# Analytical Modelling of the Slot Opening Function

^{*}

## Abstract

**:**

## 1. Introduction

_{s}and the slot pitch τ

_{s}, for example, in induction machines. However, when the equivalent air gap g is not “small” anymore compared to the slot opening b

_{s}and the slot pitch τ

_{s}, the precision of the method gets degraded, due to the effect of adjacent slots being stronger and not negligible anymore.

## 2. Laplace Equation for a Multi-Slot Disposition: Solution Structure and Properties

_{s}is the slot pitch; for this situation, the adopted reference frame xy is centred at the point c.

**z**that in the xy reference frame of Figure 1 can be written as the following:

**z**= x + j·y.

_{o}is the average flux density, calculated within the slot pitch τ

_{s}:

_{y}(x,y) distribution shape depends on the position y chosen for the horizontal exploration line in the air gap, it is possible to show that B

_{o}does not depend on y.

_{i}, its value can be calculated as

_{i}, the pu flux density components β

_{x}and β

_{y}can be written as

_{C}is the Carter’s factor [17]

- -
- Equation (8) is periodic in space, along the x axis, with a period equal to the slot pitch τ
_{s}:

- -
- The functions β
_{x}and β_{y}are symmetrical with respect to the origin O of the xy reference frame:

- -
- For x = ±τ
_{s}/2, the β_{x}component is zero:

_{k}(or Ak, from (5) and (9)), for k = 1, 2… ∞. In practice, the series should be extended up to a maximum suited term k

_{M}, as will be discussed later.

## 3. Single-Slot Air Gap Field Analysis by Conformal Transformation

_{k}, at first, the field of the single-slot system must be studied using conformal transformations [18], as resumed in this section.

_{b}y

_{b}. The new complex position variable was

**z**

_{b}= x

_{b}+ j·y

_{b}.

**z**

_{b}plane (

**z**

_{b}= x

_{b}+ j·y

_{b}) to the

**w**plane (

**w**= u + j·v) is represented by the following equation [18]:

**z**

_{b}(

**w**) was retrieved as

_{i}, the following expression was obtained, as a function of the complex variable

**w**[18]:

_{b}and y

_{b}components followed as

**w**in putting together (17) and (20) would give the slotting opening functions β

_{sx}(

**z**

_{b}) and β

_{sy}(

**z**

_{b}) for the single-slot disposition but unfortunately, (17) could not be inverted in closed form.

**z**

_{b}= x

_{b}+ j·y

_{b}was considered inside the air gap (with x

_{b}as the exploring variable and y

_{b}< g kept constant during exploration), the numerical inversion of (18) in the complex domain, as described in [9], exhibited some convergence issues.

_{k}of (8) is described in [12]. It is based on an approximated formulation of the field along the slot opening segment; however, the accuracy of the obtained distributions could be critical, depending on the air gap geometry and on the exploring line position in the air gap.

## 4. Single-Slot and Multi-Slot Normal Slotting Function along a Smooth Surface

_{b}= g), because the calculation involved just real variables (x

_{b}and u); in fact, along the smooth surface, where we had

**z**

_{b}= x

_{b}+ j·g,

**w**= u occurred.

_{b}was −b

_{s}/2 ≤ x

_{b}< ∞, corresponding to −1 ≥ u > −∞ for

**w**= u (see Figure 2); in practice, u

_{lim}= 1013 can be adopted, from which, by (17), it follows that x

_{lim}= Re[

**z**

_{b}(−u

_{lim})].

_{s}= 5 mm, τ

_{s}= 10 mm, we obtained x

_{lim}= 45.5 mm = 4.55·τ

_{s}.

_{b}+ j·g along the smooth surface in the interval −b

_{s}/2 ≤ x

_{b}< x

_{lim}, the numerical inversion of (17) gives the corresponding

**w**values:

_{g}is a guess value, here set to −1, corresponding to the point c present in Figure 1.

^{−15}).

