Synthesis of Application-Optimized Air Gap Field Distributions in Synchronous Machines

Abstract

This paper deals with the method of shaping the magnetic field distribution in the air gap of a synchronous machine. The main goal is to obtain a specific distribution of the magnetic field in the machine’s air gap to enable easier powering, to increase the torque and power density, or to limit the content of higher harmonics in the induced voltage and torque. This method can be applied for both the electrically excited machines and permanent magnets excited machines. The problem has been reduced to solving a modified system of algebraic equations obtained by the finite element method. The presented examples show the effectiveness of the proposed method both in shaping the contours of the air gap and the magnetization of permanent magnets placed on the rotor. The method can particularly be used in the design of the high-speed synchronous machines.

1. Introduction

Inverter-fed synchronous machines (SM) are increasingly used for the advanced production machinery and vehicle drives. The quality and profitability of an electrical drive is mainly determined by the magnetic circuit of the machine and the performance characteristics of the inverter. The control system should be designed to maximize the force and power density of the machine as well as reduce the cogging torque or harmonic content while maintaining the maximum efficiency. In automotive applications (e.g., cars, trains, trams) with a fixed gear transmission ratio, the drive must perform in a wide speed operating range. The above requirements are especially important for high-speed machines.

2. Optimal Induced Voltage and Force Generation

The inverter must meet all the requirements of the control unit when feeding the machine. Here, in addition to the energy conversion efficiency, the control strategy must consider a whole series of secondary conditions, of which the most important is the minimum power consumption of the inverter. From the point of view of machine design, there are various options for reducing the required inverter power. The power should always be reduced if it is not possible to use expensive switching elements designed for high reverse voltages or high switching frequencies. The high voltage induced at the start of the commutation process requires a high inverter voltage and a long duration of this voltage. The reduction in this voltage and its duration reduces the required power of the inverter. The armature current should be controlled in such a way that the machine performance, as mentioned above, can be achieved with minimum requirements put on the electronic switching components (current, voltage, and switching frequency). The possibilities for this are investigated by adapting the pole geometry of salient pole SM and/or magnetization distribution permanent magnet (PM) excited machines, respectively [1,2]. These measures can have a positive effect on all important machine parameters (inductances, flux density distribution, saturation, etc.). Many practical comments on the analytical calculation of field distributions in slotted PM-machines are also presented in [3,4]. The conclusions obtained there can be used for the initial shaping of synchronous machines with permanent magnets.

As most of the SMs are characterized by a constant air gap, there is little scope left to adapt the armature emf (electromotive force) waveform to the needs of the inverter to feed the machines without a drastic over-dimensioning of the electronic switches. Based on the dynamic simulations of the inverter-fed drives, specific demands can be imposed on the armature emf waveforms for different applications. To cope with these specifications, new machine designs with an adapted air gap contour (machines with iron poles) and/or the magnetization (amplitude and direction) of the magnets of permanent excited machines are proposed.

The machine design should be optimized in such a way that the burden on the inverter is minimized. The choice of the power semiconductors is based on their ampacity, motor reverse voltage, and the switching frequency. This can be deduced from the voltage equations of the synchronous machine. Figure 1 shows a typical three-phase machine.

The voltage equation for the ν-phase of the above machine (a magnetically smooth air gap and non-saturated iron are assumed) can be written as follows:

$${{\displaystyle u}}_{\nu }={{\displaystyle R}}_{\nu }\cdot {{\displaystyle i}}_{\nu }+{\displaystyle \sum _{\mu =1}^3\left(M_{\nu ,\mu }\frac{d\hspace{0.17em}\hspace{0.17em}{i}_{\mu }}{d\hspace{0.17em}t}+\frac{d\hspace{0.17em}{M}_{\nu ,\mu }}{d\hspace{0.17em}t}\cdot i_{\mu }\right)}$$

(1)

where R—resistance, M—self- or mutual-inductance. The external voltage at the terminals is described by ${{\displaystyle u}}_{\nu }$ and the currents are described by i_ν; t indicates the time and ϑ specifies the rotation angle (see Figure 1).

