 No. 4 https://ieeexplore. ieee.org/document/8570132 DOI: 10.1109/GloSIC.2018.8570132

*DOI: http://dx.doi.org/10.5772/intechopen.95945*

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis*

[7] Gandzha S., Aminov D., Bakhtiyor K. Application of the combined excitation submersible hydrogenerator as an alternative energy source for small and medium rivers. IEEE Russian Workshop on Power Engineering and Automation of Metallurgy Industry. 4–5 Oct. 2019 Magnitogorsk, Russia. DOI: 10.1109 /

[8] Aydin, M. S. Huang and T. A. Lipo. "Axial Flux Permanent Magnet Disc Machines: A Review", In Conf. Record of SPEEDAM, , May 2004, pp. 61–71

[9] Akatsu K. and Wakui S. "A comparison between axial and radial flux PM motor by optimum design

[10] Gandzha, S., Kiessh, I. The highspeed axial gap electric alternator is the best solution for a gas turbine engine. The high-speed axial gap electric alternator is the best solution for a gas

GeoConference Surveying Geology and Mining Ecology Management, SGEM.

International Conference on Industrial

turbine engine. International Multidisciplinary Scientific

[11] Gandzha, S. Development of engineering technique for calculating magnet systems with permanent magnets / S. Gandzha, E. Kiessh, D.S.

Aminov //Proceedings - 2018

Engineering, Applications and Manufacturing, ICIEAM 2018.–2018 No. 10.15593/2224-9397/2019.1.04 https://ieeexplore.ieee.org/document/ 8728650 DOI:10.1109/ICIEAM.2018.

[12] Gandzha S., Bakhtiyor K.,

Aminov D. Development of a system of multi-level optimization for Brushless Direct Current Electric Machines. International Ural Conference on Electrical Power Engineering (Ural

8728650

method from the required NTcharacteristics", Conference Proceeding of ICEM2004, No. 361,

Cracow-Poland, 2004.

PEAMI.2019.8915294

[2] Gandzha, S. Selecting Optimal Design of Electric Motor of Pilgrim Mill Drive for Manufacturing Techniques Seamless Pipe / S. Gandzha, B. Kosimov,

D. Aminov //2019 International Conference on Industrial Engineering, Applications and Manufacturing, ICIEAM 2019.–2019 https://ieeexplore. ieee.org/document/8742941 DOI: 10.1109/ICIEAM.2019.8742941

[3] I.E. Kiessh, S.A. Gandzha, "Application of Brushless Machines with Combine Excitation for a Small and Medium Power Windmills", Procedia Engineering.– 2016.–Vol. 129.– P.191– 194, DOI: 10.1016/j.proeng.2016.12.031 https://www.sciencedirect.com/science/ article/pii/S1877705815039156?via%

[4] Gandzha, S., Andrey, S., Andrey, M., Kiessh, I. The design of the low-speed brushless motor for the winch which operates in see-water. International

GeoConference Surveying Geology and Mining Ecology Management, SGEM

[5] Gandzha, S. The application of the double-fed alternator for the solving of the wind power problems. International

GeoConference Surveying Geology and Mining Ecology Management, SGEM

[6] Gandzha S.A., Sogrin A.I., Kiessh I.E. The Comparative Analysis of Permanent Magnet Electric Machines with Integer and Fractional Number of Slots per Pole and Phase. Procedia Engineering 129:

Multidisciplinary Scientific

Multidisciplinary Scientific

408–414, December 2015.

**33**

3Dihub


Further research will be aimed at developing methods for computer-aided design of machines of this class.

## **Author details**

Sergey Gandzha\* and Dmitry Gandzha South Ural State University, Chelyabinsk, Russia

\*Address all correspondence to: gandzhasa@susu.ru

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Brushless Electric Machines with Axial Magnetic Flux: Analysis and Synthesis DOI: http://dx.doi.org/10.5772/intechopen.95945*

## **References**

**4. Conclusion**

drawn from the presented research.

efficiency of commutation types.

supply are selected.

design of machines of this class.

Sergey Gandzha\* and Dmitry Gandzha

provided the original work is properly cited.

South Ural State University, Chelyabinsk, Russia

\*Address all correspondence to: gandzhasa@susu.ru

**Author details**

**32**

BMAMP is a new class of electric machines. Their use is expanding for electric drives for General industrial applications and special applications in medicine, space and robotics. The theory of their analysis is not fully developed. These studies expand the possibilities of this analysis. The following main conclusions can be

*Emerging Electric Machines - Advances, Perspectives and Applications*

1.Classification of the main types of commutations is carried out. For a BMAMP with an arbitrary number of phases the discrete (180–180/m)-degree commutation and 180-degree commutation with galvanically isolated phase

2.The efficiency factor of the armature winding for various types of

commutations is given. It is convenient to use this factor to compare the

4.Classification of BMAMF designs based on the shape of permanent magnets and armature winding sections is carried out. Basic models for analysis are defined.

