Antoine Joux
email: antoine.joux@m4x.org

Reynald Lercier
email: reynald.lercier@m4x.org

David Naccache
email: david.naccache@ens.fr

Emmanuel Thomé
email: emmanuel.thome@normalesup.org

Oracle-Assisted Static Diffie-Hellman Is Easier Than Discrete Logarithms

Keywords: dlp, sdhp, ffs, nfs

This paper extends Joux-Naccache-Thomé's e-th root algorithm to the static Diffie-Hellman problem (sdhp).

The new algorithm can be adapted to diverse finite fields by customizing it with an nfs-like core or an ffs-like core.

In both cases, after a number of non-adaptive sdhp oracle queries, the attacker buildsup the ability to solve new sdhp instances unknown before the query phase.

While sub-exponential, the algorithm is still significantly faster than all currently known dlp and sdhp resolution methods.

We explore the applicability of the technique to various cryptosystems.

The attacks were implemented in F 2 1025 and also in F p , for a 516-bit p.

Introduction

In [START_REF] Joux | When e-th roots become easier than factoring[END_REF], Joux, Naccache and Thomé showed that carefully interacting with an e-th root oracle improves one's ability to compute e √ t mod n for arbitrary t after the communication with the oracle. In Joux et alii's game, the attacker can only query the oracle during a learning phase that takes place before t is known. As this learning phase ends, the attacker receives the target t and outputs it's e-th root. In other words, [START_REF] Joux | When e-th roots become easier than factoring[END_REF] establishes that the release of certain modular roots leaks information that eases the calculation of all future roots.

Work partially supported by dga research grant 05.34.058 This is interesting because, while sub-exponential and oracle-assisted, [START_REF] Joux | When e-th roots become easier than factoring[END_REF] is still faster than gnfs-factoring.

A recent ePrint publication ( [START_REF] Koblitz | Another look at non-standard discrete log and Diffie-Hellman problems[END_REF]) extended [START_REF] Joux | When e-th roots become easier than factoring[END_REF] to discrete logarithms (dls). While complexities and details vary, [START_REF] Koblitz | Another look at non-standard discrete log and Diffie-Hellman problems[END_REF]'s overall scenario remains essentially the same, replacing root extractions by dlp calculations. This work tackles the static Diffie-Hellman problem (sdhp) on which many cryptosystems rely (e.g., [START_REF] Chaum | Undeniable signatures[END_REF][START_REF] El-Gamal | A public-key cryptosystem and a signature scheme based on discrete logarithms[END_REF][START_REF] Ford | Server-assisted generation of a strong secret from a password[END_REF] -to mention only a few). Namely, we show how after several carefully crafted non-adaptive sdhp oracle queries, an attacker can improve his ability to solve arbitrary sdhp instances, unknown before the last oracle query. The attacker's workload is significantly lower than [START_REF] Koblitz | Another look at non-standard discrete log and Diffie-Hellman problems[END_REF] 5 and all currently known dlp and sdhp resolution algorithms.

The method can be adapted to diverse finite fields by customizing it either with an nfs-like core or an ffs-like core.

We assume that the reader is familiar with [START_REF] Brown | The static Diffie-Hellman problem[END_REF][START_REF] Joux | When e-th roots become easier than factoring[END_REF][START_REF] Koblitz | Another look at non-standard discrete log and Diffie-Hellman problems[END_REF], whose notations, introductory descriptions and terminology we extensively borrow and recycle.

Results: Let Q = q n where q is a prime power6 and n an integer.

We present an algorithm A performing, during a learning phase, L non adaptive sdhporacle queries {x 1 , . . . , x L } ∈ F L Q to which the oracle responds with {x d 1 , . . . ,

x d L } ∈ F L Q for some secret d ∈ [0, Q -1].
After the learning phase, A is given a previously unseen challenge z and outputs

z d ∈ F Q . The L Q ( 1 3 , 3 √ x)-
Here, nfs-hd stands for high-degree nfs to emphasize the difference between this middle case and the regular nfs.

Let us recall that x values of the ffs, nfs-hd and nfs are respectively 32 9 , 128 9 and 64 9 . The following sections will detail A's inner mechanics 7 , complexity and the relation between L and {q, n}.

The attacks was implemented in F 2 1025 and also in F p , for a 516-bit p (details in the Appendix).

Conventions

Complexity: We express complexities using the following notations8 :

L Q (ν, λ) = exp λ (1 + o(1)) (log Q) ν (log log Q) 1-ν , Q (ν) = (log Q) ν (log log Q) 1-ν and D Q (ν, λ) = log q (L Q (ν, λ)) . As L Q (ν, λ) includes an o(1), D Q (ν, λ) inherently stands for D Q (ν, λ + o(1)).
Following [START_REF] Joux | The function field sieve in the medium prime case[END_REF][START_REF] Joux | The number field sieve in the medium prime case[END_REF], we guarantee that the complexity of our algorithms is L( 1 3 , O( 1)) by distinguishing the following three cases:

ffs: log q ∈ o( Q ( 1 3 )) q is asymptotically smaller than all functions in L Q ( 1 3 , γ). nfs-hd: log q ∈ [ω( Q ( 1 3 )), o( Q ( 2 3 ))] q is asymptotically between all functions in L Q ( 1 3 , γ) and L Q ( 2 3 , γ). nfs: log q ∈ ω( Q ( 2 3
)) q is asymptotically greater than all functions in L Q ( 2 3 , γ).

Assumptions: We rely on the usual heuristic assumptions [START_REF] Dickman | On the frequency of numbers containing prime factors of a certain relative magnitude[END_REF][START_REF] Panario | An analytic approach to smooth polynomials over finite fields[END_REF][START_REF] Lenstra | The number field sieve[END_REF]:

-The probability that an integer q ∼ L Q (ν q , λ q + o( 1)) is a product of prime factors of magnitude at most L Q (ν p , λ p + o(1)) is:

L Q ν q -ν p , - λ q λ p (ν q -ν p ) + o(1)
despite the fact that integers q might follow non-uniform distributions. -The probability that a polynomial q ∈ F q [X] of degree D Q (ν q , λ q ) 1 is a product of irreducible polynomials of degree at most D Q (ν p , λ p ) 1 is also:

L Q ν q -ν p , - λ q λ p (ν q -ν p ) + o(1)
3 Function Field Sieve Oracle-Assisted sdh Resolution

The Algorithm

In this section, we denote by ε the constant ε = 3 4 9 and by

ϑ n+1 = 1 3 (1 + 1 2 n ). Polynomial Construction. Let D = D Q (ϑ n+1 , ε), D tends to infinity with Q because log q ∈ o( Q ( 1 3 )) and thus D 1. Let d 1 ∼ = √ nD and d 2 ∼ = n D with d 1 d 2 n. Choose two univariate polynomials f 1 , f 2 ∈ F q [x
] of respective degrees d 1 , d 2 , such that the resultant of F 1 (x, y) = y -f 1 (x) and F 2 (x, y) = x -f 2 (y) with respect to variable y has an irreducible factor of degree n over F q .

Let f (x) denote this irreducible factor of Res

y (F 1 , F 2 ) = x -f 2 (f 1 (x)), then clearly F q n F q [x]/(f (x)).
Let α be a root of f (x) in F q n and let β = f 1 (α). Then, by construction, we have α = f 2 (β) too.

Learning Phase. We let F D be the set of all monic irreducible polynomials over F q of degree up to D and define

{x i } = {{p(α), p(β)} | p(x) ∈ F D }. After about 2q D L Q ( 1 3 , ε + o(1)) sdhp-oracle queries, we get {p(α) d , p(β) d } p∈F D .
Descent. The descent phase consists in finding a multiplicative relation between the target z and the elements of F α ∪ F β . To do so, we use a special-q sieving subroutine (hereafter SqSS) as a basic building block. SqSS is called iteratively to steadily decrease the degree of intermediate polynomials.

The SqSS works as follows:

Let q be an irreducible polynomial in x (respectively y) of degree d q .

To ease SqSS's analysis we write d q as:

d q = D Q (ν q , λ q ) = (λ q + o(1)
) n ν q log 1-ν q q n for two constants ν q , λ q such that 3 √ n d q n. Note that the values ν q , λ q evolve simultaneously -as will be seen later in further detail.

SqSS's purpose is to write an element q(α) or q(β) as a product of elements of the form p(α) and p(β), where the p-polynomials are of degrees smaller than the degree of q.

Let d ε = D Q ( 1 3 , ε), we consider q d ε L Q ( 1 3 , ε) polynomials: H(x, y) = 0 ≤ i ≤ d x 0 ≤ j ≤ dy h i,j x i y j ∈ F q [x, y]
where

d y = max( (d q + d ε )/D -1, 1) and d x = d q + d ε + 1 d y + 1 -1
We only consider polynomials H(x, y) such that

q divides H x = i,j h i,j x i f 1 (x) j (re- spectively H y = i,j h i,j f 2 (y) i y j ).
The complexity analysis (cf. infra) shows that there exists, amongst these polynomials, polynomials H such that H x and H y both admit a factorization of the form:

p ∈ F D Q (νp,λp) for p = q p with    ν p = 1 6 + ν q 2 and λ p = 2 √ λ q +ε 3ε if ν q > ϑ n+1 ν p = ν q and λ p = 3ε+λ q 6ε √ ε otherwise.
By proceeding so, we can express an element q(α) or q(β) as a product of polynomials p of degrees at most D Q (ν p , λ p ) evaluated at α or β.

Descending Using SqSS. Define the sequences:

ν k+1 = 1 6 + ν k 2 and λ k+1 = 2 √ λ k + ε 3ε with ν 0 = λ 0 = 1
We note that, indeed, ν k = ϑ k and that the limit at infinity of the sequence λ k is equal to

1+ √ 5 3 √ 18 ∼ = 1.234.
As our procedures perform a finite number of steps, these ad infinitum convergence targets are only approached. However, the limit at infinity guarantees that during the phase of the descent where ν decreases, λ remains bounded (smaller than the above limit). After that point ν remains constant and λ quickly tends to ε.

