This method is one way of computing field lines from a scalar potential formulation. It is presented in [15]. If l is a vector tangential to a field line, then, in a direct orthonormal system, the cross-product of l and E is null. The point on a field line can be calculated step by step with the following equations:

l × E = 0 (8)

l x ∗ E y − l y ∗ E x = 0

With l_x and l_y respectively the projections of l along x and y axis in a 2D-system.

From (8) it comes:

E y E x = l y l x (9)

E_x and E_y are the field components over the mesh. These are known.

Starting at any point, the field line l can be computed by incrementing (9) given an arbitrary displacement ∆_x along x axis. Thus, the vertical displacement ∆_y along y axis is expressed as follow:

Δ y = Δ x ∗ E y E x (10)

Let us express Equation (10) between two consecutive points M_i(x_i, y_i) and M_i+1(x_i + 1, y_i+1):

( y i + 1 − y i ) = ( x i + 1 − x i ) ∗ E y , i E x , i (11)

Let us define a constant step increase ∆_step at each iteration, so that:

Δ s t e p = ( y i + 1 − y i ) E y , i = ( x i + 1 − x i ) E x , i (12)

It is thus possible to express the M_i+1(x_i + 1, y_i+1) coordinates from M_i(x_i, y_i) data (coordinates and field components) and ∆_step:

x i + 1 = x i + Δ s t e p ∗ E x , i y i + 1 = y i + Δ s t e p ∗ E y , i (13)

When chosen arbitrary, the starting point may not be the start of the field line. It is then necessary to integrate (12) backward to complete the line. This method is called a ballistic method. The mesh is swept and field lines are started in elements which do not already contain a field line. The field lines are computed by integration. The integration process is stopped if one of the following condition is checked:

• The field line enters a forbidden region (for instance the limit of the domain);

• The field line reaches a null field;

• The field line loops back onto itself;

• The field line has too many segments. This is a safety measure in case the previous conditions do not work properly.

By doing so, there is the same density of field line all over the model. It does not allow to represent the electric field intensity.

As an illustration, let us consider an electrostatic problem made of two infinitely long cylinder oppositely. Due to the symmetries, only a quarter of the field of study is considered as displayed on Figure 2. The mesh is represented with black crosses. It is made of rectangular elements (dotted red line). The blue line delimits the conductor contour. Let us apply the Whittaker ballistic method on that system. The algorithm noted above is implanted in a Matlab script [11]. The orientation vector of the electric field is computed on all mesh nodes, except for the ones located inside the conductor. The starting points are chosen on the contour of the domain. Figure 2 displays the computed orientation vectors and the starting points (circles). The starting points are arbitrary chosen on the limit of the field of study. Let us designate the starting point Q(x₀, y₀) of a field line. Then, the next point on the line M₁(x₁, y₁) is computed by iteration from point Q. The iteration is backward because the starting points are on the end of the field lines:

x 1 = x 0 − Δ s t e p ∗ E x , 0 y 1 = y 0 − Δ s t e p ∗ E y , 0 (14)

With E_x,0 and E_y,0 the electric field orientation vector components along x and y axis respectively on starting point Q. The step increase ∆_step at each iteration is taken equal to the grid spacing along x axis d_x. Figure 3 displays the field lines after one iteration backward from starting points. It can be seen that the M₁ points do not coincide with a mesh node. The electric field orientation vectors are interpolated on the extra M₁ points using the interp 2 Matlab function [11]. At each iteration, the field lines extend. The ith iteration between two consecutive points M_i(x_i, y_i) and M_i+1(x_i + 1, y_i+1) correspond to:

x i + 1 = x i − Δ s t e p ∗ E x , i y i + 1 = y i − Δ s t e p ∗ E y , i (15)

The computed field lines after 11 iterations are displays on Figure 3.

The objective is to represent the strength of the electric field by the density of the electric flux lines [16]. The core of the method is the choice of the field lines starting points. A predetermined amount of flux δϕ has to be present between two adjacent lines a and b:

δ ϕ = ∫ a b E ⋅ n ⋅ d s (16)

where n is the normal vector and s is the path between the lines a and b. This path is chosen as the contour of scalar potential V. Such as in Whittaker’s method, a uniform mesh is applied. The electric field is calculated on the whole meshed domain using linear interpolations. In a simple case, all the field lines pass through a single V contour. The contour results in a series of points s i = { x i , y i } ; i = 1 , ⋯ , I . The total flux through the contour of potential V is:

