Pierre Bérard

Ricardo Sa Earp

Lindelöf's theorem for hyperbolic catenoids

Keywords: Mathematics Subject Classification (2000): 53C42, 53C21, 58C40 Minimal surfaces. Constant mean curvature surfaces. Stability. Index

Introduction

In [START_REF] Lindelöf | Sur les limites entre lesquelles le caténoïde est une surface minimale[END_REF], L. Lindelöf determines which domains of revolution on the catenoid C in R 3 are stable. More precisely, he gives the following geometric construction, see Figure 1. Take any point A on the generating catenary C = {(x, z) ∈ R 2 | x = cosh(z)}. Draw the tangent to C at the point A and let I be the intersection point of the tangent with the axis {x = 0}. From I, draw the second tangent to C. It touches C at the point B. Lindelöf's result states that the compact connected arc AB generates a stable-unstable domain on the catenoid C (see definitions in Section 2). As a consequence, the upper-half of the catenoid, C ∩ {z > 0}, is a maximal stable domain among domains invariant under rotations. We will refer to this property as Lindelöf's property.

In [START_REF] Bérard | Minimal hypersurfaces in H n ×R, total curvature and index[END_REF], we prove that minimal catenoids in H n × R do not satisfy Lindelöf's property (Theorem 3.5, Assertion 2). In this paper we consider minimal and embedded constant mean curvature one catenoids in H 3 . The global picture is as follows. Catenoid-cousins in H 3 , as their minimal counterparts in R 3 , satisfy Lindelöf's property (Theorem 4.16). This is not surprising in view of the local correspondence between minimal surfaces in R 3 and surfaces with constant mean curvature 1 in H 3 . One may also observe that the Jacobi operators look the same, namely -∆ -|A 0 | 2 , where A 0 is the second fundamental form for catenoids or its traceless analog for catenoid-cousins. On the other-hand, catenoids in H 3 divide into two families, a family of stable catenoids which foliate the space, and a family of index 1 catenoids which intersect each other and have an envelope; the hyperbolic catenoids do not satisfy Lindelöf's property (Theorem 4.7, Proposition 4.8). We finally point out that among the examples we have studied, the hypersurfaces which do not satisfy Lindelöf's property are precisely those which are vertically bounded.

The authors would like to thank M. do Carmo for pointing out Lindelöf's paper to them. The authors gratefully acknowledge the hospitality of the Mathematics Department of puc-rio (pb) and Institut Fourier (rsa), and the financial support of cnpq, faperj, Université Joseph Fourier and Région Rhône-Alpes.

Preliminaries

Let M 2 M 3 be an orientable minimal or constant mean curvature surface in an oriented Riemannian manifold ( M , g). Let N M be a unit normal field along M and let A M be the second fundamental form with respect to N M . Let Ric be the Ricci curvature of M . The second variation of the volume functional gives rise to the Jacobi operator (or stability operator) J M of M , see [START_REF] Blaine Lawson | Lectures on minimal submanifolds[END_REF],

(1)

J M := -∆ M -|A M | 2 + Ric(N M ) ,
where ∆ M is the non-positive Laplacian on M for the induced metric. Given a relatively compact regular domain Ω on the surface M , we let Ind(Ω) denote the number of negative eigenvalues of J M for the Dirichlet problem in Ω. The index of M is defined to be the supremum Ind(M ) := sup{Ind(Ω) | Ω ⋐ M } ≤ +∞, taken over all relatively compact regular domains. Let λ 1 (Ω) be the least eigenvalue of the operator J M with Dirichlet boundary conditions in Ω. We call a relatively compact regular domain Ω, stable, if λ 1 (Ω) > 0; unstable, if λ 1 (Ω) < 0; stable-unstable, if λ 1 (Ω) = 0. More generally, we say that a domain Ω is stable if any relatively compact subdomain is stable. Properties 2.1 We recall the following properties.

1.

Let Ω be a stable-unstable relatively compact domain. Then, any smaller domain is stable while any larger domain is unstable (monotonicity of Dirichlet eigenvalues).

2. We refer to the solutions of the equation J M (u) = 0 as Jacobi fields on M . Let X a : M n ( M n+1 , g) be a one-parameter family of oriented constant mean curvature H a immersions, with variation field V a = ∂Xa ∂a and unit normal N a . If H a does not depend on a, then, the function g(V a , N a ) is a Jacobi field on M ([1], Theorem 2.7).

3.

Let Ω be a relatively compact domain on a minimal or constant mean curvature manifold M . If there exists a positive function u on Ω such that J M (u) ≥ 0, then Ω is stable ( [START_REF] Fischer-Colbrie | The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature[END_REF], Theorem 1).

3 Catenoids in R 3

In this section, we briefly recall Lindelöf's results for Euclidean catenoids. We consider the family of catenoids given by parametrization

X(a, t, θ) = a cosh( t a ) cos θ, a cosh( t a ) sin θ, t , t ∈ R, a > 0,
and in particular the catenoid C given by a = 1. Let N (t, θ) be the unit normal to C, pointing towards the axis. According to Properties 2.1, the functions

(2) v(t) = tanh(t) = ∂ ∂z , N (t, θ) , and e(t) = 1 -t tanh(t) = -d da X a | a=1 (a, t, θ), N (t, θ) ,
are Jacobi fields on C.

Theorem 3.1 Let ξ 0 be the positive zero of the function e(t) = 1 -t tanh(t).

1. The domain

D(-ξ 0 , ξ 0 ) = X(1, ] -ξ 0 , ξ 0 [, [0, 2π]
) is a stable-unstable domain on the catenoid C.

The domain

D(0, ∞) = X(1, ]0, ∞[, [0, 2π]
) is a maximal stable rotation invariant domain on C. More precisely, given any α > 0, there exists some

β(α) > 0 such that the domain D α, β(α) = X(1, ] -α, β(α)[, [0, 2π]) is stable-unstable, see Figure 1. 3. The catenoid C = X(1, ] -∞, ∞[, [0, 2π]) has index 1.
Sketch of the proof of Theorem 3.1. Assertion 1. Use the Jacobi field e(t). Assertion 2. The Jacobi field v(t) is positive in D(0, ∞). It follows that this domain is stable (Properties 2.1). Take any α > 0. The function e(α, t) defined by

(3) e(α, t) = v(α)e(t) + e(α)v(t).

is a Jacobi field and satisfies e(α, -α) = 0. Because e(α, ±∞) = -∞ and ∂e ∂t (α, -α) = 0, the function e(α, •) must have another zero β(α) = -α. Observe now that the function e(α, •) cannot have two negative zeroes or two positive zeroes because the upper and lower half-catenoids C ∩ {z > < 0} are stable. Hence e(α, •) has exactly one negative zero -α, and one positive zero β(α). It follows that the domain D α, β(α) is stable-unstable. It also follows that D(0, ∞) is a maximal stable domain among rotation invariant domains. Assertion 3. It follows from Assertion 1 that the index of C is at least 1. The horizontal halfcatenoid C ∩ {x > 0} is stable. This implies that an eigenfunction of the Jacobi operator associated with a negative eigenvalue must be invariant by rotations (see [START_REF] Bérard | Minimal hypersurfaces in H n ×R, total curvature and index[END_REF], Theorem 3.5 or [START_REF] Tam | Stability properties of the higher dimensional catenoid in R n+1[END_REF]). Use Assertion 1 to conclude that there is only one negative eigenvalue. 2

Remarks.

1. Using the function e(α, t) defined by (3), one can recover Lindelöf's tangent construction, see Figure 1.

2.

The function e(α, •) is a Jacobi field arising from the variation of the oneparameter family of catenaries passing through the point (cosh(α), -α). 3. It turns out that the above proof and tangent construction of maximal stable domains work for n-dimensional catenoids in R n+1 as well, see [START_REF] Bérard | Lindeloef's theorem for catenoids, revisited[END_REF].

When n ≥ 3, these catenoids have bounded height and the tangent construction shows that they do not satisfy Lindelöf's property, see Figure 1.

Catenoids and catenoid cousins in H 3

In the sequel, we use subscripts to denote the derivative with respect to a given variable. For example, if y 0 (a, s) is a function of the variables a and s, then y 0,a (a, s) = ∂ ∂a y(a, s) and y 0,s (a, s) = ∂ ∂s y(a, s).