_{s}/2, and considering the 2nd of (13), it follows that

_{sy}

_{0}(x) of the single-slot slotting function along the smooth surface (subscript 0, because y = 0) can be written as the following:

_{sy}

_{0}(x − h·τ

_{s}). Of course, if h was negative, the displaced slot was positioned at the right of the original one.

_{sy}

_{0}(x − h·τ

_{s}) with h = −2, −1, 0, 1, 2, again for g = 5 mm, b

_{s}= 5 mm, τ

_{s}= 10 mm.

_{s}= ±0.5, β

_{sy}

_{0}(±τ

_{s}/2) appears significantly lower than 1. This means that the single-slot slotting functions of adjacent slots interfered among each other; in this situation, the air gap width could be qualified as “high”.

_{ℓ}

_{y}

_{0}(x) can be expressed as the following:

_{y}

_{0}(x) is given by the following:

_{y}

_{0}(x) along the smooth surface, according to (27) (red continuous curve), together with the FEM 2D calculated one (blue dotted curve, [19]), for g = 5 mm, b

_{s}= 5 mm, τ

_{s}= 10 mm. The agreement is excellent, confirming the correctness of the superposition principle of the single-slot lost flux density functions’ distribution; moreover, β

_{y}

_{0}(±τ

_{s}) is considerably lower than 1, confirming the appreciable interference between adjacent single-slot slotting distributions.

## 5. Calculation of the Fourier Coefficients of the Multi-Slot Slotting Function β_{y}_{0}(x)

_{y}

_{0}(x) followed from the 2nd of (8), for y = 0:

_{o}by the analytical formulation (10) and by the numerical expression ∫

_{τ}

_{s}β

_{y}

_{0}(x)dx/τ

_{s}, for g = 5 mm, b

_{s}= 5 mm, τ

_{s}= 10 mm, gave, respectively,

_{k}, they were calculated by using the multi-slot flux density function β

_{y}

_{0}(x), as the following:

_{s})/τ

_{s}] = cos(k·2π·x/τ

_{s}) for any h integer, and considering that

_{y}

_{0}(x) could be calculated by using a formulation involving the single-slot flux density function, provided that the integration was extended to infinity; in practice, it can be extended to an extreme x

_{max}= n

_{τ}·τ

_{s}, multiple of τ

_{s}, where β

_{ℓsy}

_{0}(x) becomes negligible (for example n

_{τ}= 10).

## 6. Complex Formulation of the Slotting Function in the Air Gap

**z**= x + j·y.

**β**(

**z**) can be written as

## 7. Distribution of the Slotting Functions Compared with FEM

#### 7.1. Slotting Functions for “High” Air Gap Width

_{s}, for different values of the exploring line y position in the air gap. At first, the considered geometry was g = 5 mm, b

_{s}= 5 mm, τ

_{s}= 10 mm; as observed before, this was a “high” air gap width and in this case, the maximum considered harmonic order in (35) was k

_{M}= 10.

_{fe}= 10

^{6}pu).

_{c}considered in the FEM model, the ideal normal component B

_{i}of the flux density, occurring in case of a smooth upper core, equals B

_{i}= μ

_{0}·I

_{c}/(2·g); thus, from the actual FEM-calculated distributions B

_{FEMx}(x) and B

_{FEMy}(x), the corresponding FEM slotting functions are β

_{FEMx}(x) = B

_{FEMx}(x)/B

_{i}and β

_{FEMy}(x) = B

_{FEMy}(x)/B

_{i}.

^{−6}%; Δenergy = 1.22·10

^{−5}%; CPU simulation time = 508 s; total number of mesh triangles (thousands) = 422; in the conductor = 13.5; in each slot = 10.5; and in the air-gap = 277. The particularly high mesh refinement around the tooth corners is evident, where the field changes quickly in space.

- -
- -
- as can be observed, continuous analytical curves and dashed FEM 2D curves were well superposed for any chosen y position of the exploration line.