ϕ V = ∫ s 1 s I E d s = 1 2 ∑ i = 1 I ( E i + E i + 1 ) | s i + 1 − s i | (17)

From (17) to (21) E refers to the normal electric field component. Figure 4 displays the contour V discretization in a uniform meshed domain. Here the total number of points is I = 5. The field lines starting points on contour V results in a second series of points s j = { x j , y j } ; j = 1 , ⋯ , N . Such points satisfy the following equation:

∫ s j s j + 1 E d s = δ Φ (18)

Starting from the s₁ point of the contour V, the s_i contour point which is in proximity of the next field line starting point is identified by this equation:

∫ s 1 s i E d s < δ ϕ < ∫ s 1 s i + 1 E d s (19)

The s_j field line starting point on contour V segment s_i, s_i+1 is defined as:

∫ s i s j E d s = δ ϕ − ∫ s 1 s i E d s = ϕ 0 (20)

Equation (20) can be expressed as:

1 2 ( E i + E j ) | s j − s i | = ϕ 0 (21)

Let a = | s j − s i | | s i + 1 − s i | be the fractional distance from contour point s_i to field line starting point s_j. As the electric field E is linear along the grid, one finally gets (22) [16]:

s j = { x j , y j } (22)

s j = { ( 1 − a ) ∗ x i + a ∗ x i + 1 , ( 1 − a ) ∗ y i + a ∗ y i + 1 } (23)

The field lines are then built from their starting point by integrating as in Whittaker’s method [16]. As the same flux quantity δϕ is present between two adjacent lines, the electric field strength is represented by the concentration of field lines. The same termination criteria as in Whittaker’s method are used. However, an electric machine slot filled with conductors is a more complex case. It mainly happens that all the field lines do not pass through a single V contour. Let us consider two conductors at potential of V₁ and V₂ respectively. A field line starting from V₁ contour is identified by l n 1 1 , n 1 = 1 , ⋯ , N 1 . The same procedure described in the simple case is applied to V₁ contour. However, the intersections of l n 1 1 field lines with V₂ contour have to be tracked. To do that, each segment of l n 1 1 is checked to verify whether or not it intersects with V₂ contour. The intersection points are added to V₂ contour points series. Now, one can apply the same procedure with V₂ contour adding some steps:

• Step 1: choose as a starting point of field lines from V₂ contour ( l n 2 2 , n 2 = 1 , ⋯ , N 2 ) one of the intersection point of a l n 1 1 field line with V₂ contour;

• Step 2: integrate from that point until (a) another intersection point of a l n 1 1 field line with V₂ contour is reached or (b) the integral exceeds the fixed flux quantity δϕ. In case (b), the next point as to be determined the same way s_j point is in the simple case;

• Step 3: repeat step 2 until the whole V₂ contour is swept;

• Step 4: integrate backward from the first intersection point to find the remaining starting points.

Figure 5 [16] illustrates the field lines computed between two conductors at potential V₁ and V₂ using the presented algorithm.

The 2D-finite element model on Ansys Mechanical APDL [17] uses eight nodes elements. On each element, the coordinates and the scalar potential solution are expressed as a combination of each nodes data with a defined shape function. Shape functions are expressed in the local coordinate system (u, v) of the element. It is a coordinate system attached to the element which defines the location of each node.

{ x e ( u , v ) = ∑ i = 1 8 N i ( u , v ) ⋅ x e ( i ) y e ( u , v ) = ∑ i = 1 8 N i ( u , v ) ⋅ y e ( i ) V e ( u , v ) = ∑ i = 1 8 N i ( u , v ) ⋅ V e ( i ) (24)

With:

N i ( u , v ) = 1       at   node   i N i ( u , v ) = 0     elsewhere

With x_e and y_e the coordinates in the global coordinate system (x,y) of the nodes of an element e. For eight nodes elements, the shape functions N are given in [18]. The electric field on the nodes of an element is derived from the voltage values on the node using (5). It requires the expression of partial derivatives in the global system. First, the partial derivatives of a function in the local system (u,v) can be expressed from its partial derivatives in the global system (x,y):

( ∂ ∂ u ∂ ∂ v ) = [ ∂ x ∂ u ∂ y ∂ u ∂ x ∂ v ∂ y ∂ v ] ( ∂ ∂ x ∂ ∂ y ) = J ( ∂ ∂ x ∂ ∂ y ) (25)

With:

J = [ J 11 J 12 J 21 J 22 ]