For the machine with only one phase in the stator and one phase in the rotor, Equation (1) is written explicitly as follows:

$${{\displaystyle u}}_1={{\displaystyle R}}_1\cdot {{\displaystyle i}}_1+{{\displaystyle L}}_1\frac{{{\displaystyle di}}_1}{dt}+{{\displaystyle i}}_1\frac{{{\displaystyle \partial L}}_1}{\partial \vartheta }\cdot \frac{d\vartheta }{dt}+{{\displaystyle M}}_{1,2}\frac{{{\displaystyle di}}_2}{dt}+{{\displaystyle i}}_2\frac{{{\displaystyle \partial M}}_{1,2}}{\partial \vartheta }\cdot \frac{d\vartheta }{dt}$$

(2)

$${{\displaystyle u}}_2={{\displaystyle R}}_2\cdot {{\displaystyle i}}_2+{{\displaystyle L}}_2\frac{{{\displaystyle di}}_2}{dt}+{{\displaystyle i}}_2\frac{{{\displaystyle \partial L}}_2}{\partial \vartheta }\cdot \frac{d\vartheta }{dt}+{{\displaystyle M}}_{2,1}\frac{{{\displaystyle di}}_1}{dt}+{{\displaystyle i}}_1\frac{{{\displaystyle \partial M}}_{2,1}}{\partial \vartheta }\cdot \frac{d\vartheta }{dt}$$

(3)

The no-load stator voltage for this machine can be written as:

$${{\displaystyle u}}_{20}={{\displaystyle M}}_{2,1}\frac{{{\displaystyle di}}_1}{dt}+{{\displaystyle i}}_1\frac{{{\displaystyle \partial M}}_{2,1}}{\partial \vartheta }\cdot \frac{d\vartheta }{dt}$$

(4)

With the simplifications mentioned above (a smooth air gap and linear iron), the force F caused by the rotor winding “1” interacting with the stator winding “2” reads as follows:

$$F=-½\frac{\partial L_1}{\partial \vartheta }{i}_1^2-\frac{\partial M_{1,2}}{\partial \vartheta }{i}_1{i}_2-½\frac{\partial L_2}{\partial \vartheta }{i}_2^2$$

(5)

Equation (5) indicates that the performance of an energy converter is determined by the space variation of the SM inductances and the feeding parameters. From this equation, it can be deduced that for direct drive applications with an increased torque demand at low speeds, the mutual inductance and thus their rotational gradient should be very high. Salient pole SMs feature these characteristics. On the other hand, for high-speed applications, the values of the self- and mutual inductances shall be mostly identical and with lower values. Under these conditions, the inverter benefits from the well-balanced quantities, which create the force and back-emf. Simultaneously, the reduced value of L and thus the time constant supports a fast increase in the current needed for high dynamic applications at increased speeds. These demands on the inductances of permanent magnet excited machines are mostly complied with if the machine exhibits a large pole coverage (pole width to pole pitch ratio) value.

Figure 2 shows some possible flux density distributions in the air gap (normal component) within the pole pitch, which induce appropriate no-load armature emf waveforms. Particularly, the curve with an unsymmetrical distribution characterized by a slow rise at the beginning and an increased value at the end (emf-dynamic) of the pole-pitch τ enables the machines to run with extremely high speeds.

Due to the improved commutation conditions, the curve (a) marked with squares, which initially has a slow rise in the direction of movement and an additional maximum shortly before the end of the pole pitch, supports the machine supply at very high speeds. To obtain a favorable dependence of the induced voltage in the stator winding, it is necessary to generate a suitable magnetic field inside the machine air gap. This goal can be achieved by appropriately shaping the machine air gap, as shown in Figure 3.

The above machine with the variable air gap (decreasing in the direction of rotation) generates the induced voltage given in Figure 2 (squares). The air gap magnetic flux Φ for one pole pitch τ can be calculated as follows (the x0y coordinate system is shown in Figure 3):

$$\Phi ={\displaystyle \underset{0}{\overset{\tau }{\int }}{B}_y(x)\hspace{0.17em}dx}$$

(6)

where B_y(x) is the y-component of the magnetic field induction.