The descent starts with one polynomial q(x) of maximal degree D Q (ν 0 , λ 0 ) with z = q(α). After a first invocation of SqSS we get polynomials p in x and y of degree at most

D Q (ν 1 , λ 1 )
where ν 1 = 2 3 and λ 1 = ε(1 + ε). Again, we hand over each intermediate polynomial p to SqSS to obtain new polynomials p in x and y of degree at most D Q (ν 2 , λ 2 ) etc.

After N = O(log n) iterations involving intermediate polynomials of degrees D Q (ν k , λ k ),
we eventually reach polynomials of degree at most D Q (ϑ n+1 , λ N ).

Consider the sequence λ

k+1 = 3ε+λ k 6ε √ ε starting at λ 0 = λ N = 1+ √ 5 3 √ 18 + o(1)
. We iterate SqSS O(log n) additional times; thereby generating polynomials of degree D Q (ϑ n+1 , λ k ). Eventually, we reach polynomials of degree at most D, i.e. D Q (ϑ n+1 , ε).

It remains to combine these relations to write z as a fraction of products of elements of the form p(α) or p(β) with the degrees of p bounded by D. Finally, we use the oracle's answer-list {p(α) d , p(β) d } p∈F D , to retrieve z d .

Complexity

We analyze the complexity of each step: SqSS For ν q > ϑ n+1 : The sum of the degrees of H x and H y is:

(d x + 1) (d q + d ε )/D + (d y + 1) (d q + d ε )D -2 2 (d q + d ε )n
As, in addition, q divides one of these two polynomials, our smoothness probability is identical to that of d p -smooth polynomials of degree 2 (d q + d ε )n -d q i.e.:

2 (d q + d ε )n -d q 2( λ q + ε + o(1))n 1+ν q 2 log 1-ν q 2 q n D Q 1 + ν q 2 , 2 λ q + ε
And the probability that polynomials of this degree are 1)) (cf. to section 2).

D Q ( 1 6 + νq 2 , 2 √ λq+ε 3ε )-smooth is L Q ( 1 3 , -ε + o(
SqSS For ν q ϑ n+1 : The sum of the degrees of H x and H y is:

√ nD + d q + d ε 2 + d q + d ε 2 n D + 1 √ nD + d q + d ε 2 n D
Then,

√ nD + d q + d ε 2 n D -d q √ ε + λ q 2 √ ε + √ ε 2 + o(1) n 2 3 + 1 3 2 n log 2 3 -1 3 2 n q n D Q 2 3 + 1 3 2 n , 3ε + λ q 2 √ ε
Again, the odds that polynomials of this degree are

D Q (ϑ n+1 , 3ε+λq 6ε √ ε )-smooth is L Q ( 1 3 , -ε+ o(1)).
In both cases, after enumerating L Q ( 1 3 , ε + o( 1)) polynomials, we expect to find a good polynomial H. Descent: SqSS calls involve medium-sized p-polynomials of degree larger than D. The number of these polynomials is thus bounded by O( n D ), i.e. O( 3 n 2 log 2 q n). As the whole process involves O(log n) SqSS calls, the total number of medium-size polynomials p to be considered is:

O n log q n 2 3 O(log n) = e O(log 2 n)
Running SqSS on each of these polynomials, costs L Q ( 1 3 , ε+o(1)), which is far larger than exp(O(log 2 n)); we thus conclude that the descent's complexity is

L Q ( 1 3 , ε + o(1)), having pre-computed a table of L Q (ϑ n+1 , ε + o(1)) elements {p i (α) d } i ∪ {p i (β) d } i . Since log ±2 -n q n = 1 + o(1) L Q (ϑ n+1 , ε + o(1)) exp (ε + o(1))(log 1 3 + 1 3 2 n q n )(log log 2 3 -1 3 2 n q n ) L Q ( 1 3 , ε + o(1))
Total Complexity: The attack's total complexity is hence 1)), in time, space and oracle accesses.

L Q ( 1 3 , ε + o(1)) = L Q ( 1 3 , 3 4 9 + o(
4 Number Field Sieve Oracle-Assisted sdh Resolution

The Algorithm

Due to lack of space, we only sketch the two nfs algorithms covering large and medium basefield cardinality-values q. As our algorithms preserve, mutatis mutandis, most ffs features, we will mainly focus on the specificities proper to these two cases. We also refer the reader to [START_REF] Joux | When e-th roots become easier than factoring[END_REF], whose arbitrary e-th roots procedure has to be slightly modified to fit the case under consideration.

Algebraic Side: In the ffs case, thanks to the existence of an ad-hoc construction, both sides are rational, i.e. on both side we only encounter polynomials. With the nfs matters get more complicated. Indeed, all known polynomial constructions for nfs procedures involve at least one non-trivial number field.

For example over F p the base-m construction yields two polynomials f 1 (x) = x -m and f 2 (x), such that p | f 2 (m). Since f 1 is a linear polynomial, it generates a trivial number field: the corresponding side is rational and can be addressed using integers. However, f 2 defines a number field Q(α) with non trivial unit and class groups.

Consequently, to deal with f 2 's side 9 , we need to work with factorizations into ideals: the algebraic factor base consists of ideals of small norm. This change truly complicates the approach. As the ideals in the factor base are usually non-principal, we do not know how to construct an oracle query corresponding to a given ideal.

Instead, we need to find smooth a -bα values which factor into small ideals and "translate" them to integers a -bm for the oracle.

Linear Algebra: To relate these factorizations back to multiplicative relations in F Q we must resort to linear algebra, which can be done in two ways:

-Repeating the descent of [START_REF] Joux | When e-th roots become easier than factoring[END_REF], it is possible (using the oracle's answers) to relate the challenge z ∈ F Q to some a -bm ∈ Z such that a -bα factors into small ideals. By solving a linear system over Z/ (Q -1)Z , this exact factorization can be mapped to a multiplicative combination of oracle answers.

Note that to transform the equality of ideal factorizations into an equality that makes sense in F Q , one must include additive characters in Z/ (Q -1)Z , à la Schirokauer [START_REF] Schirokauer | Discrete logarithms and local units[END_REF]. Since this methodology performs a linear algebra step that depends on the challenge, it therefore yields an algorithm where the linear algebra step has to be redone for each challenge.

-There is also a second method -very specific to our setting. It is possible to use linear algebra to compute virtual sdh values that correspond to each ideal 10 . From a theoretical standpoint, this can be justified as in [START_REF] Joux | Improvements to the general number field sieve for discrete logarithms in prime fields. A comparison with the gaussian integer method[END_REF][START_REF] Schirokauer | Virtual logarithms[END_REF]. Note that this second method also requires the inclusion of additive characters à la Schirokauer.

Note that in some cases (nfs-hd), there are two algebraic sides to consider.

We start from a large system of equations, where each equation expresses a value in F Q -the equation's generator -as a product of ideals in the number field. Moreover, we are also given the power of this value returned by the oracle. Our goal is to associate to each ideal I a number P I such that in each equation, the substitution of I by P I will yield a product whose value matches the oracle's answer for this equation.

If we can achieve this goal, P I can be looked upon as an oracle output for I, and we are back in a ffs-like scenario. However, this is not completely straightforward:

A first (costly) option would be to use linear algebra over the integers and express each ideal as a product of generators (possibly with rational exponents) but one may note that the vector of generators and the vector11 of ideals are related by a matrix multiplication in the exponent. More precisely, if ideals are indexed from 1 to B, and denoting by G i the i-th equation generator (1 ≤ i ≤ B), there exists a B × B matrix A, such that:

     G 1 G 2 . . . G B      = A      I 1 I 2 . . . I B      (1)
where denotes matrix multiplication in the exponent, i.e. where for all i:

G i = B j=1 I A i,j j ( 2
)
Moreover this matrix multiplication in the exponent is defined modulo Q -1, the cardinality of F * Q . Having observed that, we can use Wiedemann's algorithm [START_REF] Wiedemann | Solving sparse linear equations over finite fields[END_REF] to obtain the P I values. e.g. if A is invertible, Wiedemann's algorithm finds an expression A -1 = F (A) where F is a polynomial.

From this expression, we obtain:

     P I1 P I 2 . . . P I B      = F (A)      O(G 1 ) O(G 2 ) . . . O(G B )      (3)
where O(G i ) denotes the oracle's output for the generator G i .

This linear algebra step based on [START_REF] Wiedemann | Solving sparse linear equations over finite fields[END_REF] has essentially the same complexity as usual, with the small caveat that additions are replaced by multiplications and multiplications by constants are replaced by exponentiations.

Complexity

The L Q ( 3

These figures stem from the technical analysis found in the Appendix.

Cryptanalytic Applications

This section follows very closely Brown and Gallant's [START_REF] Brown | The static Diffie-Hellman problem[END_REF]. Variable were renamed to ease the identification of on parameters playing specific roles in our algorithms (e.g. secret d, oracle queries x i etc).

It is clear that any cryptographic protocol where the secret-owner plays the role of a sdhp-oracle will succumb to the new attacks. The technique can apply to primitives as diverse as public-key encryption, digital signature and key retrieval.

Textbook El-Gamal Encryption

As observed in [START_REF] Brown | The static Diffie-Hellman problem[END_REF], textbook El-Gamal encryption makes an sdhp oracle available. El-Gamal encryption of message m to a receiver with public-key

g d results in a ciphertext c = {c 1 , c 2 } = {g r , m • g dr }.
In a chosen ciphertext attack against an encryption scheme, an adversary A can select any ciphertext and obtain its decryption. For El-Gamal encryption, if A chooses c = {x, c 2 }, then he obtains m = c 2 /x d . A can thus obtain x d = c 2 /m, thereby transforming the receiver into an sdhp-oracle for the challenge x.

