Ying Tang
email: ying.tang@gipsa-lab.fr

Christophe Prieur
email: christophe.prieur@gipsa-lab.fr

Antoine Girard
email: antoine.girard@imag.fr.

Approximation of singularly perturbed linear hyperbolic systems

This paper is concerned with systems modelled by linear singularly perturbed partial differential equations. More precisely a class of linear systems of conservation laws with a small perturbation parameter is investigated. By setting the perturbation parameter to zero, the full system leads to two subsystems, the reduced system standing for the slow dynamics and the boundary-layer system representing the fast dynamics. The exponential stability for both subsystems are obtained by the stability of the overall system of conservation laws. However, the stability of the two subsystems does not imply the stability of the full system. The approximation of the solution for the overall system by the solution for the reduced system is validated via Lyapunov techniques.

I. INTRODUCTION

The singular perturbation techniques occurred at the beginning of the 20th century. The interests in this method arose from many physical problems exhibiting both fast and slow dynamics. For example, DC-motor model and Voltage regulator in [START_REF] Kokotović | Singular pertrubation methods in control: analysis and design[END_REF]. From late 1980s, the singularly perturbed partial differential equations (PDEs) have been considered in research works. This kind of systems is interesting for analysis since it describes numerous phenomenon in various fields, for instance, fluid dynamics, chemical-reactor, aerodynamics etc. (see [START_REF] Kadalbajoo | Singularly perturbed problems in partial differential equations: a survey[END_REF]).

The model of fluid transport through a constant cross section tube from [START_REF] Castillo | Contrôle de Température dans un Flux de Poiseuille[END_REF] provides the first motivation for this work. This model contains two time scales for propagation speed, which can be described by a singularly perturbed system of conservation laws. The decomposition of a singularly perturbed system into lower order subsystems, namely the reduced system and the boundary-layer system, provides a powerful tool for stability analysis in [START_REF] Habets | Stabilité asymptotique pour des problèmes de perturbations singulières[END_REF], [START_REF] Chow | Asymptotic stability of a class of nonlinear singularly perturbed systems[END_REF], [START_REF] Grujic | Uniform asymptotic stability of nonlinear singularly perturbed and large scale systems[END_REF] and [START_REF] Chow | A two-stage Lyapunov-Bellman feedback design of a class of nonlinear systems[END_REF].

In this paper, we consider a class of linear systems of conservation laws with a small perturbation parameter ǫ. By setting ǫ = 0, two subsystems, the reduced and boundarylayer systems, are computed. The exponential stability for singularly perturbed system of conservation laws implies both subsystems are exponentially stable. On the other hand the converse does not hold. The stability analysis for hyperbolic systems of conservation laws has been considered by many researchers. For instance, a stability criterion for linear hyperbolic systems by characteristics method has been given in [START_REF] Li | Global classical solutions for quasilinear hyperbolic systems[END_REF] and the stability condition considered in [START_REF] Hale | Introduction to Functional-Differential Equations[END_REF] relies on the frequency domain. In [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF], stability condition for the quasilinear systems of conservation laws is introduced by Lyapunov method. In this paper, it will be studied a Tikhonov like theorem for linear hyperbolic systems, the solutions of the full system can be approximated by that of the reduced system. The Tikhonov Theorem has been studied in many works for standard singular perturbation systems (ODEs) (e.g. [START_REF] Buckdahn | A stochastic Tikhonov theorem in infinite dimensions[END_REF], [START_REF] Verhulst | Singular perturbation methods for slow-fast dynamics[END_REF]). [START_REF] Khalil | Nonliear systems[END_REF] shows the Tikhonov theorem based on the exponential stability criterion of both the reduced and boundary-layer systems. Moreover, to the best of our knowledge, this is the first paper dealing with singularly perturbed hyperbolic systems.

The paper is organized as follows. Section II introduces the linear singularly perturbed system of conservation laws. In Section III, the stability of both subsystems is presented. Precisely, the exponential stability of the full system of conservation laws implies that each of the two subsystems is stable. However, a counter-example is given to show that the stability of the two subsystems does not guarantee the stability of the full system. Section IV shows the approximation of solutions for the full system by that of the reduced system. In Section V, a numerical example is studied to illustrate the results. Finally, concluding remarks end the paper. Due to space limitation, some proofs are omitted.

Notation. Given a matrix A, A -1 and A T represent the inverse and the transpose matrix of A respectively. For a symmetric matrix B, λ min (B) is the minimum eigenvalue of the matrix B. The symbol ⋆ in partitioned symmetric matrices stands for the symmetric block. For a positive integer n, I n is the identity matrix in R n×n . | | denotes the usual Euclidean norm in R n and is associated to the matrix norm.

L 2 denotes the associate norm in L 2 (0, 1) space, defined by ξ L 2 = 1 0 |ξ| 2 dx 1 2 for all functions ξ ∈ L 2 (0, 1). Similarly, the associate norm in H 2 (0, 1) space is denoted by H 2 , defined for all functions ψ ∈ H 2 (0, 1), by ψ

H 2 = 1 0 |ψ| 2 + |ψ x | 2 + |ψ xx | 2 dx 1 2 .

II. LINEAR SINGULARLY PERTURBED SYSTEMS OF CONSERVATION LAWS

Firstly, let us consider the following linear singularly perturbed system of conservation laws:

y t (x, t) + Λ 1 y x (x, t) = 0, (1a) ǫz t (x, t) + Λ 2 z x (x, t) = 0, (1b)
where

x ∈ [0, 1], t ∈ [0, +∞), y : [0, 1] × [0, +∞) → R n , z : [0, 1] × [0, +∞) → R m , Λ 1 is a positive diagonal matrix in R n×n , Λ 2 is a positive diagonal matrix in R m×m , the
perturbation parameter ǫ is a small positive value. Moreover, we consider the following boundary conditions:

y(0, t) z(0, t) = G y(1, t) z(1, t) , t ∈ [0, +∞), (2)
where

G = G 11 G 12 G 21 G 22 is a constant matrix in R (n+m)×(n+m) with the matrices G 11 in R n×n , G 12 in R n×m , G 21 in R m×n and G 22 in R m×m .
Given two functions y 0 : [0, 1] → R n and z 0 : [0, 1] → R m , the initial conditions are:

y(x, 0) z(x, 0) = y 0 (x) z 0 (x) , x ∈ [0, 1]. (3)
Remark 2.1: Let us recall the existence of the solutions to the Cauchy problem ( 1)-( 3). According to Section 2.1 in [START_REF] Coron | Control and nonlinearity[END_REF], for all y 0 z 0 ∈ L 2 (0, 1), there exists a unique solution y z ∈ C 0 ([0, +∞), L 2 (0, 1)) for the Cauchy problem ( 1)-

. By Proposition 2.1 in [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF], for every y 0 z 0 ∈ H 2 (0, 1) satisfying the following compatibility conditions:

y 0 (0) z 0 (0) = G y 0 (1) z 0 (1) , (4)
Λ 1 y 0 x (0) Λ 2 z 0 x (0) = G Λ 1 y 0 x (1) Λ 2 z 0 x (1)
,

the Cauchy problem ( 1)-( 3) has a unique maximal classical solution y z ∈ C 0 ([0, +∞), H 2 (0, 1)).