5.The resulting equation of the electromagnetic torque and electromagnetic power for base structures is given. The equations determine the dependence of energy indicators on the main dimensions, electromagnetic loads, design features, and type of commutation. Model efficiency factors are derived for all basic models.

electromagnetic power for the same dimensions and the same electromagnetic loads is carried out. The results of the analysis are summarized in tables for different types of switching and the number of phases, quantitatively showing the advantages of one design over another. Tables are convenient to use in engineering practice to select the best option depending on the project situation.

6.The comparative analysis of the effectiveness of the basic models for

Further research will be aimed at developing methods for computer-aided

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

3.The influence of the number of phases on the developed electromagnetic moment for (180–180/m)-degree switching and 180-degree switching is analyzed. It is proved that to increase the electromagnetic moment in the same dimensions and with the same electromagnetic loads, it is necessary to increase the number of phases and the pole arc coefficient for both types of commutation. [1] Gandzha, S.A Application of the Ansys Electronics Desktop Software Package for Analysis of Claw-Pole Synchronous Motor / S.A. Gandzha, B.I. Kosimov, D.S. Aminov //Machines.– 2019.–Vol. 7 No. 4 https://ieeexplore. ieee.org/document/8570132 DOI: 10.1109/GloSIC.2018.8570132

[2] Gandzha, S. Selecting Optimal Design of Electric Motor of Pilgrim Mill Drive for Manufacturing Techniques Seamless Pipe / S. Gandzha, B. Kosimov, D. Aminov //2019 International Conference on Industrial Engineering, Applications and Manufacturing, ICIEAM 2019.–2019 https://ieeexplore. ieee.org/document/8742941 DOI: 10.1109/ICIEAM.2019.8742941

[3] I.E. Kiessh, S.A. Gandzha, "Application of Brushless Machines with Combine Excitation for a Small and Medium Power Windmills", Procedia Engineering.– 2016.–Vol. 129.– P.191– 194, DOI: 10.1016/j.proeng.2016.12.031 https://www.sciencedirect.com/science/ article/pii/S1877705815039156?via% 3Dihub

[4] Gandzha, S., Andrey, S., Andrey, M., Kiessh, I. The design of the low-speed brushless motor for the winch which operates in see-water. International Multidisciplinary Scientific GeoConference Surveying Geology and Mining Ecology Management, SGEM

[5] Gandzha, S. The application of the double-fed alternator for the solving of the wind power problems. International Multidisciplinary Scientific GeoConference Surveying Geology and Mining Ecology Management, SGEM

[6] Gandzha S.A., Sogrin A.I., Kiessh I.E. The Comparative Analysis of Permanent Magnet Electric Machines with Integer and Fractional Number of Slots per Pole and Phase. Procedia Engineering 129: 408–414, December 2015.

[7] Gandzha S., Aminov D., Bakhtiyor K. Application of the combined excitation submersible hydrogenerator as an alternative energy source for small and medium rivers. IEEE Russian Workshop on Power Engineering and Automation of Metallurgy Industry. 4–5 Oct. 2019 Magnitogorsk, Russia. DOI: 10.1109 / PEAMI.2019.8915294

[8] Aydin, M. S. Huang and T. A. Lipo. "Axial Flux Permanent Magnet Disc Machines: A Review", In Conf. Record of SPEEDAM, , May 2004, pp. 61–71

[9] Akatsu K. and Wakui S. "A comparison between axial and radial flux PM motor by optimum design method from the required NTcharacteristics", Conference Proceeding of ICEM2004, No. 361, Cracow-Poland, 2004.

[10] Gandzha, S., Kiessh, I. The highspeed axial gap electric alternator is the best solution for a gas turbine engine. The high-speed axial gap electric alternator is the best solution for a gas turbine engine. International Multidisciplinary Scientific GeoConference Surveying Geology and Mining Ecology Management, SGEM.