Textbook El-Gamal is already known to be vulnerable to chosen ciphertext attacks. The known attacks mainly allow information to be learned about previously encrypted messages but do not leak anything about d. In the scenario described here, an attacker use the receiver as an oracle to learn how to decrypt future traffic sub-exponentially, but still significantly faster than all previously known sdhp-solving algorithms.

Ford-Kaliski Key Retrieval

In Ford-Kaliski's key retrieval protocol [START_REF] Ford | Server-assisted generation of a strong secret from a password[END_REF], a client picks a random exponent b and computes x = g b .

x is sent to the server who, replies with s = x d . The client computes t = b √ s and retrieves

the hardened password t = b √ s = g d .
The protocol's purpose is to transform a low-entropy secret password g into a hardened password g d involving a longer secret known only to the server. This prevents dictionary attacks on the client's password.

Therefore, the server plays exactly the role of the sdhp oracle considered in this paper.

Variants of this protocol are currently under discussion in two standardization groups [6, 7].

Chaum-Van Antwerpen's Undeniable Signatures

Another protocol in which an sdhp oracle is available is Chaum and van Antwerpen's undeniable signature scheme [START_REF] Chaum | Undeniable signatures[END_REF]. The protocol uses a prime modulus p = 2q + 1 such that q is prime.

Let g ∈ Z *

p be an element of order q and y = g d where d is the signer's private key (all other parameters are public). To sign a message x, Alice produces the signature s = x d mod p. Bob verifies s interactively: Bob picks {e 1 , e 2 } at random in Z q , and sends c = s e 1 y e 2 to Alice.

Alice computes h = d √ c mod p and sends it to Bob. Bob accepts s as a valid signature if h = x e1 g e2 mod p.

Note that [START_REF] Chaum | Undeniable signatures[END_REF], provides sdhp oracles for both d and 1 d mod q.

π(α) = σ( 2 1023 π )(α) = α 1024 + α 1023 + α 1020 + . . . + α 5 + α 4 + α 2
As the factors of x 18 π(x) mod f (x) have degrees 1, 4, 6, 11, 32, 73, 217, 339 and 341, listing here the entire descent tress would require too much space. Instead, we opt to zoom on the path of length 12 starting with a special-q equal to the factor of degree 341 and following recursively, at each node, the highest degree factor. This yields table 1, where the a i,j stand for the following hexadecimal values:

a 1,0 = 1de914aa624143ee2880268 a 1,1 = 5368318d2e945f69775022f a1,2 = 6b625fb7825342aecdabd80 a1,3 = 1055c12e550f64c6d8bd2e6 a 2,0 = 22ac2088c59 a 2,1 = 114043dab72 a 2,2 = 3e6c922af2e a 3,0 = 3f536e224dd a 3,1 = 1077f087dba a 4,0 = 000b1a0283e a 4,1 = 002cce1067f a5,0 = 0006a24a2be a5,1 = 000164b63c6 a6,0 = 0001bf55bcb a6,1 = 0000c949e26 a 7,0 = 00026dc2d0b a 7,1 = 0000b748064 a 8,0 = 003e4818437 a 8,1 = 0001b3671a7 a9,0 = 000138bc9c0 a9,1 = 000122f0d95 a10,0 = 000052ee9bb a10,1 = 0000e82d833 a 11,0 = 00003972dfb a 11,1 = 00069fc51c5 a 12,0 = 0006faf0133 a 12,1 = 0000d86d785
The special-q degree in x or y is underlined in the corresponding column and smoothness probabilities are expressed as 2 -ψ .

We wrote our software chain in c, relying upon the computer algebra systems pari-gp and magma for a handful of specific tasks. The whole computation was run on single Intel Core-2 at 3.6 GHz, with a 16 Gb memory. Each SqSS call claimed roughly five minutes and the entire attack took less than a week.

A.2 Number Field Side Experiments

We also experimented the nfs variant in the finite field F p , where p is the 516-bit prime 10 155 π + 88896 (p is the smallest prime above 10 155 π such that p-1 2 is also prime). We computed the secret d-th power of an element in F Q after a learning phase of 140 million sdhp-oracle queries.

We selected the following polynomials for defining respectively the rational and algebraic sides:

f 1 = x -0x5bd59a8c1dfa4580bbd2cee f 2 = 0xf6841ca54cc1267c • x 5 + 0x423b1fc0c94938f26 • x 4 -0x2495d40c188755399c7c • x 3 + 0x3b7ed50dd0329dda55051 • x 2 -0x23e3eeb7641a7d13c4b182 • x + 0x1a29246950783fbbb6b7b7c7
Sieving was done using a modified version of J. Franke and T. Kleinjung's lattice sieving code as included in the ggnfs software suite. We selected a factor base comprising the 43,321 prime ideals of norm smaller than 2 19 and obtained generators factoring over this set of ideals after four sieving minutes using 128 Intel Core-2 cpu cores clocked at 1.6 GHz.

We then used the factor base extension step detailed in [START_REF] Joux | When e-th roots become easier than factoring[END_REF] and sieved again for 24 minutes, using the same set of processors. This allowed us to relate the vast majority of ideals of norm smaller than 2 32 to smaller-norm ideals using 140 million oracle queries.

We postponed linear algebra computations after the descent (cf. to the two options explained in section 4).

The descent was coded in magma, where the SqSS was performed by a modified version of J. Franke and T. Kleinjung's lattice sieving code, including several modifications pertaining to the descent step (most notably co-factorization of norms using the gmp-ecm program).

We started from the challenge value τ = 2 log 2 N -log 3 N . The first decomposition step consisted in finding a related finite field element (e.g. 2 k τ for some k, assuming 2 was an oracle query) which factored suitably over the integers.

We searched during a few minutes for a suitable integer k. The corresponding factorization involved seven primes unlinked to the factor base, the largest of which having 27 digits. Afterwards, using 164 intermediate descent steps (139 on the algebraic side and 25 on the rational side), we managed to link our initial challenge to the factor base.

The final link between our challenge and the oracle queries was obtained by solving a linear system involving the matrix formed by the first set of relations obtained by sieving; the right hand side formed by the ideal factorization stemming from the descent. After adding character maps, we solved this 43,378 rows and columns system in eight hours on four Intel Core-2 cpu cores clocked at 2.4 GHz.

B Complexity Analysis -Regular nfs

Here, we work in F Q , with Q = p n and p > L Q ( 23 ). The polynomial construction is done through lattice reduction. First, choose an irreducible polynomial f 1 of degree n and find another polynomial of degree D, multiple of f 1 modulo p. The coefficients of f 2 have size Q 1/(D+1) , the coefficients of f 1 are essentially of the same size, just slightly larger.

The main parameter is D = δ 3 log Q log log Q . This implies that the coefficients of f 1 and f 2 have size L Q ( 2 3 , 1 δ ).

B.1 Sieving And Linear Algebra

There is a small and a large side. The complexity is clearly dominated by the large side, corresponding to f 2 . We sieve over elements a + bα, map them into the number field and compute their norm b D f 2 (-a b ). If a and b are bounded in absolute value by B = L Q ( 1 3 , β), the norm is L Q ( 2 3 , βδ + 1 δ ). As usual, the smoothness probability over a smoothness basis of ideals of prime norm up to

L Q ( 1 3 , β) is L Q ( 1 3 , -(βδ + 1 δ ) 1 3β
). To ensure that we get enough relations, we need:

β = -βδ + 1 δ 1 3β ⇒ 3β 2 -βδ - 1 δ = 0

Note, however, that[START_REF] Koblitz | Another look at non-standard discrete log and Diffie-Hellman problems[END_REF] addresses a different problem than ours.

For simplicity, we assume that q is prime although for some composite degree extensions it is sometime more efficient to choose a composite subfield as a base field.

Note that as the learning phase takes place before z is disclosed, {x 1 , . . . , x L } and L only depend on Q.

Throughout this paper log denotes the natural logarithm.

Called the algebraic side.

We assume here that the class number of Q(α) is co-prime with the relevant prime factors of Q -1. This assumption is clearly expected to hold.

In this vector, there are also a few additional components corresponding to the additive characters.

A Implementation Details

As our algorithms are sub-exponential, the assessment of their experimental behavior is essential. We hence implemented the ffs and nfs attacks as described in sections 3.1 and 4.

A.1 Function Field Sieve Experiments

The ffs-based attack acquired the capability to compute the d-th power of an arbitrary target in F 2 1025 after a learning phase of only 76,916,368 sdhp-oracle calls (these requests correspond to irreducible polynomials in α and β of degree up to 29).

We selected: f 1 (x) = x 171 + x 4 + x 3 + x 2 + 1 and f 2 (y) = y 6 + y + 1 so that an irreducible polynomial of degree 1025 divides the resultant f (x) of y -f 1 (x) and x -f 2 (y) with respect to the variable y.

Let σ be the mapping sending an integer p to a polynomial σ(p) over F 2 such that the evaluation at 2 of this polynomial yields p. For instance, σ(b) = x 3 + x + 1 where b is the hexadecimal integer 0x0b.

Denoting by π = 3.14159 . . ., the attacked instance is: which has the positive root:

This is minimized for δ = 3 3 2 i.e. β = 3 4 9 . The complexity of the sieving step is thus L Q ( 1 3 , 3 32 9 ).

B.2 Descent Initialization: The Descent's Dominating Step

In the initial descent step, we first need to go from norms of size L Q (1) down to factors of size L Q ( 2 3 ). Indeed, by opposition to the ffs setting, here smoothness testing can only be done using ecm factorization. Consequently, this step costs L factor ( 1 2 , √ 2) of the bound on the factor size.