#### 7.2. Slotting Functions for “Small” Air Gap Width

_{sy}

_{0}(x − h·τ

_{s}) with h = −1, 0, 1, for g = 2.5 mm, b

_{s}= 2.5 mm, τ

_{s}= 10 mm. For h = 0, the analytical curve is shown together with the FEM curve [19], with bold lines.

_{s}= ±0.5, β

_{sy}

_{0}(±τ

_{s}/2) appears very close to 1. This means that in practice, the single-slot slotting functions of adjacent slots do not interfere by superposition significantly, almost without reciprocal interference; in this situation, the air gap width can be qualified as “small”.

_{y}

_{0}(x) along the smooth surface (y = 0), for the considered “small” air gap geometry (g = 2.5 mm, b

_{s}= 2.5 mm, τ

_{s}= 10 mm): the red curve was analytically calculated (by (28) and the blue dashed curve, by FEM 2D [19]. Moreover, β

_{y}

_{0}(±τ

_{s}/2) is very close to 1, confirming the negligible interference between adjacent single-slot slotting distributions.

_{s}= 2.5 mm, τ

_{s}= 10 mm, Figure 10 shows the multi-slot slotting function x and y components, as a function of the peripheral position x in the slot pitch τ

_{s}, for different values of the exploring line y position in the air gap: here, the maximum considered harmonic order in (35) was k

_{M}= 21. Also in this case, continuous analytical curves and dashed FEM 2D curves are well superposed, for any y position of the exploration line.

_{s}= 2.5 mm, τ

_{s}= 10 mm:), adopted for a comparison with the analytically calculated slotting functions shown in Figure 10, while Figure 12 shows the detail of the mesh around the central slot at the right of the conductor.

^{−5}%; Δenergy = 5.76·10

^{−5}%; CPU simulation time = 218 s; total number of mesh triangles (thousands) = 165; in the conductor = 4.6; in each slot = 3.8; and in the air gap = 106.

## 8. Accuracy of the Slotting Functions with the Choice of the Maximum Harmonic Order k_{M}

_{M}of the Fourier series and their consequences on the slotting function distributions.

#### 8.1. Slotting Function Accuracy for “High” Air Gap Width

_{k}| coefficients of the Fourier series (28), calculated by (33), of the “cosh” factors and of the total factors |β

_{k}|·cosh[k·(2π/τ

_{s})·(7·g/8)] as a function of harmonic order k, for g = 5 mm, b

_{s}= 5 mm, τ

_{s}= 10 mm (“high” air gap width).

- -
- |β
_{k}| decreases with k increasing up to k = 10, while above this order, apparently the amplitude remains almost stationary; however, by observing the |β_{k}| values for k > 10, it appears that a level around the convergence tolerance TOL = 10^{−15}was reached and thus, above k > 10 the |β_{k}| values were inaccurate. - -
- As regards the factor cosh[k·(2π/τ
_{s})·y], for y = 7·g/8 it greatly increases with the increase in the harmonic order k, while the increase is lower for smaller y values. - -
- Up to the order k = 10, the total factor|β
_{k}|·cosh[k·(2π/τ_{s})·(7·g/8)] decreases, but with a reduction trend much lower than that of |β_{k}|. - -
- For k < 10, the factor |β
_{k}|·cosh[k·(2π/τ_{s})·(7·g/8)] shows the typical decreasing behaviour of any Fourier series, while for k > 10, the total harmonic factor tends to suddenly increase; however, this is caused by the numerical error in the estimation of |β_{k}|, when it falls into the convergence tolerance range.

_{M}choice on the slotting function distributions.