The relationship between the magnetic field distribution within the air gap and the voltage induced by the rotor moving with the speed v in the stator coil with span τ can be expressed by the formula:

$$U=\frac{\partial \Phi }{\partial t}=v\frac{\partial \Phi }{\partial x}.$$

(7)

The above equations show that the induced voltage has the same shape as the magnetic flux distribution within the air gap. The final voltage seen at the machine terminals is the phasor sum of individually induced voltages in successive series-connected stator windings. In this sum, all the features of individual partial voltages are visible; therefore, it is necessary to ensure that the voltage induced in each individual winding has the desired dependence (e.g., a gentle increase in the initial phase or a low content of higher harmonics).

The criteria of the quality of the voltage waveform are ambiguous. One should always consider many different and sometimes conflicting requirements, e.g., torque ripple, power losses in windings, system complexity, and the cost. An additional problem is that the emf waveform is generated by a moving pre-defined rotor structure, so the values of the voltage induced in the subsequent time steps are strictly defined. The induced voltages (b) and (c) from Figure 2 are generated by clearly defined magnetic field distributions. The generation of (a) is much complicated, and the criteria for its determination are ambiguous.

The limitation of the harmonic content in emf can be performed by appropriate selection of such a magnetic field distribution in the gap, which, by definition, is characterized by low harmonic values. For a perfectly sinusoidal distribution, the harmonic content will be equal to zero, and for a trapezoidal distribution, the harmonic content can be controlled by selecting the slope of the trapezoidal function.

Similar problems related to the generation of appropriate voltages in electrical machines from the point of view of various requirements and applications were presented, for example, in the papers [5,6,7,8,9]. In [5], the influence of PM magnetization and the stator slot and tooth structure on induced emf waveforms and torque in a brushless PM machine were discussed. The paper [6] deals with the comparison of different hybrid excited synchronous motors with sinusoidal and non-sinusoidal back-emf and their influence on different features of these machines. The optimization of back-emf waveforms for the torque ripple reduction in a flux-switching PM-motor can be found in [7] and for the doubly salient machine in [8]. To improve the torque performance of the doubly salient electromagnetic machine, paper [9] proposes an optimization method of the magnetomotive force based on the optimized rotor pole geometry. Paper [10] deals with the comparison of the back electromotive force for two different electrical machines with the same dimensions. This analysis provides some qualitative advice for the application, optimization, and control methods. Paper [11] performs the back-emf waveform analysis of different PM machines from the flux weakening point of view. Reduction in the back-emf values and its pulsations were studied in [12].

All of the above-mentioned papers examine the back-emf waveforms, pointing to the positive and negative effects of their different shapes (torque increase, harmonic reduction, etc.), but they do not offer much guidance on how to generate the desired emf-curves.

3. Shape Synthesis and Optimization

The task analyzed in this paper depends on the determination of the magnetic pole contour in such a way that the desired dependence of the induced voltage is achieved through a corresponding flux density distribution in the air gap. This is a typical field synthesis task. Field synthesis tasks are classified according to the type of sought physical quantities. The following problems can be distinguished here (see Figure 4):

• Determination of distribution of field excitors;
• Determination of the shapes of individual regions;
• Determination of boundary conditions;
• Determination of material coefficients and parameters appearing in the field equations or boundary conditions.

Specific methodologies and many unique algorithms were developed to solve various field synthesis problems. Many scientists have been dealing with the shape synthesis of parts of electrical machines or other electrical devices for many years. One can mention here, among others, Lowther [13,14,15,16], Sykulski [17,18,19,20,21,22,23,24], Di Barba [25,26,27], Mognaschi [28,29], Adamiak [30,31,32,33], and teams cooperating with them.

Some studies have analyzed various structures of electrical machines and made a preliminary selection of their optimal configurations by comparing many possible machine variants. These papers, by limiting the scope of considerations and giving the possibility of choosing solutions better than the others, constitute the starting point for “final optimization”. The initial classification of magnetic structures and their design methods were presented in [34,35]. The paper [36] presents some fundamental formulations of different optimization problems in PM synchronous machines and describes mathematical algorithms for unconstrained and constrained problems. Such preliminary recommendations make it possible to design optimal electrical machines [37,38].