With factors of size L Q ( 2 3 ), this becomes L Q ( 1 3 ) and we cannot afford more than that. We use the same strategy as described in [START_REF] Joux | When e-th roots become easier than factoring[END_REF] and simply randomize the input challenge until it becomes smooth as an integer. As the complexity of pulling-out a factor F with ecm costs L F ( 1 2 , √ 2), we can analyze the descent from arbitrary numbers with norm L Q (1, 1)

). Thus, the total workfactor per smooth value is

.

B.3 Continuing The Descent

In the middle of the descent, we consider a special-q of size L Q (λ, µ) with 1 3 < λ < 2 3 . We sieve over polynomials of degree:

and with coefficients of size

3 ) freedom should suffice, we can make st tends to µ from upward. The

Thus the total norm after dividing by the special-q, which can be ignored is,

Minimizing this under the constraint st = µ yields t = δ µ 2 . This step is clearly not the dominating task in the descent. In fact, we can adapt the descent's speed to the specific problem at hand. The only restriction is that the descent should be fast enough to only get a relatively small number of values at the final step of the descent.

B.4 Concluding The Descent

In the final descent step, we need to express special-q values of size immediately above L Q ( 1 3 , β) in the smoothness bases up to L Q ( 1 3 , β). We sieve over polynomials a + bx with a and b bounded by L Q ( 1 3 , β+ 2 ), with a total sieving space size equal to L Q ( 1 3 , ). At the f 2 side, the norm's size is computed as above and yields L Q ( 23 , (β + ) δ 2 + 1 δ ). At the f 1 side, only the constants matter and the norm has size

) and is left unchanged when taking into account the special-q that can be removed. The smoothness probability in basis L Q ( 1 3 , β) is:

This is constrained by the fact that we need enough sieving space, i.e. by the relation:

which yields 1.27 as in [START_REF] Joux | When e-th roots become easier than factoring[END_REF], which does not dominate the overall asymptotic complexity of the descent step.

C Complexity Analysis -High Degree nfs

Here, we work in

). We write p = L Q (λ, 1). The polynomial construction is as follows. First, choose a parameter 0 < α < 1 2 , which value will be determined later, on and an integer A close to p α . Using Euclidean division write p = BA + r p with B p 1-α and 0 ≤ r p < A.

Then, find an irreducible polynomial f 1 of degree n, which coefficients are either small numbers or a small multiples of A. All coefficients of f 1 have a size bounded by p α . Finally, let f 2 = Bf 1 mod p, all coefficients of f 2 are either small multiples of B or small multiples of r p . Hence, their size is at most of order p 1-α .

C.1 Sieving And Linear Algebra

Again, complexity is bounded by the large side, given by f 2 .

We sieve over polynomials of degree D defined by:

For a given sieving space size, this is minimized when δ(1 -α) = µ. With a smoothness basis bounded by

To balance the complexities of the sieving and the linear algebra we have:

i.e. 3β 2 = 2µ.

In addition, to ensure that we get enough relations, we need:

Put together, this yields β = 3 8 9 (1 -α). And, the complexity of the sieving step is

C.2 Descent Initialization

In the initial descent step, as in the regular nfs case, we first need to go from norms of size

). However, since f 1 's coefficients are of order p α , the norm of a generic element, with degree n and coefficient modulo p is larger than Q. This norm is in fact, of order L Q (1, 1 + α).

Using the ecm to go to factors of size

3 ). This is minimized for

Equilibrating the complexities of the sieving and of the descent's initialization, we get:

i.e. α = 37 91 .

For this α the complexities of the sieving and of the descent's initialization become:

C.3 Continuing the descent

In the middle of the descent, we encounter a special-q of size L Q (λ , µ ) with 1 3 < λ < 2 3 . We sieve over polynomials of degree:

and with coefficients of size S = L Q (λ s , s). The two values λ s and λ t are related and sum to λ . More precisely, we choose:

The sieving space size is L Q (λ , st -µ ). Since L Q ( 1 3 ) freedom should suffice, we can make st tends to µ from upward. Again, there is enough freedom here and this step is not limiting.

C.4 Concluding The Descent

In the final descent step, we need to express special-q values of size immediately above L( 13 , β) in the smoothness bases up to L( 1 3 , β). We sieve over polynomials of degree D defined by:

with coefficients of size L Q (λ- 1 3 , µ). The total sieving space has size L Q ( 1 3 , δµ-β). Here the total norm does not depend on α and is L Q ( 2 3 , δ + 2µ). Moreover, the norm's expression is left unchanged when removing the special-q contribution. The smoothness probability in basis L Q ( 1 3 , β) is

Again, this is constrained by the fact that we need enough sieving space, i.e. by the relation:

Choosing δ = 2µ, we get:

which yields µ 1.13. The total complexity L Q ( 1 3 , 2µ 2 -β) L Q ( 1 3 , 0.93) is, again, not dominating.


R Pusca

R Romary

A Ceban

Use of the External Magnetic Field for Supervision and Diagnosis of Electrical Traction Motors

Keywords: AC motors, electrical traction application, fault diagnosis, low frequency, magnetic field spectral analysis

The work presents a supervision method for detection of AC electrical machine faults, generally used in electrical traction, exploiting the information given by the external magnetic field in the vicinity of the machine. This field is an image of the air-gap flux density and consequently, it contains information concerning the failure presence. In this work, two methods based on the analysis of the magnetic field outside the machine are presented. They are reliable, original, inexpensive and simple to implement in electrical automotive vehicle. They have the advantage of being non-invasive and just get rid from the main drawback presented by other diagnostic methods based on a comparison with a healthy state assumed to be known. The both methods need simple coil sensors placed in particular positions of the motors for flux density measurements. These sensors are simple, non-invasive, low cost and they induce an electromotive force which derivative effect amplifies the low and medium frequencies. These methods are developed in the case of AC electrical rotating machines (induction and synchronous machines) often used in electrical traction. The analysis is focused on the low frequency components of the magnetic field which is not influenced by the presence of electronic converters always used in tractions applications. The theoretical analysis and his experimental validation are also presented.

Introduction

The work presented in this chapter is part of the current concerns in development of the electric traction whose EV (electric vehicle) and HEV (hybrid electric vehicle) are part. Although electric traction is not a new idea, its application to electrical automotive vehicle has be previously abandoned, consequence of an insufficient level of technological development and, recovery today to responds to current problems like the development of a non-polluting vehicle with sufficient reliability and autonomy. In this context, in order to have a chance to succeed, the automotive vehicle should not be a modification of the conventional one. It must be a "new vehicle".

Taking in consideration this remark, the work presented in this chapter show a non-invasive method for supervision and diagnosis of electrical motors used in automotive vehicle. The presence of new traction control systems in the automotive vehicle allows faster control of drive torque and improved dynamic performance [START_REF] Pusca | Contribution to the study of the driving of a mobile platform to test electrical power train components. Design of a simulator and real time system[END_REF]. However, these systems require new methods of monitoring and diagnosis to detect the faults because strong vibrations presented in embedded systems reduce the reliability of all system components including electrical machines (motors and generators). To improve the reliability of the "new vehicle" the proposed method, presented in this chapter, can assure a preventive diagnostic based by analyze of the magnetic flux leakage measured outside of electrical machine by non-invasive sensors.

Considering the electrical motor as part integrant of EV and HEV, this chapter presents the used sensors, their positioning around of motor and analysis of measured signals allowing an easy use of the method for supervision and diagnosis of AC (alternating current) motors often used in automotive vehicle. Generally, the electrical motor diagnosis methods require a good knowledge of the machine and its behaviour in the presence of internal faults. For diagnosis of electrical machines used in classical applications different methods are distinguished [START_REF] Bellini | Advances in diagnostic Techniques for induction machines[END_REF]:

-diagnostic techniques that rely on knowledge methods and which use usually the artificial intelligence. These methods are enriched by feedback and not require the mathematical model of a failure machine.

-methods of identification and estimation of the physical parameters of the machine. In this case, monitoring is established by comparing the measurement results with those obtained by analytical modelling or by calculation of a state vector.

-methods based on signal modelling. These methods analyze the time evolution or the spectral content of the different measured signals in order to detect a fault of the electrical machine.

Today many works are dedicated to motor fault analysis using different methods as the current signature analysis [START_REF] Thomson | Current signature analysis to detect induction motor faults[END_REF][START_REF] Kliman | Noninvasive detection of broken rotor bars in operating induction motors[END_REF][START_REF] Kliman | Methods of motor current signature analysis[END_REF], exploitation of the slot harmonics [START_REF] Khezzar | On the use of slot harmonics as a potential indicator of rotor bar breakage in the induction machine[END_REF][START_REF] Romary | Induction machine fault diagnosis using an external radial flux sensor[END_REF], artificial intelligence techniques [START_REF] Filippetti | AI techniques in induction machines diagnosis including the speed ripple effect[END_REF], vibrations and noise measurement [START_REF] Arthur | Induction Machine Condition Monitoring with Higher Order Spectra[END_REF][START_REF] Menacer | Stator current analysis of incipient fault into induction machine rotor bars[END_REF][START_REF] Li | Detection of induction motor faults: A comparison of stator current, vibration and acoustic methods[END_REF] or flux analysis [START_REF] Thailly | Synchronous machine diagnosis using an external magnetic flux sensor[END_REF][START_REF] Yazidi | Flux signature analysis: an alternative method for the fault diagnosis of induction machines[END_REF][START_REF] Ceban | Eccentricity and broken rotor bars faults -Effects on the external axial field[END_REF]] and all of these techniques have been validated beyond the research laboratories in order to fiend their place in industry [START_REF] Tavner | Condition Monitoring of Rotating Electrical Machines[END_REF]. It could be interesting to know if these methods can be implemented in electric traction motors in order to diagnose incipient faults because, in electric traction, the speed variation is often assured by static converters which feed the motors with adapted waveforms of the currents. These signals are generally not perfectly sinusoidal and are source for electromagnetic disturbances. Moreover, the interpretation of the results obtained from these methods requires a level of expertise advanced which makes it difficult a real democratization of these techniques among users of electrical machines. Indeed, only systems where a breakdown of the machine can have disastrous consequences (e.g. the electricity generating plant) are equipped with a supervision system.