[11] Gandzha, S. Development of engineering technique for calculating magnet systems with permanent magnets / S. Gandzha, E. Kiessh, D.S. Aminov //Proceedings - 2018 International Conference on Industrial Engineering, Applications and Manufacturing, ICIEAM 2018.–2018 No. 10.15593/2224-9397/2019.1.04 https://ieeexplore.ieee.org/document/ 8728650 DOI:10.1109/ICIEAM.2018. 8728650

[12] Gandzha S., Bakhtiyor K., Aminov D. Development of a system of multi-level optimization for Brushless Direct Current Electric Machines. International Ural Conference on Electrical Power Engineering (Ural

**Chapter 3**

**Abstract**

signal processing

**1. Introduction**

**35**

Detection of Stator and Rotor

Rotor Induction Machines:

Implementation

*Shahin Hedayati Kia*

Asymmetries Faults in Wound

Modeling, Test and Real-Time

This chapter deals with detection of stator and rotor asymmetries faults in wound rotor induction machines using rotor and stator currents signatures analysis. This is proposed as the experimental part of *fault diagnosis in electrical machines* course for master's degree students in electrical engineering at University of Picardie "Jules Verne". The aim is to demonstrate the main steps of real-time condition monitoring development for wound rotor induction machines. In this regard, the related parameters of classical model of wound rotor induction machine under study are initially estimated. Then, the latter model is validated through experiments in both healthy

and faulty conditions at different levels of the load. Finally, an algorithm is

Fourier transform, hardware-in-the-loop, induction motors, monitoring,

implemented in a real-time data acquisition system for online detection of stator and rotor asymmetries faults. An experimental test bench based on a three-phase 90 W wound rotor induction machine and a real-time platform for hardware-in-the-loop test are utilized for validation of the proposed condition monitoring techniques.

**Keywords:** AC motor protection, asynchronous rotating machines, fault diagnosis,

Fault diagnosis of electrical machines is a very active topic of research and several books have been published, which detail new developed techniques for efficient condition monitoring of electrical machines. The run-to-break is an unplanned strategy of maintenance that needs to be avoided at the expense of high emergency repair cost. By means of preventive maintenance at regular intervals, which is commonly shorter than the expected time between failures, the maintenance actions can be planned in advance. Any potential breakdown in industrial systems can be predicted through the condition based maintenance (CBM) so called 'predictive maintenance' which gives a reasonable remaining useful life and leads consequently to the optimum time maintenance planning [1]. Since the electrical machines are the key components of the majority of industrial processes, it is essential to setup a CBM in order to minimize their downtime and consequently

Con) 2019. 1–3 Oct. 2019 Chelyabinsk, Russia. DOI: 10.1109/URALCON.2019. 8877650

[13] Gandzha, S., Kiessh, I. Selection of winding commutation for axial gap machines with any phases. Proceedings - 2018 International Conference on Industrial Engineering, Applications and Manufacturing, ICIEAM 2018

[14] Gandzha S.A. Application of Digital Twins Technology for Analysis of Brushless Electric Machines with Axial Magnetic Flux / Gandzha, S. // Proceedings - 2018 Global Smart Industry Conference, GloSIC 2018.– 2018 https://ieeexplore.ieee.org/docume nt/8570132 DOI:10.1109/ GloSIC.2018.8570132

[15] Gandzha, S. Design of Brushless Electric Machine with Axial Magnetic Flux Based on the Use of Nomograms / S. Gandzha, D. Aminov, B. Kosimov // Proceedings - 2018 International Ural Conference on Green Energy, UralCon 2018.–2018.– P.282–287 https:// ieeexplore.ieee.org/document/8544320 DOI: 10.1109/URALCON.2018.8544320

## **Chapter 3**

Con) 2019. 1–3 Oct. 2019 Chelyabinsk, Russia. DOI: 10.1109/URALCON.2019.

*Emerging Electric Machines - Advances, Perspectives and Applications*

[13] Gandzha, S., Kiessh, I. Selection of winding commutation for axial gap machines with any phases. Proceedings - 2018 International Conference on Industrial Engineering, Applications and Manufacturing, ICIEAM 2018

[14] Gandzha S.A. Application of Digital Twins Technology for Analysis of Brushless Electric Machines with Axial

Magnetic Flux / Gandzha, S. // Proceedings - 2018 Global Smart Industry Conference, GloSIC 2018.– 2018 https://ieeexplore.ieee.org/docume

[15] Gandzha, S. Design of Brushless Electric Machine with Axial Magnetic Flux Based on the Use of Nomograms / S. Gandzha, D. Aminov, B. Kosimov // Proceedings - 2018 International Ural

UralCon 2018.–2018.– P.282–287 https:// ieeexplore.ieee.org/document/8544320 DOI: 10.1109/URALCON.2018.8544320

Conference on Green Energy,

**34**

nt/8570132 DOI:10.1109/ GloSIC.2018.8570132

8877650