In [39,40,41,42,43,44,45,46], sensitivity analysis for the shape optimization of various electromagnetic devices has been shown. Application of topological gradient and adjoint sensitivity analysis to the multi-objective design optimization of different PM-excited and hybrid excited synchronous machines was presented in [47,48,49,50]. In [51], the Taguchi algorithm was used to analyze the sensitivity of design variables for the multi-objective optimization of switched reluctance machines. Genetic algorithms for the optimization of electrical machines have been discussed in [52,53]. With the help of the above algorithms, it is possible to optimally select the geometrical sizes of machines with a given structure, so they are not direct algorithms of the shape synthesis. Response surface methods (RSM) for the shape optimization were used in [20]. In [21,24], kriging surrogate models for electromagnetic design were applied. In [54,55], the influence of magnet shape on the cogging torque and back-emf waveform of PM-motors has been analyzed.

The problems of shape synthesis in which the optimization of geometric coordinates of boundary areas is carried out were considered, among others, in [15,20,22,40,44,45,53].

The list of papers on shape optimization and synthesis could be significantly extended, and the references presented here do not exhaust the issue. A summary of the methods used in optimization in electromagnetism can be found, among others, in [19,24,27].

4. Shape Optimization of the SM Pole

Initially, the problem of generating a symmetrical (sinusoidal or trapezoidal) field in the machine gap will be considered as completely unambiguous. This leads to the geometric symmetry of the machine pole (Figure 5).

The above task belongs to the class of ill-posed problems, which means that its solution can either be ambiguous or discontinuous [19,56,57]. The magnetic field distribution within the region is given as the solution of the following equation for the vector potential A (linear 2D case):

$${\nabla }^{2}A=\hspace{0.17em}-\mu J-\mathrm{rot}P$$

(8)

where J—current density, P—magnetic polarization, and μ—permeability.

Application of the finite element method (FEM) for solving such a problem leads to the standard system of n algebraic equations with n unknown values of the vector potential (n—the number of points in the region of investigation G):

$$WC=\hspace{0.17em}U$$

(9)

where W—the main matrix of a system, C—matrix of unknowns, U—right-side matrix.

To formulate the field synthesis problem, the fundamental FEM equation set describing the magnetic field distribution within the analyzed region must be extended by equations defining additional requirements on the field distribution in the field synthesis region G₀:

$$B\hspace{0.17em}(x,y)=B_0(x,y)\hspace{1em}in\hspace{1em}{G}_0$$

(10)

The components of the magnetic field vector in the triangular finite element with nodes i, j, k are calculated by using the formula B = rot A:

$$B_x=\hspace{0.17em}\frac{\partial A}{\partial y}=\left[a_i,a_i,a_k\right]\left\{A_i,A_j,A_k\right\}$$

(11)

$$B_y=\hspace{0.17em}-\frac{\partial A}{\partial x}=\left[b_i,b_i,b_k\right]\left\{A_i,A_j,A_k\right\}$$

(12)

where a_i, a_j, a_k, b_i, b_j, and b_k are known coefficients depending only on the geometry of the investigated element.

If in the region G₀ there are k finite elements, the condition of Equation (10) additionally generates 2k equations of the form given in Equations (11) and (12). Let us suppose that there are m moving points lying on the pole contour. If its shape is unknown, the system of Equation (9) additionally contains 2m unknown x- and y-coordinates of points lying on this contour (some coefficients of the matrix W and U are also unknown), so the system of Equation (9) becomes the set of n + 2k nonlinear and ill-conditioned equations with n + 2m unknowns (vector potential values and coordinates of points on the magnetic pole surface):

$$v=\hspace{0.17em}f(z)$$

(13)

In all the shape synthesis problems, the number of finite elements in the synthesis region G₀ is bigger than the number of moving points (k > m) so that the system of Equation (13) is overdetermined. The solution z₀ of the system (13) in the sense of the least-squares minimizes the following function Q:

$$Q=\hspace{0.17em}{e}^{T}e$$

(14)

where $\mathbf{e}=\mathbf{v}-f\left(\mathbf{z}\right)$ is the error vector. The iterative solution of the system (13) can be written as:

$$z^{k+1}=\hspace{0.17em}{z}^k+{\alpha }_k{D}^k$$

(15)

where D^k is the direction vector in the k step and ${\alpha }_k$ is a step-length factor. The direction vector D^k in each iteration of Equation (15) can be expressed as the solution of the following system of equations:

$$J^k{D}^k=e^k$$

(16)

where J^k is the Jacobi matrix:

$$J^k=\left[\partial f_i/\partial z_j\right],i=1,2,n+2k,j=1,2,n+2m$$

(17)

The solution $D_0^k$ of (16) in the sense of the least squares can be written as:

$$D_0^k=R^k{e}^k$$

(18)

where R^k denotes the Moore–Penrose pseudo-inverse of J^k [58,59].