Preliminaries

We work in the half-space model for the hyperbolic space,

H 3 {x1,x2,x3} = {(x 1 , x 2 , x 3 ) ∈ R 3 | x 3 > 0}, g h = x -2 3 dx 2 1 + dx 2 2 + dx 2 3 .
In the hyperbolic plane

H 2 {x1,x3} = {(x 1 , x 3 ) ∈ R 2 | x 3 > 0},
we consider the Fermi coordinates (u, v) associated with the geodesic s → (0, e s ). They are defined by R 2 ∋ (u, v) → x 1 = e v tanh(u), x 3 = e v / cosh(u) . Given a function f , we consider the curve t → (t, f (t)) in the {u, v}-plane and the associated rotation surface

F : M H 3 {x1,x2,x3} ,
F (t, θ) =    e f (t) tanh(t) ω θ e f (t) cosh(t)    ,
where ω θ = (cos θ, sin θ) for short.

The principal directions of curvature are the tangents to the generating curve and to the horizontal circle. Using [START_REF] Sa | Introduction à la géométrie hyperbolique et aux surfaces de Riemann[END_REF] Theorem 2.6.18, we easily compute the respective principal curvatures

         k g (t) = f tt (t) cosh(t) + 2f t (t) sinh(t) + f 3 t (t) cosh 2 (t) sinh(t) 1 + cosh 2 (t)f 2 t (t) 3/2 , k c (t) = f t (t) cosh 2 (t) sinh(t) 1 + cosh 2 (t)f 2 t (t) 1/2 .
and the mean curvature ( 4)

H(t) sinh(2t) = d dt f t (t) sinh(t) cosh 2 (t) 1 + cosh 2 (t)f 2 t (t) 1/2 .
of the immersion F . Integrating Equation ( 4) when H = 0 or H = 1 in the next sections, we will find graphs, ϕ(a, t) = (t, λ(a, t)), t ≥ a, in the plane H 2 {u,v} . They extend by symmetry with respect to the u-axis as smooth curves with an arc-length parametrization of the form Φ(a, s) = y(a, s), Λ(a, s) , where y(a, s) is a smooth even function of s and Λ(a, s) a smooth odd function of s, such that Λ(a, s) = λ a, y(a, s) for s ≥ 0.

The corresponding constant mean curvature rotation surfaces in H 3 are given by the parametrizations ( 5)

Y (a, s, θ) =    e Λ(a,s) tanh(y(a, s)) ω θ e Λ(a,s) cosh(y(a, s))    .
The Killing field associated with the hyperbolic translations along the vertical geodesic t → (0, 0, e t ) in H

= g h (Y a , N Y ) = cosh(y(a, s)) Λ a y s -Λ s y a .

Minimal catenoids in H 3

When H = 0, Equation (4) yields the solutions curves {C 0,a } a>0 , [START_REF] Blaine Lawson | Lectures on minimal submanifolds[END_REF] λ 0 (a, t) = sinh(2a)

t a dτ cosh(τ ) sinh 2 (2τ ) -sinh 2 (2a) 1/2
which are defined for t ≥ a (here, the lower index 0 refers to the value of H).

Notice that this parametrization only covers a half-catenary and that we work up to a v-translation in H 2 {u,v} , i.e. up to a hyperbolic translation with respect to the vertical geodesic in H 2 {x1,x3} . The arc-length parameter along the curve is given by [START_REF] Lopes | On the index of constant mean curvature 1 surfaces in hyperbolic space[END_REF] S 0 (a, t)

= t a sinh(2τ ) cosh 2 (2τ ) -cosh 2 (2a) -1/2 dτ, cosh(2a) cosh 2S 0 (a, t) = cosh(2t), t ≥ a.
Proposition 4.3 For s ∈ R, define the functions y 0 (a, s) and Λ 0 (a, s) by the formulas (8)

         y 0 (a, s) = a + s 0 cosh(2a) sinh(2t) cosh 2 (2a) cosh 2 (2t) -1 1/2 dt, and Λ 0 (a, s) = √ 2 sinh(2a) s 0 cosh(2a) cosh(2t) -1 1/2 cosh 2 (2a) cosh 2 (2t) -1 dt.
1. The function y 0 is an even function of s, and Λ 0 an odd function of s.

2.

For s ≥ 0, the function y 0 (a, •) is the inverse function of the function S 0 (a, •). In particular, cosh 2y 0 (a, s) = cosh(2a) cosh(2s).

3. For s ≥ 0, we have Λ 0 (a, s) = λ 0 (a, y 0 (a, s)).

4.

For s ∈ R, the functions s → y 0 (a, s), Λ 0 (a, s) are arc-length parametrizations of the family of catenaries {C 0,a } a>0 .

Proof. The proof is straightforward. 2

For later reference, we introduce the function

(9) J 0 (a, t) = sinh(2a)(cosh(2a) cosh(2t) + 1) -1 (cosh(2a) cosh(2t) -1) -1/2 , so that Λ 0 (a, s) = √ 2 s 0 J 0 (a, t) dt. We compute ∂J0 ∂a (a, t) and find, (10)
       I 0 (a, t) = ∂J 0 ∂a (a, t) = n(cosh(2a), cosh(2t)) d(cosh(2a), cosh(2t))
, where

n(A, T ) = A(3 -A 2 )T 2 + (A 2 -1)T -2A, d(A, T ) = (AT + 1) 2 (AT -1) 3/2 .
We note that n(A, T ) is a polynomial of degree 2 in T .

Lemma 4.4 Let a 1 > 0 be such that cosh 2 (2a 1 ) = 11+8 √ 2 7 ≈ 3.1876, i.e. a 1 ≈ 0.5915. For a ≥ a 1 and for all t, we have n(cosh(2a), cosh(2t)) ≤ 0.

To the above family {C 0,a } a>0 of catenaries corresponds a family {C 0,a } a>0 of catenoids in H 3 with the arc-length parametrization Y 0 (a, s, θ),

(11) Y 0 (a, s, θ) = e Λ0 tanh(y 0 ) ω θ e Λ0 / cosh(y 0 ) ,
where the functions Λ 0 (a, s) and y 0 (a, s) are given by Proposition 4.3.

Catenoids in H 3 have been considered in [START_REF] Mori | Minimal surfaces of revolution in H 3 and their stability properties[END_REF][START_REF] Do | Hypersurfaces in spaces of constant curvature[END_REF] and more recently in [START_REF] Seo | Stable minimal hypersurfaces in the hyperbolic space[END_REF]. As pointed out by these authors, among the family {C 0,a } of catenoids in H 3 , there are stable and index one catenoids. We now give a precise analysis of this phenomenon and we also consider Lindelöf's property for these catenoids.

Jacobi fields on C 0,a

We introduce Jacobi fields on C 0,a , having Properties 4.1 and 4.2 in mind. The variation Jacobi field e 0 (a, s) is given by e 0 (a, s) = -g h (Y 0,a (a, s, θ), N 0 (a, s, θ)) =cosh(y 0 ) Λ 0,a y 0,s -Λ 0,s y 0,a .

We have [START_REF] Tam | Stability properties of the higher dimensional catenoid in R n+1[END_REF] e 0 (a, s) = sinh 2 (2a) cosh(2s) cosh 2 (2a) cosh 2 (2s) -1

-1 - cosh(2a) sinh(2s) cosh(2a) cosh(2s) -1 1/2 s 0 I 0 (a, t) dt,
where I 0 (a, t) is defined by [START_REF] Sa | Introduction à la géométrie hyperbolique et aux surfaces de Riemann[END_REF]. The vertical Jacobi field v 0 (a, s) is defined by

(13) v 0 (a, s) = √ 2 cosh(y 0 )y 0,s = cosh(2a) sinh(2s) cosh(2a) cosh(2s) -1 1/2 . Define (14) f 0 (a, s) = sinh 2 (2a) cosh(2s) cosh 2 (2a) cosh 2 (2s) -1 -1
an even function of s which goes to 0 at infinity. In view of Equations ( 12), ( 13) and ( 14), we have the relations

(15) e 0 (a, s) = f 0 (a, s) -v 0 (a, s) s 0 I 0 (a, t) dt.
Observe that the integral

(16) E 0 (a) := ∞ 0 I 0 (a, t) dt
exists for all values of a.

Stable domains on C 0,a

We can now investigate the stability properties of the catenoids C 0,a in H 3 .

Lemma 4.5 The half-catenoids D 0,a,± = Y 0 (a, R • ± , [0, 2π]) are stable. As a consequence, a Jacobi field w(a, s), which only depends on the radial variable s on C 0,a , can have at most one zero on R • + and on R • -.