•

Adapting the approach in [START_REF] Saberi | Quadratic-type Lyapunov Functions for Singularly Perturbed Systems[END_REF] and [START_REF] Khalil | Nonliear systems[END_REF] to the infinite dimensional case, let us compute the two subsystems, reduced and boundary-layer systems, for system (1)- [START_REF] Castillo | Contrôle de Température dans un Flux de Poiseuille[END_REF]. Setting ǫ = 0 in system (1) yields

y t (x, t) + Λ 1 y x (x, t) = 0, (6a) z x (x, t) = 0. (6b)
Substituting (6b) into the boundary conditions (2) yields

y(0, t) = (G 11 + G 12 (I m -G 22 ) -1 G 21 )y(1, t), z(., t) = (I m -G 22 ) -1 G 21 y(1, t). (7)
Then, the reduced system is defined as

ȳt (x, t) + Λ 1 ȳx (x, t) = 0, x ∈ [0, 1], t ∈ [0, +∞), (8)
with the boundary condition

ȳ(0, t) = G r ȳ(1, t), t ∈ [0, +∞), (9)
where G r = G 11 +G 12 (I m -G 22 ) -1 G 21 , whereas the initial condition is given as

ȳ(x, 0) = y 0 (x), x ∈ [0, 1]. (10)
To define the boundary-layer system, let first perform the change of variable

z = z -(I m -G 22 ) -1 G 21 y(1, t). (11)
This shifts the equilibrium of z to the origin. Let us use a new time variable τ = t ǫ . In the τ time scale, y(1, t) in ( 11) is considered as a fixed parameter with respect to time. Then, the boundary-layer system is defined as

zτ (x, τ ) + Λ 2 zx (x, τ ) = 0, x ∈ [0, 1], τ ∈ [0, +∞), (12)
with the boundary condition:

z(0, τ ) = G 22 z(1, τ ), τ ∈ [0, +∞), (13)
whereas the initial condition is given as

z(x, 0) = z 0 (x) -(I m -G 22 ) -1 G 21 y 0 (1), x ∈ [0, 1]. (14)

III. STABILITY OF REDUCED AND BOUNDARY-LAYER

SYSTEMS

In this section, we will show how the stability of the singularly perturbed system of conservation laws (1)-( 2) is related to the stability of the two subsystems, the reduced system ( 8)-( 9) and the boundary-layer system ( 12)- [START_REF] Kadalbajoo | Singularly perturbed problems in partial differential equations: a survey[END_REF].

Let us recall the following definition introduced in [6]:

Definition 3.1: For all matrices G ∈ R (n+m)×(n+m) , ρ 1 (G) = inf{ ∆G∆ -1 , ∆ ∈ D (n+m),+ }, ( 15
)
where D (n+m),+ denotes the set of diagonal positive matrix in R (n+m)×(n+m) .

The following definition is adopted for the exponential stability of the linear singularly perturbed system of conservation laws (1)-( 2) in L 2 -norm.

Definition 3.2: The linear system of conservation laws (1)-( 2) is exponentially stable to the origin in L 2 -norm if there exist γ 1 > 0 and C 1 > 0, such that for every y 0 z 0 ∈ L 2 (0, 1), the solution to the system (1)-( 2) satisfies

y(., t) z(., t) L 2 C 1 e -γ1t y 0 z 0 L 2 , t ∈ [0, +∞).
Similarly the exponential stability of the linear system of conservation laws (1)-( 2) in H 2 -norm is defined as follows Definition 3.3: The linear system of conservation laws (1)-( 2) is exponentially stable to the origin in H 2 -norm if there exist γ 2 > 0 and C 2 > 0, such that for every y 0 z 0 ∈ H 2 (0, 1) satisfying the compatibility conditions ( 4)-( 5), the solution to the system (1)-( 2) satisfies

y(., t) z(., t) H 2 C 2 e -γ2t y 0 z 0 H 2 , t ∈ [0, +∞).
In a similar way, we can define the exponential stability in L 2 -norm and H 2 -norm for the reduced and boundary-layer systems.

Let recall the following result for quasilinear hyperbolic system: Theorem 1 ([6] and [START_REF] Diagne | Lyapunov exponential stability of 1-D liear hyperbolic systems of balance laws[END_REF]):

If ρ 1 (G) < 1 (resp. ρ 1 (G r ) < 1, ρ 1 (G 22 ) < 1)
, then the linear system (1)-( 2) (resp. the reduced system ( 8)-( 9), the boundary-layer system ( 12)-( 13)) is exponentially stable to the origin in L 2 -norm and H 2norm.

With the above theorem, we are ready to give a proposition which is about the stability of the reduced and the boundarylayer systems.

Proposition 3.4: If ρ 1 (G) < 1, then the reduced system ( 8)-( 9) and the boundary-layer system ( 12)-( 13) are exponentially stable to the origin in L 2 -norm and H 2 -norm.

The stability criterion ρ 1 (G) < 1 is thus a sufficient condition for stability of the reduced system (8)-( 9) and the boundary-layer system ( 12)-( 13). On the contrary, the stability of the two subsystems does not guarantee the stability of the overall system (1)-(2). To see this, let us consider the following counter-example.

Counter-example:

Let Λ 1 = Λ 2 = 1 in (1) with n = m = 1.
The boundary condition of the singularly perturbed system in ( 2) is chosen as G 11 = 2.5, G 12 = -1, G 21 = 1, G 22 = 0.5. The boundary condition of the reduced system in ( 9) is computed as G r = 0.5. Considering ∆ = 1 and (15), it holds ρ 1 (G r ) < 1. By Theorem 1 the reduced system ( 8)-( 9) is exponentially stable in L 2 -norm and H 2 -norm. The boundary condition of the boundary-layer system in ( 13) is G 22 = 0.5. Considering the same ∆ = 1 and ( 15), it holds ρ 1 (G 22 ) < 1. The boundary-layer system ( 12)-( 13) is exponentially stable in L 2 -norm and H 2 -norm according to Theorem 1. Now let us check the stability condition ρ 1 (G) < 1, which is equivalent to find a diagonal positive matrix ∆ such that the following condition is satisfied

G T ∆ 2 G < ∆ 2 . ( 16
)
There is no loss of generality to consider ∆ = 1 0 0 b .

Straightforward computation shows that there is no such matrix ∆ which satisfies the condition ( 16), thus ρ 1 (G) 1.

Note that, Proposition 3.7 in [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF] implies that ρ 1 (G) < 1 is a necessary and sufficient condition for stability of linear hyperbolic systems with dimension 1 to 5. As this example is a linear singularly perturbed system of two conservation laws, it is not exponentially stable neither in L 2 -norm nor in H 2norm, although the reduced and boundary-layer systems are both exponentially stable. •

IV. APPROXIMATION THEOREM FOR LINEAR SINGULARLY PERTURBED SYSTEM OF CONSERVATION LAWS

In this section, we present how solutions to the linear singularly perturbed system of conservation laws ( 1)-( 2) can be approximated by solutions to the reduced system (8)- [START_REF] Santos | Boundary control of open channels with numerical and expermental validations[END_REF]. It is based on the stability condition we considered in the previous section.

