**2.1 Modeling of the voltage source**

To model the voltage source (*E0*) of the battery-pack, the relationship between the open circuit voltage (OCV) and the state of charge is calculated by means of current interruption tests. First, the battery-pack is totally charged applying the constant current-constant voltage method (25 A until reaching the maximum voltage). After charging process, the device is discharged at current pulses of 10 A for 30 min (0.1 SOC variation) followed by 90 min of relaxation time. The OCV value for each test point is recorded when the relaxation time ends. Finally, the batterypack is recharged at current pulses of 10 A for 30 min (0.1 SOC variation) followed by 90 min of relaxation time as in the case of discharge process. In the same manner, the OCV values are recorded at the end of the relaxation time. The results of these tests are shown in **Figure 2** (discharging test) and **Figure 3** (charging test). **Table 2** presents the OCV values associated with each SOC test point, and this relationship is also sketched in **Figure 4**. As it can be seen, at the end of charge test, the final value of 100% of SOC is not reached because the BMS limits the applied current during the two last pulses. It is important to highlight that this situation does not occur when a single cell is tested, because the cell is charged and discharged from 100% SOC to 0% SOC without protection of BMS. In addition, test results do not show a high deviation of the average values as is reported in literature [25]; therefore, these values are used to evaluate the OCV-SOC relationship, which is presented in Eq. (1).

$$E\_o \text{(SOC,} t) = 26.05 - 0.15 \cdot \text{SOC} + 3.51 \cdot \text{SOC}^2 \tag{1}$$

**147**

**Figure 3.** *Charge test result.*

**Table 2.**

*OCV values at each test point.*

*Hybrid Modeling Procedure of Li-Ion Battery Modules for Reproducing Wide Frequency…*

**Discharge results Charge results**

**OCV SOC (%) OCV SOC (%)** 29.40 100 29.28 90.27 28.77 90 29.06 87.64 28.23 80 28.65 80 27.79 70 28.09 70 27.40 60 27.62 60 26.89 50 27.08 50 26.65 40 26.76 40 26.49 30 26.57 30 26.33 20 26.33 20

*DOI: http://dx.doi.org/10.5772/intechopen.88718*

**2.2 Modeling of the battery-pack's complex impedance**

experimental test bench deeply explained in [26].

To carry out EIS tests, an impedance analyzer is generally used. This device generates a frequency sweep signal and measures the voltage and current in each test point. As a result, the complex impedance is calculated. Because most of commercial impedance analyzer generates AC signals less than 100 mA (suitable for cell testing), in this work, this signal is amplified and controlled by means of the

EIS tests have been performed at different SOC values (20, 40, 60, 80, and 90% SOC) to analyze the effects of these SOC variations. The frequency sweep has been set from 1 mHz to 5 kHz (typically test range), with an AC ripple of 5 A (10% *Imax*). **Figure 5** shows the Nyquist plots of the EIS results, which has been used to analyze the impedance behavior of the tested battery module. In this graph, the real part of the complex impedance (Z') is represented along the x-axis and the imaginary part (Z") along the y-axis. The capacitive behavior corresponds to negative values of Z" and the inductive behavior to the positive ones. In this way, it is easy to identify the parameters of the electrical circuit. According to EIS results, the impedance

**Figure 2.** *Discharge test result.*

*Hybrid Modeling Procedure of Li-Ion Battery Modules for Reproducing Wide Frequency… DOI: http://dx.doi.org/10.5772/intechopen.88718*

#### **Figure 3.** *Charge test result.*

*Research Trends and Challenges in Smart Grids*

**2.1 Modeling of the voltage source**

**Table 1.**

*Battery-pack technical data.*

To model the voltage source (*E0*) of the battery-pack, the relationship between the open circuit voltage (OCV) and the state of charge is calculated by means of current interruption tests. First, the battery-pack is totally charged applying the constant current-constant voltage method (25 A until reaching the maximum voltage). After charging process, the device is discharged at current pulses of 10 A for 30 min (0.1 SOC variation) followed by 90 min of relaxation time. The OCV value for each test point is recorded when the relaxation time ends. Finally, the batterypack is recharged at current pulses of 10 A for 30 min (0.1 SOC variation) followed by 90 min of relaxation time as in the case of discharge process. In the same manner, the OCV values are recorded at the end of the relaxation time. The results of these tests are shown in **Figure 2** (discharging test) and **Figure 3** (charging test). **Table 2** presents the OCV values associated with each SOC test point, and this relationship is also sketched in **Figure 4**. As it can be seen, at the end of charge test, the final value of 100% of SOC is not reached because the BMS limits the applied current during the two last pulses. It is important to highlight that this situation does not occur when a single cell is tested, because the cell is charged and discharged from 100% SOC to 0% SOC without protection of BMS. In addition, test results do not show a high deviation of the average values as is reported in literature [25]; therefore, these values are used to evaluate the OCV-SOC relationship, which is presented in Eq. (1).

Cells reference MP 176065 Int (Saft batteries)

Pack rated voltage 25.9 V Pack maximum voltage 29.4 V (4.2 V per cell) Pack minimum cutoff voltage 20.3 V (2.9 V per cell) Pack capacity, *Cn* 50 Ah Pack maximum current 50 A Range of temperature (charge) −20 to 60°C Range of temperature (discharge) −30 to 55°C

*Eo* (*SOC*,*t*) = 26.05 − 0.15 ⋅ *SOC* + 3.51 ⋅ *SOC*2 (1)

**146**

**Figure 2.** *Discharge test result.*

### **Table 2.**

*OCV values at each test point.*


*2.8. TreeNet Regression*

We used the TreeNet regression algorithm to determine influential features on the positioning accuracy of the u-blox ZED-F9P. Numerous advantages have been reported for TreeNet, in comparison with other machine learning based approaches. In addition to its highly accurate predictions, TreeNet is not sensitive to errors in data or missing data. No data pre-processing (e.g., transformation, normalisation, or reduction) or preselection of the variables is required. TreeNet is strong against overfitting, and the process of growing trees is extraordinarily fast [54,55].

TreeNet begins with an initial model, which consists of a very small tree. This simple model is deliberately weak. The residuals are computed for each data in the first model and are used to grow the second tree. The residuals of the second tree are then computed and used to grow the third tree. Likewise, this process repeats to generate a sequence of hundreds or thousands of trees, in order to achieve an optimal tree. All trees contribute to the optimal model. The final model prediction is based on the total contribution of the individual trees, which is known as score. The accuracy of the TreeNet score will improve steadily by increasing the number of trees until to reach an optimal number of trees [55].

We used a dataset including 2000 sample objects. Twenty percent of the sample objects were randomly assigned as the testing set and the remaining 80% as the learning set. The TreeNet loss function was set on Huber-M. The mean square error (MSE) was chosen as the criterion for determining the number of optimal trees. We set the initial tree size at 10,000 and generated 12 TreeNet models based on the different learning rates and tree complexity levels. The optimal model was selected based on the one that recorded the minimum MSE, and its parameters were tuned for the final TreeNet model. The performance of the model was tested using the area under the receiver operating characteristic (ROC) curve. A value greater than 0.9 represents high performance, while values less than 0.7 indicate low performance [56] of the TreeNet model.

The influence of features on the positioning accuracy of the u-blox ZED-F9P was determined via relative importance [55]. The importance values of the features are ranged between 0 and 100. The most influential feature gains a value of 100 and the remaining features are rescaled to reflect their importance relative to this feature. We produced partial dependence (PD) plots for individual and pairs of features that contributed to the predicted positioning accuracy in the model. The PD plots represent the response of the target variable to individual or pairs of features, as all remaining features are taken into account [57].

**Table 1.** Object features affecting on the accuracy of positioning by u-blox ZED-F9P, derived from different metrics of high-density LiDAR data.

### **3. Results**


*4.2. Checking the Assumptions of Multiple Regression*

Before analysis, the data were verified, assessing whether they were suitable for analysis concerning multicollinearity outliers, normality, linearity, and homoscedasticity.

**Figure 2.** Characteristics of the respondents: Gender, age, education, ethnicity, and employment.


**Theory of Fuzzy Logic**

**Chapter 1**

**Provisional chapter**

**Introductory Chapter: Which Membership Function is**

**Introductory Chapter: Which Membership Function is**

Fuzzy logic systems are widely used for control, system identification, pattern recognition problems, and many more applications from industry to academia. The membership functions (MFs) play vital role in the overall performance of fuzzy representation. The MFs are the building blocks of fuzzy set theory, that is, fuzziness in a fuzzy set is determined by its MF. Accordingly, the shapes of MFs are important for a particular problem since they effect on a fuzzy inference system. They may have different shapes such as triangular, trapezoidal, Gaussian, and so forth. The only condition a MF must really satisfy is that it must vary

The MFs can be of any shape and form as long as it maps the given data with desirable degree of memberships. As far as choice of MFs is concerned, it is us to decide. This is where fuzzy system offers individual degrees of freedom. With experience, one will come to know which

As there are infinite number of ways to characterize fuzziness, there are infinite number of ways to graphically depict the MFs that describe this fuzziness. The choice of which of the methods to use depends entirely on the problem size and problem type. Instead of choosing the shape of MF, setting the interval and number of MFs are also very important. For instance, to model a temperature control system by fuzzy logic, it is really important to know how

**2. How to choose an appropriate membership function?**

shape of MF is good for the application under consideration.

© 2016 The Author(s). Licensee InTech. 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.

© 2018 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.

DOI: 10.5772/intechopen.79552

**Appropriate in Fuzzy System?**

**Appropriate in Fuzzy System?**

Additional information is available at the end of the chapter

Ali SadollahAdditional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.79552

**1. Role of membership functions**

Ali Sadollah

between 0 and 1.

#### **Introductory Chapter: Which Membership Function is Appropriate in Fuzzy System? Introductory Chapter: Which Membership Function is Appropriate in Fuzzy System?**

DOI: 10.5772/intechopen.79552

Ali Sadollah

Additional information is available at the end of the chapter Ali SadollahAdditional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.79552


We are IntechOpen, the world's leading publisher of Open Access books Built by scientists, for scientists

5,400+

Open access books available

132,000+

International authors and editors

160M+

Downloads

156 Countries delivered to Our authors are among the

Top 1% most cited scientists

12.2%

Contributors from top 500 universities

Selection of our books indexed in the Book Citation Index in Web of Science™ Core Collection (BKCI)


**4. Three types of terrorist attack scenarios**

to probability analysis, it is also necessary to apply the tools of game theory, which makes it

*Complex nature of the terrorist threat*: The presence of a terrorist organization in a region may give rise to the possibility of a broad spectrum of attack scenarios. Thus, to counter terrorist threats and terrorist mechanisms for initiating emergency situations to an even greater degree than for natural and man-made risks, a systemic approach is needed for ensuring security and developing an optimal strategy for counterterrorism force and resource deployment. Inasmuch as concentrating resources on protecting one system element (or protecting a target from one scenario of terrorist action) could prove useless because, after evaluating the situation, the terrorists could redirect the attack against another element of the system or switch to a different attack scenario. In this case, counterterrorism efforts will fail to reduce risk and increase the system's level of protection. *Presence of two-way linkages between the terrorist threat and system vulnerability*: The structure of linkages among the risk factors for the given *CES* in case of natural or manmade catastrophes is presented in **Figure** 3**a**. One differentiating feature of a terrorist risk assessment is the presence of two-way linkages (feedbacks) between the terrorist threat and (a) vulnerability of the system to the threat and (b) the magnitude of expected losses if the threat is successfully realized (see **Figure** 3**b**). This characteristic of terrorism must be examined in detail. In particular, reducing the vulnerability of a given system makes it possible to reduce substantially the level

In terrorist risk assessment framework, the main challenge is to estimate the probability of a terrorist attack. Some specialists believe that probabilistic measure is not adequate for the terrorist risk assessment since terrorist attack is not a stochastic event but a deliberate action based on the assessment made by terrorists regarding their skills and capabilities and the

Assignment of probabilities to the terrorist attack is a task which has a substantial human and behavioral dimension. The main problem is to describe the intentions of terrorists, their preferences, system of values (i.e., utility function), and decision rule. This allows assessing

**Figure 3.** (a) System of linkages among risk factors for natural or man-made hazards (safety context). (b) System of

possible to take into account the intentional actions of terrorists.

of the terrorist threat it faces.

262 Probabilistic Modeling in System Engineering

system's vulnerabilities.

probability of different attack scenarios.

linkages among risk factors for terrorist threat (security context).

Scenarios of terrorist attacks can be divided into three types, scenarios of ordinary, technological, and intelligent terrorism, that differ in resources used by terrorists to carry out the attacks and structure of losses inflicted by the attacks (**Figure** 1) [17–19].

Scenarios of ordinary terrorism imply organization of explosions, fires, and assassinations of officials, public figures, and people at large in order to intimidate people and destabilize political situation in the country or region. Scenarios of ordinary terrorism are not considered in this paper since these scenarios are not focused on complex engineering systems. We are going to deal with two other types of terrorist attack scenarios that are directly related to *CES.*


**3. How do constructed wetlands treat stormwater?**

To enable efficient operation and maintenance of constructed wetlands, the current practice is to split the treatment processes into separate zones; the sediment pond and macrophyte zone. **Figure 1** shows the design layout for constructed wetlands within Melbourne as per the guideline and **Figure 2** indicates the sediment storage pond.

#### **3.1 Sediment removal**

The sediment pond is utilized to remove sediments before the wetland. Current design guidelines require that sediment ponds shall be sized to retain 95% of sand particles (i.e., particles down to 125 μm) during a 1 in 3-month ARI rainfall event [16].