Proof. Use Property 2.1 (3) and the fact that v 0 (a, s) is a Jacobi field which only vanishes at s = 0. 2 Lemma 4.6 The catenoid C 0,a has index at most 1.

Proof. The fact that the index of C a is at most 1 has been proved by [START_REF] Seo | Stable minimal hypersurfaces in the hyperbolic space[END_REF] using the same method as in [START_REF] Tam | Stability properties of the higher dimensional catenoid in R n+1[END_REF]. Alternatively, one could use Jacobi fields associated to geodesics orthogonal to the axis of the catenoids to prove that halfcatenoids Y 0 (a, R, ]ϕ, ϕ + π[) are stable and to show that negative eigenvalues of the Jacobi operator J C0,a on domains of revolution are necessarily associated with eigenfunctions depending only on the parameter s, see [START_REF] Bérard | Minimal hypersurfaces in H n ×R, total curvature and index[END_REF], Theorem 3.5. 2

We can now state the main theorem of this section. Recall that the number E 0 (a) is defined by ( 16) and that the Jacobi fields v 0 (a, s) and e 0 (a, s) are given respectively by ( 13) and [START_REF] Tam | Stability properties of the higher dimensional catenoid in R n+1[END_REF], with the relation (15). Theorem 4.7 Let {C 0,a } a>0 be the family of catenoids in H 3 given by [START_REF] Seo | Stable minimal hypersurfaces in the hyperbolic space[END_REF].

1. The index of the catenoid C 0,a depends on the value of the integral E 0 (a) defined by ( 16). More precisely, if E 0 (a) ≤ 0 then the catenoid C 0,a is stable, if E 0 (a) > 0, then the catenoid C 0,a has index 1.

2. When C 0,a has index 1, there exists a positive z(a) such that the domain

D 0,z(a) = Y 0 (a, ] -z(a), z(a)[, [0, 2π]
) is a stable-unstable.

3. When C 0,a has index 1, there exists some ℓ(a), 0 < ℓ(a) < z(a), such that

D 0,ℓ(a) = Y 0 (a, ] -ℓ(a), ∞[, [0, 2π]
) is a maximal stable, rotation invariant domain.

4. The catenoids {C 0,a } a>0 do not satisfy Lindelöf's property.

5. There exist two numbers 0 < a 2 < a 1 such that for all a > a 1 , the catenoids C 0,a are stable, and for all a < a 2 , the catenoids C 0,a have index 1.

Proof. Assertion 1. As stated in Lemma 4.5, the function e 0 (a, s) can have at most one zero on ]0, ∞[ and at most one zero on ] -∞, 0[. Observe that the function e 0 (a, s) is even and that e 0 (a, 0) = 1. To determine whether e 0 has a zero, it suffices to look at its behaviour at infinity. If E 0 (a) > 0, the function e 0 (a, s) tends to -∞ at infinity so that it has exactly two symmetric zeroes in R. This implies that the index of C 0,a is at least 1. Using Lemma 4.6, we conclude that C 0,a has index 1. If E 0 (a) < 0, the function e 0 (a, s) tends to +∞ at infinity so that it is always positive and the catenoid C 0,a is stable. Assume now that E 0 (a) = 0. We then have the relation

e 0 (a, s) = f 0 (a, s) + v 0 (a, s) ∞ s I 0 (a, t) dt.
Using Equation [START_REF] Sa | Introduction à la géométrie hyperbolique et aux surfaces de Riemann[END_REF], we see that I 0 (a, t) is positive for t large enough provided that cosh 2 (2a) ≤ 3. In that case, it follows that e 0 (a, s) is positive at infinity and hence that C 0,a is stable. If E(a) = 0 and cosh 2 (2a) > 3, we need to look at the behaviour of e 0 (a, s) at infinity more precisely. When s tends to +∞, we have

f 0 (a, s) ∼ 2 tanh 2 (2a)e -2s , v 0 (a, s) ∼ cosh(2a) 2 e s , and
∞ s I 0 (a, t) dt ∼ 2 3/2 (3 -cosh 2 (2a)) 3 cosh 5/2 (2a) e -3s .
It follows that e 0 (a, s) ∼ 4 3 e -2s is positive at infinity and hence that C 0,a is stable. Assertion 2. Saying that C 0,a has index 1 is equivalent to saying that E 0 (a) > 0 and hence that e 0 has two symmetric zeroes. Assertion 3. Given any α > 0, we introduce the Jacobi field e 0 (a, α, s), (17) e 0 (a, α, s) = v 0 (a, α)e 0 (a, s) + e 0 (a, α)v 0 (a, s).

This Jacobi field vanishes at s = -α < 0 so that it cannot vanish elsewhere in ] -∞, 0[ and can at most vanish once in ]0, ∞[. Using Equations ( 17) and ( 15), we can write e 0 (a, α, s) = v 0 (a, α)f 0 (a, s) + v 0 (a, s) e 0 (a, α) -v 0 (a, α)

s 0 I 0 (a, t) dt .
We have e 0 (a, α, -α) = 0 and e 0 (a, α, 0) = v 0 (a, α) > 0 so that e 0 (a, α, 1. Let S be an open interval on which E 0 < 0 (hence the catenoid C 0,a is stable for all a ∈ S). For a ∈ S, the catenaries C 0,a locally foliate the hyperbolic plane H 2 {x1,x3} , see Figure 2 (half-catenaries).

•) van- ishes in ]0, ∞[ if and only if e 0 (a, α) -v 0 (a, α)E 0 (a) < 0 (recall that E 0 (a) = ∞ 0 I 0 (a, t) dt). If C 0,

Let

U be an open interval on which E 0 > 0 (hence the catenoid C 0,a has index 1 for all a ∈ U). For a, b ∈ U, the catenaries C 0,a and C 0,b in H 2 {x1,x3}

intersect exactly at two points and the family {C 0,a } a∈U has an envelope. Furthermore, the points at which C 0,a touches the envelope correspond to the stable-unstable domain D 0,z(a) , see Figure 3 (half-catenaries).

Proof. Define the v-height function of the catenoid C 0,a in H 2 {u,v} by

(18) V 0 (a) = lim t→∞ λ 0 (a, t) = lim s→∞ Λ 0 (a, s).
Lemma 4.9 Let a 2 > a 1 > 0 be two values of the parameter a. The catenaries C 0,a1 and C 0,a2 intersect at most at two symmetric points and they do so if and only if V 0 (a 2 ) > V 0 (a 1 ).

Proof. To prove the Lemma, consider the difference w(t) := λ 0 (a 2 , t) -λ 0 (a 1 , t) for t ≥ a 2 > a 1 . A straightforward computation shows that this function increases from the negative value -λ 0 (a 1 , a 2 ) (achieved for t = a 2 ) to V 0 (a 2 ) -V 0 (a 1 ) (the limit at t = ∞). It follows that w has at most one zero and does so if and only if V 0 (a 2 ) -V 0 (a 1 ) > 0. The Proposition follows from the fact that

V 0 (a) = √ 2 ∞ 0 J 0 (a, t) dt and that V ′ 0 (a) = √ 2E 0 (a)
where E 0 (a) is defined by (16).

2

Observation 1. One can also define the x-height function of the catenoid C 0,a by X 0 (a) = lim s ∞ e Λ0(a,s) tanh(y 0 (a, s)

) = e √ 2 R
∞ 0 J0(a,t) dt , where J 0 (a, t) is defined by [START_REF] Mori | Minimal surfaces of revolution in H 3 and their stability properties[END_REF]. Then, the critical points of X 0 (a) correspond to the zeroes of the function E 0 (a). Observation 2. We point out that Theorem 4.7, Assertion 1, provides a criterion (the sign of E 0 (a)) to determine whether the catenoid C 0,a has index 0 or 1 whereas [START_REF] Mori | Minimal surfaces of revolution in H 3 and their stability properties[END_REF][START_REF] Do | Hypersurfaces in spaces of constant curvature[END_REF][START_REF] Seo | Stable minimal hypersurfaces in the hyperbolic space[END_REF] only give sufficient conditions to insure that the index is 0 or 1. Numerical computations show that the following result holds (see [START_REF] Bérard | Lindeloef's theorem for catenoids, revisited[END_REF] and the graph in Figure 4). Proposition 4.10 The function E 0 (a) has exactly one zero a 0 ≈ 0.4955 on ]0, ∞[ which is also the unique critical point of the x-height function X 0 (a).