Generally, all diagnostic methods require knowledge of the healthy state of the machine regardless of the physical exploited parameter. The fault detection is then based on the comparison of a given state signature with that of the healthy state assumed to be known, considering an indicator sensitive to a particular defect. The difficulty with this approach is that, often, the sensitive indicators already exist for the healthy machine (case of a spectrum lines). It is then their change, often the increase, which is likely to provide information on the presence of a fault. On the other hand, the load of the machine can be considered for diagnostic as a disturbing factor, because it induces several healthy states. Additional difficulty lies in the fact that the supposed healthy state is almost never known before that the failure appears, because the operator of the machine has not taken the precaution to record the parameters that characterize the healthy machine.

Recently, methods based on the analysis of the external magnetic field have appeared, they have the advantage of being non-invasive and easy to implement [START_REF] Henao | A frequency-domain detection of stator winding faults in induction machines using an external flux sensor[END_REF][START_REF] Yazidi | Flux signature analysis: An alternative method for the fault diagnosis of induction machines[END_REF][START_REF] Kokko | Condition monitoring of squirrel-cage motors by axial magnetic flux measurements[END_REF]. These techniques exploit information in the stray field around the machine in order to respond to industrial problems concerning the diagnosis [START_REF] Negrea | Electromagnetic flux monitoring for detecting faults in electrical machines[END_REF], but also on monitoring the aging of insulation or the monitoring of eco efficiency [START_REF] Werynski | Vieillissement des diélectriques et surveillance in situ des machines électriques[END_REF][START_REF] Zidat | Conception d'une cellule de suivi des performances d'éco-efficacité énergétique des machines électriques tournantes à courant alternatif[END_REF]. The different applications differ in the frequency range to be analyzed.

The current diagnosis techniques exploiting the range of low and medium frequencies, up to the slotting effect of the electric machine, give the possibility to detect the stator faults (inter-turn short circuit) and rotor faults (broken bars, inter-turn short circuit) by analysis of the external magnetic field of induction and synchronous machines. At the base of these techniques, there is an analytical model, which shows the decomposition of the external magnetic field into a radial and axial field does not have the same properties, because each has a different origin. This model predicts the spectral components of the stray field affected by different faults.

The work presented in this chapter is part of this category, mainly considering the effects of a fault on the magnetic stray field and it is adapted toward the diagnosis of rotating electrical machines including electric traction motors since it is not influenced by the static converters and use the load working for the detection of the faults. Nowadays, rotating electrical machines are presented in all areas of modern life including the electrical vehicle and one of the most important objectives of the maintenance systems is the detection of incipient faults in order to anticipate a shutdown of the system. In this work, two methods based on the analysis of the magnetic field outside the machine are presented. They are reliable, inexpensive, simple to implement and non-invasive. These methods are developed in the case of AC electrical rotating machines (induction and synchronous machines). The analysis focuses on the magnetic field dispersion in the vicinity of the machine, especially its radial an axial distribution which presents different sensitivity according to various faults.

For application of these methods in electrical traction, two coil sensors are used to measure the stray field in the motor vicinity (Fig. 1). These sensors induce an electromotive force (emf) which derivative effect amplifies the low and medium frequencies. They are simple to carry out, non-invasive and low cost. Moreover, they can be set up outside the motor without needing large place in electrical system of hybrid vehicles. The diagnosis procedure is able to indicate the failure presence and can be implemented within a simple microcontroller. For diagnosis, each fault gives specific signature that are identified and analysed. Two diagnosis methods are developed in this work. The first method concern the inter-turn shortcircuit faults in the stator winding. It uses two external flux coil sensors placed in particular positions that measure the external magnetic field, the signal delivered by the both sensors being exploited for diagnosis. The analysis concerns the magnitude of specific harmonics of the electromotive force delivered by the coil. One will be interested in the magnitude of the third rank harmonic for Salient Synchronous Machines and the rotor slotting harmonics for Induction Machines. This method does not require any knowledge of a presumed machine's healthy former state, it's originality is the use of the load to improve the diagnostic. Here, the load is not longer a disturbing factor but it is used for the fault discrimination. Actually a comparison between machine operating conditions (no-load and load) enables to detect a stator fault.

The second method is based on the variations of the axial flux density in the presence of rotor faults. Any changes of magnetic circuit conditions, such as a broken bar or the formation of an eccentric air-gap due to bearing wear, will, by the same token, be reflected with a corresponding change in the axial flux. The analysis is focused on the low frequency components of the magnetic field, which are not influenced by the presence of electronic converters usually used in industrial and traction applications and which are less attenuated during their penetration through the stator magnetic core or the machine external frame. The both methods need simple coil sensors for flux density measurement.

In the end of the chapter experimental implementation of the sensors and results are presented. The faults have been artificially created in the rotor and stator on different AC machines, special designed for the tests in order to present the experimental results of the suggested diagnosis methods.

Magnetic field

From a diagnostic point of view, the advantage of methods based on measurement of the external magnetic field is that they are non-invasive. These methods are simple to implement and they use a sensor placed near the electrical machine, able to measure the external magnetic field. Considering the position of the sensor, the measured of dispersion field does not originate from the same source, and not result, necessarily from the same physical phenomenon [START_REF] Thailly | Etude du champ de dispersion des machines à courant alternatif: Application au diagnostic[END_REF].

Measurement of external magnetic field

In a rotating electrical machine, the leakage flux is created by the different elements of the machine. A distribution of these field lines is shown in Fig. 2 [START_REF] Thomson | A review of on-line condition monitoring techniques for three-phase squirrelcage induction motors-past, present and future[END_REF]. From this distribution, we can decompose the external field in two parts called axial and radial field. These two fields are combined to form the leakage field: the axial field in a plane that includes the axis of the ma-chine and the radial field in a plane perpendicular to the axis of the ma-chine.The leakage field may give information about the presence of a fault, and sometimes its location. Figure 3 shows the different position for measurement of the leakage field using a coil sensor. Assuming the field lines shown in Fig. 3, the sensor will be sensitive to different components of the field and the measured signal will depend on the position of the coil: Fig. 3. Measure of leakage field in near vicinity of the machine -in position 1 (Pos. 1), the coil measures the axial field. In this case the amplitude of the measured flux is maximum at the motor endshield (around heads of the windings), and minimum at the middle of the machine length; this minimum is practically null.

-in position 2 (Pos. 2), the coil measures the normal component of the radial field. Taking the coil sensor away from the motor endshields increases the sensitivity to the air gap flux density.

-in position 3 (Pos. 3), the coil is used to measure the external field called "pure radial" sensitive to the tangential component. In theory, field lines generated by the winding heads cannot cross the section of the coil sensor in this position.

Measurement sensors

Several types of sensors can be used to measure the external field (magnetoresistive sensor, Hall effect sensor, coil sensor ...).

In the diagnostic method presented in this chapter, the coil sensor is used. The principle of the coil sensor is to measure the field embraced by a coil located in the vicinity of the electrical machine [START_REF] Tumanski | Induction coil sensors -a review[END_REF]. The sensor provides an emf equal to the derivative of the magnetic field with respect to the time; however, the measurement is only possible for the variable fields. With this type of sensor, the measurement is simple and direct because it does not require associated electronics, unless it is necessary to amplify the induced emf signal [START_REF] Thailly | Etude du champ de dispersion des machines à courant alternatif: Application au diagnostic[END_REF]. The performance of the measurement depends on the size of the sensor. The main advantage of these sensors is their simple design which allows users to make it according to their needs [START_REF] Ceban | Supervision of electric traction motors by external magnetic field analysis[END_REF]. The sensor used for experimental tests and its frequency response are represented in Fig. 4. This small sensor will be used for all experimental measurements presented in this work. It was been built for small and low power machines. The coil diameter is about 3.2 cm and it has 1200 turns. Its frequency response shows a resonance at 93 kHz, so, the sensor is suitable to be used for measurement of the field in a frequency range lower than this frequency, what actually corresponds to the frequency range exploited by proposed diagnosis methods. This type of sensor is used for flux measurements because it is simple, non-invasive, low cost, induces an emf with a natural derivative effect that amplifies the low and medium frequencies [START_REF] Pusca | An online universal diagnosis procedure using two external flux sensors applied to the ac electrical rotating machines[END_REF].

Stator faults in AC electrical rotating machines

The most of stator faults are attributed to the degradation of insulation. They manifest themselves as a short-circuit between turns, by a short circuit between two phases or by a short-circuit between a phase and the ground. In this paragraph the analytical model presented is developed for the shortcircuit between the turns of the same phase because it is a common defect that can occur either at the coil ends or in the notches, causing a decrease of effective winding turns. On the other hand, it leads to increase the stator current in the affected phase and to a slight variation of the amplitude in the others. In the case of induction machines, it amplifies also the current in the rotor circuit [START_REF] Bonnett | Cause and analysis of stator and rotor failures in three-phase squirrelcage induction motors[END_REF][START_REF] Nandi | Condition monitoring and fault diagnosis of electrical motors-a review[END_REF]. Thermal stress caused by short-circuit can cause the spread of the fault to other coils and the complete destruction of the machine.

The developed analytical approach makes it possible to identify the properties of the airgap flux density waves generated by the fault and proposes a diagnostic method based on those.