**Figure 2.** *Sediment pond storage [17].*

Regular cleanout of retained sediment is required whenever retained sediment levels reach 500 mm below the normal water level (NWL) [17]. There is conjecture as to how well this maintenance is carried out, as many existing constructed wetlands do not receive regular sediment maintenance.

Secondly, the Macrophyte Zone (shallow water-filled ponds planted with aquatic plants) is utilized to contain the remaining sediments not captured by the sediment pond by the velocity reducing and filtering effect of Macrophyte (aquatic plant) stems and root systems [18].

*Understanding the Role of Constructed Wetlands in Stormwater Management DOI: http://dx.doi.org/10.5772/intechopen.102912*

### *3.1.1 Nitrogen removal*

The macrophyte zone removes nitrogen through the Nitrification–denitrification process [19, 20]. Sediment ponds also provide limited nutrient treatment. This includes an attachment of approx. 15% of Total Nitrogen (TN) and Total Phosphorus (TP) to particles larger than 300 um [21], and small amounts via anoxic biochemical processes [3]. Studies on the Prado wetlands in California USA suggest that constructed wetlands have, on average, 50–60% TN removal [22].


**Exact Two-Dimensional Analytical Calculations for Magnetic Field, Electromagnetic Torque, UMF, Back-EMF, and Inductance of Outer Rotor Surface Inset Permanent Magnet Machines**

#### **AmirAbbas Vahaj \*, Akbar Rahideh, Hossein Moayed-Jahromi and AliReza Ghaffari**

Department of Electrical and Electronics Engineering, Shiraz University of Technology, Shiraz 13876-71557, Iran<EMAIL_ADDRESS>(A.R<EMAIL_ADDRESS>(H.M.-J<EMAIL_ADDRESS>(A.G.)

**\*** Correspondence<EMAIL_ADDRESS>
Received: 7 January 2019; Accepted: 13 February 2019; Published: 17 February 2019

**Abstract:** This paper presents a two-dimensional analytical model of outer rotor permanent magnet machines equipped with surface inset permanent magnets. To obtain the analytical model, the whole model is divided into the sub-domains, according to the magnetic properties and geometries. Maxwell equations in each sub-domain are expressed and analytically solved. By using the boundary/interface conditions between adjacent sub-regions, integral coefficients in the general solutions are obtained. At the end, the analytically calculated results of the air-gap magnetic flux density, electromagnetic torque, unbalanced magnetic force (UMF), back-electromotive force (EMF) and inductances are verified by comparing them with those obtained from finite element method (FEM). One of the merits of this method in comparison with the numerical model is the capability of rapid calculation with the highest precision, which made it suitable for optimization problems.

**Keywords:** analytical model; partial differential equations; separation of variable technique; electrical machines; surface inset permanent magnet

#### **1. Introduction**

The existence of different types of PM brushless machines (PMBLM) made them applicable for a wide range of applications. PMBLMs have superiorities in comparison with other rivals like induction machines or reluctance synchronous machines due to higher efficiency, high torque per volume, lower torque ripple, lower vibration, and lower acoustic noise.

PMBLM can be categorized in terms of various criteria such as the topology of PMs, the relative position of the rotor and stator, the slotted or slotless stator structure, etc.

Various PM topologies such as surface-mounted, surface-inset, and interior are used where each has its own advantages and disadvantages. Among these topologies, surface-inset can provide a compromise between the other two topologies.

Electric machines with single rotor and single stator can be either inner rotor or outer rotor. The outer rotor motors can develop more output torque than the inner ones for the same volume of the machine. Usually, inner rotor machines are used for applications, which need rapid acceleration and deceleration. Outer rotor machines usually are used for applications which need constant speed. Also, the mechanical robustness of the PMs in the outer rotor configuration is higher than the inner one.

In this paper because of the aforementioned advantages of the outer rotor machines and surface inset PMs, an exact two-dimensional electromagnetic model for this type of machines is extracted.

In the design procedure, the static model is normally considered. Numerous static models have been presented for electric machines, in which some of them are based on the analytical approaches [1–52], and the others are based on numerical methods [53–55].

The presented analytical model for electric machines are based on permeance model [1] or magnetic equivalent circuit, also known as (a.k.a.) 0-D analytic model [2], or resolution of the Maxwell's equations in 2-D plane (a.k.a. 2-D analytic model) [3–48] or 3-D plane (a.k.a. 3-D analytic model) [49,50]. Also, other methods based on mapping techniques such as Schwarz-Christoffel have been used to extract the model of some machine with the analytical approach [51,52]. The most accurate presented models among the analytical methods are 3-D analytic and 2-D analytic. The 2-D analytic method can be used instead of 3-D when the model is symmetric and has no skewing. Two-dimensional analytical models are not only capable of considering a high number of harmonics which elevate the precision of the models, but also have less computational time in comparison with the numerical methods and made them appropriate for optimal design problems.

Two-dimensional analytical models are presented for the slotted [3–17,19,20,29–38,41,42,44,45] and slotless [21–28] machines, equipped with surface mounted [3–6,10–12,14–20,23,26,32,34,37,41,52] and surface inset [9,13,21,22,24,25,35,36] or spoke type magnets [7,8], in order to obtain the important quantities like magnetic flux density, electromagnetic torque, unbalanced magnetic force, back-EMF, and inductances. Also, the 2-D analytical model is used to calculate the eddy current effect [11,18,22,31,32,49,50] in electrical machines. Most of the abovementioned publications are focused on the inner rotor structure [5–18,22–27,31–38,42,44–46] and only a few of them present the 2-D model for outer rotor machines with surface mounted PMs [4,28–30,41]. Therefore, to the best knowledge of the authors, it is for the first time that 2-D analytical model of brushless PM machines with outer rotor and surface inset PM is presented using the subdomain technique.

Most of the developed 2-D or 3-D analytic model are assumed with the infinite permeability of the iron parts. New techniques to account for finite soft-magnetic permeability have been recently developed, i.e., the multi-layer model using the Cauchy's product theorem is presented in [38], and the subdomain technique by applying the superposition principle in both directions is proposed in [39–46] which can be used to calculate the core losses and saturation phenomena.

An overview of the analytical models in the Maxwell-Fourier method with a global or local saturation effect has been realized in [40]. According to [48], Dubas' superposition technique [39,40] is very interesting since it enables the magnetic field calculation in the material of slotted geometries. This superposition technique has been implemented in radial-flux electrical machines with(out) PMs supplied by a direct or alternate current [44,45].

The presented technique in [39–46] is not only used to predict the magnetic field in all parts of the electrical machines, but also it is used to obtain a 2-D analytical model of the steady state heat transfer of the electrical machines by solving the heat equations [47].

The aim of this paper is to extract a 2-D analytical model of PM brushless outer rotor machines equipped with surface inset PMs. The model is used to analytically compute the electromagnetic torque, torque ripple, back-electromotive force (EMF), inductances and unbalanced magnetic forces (UMF).

Therefore, this paper is organized as follows. In Section 2 the procedure of extracting the 2-D model is explained. Section 3 is dedicated to the calculation of the electromagnetic quantities. In Section 4 the analytical results of the case study are presented and compared with those of the numerical method. In the final part this paper is concluded.

#### **2. Extracting the Magnetic Model**

#### *2.1. Assumptions*

Figure 1 shows the topology of an outer rotor surface inset brushless permanent magnet machine.

**Figure 1.** Outer rotor surface inset brushless permanent magnet machine illustration.

In order to make the problem solvable, below assumptions are made:

#### *2.2. Dividing Region into Sub-Regions*

According to the shape and material characteristics, the whole domain is divided into a number of sub-domains. All the sub-domains are illustrated in Figure 1 and listed in Table 1 for a PMBLM with *Q* slots and *p* pole-pairs.

When the winding is single-layer or double-layer non-overlapping, as shown respectively in Figure 2a,b, each slot is a single subdomain; however, if the winding is two-layer overlapping, as shown in Figure 2c, each slot is divided into two subdomains.

**Table 1.** The sub-domains and related symbols.

**Figure 2.** Winding topologies (**a**) single layer alternate teeth wound (**b**) double layer all teeth wound non-overlapping (**c**) double-layer overlapping.

#### *2.3. Extracting the Magnetic Model*

In this part, for each sub-domain a partial differential equation is extracted based on Maxwell's equations.

Maxwell's equations in quasi-static form are as follows:

$$
\nabla \cdot \mathbf{B} = 0 \tag{1}
$$

$$
\nabla \times \mathbf{H} = \mathbf{J} + \hat{\mathbf{J}} \tag{2}
$$

where **B** is the magnetic flux density vector. Ampere's law represents the relation between the magnetic field intensity vector (**H**), external current density vector (**J**), and current density vector in media (**^ J**). In this investigation the current density vector in media is assumed to be negligible, i.e., **^ J** = 0.

The relation between the magnetic flux density vector and the magnetic field intensity vector in permanent magnet media with linear demagnetizing curve is as follows:

$$\mathbf{B} = \mu\_0 \mu\_\mathbf{r} \mathbf{H} + \mu\_0 \mathbf{M} \tag{3}$$

where **M** is the magnetization vector.

Substituting (3) in (2) yields

$$
\nabla \times \mathbf{B} = \mu\_0 \mu\_r \mathbf{J} + \mu\_0 \nabla \times \mathbf{M} \tag{4}
$$

The magnetic flux density vector can be represented as the curl of the magnetic vector potential (**A**):

$$\mathbf{B} = \nabla \times \mathbf{A} \tag{5}$$

Using (4) and (5) the following expression is obtained:

$$
\nabla^2 \mathbf{A} = -\mu\_0 \mu\_r \mathbf{J} - \mu\_0 \nabla \times \mathbf{M} \tag{6}
$$

For each sub-domain, Equation (6) results in Poisson equations for the magnet and slot regions, and Laplace equations for the air-gap and slot-opening sub-domains, as represented below.

$$
\nabla^2 \mathbf{A}^{sl,j} = -\mu\_0 \mu\_r \mathbf{J} \tag{7}
$$

$$
\nabla^2 \mathbf{A}^{m,k} = -\mu\_0 \nabla \times \mathbf{M} \tag{8}
$$

$$
\nabla^2 \mathbf{A}^\chi = 0,\ \chi = \{ (a), (\text{so}, j) \}\tag{9}
$$

In 2-D polar coordinates, the magnetic vector potential and the current density vector have just a component along *z*, i.e., **A** = [0, 0, *Az*(*r*, *θ*)] and **J** = [0, 0, *Jz*(*θ*, *t*)]. Also, the magnetic flux density vector and the magnetization vector have radial and tangential components as below:

$$\mathbf{B} = \left[ B\_r(r, \theta), B\_\theta(r, \theta), 0 \right]^2$$

$$\mathbf{M} = \left[ M\_{\mathcal{I}}(r, \theta), M\_{\theta}(r, \theta), 0 \right]$$

Therefore, Equations (7)–(9) are rewritten as

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial A\_z^{sl,j}}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 A\_z^{sl,j}}{\partial \theta^2} = -\mu\_0 I\_z^{sl,j} \tag{10}$$

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial A\_z^{m,k}}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 A\_z^{m,k}}{\partial \theta^2} = -\frac{\mu\_0}{r}\left(M\_\theta^k - \frac{\partial M\_r^k}{\partial \theta}\right) \tag{11}$$

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial A\_z^{\chi}}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 A\_z^{\chi}}{\partial \theta^2} = 0,\\ \chi = \{(a)\_\prime (so, j)\}\tag{12}$$

#### *2.4. Boundary Conditions*

The perpendicular magnetic flux density in two adjacent sub-domains must be equal as mathematically represented as follows:

$$\mathbf{n}. (\mathbf{B}^i - \mathbf{B}^{i+}) = 0\tag{13}$$

In this equation, **B***i* is the magnetic flux density in sub-domain *i*, and **B***i*+ is the magnetic flux density in the sub-domain *i*+.

Also, if there is no current between the two adjacent sub-domains, the tangential components of the magnetic field intensity at the boundary of the two sub-domains are equal; this expression is shown mathematically by Relation (14).

$$\mathbf{n} \times (\mathbf{H}^i - \mathbf{H}^{i+}) = 0 \tag{14}$$

In this equation **H***i* is the magnetic field intensity of the sub-domain *i* and **H***i*+ is the magnetic field intensity of sub-domain *i*+.

In both (13) and (14), **n** is the perpendicular unit vector to the interface between two adjacent sub-domains.

According to Figure 1, all boundary/interface conditions between sub-domains have been shown from (15) to (25) where *αr*, *β*, and *δ* are respectively the magnet arc per pole pitch ratio, the span angle of slot-openings, and the span angle of slots as shown in Figure 1.

#### *2.5. Extracting the Fourier Series of the Armature Reaction*

To consider the effect of the armature reaction, the current density of each slot should be represented as Fourier series. The current of each phase varies with time and can be represented as Equation (26).

$$I\_k(t) = \sum\_{\upsilon} I\_{\upsilon} \sin[\upsilon(p\omega t - \gamma\_k) + \theta\_{\upsilon}], \ k = 1, 2, \cdots, q \tag{26}$$

where *q* is the number of the phases, *v* shows the order of the harmonics, *ω* is the angular velocity of the rotor, *p* is the number of the pole-pairs, *γk* = 2*π*(*k* − 1)/*q* is the time offset of the phase k*th* respect to the first phase. Also *Iv* and *θv* are the magnitude and phase offset of *vth* harmonic, respectively.