Catenoid cousins in H 3

Basic formulas

We now consider catenoid cousins, i.e. rotation surfaces with constant mean curvature 1 in H 3 (-1). In this case, the mean curvature equation ( 4) reads

(19) sinh(2t) = d dt f t (t) sinh(t) cosh 2 (t) 1 + cosh 2 (t)f 2 t (t) 1/2 .
which yields for some constant d ∈ R. For a solution to exist, d needs to be positive so that we may assume that 2d = e -2a for some a ∈ R and we obtain (20)

f t (t) sinh(t) cosh 2 (t) 1 + cosh 2 (t)f 2 t (t) 1/2 = 1 2 cosh(2t) -d,
f t = e a cosh(2t) -e -2a √
2 cosh(t) cosh(2t)cosh(2a) , t ≥ |a|.

◮ We now limit ourselves to the embedded case and assume that a > 0. ◭ Equation (20) yields embedded catenary cousins {C 1,a } a>0 , given by (21) λ 1 (a, t) = t a e a cosh(2τ ) -e -2a √ 2 cosh(τ ) cosh(2τ )cosh(2a) dτ, for t ≥ a,

where the lower index 1 refers to H = 1. Notice that the function λ 1 describes the upper halves of catenary-like curves. We compute the arc-length function

(22) S 1 (a, t) = e a √ 2 cosh(2t) -cosh(2a), cosh(2t) = 2e -2a S 2 1 (a, t) + cosh(2a), t ≥ a.
As in Section 4.2, we obtain the following result. 1. The function y 1 is smooth, even, and satisfies cosh(2y 1 (a, s)) = 2e -2a s 2 + cosh(2a).

2. The function Λ 1 is smooth, odd, and satisfies Λ 1 (a, s) = λ 1 (a, y 1 (a, s)) for s ≥ 0.

3. For a > 0, the maps R ∋ s → y 1 (a, s), Λ 1 (a, s) ∈ H 2 {u,v} are arc-length parametrizations of the family of embedded catenary cousins {C 1,a } a>0 which generate the family {C 1,a } a>0 of embedded catenoid cousins.

4. The parametrization of the family {C 1,a } a>0 in H 3 {x1,x2,x3} is given by (25) Y 1 (a, s) = e Λ1(a,s) tanh(y 1 (a, s)) ω θ e Λ1(a,s) 1 cosh(y1(a,s)) .

Jacobi fields on C 1,a

As in Section 4.2, we define the vertical and variation Jacobi fields on C 1,a . Lemma 4.12 The vertical Jacobi field v 1 is a smooth odd function of s. It is given by v 1 (a, s) = cosh(y 1 (a, s))y 1,s (a, s) = e -a s s 2 + e 2a sinh 2 (a)

-1/2 and satisfies v 1 (a, 0

) = 0, v 1 (a, ∞) = e -a .
The variation Jacobi field e 1 (a, s) on C 1,a is given by e 1 (a, s) = cosh y 1 (a, s) Λ 1,a y 1,s -Λ 1,s y 1,a (a, s).

which we can write as e 1 (a, s) = v 1 (a, s)Λ 1,a (a, s)cosh(y 1 (a, s))y 1,a (a, s)Λ 1,s (a, s).

Using Proposition 4.11 and Lemma 4.12, we find the formula (26) cosh(y 1 ) Λ 1,s y 1,a = sinh 2 (2a) -4e -4a s 4 4 e -2a s 2 + cosh 2 (a) e -2a s 2 + sinh 2 (a)

.

By (24), we can write Λ 1 (a, s) as s 0 A(a, t) dt, where the integrand A(a, t) is given by ( 27)

         A(a, t) = 2e a t 2 + e 3a sinh(2a) 2 t 2 + e 2a cosh 2 (a) t 2 + e 2a sinh 2 (a) 1/2 , = A 1 (a, t) 2A 2 (a, t)A 1/2 3 (a, t)
, where the second equality defines the functions A i . One can now compute the derivative of A(a, t) with respect to the variable a.

A a (a, t) = A 1,a (a, t)

2A 2 (a, t)A 1/2 3 (a, t) - A 1 (a, t)B 2 (a) 2A 2 2 (a, t)A 1/2 3 (a, t) - A 1 (a, t)B 3 (a)
4A 2 (a, t)A 2A 2 (a, t)A

1/2 3 (a, t) - A 1 (a, t)B 2 (a) 2A 2 2 (a, t)A 1/2 3 (a, t) • • • - A 1 (a, t)B 3 (a) 4A 2 (a, t)A 3/2 3 (a, t) , i.e. ( 28
)        A a (a, t) = B(a, t) -C(a, t), where B(a, t), C(a, t) > 0, for a > 0, t ∈ R, B(a, t) ∼ e a |t| , at infinity, C(a, t) = O( 1 |t| 3 ), at infinity.
Finally, with the above notations, we can write the variation Jacobi field as e 1 (a, s) = -e 4a sinh 2 (a) cosh 2 (a) -s 4 s 2 + e 2a cosh 2 (a) s 2 + e 2a sinh 2 (a)

-v 1 (a, s) s 0 C(a, t) dt + v 1 (a, t) s 0 B(a, t) dt
We have proved, Lemma 4.13 The variation Jacobi field e 1 is a smooth, even function of s which can be written as

(29) e 1 (a, s) = -f 1 (a, s) + v 1 (a, s) s 0 B(a, t) dt,
where the function f 1 is a smooth, even function of s, such that f 1 (a, 0) =

We can now state the main theorem of this section. Recall that the Jacobi fields v 1 (a, s) and e 1 (a, s) are given respectively by Lemmas 4.12 and 4.13. Theorem 4.16 Let {C 1,a , a > 0} be the family of embedded catenoid cousins in H 3 given by the parametrization Y 1 , Equation (25). are stable-unstable.

3. In particular, the embedded catenoid cousins {C 1,a } a>0 satisfy Lindelöf's property: the upper and lower halves of the embedded catenoid cousins D 1,a,± are maximal rotationally symmetric domains.

4. The index of the catenoid C 1,a is equal to 1.

Proof. Assertion 1. As we have seen in Lemma 4.14, the function e 1 (a, s) can have at most one zero on ]0, ∞[ and at most one zero on ] -∞, 0[. By Lemma 4.13, the function e 1 (a, s) is even, e 1 (a, 0) = -1 and e 1 (a, ∞) = ∞. It follows that e 1 (a, s) has exactly two symmetric zeroes in R. Assertion 2. Given any α > 0, we introduce the Jacobi field e 1 (a, α, s), (30) e 1 (a, α, s) = v 1 (a, α)e 1 (a, s) + e 1 (a, α)v 1 (a, s).

This Jacobi field vanishes at s = -α < 0 so that it cannot vanish elsewhere in ] -∞, 0[ and can at most vanish once in ]0, ∞[. Using Lemma 4.13, we can write e 1 (a, α, s) = -v 1 (a, α)f 1 (a, s) + v 1 (a, s) e 1 (a, α) + v 1 (a, α) s 0 B(a, t) dt .

It follows that e 1 (a, α, -α) = 0, e 1 (a, α, 0) < 0 and lim s→∞ e 1 (a, α, s) = +∞, and hence that e 1 (a, α, •) must vanish at least once. Assertion 3. This is a consequence of Assertion 2. Assertion 4. This assertion follows from Assertion 1 and from Lemma 4.15. This has also been proved, using different methods, by Lima and Rossman [START_REF] Lopes | On the index of constant mean curvature 1 surfaces in hyperbolic space[END_REF]. 2

Further results

One can also study rotation surfaces with constant mean curvature H, 0 ≤ H < 1 in H 3 (-1). This is similar to the case of minimal surfaces. More precisely, H-rotation surfaces in H 3 (-1), with 0 ≤ H < 1, come in a one-parameter family C H,a . For some values of a the surfaces are stable, for other values of a they have index 1. Furthermore, they do not satisfy Lindelöf's property. The computations are much more complicated but similar to the minimal case. The functions involved depend continuously on the parameter H, for 0 ≤ H < 1.

The method described in the previous sections can be applied to study the stable domains on higher dimensional catenoids (minimal rotation hypersurfaces or constant mean curvature 1 rotation hypersurfaces) in H n+1 or other homogeneous spaces.