Theorem 2: Consider the linear system of conservation laws (1)- [START_REF] Castillo | Contrôle de Température dans un Flux de Poiseuille[END_REF]. Assume that the boundary conditions G satisfy ρ 1 (G) < 1, then, for all initial conditions y ∈ H 2 (0, 1) satisfying the compatibility conditions y 0 (0) = G r y 0 (1), Λ 1 y 0

x (0) = G r Λ 1 y 0 x (1), and z ∈ L 2 (0, 1), there exist positive values ǫ * , C, C

′ and ω such that for all 0 < ǫ < ǫ * and for all t 0, y(., t) -ȳ(., t) 2

L 2 Cǫe -ωt , (17)
∞ 0 z(., t) -(I m -G 22 ) -1 G 21 ȳ(1, t) 2 L 2 dt C ′ ǫ. ( 18
)
Sketch of proof: First, let us perform the change of variables: η(x, t) = y(x, t) -ȳ(x, t), δ(x, t) = z(x, t) -

(I m -G 22 ) -1 G 21 ȳ(1, t).
In the new variables (η, δ), the system is written as

η t + Λ 1 η x = 0, (19a) ǫδ t + Λ 2 δ x = ǫ(I m -G 22 ) -1 G 21 Λ 1 ȳx (1, t), (19b)
with the boundary conditions

η(0, t) δ(0, t) = G η(1, t) δ(1, t) . (20)
We consider the following Lyapunov function candidate for system ( 19)-( 20)

V ǫ (η, δ) = 1 0 e -µx (η T Qη + ǫδ T P δ)dx, (21)
with µ > 0, Q a diagonal positive matrix in R n×n and P a diagonal positive matrix in R m×m . After computing the time derivative of V ǫ (η, δ) along ( 19)-( 20) and integrating by parts, we obtain

Vǫ (η, δ) BC -αV ǫ (η, δ) + ǫ 2 β|ȳ x (1, t)| 2 , ( 22
)
where BC is the boundary term and α, β are positive constants. Since ρ 1 (G) < 1, there exists a positive diagonal matrix ∆ such that ∆G∆ -1 < 1, thus the boundary term BC is always non positive. Due to Proposition 3.4, ρ 1 (G) < 1 implies the reduced system (8)-( 9) is exponentially stable in H 2 -norm. It is deduced from (22)

V ǫ (η, δ) e -αt V ǫ (η 0 , δ 0 ) + ǫ 2 γe -αt ȳ0 2 H 2 , ( 23
)
where γ is a positive constant value. Note that there exist positive values a, b, a ′ and b ′ such that

a η 2 L 2 + ǫb δ 2 L 2 V ǫ (η, δ) a ′ η 2 L 2 + ǫb ′ δ 2 L 2 . ( 24
)
Choosing the initial condition y 0 = ȳ0 (i.e. η 0 = 0) yields η(., t) 2

L 2 ǫC 1 e -αt δ 0 2 L 2 + ǫ 2 C 2 e -αt ȳ0 2 H 2 , (25)
C 1 and C 2 are given positive values. Thus ( 17) is proved. Next, we rewrite (22) as follows

Vǫ (η, δ) -ρ δ(., t) 2 L 2 + ǫ 2 ι ȳ0 2 H 2 , (26)
for suitable positive values ρ and ι. Performing the time integration of both sides of (26), using lim t→+∞ V ǫ (η, δ) = 0, according to (24) and choosing initial condition y 0 = ȳ0 , we get

∞ 0 δ(., t) 2 L 2 dt ǫC 3 δ 0 2 L 2 + ǫ 2 C 4 ȳ0 2 H 2 , (27)
where C 3 and C 4 are given positive constants. Thus [START_REF] Shampine | Solving hyperbolic PDEs in Matlab[END_REF] holds.

V. NUMERICAL EXAMPLE

In this section, we use a numerical example to illustrate the results that we get in the previous sections.

Let us consider a singularly perturbed system of two conservations laws (1) with Λ 1 = Λ 2 = 1, and the boundary conditions G = 0.5 0.5 -0.5 -0.2 in (2).

Use a two-step variant of the Lax-Wendroff method which is presented in [START_REF] Shampine | Two-step Lax-Friedrichs method[END_REF] and the solver on Matlab in [START_REF] Shampine | Solving hyperbolic PDEs in Matlab[END_REF] to discretize the system. More precisely, we divide the space domain [0, 1] into 100 intervals of identical length, and 10 as final time. We choose a time-step that satisfies the CFL condition for the stability and select the following initial functions:

y(x, 0) = 1 -cos(6πx), z(x, 0) = sin(5πx),
for all x ∈ [0, 1]. Choosing ǫ = 0.003, Figure 1 shows that the solution of the reduced system ȳ converges to the origin as time increases which as expected from Proposition 3.4. Figures 2a and2b give the time evolutions of η and δ which are the error between y of the full system and ȳ of the reduced system, and the error between z and its equilibrium respectively. They decrease to 0 as time increases.

VI. CONCLUSION

In this paper, a linear singularly perturbed system of conservation laws has been studied. The stability condition for the whole singularly perturbed system ρ 1 (G) < 1 implies the two subsystems are exponentially stable. However, a counter-example has been given to show that the stability of the two subsystems does not guarantee the stability of the full system. A Tikhonov like theorem has been given under the stability condition ρ 1 (G) < 1. The solution of the linear singularly perturbed system of conservation laws can be approximated by the solution of the reduced system.

This work leaves many open questions. It is natural to extend this work to systems with source terms. Another interesting point is to consider some physical applications, like open channels as considered in [START_REF] Santos | Boundary control of open channels with numerical and expermental validations[END_REF] and gas flow through pipelines in [START_REF] Dick | Classical solutions and feedback stabilization for the gas flow in a sequence of pipes[END_REF] or [START_REF] Castillo | Contrôle de Température dans un Flux de Poiseuille[END_REF].

Fig. 1 :Fig. 2 :

 12 Fig. 1: Time evolution of the solution ȳ of the reduced system

TABLE I :

 I Estimations of η 2

		L 2 and	10 0 ||δ|| 2 L 2 dt for different ǫ
	ǫ	0.001	0.002	0.003
	||η(., t = 1)|| 2 L 2 10 0 ||δ|| 2 L 2 dt	4.7 × 10 -6 6.9 × 10 -6	1.8 × 10 -5 2.8 × 10 -5	4.2 × 10 -5 6.2 × 10 -5

  Table I gives estimations of square of L 2 -norm of η and of the time integral of square of L 2 -norm of δ with the different values of ǫ. It indicates that these two values are near to zero and decrease as ǫ decreases, as expected from Theorem 2.


Mike Boyle

J Ér

AND Ôme Buzzi
email: jerome.buzzi@math.u-psud.fr

Ricardo G Ómez
email: rgomez@math.unam.mx

BOREL ISOMORPHISM OF SPR MARKOV SHIFTS

We show that strongly positively recurrent Markov shifts (including shifts of finite type) are classified up to Borel conjugacy by their entropy, period and their numbers of periodic points.

Introduction

Theorem 1.1 below is one of the results in the "full sets" paper of Hochman [START_REF] Hochman | Isomorphism and embedding of Borel systems on full sets[END_REF]. In the statement, 'Markov shift' means countable state Markov shift. The free part of a Borel system is the subsystem obtained by restriction to the nonperiodic points, and a full subset is an invariant subset of measure one for every invariant Borel probability measure. Two Borel systems are almost-Borel isomorphic if they are Borel isomorphic after restriction to full subsets of their free parts. Detailed definitions for the Introduction are given in the next section.