It is obvious that the relation between the current density in each slot and phase current is dependent on the winding configuration. If the winding configuration is like Figure 2a,b, each slot is considered as one sub-region, like Figure 3a,b. But, if the configuration is similar to Figure 2c, each slot consists of two sub-regions (upper and lower sub-regions), as represented in Figure 3c.

**Figure 3.** Sub-region division according to the winding configuration: (**a**) whole slot is considered as one region and belongs to one coil, (**b**) whole slot is considered as one region and left side and right side of the slot belongs to different coils, (**c**) whole slot divided into upper and lower sub-regions and each part belongs to one coil.

The current density in each sub-region of a slot can be represented in Fourier series form as in Equation (27).

$$J\_z^{\ j}(\theta, t) = J\_0^{\dot{j}}(t) + \sum\_{\substack{v=1 \\ v \neq 1}}^{\infty} J\_v^{\dot{j}}(t) \cos\left(\frac{\pi v}{\delta} (\theta - \theta\_{\dot{j}} + \delta/2)\right) \tag{27}$$

$$\theta\_{\dot{j}} - \delta/2 \le \theta \le \theta\_{\dot{j}} + \delta/2.$$

where *J j* 0(*t*) and *J j v*(*t*) are as follows:

$$J\_0^j(t) = \frac{J\_\ell^j(t) + J\_r^j(t)}{2} \tag{28}$$

$$J\_{v}^{\dot{j}}(t) = \frac{l\_{\ell}^{\dot{j}}(t) - l\_{r}^{\dot{j}}(t)}{\pi v/2} \sin(\pi v/2) \tag{29}$$

In order to complete the Fourier series of each sub-region of a slot, it is necessary to obtain the current density of phases in each sub-region of a slot at a specific time by Equations (30) and (31):

$$J = \frac{I}{K\_f A\_{slot}}\tag{30}$$

$$J = \frac{I}{K\_f A\_{\text{slot}} / 2} \tag{31}$$

For instance, the current density in each sub-domain of a slot could be as Figure 4a,b.

If the winding configuration is as shown in Figure 2a,c, the figure of the current density in each sub-region of slot, at a time instant will be as Figure 4a, and if the winding configuration is as shown in Figure 2b, the figure of the current density in a time instant will be as Figure 4b. Also if the configuration of the winding is similar to Figure 2c, the current density in each sub-region of the slot will be as Figure 4a.

**Figure 4.** Value of the current density in each sub-region of slot, according to represented winding configurations.

#### *2.6. Extracting the Fourier Series of the Magnetization*

In the 2-D polar coordinate system, the magnetization vector only has the radial and tangential components as Equation (32).

$$\mathbf{M} = M\_r \mathbf{r} + M\_\theta \mathbf{\hat{e}} \tag{32}$$

where **r** and θ are the radial and tangential unit vectors. *Mr* and *Mθ* are the components of magnetization vector which can be represented as Fourier series expansion of Equations (33) and (34).

$$M\_{\rm I}(\theta\_{\rm I}) = \sum\_{n=1,3,5,\dots}^{\infty} M\_{\rm Im} \cos(np\theta\_{\rm I}) \tag{33}$$

$$M\_{\theta}(\theta\_{\mathcal{I}}) = \sum\_{n=1,3,5,\dots}^{\infty} M\_{\theta \text{n}} \sin(np\theta\_{\mathcal{I}}) \tag{34}$$

where *Mr*n and *Mθ*n respectively are the radial and tangential coefficient of the Fourier series and will be determined according to the magnetization pattern (Table 2). In this paper, only the radial magnetization pattern has been used and represented in Figure 5.

**Table 2.** Radial magnetization pattern and its Fourier series components.

#### *2.7. Finding the General Solution*

The overall format for the general solution in all sub-domains can be represented as Equation (35):

$$A(\mathbf{r},\theta) = \sum\_{n'=1}^{\infty} \left( A\_{n'} r^{n'} + B\_{n'} r^{n'} \right) \left( \mathbb{C}\_{n'} \cos \mathbf{n}' \theta + D\_{n'} \sin \mathbf{n}' \theta \right) + \left( A\_0 \text{lnr} + B\_0 \right) \left( \mathbb{C}\_0 \theta + D\_0 \right) \tag{35}$$

The general solution not only has the capability to satisfy the related PDE, but must satisfy the boundary conditions of the related sub-domain, especially Equations (16), (19), and (24). So the general solutions for sub-domains are as Equations (36)–(40).

The general solution of Poisson equation in slot sub-domain will be as follows:

$$\begin{array}{rcl} A\_z^{sl,j}(r,\theta) &= b\_0^{sl,j} \ln r + A\_p^{sl,j}(r,\theta) + \sum\_{v=1}^{\infty} \left[ a\_v^{sl,j} \left( \frac{r}{\mathbb{R}\_{sv}} \right)^{\frac{\pi v}{\delta}} + b\_v^{sl,j} \left( \frac{\mathbb{R}\_{sl}}{r} \right)^{\frac{\pi v}{\delta}} \right] \\ & \times \cos \left( \frac{\pi \underline{v}}{\delta} \left( \theta - \theta\_{\bar{j}} + \frac{\delta}{2} \right) \right) \end{array} \tag{36}$$

The particular solution is as follows:

$$A\_p^{sl,j}(r,\theta) = -\frac{\mu\_0}{4}l\_0^j r^2 + \sum\_{v=1}^{\infty} \frac{\mu\_0 l\_v^j r^2}{\left(\frac{\pi \cdot v}{\delta}\right)^2 - 4} \cos\left(\frac{\pi \cdot v}{\delta} \left(\theta - \theta\_j + \frac{\delta}{2}\right)\right) \tag{37}$$

Also the general solution for the slot-opening sub-domain is

$$\begin{aligned} A\_z^{so,j}(r,\theta) &= b\_0^{so,j} \ln r + \sum\_{u=1}^{\infty} \left[ a\_u^{so,j} \left( \frac{r}{\mathbb{R}\_s} \right)^{\frac{\pi u}{\beta}} + b\_u^{so,j} \left( \frac{\mathbb{R}\_{sa}}{r} \right)^{\frac{\pi u}{\beta}} \right] \\ &\times \cos \left( \frac{\pi u}{\beta} \left( \theta - \theta\_j + \frac{\beta}{2} \right) \right) \end{aligned} \tag{38}$$

The general solution for the air-gap sub-domain is

$$\begin{split} \left( A\_{z}^{so,j}(r,\theta) \right) \ = b\_{0}^{so,j} \ln r + \sum\_{u=1}^{\infty} \left[ a\_{u}^{so,j} \left( \frac{r}{R\_{v}} \right)^{\frac{\pi u}{\beta}} + b\_{u}^{so,j} \left( \frac{R\_{u}}{r} \right)^{\frac{2\pi u}{\beta}} \right] \\ \ \times \cos \left( \frac{\pi u}{\beta} \left( \theta - \theta\_{j} + \frac{\beta}{2} \right) \right) \end{split} \tag{39}$$

And finally, Poisson equation in the PMs sub-domain has the general solution as Equation (40).

$$\begin{array}{rcl} A\_z^{w,k}(r,\theta) &= b\_0^{w,k} \ln r + \sum\_{w=1}^{\infty} \left[ a\_w^{m,k} \left( \frac{r}{k\_r} \right)^{\frac{wp}{a\_r}} + b\_w^{m,k} \left( \frac{R\_m}{r} \right)^{\frac{wp}{a\_r}} + k\_w^k r \right] \\ & \times \cos \left( \frac{wp}{a\_r} \left( \theta - \pi - \frac{k\pi}{p} + \frac{a\_r\pi}{2p} \right) \right) \end{array} \tag{40}$$

where

$$k\_w^k = -\mu\_0 \chi\_w \begin{cases} \frac{\frac{wp}{a\_r} M\_{rw}^k - M\_{\vartheta\_w}^k}{\left(\frac{wp}{a\_r}\right)^2 - 1} & wp \neq \mathfrak{a}\_r\\ \frac{M\_{rw}^k - M\_{\vartheta\_w}^k}{2} \ln r & wp = \mathfrak{a}\_r \end{cases} \tag{41}$$

$$\chi\_w = \frac{1 - (-1)^w}{2} \tag{42}$$

In order to simplify the general solution in PM sub-domain, boundary condition (15) has been implemented.

$$\begin{array}{rcl} A\_z^{m,k}(r,\theta) &=& \sum\_{w=1}^{\infty} \left\{ h\_{w}^{m,k} \left[ \left( \frac{R\_{\text{m}}}{R\_{r}} \right)^{\frac{wp}{a\_{r}}} \left( \frac{r}{R\_{r}} \right)^{\frac{wp}{a\_{r}}} + \left( \frac{R\_{\text{m}}}{r} \right)^{\frac{wp}{a\_{r}}} \right] \\ &+ R\_{r} \zeta\_{w1}^{k} \left( \frac{r}{R\_{r}} \right)^{\frac{wp}{a\_{r}}} + k\_{w}^{k} r \right\} \times \cos \left( \frac{wp}{a\_{r}} \left( \theta - a - \frac{k \pi}{p} + \frac{a\_{r} \pi}{2p} \right) \right) \end{array} \tag{43}$$

where

$$\begin{cases} \xi\_{w1}^{k} = \frac{a\_{r}}{pw} \left( \frac{dk\_{w}^{k}r}{dr} \Big|\_{r=R\_{r}} + \mu\_{0} \chi\_{w} M\_{\theta}^{k} w \right) = \\ \eta - \mu\_{0} \chi\_{w} \begin{cases} \frac{M\_{rw}^{k} - \frac{wp}{4r} M\_{\theta w}^{k}}{\left(\frac{wp}{4r}\right)^{2} - 1} & wp \neq a\_{r} \\\ \frac{M\_{rw}^{k} - M\_{\theta w}^{k}}{2} (1 + \ln R\_{r}) - M\_{\theta w}^{k} & wp = a\_{r} \end{cases} \end{cases} \tag{44}$$

Also, by implementing boundary condition (25), the general solution in slot sub-domain will be simplified as in Equation (45).

$$\begin{split} \left(A\_{z}^{sl,j}(r,\theta)\right) &= \frac{\mu\_{0}}{4}f\_{0}^{sl,j}\left(2R\_{sl}^{2}\ln(r)-r^{2}\right)+\sum\_{v=1}^{V}\left\{b\_{v}^{sl,j}\left[\left(\frac{r}{R\_{sv}}\right)^{\frac{\pi v}{\delta}}+\left(\frac{R\_{sl}}{r}\right)^{\frac{\pi v}{\delta}}\left(\frac{R\_{sl}}{R\_{sv}}\right)^{\frac{\pi v}{\delta}}\right]\right\} \\ &+\frac{\mu\_{0}l\_{v}^{j}}{\left(\frac{\pi v}{\delta}\right)^{2}-4}\left[r^{2}-\frac{2R\_{sl}}{r^{2}}\left(\frac{R\_{sl}}{r}\right)^{\frac{\pi v}{\delta}}\right]\right\} \times \cos\left(\frac{\pi v}{\delta}\left(\theta-\theta\_{j}+\frac{\delta}{2}\right)\right) \end{split} \tag{45}$$

#### *2.8. Obtaining Integral Coefficients*

For implementing boundary condition (17), the correlation technique must be used [31]. By multiplying Equation (17) in 2*p αrπ* sin *wp αr* (*θ* − *α* −*kπ p* + *αrπ* 2*p* and integration over ! *α* + *kπ p* − *παr* 2*p* , *α* + *kπ p* + *παr* 2*p* " , Equation (46) will be obtained.

$$\begin{aligned} &\frac{2p}{a\_r\pi} \int\_{a+k\pi/p-a\_r\pi/2p}^{a+k\pi/p+a\_r\pi/2p} B\_r^a \Big|\_{r=R\_W} \sin\left(\frac{wp}{a\_r} \left(\theta-\alpha-\frac{k\pi}{p}+\frac{a\_r\pi}{2p}\right)\right) d\theta\\ &= \frac{2p}{a\_r\pi} \int\_{a+k\pi/p-a\_r\pi/2p}^{a+k\pi/p+a\_r\pi/2p} B\_r^{m,k} \Big|\_{r=R\_W} \sin\left(\frac{wp}{a\_r} \left(\theta-\alpha-\frac{k\pi}{p}+\frac{a\_r\pi}{2p}\right)\right) d\theta \end{aligned} \tag{46}$$

From Equation (46), Equation (47) will be deduced.

$$\begin{split} & \frac{w\rho}{a\_r} \left[ 1 + \left( \frac{R\_{\text{ff}}}{R\_r} \right)^{\frac{2w\rho}{a\_r}} \right] a\_w^{w,k} - \sum\_{n=1}^N n \left\{ \left[ a\_n^a \left( \frac{R\_a}{R\_m} \right)^n + b\_n^a \right] \sigma\_s(n, w, k) \\ & - \left[ c\_n^a \left( \frac{R\_s}{R\_m} \right)^n + d\_n^a \right] \sigma\_c(n, w, k) \right\} \\ & - \left[ c\_n^a \left( \frac{R\_s}{R\_m} \right)^n + d\_n^a \right] \sigma\_c(n, w, k) \left\{ \right. \left. \left. - \left( \frac{R\_s}{R\_m} \right)^{\frac{w\rho}{a\_r} + 1} + \zeta\_{w2}^k \right] \right. \\ & \left. - \left( \frac{R\_s}{R\_m} \right)^{\frac{w\rho}{a\_r} + 1} + \zeta\_{w2}^k \right] \end{split} \tag{47}$$

where

$$\mathcal{Z}\_{w2}^{k} = \left. k\_{w}^{k} \right|\_{r=R\_{m}} = -\mu\_{0} \chi\_{w} \begin{cases} \frac{\frac{wp}{a\_{r}} M\_{rw}^{k} - M\_{\theta w}^{k}}{\left(\frac{wp}{a\_{r}}\right)^{2} - 1} & wp \neq a\_{r} \\\ \frac{M\_{rw}^{k} - M\_{\theta w}^{k}}{2} \ln R\_{m} & wp = a\_{r} \end{cases} \tag{48}$$

$$\sigma\_{\varepsilon}(n,\ w,k) = \frac{2p}{a\_{\mathrm{r}}\pi t} \int\_{a+k\pi/p-a\_{\mathrm{r}}\pi/2p}^{a+k\pi/p+a\_{\mathrm{r}}\pi/2p} \cos\left(n\theta\right) \sin\left(\frac{wp}{a\_{\mathrm{r}}}\left(\theta - a - \frac{k\pi}{p} + \frac{a\_{\mathrm{r}}\pi}{2p}\right)\right) d\theta \tag{49}$$

$$\sigma\_s(n, w, k) = \frac{2p}{\mathfrak{a}\_r \pi \mathfrak{T}} \int\_{a + k\pi/p - a\_r \pi / 2p}^{a + k\pi/p + a\_r \pi / 2p} \sin(n\theta) \sin\left(\frac{wp}{a\_r} \left(\theta - a - \frac{k\pi}{p} + \frac{a\_r \pi}{2p}\right)\right) d\theta \tag{50}$$

The solutions of the integrals have been given in the Appendix A.