Figure 1 :

 1 Figure 1: Catenaries, n ≥ 2 and n ≥ 3

3

 3 {x1,x2,x3} is just the position vector. The vertical Jacobi field is the function v Y (a, s) = g h (Y, N Y ), where N Y is the unit normal vector to the immersion Y . The following results are straightforward. Property 4.1 The vertical Jacobi field v Y (a, s) = g h (Y, N Y ) is an odd function of s given by v Y (a, s) = cosh(y(a, s))y s (a, s).Property 4.2 The variation Jacobi field e Y (a, s) is an even function of s given by e Y (a, s)

  a is stable, then clearly e 0 (a, α, •) cannot vanish twice in R. Assume that C 0,a has index 1 or, equivalently, that E 0 (a) > 0. In that case, e 0 (a, •) has exactly one positive zero z(a). (i) For α > z(a), e 0 (a, α) < 0 so that e 0 (a, α) -v 0 (a, α)E 0 (a) < 0 and e 0 (a, α, •) has a positive zero β (which must satisfy β < z(a)). (ii) For α = z(a), e 0 (a, α, s) = v 0 (a, α)e 0 (a, s) has two zeroes ±z(a). (iii) For 0 < α < z(a), we can argue as follows. Consider the Jacobi field w(a, t) = e 0 (a, t) -E 0 (a)v 0 (a, t). At t = 0, we have w(a, 0) = 1 and at t = z(a), we have w(a, z(a)) < 0 because e 0 (a, z(a)) = 0, E 0 (a) > 0 and v 0 (a, z(a)) > 0. It follows that w(a, t) has a unique zero in ]0, z(a)[ and hence that there exists a value ℓ(a) > 0 such that D 0,ℓ(a) = Y 0 (a, ] -ℓ(a), ∞[, [0, 2π]) is a maximal stable rotation invariant domain. Assertion 4. This follows immediately from the previous assertion. Assertion 5. The first part of the Assertion follows from Lemma 4.4 which implies that e(a, s) never vanishes when a > a 1 . To prove the second part of Assertion 3, we can either use the fact that E 0 (a) tends to +∞ when a tends to zero from above or use the criteria given in[START_REF] Do | Hypersurfaces in spaces of constant curvature[END_REF] (Corollary 5.13, p. 708) or[START_REF] Seo | Stable minimal hypersurfaces in the hyperbolic space[END_REF] (Corollary 4.2).2We have the following geometric interpretation of Theorem 4.7.

Figure 2

 2 Figure 2: Foliating Figure 3: Intersecting

Figure 4 :

 4 Figure 4: Graph of E 0 (a)

Proposition 4 . 0 2e -2a t dt 2e -2a t 2 +e a 2t 2 +

 4022 [START_REF] Seo | Stable minimal hypersurfaces in the hyperbolic space[END_REF] For a > 0 and s ∈ R, define the functions y 1 (a, s) and Λ 1 (a, s) by the formulas (23) y 1 (a, s) = a + s e 2a sinh(2a) dt 2 t 2 + e 2a cosh 2 (a) t 2 + e 2a sinh 2 (a).

3 / 2 3

 32 (a, t) , where B 2 (a) = ∂ a e 2a cosh 2 (a) , B 3 (a) = ∂ a e 2a sinh 2 (a) . It follows that A a (a, t) = 2e a t 2 + e 3a (3 sinh(2a) + 2 cosh(2a))

1 . 2 .

 12 The Jacobi field e 1 (a, s) has exactly one positive zero z 1 (a) and the domains D 1,a,z1(a) = Y 1 (a, ] -z 1 (a), z 1 (a)[, [0, 2π]) are stable-unstable domains. For any α > 0, there exists a β(α) > 0 such that the domainsD 1,a,-α,β(α) = Y 1 (a, ] -α, β(α)[, [0, 2π])

  Stable domains on the embedded catenoid cousinsWe can now investigate the stability properties of the embedded catenoids cousins {C 1,a } a>0 in H 3 (-1). Lemma 4.14 The upper and lower halves D 1,a,± = Y 1 (a, R • ± , [0, 2π]) of the embedded catenoid cousins are stable. As a consequence, a Jacobi field w(a, s), which only depends on the radial variable s, on C 1,a can have at most one zero on R • + and on R • -.

				1
	and f 1 (a, ∞) finite. Furthermore,	s
	lim s→∞	v 1 (a, s)	0	B(a, t) dt = +∞.
	4.3.3 Proof. Use Property 2.1 (3) and the fact that v 1 (a, s) is a Jacobi field which
	only vanishes at s = 0.			2
	Lemma 4.15 The embedded catenoid cousins C 1,a have at most index 1.
	Proof. Same proof as for Lemma 4.6


Daniel Roger

Member, IEEE Sonia Ait-Amar

Ewa Napieralska-Juszczak

Piotr Napieralski

Senior Member, IEEE Ewa Napieralska

A Proposition for Improving the Design

Keywords: Resistive layer, Electrical machine windings, Aircraft environment, Partial Discharge Inception Voltage (PDIV), Paschen's law

whether they are published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

I N more electric aircrafts, the electrical energy brings many advantages comparing to hydraulic and pneumatic ones. For increasing the electric power with the same copper wire mass, the aircraft industry uses higher and higher voltages, switching from 115V ac to 540V dc. Studies are under progress for designing aircrafts with an electrical assistance during take off, requiring higher voltage grids [START_REF] Zhang | Development of megawatt-scale medium-voltage high efficiency high power density power converters for aircraft hybrid-electric propulsion systems[END_REF]- [START_REF] Chen | Investigation on the selection of a more suitable power system architecture for future more electric aircraft from the prospective of system stability[END_REF]. Generally speaking, high power density electric actuators are connected to the dc grid by inverters that provide an efficient machine control and power reversibility. However, the electronic switches of inverters impose steep-fronted voltage pulses causing repetitive high magnitude short voltage spikes that stress the machine insulation. These problems were detected more than two decades ago when IGBT converters spread in industry [START_REF] Bell | Will your motor insulation survive a new adjustable-frequency drive?[END_REF], [START_REF] Manz | Motor insulation system quality for IGBT drives[END_REF]. These spikes may cause PDs in the small remaining air-voids in the machine coils. It is now known that the space charges trapped in the polymer layers makes the analysis more complex [START_REF] Banda | Simulation of space charge impact on partial discharge inception voltage in power busbars dedicated to future hybrid aircrafts[END_REF]. Measurement method of the Partial Discharge Inception Voltage (PDIV) inside low voltage machines has been developed [START_REF] Bogh | Partial-dischargeinception testing on low-voltage motors[END_REF], [START_REF] Abadie | Partial discharges in motor fed by inverter: From detection to winding configuration[END_REF]. PDs cause an earlier aging of the Electrical Insulation System (EIS) [START_REF] Maussion | Lifespan and aging modeling methods for insulation systems in electrical machines: A survey[END_REF]- [START_REF] Barater | Multistress characterization of fault mechanisms in aerospace electric actuators[END_REF]. These problems are more critical at low pressures because PDs appear at lower voltages [START_REF] Lusuardi | The problem of altitude when qualifying the insulating system of actuators for more electrical aircraft[END_REF], [START_REF] Koliatene | Impact of the aeronautic environment on the partial discharges ignition: A basic study[END_REF].

Many machine topologies can be used in aircraft motors and generators [START_REF] Cao | Overview of electric motor technologies used for more electric aircraft (mea)[END_REF], [START_REF] Jia | An induction generator-based ac/dc hybrid electric power generation system for more electric aircraft[END_REF]. A solution consists in using prefabricated coils with a careful design of the turn deterministic D. Roger, S. Ait-Amar and E. Napieralska are in the Artois University (EA4025), Laboratoire Systèmes Electrotechniques et Environnement (LSEE) 62400, Bethune, France.

P. Napieralski is in the Institute of Information Technology, Lodz University of Technology, ul. Stefanowskiego 18/22, 90-924 Lodz, Poland.

arrangement for keeping the turn-to-turn voltage stress under the PDIV everywhere in the machine [START_REF] Iosif | Voltage stress predetermination for long-life design of windings for electric actuators in aircrafts[END_REF]. An estimation of the PDIV can be made using 2D electrostatic simulations and Paschen's law [START_REF] Duchesne | Prediction of pdiv in motor coils using finite element method[END_REF]. This coil technology can only be applied to permanent magnet synchronous machines (PMSM) or synchronous reluctant machines (SRM) made with concentrated windings and opened slots (one coil per stator tooth). With such prefabricated coils, the slot filling factor is lower than for standard random coils directly wound in the stator slots.