With J the Jacobian matrix. It is computed from the known shape functions partial derivatives in the local system when putting (24) in (25). Then, the partial derivatives of a function in the global system (x, y) can be expressed from its partial derivatives in the local system (u, v):

( ∂ ∂ x ∂ ∂ y ) = [ ∂ u ∂ x ∂ v ∂ x ∂ u ∂ y ∂ v ∂ y ] ( ∂ ∂ u ∂ ∂ v ) = I ( ∂ ∂ u ∂ ∂ v ) (26)

With:

I = [ I 11 I 12 I 21 I 22 ]

I = J − 1

It is matrix I which is used in practice because the data have to be expressed in the global system. It is computed as follow:

I = J − 1 = 1 det ( J ) [ J 22 − J 12 − J 21 J 11 ] (27)

With:

det ( J ) = J 11 J 22 − J 12 J 21

The electric field components on the nodes of an element e can finally be expressed in the global system by combining (24) and (26):

{ E x , e = − ∂ V e ∂ x = − ∑ i = 1 8 ( I 11 ∂ N i ∂ u + I 12 ∂ N i ∂ v ) ⋅ V e ( i ) E y , e = − ∂ V e ∂ y = − ∑ i = 1 8 ( I 21 ∂ N i ∂ u + I 22 ∂ N i ∂ v ) ⋅ V e ( i ) (28)

The electric flux components are computed from electric field components using (6).

The next step consists of forming equipotential lines. These are contours on which the voltage value is constant. Equipotential lines are obtained by regrouping nodes with the same voltage value. Let us see the steps to compute an equipotential line at voltage V. A test is done on each edge of each element to check whether or not it is crossed by the equipotential line V. As can be seen on Figure 6, each edge is composed of three nodes. There are three possible outcomes:

• The voltage V is lower or bigger than any of the three nodes voltage. The edge is not intersected by the equipotential line V;

• The voltage V is equal to one of the three nodes voltage. The equipotential line intersects the edge at the corresponding node of voltage V;

• The voltage V is between any of the three nodes voltage. The edge is intersected by the equipotential line at a point which is interpolated.

Figure 6 illustrates an equipotential intersecting two edges of an element. On one edge the intersection point is an already existing node (node 6). The second edge intersection point has to be interpolated (red cross). The local numbering of the nodes is rearranged compared to Ansys eight nodes reference element to facilitate the programming. A second order polynomial is used for the interpolation. The following equation gives the parametric expression of the polynomial:

z ( t ) = a 1 , z + a 2 , z ∗ t + a 3 , z ∗ t 2 (29)

With:

t ∈ [ 0 , 1 ]

Figure 6 displays (29) along a parameterized edge. For instance, on Figure 6 the interpolation is done on the edge containing the nodes (1, 2, 3). The nodes are parameterized according to the local numbering order: t = 0 for node 1, t = 0.5 for node 2 and t = 1 for node 3. By solving (29) in term of voltage it is possible to derive the polynomial coefficients:

V ( t ) = a 1 , V + a 2 , V ∗ t + a 3 , V ∗ t 2 (30)

With:

{ V ( 0 ) = a 1 , V = V 0 V ( 0.5 ) = a 1 , V + a 2 , V ∗ 0.5 + a 3 , V ∗ 0.25 = V 0.5 V ( 1 ) = a 1 , V + a 2 , V + a 3 , V = V 1

In the considered example, (V₀, V_0.5, V₁) are respectively the finite elements voltage solution on nodes (1, 2, 3) in Figure 6. The three polynomial coefficients (a_1,V, a_2,V, a_3,V) are thus determined.

V ( t 0 ) = V = a 1 , V + a 2 , V ∗ t 0 + a 3 , V ∗ t 0 2 (31)

With V the voltage of the considered equipotential line V. The obtained parameter t₀ is injected back into (29) to determine the interpolated node coordinates, electric field components and electric flux components. As in (30), the associated polynomial coefficients are deduced from the known nodes data.