Theorem 1.1. [START_REF] Hochman | Isomorphism and embedding of Borel systems on full sets[END_REF] Two mixing Markov shifts are almost-Borel isomorphic if and only if [START_REF] Boyle | Lower entropy factors of sofic systems[END_REF] they have equal entropy and [START_REF] Boyle | The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors[END_REF] one has a measure of maximum entropy if and only the other does.

An important observation [START_REF] Hochman | Isomorphism and embedding of Borel systems on full sets[END_REF] in this setting is that two Borel systems that embed each into the other are Borel isomorphic, by a Borel variant of Cantor-Bernstein Theorem (a.k.a. the measurable Schröder-Bernstein Theorem). Consequently Theorem 1.1 was an immediate corollary of the following embedding theorem. Theorem 1.2. [START_REF] Hochman | Isomorphism and embedding of Borel systems on full sets[END_REF] Suppose (Y, T ) is a mixing Markov shift and (X, S) is a Borel system such that h(S, µ) < h(T ) for every ergodic invariant Borel probability µ on X. Then there is an almost-Borel embedding of (X, S) into (Y, T ). This theorem easily leads to a decisive almost-Borel classification of Markov shifts, and has implications for other systems [START_REF] Hochman | Isomorphism and embedding of Borel systems on full sets[END_REF][START_REF] Boyle | The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors[END_REF].

The study of Borel dynamics, adopting weakly wandering sets as the relevant notion of negligible sets, was initated by Shelah and Weiss [START_REF] Shelah | Measurable recurrence and quasi-invariant measures[END_REF][START_REF] Weiss | Measurable dynamics[END_REF][START_REF] Weiss | Countable generators in dynamics-universal minimal models[END_REF]. Here that notion of isomorphism preserves additionally the infinite and quasi-invariant measures (and again it is natural to restrict to free parts). Whether there is a theorem for Borel dynamics like Theorem 1.2 is a difficult open problem, discussed in [START_REF] Hochman | Isomorphism and embedding of Borel systems on full sets[END_REF]. Our purpose in this paper is to show that a generalization of Theorem 1.1 to this richer category holds in at least one meaningful case.

Theorem 1.3. The free parts of mixing SPR Markov shifts are Borel isomorphic if and only if they have equal entropy.

We note that Hochman [START_REF] Hochman | Isomorphism and embedding of Borel systems on full sets[END_REF] has asked if those free parts are in fact topologically conjugate, at least in the case of subshifts of finite type.

As in the almost-Borel case, Theorem 1.3 is an immediate corollary of an embedding result, stated next.

Theorem 1.4. Suppose (Y, T ) is a mixing SPR Markov shift and (X, S) is a Markov shift such that h(X) = h(Y ) and X has a unique irreducible component of full entropy and this component is a mixing SPR Markov shift. Then there is a Borel embedding of (X, S) into (Y, T ).

The proof is independent of Hochman's result and techniques. Roughly speaking, Hochman builds almost-Borel embeddings from the bottom up with a uniform version of the Krieger Generator Theorem [START_REF] Krieger | On entropy and generators of measure-preserving transformations[END_REF]. In our much more special situation, we can build Borel embeddings with the following offshoot of the Krieger Embedding Theorem.

Theorem 1.5. Suppose (Y, T ) is a mixing Markov shift and (X, S) is a Markov shift such that h(X) < h(Y ). Then there is a Borel embedding of the free part of (X, S) into (Y, T ). Theorem 1.5, though not completely trivial, is completely unsurprising. (The question of when a Markov shift embeds continuously into a mixing Markov shift is much harder [START_REF] Fiebig | Entropy and finite generators for locally compact subshifts[END_REF][START_REF] Fiebig | Embedding theorems for locally compact Markov shifts[END_REF].) The novel feature in the proof of Theorem 1.4 is the use of a "top-down" embedding given by the almost isomorphism theorem of [START_REF] Boyle | Almost isomorphism for countable state Markov shifts[END_REF] to reduce the problem to embeddings of lower entropy systems.

At the end of the paper we state the Borel classification of the free parts of irreducible SPR Markov shifts, which follows from the mixing case.

Acknowledgments. We thank Mike Hochman for the stimulating discussions out of which this paper emerged. M. Boyle gratefully acknowledges the financial support of ANR project DynNonHyp BLAN08-2 313375 and the hospitality of the Mathematics Department of Université Paris-Sud in Orsay.

Definitions and background

A Borel system (X, X , T ) is a standard Borel space 1 (X, X ) together with a Borel automorphism 2 T : X → X. We often abbreviate (X, X , T ) to (X, T ) or X or T if it does not create confusion. A Borel factor map is a homomorphism of Borel systems: a (not necessarily onto) Borel measurable map intertwining the actions. An isomorphism or conjugacy of Borel systems 1 X is a σ-algebra of subsets of X such that there is distance on X which turns it into a complete separable space whose collection of Borel subsets is X .

2 A bijection such that T -1 X := {T -1 E : E ∈ X } = T X = X .
is a bijective Borel factor map; an embedding of Borel systems is an injective Borel factor map. By an easy exercise in descriptive set theory (see [16, p.399]), there is a Borel conjugacy of two systems if and only if there is a Borel conjugacy between their free parts and for each n the cardinalities of their sets of periodic orbits of size n is the same. Given a Borel system (X, T ), we use P(X) ⊃ P erg (X) ⊃ P ′ erg (X) respectively to denote the sets of all measures3 , all ergodic measures, and all ergodic nonatomic measures. Recall from [START_REF] Weiss | Measurable dynamics[END_REF] that a set W is wandering if it is Borel and if k∈Z T k W is a disjoint union (which we denote k∈Z T k W ). A set is weakly wandering if it is a Borel subset of a countable union of wandering sets. Such a set has measure zero for all quasi-invariant measures [START_REF] Shelah | Measurable recurrence and quasi-invariant measures[END_REF][START_REF] Weiss | Measurable dynamics[END_REF], not only for measures in P(X). To avoid any mystery, we record a simple remark.

Remark 2.1. Suppose (X, S) and (Y, T ) are Borel systems and each contains an uncountable Borel set which is wandering. Then the systems are Borel isomorphic if and only if they are Borel isomorphic modulo wandering sets.

The basis of the remark is the following. Any weakly wandering set is contained in the orbit of a wandering set. Under the assumption, such wandering sets in X and Y can be enlarged to uncountable Borel subsets of the ambient Polish space. Any two such sets are Borel isomorphic.

A Markov shift (X, S) is a topological system Σ(G) defined by the action of the left shift σ : (x n ) n∈Z → (x n+1 ) n∈Z on the set Σ(G) of paths on some oriented graph G with countably (possibly finitely) many vertices and edges. We will use the edge shift (rather than the vertex shift) presentation. The domain X is the set of x = (x n ) n∈Z ∈ E Z (where E is the set of oriented edges) such that for all n, the terminal vertex of x n equals the initial vertex of x n+1 . The (Polish) topology on X is the relative topology of the product of the discrete topologies. When G is finite, Σ(G) is a shift of finite type (SFT). Σ(G) is irreducible if G contains a unique strongly connected component, i.e., a maximal set of the vertices such that for any pair, there is a loop containing both. An arbitrary Markov shift is the disjoint union of a wandering set and countably many disjoint irreducible Markov shifts. An irreducible Markov shift is mixing if and only if the g.c.d. of the periods of its periodic points is 1.