For compact machines that require a high slot feeling factor, the random winding method is better. The exact position of each turn remains unknown. The probability to have the input turn near the output one is not negligible [START_REF] Duchesne | Study of wire distribution in a slot of a motor fed by steep fronted pulses for lifetime extension[END_REF]; the turn-to-turn voltage may be large and PDs may appear.

A solution consists in accepting a reasonable number of PDs during the whole expected life time of the machine using corona resistant wires. This enameled wire is made with an external polymer layer charged with micro or nano hard particles of inorganic materials that resist to PD abrasion [START_REF] Cavallini | Evaluation of corona-resistant magnet wires through partial discharge and space charge measurements[END_REF]- [START_REF] Honag | Electrical characterization of partial discharge resistant enamel insulation[END_REF].

Our research shows that another approach can be used; it consists in adding a thin resistive layer on the outer surface of enameled wire. This layer changes the electric field distribution in the EIS. It concentrates the field in the thin polymer layer of the enameled wire, which can withstand high field strengths. For a voltage stress over the PDIV, PDs appears only in specific zone situated near the end-connections of the coil which can easily be reinforced. Some of these issues were presented at the ITEC 2018 conference in California [START_REF] Roger | A proposition for improving the design of motor windings for low-pressure environment[END_REF]. The paper proposes an extension of the theoretical approach for estimating the resistivity of this layer able to reduce the voltage turn-to-turn stress, with limited extra losses inside the additional resistive layers. The paper uses a standard compact machine design for defining the electrical properties of the resistive layer that can be easily added by enameled wire manufacturers without changing the external diameter. Experiments are performed on twisted pairs and random coils made with a wire covered with a resistive paint. They prove the effectiveness of the proposed principle before convincing an enameled wire manufacturer to test the new wire.

II. ENVIRONMENT OF AIRCRAFT ELECTRICAL MACHINES

For studying PDs in aircraft embedded machines, it is necessary to consider temperature and pressure in the small remaining air-voids between the enameled wires of the machine coils. Several cases are possible: the motor can be placed inside the cabin, where the air-control system provides classical temperatures and pressures (80% of the sea-level pressure). For this case, the motor operating conditions are very near the classical ones and no specific investigations are necessary.

For electrical machines placed outside the cabin, two cases must be analysed: the air-void can be a closed bubble without any contact with the air outside the machine (closed air-void) or a small volume open towards the outside atmosphere (open air-void). For the first case, the air pressure inside the closed air-void does not depend on the ambient pressure but only of the temperature. For the second case, the air-void pressure is equal to the ambient one at the considered altitude, which can be estimated by the standard atmosphere table [START_REF]Properties of the U.S. standard atmosphere[END_REF] giving temperatures and pressures at any point in the world .

The pressure in closed air-void, depends on the machine manufacturing process. A varnish always impregnates machine windings.Whatever the impregnation process, it ends by a polymerization thermal cycle in an oven. At the beginning of this cycle, the varnish is liquid. When the temperature increases, the closed air-void volume changes for getting a global pressure balance. During the polymerization process at a constant temperature, the varnish viscosity increase slowly. At the end of the polymerization, the varnish is a hard thermoplastic material, the volume of the closed air voids does not change anymore.When the temperature of the oven decreases, the pressure in the constant volume void decreases following the perfect gas law.

Table I Paschen's formulae [START_REF] Zhang | Development of megawatt-scale medium-voltage high efficiency high power density power converters for aircraft hybrid-electric propulsion systems[END_REF], based on Townsend's gas ionization theory [START_REF] Townsend | The theory of ionization of gases by collision[END_REF], can be used for getting an estimation of the electronic avalanche threshold voltage V T h between the airvoid border that create an ionization of the air. It results of experimental works on ionization of a gas, at a given pressure p, placed between two flat electrodes at a distance d. This expression yields the threshold voltage from the pressure distance product (pd). For dry air at 20 o C, B = 365 V.T orr -1 .cm -1 . C is defined by [START_REF] Thauvin | Hybrid propulsion for regional aircraft: A comparative analysis based on energy efficiency[END_REF] where A = 15 T orr -1 .cm -1 ; γ is the second Townsend coefficient γ ≈ 0.01 for polymer.

V T h = B (p.d) C + ln((p.d) (1)
C = ln A 1 + 1/γ (2)
The correction (3) considers the gas density at a given temperature improves the Paschen's law [START_REF]High voltage measurement techniques part 1[END_REF]. In (3), V T h0 is the Paschen's threshold voltage computed at room temperature (20 o C, 293 o K); the pressure p is expressed in Torr (760 T orr = 1013 hP a).

V T h (p, T ) = 293 T p 760 V T h0 (p.d) (3)
Figure 1 shows the Paschen's curves for the 3 temperatures of table I. The horizontal axis is the pd product with standard units. For instance, the vertical dotted line at pd = 0. Let us remind that Paschen's experiences were made with uniform fields. In a motor winding aircois, the electric field is not uniform. A more detailed analysis of Paschen's hypothesis shows that, at sea-level pressure, Paschen's law can be used with a reasonable error when the Pashen's distance d is replaced by the length of the part of the electric field lines in air-voids [START_REF] Roger | A method to reduce partial discharges in motor windings fed by PWM inverter[END_REF].

Recent experimental works confirm the Paschen's law for distances and pressure over the Paschen's minimum [START_REF] Allen | Effects of temperature and pressure change on positive corona and sparkover under direct voltage in short airgaps[END_REF]- [START_REF] Sili | Temperature dependence of electrical breakdown mechanism on the left of the paschen minimum[END_REF]. Another recent work confirms the Peek's correction for very cold Nitrogen [START_REF] Massarczyk | Paschen's law studies in cold gases[END_REF].

III. DRAWBACK OF PDS IN MACHINE WINDINGS AND PROPOSED IMPROVEMENT

The most widely used enameled wires are made with two polymers. The polyester-imide (PEI) in contact with the copper provides a good adherence. The polyamide-imide (PAI) provides the required mechanical and thermal properties. The global thicknesses of the insulating layers is defined in the IEC 60317 standard [START_REF]Specifications for particular types of winding wirespart 0-1: General requirements -enamelled round copper wire[END_REF]; it depends on the wire diameter and its grade.

The enameled wire is manufactured with a complex industrial process, which consists in depositing many thin layers of 3 µm that are polymerized separately in an oven. This operation is repeated 10-15 times for getting the insulating thickness defined by the standards. With this manufacturing process, no significant air-void can exist inside the insulation layer of the enameled wire. Since the polymer has a much higher breakdown electric field than air, PDs occur in the air voids between wires. PDs cause a slow erosion polymer insulation layer. Figure 2 shows a microscopy of the crosssection of a twisted pair made of standard enameled wire eroded by many PDs. The abrasion near the contact point is clearly visible. The phenomena are similar with coronaresistant enameled wire but the abrasion is much slower; the time before failure is much longer [START_REF] Cavallini | Evaluation of corona-resistant magnet wires through partial discharge and space charge measurements[END_REF].

IV. MOTOR COIL MADE WITH AN ENAMELED WIRE COATED BY A RESISTIVE LAYER

A. Example of compact machine

Compact machine are very often made with round enameled copper wire and random coils. For high currents, coils are wound with several small wires in hand. They are placed in the slots by their opening. This winding technology is used with semi-closed slots that reduce the cogging torque. Let us consider a compact 3-phase 12-teeth 10-pole PMSM widely used in more electric aircrafts for cabin pressurisation systems. For this example, the air-gap diameter is 80 mm, the stator length is also 80 mm. This machine is made with 12 coils, one per stator tooth. The rated values are: 8000 rpm; 400 V RM S ; 25 A RM S and 17 kW .

In each slot, polymer sheets provide the common mode insulation and the insulation between the two coils. The 3 phases are made of 4 coils connected in series with the appropriate connections for producing a 10-pole rotating field. Each coil is made of 20 turns wound with 5 enameled wires of diameter 1 mm in hand. For such coils, the probability to have a wire of the input turn adjacent to a wire of the output one is large.

The turn-to-turn insulation must be designed for this worst case. In a more electric aircrafts, the inverters connected to the 540 V on-board dc bus feed the machines; their EISs must be designed considering the electrical stresses due to the inverter fast fronted voltages.

Our proposition consist in changing the varnish bath of the last layers of the enameled wire manufacturing process, using PAI charged with micro particles of carbon, for getting a thin resistive outer layer. The theoretical approach is aimed at defining a method able to find the suitable resistivity of the outer resistive layer. The influence of this layer on the machine insulation analysed considering the two adjacent wires presented in figure 3.