At this point, equipotential lines made of nodes at the same scalar potential V are determined. The coordinates and fields components are determined on all these nodes. The number of equipotential lines depends on the accuracy from which the electric scalar potential problem is solved. The finer the mesh used to solve the problem, the higher the number of equipotential lines that can be accurately determined. For each equipotential line, a geometrical reference is defined on its barycenter. The points on the line are located by using polar coordinates in this reference. A starting point Q is chosen for instance by means of the angular coordinates. Figure 7 illustrates the barycenter reference attached to an equipotential line V. Point G is the barycenter. A point M located on the equipotential line is identified by its curvilinear abscissa s(M). A flux function ϕ(M) of points on each equipotential is defined as the flux per meter crossing the line between the starting point Q and point M:

ϕ ( M ) = ∫ Q M D n d s (32)

D_n is the normal electric flux density on the equipotential line, ds is the elemental curvilinear length of the equipotential line. Each point M on the line is parameterized by the curvilinear abscissa s(M) and:

{ s ( M ) = ∫ Q M d s x ( M ) = x ( s ) y ( M ) = y ( s ) ϕ ( M ) = ϕ ( s ) (33)

The trace of a flux tube on the equipotential line is delimited by two points P_i and P_i+1 which are given by the predetermined flux per meter δϕ:

δ ϕ = ∫ P i P i + 1 D n d s (34)

The properties of these points are:

{ P 1 = Q ϕ ( P 1 ) = 0 ϕ ( P i ) = ϕ ( P i − 1 ) + δ ϕ = ( i − 1 ) δ ϕ (35)

All the points P_i on the equipotential line are determined by inversing the previous functions:

s ( P i ) = ϕ − 1 ( ( i − 1 ) δ ϕ ) (36)

Figure 7 illustrates the flux tubes computation using (33), (34), (35) and (36). The determination of points P_i is done on all equipotential lines. If the starting point Q on each equipotential line is correctly chosen, then the electric flux lines are defined by iso-ϕ lines. The starting point has to be chosen on a field line crossing all the equipotential lines present in the domain.

A voltage amplitude of 1000 V_peak is applied as boundary conditions on the copper contour of the left wire (Figure 8). A null potential is imposed on the copper contour of the right wire. The voltage drop in the domain is subdivided into one hundred equipotential lines. The starting point of each equipotential line is taken on the symmetry axis. That because the symmetry axis is an electric field line. The flux tubes contours are determined following the process presented on paragraph 4. Figure 10 shows that the electric flux crossing the equipotential lines is constant. It has been arbitrary chosen to plot twenty field lines. Each flux tubes limit is thus one nineteenth of the electric flux. That because the first field line is the symmetry axis. Figure 11 displays the one hundred equipotential lines (colored dotted lines) and the twenty electric field lines (colored solid lines). The black contours are the enamel external layer. The field lines are computed from the finite elements solution in 9.11 seconds.

A ballistic method is also used to compute the electric field lines. The voltage drop is also taken equals to 1000 V_peak between the copper cores. This method is

implanted on Matlab function stream2 [11]. That function requires a uniform mesh. However, the finite elements mesh on Figure 9 is not uniform. A second mesh is thus created on Matlab using meshgrid function [11]. Axis vectors X_a, Y_a are used as inputs of meshgrid. They are built from X, Y finite elements nodes coordinates lists such that:

• X a ∈ [ X min , X max ] and Y a ∈ [ Y min , Y max ]

• X_a and Y_a results in an N uniformly spaced values.

In this paper N is taken equals to 600. The mesh generated by meshgrid is referred as Matalb mesh. The Matlab mesh is different from the one generated with the finite elements software. Multiple points of the finite elements mesh are deleted. Besides, the Matlab mesh adds some points. The voltage solution on these points is linearly interpolated from the existing ones using griddata function [11]. Then, the voltage gradient over the Matlab mesh is computed using gradient function [11]. We now have all the inputs necessary for using stream2 functions. Figure 12 displays twenty electric field lines in air (black solid lines). Only the contours of the enamel external layer are represented. The field lines are computed in 1.4 seconds with the Matlab ballistic method.

In PD evaluation, only the part of the electric field line in the air gap is considered. Besides, the electric field along the field lines has to be uniform. Figure 13 displays the electric field amplitude. The field lines from Figure 12 are superposed. It can be seen that the electric field in the air gap is quite constant along the obtained field lines. So the Paschen’s criterion can be applied on the part of the field lines in the air gap. To take into account the impact of the enamel layer on PDIV, the Paschen’s criterion is corrected as in [19]. The secondary electron emission coefficient γ is equal to 9 × 10⁻⁴. The voltage on the enamel overcoat contours at the interface with the air is picked up. Figure 14 displays the computed voltage drops along each part of field lines in air for the two approaches at

evaluated PDIV (1260 V_peak). The points intersecting with the Paschen’s curve indicate field lines on which PD activity is likely to occur. Both methods give close results for field lines in air longer than 10 µm. PD is evaluated to take place on field line of 37 µm length with a voltage drop in air of 561 V.