By using correlation technique, Relation (18) must be multiplied by 1 *π* sin (*nθ*) and integration on the interval [*α* − *π*, *α* + *π*], Equation (51) will be obtained.

$$\frac{1}{2\pi} \int\_{a-\pi}^{a+\pi} H\_{\theta}^{a} |\_{r=R\_{m}} \sin(n\theta) \, d\theta \, = \frac{1}{\pi} \sum\_{k=0}^{2p-1} \int\_{a+k\pi/p-a, \pi/2p}^{a+k\pi/p+a, \pi/2p} H\_{\theta}^{m,k} \Big|\_{r=R\_{m}} \sin(n\theta) \, d\theta \tag{51}$$

Equation (51) results in Equation (52):

$$\begin{aligned} &\sum\_{k=0}^{2p-1} \sum\_{w=1}^{W} \frac{\text{up}}{\mu\_r a\_r} \left[ \left( \frac{\text{R}\_{\text{m}}}{\text{R}\_r} \right)^{\frac{2wp}{a\_r}} - 1 \right] \rho\_s(n, \, w, k) \, a\_w^{m, k} + n \left[ c\_n^a \left( \frac{\text{R}\_r}{\text{R}\_w} \right)^n - d\_n^a \right] \\ &= \sum\_{k=0}^{2p-1} \sum\_{w=1}^{W} \frac{\text{up} \rho\_{\text{m}}}{\mu\_r a\_r} \left[ -\zeta\_{w1}^k \left( \frac{\text{R}\_{\text{m}}}{\text{R}\_r} \right)^{\frac{wp}{a\_r} - 1} + \zeta\_{w3}^k \right] \rho\_s(n, \, w, k) \end{aligned} \tag{52}$$

where

$$\begin{aligned} \xi\_{w3}^{k} &= \begin{array}{cc} \frac{\mu\_{r}}{wp} \left( \frac{dk\_{w}^{k}r}{dr} \Big|\_{r=R\_{m}} + \mu\_{0} \chi\_{w} \mathcal{M}\_{\theta}^{k}w \right) &= \\ -\mu\_{0} \chi\_{w} \begin{cases} \frac{\mathcal{M}\_{rw}^{k} - \frac{wp}{a\_{r}} \mathcal{M}\_{\theta}^{k}w}{\left(\frac{wp}{a\_{r}}\right)^{2} - 1} & wp \neq a\_{r} \\ \frac{\mathcal{M}\_{w}^{k} - \mathcal{M}\_{\theta w}^{k}}{2} (1 + \ln R\_{m}) - \mathcal{M}\_{\theta}^{k}w & wp = a\_{r} \end{cases} \end{aligned} \tag{53}$$

$$\rho\_s(n, w, k) = \frac{1}{\pi} \int\_{\pi + k\pi/p - a\_r\pi/2p}^{a + k\pi/p + a\_r\pi/2p} \cos\left(\frac{wp}{a\_r} \left(\theta - a - \frac{k\pi}{p} + \frac{a\_r\pi}{2p}\right)\right) \sin(n\theta) \,d\theta \tag{54}$$

The solution of the integral has been given in the Appendix A.

Again Equation (18) must be multiplied by 1 *π* cos (*nθ*), then integration on the interval [*α* − *π*, *α* + *π*] causes:

$$\frac{1}{2\pi} \int\_{a-\pi}^{a+\pi} H\_{\theta}^{a} |\_{r=R\_{m}} \cos(n\theta) \, d\theta \, = \frac{1}{\pi} \sum\_{k=0}^{2p-1} \int\_{a+k\pi/p-a\_{r}\pi/2p}^{a+k\pi/p+a\_{r}\pi/2p} H\_{\theta}^{m,k} \Big|\_{r=R\_{m}} \cos(n\theta) \, d\theta \tag{55}$$

Equation (55) results in Equation (56).

$$\begin{split} & \sum\_{k=0}^{2p-1} \sum\_{w=1}^{W} \frac{\text{up}}{\mu\_r a\_r} \left[ \left( \frac{R\_{\text{m}}}{R\_r} \right)^{\frac{2wp}{a\_r}} - 1 \right] \rho\_c(n, w, k) a\_w^{m, k} + n \left[ a\_n^a \left( \frac{R\_s}{R\_w} \right)^n - b\_n^a \right] \\ & = \sum\_{k=0}^{2p-1} \sum\_{w=1}^{W} \frac{\text{up} \, R\_{\text{m}}}{\mu\_r a\_r} \left[ - \zeta\_{w1}^k \left( \frac{R\_{\text{m}}}{R\_r} \right)^{\frac{wp}{a\_r} - 1} + \zeta\_{w3}^k \right] \rho\_c(n, w, k) \end{split} \tag{56}$$

where

$$\rho\_{\varepsilon}(n, w, k) = \frac{1}{\pi} \int\_{\pi + k\pi/p - u, \pi/2p}^{a + k\pi/p + u, \pi/2p} \cos\left(\frac{wp}{a\_r} \left(\theta - a - \frac{k\pi}{p} + \frac{a\_r\pi}{2p}\right)\right) \cos\left(n\theta\right) d\theta \tag{57}$$

The solution of the integral has been given in the Appendix A.

For implementing boundary condition (20), multiplying 2 *β* sin *π u β θ* − *θj* + *β* 2 to Equation (20) and integration over ! *θj* − *β* 2 , *θj* + *β* 2 " yields

$$\begin{aligned} \frac{\theta\_j + \theta/2}{\frac{\theta}{\theta} \int\_{\theta\_j - \theta/2}^{\theta} B\_r^{so, j}(r, \theta) \Big|\_{r=R\_s} \sin\left(\frac{\pi \underline{u}}{\overline{\theta}} \left(\theta - \theta\_j + \frac{\theta}{2}\right)\right) d\theta &= \\ \frac{\theta\_j + \theta/2}{\frac{\theta}{\theta} \int\_{\theta\_j - \theta/2}^{\theta} B\_r^q(r, \theta) \Big|\_{r=R\_s} \sin\left(\frac{\pi \underline{u}}{\overline{\theta}} \left(\theta - \theta\_j + \frac{\theta}{2}\right)\right) d\theta \end{aligned} \tag{58}$$

Equation (59) is obtained via the simplification of Equation (58).

$$\begin{cases} -\sum\_{n=1}^{N} n \left\{ \left[ a\_n^a + b\_n^a \left( \frac{R\_s}{R\_m} \right)^n \right] \varepsilon\_s (n, u, j) - \left[ c\_n^a + d\_n^a \left( \frac{R\_s}{R\_m} \right)^n \right] \varepsilon\_c (n, u, j) \right\} \\ + \frac{\pi \underline{u}}{\overline{\beta}^l} \left( \frac{R\_{sa}}{R\_s} \right)^{\frac{\pi \underline{u}}{\overline{\beta}}} a\_{u}^{so, j} + \frac{\pi \underline{u}}{\overline{\beta}^l} b\_{u}^{so, j} = 0 \end{cases} \tag{59}$$

where

$$\varepsilon\_s(u, u, j) = \frac{2}{\beta} \int\_{\theta\_j - \beta/2}^{\theta\_j + \beta/2} \sin(n\theta) \sin\left(\frac{\pi u}{\beta} \left(\theta - \theta\_j + \frac{\beta}{2}\right)\right) d\theta \tag{60}$$

$$\varepsilon\_{\varepsilon}(\boldsymbol{u},\boldsymbol{u},\boldsymbol{j}) = \frac{2}{\beta} \int\_{\theta\_{\hat{j}} - \theta/2}^{\theta\_{\hat{j}} + \theta/2} \cos(n\theta) \sin\left(\frac{\pi\,\boldsymbol{u}}{\beta} \left(\theta - \theta\_{\hat{j}} + \frac{\beta}{2}\right)\right) d\theta \tag{61}$$

The solution of the integral has been given in the Appendix A.

The correlation technique is used for boundary condition (21) and 1 *π* cos (*nθ*) is multiplied to Equation (21) and integration is taken over [−*π*, *π*].

$$\frac{1}{\pi} \int\_{-\pi}^{\pi} H\_{\theta}^{a}(r, \theta)|\_{r=R\_{s}} \cos(n\theta) \, d\theta \, = \frac{1}{\pi} \sum\_{j=1}^{Q} \int\_{\theta\_{j} - \theta/2}^{\theta\_{j} + \theta/2} H\_{\theta}^{so, j}(r, \theta) \Big|\_{r=R\_{s}} \cos(n\theta) \, d\theta \tag{62}$$

Simplifying Equation (62) yields (63).

$$\begin{aligned} \ln \left[ a\_n^d - b\_n^d \left( \frac{R\_s}{R\_m} \right)^n \right] - \sum\_{j=1}^Q \sum\_{u=1}^{\underline{U}} \frac{\pi\_u}{\overline{\rho}} \left[ \left( \frac{R\_{sa}}{R\_s} \right)^{\frac{\underline{u}\underline{u}}{\overline{\rho}}} a\_u^{sa\_s j} - b\_u^{sa\_s j} \right] \eta\_c(n, u, j) \\\ - \sum\_{j=1}^Q \eta\_c(n, 0, j) \left. b\_0^{sa\_s j} \right| = 0 \end{aligned} \tag{63}$$

where

$$\eta\_c(n, u, j) = \frac{1}{\pi} \int\_{\theta\_j - \beta/2}^{\theta\_j + \beta/2} \cos(n\theta) \cos\left(\frac{\pi u}{\beta} \left(\theta - \theta\_j + \frac{\beta}{2}\right)\right) d\theta \tag{64}$$

Also, by multiplying Equation (21) to 1 *π* sin(*nθ*) and integration over interval [−*π*, *π*] the following expression is obtained:

$$\frac{1}{\pi t} \int\_{-\pi}^{\pi} H\_{\theta}^{a}(r, \theta) |\_{r=R\_s} \sin(n\theta) \, d\theta \, = \frac{1}{\pi} \sum\_{j=1}^{Q} \int\_{\theta\_j - \theta/2}^{\theta\_j + \theta/2} H\_{\theta}^{a, j}(r, \theta) \Big|\_{r=R\_s} \sin(n\theta) \, d\theta \tag{65}$$

Simplification of Equation (65) causes the formation of Equation (66).

$$\begin{aligned} n \left[ c\_n^a - \left( \frac{R\_s}{R\_m} \right)^n d\_n^a \right] - \sum\_{j=1}^Q \sum\_{u=1}^{\underline{U}} \frac{\pi\_u \underline{u}}{\overline{\rho}^2} \left[ \left( \frac{R\_{sa}}{R\_s} \right)^{\frac{\underline{X} \cdot \underline{u}}{\overline{\rho}}} a\_u^{so,j} - b\_u^{so,j} \right] \eta\_s(n, u, j) \\\ - \sum\_{j=1}^Q \eta\_s(n, 0, j) \left. b\_0^{so,j} \right| = 0 \end{aligned} \tag{66}$$

where

$$\eta\_s(\mathbf{u}, u, j) = \frac{1}{\pi} \int\_{\theta\_j - \beta/2}^{\theta\_j + \theta/2} \sin(n\theta) \cos\left(\frac{\pi u}{\beta} \left(\theta - \theta\_j + \frac{\beta}{2}\right)\right) d\theta \tag{67}$$

The correlation technique is used and boundary condition (22) is multiplied by 2 *β* sin *π u β θ* − *θj* + *β* 2 . Integration over interval ! *θj* − *β* 2 , *θj* + *β* 2 " results in Equation (68).

