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The sizing of the Electrical Insulation System (EIS) is an important challenge in electric motors of higher specific power driven by faster inverters. That keeps increasing the electric stress which the winding is submitted in the stator slot. Consequently, Partial Discharges (PD) are more likely to occur. Nowadays, the Paschen’s criterion is widely used to evaluate the risk of partial discharge. It requires the knowledge of electric field lines. This paper presents a method to precisely compute the electric field lines in a two-dimensional (2D) electrostatic problem. The field of study is composed of two magnet wires in close contact. Such configuration is representative of the turn - to - turn interaction in an electric motor slot. The problem is solved using the scalar potential formulation only. The notion of flux tubes is used for the post process of the electric field lines in a developed numerical code on Matlab. The developed method is compared to a ballistic method already included on Matlab. The work presented here is included in an automatic tool to suppress or reduce the partial discharge risk in a stator slot of high power density motor destined for future transportation systems.

The role of transportation in the world sustainable development was firstly pointed out during the 1992 United Nation’s Earth Summit at Rio de Janeiro [_{2} emissions [_{2} emissions, while both aviation and shipping reach 22% and rail 1.3%. The transportation sector remains the largest consumer of oil: 57% of the global demand [

In order to reduce the impact of transportation on global warming, human health and environmental issues, different efforts must be undertaken or pursued in all kinds of transport: increasing even more the efficiency of existing, Internal Combustion Engine (ICE) powertrains: lower fuel consumption, use of bio- or low carbon-fuels, better decontamination of exhaust gas; increasing the electrification of ICE powertrains: hybridization or full-electrification and increasing the capacity, the efficiency, the lifetime and the hybridization of embedded power sources. Such measures are essentially dependent on R & D efforts in the field of both Electrical and Electrochemical (applied to energy) Engineering.

Aircraft manufacturers such as Airbus, Boeing and Bombardier are engaged in the competition to develop more- and full-electric aircrafts. That incoming revolution takes place in a context where more and more people and countries are expecting a much greener air transportation. The proportion of on board electric power has continuously increased in aircrafts. The electric power tends to replace more and more systems which were powered by either pneumatic or hydraulic power.

demand in commercial aviation [

Since the introduction of power electronic power supplies, that provides easy control of the machine rotational speed, the electrical insulation of inverter fed motors faces new hazards. Fast changing supply voltage, with high dV/dt, may cause the apparition of Partial Discharges (PD), that results in accelerated insulation aging [

The Paschen’s criterion is widely used to evaluate the PD risk. It is essential to get the electric field lines precise computation. This paper presents a general method to get precise electric field lines for an electrostatic problem made of two magnet wires in close contact. The advantages of that method are that it, first, only use the electric scalar formulation, second, uses the same mesh defined for solving the problem with finite elements. First, the scalar potential formulation is presented. Then, the classic ballistic method already included on Matlab [

The basis of electromagnetism is the Maxwell’s equations. The scalar potential formulation used to solve the problem with the finite elements method is derived from the following Maxwell’s equations:

c u r l E = − ( ∂ B ) / ∂ t (1)

d i v D = ρ (2)

With E the electric field intensity, B the magnetic flux density, D the electric flux density and ρ the volume charge density of the dielectric medium. The fields produced by a power frequency alternating voltage are not electrostatic. However, they are considered quasi-stationary: the fields time variation is neglected. Besides, in this work both air and polymer insulation materials are considered free of charge carrier. The simplified Maxwell’s equations which come into play in the considered electrostatic problem are given below:

c u r l E = 0 (3)

d i v D = 0 (4)

Because E is curl free, it can be expressed as a gradient of a scalar potential. That is the electric potential V:

E = − g r a d V (5)

The electric flux density D is derived from the electric field intensity E and the medium properties:

D = ε 0 ∗ ε r ∗ E (6)

With ε_{0} and ε_{r} the permittivity of the air and the dielectric constant of the medium respectively.

By combining (3), (4), (5) and (6) it comes the so-called Poisson’s equation:

∇ 2 V = 0 (7)

Most of commercial finite elements software solve (7). That formulation is easy to handle and gives a unique solution. The main disadvantage of that formulation is that it does not give directly the electric field lines. Additional steps are required and are presented in the following paragraphs. By analogy with the magnetic vector potential, there also exists a vector potential formulation. Such formulation is complex to execute in an electrostatic problem with multiple conductors. It requires to put in place a network of branches and cuts all across the field of study [

In this paragraph, two methods are presented. The first one is the ballistic method. It is used by Matlab stream [

This method is one way of computing field lines from a scalar potential formulation. It is presented in [

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 _{0}, y_{0}) of a field line. Then, the next point on the line M_{1}(x_{1}, y_{1}) 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}. _{1} points do not coincide with a mesh node. The electric field orientation vectors are interpolated on the extra M_{1} points using the interp 2 Matlab function [_{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

The objective is to represent the strength of the electric field by the density of the electric flux lines [

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.