- -
- for limited values of the y position of the exploring line (y = g/8, 3·g/8, or 5·g/8), the value of k
_{M}has a weak effect on the distribution shape, and the analytically calculated slotting functions appear well superposed with the FEM 2D distributions; - -
- if the exploration line inside the air gap is close to the slotted surface (as for y = 7·g/8), the analytically calculated slotting function shape depends on the choice of k
_{M}; - -
- if the k
_{M}is too low (k_{M}= 4, Figure 14), the distribution for y = 7·g/8 is distorted, because the number of harmonics is not enough to reproduce the correct distribution; - -
- if the k
_{M}is intermediate (k_{M}= 7, Figure 15), the distribution for y = 7·g/8 is less distorted, because the number of included harmonics is higher, although not enough to avoid some oscillations; - -
- -

#### 8.2. Slotting Function Accuracy for “Small” Air Gap Width

_{k}| coefficients of the Fourier series (28), calculated by (33), of the “cosh” factors and of the total factors |β

_{k}|·cosh[k·(2π/τ

_{s})·(7·g/8)] as a function of harmonic order k, for g = 2.5 mm, b

_{s}= 2.5 mm, τ

_{s}= 10 mm.

- -
- |β
_{k}| decreases with k increases up to k = 21, while above this order, apparently the amplitude increases again or remains almost stationary; however, by observing the |β_{k}| values for k > 21, it appears that a level around the convergence tolerance TOL = 10^{−15}was reached and thus, above k > 21 the |β_{k}| values were inaccurate. - -
- As regards the factor cosh[k·(2π/τ
_{s})·y], for y = 7·g/8, it greatly increases with the increase in the harmonic order k, while the increase is lower for smaller y values. - -
- Up to the order k = 21, the total factor|β
_{k}|·cosh[k·(2π/τ_{s})·(7·g/8)] generally decreases, but with a reduction trend much lower than that of |β_{k}|. - -
- For k < 21, the factor |β
_{k}|·cosh[k·(2π/τ_{s})·(7·g/8)] shows the typical decreasing behaviour of any Fourier series, while for k > 21, the total harmonic factor tends to suddenly increase; however, this is the wrong effect of the inaccurate estimation of |β_{k}| when it falls into the convergence tolerance range.

_{M}choice on the slotting function distributions.

- -
- for limited values of the y position of the exploring line (y = g/8, 3·g/8, or 5·g/8), the value of k
_{M}has a weak effect on the distribution shape, and the analytically calculated slotting functions appear well superposed with the FEM 2D distributions; - -
- if the exploration line inside the air gap is close to the slotted surface (as for y = 7·g/8), the analytically calculated slotting function shape depends on the choice of k
_{M}; - -
- if the k
_{M}is too low (k_{M}= 7, Figure 18), the distribution for y = 7·g/8 is distorted because the number of harmonics is not enough to reproduce the correct distribution; - -
- if the k
_{M}is intermediate (k_{M}= 15, Figure 19)), the distribution for y = 7·g/8 is less distorted, because the number of harmonics is higher, although not enough to avoid oscillations; - -
- -

**×**) and with a single-slot approach (Equation (33), subscript ss, □). On the left is a histogram for the case of g = 5 mm, b

_{s}= 5 mm, τ

_{s}= 10 mm (“high” air gap width) and on the right is a histogram for the case of g = 2.5 mm, b

_{s}= 2.5 mm, τ

_{s}= 10 mm (“small” air gap width). All the harmonic amplitudes are referred to as the amplitude of the fundamental components.

- -
- In the low harmonic order range (below k
_{M}), the coefficients calculated with the multi-slot approach (Equation (30)) and with the single-slot approach (Equation (33)) had the same values, confirming the correctness of (33). - -
- For orders approaching k
_{M}, the two formulas started to give different results, due to the numerical issues about the TOL limit; these issues appeared more critical for the multi-slot approach, because of the superposition of several single-slot distributions in (33), although each distribution had its own inaccuracies. - -
- -

## 9. Conclusions and Perspectives

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Slot in a multi-slot structure, with infinitely deep slot height, air gap width g and slot pitch τ

_{s}, slot opening b

_{s}(here equal to the slot width); the reference frame, xy, is centred at the point c.