$$\begin{array}{l} \begin{array}{l} \frac{\theta\_{j}+\beta/2}{\overline{\beta}} \int\_{0}^{\xi\_{j}} B\_{r}^{\xi,\bar{j}}(r,\theta) \Big|\_{r=R\_{\infty}} \sin\left(\frac{\pi u}{\overline{\beta}}\left(\theta-\theta\_{j}+\frac{\beta}{2}\right)\right) d\theta =\\ \frac{\theta\_{j}+\beta/2}{\theta\_{j}+\beta/2} \\ \frac{2}{\overline{\beta}} \int\_{\theta\_{j}-\overline{\beta}/2} B\_{r}^{\alpha,\bar{j}}(r,\theta) \Big|\_{r=R\_{\infty}} \sin\left(\frac{\pi u}{\overline{\beta}}\left(\theta-\theta\_{j}+\frac{\beta}{2}\right)\right) d\theta \end{array} \end{array} \tag{68}$$

Simplifying (68) results in (69).

$$\begin{aligned} &\frac{\mu\_{\frac{\pi}{\beta}}}{\beta} \left[ a\_{u}^{so\_{r}} + \left( \frac{R\_{su}}{R\_{u}} \right)^{\frac{\pi\_{\rm ul}}{\beta}} b\_{u}^{so\_{r}} \right] - \sum\_{v=1}^{V} \frac{\pi\_{\rm v}}{\delta} \left[ \left( \frac{R\_{u}}{R\_{w}} \right)^{\frac{2q\omega}{\delta}} + 1 \right] \gamma\_{s}(u\_{\prime}v) \left. b\_{v}^{sl\_{\prime}} \right| \\ &= \sum\_{v=1}^{V} \frac{\mu\_{\rm o} l\_{v}^{j}}{\left( \frac{\pi v}{\delta} \right)^{2} - 4} \left[ \frac{\pi v}{\delta} R\_{sl}^{2} - 2 R\_{so}^{2} \left( \frac{R\_{sl}}{R\_{w}} \right)^{\frac{2q\omega}{\delta}} \right] \gamma\_{s}(u\_{\prime}v) \end{aligned} \tag{69}$$

where

$$\gamma\_s(u,v) = \frac{2}{\beta} \int\_{\theta\_j - \beta/2}^{\theta\_j + \beta/2} \sin\left(\frac{\pi v}{\delta} \left(\theta - \theta\_j + \frac{\delta}{2}\right)\right) \sin\left(\frac{\pi u}{\beta} \left(\theta - \theta\_j + \frac{\beta}{2}\right)\right) d\theta \tag{70}$$

Solution of the integral are given in the Appendix A.

Correlation technique is implemented for boundary condition (23) and it is multiplied by 2 *δ* cos *π v δ θ* − *θj* + *δ* 2 . Integration over interval ! *θj* − *δ* 2 , *θj* + *δ* 2 " yields Equation (71).

$$\begin{aligned} \frac{2}{\delta} \int\_{\theta\_{j} - \delta/2}^{\theta\_{j} + \delta/2} H\_{\theta}^{sl, \bar{j}}(r, \theta) \Big|\_{r = R\_{so}} \cos\left(\frac{\pi v}{\delta} \left(\theta - \theta\_{\bar{j}} + \frac{\delta}{2}\right)\right) d\theta &= \\ \frac{\theta\_{j} + \delta/2}{\theta\_{j} + \delta/2} \int\_{\theta\_{j} - \delta/2}^{\theta\_{j} + \delta/2} H\_{\theta}^{sa, \bar{j}}(r, \theta) \Big|\_{r = R\_{so}} \cos\left(\frac{\pi v}{\delta} \left(\theta - \theta\_{\bar{j}} + \frac{\delta}{2}\right)\right) d\theta \end{aligned} \tag{71}$$

Simplifying Equation (71) results in Equation (72).

$$\begin{split} & \sum\_{u=1}^{L} \frac{\pi \underline{u}}{\overline{\rho}^{2}} \left[ -a\_{\rm u}^{\rm iso,j} + \left( \frac{R\_{\rm su}}{R\_{\rm s}} \right)^{\frac{\pi \underline{u}}{\overline{\rho}}} b\_{\rm u}^{\rm iso,j} \right] \gamma\_{\rm c}(\mu, \upsilon) - \gamma\_{\rm c}(0, \upsilon) \left. b\_{0}^{\rm iso,j} + \frac{\pi \underline{v}}{\overline{\delta}} \left[ \left( \frac{R\_{\rm u}}{R\_{\rm u}} \right)^{\frac{2 \pi \underline{u}}{\overline{\delta}}} - 1 \right] b\_{\upsilon}^{\rm el,j} \right] \\ & = \frac{-2\mu\_{0} \overline{\rho}\_{\rm c}^{2}}{\left( \frac{\pi \underline{v}}{\overline{\delta}} \right)^{2} - 4} \left[ R\_{\rm s}^{2} - R\_{\rm s\alpha}^{2} \left( \frac{R\_{\rm u}}{R\_{\rm s\alpha}} \right)^{\frac{\pi \underline{v}}{\overline{\delta}}} \right] \end{split} \tag{72}$$

where

$$\gamma\_{\varepsilon}(u,v) = \frac{2}{\delta} \int\_{\theta\_{\parallel}-\theta/2}^{\theta\_{\parallel}+\beta/2} \cos\left(\frac{\pi v}{\delta}\left(\theta-\theta\_{\parallel}+\frac{\delta}{2}\right)\right) \cos\left(\frac{\pi u}{\beta}\left(\theta-\theta\_{\parallel}+\frac{\beta}{2}\right)\right) d\theta \tag{73}$$

Integration over interval ! *θj* − *δ* 2 , *θj* + *δ* 2 " on boundary condition (23) causes Equation (74).

$$\int\_{\theta\_j - \delta/2}^{\theta\_j + \delta/2} H\_{\theta}^{sl, j}(r, \theta) \Big|\_{r = R\_{\text{os}}} d\theta = \int\_{\theta\_j - \delta/2}^{\theta\_j + \delta/2} H\_{\theta}^{so, j}(r, \theta) \Big|\_{r = R\_{\text{os}}} d\theta \tag{74}$$

From Equation (74), Equation (75) will be deduced.

$$h\_0^{so,\dot{j}} = \frac{\mu\_0 l\_0^{\dot{j}}}{2} \left(R\_{so}^2 - R\_{sl}^2\right) \frac{\delta}{\beta} \tag{75}$$

#### *2.9. Overlapping Winding*

In overlapping winding, each slot is divided into two sub-domains as represented in Figure 3c. The upper and lower sub-domains respectively are indicated with *slb* and *slt* indices. Hence Equation (69) will be as follows:

$$\frac{\pi \,\mathrm{u}}{\beta} \left[ a\_{\mathrm{u}}^{\mathrm{so},j} + \left( \frac{\mathrm{R}\_{\mathrm{o}o}}{\mathrm{R}\_{\mathrm{s}}} \right)^{\frac{\pi \,\mathrm{u}}{\beta}} b\_{\mathrm{u}}^{\mathrm{so},j} \right] - \sum\_{v=1}^{V} \frac{\pi \,\mathrm{v}}{\delta} \left[ \frac{\left( \mathrm{R}\_{\mathrm{sl}} / \mathrm{R}\_{\mathrm{s}o} \right)^{\frac{2\pi v}{\delta}} + 1}{\left( \mathrm{R}\_{\mathrm{sl}m} / \mathrm{R}\_{\mathrm{s}o} \right)^{\frac{\pi \,\mathrm{u}}{\delta}}} \right] \gamma\_{s}(\mathrm{u},\mathrm{v}) b\_{v}^{\mathrm{slb},j} = 0 \tag{76}$$

Also, Equations (72) and (75) will be modified as (77) and (78) respectively.

$$\begin{cases} \sum\_{\mu=1}^{l\text{L}} \frac{\pi \,\text{u}}{\beta^{\text{s}}} \left[ -a\_{\mu}^{\text{so},j} + \left( \frac{R\_{\text{us}}}{R\_{\text{s}}} \right)^{\frac{\text{L}\cdot\text{u}}{\beta}} b\_{\text{u}}^{\text{so},j} \right] \gamma\_{\text{c}}(\mu, \upsilon) + \frac{\pi \,\upsilon}{\delta} \left[ \frac{(R\_{\text{s}}/R\_{\text{as}})^{\frac{2\text{s}\cdot\text{s}}{\beta}} - 1}{(R\_{\text{s}}/R\_{\text{as}})^{\frac{\text{s}\cdot\text{s}}{\beta}}} \right] b\_{\upsilon}^{\text{slb},j} \\ \left. - \gamma\_{\text{c}}(0, \upsilon) \right\, b\_{0}^{\text{so},j} = 0 \end{cases} \tag{77}$$

$$b\_0^{\rm so,j} = \frac{\mu\_0}{2} \left[ J\_{l0}^j \left( R\_{slm}^2 - R\_{sl}^2 \right) + J\_{l0}^j \left( R\_{so}^2 - R\_{slm}^2 \right) \right] \frac{\delta}{\beta} \tag{78}$$

where *Rslm* = / *R*2 *sl* + *R*2 *so* /2 is the radii of middle of the slot which divides it into two equal areas. *J j b*0 and *J j t*0 are current densities in the lower and upper sub-domains in a slot.

#### **3. Quantities**

#### *3.1. Flux Density*

The air-gap flux density vector is one of the most important quantities required for the calculation of other quantities. For obtaining the air-gap flux density, Equation (5) is expanded and Relations (78) and (79) in 2-D polar coordinates are deduced.

$$\begin{cases} B\_r^a(r,\theta) = -\frac{1}{r}\frac{\partial A\_r^a}{\partial \theta} = -\sum\_{n=1}^N n \left\{ \left[ \frac{a\_n^a}{\mathbb{R}\_m^a} \left( \frac{r}{\mathbb{R}\_m} \right)^{n-1} + \frac{b\_n^a}{\mathbb{R}\_s} \left( \frac{\mathbb{R}\_s}{r} \right)^{n+1} \right] \sin(n\theta) \\\ -\left[ \frac{c\_n^a}{\mathbb{R}\_m^a} \left( \frac{r}{\mathbb{R}\_m} \right)^{n-1} + \frac{d\_n^a}{\mathbb{R}\_s} \left( \frac{\mathbb{R}\_s}{r} \right)^{n+1} \right] \cos(n\theta) \right\} \end{cases} \tag{79}$$

$$\begin{array}{rcl}B^{a}\_{\theta}(r,\theta)&=&-\frac{\partial A^{a}\_{r}}{\partial r}=-\sum\_{n=1}^{N}n\left\{\left[\left[\frac{d^{a}\_{n}}{\mathcal{R}\_{n}}\left(\frac{r}{\mathcal{R}\_{n}}\right)^{n-1}-\frac{b^{a}\_{n}}{\mathcal{R}\_{n}}\left(\frac{\mathcal{R}\_{n}}{r}\right)^{n+1}\right]\right]\cos\left(n\theta\right)\\ &+\left[\frac{c^{a}\_{n}}{\mathcal{R}\_{n}}\left(\frac{r}{\mathcal{R}\_{n}}\right)^{n-1}-\frac{d^{a}\_{n}}{\mathcal{R}\_{n}}\left(\frac{\mathcal{R}\_{n}}{r}\right)^{n+1}\right]\sin(n\theta)\end{array} \tag{80}$$

#### *3.2. Inductances*

For calculation of the inductances, just the flux produced by armature is considered. Inductances between phase *k* and *k* are obtained as follows:

$$L\_{k,k'} = \sum\_{\substack{j \in k \ \text{else} \ j' \in k'}} \frac{\lambda\_{j,j'}}{\dot{\mathbf{i}}\_{j'}} \tag{81}$$

where *j* = 1, 2, ... , *Q* and *j* = 1, 2, ... , *Q* are the indices of the coils. *ij* is the current of phase *k* and *λj*,*j* is the flux linked by coil *j*, which is produced by coil *j* . If *k* = *k* , self-inductance of the phase is calculated.

#### *3.3. Back-EMF*

In order to calculate the no-load back-EMF of a phase, the permanent magnet flux linked by coils must be calculated by Equation (82).

$$
\varphi = \int \mathbf{B} \cdot d\mathbf{S} \tag{82}
$$

According to Faraday's law, induced voltage in *j*th coil obtain by Equation (83).

$$E\_{\bar{j}} = -N\iota\omega \frac{d\varphi\_{\bar{j}}}{d\alpha} \tag{83}$$

where *Nt* is the number of turns of the coil, *ω* and *α* are the angular velocity and rotor position respectively. The total back-EMF of a phase depends on coils connections.

#### *3.4. Instantaneous Electromagnetic Torque*

Instantaneous torque consists of cogging torque (*Tcog*), electromagnetic torque (*Tem*) and reluctance torque (*Trel*).

$$T(t) = T\_{\text{copy}}(t) + T\_{\text{em}}(t) + T\_{\text{rel}}(t) \tag{84}$$

By using Maxwell stress tensor, the instantaneous electromagnetic torque can be obtained as follows:

$$T(t) = \int \int \frac{1}{\mu\_0} B\_r B\_\theta \, ds\tag{85}$$

By expanding Equation (85), Relations (86) and (87) will be obtained.

$$T(t) = L\_s \int\_{-\pi}^{\pi} \frac{1}{\mu\_0} \left( B\_{r,PM}^a + B\_{r,AR}^a \right) \left( B\_{\theta,PM}^a + B\_{\theta,AR}^a \right) \mid\_{r = R\_c} R\_c^2 \, d\theta \tag{86}$$

$$T(t) = \frac{L\_s R\_c^2}{\mu\_0} \int\_{-\pi}^{\pi} \left( B\_{r, PM}^a B\_{\theta, PM}^a + B\_{r, AR}^a B\_{\theta, PM}^a + B\_{r, PM}^a B\_{\theta, AR}^a + B\_{r, AR}^a B\_{\theta, AR}^a \right) \Big|\_{r = R\_c} d\theta \tag{87}$$

where *Ba r*,*PM* and *Ba θ*,*PM* respectively are the radial and tangential flux density components in the air-gap, due to the PMs. Also *Ba r*,*AR* and *Ba θ*,*AR* are the radial and tangential magnetic flux density components due to the armature reaction in the air-gap. The parameter *Rc* is the radius of the middle of inner air-gap.