∫ s j s j + 1 E d s = δ Φ (18)

Starting from the s_{1} 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) [

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 [_{1} and V_{2} respectively. A field line starting from V_{1} contour is identified by l n 1 1 , n 1 = 1 , ⋯ , N 1 . The same procedure described in the simple case is applied to V_{1} contour. However, the intersections of l n 1 1 field lines with V_{2} 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 V_{2} contour. The intersection points are added to V_{2} contour points series. Now, one can apply the same procedure with V_{2} contour adding some steps:

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

• Step 2: integrate from that point until (a) another intersection point of a l n 1 1 field line with V_{2} 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 V_{2} contour is swept;

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

_{1} and V_{2} using the presented algorithm.

The method we developed is an upgrade of the method presented by Horowitz [

The 2D-finite element model on Ansys Mechanical APDL [

{ 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 [

( ∂ ∂ 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

• 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.

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

With:

t ∈ [ 0 , 1 ]

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, (V_{0}, V_{0.5}, V_{1}) are respectively the finite elements voltage solution on nodes (1, 2, 3) in _{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 t_{0} 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.

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)

_{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.

The problem consists of two enameled magnet wires in close contact in air. Such configuration is representative of a contact between two adjacent wires in a stator slot in the presence of a default (air bubbles or bad impregnation) in the surrounding impregnation resin. The copper areas are obviously not modelled since there is no electric field in conductor materials. Boundary conditions (i.e.: voltages) are applied on the copper areas contour. The air and the enamel are considered charge carrier free. The wires are considered infinitely long in the machine active length dimension and the voltage drops are neglected. Due to invariance along this dimension and symmetries, the problem is reduced to a two dimensional (2D) electrostatic problem with only a quarter of the wires being modelled. The described 2D problem is displayed on

Parameter | Description | Value |
---|---|---|

Rint | Round wire copper radius | 0.75 mm |

e | Enamel overcoat thickness | 70 µm |

ε_{r} | Enamel dielectric constant | 3.5 |

The model is realized on Ansys Mechanical APDL [

A voltage amplitude of 1000 V_{peak} is applied as boundary conditions on the copper contour of the left wire (

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 [_{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 [

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. ^{−}^{4}. The voltage on the enamel overcoat contours at the interface with the air is picked up.

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.

Partial Discharges (PD) phenomenon represents a great deal in the design of future rotating machines fed by inverter. On one hand, new faster components made out of SiC and GaN technologies will considerably improve the performances of the inverters. On the other hand, they will generate harder voltage fronts at the motor terminals which lead to higher transient voltage overshoots. PD will be more likely to appear and the insulation lifespan will be reduced due to both voltage overshoots and high switching frequency. That is the reason why it is absolutely necessary to take into account such a phenomenon when designing the motor to avoid any PD appearance, rather than searching for solutions in a motor already produced to remove them. The Paschen’s criterion is used to evaluate PD activity. This criterion is corrected for taking into account the impact of the enamel layer. The Paschen’s criterion requires the complete knowledge of electric field lines geometry. A numerical code has then been developed on Matlab to drawn electric field lines from a 2D-Finite Elements (2D-FEM) solution of an electrostatic problem. The originality of the proposed method is that only the scalar potential solution is required and the density of displayed lines is analogous to the field intensity. It has been compared to a ballistic approach already implanted on Matlab. It results that the Matlab function is much faster. Both methods give similar results for the considered 2D-electrostatic problem. The development of the proposed method provided better understanding on scalar potential formulation and the notions of electric flux tubes. These methods have been used to evaluate the PDIV of several magnet wires. The ultimate goal is the computation of insulation design graphs for suppressing PD risk between turns in a stator slot.

This project has received funding from the Clean Sky 2 Joint Undertaking under the European Union’s Horizon 2020 research and innovation program under grant agreement No 715483.

The authors declare no conflicts of interest regarding the publication of this paper.

Collin, P., Malec, D. and Lefevre, Y. (2021) A General Method to Compute the Electric Flux Lines between Two Magnet Wires in Close Contact and Its Application for the Evaluation of Partial Discharge Risks in the Slots of Electric Machines Embedded in Future Transportation Systems. Advances in Aerospace Science and Technology, 6, 24-42. https://doi.org/10.4236/aast.2021.61003