B. General assumptions and simplifications

The theoretical analysis of the influence of the outer resistive layer is made considering several assumptions: -PDs may appear only in the residual air bubbles between the outer resistive layers of the wires.

-The study is made for the worst case in a random winding made of turns with several round wires in hand. A wire of the input turn is supposed to be adjacent to a wire or the output turn.

-The Y-connected machine is fed by a PWM inverter with fast fronted voltages. The first coil of each phase is supposed to withstand the full voltage stress during the fast transients.

-Only short times (< 1µs) corresponding to fast transients are considered. The complete study of the transients in the machine and the feeding cable is out of the scope: the Ph-N voltage waveform is an input data.

-The analysis uses the equivalent circuit presented in figure 4. This circuit is based on a regular mesh of the two resistive layers of adjacent wires. Each resistive layer is cut in a set of half cylinders of length ∆l = 2.5 mm as presented in the left part of figure 3. The red points A1x, A2x, B1x, and B2x are placed at the centre of the half cylinders, they corresponds yo the x th column of the equivalent circuit of figure 4.

-The insulating and the resistive layers are supposed to be perfect cylinders of constant thicknesses made of homogenous and isotrope materials. Their sizes are used for computing the resistance and the capacitances of the equivalent circuit of figure 4. The capacitances between the copper and the resistive layer ∆C W are drawn in black. The capacitances between the resistive layers ∆C S are drawn in green; they are estimated to ∆C W /10. This value is estimated form capacitance measurements on twisted pairs. -For a random winding in a slot, the resistive outer layers of adjacent wires are in contact in limited number of points because the wires are not perfectly straight. Several simulations have been made for understanding the influence of the number of contact points between adjacent wires and their resistances. As expected, the voltage stress in the air between wires is all the weaker as the contacts are numerous and their resistances small. A pessimistic case is considered with only 8 contact points on the length of a stator slot (10 mm between contacts ), with a resistance ∆R S equal to the longitudinal resistance between the centre of two consecutive half cylinder ∆R L .

-The simulation of the electric field in the connection zone of a wire is made in 2D with linear conditions. The additional hypothesis are explained in section IV-D.

C. Maximum voltage stress between adjacent wires of a coil

The voltage stress are computed with the equivalent circuit of figure 4 with a Spice processor. The parameters are estimated considering the wire sizes (R CU = 0.5mm; a polymer layer of 30 µm and a resistive one of 6 µm). The polymer relative permittivity is R = 3.5. The resistive layer resistivity is ρ = 10.10 -3 Ω.m. For these values ∆R L = 396 Ω, ∆R T = 560 Ω and ∆C W = 4.2 pF .

The input voltage v(t) is representative of a transient state observed on the Ph-to-N voltage of a machine connected to an inverter with a 4-meter long cable; the voltage spike is 550 V .

Considering the pessimistic case define above, the peak voltage stress on the air layer is only 100 V , which is under the Paschen minimum at low pressures of the airliner flying altitude. With more contact points between the resistive layers, the peak voltage stress is lower. The peak voltage is located halfway between two contact points.

For a motor coil made with the proposed enameled wire, the resistive layers in contact together form closed loops around each stator tooth that create eddy currents and additional losses.

The global resistance of such a close loop is computed for the worst case that suppose a perfect contact between the external surface of wires. All the resistive layers of a coil are connected in parallel (one resistance per wire, 5 wires per turn, 20 turns = 100 resistances connected in parallel). For each coil this closed resistive loop creates extra Joule losses that are estimated to 32 W per coil for the motor steady state at its rates power.

It is well known that the voltage stress due to the inverter fast fronted voltages is concentrated on the first turns of the first coils of each phase. For the other coils, the common-mode capacitances mitigate the voltage spikes. Only the first coil of each phase should use the proposed solution; the other ones can be wound with a standard enameled wire without any resistive layer . The additional losses estimation is 128 W , which represents only 0.75% of the motor rated power.

D. Finite element simulation in the connection zones

A special care must be paid in the vicinity of the coil endconnections: the outer resistive layer must be stopped far form the nude copper wire as shown in figure 6. Consequently, a high electric field exists in the air of this specific zone and PDs appear. The electric field is computed in the polymer and in the ambient air using a 2D finite-element (FE) software. The mesh is built at the micrometer scale (figure 7). The exact shape of the external conducting layer is supposed to be smooth. The field is computed in linear conditions with a constant polymer relative permittivity of 3.5 and for 1 kV imposed between the resistive layer and central copper wire. The simulation results can be applied to other voltages because of the problem linearity. The FE simulation yields the field lines of figure 8. They are plotted in the air of the specific zone near the connections. PDs may occur in the air but not in the polymer because of its higher dielectric strength. The numbers added at the bottom of each line is a simple serial number, from the longest line to the shortest one. The global shape of these field lines is far from the straight lines of the uniform field of the Paschen's theory. A deeper analysis must be made before applying Paschen's results.

Let us remind that Paschen's works were mainly based on the Townsend's gas ionization theory [START_REF] Townsend | The theory of ionization of gases by collision[END_REF] and on a free electrons accelerated by the Coulomb's force. This theory is valid only before ionization for estimating the voltage threshold of the electronic avalanche. Following Townsend's theory, an electronic avalanche appears when an accelerated free electron acquires enough energy for ionizing a gas molecules when a collision occurs. After the electronic avalanche ignition, the development and extinction of PDs require more complex analysis based on plasma theory. The PDIV corresponds to the voltage at the very beginning of the electronic avalanche. In Paschen's theory, the free electrons are accelerated following the straight field lines: the speed ( v) and the Coulomb force ( F ) are collinear vectors. The free electron trajectories are straight lines superimposed with the field lines, which have the same length d.

With a non-uniform field, free electrons do not follow exactly the field lines because their mass influence the trajectories. The proposed analysis adds the inertial equation due to the electron mass m. The input data is the force field F = q E defined by a square mesh, which is 0.5 µm. The electric field E is computed in air, with the FE simulation described above. The hypothesis are similar to Townsend's and Paschen's ones based on a mechanical interpretation of the physical phenomena in gas: -The electronic avalanche is a succession of ionizing collisions of gas molecules by accelerated free electrons.

-When the applied voltage is exactly equal to the electronic avalanche ignition voltage, the changes in the local field due to ions are not considered.

-When an accelerated free electron collides an atom of a gas molecule with enough energy, the atom absorbs the kinetic energy for creating a new free electron at zero speed.

-The air of the connection zone is only composed of molecule that ionization energy is 14 eV (this value is 14.5 eV for Nitrogen and 13.6 eV for oxygen).

-Elastic collisions and thermal agitation are not considered.

The mechanical time differential equation is simulated with Euler method and a very short time step (0.01 ps) because of the low value of the electron mass (10 -30 kg). Each trajectory is computed using a starting point defined at zero speed v = 0 at the beginning point of a field line situated on the border between polymer and air. Figure 9 considers two starting points at the beginning of field lines 10 and 15 in figure 8. For each time step the free electron speed ( v) and the kinetic energy w = 1 2 m| v| 2 are computed. An ionizing collision is supposed to occur when this kinetic energy is 14eV , which initialize the speed to v = 0 at the current position represented by a small circle in Figure 9. The trajectory ends when it reaches the external resistive layer. The script computes also the trajectory length.

Figure 9 shows that, for the considered geometry and the given voltage, the free electron trajectories (black lines) are very close to the corresponding field lines (red lines). It is not exactly the case for a lower voltage (green lines) estimated to 1/3 of the previous voltage. The first voltage corresponds to the PDIV at sea-level pressure; the second one correspond to the PDIV at an airliner flying altitude. The distance between two successive ionizing collisions is larger because of the lower Coulomb accelerating force. The trajectory and field line lengths are compared in table II. For starting points situated at the beginning of a longer field line, the curves are similar but the acceleration is weaker; the distance between two ionizing collisions is larger and the difference between the field line length and the trajectory one is a bit larger. Accepting an error under 10%, the field line lines length can be used instead successive free electrons trajectories one for estimating the threshold voltage of an electronic avalanche in the specific connection zones.