#### *3.5. Unbalanced Magnetic Force*

The radial and tangential components of the local traction exerted on the rotor surface can be obtained by Maxwell stress tensor as follows:

$$f\_r = \frac{1}{2\mu\_0} \left( B\_r^2 - B\_\theta^2 \right) \tag{88}$$

$$f\_{\theta} = \frac{1}{\mu\_0} B\_r B\_\theta \tag{89}$$

By transforming these local tractions to the Cartesian plane, and summation of the same directions, Equations (90)–(93) will be obtained.

$$f\_x = f\_r \cos\theta - f\_\theta \sin\theta \tag{90}$$

$$f\_y = f\_r \sin \theta + f\_\theta \cos \theta \tag{91}$$

$$F\_{\mathbf{x}}(t) = \int\_{-1/2}^{1/2} \int\_{-\pi}^{\pi} f\_{\mathbf{x}} \operatorname{r} \, d\theta \, dz = \operatorname{L} \int\_{-\pi}^{\pi} f\_{\mathbf{x}} \operatorname{r} \, d\theta \tag{92}$$

$$F\_y(t) = \int\_{-L/2}^{L/2} \int\_{-\pi}^{\pi} f\_y \tau \, d\theta \, dz = L \int\_{-\pi}^{\pi} f\_y \tau \, d\theta \tag{93}$$

Finally, the amplitude of the unbalanced magnetic force can be obtained by Equation (94).

$$F\_{\mathbf{r}} = |F(t)| = \sqrt{F\_{\mathbf{x}}^2(t) + F\_{\mathbf{y}}^2(t)}\tag{94}$$

#### **4. Results**

In order to investigate the efficacy of the model, a case study with the parameters listed in Table 3 has been used.

Also, the winding configuration for this case study has been shown in Figure 5.

**Table 3.** Parameters used in the case study.

**Figure 5.** The machine topology and winding configuration.

#### *4.1. Flux Density*

The radial and tangential magnetic flux density components due to the open-circuit and armature reaction are respectively depicted in Figures 6–9. The numerical results obtained from FEM are shown respectively in each figure and confirm the accuracy of the proposed model.

**Figure 6.** Radial magnetic flux density due to just PMs in the middle of the air-gap, when the rotor position is set to zero.

**Figure 7.** Tangential magnetic flux density due to just PMs in the middle of the air-gap, when the rotor position is set to zero.

**Figure 8.** Radial magnetic flux density due to just armature winding in the middle of the air-gap, when the current density of phase A is zero, current density of phase B is 4.33 A/mm2 and phase C is 4.33 A/mm2.

**Figure 9.** Tangential magnetic flux density due to just armature winding in the middle of the air-gap, when the current density of phase A is zero, current density of phase B is 4.33 A/mm2 and phase C is 4.33 A/mm2.

#### *4.2. Torque*

Instantaneous electromagnetic torque, reluctance torque and cogging torque of the machine have been depicted in Figures 10–12. Both analytic and numeric methods show good agreement which confirms the efficacy of the proposed model.

**Figure 10.** Electromagnetic torque vs. rotor position.

**Figure 12.** Cogging torque vs. rotor position.

#### *4.3. Back-EMF and Inductance*

Results of the phase back-EMF and line back-EMF are shown in Figure 13. Again it is shown that both analytic and numeric results have good conformity.

Also, the self and mutual inductances have been depicted in Figure 14.

**Figure 13.** Back-electromotive force (EMF) of the first phase and first line.

**Figure 14.** Self and mutual inductance of a phase.

#### *4.4. Unbalanced Magnetic Force (UMF)*

Unbalanced magnetic forces due to the open-circuit, armature reaction, and both of them have been depicted in Figures 15–17. As evident from these figures, unbalanced forces due to the armature reaction exert considerable forces compared to those of the open circuit.

**Figure 15.** Unbalanced magnetic forces just due to the PMs.

**Figure 16.** Unbalanced magnetic forces just due to the armature reaction.

**Figure 17.** Total unbalanced magnetic forces exerted on the rotor.

Both analytic and numeric results are shown in Figures 15–17 and confirm the correctness of the proposed model.

#### **5. Conclusions**

A 2-D analytical magnetic model is presented for brushless synchronous outer rotor machines with surface inset PMs. For this purpose, Maxwell's equations in the form of the Laplace and Poisson equations are solved in predefined sub-domains of the 2-D polar coordinates. The general and particular solutions for each sub-region are presented so that they have the capability to satisfy the governing PDE and related boundary conditions. Finally, by imposing boundary conditions and solving simultaneous linear algebraic equations, all important quantities such as magnetic flux density, electromagnetic torque, UMF, back-EMF, and inductances are calculated and validated by those obtained by FEM. The results of the analytical model show the efficacy of the proposed approach.

**Author Contributions:** A.V. has done formal analysis, investigation, methodology, writing and editing the original draft. A.R. has done supervision, conceptualization, project administration and reviewing the original draft. H.M.-J. has done formal analysis, investigation, methodology, software, and validation. A.G. has done review and editing.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

To define the left and right side current density in each slot for the two layers non-overlapping concentrated winding topology, we have:

$$J\_r = \frac{i(t)}{\mathcal{K}\_f A\_W / 2} \mathbf{C}\_r \tag{A1}$$

$$J\_l = \frac{i(t)}{K\_f A\_w/2} \mathbf{C}\_l \tag{A2}$$

where *Jr* = [*J*1 *r* ... *J j r* ... *J Q r* ], *Jl* = [*J*1 *l* ... *J j l* ... *J Q l* ], *i*(*t*)=[*ia ib ic*], *Kf* is the filling factor and *Aw* is the slot area. On the other hand, **C***r* and **C***l* are as follows:

$$\mathbf{C}\_{\mathbf{f}}(i,j) = \begin{cases} \frac{\mathbf{C}(i,j) + |\mathbf{C}(i,j)|}{2} & if \quad -1 < \mathbf{C}(i,j) < 1\\ 1 & if \quad \mathbf{C}(i,j) = 2\\ -1 & if \quad \mathbf{C}(i,j) = -2 \end{cases} \tag{A3}$$

$$\mathbf{C}\_{l}(i,j) = \begin{cases} \frac{\mathbf{C}(i,j) - |\mathbf{C}(i,j)|}{2} & if \quad -1 < \mathbf{C}(i,j) < 1\\ 1 & if \quad \mathbf{C}(i,j) = 2\\ -1 & if \quad \mathbf{C}(i,j) = -2 \end{cases} \tag{A4}$$

According to the winding topology, **C** is as follows:

$$\mathbf{C} = \begin{bmatrix} 2 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -2 & 1 \end{bmatrix}\_{3 \times Q = 3 \times 9} \tag{A5}$$

where **C**(*i*, *j*) = 2 or −2 means slot *j* accommodates two sides of two coils of phase *i*, which respectively carry positive or negative current. **C**(*i*, *j*) = 1 or −1 means slot *j* accommodates one side of one coil of phase *i*, which respectively carry positive or negative current. Also, 0 means there is no coil of the phase *i* th in the slot.

For *αrn* = *wp*:

$$\begin{split} \rho\_s(\mathbf{u}, w, k) &= \frac{a\_r}{2\pi} \left\{ \frac{-\cos\left(w\pi + \frac{n\pi a\_r}{2p} + na + \frac{k\pi x}{p}\right) + \cos\left(\frac{n\pi a\_r}{2p} - na - \frac{k\pi x}{p}\right)}{a\_r n + wp} \\ &- \frac{\cos\left(w\pi - \frac{n\pi a\_r}{2p} - na - \frac{k\pi x}{p}\right) - \cos\left(\frac{n\pi a\_r}{2p} - na - \frac{k\pi x}{p}\right)}{a\_r n - wp} \right\} \end{split} \tag{A6}$$

$$\begin{split} \rho\_c(n, w, k) &= \frac{a\_r}{2\pi} \left\{ \frac{\sin\left(w\pi + \frac{n \pi a\_r}{2p} + na + \frac{k \pi v}{p}\right) + \sin\left(\frac{n \pi a\_r}{2p} - na - \frac{k \pi v}{p}\right)}{a\_r n + w\eta} \\ &- \frac{\sin\left(w\pi - \frac{n \pi a\_r}{2p} - na - \frac{k \pi v}{p}\right) - \sin\left(\frac{n \pi a\_r}{2p} - na - \frac{k \pi v}{p}\right)}{a\_r n - w\eta} \end{split} \tag{A7}$$

$$\begin{array}{rcl} \sigma\_{s}(\mathsf{u},w,k) &=& \frac{p}{\pi} \left\{ \frac{-\sin\left(w\pi + \frac{n\pi a}{2p} + na + \frac{kwn}{p}\right) - \sin\left(\frac{n\pi a\_{r}}{2p} - na - \frac{kwn}{p}\right)}{a\_{r}n + wp} \\ & - \frac{\sin\left(w\pi - \frac{n\pi a\_{r}}{2p} - na - \frac{kwn}{p}\right) - \sin\left(\frac{n\pi a\_{r}}{2p} - na - \frac{kwn}{p}\right)}{a\_{r}n - wp} \end{array} \tag{A8}$$

$$\begin{split} \sigma\_{\mathsf{c}}(\mathsf{u},\mathsf{w},k) &= \frac{p}{\pi} \left\{ \frac{-\cos\left(w\pi + \frac{n\mathsf{u}\mathsf{u}\mathsf{z}}{2p} + na + \frac{k\mathsf{u}\mathsf{z}}{p}\right) + \cos\left(\frac{n\mathsf{u}\mathsf{z}}{2p} - na - \frac{k\mathsf{u}\mathsf{z}}{p}\right)}{a\_{\mathsf{r}}n + wp} \\ &+ \frac{\cos\left(w\pi - \frac{n\mathsf{u}\mathsf{z}}{2p} - na - \frac{k\mathsf{u}\mathsf{z}}{p}\right) - \cos\left(\frac{n\mathsf{u}\mathsf{z}}{2p} - na - \frac{k\mathsf{u}\mathsf{z}}{p}\right)}{a\_{\mathsf{r}}n - wp} \right\} \end{split} \tag{A9}$$

For *αrn* = *wp*:

$$\begin{array}{rcl}\rho\_{5}(n,w,k) &= \frac{-1}{4n\pi} \left[ \cos\left(3w\pi\pi/2 + na + \frac{kw\pi}{p}\right) - \cos\left(w\pi/2 - na - \frac{kw\pi}{p}\right) \right] \\ & - \frac{a\_{\rm f}}{2p} \sin\left(w\pi/2 - na - \frac{kw\pi}{p}\right) \end{array} \tag{A10}$$

$$\begin{array}{rcl}\rho\_{\varepsilon}(n,w,k) &=& \frac{1}{4n\pi} \left[ \sin\left(3w\pi/2 + na + \frac{\text{kux}}{p}\right) + \sin\left(w\pi/2 - na - \frac{\text{kux}}{p}\right) \right] \\ &+ \frac{a\_{\varepsilon}}{2p} \cos\left(w\pi/2 - na - \frac{\text{k}n\pi}{p}\right) \end{array} \tag{A11}$$

$$\begin{array}{rcl} \sigma\_{s}(\boldsymbol{n},\boldsymbol{w},\boldsymbol{k}) &=& \frac{-1}{2w\pi} \Big[ \sin \left( 3w\pi\pi/2 + na + \frac{kux}{p} \right) \\ &+ \cos \left( w\pi\pi/2 - na - \frac{kux}{p} \right) \Big] \\ \end{array} \tag{A12}$$

$$\begin{array}{rcl} \sigma\_{\varepsilon}(\boldsymbol{u}, \boldsymbol{w}, k) &=& \frac{-1}{2n\pi} \Biggl[ \cos\left(3w\pi\tau/2 + n\boldsymbol{a} + \frac{k n \pi}{p}\right) - \cos\left(w\pi/2 - n\boldsymbol{a} - \frac{k n \pi}{p}\right) \Biggr] \\ &+ \sin\left(w\pi/2 - n\boldsymbol{a} - \frac{k n \pi}{p}\right) \end{array} \tag{A13}$$

For *π u* = *β n*:

$$\varepsilon\_{\rm s}(\nu, u, j) = 2\pi \ln \frac{(-1)^{u+1} \sin \left( n \left( \theta\_j + \frac{\beta}{2} \right) \right) + \sin \left( n \left( \theta\_j - \frac{\beta}{2} \right) \right)}{\pi^2 u^2 - \beta^2 n^2} \tag{A14}$$

$$\varepsilon\_{\rm c}(\boldsymbol{u},\boldsymbol{u},\boldsymbol{j}) = 2\pi \ln \frac{(-1)^{\boldsymbol{u}+1}\cos\left(n\left(\theta\_{\hat{\boldsymbol{\beta}}} + \frac{\beta}{2}\right)\right) + \cos\left(n\left(\theta\_{\hat{\boldsymbol{\beta}}} - \frac{\beta}{2}\right)\right)}{\pi^2 \boldsymbol{u}^2 - \beta^2 n^2} \tag{A15}$$

$$\eta\_s(n, u, j) = \frac{\beta^2 n}{\pi} \frac{(-1)^u \cos\left(n\left(\theta\_j + \frac{\theta}{2}\right)\right) - \cos\left(n\left(\theta\_j - \frac{\theta}{2}\right)\right)}{\pi^2 u^2 - \beta^2 n^2} \tag{A16}$$

$$\eta\_{\varepsilon}(\boldsymbol{u},\boldsymbol{u},\boldsymbol{j}) = \frac{\beta^2 n}{\pi} \frac{\left(-1\right)^{\boldsymbol{u}+1} \sin\left(\boldsymbol{u}\left(\theta\_{\boldsymbol{j}} + \frac{\theta}{2}\right)\right) + \sin\left(\boldsymbol{u}\left(\theta\_{\boldsymbol{j}} - \frac{\theta}{2}\right)\right)}{\pi^2 \boldsymbol{u}^2 - \beta^2 n^2} \tag{A17}$$