The Borel entropy of a system (X, S) is the supremum of the Kolmogorov-Sinai entropies h(S, µ), µ ∈ P(X). Markov shifts of positive entropy contain uncountable wandering sets; so, by the Remark 2.1, for simplicity we can neglect weakly wandering sets in both statements and proofs. An irreducible Markov shift (X, S) (more generally, an irreducible component) has at most one measure of maximum (necessarily finite) entropy [START_REF] Gurevič | Shift entropy and Markov measures in the space of paths of a countable graph[END_REF]; if this measure µ exists, then (S, µ) is measure-preservingly isomorphic to the product of a finite entropy Bernoulli shift and a finite cyclic rotation (see [START_REF] Boyle | The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors[END_REF] for comment and references).

An irreducible Markov shift Σ is strongly positively recurrent (or stably positive recurrent or just SPR) if it admits a measure µ of maximal entropy which is exponentially recurrent: for every non-empty open subset

U ⊂ Σ, lim sup n→∞ 1 n log µ Σ \ n-1 k=0 σ -k U < 0 .
We refer to [START_REF] Boyle | Almost isomorphism for countable state Markov shifts[END_REF][START_REF] Gurevich | Stably recurrent nonnegative matrices[END_REF][START_REF] Gurevich | Thermodynamic formalism for symbolic Markov chains with a countable number of states[END_REF] for more on SPR shifts. In the language of [START_REF] Gurevich | Stably recurrent nonnegative matrices[END_REF][START_REF] Gurevich | Thermodynamic formalism for symbolic Markov chains with a countable number of states[END_REF], the SPR Markov shifts are the positively recurrent symbolic Markov chains defined by stably recurrent matrices (further developed in [START_REF] Gurevich | Thermodynamic formalism for symbolic Markov chains with a countable number of states[END_REF] as the fundamental class of "stably positive" matrices). The SPR Markov shifts are a natural subclass preserving some of the significant properties of finite state shifts [3, Sec.2].

Embedding a Markov with smaller entropy

In this section we will prove Theorem 1.5. First we recall and adapt some standard finite-state symbolic dynamics (for more detail on this, see [START_REF] Boyle | Lower entropy factors of sofic systems[END_REF] or [START_REF] Lind | An Introduction to Symbolic Dynamics and Coding[END_REF]). Lemma 3.1. Suppose ǫ > 0 and X is a mixing Markov shift with entropy h(X) > 0. Then X contains infinitely many mixing SFTs S n , pairwise disjoint, such that h(S n ) > h(X) -ǫ for all n.

Proof. X contains an SFT S with entropy greater than h(X) -ǫ [START_REF] Gurevič | Shift entropy and Markov measures in the space of paths of a countable graph[END_REF]; S is easily enlarged to a mixing SFT S ′ in X. The complement of a given proper subshift of S ′ contains a mixing SFT with entropy arbitrarily close to h(S ′ ) [4, Lemma 26.17]. Thus one can construct the required family inductively. Definition 3.2. For a system (X, S), |P o n (X)| denotes the cardinality of the set of points in S-orbits of length n. Theorem 3.3 (Krieger Embedding Theorem [START_REF] Krieger | On the subsystems of topological Markov chains[END_REF]). Let X be a subshift on a finite alphabet and Y a mixing SFT such that h(X) < h(Y ) and

|P o n (X)| ≤ |P o n (Y )| for all n.
Then there is a continuous embedding of X into Y . Proposition 3.4. [1, Lemma 2.1 and p.546] Suppose X is a mixing SFT and M is a positive integer. Let O 1 , . . . , O r be distinct finite orbits in X. Let W i be the set of points whose positive iterates are positively asymptotic to O i , and let W = ∪ i W i . Then there exist a mixing SFT Z and a continuous surjection p : Z → X such that:

(1)

|p -1 (x)| = 1 for all x outside W (2) The preimage of O i is an orbit O i of length M |O i |.
(3) p -1 (W i ) is the set of points positively asymptotic to O i .

Corollary 3.5. Let X and Y be SFTs such that h(X) < h(Y ) and Y is mixing. Then there is a continuous embedding of X \ X 0 into Y where X 0 is the union of a weakly wandering set and a finite set of periodic points.

Proof. We have that lim

n ( |P o n (Y )| -|P o n (X)| ) = ∞.
Thus we may choose M to build Z as in Proposition 3.4 such that Z, by Theorem 3.3, embeds into Y . The map Z → X is a Borel isomorphism on the complement of a set X 0 of points positively asymptotic to finitely many periodic points.

To reduce Theorem 1.5 to this corollary, we use reductions stated as three lemmas. A loop system is a Markov shift defined by a loop graph: a graph made of simple loops which are based at a common vertex and otherwise do not intersect. Given a power series f = ∞ n=1 f n z n with coefficients in Z + , we let Σ f denote the loop system with exactly f n simple loops of length n in the loop graph. If

h(Σ f ) = log λ < ∞, then (1) 0 < f (1/λ) ≤ 1, (2) α < λ =⇒ f (1/α) = ∞ and (3) f (1/λ) = 1 if Σ f
has a measure of maximum entropy (i.e. is positive recurrent). For more on loop systems and Markov shifts, see [START_REF] Boyle | Almost isomorphism for countable state Markov shifts[END_REF][START_REF] Gurevich | Thermodynamic formalism for symbolic Markov chains with a countable number of states[END_REF][START_REF] Bruce | Symbolic Dynamics. Universitext[END_REF] and their references.

Lemma 3.6. Any Markov shift X is Borel isomorphic to a Borel system

W ⊔ n∈N Σ(L n )
where W is weakly wandering and for each n, L n is a loop graph. Lemma 3.7. Let Σ be a loop system and h > h(Σ). Then there is a SFT S with h(S) < h such that Σ has a continuous embedding into S.

Before proving the lemmas, we deduce the lower-entropy embedding theorem from them.

Proof of Theorem 1.5. According to Remark 2.1 and Lemma 3.6, we may assume that X is a disjoint union of loop systems Σ(L n ). Let h = (h(Y ) + h(X))/2 > h(X). By Lemma 3.7, each loop system Σ(L n ) can be (continuously) embedded into some SFT W n with entropy less than h. Let

ǫ = h(Y ) -h > 0. By Lemma 3.1 (with ǫ = (h(Y ) -h))/2)), there are pairwise disjoint mixing SFTs Y n in Y ′ with h(Y n ) > h.
Finally, Corollary 3.5 shows that each W n (apart from finitely many periodic points) can be Borel embedded into Y n ⊂ Y . Altogether, apart from a countable set of periodic points, X has been Borel embedded into Y .

We now prove the lemmas.

Proof of Lemma 3.6. Let G be some graph presenting X. For convenience, we identify its vertices with 1, 2, . . . . Observe that each W ǫ n := {x ∈ X : x 0 = n and ∀i > 0 x ǫi = n} (n ∈ N * , ǫ ∈ {-1, +1}) is wandering. Consider the loop graphs L n defined by the first return loops of G at vertex n which avoid the vertices k < n.