Analytical model

For analytical study of stator faults in AC motors, a simplified geometry of the machine has been considered (Fig. 5) to study the external magnetic field. The point M' outside the machine has polar coordinates x, s . The stator is composed of a magnetic yoke and an external frame; the gap which can possibly exist between the stator yoke and the external frame is neglected. To simplify, the considered machine presents smooth airgap where the flux density is imposed. Let us suppose that y turns among n s / m s of an elementary coil of the phase q are short-circuited. If y is small compared to pn s , it is possible to consider that the s q i currents are unchanged. This assumption makes it possible to characterize inter-turn short circuit by implementing a model which preserves the initial structure of the machine. This model considers that the faulty stator winding is equivalent to the healthy winding, plus y short-circuited turn coil which is denoted "additional coil".

A s qf i fictitious current circulates in this coil, which is composed of inter-turn short circuit induced cur- rent s sc i and the current in opposite direction s q i :

s s s qf sc q i i i = - (1)
A damaged elementary coil is shown in Fig. 7. This approach makes it possible to separate the effects of the fault while preserving the initial structure, what simplifies the fault analysis. In this way, the resulting air-gap flux density b * can be expressed as: The particularity of this result concerns the s qsc ε mmf generated by the additional coil which is a unidirectional wave, although s ε is a rotational one. Also, its polarity H is equal to 1 that produces a dissymmetry in the stator mmf wave.

s qsc b b b * = + (2) with , , K H K H b b = , where ( ) , , , ˆcos s K H K H K H b b K t H ω α ϕ = - - (
In the stator reference frame s d which is confounded with the stator phase 1 axis, s qsc ε can be written as the sum of rotational mmf waves evolving per pair in opposite directions:

( ) cos s s s s qsc f h h h I A t h ε ω α ϕ = - - (3)
where s f I is the rms value of s qf i , ω is the angular frequency, h is a not null relative integer, which takes among others, all the values of s h which is defined by: 6 1 ( )

s h k = + ( k is an integer which varies between -∞ to +∞ ).
cos +∞ +∞ =-∞ =-∞ = Λ + - s r s r s s r r s r r k k k k k N k N p pk N λ α θ (4)
where Λ s r k k is a coefficient which depends on the ranks s k and r k and the air-gap geometry, s N is the stator per pole pair tooth numbers.

Model for salient-pole synchronous machine

For a healthy salient-pole synchronous machine, K and H are defined as:

( )

1 2 2 = + = + + r sr s s r K k H p h k N k (5)
where sr h includes all the values taken by s h and r h :

sr s h h = U r h and ( ) 6 1; 1,3,5,
, = + ∈ +∞ s r h k h .
If a fault exists, the flux density s qsc b can be expressed as:

( ) , , , ˆcos K H K H sc sc sc sc sc sc s s qsc sc sc sc sc K H b b K t H ω α ϕ = - - (6)
with:

1 2 2 r sc s s r sc K k H h pk N pk ′ = + ′ ′ = + + (7)
s k′ and r k′ are equivalent to s k and r k . Consequently they vary from -∞ to +∞ . The resulting flux density appears, after attenuation, at the level of the external transverse field. That results in the presence of harmonics at Kω and sc K ω angular frequencies in the signal delivered by the flux sensor. Considering K and sc K the values calculated by ( 5) and [START_REF] Romary | Induction machine fault diagnosis using an external radial flux sensor[END_REF], it is possible to make the following remarks: • sc K does not bring new frequencies and the failure presence will be appreciated through the variation of the amplitudes of existing lines in the healthy machine spectrum.

• the sensitive spectral lines are in low frequencies. Indeed, the corresponding harmonics for 1 r k′ = ± , are ω and 3ω . Consequently, one of these components is confused with the fundamen- tal [START_REF] Thailly | Synchronous machine diagnosis using an external magnetic flux sensor[END_REF].

• Other sensitive spectral lines ( )

2, 3, r r k k = ±
= ± are superimposed on the components at angular frequencies 3ω , 5ω , 7ω , ...; that may exist for a healthy machine (supply harmonic, satura- tion). These properties make difficult the diagnosis by analysis of the changes in the amplitudes of the measured components. In the following the properties relating to the dissymmetry generated by the fault will be exploited.

Model for induction machine

Under healthy conditions the quantities K and H in the case of induction machine can be expressed by: ( ) ( )

1 1 r r s s s r r K k N s H p h k N k N = + - = + + (8)
The sensitive lines to the fault are characterized by the quantities sc K and sc H defined as follows:

( ) ( )

1 1 r r sc s s r r sc K k N s H h p k N k N ′ = + - ′ ′ = + + (9)
A numerical application for induction machine is presented in annexe. The results obtained by compilation show that each harmonic of the external field, at given r k is essentially created by components with the lowest possible polarities. The analysis will focus on the diagnosis harmonic at 850 Hz, because it is generated by a component with the lowest polarity both for the healthy machine and for the faulty machine ( )

2, 1 cc H H = - = -.
As shows the numerical application (annexe 1) this harmonic has a high amplitude value and can be more easily measured at the outside of motor frame.

Principle of the tests

For the new diagnosis method, which exploits the change of external magnetic field in presence of the faults, the measurements are carried out in two diametrically opposed positions: The principle of the method consists in the analysis of the variations induced by the fault at the level of particular harmonic components measured in the both positions. The corresponding flux density components must have significant amplitude in the external magnetic field. Consequently, taking into account the magnitude of the air-gap flux density and considering the attenuation of these components in the stator core, one will be interested in the components defined by the lowest values of , , , , , will be not very sensitive to the components defined for 1 > H .

In the case of salient-pole synchronous machine, the analysis will be focused on the spectral lines related to 3 K = , ( line at 150Hz). For a four pole machine ( ) 2 p = , the flux density components that contribute to this harmonic are defined as follows:

• for healthy machine:

1, 0 5), which gives 3 K = and 3 6 H p = = ;

r s h k = = , 1, 1 r sr k h = = in (
• for faulty machine: 3, 0, 1 [START_REF] Romary | Induction machine fault diagnosis using an external radial flux sensor[END_REF], that leads to 3

s r h k k ′ ′ = - = = in
sc K K = = , 1 sc H = -.
The flux density component 3

x b * at 3ω angular frequency on the outside of the machine is the sum of a term relative to the healthy machine (of amplitude 3 ˆx b ) and a term relative to the faulty machine (of amplitude 3 ˆx cc b ). In Pos.1, the resulting component can be expressed as:

( ) ( ) * (1) 3 3 0 3 3 0 3 ˆĉos 3 6 cos 3 = - + + + + x x x x x cc sc b b t b t ω β ϕ ω β ϕ (10)
In Pos.2, it becomes:

( ) ( ) * (1) 3 3 0 3 3 0 3 ˆĉos 3 6 cos 3 = - + - + + x x x x x cc sc b b t b t ω β ϕ ω β ϕ (11)
The difference between the both positions is tied to the sign which precedes the component due to the fault. For a healthy machine such as 3 ˆ0 x sc b = , the components defined by ( 10) and ( 11) present the same amplitude. Under fault conditions, the amplitude of the component in one of the positions increases whereas the other one decreases. Practically, it can observe differences between measurements for the both positions, even in the healthy case: (1) 3 ˆx b in Pos.1 is different from (2) 3 ˆx b in Pos.2. Two external causes can be responsible for this dissymmetry: • Pos.1 and Pos.2 cannot be perfectly symmetrical from each other.

• the machine realization techniques can lead to a dissymmetry causing different attenuation coefficients at the level of the two sensor positions. In this case, the method can be extended thanks to the exploitation of a test in load. The method can be refined as following:

Under healthy conditions, at load, the stator current increases but, at given supply voltage, the air-gap flux density stays practically identical to that obtained at no-load. As the external elements responsible of the attenuation act in the same way in Pos.1 and Pos.2 for the no-load and for the load tests, the amplitudes (1) 3 ˆx b and (2) 3 ˆx b keep similar values or at least evolve in the same way when the machine is loaded.

Under faulty conditions, at load, 3 ˆx sc b varies taking into account the model chosen to characterize short-circuit (the loading current contributes to the definition of the fictitious current which circulates in the damaged coil). Consequently, according to [START_REF] Menacer | Stator current analysis of incipient fault into induction machine rotor bars[END_REF] and [START_REF] Li | Detection of induction motor faults: A comparison of stator current, vibration and acoustic methods[END_REF], the magnitude of the component at 3ω angular frequency in Pos.1 and Pos.2 will evolve in opposed directions. The variations of the lines at 3ω are thus an indicator of defect. The advantage of the method is that it does not require any knowledge of a healthy state to detect the fault.

In the case of induction machine, the lines are located in a frequency range where they can be easily identified. The analysis is focused on the line at 850 Hz because it is relative to a flux density component having a low pole pair number as well for the healthy case as for the faulty one.

• for healthy machine:

1, 1, 7

r s s k k h = = - = lead to 17, 2 K H = = -in (8)
• for faulty machine: 1, 1, 15

r s s k k h ′ ′ = = - = lead to 17, 1 sc sc K H = = -in (9).
Relations similar to [START_REF] Menacer | Stator current analysis of incipient fault into induction machine rotor bars[END_REF] and ( 11) are obtained, but by taking 17

sc K K = = (instead of 3) and 2 H = - (instead of 6).

Rotor faults in AC electrical rotating machines

This section briefly presents the main phenomena characterizing the rotor faults. It will focus on the influence of a broken bar on the change of the external magnetic field of an induction machine.

An analytical approach requires some simplifying assumptions but will allow one to give the specific harmonics of the fault, including their frequency and polarity. The analytical study will highlight the need to consider the axial field as part of leakage flow for which a simplified model will be proposed and a specific signature for this fault type, visible in the spectrum of the external magnetic field.