For *π u* = *β n*:

$$\varepsilon\_s(n, u, j) = \cos\left(n\left(\theta\_j - \frac{\beta}{2}\right)\right) - \frac{\sin\left(n\left(\theta\_j + \frac{3\beta}{2}\right)\right) - \sin\left(n\left(\theta\_j - \frac{\beta}{2}\right)\right)}{2n\beta} \tag{A18}$$

$$\varepsilon\_{\varepsilon}(n, u, j) = -\sin\left(n\left(\theta\_{\hat{j}} - \frac{\beta}{2}\right)\right) - \frac{\cos\left(n\left(\theta\_{\hat{j}} + \frac{3\theta}{2}\right)\right) - \cos\left(n\left(\theta\_{\hat{j}} - \frac{\beta}{2}\right)\right)}{2n\beta} \tag{A19}$$

$$\eta\_{\rm s}(n, u, j) = \frac{\sin\left(n\left(\theta\_{\hat{j}} - \frac{\beta}{2}\right)\right)}{2\pi/\beta} - \frac{\cos\left(n\left(\theta\_{\hat{j}} + \frac{3\beta}{2}\right)\right) - \cos\left(n\left(\theta\_{\hat{j}} - \frac{\beta}{2}\right)\right)}{4n\pi} \tag{A20}$$

$$\eta\_{\mathbb{C}}(n,\mu,j) = \frac{\cos\left(n\left(\theta\_{j}-\frac{\beta}{2}\right)\right)}{2\pi/\beta} + \frac{\sin\left(n\left(\theta\_{j}+\frac{3\beta}{2}\right)\right) - \sin\left(n\left(\theta\_{j}-\frac{\beta}{2}\right)\right)}{4n\pi} \tag{A21}$$

For *δ u* = *β v*:

$$\gamma\_s(\boldsymbol{u}, \boldsymbol{v}) = \frac{2\delta^2 \boldsymbol{u}}{\pi} \frac{(-1)^{\boldsymbol{u}+1} \sin\left(\frac{\pi \boldsymbol{v}}{2\delta}(\delta + \beta)\right) + \sin\left(\frac{\pi \boldsymbol{v}}{2\delta}(\delta - \beta)\right)}{\delta^2 \boldsymbol{u}^2 - \beta^2 \boldsymbol{v}^2} \tag{A22}$$

$$\gamma\_{\varepsilon}(\boldsymbol{u}, \boldsymbol{v}) = \frac{2\beta^2 \boldsymbol{v}}{\pi} \frac{(-1)^{\boldsymbol{u} + 1} \sin\left(\frac{\pi \boldsymbol{v}}{2\delta}(\delta + \beta)\right) + \sin\left(\frac{\pi \boldsymbol{v}}{2\delta}(\delta - \beta)\right)}{\delta^2 \boldsymbol{u}^2 - \beta^2 \boldsymbol{v}^2} \tag{A23}$$

For *δ u* = *β v*:

$$\gamma\_{\mathbb{S}}(\boldsymbol{u},\boldsymbol{v}) = \frac{2\pi \,\boldsymbol{u} \cos\left(\frac{\pi}{2}\left(\boldsymbol{u} - \boldsymbol{v}\right)\right) - \sin\left(\frac{\pi}{2}\left(3\boldsymbol{u} + \boldsymbol{v}\right)\right) - \sin\left(\frac{\pi}{2}\left(\boldsymbol{u} - \boldsymbol{v}\right)\right)}{2\pi \,\boldsymbol{u}}\tag{A24}$$

$$\gamma\_{\varepsilon}(\boldsymbol{u},\boldsymbol{v}) = \frac{2\pi\,\boldsymbol{u}\cos\left(\frac{\pi}{2}\left(\boldsymbol{u}-\boldsymbol{v}\right)\right) + \sin\left(\frac{\pi}{2}\left(3\boldsymbol{u}+\boldsymbol{v}\right)\right) + \sin\left(\frac{\pi}{2}\left(\boldsymbol{u}-\boldsymbol{v}\right)\right)}{2\pi\,\boldsymbol{v}}\tag{A25}$$

#### **References**

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).


**2. Similarities and differences between LCF and TMF**

By their very nature, cyclic thermal loads appear with relatively low numbers of cycles in the low cycle fatigue (LCF) region so that the application of strain-based concepts (e.g. strain life diagrams etc.) is self-evident. If the loading is large enough to produce plastic strain, the number of cycles to failure is relatively low, in the order of less than 10,000 cycles. This total strain predominantly consists of plastic strain, which dominates the fatigue life. Widely used methods to determine the material behaviour are total strain based fatigue tests, whereby the resulting cyclic stress-strain hystereses are investigated. The resulting cyclic stress-strain curves as well as strain S-N curves are the basis for further lifetime evaluation where, depending on the material behaviour, softening or/and hardening effects can be found.

Depending on the application, further influences like temperature, mean strain, strain rate, atmosphere or aging-conditions must be considered. The components are primarily obtained by casting and defects such as pores, shrink holes or oxide inclusions ensued during this process have a negative influence on the lifetime. While these influences are extensively studied for isothermal conditions (Fagschlunger et al., 2006, Oberwinkler et al., 2010, Powazka et al, 2010), scientific understanding of the same for TMF is very limited. While LCF tests are always conducted under isothermal conditions, TMF tests are additionally loaded by thermal cycles, normally defined by a minimum and maximum temperature, dwell time and heating/cooling rates.

As TMF experiments are both very cost-intensive and time-consuming, it is often attempted in practice to estimate the fatigue life of components under thermo-mechanical load by means of more common isothermal LCF experiments. However, this approach may lead to non-conservative fatigue life estimates if the cyclic stress-strain behaviour or the effective damaging mechanisms under TMF loading differ considerably from the material behaviour under isothermal conditions. Furthermore LCF and TMF test results might not correlate due to differing methods used for recording and interpreting the deformation behaviour. In order to avoid misinterpretations it is crucial to pay close attention to the locally and temporally fluctuating temperature field, in particular when recording the TMF deformation behaviour. Thus a fundamental examination of the stress-strain behaviour and the predominant damage mechanisms under TMF conditions is crucial in order to enable accurate fatigue life predictions under thermo-mechanical fatigue loading. This approach can also clarify to which extent the employment of isothermal data is justified (Riedler et al., 2004; Riedler, 2005).

Differences may result from the fact that under TMF loading, as opposed to isothermal LCF loading, during every cycle a broad temperature range is experienced, in which the material properties can change and the material response may differ. The key to a comparison of LCF and TMF data thus lies in the evolution of the microstructure, whose integral behaviour is reflected in the shape of the stress-strain hysteresis loops.


**5. Results**

Management strategies in cases of functional single ventricle have come to a group of procedures where the goal is to obtain a ventricular pressure and volume close to normal. Our study demonstrated our experience in the management protocol of univentricular patients in total cavopulmonary connection (Fontan operation). As in other centers, the most commonly used technique has been the fenestrated, extra-cardiac Fontan, which in our study represent 58.9% of the total sample (30 patients) - Table No.1. The hospital stay was 13.6 days on average.

Table 1. Fontan type

256 Congenital Heart Disease – Selected Aspects

All patients were operated through a median sternotomy with extracorporeal circulation and moderate hypothermia. We used two techniques of total cavopulmonary anastomosis, according to anatomical characteristics of each patient. The technique I corresponds to the construction of an intracardiac lateral tunnel constructed with a patch of polytetrafluoroethylene (PTFE) sutured to the lateral aspect of the right atrium, thereby building a tunnel that diverts intracardiac blood from the inferior cava to the right pulmonary artery, thus completing total cavo-pulmonary connection. For these patients we routinely use aortic clamping and cardioplegia with blood. The technique II is interposition of a PTFE tube between the transected inferior cava and pulmonary artery, ipsilateral to the inferior cava vein, also completing total cavo-pulmonary connection. This is known as extracardiac Fontan. Use of aortic clamping and cardioplegia is optional according to surgeon's preference. Both techniques are performed with almost routine fenestration of 4 to 5 mm. In the technique I, a circular punch incision in the PTFE patch is made, so to communicate intracardiac lateral tunnel with atrial mass receiving the pulmonary venous return. In technique II, the fenestration is created by a similar punch incision in the lateral PTFE inner tube and a similar incision on the lateral aspect of the right atrium, proceeding

We included all patients with univentricular hearts who underwent Fontan operation. We reviewed the records of the patients included in the univentricular heart protocol and database service Congenital Cardiac Surgery Cardiovascular Clinic of Santa Maria de Medellin (Colombia), identifying all patients undergoing the Fontan operation with

**Monitoring.** All patients who survived the Fontan surgery underwent clinical and echocardiographic follow-up. We also obtained information about procedures performed after surgery, medication use and functional ability (classified according to New York Heart Association guidelines), interviewing parents and cardiologists caring for for each patient.

Management strategies in cases of functional single ventricle have come to a group of procedures where the goal is to obtain a ventricular pressure and volume close to normal. Our study demonstrated our experience in the management protocol of univentricular patients in total cavopulmonary connection (Fontan operation). As in other centers, the most commonly used technique has been the fenestrated, extra-cardiac Fontan, which in our study represent 58.9% of the total sample (30 patients) - Table No.1. The hospital stay was

**2. Surgical technique**

**3. Type of study**

**5. Results**

13.6 days on average.

**4. Population and sample**

to be anastomosed both holes as Luther-lateral (7.8).

Analytical study of cross-sectional, cohort analysis.

technique total cavopulmonary anastomosis between 1994 and 2010. We reviewed the medical records, operative reports and echocardiograms.

#### Table 2. Death

The number of patients who failed was a total of 8; the cause in 4 of them (50% of total deaths) was low output syndrome which was present in the first 72 hours postoperatively, and the remaining 4 patients died after 72 hours after Fontan. These patients had a progressive deterioration associated with symptoms and signs of heart failure, proteinlosing enteropathy, persistent chylothorax, affecting directly their functional status and no response to medical management. These data are shown in Table No. 3.

#### Table 3. Cause of death

The end result is shown in Table No. 4, where 47 of them (84%) were discharged and continued in a functional stage I 9n 82.1% of them (Table No. 5). One patient (1.8%) sample was transferred from the Cardiovascular Clinic to another facility outside the country.

Table 4. Cause of discharge

Fontan Surgery: Experience of One Cardiovacular Center 259

have been implemented for patients classically diagnosed with tricuspid atresia, but now with the increased survival of patients diagnosed with hypoplastic left heart syndrome, the question arises about the reduced ability of right ventricle to be able in time to support adequately the work of a single ventricle physiology to complete the process to the Fontan (3,17,18). This perception has not been recently supported (19.20). Mosca and colleagues show their results in 100 patients with Fontan performed between 1992 and 1998. They found no significant difference in the outcome compared to other types of single ventricle (21). Pizarro subsequently concluded that the Fontan can be performed safely in patients with SHIV making some modifications in surgical technique according to ventricular morphology, the mass ratio - volume and hemodynamic parameters, further suggests that Fontan surgery can be successful at earlier ages avoiding long exposure to hypoxia and risk

Our results regarding operative mortality and mid-term survival are comparable to those reported by other groups. In these studies, the mortality rate varied between 0 and 27% with an average of 10.5% and the 5-year survival varied between 81% and 93%, with an average

In relation to our surgical protocol in stages, 73.9% of patients had a bidirectional Glenn operation around 6 months of age, preparatory for the Fontan operation. The advantages of

Regarding the use of anticoagulation, we decided to keep our patients with oral anticoagulation for about 6 months with the objective of preventing thrombosis at the site of

Management strategies in cases of functional single ventricle have come to a group of procedures where the goal is to obtain a ventricular pressure and volume close to normal. This analytical cross-sectional, cohort analysis is meant to show expertise in the management protocol of univentricular patients by total cavopulmonary connection (Fontan operation.) in Cardiovascular Clinic Santa Maria in the city of Medellin. Based on this study we conclude that the Fontan operation is safe with a mortality rate comparable to previously published large series (14.3%); the results are independent of the type of ventricle and the hospital length of stay is short (average hospital stay of 13.9 days). Postoperatively, is that

[1] Pizarro Ch, Mroczek T, Gidding S, Murphy J, and Norwood W. Fontan Completion in

[3] Kreutzer G, Galindez E, Bono H, de Palma C, Laura JP. An operation for the correction

[2] Fontan F, Baudet E. Surgical repair of tricuspid atresia. Thorax 1971;26:240–8.

of tricuspid atresia. J Thorac Cardiovasc Surg 1973; 66: 613–21. [4] Andropoulus D, Stayer S, Russell I. Anesthesia for congenital heart disease 2010. [5] Anderson R, Pozzi M, Hutchinson S. Hypoplastic left heart syndrome 2005.