For each x ∈ X, let

N := inf{n ≥ 1 : ∃a k , b k → ∞ x -a k = x b k =
n} and consider the following three cases.

(1) N = ∞. Then there exists ǫ ∈ {-1, +1} such that x ∈ σ -j W ǫ x 0 , where j := ǫ sup{ǫi ∈ Z :

x i = x 0 } ∈ Z. (2) N < ∞ and {x m : m ∈ Z} ∩ [1, N ) = ∅. Then there exist k ∈ [1, N )
and ǫ ∈ {-1, +1} such that j := ǫ sup{ǫi ∈ Z :

x i = k} ∈ Z, so x ∈ σ -j W ǫ k . (3) Otherwise, x ∈ Σ(L N ).
To conclude, observe that k∈N * ,j∈Z,ǫ∈{-1,+1} σ -j W ǫ k is a weakly wandering set.

Proof of Lemma 3.7. Let Σ = Σ f , a loop system described by a power series f = ∞ n=1 f n z n . If f is a polynomial, then Σ f is itself an SFT. From now on, we assume f to have infinitely many non-zero terms.

We are going to build the SFT as a finite loop system Σ p , with a polynomial p obtained by truncating the power series f and then adding some monomials to ensure enough space for the embedding while keeping the entropy < h.

Let β ∈ (h(Σ), h). Given a positive integer N , let f (N ) denote the truncation of f to the polynomial

f 1 z + f 2 z 2 + • • • + f N z N . As f (e -h(Σ) ) ≤ 1 and h(Σ) < β we have f n < e nβ for all n ≥ 1. Let g <N > denote the polyno- mial g N +1 z N +1 + g N +2 z N +2 + • • • + g 2N z 2N
, where g n = ⌈e nβ ⌉ (the integer ceiling). Then

|g <N > (z)| ≤ (e (N +1)β + 1) + • • • + (e 2N β + 1)|z| N -1 |z| N +1 = e (N +1)β |z| N +1 1 -(e β |z|) N 1 -e β |z| + |z| N +1 1 -|z| N 1 -|z| .
As β > 0, we see that lim N →∞ g <N > (z) = 0 uniformly for |z| fixed, smaller than e -β . Recall that f (r) < 1 for r < e -h(Σ) . Also if r > 0 and |z| = r and

f (N ) (r) < f (r) < 1, then |1 -f (N ) (z)| ≥ 1 -f (N ) (r) > 1 -f (r) > 0.
Fix some γ ∈ (β, h) and then N sufficiently large that the following hold:

(1)

|2g <N > (z)| < 1 -f (e -γ ) < 1 -f (N ) (e -γ ) ≤ |1 -f (N ) (z)|;
(2) both 1 -f (N ) (z) and 1 -f (N ) (z) -2g <N > (z) are non-zero. It follows from Rouché's Theorem that 1 -f (N ) and 1 -f (N ) -2g <N > have the same number of zeros inside the circle |z| = e -γ , i.e. no zeros. Thus, setting p := f (N ) + 2g <N > , we get h(σ p ) < γ < h. Now, set k = g <N > and split p as p = (f (N ) + g <N > ) + g <N > =: h + k and let q := h(1 + k + k 2 + • • • ). σ q is the loop system defined from σ h+k by replacing the loops from k by all the loops made by concatenating a copy of a loop from h with an arbitrary positive number of copies of loops from k (see [START_REF] Boyle | Almost isomorphism for countable state Markov shifts[END_REF]Lemma 5.1] for detail). It follows that σ q can be identified to the subset of σ p obtained by removing a copy of σ k with the points asymptotic to it. Hence, there is a continuous embedding of σ q into σ p .

Note that for n ≤ N we have

f n = p n = q n . Also, for n > N , f n < e nβ ≤ (1 + k + k 2 + • • • ) n ≤ q n .
This yields an embedding σ f → σ q and concludes the proof.

The SPR case

We now give the proof of Theorem 1.4. Let X ′ be the mixing SPR component of X with h(X) = h(Y ). Equal entropy mixing SPR Markov shifts are almost isomorphic as defined and proved in [START_REF] Boyle | Almost isomorphism for countable state Markov shifts[END_REF]. Consequently there will be a word w and a subsystem Σ w of X ′ (consisting of the points which see w infinitely often in the past and in the future) such that there is a continuous embedding ψ 0 from X 0 = Σ w onto a subsystem Y 0 of Y and ǫ > 0 such that the complements X ′ \ X 0 and Y \ Y 0 have Borel entropy less than h(Y ) -ǫ.

The Borel subsystem X \ X 0 is (after passing to a higher block presentation) the union of a Markov shift X 1 (the subsystem of X avoiding the word w) and a weakly wandering set W (defined by the occurence of w, with a failure of infinite recurrence in the past or future). By Remark 2.1, we can forget about W . We cannot expect X 1 to have entropy less than h(Y \ Y 0 ), and therefore we cannot apply Theorem 1.5 to embed X 1 into a subsystem of Y \ Y 0 . Instead, we will push X 1 into the image of X 0 , and adjust the definition on X 0 to keep injectivity.

For L large enough, Σ w,L := {x ∈ Σ : ∀n ∈ Z ∃k ∈ {0, . . . , L} x n+k . . . x n+k+|w|-1 = w} is a mixing Markov subshift with h(Σ w,L ) > h(X 1 ). We apply Lemma 3.1 to get pairwise disjoint mixing SFTs Y 1 , Y 2 , . . . in Σ w,L satisfying h(Y i ) > h(X 1 ) for all i ∈ N.

Let C denote the complement in X 1 of the periodic points. Theorem 1.5 gives Borel embeddings γ i : C → Y i . Let Z i := γ i (C) ⊂ Y i and let φ i be the conjugacy γ i+1 • γ -1 i : Z i → Z i+1 . We define ψ :

X 0 ∪ C → Σ ′ by ψ : x → γ 1 (x) ∈ Z 1 if x ∈ C → φ i (ψ 0 (x)) ∈ Z i+1 if ψ 0 (x) ∈ Z i → ψ 0 (x) otherwise .
This ψ is a Borel embedding. This finishes the proof of Theorem 1.4. Lastly we record the obvious corollary of Theorem 1.3.

Theorem 4 . 1 .

 41 The free parts of two irreducible SPR Markov shifts are Borel isomorphic if and only if they have the same entropy and period.

Unless specified otherwise, the word measure will denote an invariant Borel probability.


Cyril Buttay
email: cyril.buttay@insa-lyon.fr

El Khalil

Rémi Falahi

Stanislas Robutel
email: robutel@gmail.com

Christian Hascoët
email: christian.martin@univ-lyon1.fr

Bruno Martin

Mark C Allard

Johnson

Khalil El

Bruno Allard
email: bruno.allard@insa-lyon.fr

Mark Johnson
email: mark.johnson@nottingham.ac.uk

Integrated packaging allows for improvement in switching characteristics of silicon carbide devices

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Silicon carbide (SiC) power devices offer many advantages over their ubiquitous silicon-based counterparts. They can operate at much higher temperature [START_REF] Raynaud | Comparison of high voltage and high temperature performances of wide bandgap semiconductors for vertical power devices[END_REF], at higher voltages [START_REF] Baliga | The Future of Power Semiconductor Device Technology[END_REF] and higher switching frequency [START_REF] Jos É Mill Án | A Survey of Wide Bandgap Power Semiconductor Devices[END_REF].