The minimal least squares solution, i.e., the solution of the minimum length that minimizes the error vector e^k, can be determined from the singular values decomposition of J^k. The calculation algorithm consists of two steps: at first, the sparse matrix J^k is reduced to the tri-diagonal form by the aid of Lanczos transformations, and in the second step, the singular values of this matrix are calculated using the QR algorithm [59]. This universal gradient method enables iterative determination of the matrix R^k and the minimal least squares solution $D_0^k$ as in Equation (18).

The above algorithm for solving overdetermined systems of algebraic equations has been implemented into the finite element method. It should be noted here that (because of possible numerical instabilities) the above algorithm is very sensitive. Therefore, it was necessary to define several accuracy criteria to prevent obtaining solutions with physically unacceptable values. As the main solution procedure, the routine F04JAF from the NAG Fortran Library has been applied [60]. Using the above algorithms, the solution to the analyzed problem can be carried out iteratively [57]. The numerical procedures discussed here, as well as the graphic algorithms, are proprietary software that has been extensively tested in many industrial projects.

Figure 5 illustrates the initial magnetic field in the machine. The basic geometric dimensions of the machine (the pole pitch τ = 80 mm, the pole width w = τ/2 and other pole shoe dimensions) have been arbitrarily assumed as typical for such a machine, and the main design variable is the size of the air gap δ(x). The task depends on the determination of the pole contour to achieve the sinusoidal distribution of the magnetic flux density along the air gap:

$$B_y(x)=B_{max}\sin (\pi x/\tau )\hspace{1em}in\hspace{1em}{G}_0$$

(19)

where B_max denotes the air gap magnetic flux density in the middle of the region G₀ (x = τ/2) calculated for the initial pole shape. Such an assumption of the B_max value means that the optimal pole shape sought will be only a small correction of the initial shape, which stabilizes and simplifies the calculations.

The results are illustrated in Figure 6 and Figure 7. Figure 6 shows the relative magnetic flux density B_y/B_max in the air gap before and after synthesis and gives the comparison with the required distribution of Equation (19). Figure 7a shows a part of the initial FEM mesh where the moving points were placed (regular finite element mesh with pre-defined connections), and Figure 7b shows the final optimized mesh [44,57].

In order to implement the above-described methodology, the FEM algorithm was modified to be able to consider the displacement of the FEM mesh nodes without destroying its structure by the proper choice of the step-length factor α in Equation (15). This required the development of algorithms for shifting entire groups of finite elements, as well as the thickening and densification of elements without generating additional points or elements in the area (dynamic mesh adaption) [44,57], as shown in Figure 7. The application of the techniques using the known FEM sparse matrix topology allowed for quick iterative determination of successive approximate solutions without changing the structure of all FEM matrices. The accuracy of the final solution is relatively high (especially in the region above the pole). It was difficult to obtain the assumed field distribution over the entire area due to the large difference between the initial and the required flux density distribution. This also demonstrates the importance of the appropriate choice of the primary configuration where the field synthesis must be performed.

As the quality criterion, the total error between the required (given by Equation (19)) and calculated B_calc magnetic field distribution has been used:

$$\epsilon ={\displaystyle {\int }_0^{\tau }(B_{max}\sin (\pi x/\tau )}-B_{calc})\hspace{0.17em}dx$$

(20)

In its most important part (the vicinity of the air gap), the mesh has a regular character of point-to-point connections (see Figure 7). This allows full control of edge point movements. In each iteration step (usually five to seven steps), the shift of the area boundary points should be controlled so as not to disturb the existing FEM mesh structure. This means a limitation in the values of the shift of the coordinates of these points (Equation (15)). After each step, the FEM mesh was modified, and the mesh points were distributed evenly (see Figure 7). In each iteration step, the value of the quality factor (Equation (20)) was calculated and compared with the value in the previous step. The lack of visible changes of this indicator was a sign to interrupt the correction process. Changes to the mesh in subsequent iteration steps are small. This causes a slow convergence of the calculation algorithm and the need to carry out many iterative steps when calculating the new pole contour, which, however, does not complicate but stabilizes the calculations.