**Figure 2.**Single slot with infinitely deep height, air gap width g, slot opening b

_{s}(assumed equal to the slot width); here, the adopted reference frame, x

_{b}y

_{b}, is positioned in corner b.

**Figure 3.**Single-slot slotting functions along the smooth surface β

_{sy}

_{0}(x − h·τ

_{s}) with h = −2, −1, 0, 1, 2 (continuous lines); central (h = 0) single-slot slotting function β

_{sy}

_{0}(x) (red bold line = analytical, by (23); blue dotted line = FEM 2D [19]); slotting parameters: g = 5 mm, b

_{s}= 5 mm, τ

_{s}= 10 mm (“high” air gap width).

**Figure 4.**Multi-slot slotting function β

_{y}

_{0}(x) along the smooth surface (y = 0), for g = 5 mm, b

_{s}= 5 mm, τ

_{s}= 10 mm: analytically calculated (red curve, by (27)); FEM 2D (blue dashed curve) [19].

**Figure 5.**Multi-slot configuration used for FEM numerical calculation of the slotting functions, with a “high” air gap condition (g = 5 mm, b

_{s}= 5 mm, τ

_{s}= 10 mm): the device consists of 10 slots, with the orange, central one equipped with a current-fed rectangular conductor.

**Figure 6.**Detail of the multi-slot configuration of Figure 5, around the central slot at the right of the conductor, with the aspect of the mesh and a few field lines.

**Figure 7.**Multi-slot slotting function x and y components as a function of the peripheral position x within the slot pitch τ

_{s}, for a few values y of the air gap exploring line, for g = 5 mm, b

_{s}= 5 mm, τ

_{s}= 10 mm (“high” air gap width): continuous line = analytical (Equations (35) and (36), max harmonic order k

_{M}= 10); dotted lines = FEM [19].

**Figure 8.**Single-slot slotting functions along the smooth surface β

_{sy}

_{0}(x − h·τ

_{s}) with h = −1, 0, 1, (continuous lines); central (h = 0) single-slot slotting function β

_{sy}

_{0}(x) (red bold line = analytical, by (24); blue dotted line = FEM 2D [19]); slotting geometric parameters: g = 2.5 mm, b

_{s}= 2.5 mm, τ

_{s}= 10 mm (“small” air gap width).

**Figure 9.**Multi-slot slotting function β

_{y}

_{0}(x) along the smooth surface (y = 0), for g = 2.5 mm, b

_{s}= 2.5 mm, τ

_{s}= 10 mm: analytically calculated (red curve (by (28)); FEM 2D (blue dashed curve) [19].

**Figure 10.**Multi-slot slotting function x and y components as a function of the position x in the slot pitch τ

_{s}, for a few values y of the exploring line, for g = 2.5 mm, b

_{s}= 2.5 mm, τ

_{s}= 10 mm (“small” air-gap): continuous line = analytical (Equations (35) and (36), max harmonic order k

_{M}= 21); dotted lines = FEM [19].

**Figure 11.**Multi-slot configuration used for FEM numerical calculation of the slotting functions, with a “small” air gap condition (g = 2.5 mm, b

_{s}= 2.5 mm, τ

_{s}= 10 mm): the device consists of 10 slots, with the orange, central one equipped with a current fed rectangular conductor.

**Figure 12.**Detail of the multi-slot configuration of Figure 11, around the central slot at the right of the conductor, with the aspect of the mesh and a few field lines.

**Figure 13.**Amplitudes of the β

_{k}cosine coefficients of the Fourier series (28), calculated by (33), of the “cosh” factors and of the total coefficients |β

_{k}|·cosh[k·(2π/τ

_{s})·(7·g/8)] as a function of the harmonic order k, for g = 5 mm, b

_{s}= 5 mm, τ

_{s}= 10 mm (“high” air gap width).