However, special packaging techniques are required to use SiC devices at their full potential (this is also true for GaN devices). Increasing the switching speed to reduce switching losses might cause voltage spikes that could either damage the devices, or generate electro-magnetic interferences (EMI). In many cases the switching speed of the transistors is limited by the impedance of the gate loop.

In many ways, these considerations about packaging parasitics are not different from those associated to the emergence of silicon MOSFETs and IGBTs, a few years ago. The physical principles are identical with wide-bandgap devices, but their magnitude differ. High voltage silicon carbide (1200 V and more) switches are comparable in performance (switching speed, recovery, on-resistance) to low-voltage (30-50 V) silicon devices. A few years ago, we demonstrated in [START_REF] Buttay | Model requirements for simulation of low-voltage MOSFET in automotive applications[END_REF] that the switching behaviour of low voltage, MOSFET-based converters was more dictated by the parasitic devices (stray inductances) than by the MOSFETs themselves. We now expect the same to be true for high voltage, SiC based architectures.

Integration is a powerful solution to design more compact converters, in particular with lower parasitic inductances. In this paper, we present some converters that have been designed in our research labs to take full advantage of SiC devices by integrating more functions in the power module.

R Gl L Gl C GDl C GSl C DSl T l L Dl L Sl C Out V DRl R Gh L Gh C GDh C GSh C DSh T h L Sh V DRh V In C CM1 C DC C CM2 L Cdc L DC1 L DC3 I Out L DC2 L DC4
Fig. 1: Circuit diagram of a half-bridge including the main parasitic devices (resistors and inductor coupling not shown for the sake of clarity).

Analysis of the switching cell

Parasitic elements and switching behaviour

The circuit diagram of a half-bridge, the basic cell that will be considered here, is visible in fig. 1. In addition to the two transistors (we assume they have an internal body diode), each interconnect can be associated with parasitic resistors, inductors and capacitors. However, for the sake of clarity, only the most relevant parasitic devices are represented in fig. 1. The resistors are not shown, nor are the coupling between the inductors. Only the inductors and capacitors that are submitted to (respectively) strong current and voltage transients are depicted. These parasitic devices caused by the layout of the converter are not the only unwanted elements. In particular, the capacitances of the power devices must be considered, as they are usually much larger than those due to the layout (in the nano-farad range, as compared to tens or hundreds of pico-farads). However, these devices capacitances are strongly non-linear (they are voltage-dependent), requiring specific characterization setup and models [START_REF] Funaki | Evaluation of capacitance-voltage characteristics for high voltage SiC-JFET[END_REF]. Some power semiconductor devices have an internal PiN body diode (as it is the case with the JFETs described below). The reverse recovery of this diode also has a strong influence on the switching performance of the circuit.

Influence of the parasitic elements

The gate inductance L G is often relatively large (tens to hundreds of nH), because the gate drivers are kept separated from the power devices (often on a PCB located above the power module). A large L G can slow down the commutation, or it can cause oscillations with the input capacitance of the power device. This requires a larger R G to control the ringing, resulting in an even lower switching speed.

However, the impedance of the gate circuit must be kept low to avoid spurious turn-ons: in the case of Fig 1, when T h turns on, the drain voltage of T l (V Dl ) rises rapidly. A current proportional to dV Dl dt flows through C GDl . If the gate circuit of T l has a low enough inductance, it will absorb most of this current, and the gate voltage of T l will remain mostly unaffected. In the opposite, if the impedance of the gate circuit is too high, the current will flow through C GSl , causing a rise in the gate voltage of T l . If V GSl exceeds the threshold voltage of the transistor, T l turns-on, resulting in a short-circuit of V In .

The drain inductance (L Dl , L DC2 ) stores energy ( 12 LI 2 ) when the corresponding transistor is on. In a hard-switching structure, this energy must be dissipated at turn-off, causing switching losses and voltage spikes. L D also causes ringing with the capacitances of the power devices. L D must therefore remain as small as possible. Much like for the gate drivers, the DC capacitors are kept separated from the semiconductor devices causing L DC2 (see fig. 1) and L DC4 to reach tens of nH.

The source inductance L S is common to the power path (drain current) and to the control path (gate current). Not only does it have the same effect as described above for L G and L D , but it also introduces a negative feedback that opposes (slows down) both the turn-on and turnoff. Therefore, even minute values of L S have a dramatic effect on the switching speed of the power device. Sophisticated solutions to cancel this negative feedback, for example by forcing a magnetic coupling between L G and L S were investigated [START_REF] Akhbari | An original design of MOSFET/IGBT gate circuit layout to suppress power/drive interaction[END_REF], but they require an increase in L G , which is not desirable. In any case, L S is usually smaller than L G or L D , in the order of a few nH (for a half-bridge power module).

It is also worth noting that there is an optimal L D L S ratio [START_REF] Xiao | Analytical modeling and experimental evaluation of interconnect parasitic inductance on MOSFET switching characteristics[END_REF]: a large value means that the devices switches rapidly (small L S ), triggering large drain voltage ringing caused by the large L D (underdamping). A small ratio means that the transistor switches more slowly (large L S ) than required to prevent drain voltage ringing to occur (over-damping).

The parasitic output capacitance C Out , formed between the output conductor and the surrounding ground, experiences fast voltage changes when the transistors switch. A current (the "common-mode current") will flow through this capacitor and return to the voltage source "externally", via any conductive path available (for example an metallic enclosure). This is depicted in fig. 1 by the "ground" symbol. For a standard power module, using a ceramic substrate, C Out is in the order of a few tens or hundreds of pF, but can be much larger depending on the load (for example if a shielded cable is used between the converter and an electrical motor). Larger C out values can also result in slower switching.

To contain the common-mode current within the converter and to prevent it from disturbing other systems, common-mode filtering capacitors (C CM 1 and C CM 2 ) can be used. They offer a shorter path for the common-mode current, especially if L DC1 and L DC3 are large compared to L DC2 and L DC4 .

Design improvements

In the previous section, we estimated the orders of magnitude of the parasitic devices associated with a "standard" packaging (i.e. a power module hosting the semiconductors, with the gate drivers and the DC-bus capacitors on separate boards). Various concepts can be developed to overcome the limitations of such packaging. Here, we describe three different approaches:

• to reduce the size of the control and power loops, by integrating the gate drivers and (part of) the DC bus capacitance directly on the same substrate as the power devices;

• to reduce the parasitic inductance values by using a busbar structure over the entire converter

• to integrate some common-mode filtering directly in the power module, as close as possible to the source of common-mode current.

It is worth noting that these concepts were developed for high temperature applications (120 to 300 °C ambient) more than for fast switching performance. In many aspects, however, high temperature converters must be optimized for fast switching. For example, the proper thermal management of a system operating at elevated ambient temperature is often difficult, so a high efficiency is desirable to reduce the amount of heat to be managed. Regarding Electro-Magnetic Compatibility (EMC), high temperature passive devices are not common and very expensive, so an efficient and smaller EMC filter is preferred.