All of the considerations presented above can also be performed for the nonlinear magnetic circuit of the machine. This is due to the fact that changes in the pole structure and the associated magnetic field changes during the optimization process are very small. Calculation of the initial field distribution in the machine should be performed taking into account the magnetization characteristics of all magnetic materials used in its construction (Newton–Raphson iterative process). The obtained local values of magnetic permeability of nonlinear regions should be remembered, and the next optimization calculations should be carried out for these remembered values. To simplify the situation, the examples shown in this paper were carried out for a constant value of magnetic permeability of iron (µ = 1000 µ₀).

The FEM grids of the configurations presented in the paper usually contain about tens of thousands of calculation points (for symmetrical configurations, it was enough to calculate for half of the presented regions—zero Dirichlet boundary condition for the vector potential in the middle of the pole). A remanence induction of 1.2 T permanent magnets was assumed, and their magnetic permeability µ = 1.05 µ₀.

5. Magnetization Optimization of the Symmetrical SM Pole

The described algorithm can also be used to calculate the optimal magnetic pole magnetization of the PM-excited SM. Figure 8 shows one-half of the symmetrical magnetic pole of a synchronous machine with surface mounted PMs. The task is to determine the magnetization of this pole in such a way that the magnetic flux density in the air gap has a certain profile, such as, for example, in Equation (19).

In this case, the optimization task can be reduced to the solution of an overdetermined, linear, and poorly conditioned system of equations, the unknowns of which are the vector potentials and magnetizations of PM region. The solution of this problem can be obtained in one step because of its linearity (Equation (13) is linear).

In practical applications, there is no need to determine the local magnetization of the pole (magnetization of all finite elements of the pole), but the magnetization of several larger parts of it should be determined. This is due to the complication and high price of the magnetizing devices. For this reason, the surface PM was divided exemplary into eight independent parts where the magnetization should be determined. Since the calculations are carried out for the half of the pole, eight additional unknowns (magnetization components in the x- and y- directions) appear in the final formulation. In this case, the resulting system of equations is heavily overdetermined, which enables its easy and stable solution. Assuming uniform magnetization of the PM in the y-direction (initial field distribution), the value of B_max in Equation (19) was determined. Figure 9a shows the initial magnetic field distribution, while Figure 9b,c show the magnetic field distribution obtained for the optimized magnetization in y-direction (P_x = 0 T) and for the optimized magnetization in both directions, respectively.

The comparison of the initial, required, and obtained field distributions is given in Figure 10.

As shown in Figure 10, it was possible to improve the magnetic field distribution in the machine gap by selecting the appropriate magnetization of individual pole segments (increasing the value of magnetization at the ends of the magnetic pole). The characteristic wave patterns of the field distribution ((b) and (c) in Figure 10) are related to the assumed sizes of individual magnetic domains in which the optimal magnetization is determined. This can be avoided by performing a magnetization synthesis in more PM areas. However, it is associated with evident difficulties in the practical implementation of such magnetization. Of course, a much cheaper solution is to synthesize the shape of a pole with a given simple magnetization, e.g., parallel or radial.

The performed optimization allowed for a significant improvement in the quality of the field in the machine air gap.

6. Shape and Magnetization Optimization of the Non-Symmetrical SM Pole

Using the presented method, it is possible to simultaneously synthesize the shape of the pole and its magnetization. However, this algorithm is associated with some difficulties, because the limitations imposed on the coordinates of the movable points cause large changes in the calculated pole magnetization (for the stability of the solution, drastic limitations on the obtained magnetization values should be introduced). For this reason, a combination of the two methods presented is preferable. In the first step, it is possible to optimize the shape of the electrically excited SM pole, and in the second step, it is possible to additionally improve its magnetization. The same procedure can be used for the PM-excited machine (with a given magnetization) where the magnetization of the pole can additionally be optimized or corrected. Initial optimization of the pole shape and then determination of the improved magnetization of the pole is a procedure that facilitates calculations, as it finally reduces the problem to a linear task.