[7] Jonas R. Comprehensive surgical management of congenital heart disease 2004.

fenestration, after this period of time, we defined the need to close the fenestration.

This study determined that the average hospital stay was 13.9 days.

of paradoxical embolism (1).

this strategy have been previously described (24,25,26).

over 90% of patients were in functional class I - II.

[6] Lake C. Pediatric cardiac anesthesia 2005.

Infants. Ann Thorac Surg 2006; 81:2243–9.

[8] Nichols D. Critical heart disease in infants and children 2006.

of 87.5% (9,10,22, 23).

**7. Conclusions**

**8. References**

Table 5. Current functional class

Based on this study we conclude that the Fontan operation is a safe with a mortality rate comparable to that reported in previously published large series (Our series 14.3%), which is shown in the table No. 2. The Mayo Clinic experience, shows a overall mortality after Fontan, of 16%, but Many factors may have contributed to decreased early mortality after Fontan. Improved patient selection, younger age at time of operation, refinements in surgical techniques and postoperative management may all have had important roles.


*Aerodynamics of Low-Rise Buildings: Challenges and Recent Advances in Experimental… DOI: http://dx.doi.org/10.5772/intechopen.92794*

to flow separation, especially at roof edges and corners. **Figure 2** represents wind flow around a residential building [13]. The flow separates at sharp edges and re-attaches again in a fluctuating manner within the separation zones at a distance that is called separation bubble length, leading to uplift forces on the roof surface. The stagnation point is also specified in the windward wall, where the along-wind velocity is zero. **Figure 3** shows homes damaged by Hurricane Andrew in 1992 as a result of low pressures on the roof; and as a result, the shingles and sheathings were blown off due to high uplift forces. Referring to **Figure 2**, now it is shown that the separation bubble effects and the flow detachment are the main causes of these damage patterns of roof coverings which are a representation of roof areas under uplift forces. To fully understand windstorm effects on low-rise and residential buildings, it is essential to replicate the physics by experimental and computational methods. There are two important requirements: (1) correct reproduction of the main characteristic in the atmospheric boundary layer (ABL) and (2) aerodynamic testing at proper scales.

**Figure 2.** *Fluctuating flow separation and re-attachment (adapted from Ref. [14]).*

**Figure 3.** *Homes damaged by hurricane Andrew in 1992 [15].*

hurricane active regions. To put it into perspective, in most parts of the United States, especially in the east coast and the southern region, hurricanes and severe windstorms hit and bring widespread damage to buildings and other types of structures. The associated losses are estimated in billion dollars. The normalized hurricane-induced damage in the United States, between 1900 and 2005 (106 years of record), was estimated at about \$10 billion (normalized to 2005 USD) [4]. Damage records totaling \$265 billion were set by hurricanes Maria, Harvey, and Irma [5]. Due to the population growth, coastal zones are being more and more concentrated with residential buildings. These buildings are mostly light and low-rise, constructed from wooden materials, with different aerodynamic performance compared to high-rise buildings and residential homes. The American Society of Civil Engineers (ASCE) design standard defines a low-rise building to have an average roof height that is less than its lateral dimension; however the building should not exceed 18.3 m [6]. The majority of failures in low-rise buildings are reported because of strong wind effects on their envelope and specifically on roof panels [7]. **Figure 1(a)** shows a total failure of a low-rise building induced by hurricane Sandy in New York in 2012 [8]. The building envelope experienced significant loads from hurricane winds and lost its load path connections. In other scenarios, once part of a roof is breached during high winds, it facilitates the penetration of rainwater which can be harmful to interior properties and may cause serious problems to the building and loss of contents. **Figure 1(b)** shows severe roof damage during Hurricane Katrina in Lake Charles, New Orleans, in 2005 [9]. Examination of post-disaster surveys indicates initiation of damage through failure of roof components under extreme wind events. Earlier studies confirm the presence of extreme negative pressures at corners, ridges, and leading edges of roofs. The performance of roofs in low-rise buildings can differ significantly during a windstorm according to the shape of roof and its dimension. For instance, large roofs in industrial buildings may behave differently, compared to those of small roofs in a single-family low-rise building which can lead to different damage patterns to the building envelope [10–12]. In large roofs, the correlations among pressures acting at different roof locations are usually low [13]. In large roofs of light metal industrial buildings, leading edge failure usually occurs due to poor attachment of metal sheathing in areas that are exposed to uplift wind forces. This weakness eventuates to progressive peeling of the roof membrane causing further

damage to the whole integrity of the building envelope.

**Figure 1.**

*Aerodynamics*

**32**

The components and claddings in small roofs are usually exposed to damage during windstorms, due to local fluctuating negative pressures (uplift effects) due

*Hurricane-induced damage: (a) complete collapse of a residential home induced by hurricane Sandy, New*

*York, 2012 [8] and (b) severe roof damage by hurricane Rita in Lake Charles, in 2005 [9].*


**6. Proposal for structural stabilization**

*"utilitas, venustas e firmitas",* Vitrúvius (1st century BC): Its standards of proportions and its conceptual principles - utility, beauty and solidity.

Since antiquity, although restricted to a limited territory from Roman times, there are registers of regulatory norms on heritage protection: Decree of about the year 44 AD, discovered in the historic city of Herculano, Italy - forcing anyone to demolish a building for speculative purposes to pay the authorities twice the purchase price. During the period of the Roman empire, an Edict dated 17 July of the year 389 appears - *"It is forbidden to disfigure the exterior ornaments of private buildings with modern additions and damage the historic buildings of an important city for reasons of greed, for the sake of profit".*

Pope Gregory I (590–604) proposes and practices a policy of reusing the immense Roman legacy abandoned after fall of the western Roman Empire. The great patrician domus are turned into monasteries, their halls of reception in churches. He warns his missionaries not to destroy the pagan temples and buildings, but rather by the otherwise they must be preserved and prepared and properly adapted - placing their altars and their relics, for Christian worship [8].

Eurocode EC 8, part 3 - *"Assessment and rehabilitation of buildings",* establishes the criteria for assessing the seismic performance of existing buildings, describing the approach regarding the corrective measures to be taken. It also establishes criteria for measures to repair and / or reinforce structural elements in the design

*Structural Consolidation of Architectural Heritage DOI: http://dx.doi.org/10.5772/intechopen.99602*

and final dimensioning of the new elements to be introduced and their necessary connections to the original structural system.

The structural elements in wood, together with the masonry, are of equal importance for the stability of the building. Due to its great vulnerability to deterioration agents and in view of the need to adapt the construction to the new uses and safety criteria currently required, it is absolutely necessary to provide for an adequate intervention, aiming at increasing the resistance of the deteriorated element, either through its reinforcement using new materials and / or by reconstructing the section with anomalies, using the same material - with or without connecting elements [9].

Using a basic model that intends to represent the genesis of structures designed by man, an attempt was made to conceive an Archetype, which in a simple and schematic way summarizes the structure of the study.

Prostheses introduction - shown in black in the photograph, for the damaged sections of the element: Type A repair: in its lower support (**Figure 15**); Type B repair: on its upper support (**Figure 16**); Type C repair: wherever there is an anomaly (**Figure 17**); Type D repair: integral replacement of the element (**Figure 18**).

**Figure 15.** *Prosthesis on the lower support.*

**Figure 16.** *Prosthesis on the upper support.*

**Figure 17.** *Prosthesis with variable location.*

**Figure 18.** *New replacement element.*


# **Flash Memory Devices**

Edited by Cristian Zambelli and Rino Micheloni Printed Edition of the Special Issue Published in *Micromachines*

www.mdpi.com/journal/micromachines

**Flash Memory Devices**


Current Drying Processes *Edited by Israel Pala-Rosas*

Published in London, United Kingdom

## *Supporting open minds since 2005*

Current Drying Processes http://dx.doi.org/10.5772/intechopen.78916 Edited by Israel Pala-Rosas

#### Contributors

Le Anh Duc, Keum Dong Hyuk, Chang Peng, Saeed Moghaddam, Artem Artyukhov, Jozef Bocko, Jan Krmela, Nadiia Artyukhova, Ruslan Ostroha, Mykola Yukhymenko, Margarita Castillo Téllez, Beatriz Castillo Téllez, José Andrés Alanís Navarro, Juan Carlos Ovando Sierra, Gerardo A. Mejia Pérez, Italo Pedro Bello Moreira, Edgar Ruperto Macías Ganchozo, Xavier Enrique Anchundia Muentes, Celio Danilo Bravo Moreira, Manuel Eduardo Anchundia Muentes, Hebert Edison Vera Delgado, Carlos Eduardo Anchundia Betancourt, Trung Thanh Bui

#### © The Editor(s) and the Author(s) 2020

The rights of the editor(s) and the author(s) have been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights to the book as a whole are reserved by INTECHOPEN LIMITED. The book as a whole (compilation) cannot be reproduced, distributed or used for commercial or non-commercial purposes without INTECHOPEN LIMITED's written permission. Enquiries concerning the use of the book should be directed to INTECHOPEN LIMITED rights and permissions department (permissions@intechopen.com).

Violations are liable to prosecution under the governing Copyright Law.

Individual chapters of this publication are distributed under the terms of the Creative Commons Attribution 3.0 Unported License which permits commercial use, distribution and reproduction of the individual chapters, provided the original author(s) and source publication are appropriately acknowledged. If so indicated, certain images may not be included under the Creative Commons license. In such cases users will need to obtain permission from the license holder to reproduce the material. More details and guidelines concerning content reuse and adaptation can be found at http://www.intechopen.com/copyright-policy.html.

#### Notice

Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book.

First published in London, United Kingdom, 2020 by IntechOpen IntechOpen is the global imprint of INTECHOPEN LIMITED, registered in England and Wales, registration number: 11086078, 7th floor, 10 Lower Thames Street, London, EC3R 6AF, United Kingdom Printed in Croatia

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library

Additional hard and PDF copies can be obtained from<EMAIL_ADDRESS>
Current Drying Processes Edited by Israel Pala-Rosas p. cm. Print ISBN 978-1-83880-109-0 Online ISBN 978-1-83880-110-6 eBook (PDF) ISBN 978-1-78985-861-7


Chapter 7

the Bottom Up. Cambridge, MA: The

Computer Architecture in Industrial, Biomechanical and Biomedical Engineering

[44] Taleb N. The Black Swan. London,

Individual Values. 2nd ed. New Haven,

[46] Festinger L, Rieckin H, Schachter S. Where Prophecy Fails: A Social and Psychological Study. New York, NY:

[48] Tarvis C, Aronson E. Mistakes Were Made (but Not by Me). San Diego, CA:

[49] Simons D, Chabris C. Gorillas in our midst: sustained inattentional blindness for dynamic events. Perception. 1999;

[50] Pronin E, Gilovich T, Ross L. Objectivity in the eye of the beholder: Divergent perceptions of bias in self versus others. Psychology Review.

[51] Kruger J. Personal beliefs and cultural stereotypes about racial characteristics. Journal of Personality and Social Psychology. 1996;71:536-548

[52] Nesbitt R, Ross L. Human Inference, Strategies and Shortcomings of Social Judgment. Upper Saddle River, NJ:

MIT Press; 1996. pp. 130-135

UK: Random House; 2007

HarperCollins; 1964

Harcourt; 2007

28:1059-1074

2004;111:781-799

Prentice-Hall; 1980

102

[45] Arrow K. Social Choice and

CT: Yale University Press; 1963

[47] Tumminia D. When Prophecy Never Fails: Myth and Reality in a Flying-Saucer Group. Cambridge, UK:

Oxford University Press; 2006


Bioactive Peptides from Legumes and Their Bioavailability

*Retno Indrati*

## **Abstract**

Bioactive peptides (BPs) isolated from legumes have functional properties as healthy foods. These functional effects depend on their stability and bioavailability in the gastrointestinal tract before reaching the target organs. Therefore, it is necessary to disclose the factors that influence it and discuss the technical processing to develop its utilisation. This chapter discusses and summarises the bioactive activities of BPs from various legumes, factors and mechanisms related to the bio-assessability, stability, bio-availability and bioactivity of BPs. Furthermore, the development of BPs bioseparation was also discussed. The results show that the nature of BPs varies greatly depending on the legume source and the production method. Factors that influenced the bio-availability of BPs include molecular weight, charge, amino acid sequence, the presence of specific residues and hydrophobic amino acids, and resistance to the action of peptidase while in the digestive tract. However, some BPs showed increased bio-accessibility and bio-availability after being hydrolyzed by digestive enzymes. Processing technologies such as encapsulation allowing BPs to enter the body and undergo release and degradation by enzymes digestion. Further studies are required to understand the increase in the bioavailability of BPs, the safety of the food components produced, and their use in producing functional foods.

**Keywords:** Legumes, bioactive peptides, fermentation, germination, health benefits, bioavailability


**4. Different types of SW-GW interactions**

There are several types of interactions between GW and SW. A losing river does not lose water as it flows downstream by percolation, but it can also lose water through evaporation, use of plants, and consumption of human activities.

#### **Figure 7.**

*Types of stream water and GW interactions, while (A) a stream water gains from GW, known as gaining stream, (B) a stream loses water to GW at shallow depth known as disconnected or losing or transition stream, (C) a stream loses water to GW at great depth known as a disconnected stream, (D) a stream loses water, but connected with GW known as a connected stream (E) a stream stage and the GW head are equally known as the parallel Stream, and (F) a stream shows the gain on one side and the loss on the other side to GW known as flow-through stream.*