Types of the rotor faults

Bearings faults

The majority of faults in electrical machines are relative to bearing defects that have mechanical causes (lubrication, fatigue, etc.) or electrical (circulation of leakage currents induced by the converters [START_REF] Busse | The effects of PWM voltage source inverters on the mechanical performance of rolling bearings[END_REF]) and leads to more general mechanical effects such as increased noise and vibration by the appearance of the movements of the rotor about the longitudinal axis of the machine [START_REF] Vaseghi | Contribution à l' étude des machines électriques en présence de défaut entre-spires[END_REF]. This type of faults induces also variations (oscillations) of the load torque. The ultimate point of the bearing defect is the rotor blocking.

Eccentricity

The eccentricity of an electric machine is a phenomenon that evolves over time and exists from its manufacture. It passes through various stages of fabrication and assembly tolerances, which induces to decentre the rotor relative to the stator. During operation of the machine, two main causes increase rotor eccentricity. The first is inherent to the system shaft in which the machine operates and can impose a radial force on the shaft of the machine. This can lead to bearing wear and amplification of the eccentricity. The second phenomenon that may exacerbate the eccentricity is itself inherent to the operation of the machine. Actually, the shift creates an unbalance in the distribution of radial forces between the stator and the rotor [START_REF] Dorrell | Calculation of UMP in induction motors with series or parallel winding connections[END_REF][START_REF] Dorrell | Analysis of airgap flux, current, and vibration signals as a function of the combination of static and dynamic airgap eccentricity in 3-phase induction motors[END_REF]. The ultimate point of eccentricity is the friction between stator and rotor, which is synonymous of fast destruction of the machine.

Broken bar

Inter-turn short circuit fault may also occur in a wound rotor. For squirrel cage induction machines, broken rotor bars and the breaking of the short-circuit rings are the most common rotor failures (Fig. 10). The broken bar or the breaking of the short-circuit rings may be due, for example, to mechanical overload (frequent starts, etc..), to excessive local heating or to a manufacturing defect ( air bubble or bad welds). This generates oscillations of currents and electromagnetic torque even more apparent when the inertia of the system has a high value (constant velocity) [START_REF] Ceban | Study of rotor faults in induction motors using external magnetic field analysis[END_REF][START_REF] Bossio | Separating Broken Rotor Bars and Load Oscillations on IM Fault Diagnosis Through the Instantaneous Active and Reactive Currents[END_REF]. If the system inertia is lower, oscillations appear on the angular speed and the amplitude of the stator currents.The occurrence of a broken bar does not cause an immediate breakage of the machine, because the current crossing the broken bar spreads over the adjacent bars. Therefore, these bars are overloaded and the thermal and electromechanical constraints can lead to their breaking and so on until the breakage of a sufficiently large number of bars to stop the machine.

Rotor broken bar

Influence on the speed variation

The presentation of the phenomenon will consider only the flux density component of p pole pairs. Under healthy operating conditions, the balanced three-phase current system creates a fundamental clockwise rotating magnetic field in the air-gap. This field induces a current in the rotor bars with a frequency proportional to the rotor slip s . The rotor bars then generate a clockwise field rotating at Ω s s in a rotor referential (Fig. 11 (a)). For a symmetrical motor (healthy machine), there is not any anticlockwise field in the air-gap [START_REF] Filippetti | AI techniques in induction machines diagnosis including the speed ripple effect[END_REF][START_REF] Didier | An induction motor model including the first space harmonics for broken rotor bar diagnosis[END_REF]. Under rotor faults, in addition to the clockwise field an anticlockwise field appears, created by the unbalance of rotor bar currents and which rotates at an angular velocity -Ω s s shown in Fig. 11 (b). It is possible to associate with this anticlockwise field a negative-sequence current system with amplitude rn I , which is added to the positive-sequence current system with amplitude rp I . ( )

1 2 Ω = Ω -Ω = Ω - fault r s s s s ( 12
)
which leads to the well-known frequency in literature fault f , appearing in the radial external magnetic field and in the line current:

( )

1 2 = - fault f s f ( 13
)
The interaction of the rotor currents with the clockwise stator field produces the electromagnetic torque. In case of broken rotor bar fault, the total torque can be expressed as follows: ϕ are respectively the phases relative to the stator phase one current of the stator flux density, the rotor flux densities in the clockwise direction and in the anticlockwise direction; s k is a coefficient that depends on the stator winding arrangement and r k is a coefficient that depends on the rotor cage constitution.

The torque oscillation at frequency 2sf , also presented in [START_REF] Filippetti | AI techniques in induction machines diagnosis including the speed ripple effect[END_REF], generates a speed oscillation at this frequency whose amplitude depends on the rotor asymmetry, the motor load level [START_REF] Didier | An induction motor model including the first space harmonics for broken rotor bar diagnosis[END_REF][START_REF] Bellini | Quantitative evaluation of induction motor broken bars by means of electrical signature analysis[END_REF], and the total moment of inertia J . The speed is calculated with the relation:

_ _ Ω = -= r e tot r e srn d J C C C dt (15)
where r C is the load torque:

_ 0 1 Ω = + Ω r e srn C dt J (16)
By calculating the integral (11) one obtains: 0 cos( 2)

Ω = Ω + ∆Ω ⋅ + r r s t ω ϕ (17)
where

0 1 2 m r s rn rn RLe k Ik I sJ π µ ∆Ω = - ⋅
is the magnitude of the speed oscillation, 0 Ω is the mean value of r Ω which changes in the dependence on the number of broken bars at given load torque, and

s rn B B ϕ ϕ ϕ = - .
It is well known that the speed oscillation also generates an additional air-gap flux density component at ( )

1 2s f + .

Influence on the axial flux

For the modeling of the axial field, it is necessary to consider the field lines existing at the machine extremities. The distribution of these ones strongly depends on the geometry and the mechanical components that exist at the machine extremities. As mentioned in [START_REF] Ceban | Study of rotor faults in induction motors using external magnetic field analysis[END_REF], the total axial flux AT Φ outside the induction machine with an eccentricity fault can be written in the following form:

( )

ˆcos Φ = Φ - AT t p ω β (18)
with 0 r t β β = + Ω , β indicates the position of the minimum air-gap thickness. Under broken rotor bar fault, a speed variation appears and will influence the external axial flux. In this case ( 18) can be written as follows:

( )

0 ˆcos Φ = Φ - + Ω AT r t p t ω β (19)
Developing the argument of the cosine, using (17) one obtains:

[ ] 0 ˆco cos( s 2 ) Φ = Φ -∆Ω + - r AT s t p t t s p ω ϕ β ω (20)
A frequency modulation appears in AT Φ . The time variation of (20) for ˆ1Wb

Φ = , 1 rad/s r ∆Ω = and 2.5% s =
is presented in Fig. 12 (a). A direct analysis of the signal is difficult; hence the change in the frequency domain is performed by Fast Fourier Transform (FFT). The obtained spectrum is given in Fig. 12 (b). In the spectrum it can be observed the appearance of the harmonic 3sf due to the speed variation and sf due to dynamic eccentricity on the axial flux. One can notice that a small speed variation can produce an important component at frequency 3sf . With the considered speed variation that corresponds approximately to 2 broken bars; the component 3sf is higher than the initial component sf .

Experimental tests 5.1 Achievements of the tests

The application of proposed methods for stator faults is presented considering the test carried out on a salient-pole synchronous machine and an induction machine. For rotor faults an experimental bench using two identically squirrel-cage induction motors (one healthy and another with a broken bar) is used. The principle of measurement is shown in Fig. 13 Fig. 13. Measurement and treatment of emf signal

Stator tests

Inter-turn short circuit in synchronous machines

The application of the proposed method is presented considering a test carried out on a salientpole synchronous machine characterized by: 7.5 kW, 50 Hz, 2 p = , 230 / 400 , 18 s V N = , grid connected. This machine has been rewound so that the terminals of the different stator elementary sections are extracted from the winding and are brought back to a connector block which is fixed above the machine as indicated in Fig. 14. So it is possible to produce short-circuits between elementary coils. The tests consist in measuring and analyse the emf delivered by the sensors in order to validate the suggested procedure when an elementary coil is short-circuited (what corresponds to 16,6% of the whole winding of a phase). In order to avoid total damage of the winding, the short-circuit current is limited by an external resistance R . Verification have also be done in order to make sure that the currents in the wires going up from the machine towards the connector block do not disturb the measurements. Pos. It can be observed here that the magnitudes vary in the same way (decrease) for the two positions. Under faulty conditions at full load (Fig. 16) it can be observe that the magnitudes in both positions vary in the opposite ways. This indicates the failure presence and confirms the proposed method.

Fig. 16. Harmonic components at 150 Hz for faulty salient-pole synchronous machine (motor) at no-load and load operations.

Another interest is the limit of the method. The tests are realised on the machine which operates as generator. The corresponding results are shown in Fig. 17 and Fig. 18. One can remark the same phenomenon: under healthy case the magnitudes varies in the same direction, but in the faulty case they vary in opposite direction. The next test has been realized on a four pole, 4,4 kW, 50 Hz 3 × 380 V permanent magnet synchronous motor. The obtained results under healthy and faulty conditions are given in Fig. 19 and Fig 20 . It can be observed that under healthy case the harmonic magnitudes at 150 Hz vary in the same direction, but when the inter-turn short circuit exists in the stator winding these harmonic magnitudes vary in different direction. Fig. 18. Harmonic components at 150 Hz for faulty salient-pole synchronous machine (generator) at no-load and load operations Fig. 19. Harmonic components at 150 Hz for a healthy permanent magnet synchronous motor at no-load and load operations Fig. 20. Harmonic components at 150 Hz for a healthy permanent magnet synchronous motor at no-load and load operations

Inter-turn short circuit in induction machines

In order to generalize the proposed method, the tests have been also realized on a 2, 50 Hz, 11kW p = , 380 / 660 V, 24 This machine, connected to the grid, has been rewound so that the terminals of the different stator elementary sections are extracted from the winding and are brought back to a connector block which is fixed above the machine. So it is possible to produce short-circuits between elementary coils. Verification have also be done in order to make sure that the currents in the wires going up from the machine towards the connector block do not disturb the measurements. Pos.1 and Pos.2 correspond respectively to 0 β and 0 β π + , 0 β is chosen "arbitrarily". One will be interested to the spectral line at 850 Hz ( ) The phenomena are similar than that obtained in the case of synchronous machines: • under healthy condition -the lines relative to 17 sc K K = = evolve in the same way with the load (Fig. 22),

• under faulty condition -these lines evolve in opposite way with the load (Fig. 23). This method is thus applicable to salient-pole synchronous machine, permanent magnet synchronous motor and induction motor often used in electrical traction applications and that confirm the universal quality of proposed diagnosis method.