Figure 11 shows the possible stages of optimization of the pole shape of an electrically excited synchronous machine.

Figure 12 shows a comparison of magnetic field distributions in the middle of the air gap for symmetrical and asymmetrical SM poles from Figure 11. The most important for all applications is the shape of these functions above the magnetic pole of the machine. Their course outside this area depends primarily on the geometrical proportions of the machine (the mutual distance between the poles) and not on the shape of the pole or the air gap.

The field distribution over the pole even for symmetrical air gaps (curves (a) and (b)) is far from sinusoidal or trapezoidal. This is due to the arbitrary assumption of the shape of the pole. This also shows how important it is to choose the right initial shape of the pole. The maximum obtained for the curves (c) and (d) may significantly exceed the maximum obtained for the symmetrical shapes of the pole; however, it relates to a significant local reduction in the air gap size. The optimized curve (d) makes it possible additionally to obtain a more favorable increase in the value of the magnetic field.

The comparison of the suitability of different air gap field distributions for the force generation in the machine can be made, among others, assuming the same area in the air gap pole zone S:

$$S={\displaystyle {\int }_0^w\delta (x)}\hspace{0.17em}dx$$

(21)

The distributions in Figure 12 can still be improved by placing the PMs on the surface of the pole shoe (or inside it), because in this way, additional possibilities of shaping the field in the gap are obtained. If magnets with any magnetization were available, such a magnet could be placed on the geometrically pre-optimized pole shoe of an electrically excited machine.

Figure 13a shows the unsymmetrical pre-optimized magnetic pole of SM with PMs uniformly magnetized in the y-direction. The shape of this pole was initially optimized according to the defined requirements as in Section 4. The distribution of the magnetic flux density in the air gap from Figure 13a is similar to the distribution desired in Figure 2a and should be further improved. Figure 13b,c show the magnetic field distribution in the configuration from Figure 13a after calculating the corrected magnetization of four magnetic parts of the pole shoe.

The comparison of obtained field distributions is given in Figure 14.

Figure 13b shows a characteristic increase in the value of the pole magnetization. In this way, a too large gap dimension for increasing values of the x-coordinate is compensated. The distribution of the magnetic field component in the y-direction presented for the configuration in Figure 13c gives the result identical to that shown in Figure 13b. The difference is that in the configuration (c), there is an additional possibility to eliminate the field component in the x-direction: B_x(x) = 0 in G₀ (flux lines in the air gap are parallel to the y-axis); hence, the characteristic magnetization values in the x-direction of individual parts of the pole. The curve (c) has the same characteristic wave pattern as the curves (b) and (c) in Figure 10. It is related to the arbitrary division of the PM area into four sub-areas in which the optimal magnetization is determined. The values of the magnetic field component in the y-direction are initially even lower than for an electrically excited SM and reach their maximum faster. As shown in Figure 14, the performed optimization allowed also in this case for a significant improvement in the quality of the field in the machine air gap.

The obtained optimal field distributions allow for easy determination of the self and mutual inductances of the machine windings that meet the assumed design requirements, which allows them to be used for the standard simulation of all machine quantities such as power, torque, current, and voltage (Equations (1)–(5)).

7. Conclusions

The paper presents the methodology of shaping the magnetic field distribution in the air gap of synchronous machines. The proposed method makes it possible to obtain a given magnetic field distribution both in the electrically excited machines as well as the machines excited by permanent magnets. The problem has been reduced to solving the appropriate overdetermined, nonlinear, and ill-conditioned system of equations. The presented examples show the effectiveness of the proposed method both in shaping the contours of the air gap and magnetizing the permanent magnets placed on the rotor. As a result of the proposed method, it is possible to modify the field in the machine gap as desired, which is a permanent problem for designers of electrical machines. This leads to the design of machines in which the induced voltage waveform has the optimal shape, e.g., by smoother build-up of this voltage when the machine pole moves into the stator winding zone. Additionally, it is possible to precisely shape the magnetic field so that the harmonic content in the induced voltage is very low (ideally equal to zero). The method can also be used in highly saturated machines and can be extended to three-dimensional cases. This can particularly be used to design high-speed synchronous machines.