Figure 10 presents the Paschen's curves for the 3 altitudes of Table I and the FE simulation results. The horizontal axis is the length of the field lines in the external air of the specific connection zone. Each red + sign is associated with the field line number defined in figure 8. The FE simulation is made with a constant permittivity for each material: the electric field modulus at any point of the field map is proportional to the input voltage imposed to the copper wire. The field line positions and lengths (x-axis) remain the same whatever the input voltage. The potential difference between the ends of each field line (y-axis) is proportional to the input voltage. The red + signs move up and down with the input voltage. In Figure 10, the red + signs are drawn for an input voltage tuned to 650 V peak . The voltage has been tuned for getting red points slightly under the Paschen's curve obtained for 5000 m. The very first electronic avalanches creating PDs appear along field lines 14-16, which length are about 30 µm. At 10000 m and for the same voltage, the red + signs are much over the Paschen's curve; many PDs exist. At sea-level pressure, the voltage can be higher (1300 V peak ) for getting + signs on the Paschen's curve. PDs appear along shorter lines 17 -18.

V. EXPERIMENTAL VERIFICATION OF THE CONCEPT

VALIDITY

The experimental verifications were made on twisted pairs following the IEC 60172 standard [33] and placed in the frame of figure 11 for their immobilization. Samples 1,3,5 are made with a standard enameled wire; samples 2,4,6 are made with the same enameled wire covered with a black resistive paint that resistivity is 2750 µΩ.m. The samples are handmade and the actual paint thickness is not really known, but the experience is only aimed at verifying the principle. The PDIV and the PDs charges are measured using a wide spread commercial equipment. The PDIV is measured at 50Hz following the IEC 60270 standard [START_REF]High-voltage test techniques[END_REF], with a threshold of 5 pC at room temperature and pressure. The average value is 684 V RM S for the specimens made with the standard enameled wire and 772 V RM S for the wires covered with the resistive paint.

The twisted pairs of the frame were connected in parallel and fed by a voltage source at 50 Hz tuned at the double of the PDIV measured on standard samples for observing PDs. The frame was placed vertically in a dark cabinet and observed with a night vision camera called "Coronafinder" [START_REF] Syntronics | UV direct view system coronafinder[END_REF]. This system makes a visible image from UV radiations of PDs. Figure 12 presents the picture of samples 1-4. It can be seen that PDs exist along the body of the standard 1 and 3 and only at the ends of the resistive layers for samples 2 and 4. A similar experience has been made with small coils with classical wire of 0.85 mm diameter. Each coils in wound with two wires in hand and 20 turns. One of them is wound using wires covered with the same resistive paint. The same test bench has been used for electrical measurements and optical monitoring. The PDIVs are similar as those determined for the twisted pairs. For an applied voltage of 1500 V RM S the PDs UV emission are illustrated on figure 13 for both coils. PDs appear along the wire for the classical coil and only on the ends of the conducting paint layer for the second coil. The measured PDs apparent charge is 40 pC for the first coil and 25 pC for the second one. A. Reinforcement of the specific zone where PDs may appear For getting higher PDIV, it is possible to add a varnish drop at the end on the external conducting layer. A 2D FE simulation explains this effect considering an example of varnish drop represented in 2D by a 40 µm radius circle. The varnish relative permittivity is supposed to be the same as the polymer; the mesh remains the same. Figure 14 presents the field lines in air and in the polymer drop for the same starting points as in figure 10. It can be seen that the relative permittivity of the polymer drop extends the field lines.

Figure 15 presents Paschen's curves at 3 altitudes. The red circles correspond to the field lines in air drawn in figure 14 (length and potential difference between their ends). It can be seen that the electronic avalanche initiating PDs, at an altitude of 10000 m, is obtained for the longest field lines 1-3 corresponding to higher potential differences between their ends.

The PDIV depends on the polymer drop geometry: for a bigger polymer drop, the red circles are shifted down. The contact with Paschen's curve is obtained for longer field lines that corresponds to higher potential differences between their ends.

VI. CONCLUSION

A new approach for designing the windings of compact PMSM fed by PWM inverters, free of partial discharges at high altitudes is proposed. The method consists of adding a thin resistive layer on the outer surface of the enameled wire used to built the first coil of each phase. In a random coil, the multiple contacts between the external resistive layers reduce strongly the electrical field in the residual air-voids existing between wires. With the external resistive layer, PDs appear in specific zones situated near the wire connections. It is possible to reduce strongly the PDs activity in these zones by adding small quantities of varnish at the end of the resistive layer.

After studying the environment of electrical machines embedded in aircrafts, the influence of pressure and temperature on the PDIV is explained. Then, the influence of the outer resistive layer of the proposed new enameled wire is explained in detail, considering a compact motor design. The proposed analysis is made with a HF equivalent circuit and a Spice processor that reproduces the fast transient voltage measured on a motor fed by a PWM inverter. This analysis shows that the contact points number between wires ant their resistances have a major influence on the voltage stress in the residual air voids between wires. A very pessimist case of the random winding is considered. It shows that a reasonable resistivity of 20.10 -3 Ω.m (more than 100, 000 times the copper one) brings interesting results with a voltage spikes under the Paschen's minimum. This external resistive layer concentrates the electric field in the polymer of the enameled wire. Consequently, PDs appears in very small zones near the connections.

FE simulations of such a specific zone show that the classical Paschen's law can be used for estimating the PDIV with an error under 10%. They show also that the PDIV can be increased by adding a polymer drop in the small zones near the connections.

Experimental verifications using an wire covered with a resistive paint, have been made with twisted pairs and random coils. They confirm the validity of the concept. The electrical properties of the resistive layer are now defined. Works in progress concerns the industrial development of the new enameled wire compatible with the thermal and mechanical requirement of compact motor design.

1

 1 Figure 1 shows the Paschen's curves for the 3 temperatures of table I. The horizontal axis is the pd product with standard units. For instance, the vertical dotted line at pd = 0.1 hP a.m corresponds to an air-void thickness d ≈ 100 µm and a sealevel pressure (p = 1013 hP a). For this case, the Partial Discharge Inception Voltage (PDIV) is about 850 V at 15 o C. It slightly increases at low temperatures.

Fig. 1 .

 1 Fig. 1. Paschen's threshold voltage in air versus distance d between flat parallel electrodes for the 3 temperatures of table I.

Fig. 2 .

 2 Fig. 2. Microscopy of the cross section of a twisted pair aged by a strong PD activity where the polymer layer abrasion is visible.

Fig. 3 .

 3 Fig. 3. Enameled wire with its external resistive layer and transversal section of two adjacent wires.

Fig. 4 .

 4 Fig.[START_REF] Bell | Will your motor insulation survive a new adjustable-frequency drive?[END_REF]. Equivalent HF circuit able to estimate the voltage stress in the air between the wires with a SPICE processor.

Fig. 5 .

 5 Fig. 5. Voltage between several points of the adjacent wire resistive layers.

Fig. 6 .

 6 Fig.[START_REF] Banda | Simulation of space charge impact on partial discharge inception voltage in power busbars dedicated to future hybrid aircrafts[END_REF]. Longitudinal section of the end-connection of the enameled wire with an external resistive layer. A specific zone with a high electric field appears.

Fig. 7 .

 7 Fig. 7. 2D mesh of the air and polymer in the specific zones at the end of the external conducting layer.

Fig. 8 .

 8 Fig. 8. Field lines in the air of the connection zone.

Fig. 9 .

 9 Fig. 9. Field lines in the air of the connection zone.

Fig. 10 .

 10 Fig. 10. FE results reported on the Paschen's curves.

Fig. 11 .

 11 Fig. 11. Preparation of twisted pairs in a frame for their immobilization.

Fig. 12 .

 12 Fig. 12. Comparison of PDs activity. For samples made with a wire covered of conducting paint, the center of twisted pairs is free of PD; PDs appear only on the ends of the resistive layers.

Fig. 13 .

 13 Fig. 13. Standard coil with a high voltage stress (left) and identical coil made with an enameled wire covered with a resistive paint (right).

Fig. 14 .

 14 Fig. 14. Field lines in the reinforced connection zone.

Fig. 15 .

 15 Fig. 15. FE results reported on the Paschen's curves.

  gives 3 typical altitudes and the corresponding temperature and pressures in open and closed air-voids.

TABLE II COMPARISON

 II OF TRAJECTORIES AND FIELD LINES LENGTH.

	Field line	Field line	Tr. length	Diff.	Tr. length	Diff.
	Nb.	length	for 625 V		for 210 V
		(µm)	(µm)	(%)	(µm)	(%)
	10	84.3	84.8	0.6	88	4.3
	15	40	40	0	41	2.5