Of course, a system that would be designed from the start for high switching speed, without the constraints of high temperature operation could take advantage of many "low temperature" technologies, such as those based on printed circuit board (PCB) [START_REF] Hoene | Ultra Low Inductance Package for SiC[END_REF]. Nevertheless, the concepts presented here demonstrate the improvements that can be expected by integrating more functions in the power module

Integration of gate driver

The module depicted in figure 2a integrates, on the same substrate, the gate driver (custom design using Silicon On Insulator -SOI-technology), the power devices (1200 V, 50 mΩ SiC JFETs from SiCED), and some passive components (including a -small, 10 nF-600 V DC bus capacitor). This demonstrator was designed for very high temperature operation (it was suc-cessfully tested at 310°C ambient), leaving only little technological choices to the designer (few available components and mounting technologies). More technical details are given in [START_REF] Khalil El Falahi | High temperature, Smart Power Module for aircraft actuators[END_REF]. SOI is commonly used for high voltage integrated circuits such as the gate drivers, as the isolating layer limits the leakage currents that would be too high with pure silicon. Another advantage associated with SOI is a higher operating temperature [START_REF] Khalil El Falahi | Evaluation of commercial SOI Driver performances While Operated in Extreme Conditions (up to 200C)[END_REF] (usually more than 150 °C), meaning that the gate driver die can be located near the hot power devices.

However, despite the use of wirebonds to connect the dies (and to connect the dies to the long terminals of the package, as a hermetic case was required for this inverter leg), it can be seen in figure 2b that the switching transient is very fast (<15 ns rising), and with little ringing. Some commercially available power modules have an integrated gate driver [START_REF] Zurnaci | MiniSKiiP IPM -Intelligent Power Modules Family -Technical Explanations[END_REF]. It is unclear whether this development was driven solely by the need of better switching performance, or to make the module easier to use, but it proves that the integration of the gate driver circuit is possible on an industrial scale.

Low inductance packaging

Another approach to improve switching performance is to reduce the value of the parasitic inductances. As inductance is related to the length of the conductors and the surface area of the commutation loop, This requires to modify the layout of the power module. The busbar structure (flat conductors separated by a thin dielectric layer) is a way to reduce the surface area of the commutation loop [START_REF] Chiad Ò Caponet | Low stray inductance bus bar design and construction for good EMC performance in power electronic circuits[END_REF] and is attractive to connect the power module.

A further improvement is obtained by replacing the wirebonds interconnects. A technology called SiPLIT (Siemens Planar Interconnect Technology) uses the same principle as the busbar, with a thin dielectric layer separating the conductors in the module. This allows for a 50% reduction in parasitic inductances [START_REF] Weidner | Planar Interconnect Technology for Power Module System Integration[END_REF]. Another approach is the "sandwich" structure [START_REF] Liang | Reducing Parasitic Electrical Parameters with a Planar Interconnection Packaging Structure[END_REF],

where the power devices are placed between two metal-ceramic substrates. In addition to an improved inductive behaviour (although probably not as reduced as with the SiPLIT), the sandwich structure allows to cool the power devices by both sides (double side cooling) [START_REF] Calata | Three-Dimensional Packaging for Power Semiconductor Devices and Modules[END_REF].

The power converter described here uses both busbar interconnects and sandwich structure.

It was designed to operate in the vicinity of an internal combustion engine, in a hybrid car, with a cooling fluid temperature of up to 120°C [START_REF] Kassakian | The Future of Electronics in Automobiles[END_REF]). For this converter, a direct cooling technique was selected, where the cooling fluid is in direct contact with the back of the power module substrates (fig. 3a). This allows both for low parasitic inductance and very low thermal resistance from junction to fluid (<0.4 K/W, [START_REF] Buttay | Compact Inverter Designed for High-Temperature Operation[END_REF]).

A laminated busbar (fig. 3b) is used to provide a low-inductance connection between the DC capacitor board and the "sandwich" modules. The capacitor board itself uses insulated metal substrate (IMS), which by design generates very small parasitic inductances. Overall, from the terminals of the inverter to the sandwich modules, a parasitic inductance of 10 nH was measured [START_REF] Buttay | Compact Inverter Designed for High-Temperature Operation[END_REF].

The complete inverter, visible figure 4a is equiped with 6 1200 V silicon IGBTs and 6 Schottky Barrier Diodes (SBDs). An example of waveform captured at the output terminal of one of the sandwich modules is visible in figure 4b. Thanks to the absence of recovery phenomenon (use of SBDs) and to the low-inductance design, the waveforms show no ringing at the nominal voltage and current.

Integration of common-mode filtering

Increasing the switching speed is usually an advantage from the thermal and efficiency points of view, but has detrimental effects on the EMC. In particular, as explained in section 2.2, higher dv/dt at the output of the bridge causes an increase in the common-mode (CM) current flowing through the parasitic capacitor C out (see figure 1). Although filtering higher frequency harmonics is in theory easier because it requires smaller passive devices, it is not so simple in the case of CM current: the location of the filter is also important, to provide a short path to keep the CM current within the converter or within the module.

A solution is to directly integrate the CM filter capacitors (C CM 1 and C CM 2 in figure 1) directly in the power module, as visible in figure 5a [START_REF] Robutel | Design and Implementation of Integrated Common Mode Capacitors for SiC JFET Inverters[END_REF]. This makes it possible to "capture" the high frequency harmonics directly at the source. In this way, a simpler input filter is required (for example a first order filter instead of a second order). An example of the EMI spectrum emit- ted with and without the integrated filtering is visible in figure 5b. The waveforms in figure 6 show that there is no noticeable effect of the integrated capacitors on the switching speed, and therefore on the efficiency of the converter.

Discussion and Conclusion

The three approaches developed here offer improvements over standard packaging, allowing switching times as small as 15 ns. If the "sandwich" structure requires a large alteration of the manufacturing process (in particular, power devices with a suitable topside metallization are needed), this is not the case for the other modules. Gate driver or common-mode capacitors are integrated using the standard "planar" technology.

The main downside of this approach is the additional module space required for the integrated functions. For a silicon-based module, this might be an issue as the size of the module is often dictated by the number of power dies to accommodate (the dies are placed as close as possible to each other). In a SiC-based module, however, the dies are smaller. They are placed further from each other for thermal considerations, so integrating additional functions might be possible with limited consequences on the the size of the module.

In any case, a specifically-designed packaging is required to make the most of the excellent existing SiC Devices.

Fig. 2 :

 2 Fig. 2: The populated power substrate (before wirebonding of the gate driver circuits) in its hermetic case (without lead) (a) and the output voltage of the inverter leg (50 Ω/1 mH load, 200 V DC bus) (b).

Fig. 3 :

 3 Fig. 3: Cross-section of the "double-side" cooling structure (a) and photograph of the laminated busbar used to connect the capacitor board to the power modules (b).

Fig. 4 :

 4 Fig. 4: The complete inverter, with the top cover removed, showing the ceramic capacitors (a). The three ribbons visible on top are the signal and power supply connections to the gate driver boards (not visible). Switching waveform captured at the output of one of the inverter legs, for a 600 V DC bus and 7 A phase current (on an inductive load) (b).

Fig. 5 :Fig. 6 :

 56 Fig.5: The power module with integrated common mode filtering (capacitors). This module is an inverter leg, using two SiC JFETs (SiCED) (a). Effect of the Common-Mode (ccm) capacitors on the EMI spectrum, compared to a module with the same layout, but without any capacitor mounted (b).