**Figure 14.**Multi-slot slotting function x and y components as a function of the peripheral position x within the slot pitch τ

_{s}, for a few values y of the air gap exploring line, for g = 5 mm, b

_{s}= 5 mm, τ

_{s}= 10 mm (“high” air gap width): continuous line = analytical (Equations (35) and (36), maximum harmonic order k

_{M}= 4); dotted lines = FEM [19].

**Figure 15.**Multi-slot slotting function x and y components as a function of the peripheral position x within the slot pitch τ

_{s}, for a few values y of the air gap exploring line, for g = 5 mm, b

_{s}= 5 mm, τ

_{s}= 10 mm (“high” air gap width): continuous line = analytical (Equations (35) and (36), maximum harmonic order k

_{M}= 7); dotted lines = FEM [19].

**Figure 16.**Multi-slot slotting function x and y components as a function of the peripheral position x within the slot pitch τ

_{s}, for a few values y of the air gap exploring line, for g = 5 mm, b

_{s}= 5 mm, τ

_{s}= 10 mm (“high” air gap width): continuous line = analytical (Equations (35) and (36), maximum harmonic order k

_{M}= 11); dotted lines = FEM [19].

**Figure 17.**Amplitudes of the β

_{k}cosine coefficients of the Fourier series (28), calculated by (33), of the “cosh” factors and of the total coefficients|β

_{k}|·cosh[k·(2π/τ

_{s})·(7·g/8)] as a function of harmonic order k, for g = 2.5 mm, b

_{s}= 2.5 mm, τ

_{s}= 10 mm (“small” air gap width).

**Figure 18.**Multi-slot slotting function x and y components as a function of the peripheral position x within the slot pitch τ

_{s}, for a few values y of the air gap exploring line, for g = 2.5 mm, b

_{s}= 2.5 mm, τ

_{s}= 10 mm (“small” air gap width): continuous line = analytical (Equations (35) and (36), maximum harmonic order k

_{M}= 7); dotted lines = FEM [19].

**Figure 19.**Multi-slot slotting function x and y components as a function of the peripheral position x within the slot pitch τ

_{s}, for a few values y of the air gap exploring line, for g = 2.5 mm, b

_{s}= 2.5 mm, τ

_{s}= 10 mm (“small” air gap width): continuous line = analytical (Equations (35) and (36), maximum harmonic order k

_{M}= 15); dotted lines = FEM [19].

**Figure 20.**Multi-slot slotting function x and y components as a function of the peripheral position x within the slot pitch τ

_{s}, for a few values y of the air gap exploring line, for g = 2.5 mm, b

_{s}= 2.5 mm, τ

_{s}= 10 mm (“small” air gap width): continuous line = analytical (Equations (35) and (36), maximum harmonic order k

_{M}= 22); dotted lines = FEM [19].

**Figure 21.**Histograms of the harmonic amplitudes of the slotting function Fourier series (28), calculated with a multi-slot approach (Equation (30), subscript ms,

**×**) and with a single-slot approach (Equation (33), subscript ss, □). Left: histogram for the case of g = 5 mm, b

_{s}= 5 mm, τ

_{s}= 10 mm (“high” air gap width); right: histogram for the case of g = 2.5 mm, b

_{s}= 2.5 mm, τ

_{s}= 10 mm (“small” air gap width). All the harmonic amplitudes are referred to as the amplitude of the fundamental components.

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**MDPI and ACS Style**

Di Gerlando, A.; Ricca, C.
Analytical Modelling of the Slot Opening Function. *Magnetism* **2023**, *3*, 308-326.
https://doi.org/10.3390/magnetism3040024

**AMA Style**

Di Gerlando A, Ricca C.
Analytical Modelling of the Slot Opening Function. *Magnetism*. 2023; 3(4):308-326.
https://doi.org/10.3390/magnetism3040024

**Chicago/Turabian Style**

Di Gerlando, Antonino, and Claudio Ricca.
2023. "Analytical Modelling of the Slot Opening Function" *Magnetism* 3, no. 4: 308-326.
https://doi.org/10.3390/magnetism3040024