Broken rotor bar in induction machine

The tests are performed on 4 poles, 4 kW / 50 Hz, 3 × 380 V squirrel cage induction motor under healthy and faulty (broken rotor bar) conditions. The obtained results at full load are shown in Figs. 24 and25. If the rotor has a broken bar, the frequency component 3sf is more sensitive than other components and it denotes the speed variation at frequency 2sf , caused by a fault. Figure 25 shows that the amplitude of this harmonic is higher in comparison with the component sf (eccentricity indicator).

Also, one can observe in the case of healthy machine, the presence of frequency components sf and 3sf with low amplitude. It is produced because no machine can be constructed with perfect symmetry.

Conclusions

The paper presents a simple method to insure the supervision of AC machines used in electrical traction. This supervision can detect the stator and the rotor faults using original signatures presented in the paper. Under stator winding faults the signature concerns the harmonic at 150 Hz for synchronous machines and the slotting harmonic at 850 Hz for induction machines. This diagnosis method does not require the knowledge of the healthy signature. It is based on comparison that concerns the state at no-load and load operations.

The indicators of rotor faults are the low frequency components of the flux spectrum at sf and 3sf frequencies. The diagnosis method consists in checking the magnitude of the component 3sf generated by the speed variation that can be compared to this of the component sf .

The advantages of proposed non invasive methods compared to other existing methods are their simplicity to implement, to interpret and a low cost set-up. The processed emf signal, delivered by a simple coil sensor, is able to indicate the presence of the fault. Moreover, the low frequency analysis is not disturbed by other sources of magnetic field presented in vicinity. Finally, the experimental tests realized on the different AC machines confirm the validation of used models and proposed methods for stator and rotor fault detection. The simulation and experimental analyses confirm the models and used approaches.

Annexe

For a numerical application it is considered an induction machine with p=2, [START_REF] Tumanski | Induction coil sensors -a review[END_REF] In order to have

Fig. 1 .

 1 Fig. 1. Positions of sensors for on-line supervision of HEV traction motor.

Fig. 2 .

 2 Fig. 2. Distribution of leakage flux lines in electrical machine

Fig. 4 .

 4 Fig. 4. Coil sensor: (a) photo; (b) frequency answer

Fig. 5 .

 5 Fig. 5. Transverse section and considered parameters of simplified machine

Fig. 5 Fig. 6 .

 56 Fig. 6. Three phase stator windings ( ) 2 = p

Fig. 7 .

 7 Fig. 7. Elementary coil with y short-circuited turns

Fig. 8 .

 8 Fig. 8. Mmf components generated by elementary damaged coil created by: (a) two healthy elementary sections; (b) y short-circuited turns

  s h A is a function that takes into account the winding coefficient tied to the rank s h . h ϕ is the phase which depends on the position of the elementary coil concerned by the short-circuit but also of the phase where the short-circuit occurs. The flux density s qsc b generated by the fault is obtained by multiplying s qsc ε with air-gap permeance λ expressed by:

Fig. 9 .

 9 Fig. 9. Sensors position for measurement of external magnetic field

  the smallest as possible value of H . The respect of these constraints leads to consider the spectral line at ω angular frequency ( )1 = K of magnitude1,1 b . However, this component corresponds to the fundamental of the air-gap flux density which generates the main effects: 1,1 1, ˆ>> H b b with H different from 1. It results that 1 b

Fig. 10 .

 10 Fig. 10. Fault of squirrel rotor cage: (a) broken bar ; (b) breaking of the short-circuiting ring (a) (b Rotor bars

Fig. 11 .

 11 Fig. 11. Operation schema for: (a) healthy motor; (b) faulty motor

Figure 11 (

 11 Figure 11 (b) shows the phenomenon due to the appearance of rotor failure. A flux density component rotating at Ω fault appears in the stator referential:

  torque due to the interaction of the stator and the rotor fields in the clockwise direction, _ e srn C is the pulsating torque of pulsation 2sω due to the interaction of the stator clockwise field with the anticlockwise rotor field;

Fig. 12 .

 12 Fig. 12. Simulation of axial flux variation: (a) axial flux signal; (b) FFT for the axial flux

Fig. 14 .

 14 Fig. 14. Modified salient-pole synchronous machine and the sensors' positions

  1 and Pos.2 correspond respectively to 0 β and 0 β π + , 0 β is chosen in the axis of the tested phase. The interest is focused on the harmonic at 150 Hz. Figures 15 a and b shows the variations of the harmonic at 150 Hz relative to the healthy machine (motor) when the state (marked with arrow) varies from no-load to load.

Fig. 15 .

 15 Fig.[START_REF] Tavner | Condition Monitoring of Rotating Electrical Machines[END_REF]. Harmonic components at 150 Hz for a healthy salient-pole synchronous machine (motor) at no-load and load operations.

Fig. 17 .

 17 Fig. 17. Harmonic components at150 Hz for a healthy salient-pole synchronous machine (generator) at no-load and load operations

  cage induction motor (Fig.21).

Fig. 21 .

 21 Fig. 21. Modified squirrel cage induction motor and sensors' positions

1 rk

 1 = ± . The results obtained by considering the harmonic of rank 17 and load), are presented in Figs.22 and 23.

Fig. 22 .Fig. 23 .

 2223 Fig. 22. Harmonic components relative to 17 = = sc K K for healthy induction motor at no-load and load operations

Fig. 24 .Fig. 25 .

 2425 Fig.[START_REF] Tumanski | Induction coil sensors -a review[END_REF]. Frequency spectrum of the external magnetic field (Pos.1) for healthy squirrel cage induction motor (s = 5%)

  the study for the first lines of rotor teeth obtained for 1 r k = ± , for one healthy induction machine with a slip null and considering 1

  The values that can take the coefficients that contribute to the description of predominant components are given by:

	N	s	=	,	N	r	=	16,	ts int R	=	0.0825 m,	ts ext R	=	0.121 m	and a short circuit in the phase 1. Here 12.5% of
	the phase coils are short-circuited and the short-circuit current is i . ( 3 = s s sc i ) ( ) ( ) ( ) ; ; ; ; ; ; ; ;
				6 1;		( 1,3,5, ,	)

Table 1 :

 1 possible for the healthy machine and for the faulty coils are gives respectively in Table1and Table2with the amplitude in Components generated by the healthy IM

						s s k N k N r +	r	the lowest possible (here equal to -6), we must have	1 k = . Also, s
	we find that the dominant mode is obtained for	5 h = -, s	6 H = (in absolute terms and because
	2 p = ). For	1 k = we obtain : r
	r k N = r	16	et	17 K = ,	f =	850 Hz	.
	As s s k N k N r +	r	must be the lowest possible (here equal to -2), we must have	1 k = -. Also, we s
	find that the dominant mode is obtained for	7 h = , s	2 H = (in absolute terms).
	The other components defined for values , , s r k k k k h h , , , s r s ′ ′ of the lowest s r ext x R = gives relative to fundamental.
					k	r	k	s	h	s	K	H	Kω	( ) Hz	κ	ts H	, ˆ/ x K H b	11 x b
				-1	1		-5	-15 6	-750	5.09e-05	2.99e-05
				-1	1		-11 -15 -6	-750	5.09e-05	8.16e-06
				1		-1 7			17 -2	850	8.16e-04	2.62e-04
				1		-1 13 17 10 850	3.97e-06	5.32e-07
				-2	1		7			-31 -2	-1550	8.16e-04	-2.49e-04
				-2	2		-17 -31 -2	-1550	8.16e-04	5.66e-05
				2		-1 -5	33 6	1650	5.09e-05	-2.85e-05
				2		-1 -11 33 -6	1650	5.09e-05	-7.76e-06
				2		-2 13 33 -6	1650	5.09e-05	3.81e-06
				2		-2 19 33 6	1650	5.09e-05	4.04e-06

Table 2 :

 2 Components generated by the faulty coils

	k′	r	k′	s	h	K	cc	H	cc	cc K ω	( ) Hz	κ	ts H	cc	, ˆ/ cc cc cc K H x b	11 x b
	-1		1		-10 -15 6		-750		5.09e-05 1.82e-05
	-1		1		-13 -15 3		-750		3.76e-04 -7.31e-05
	-1		1		-14 -15 2		-750		8.16e-04 -2.07e-04
	-1		1		-15 -15 1		-750		0.0023	-3.78e-04
	-1		1		-18 -15 -2		-750		8.16e-04 1.60e-04
	-1		1		-19 -15 -3		-750		3.76e-04 4.94e-05
	-1		1		-21 -15 -5		-750		9.72e-05 -1.15e-05
	-1		1		-22 -15 -6		-750		5.09e-05 -8.09e-06
	1		-1		10	17	-6		850		5.09e-05 1.82e-05
	1		-1		11	17	-5		850		9.72e-05 2.23e-05
	1		-1		14	17	-2		850		8.16e-04 -2.07e-04