Olivier Coquelin

Class Struggle in the 1916-23 Irish Revolution: A Reappraisal

Keywords: Ireland, revolution, class struggle, social disputes, trade unions, soviets, conservatism

In line with the view generally held by most historians over the past decades, Peter Hart's The IRA at War, 1916-1923 (2003) tends to play down significantly the importance of the social disputes in the 1916-23 Irish Revolution. Hart justifies his approach by the fact that those disputes did not pave the way for radical social change in the new Irish Free State. However respectable this statement may be, it remains nonetheless disputable given that, in Terence Dooley's own words, "other than acknowledging some contribution of agrarian issues to the revolution, historians have failed to take up the challenge of exploring them in greater detail or, indeed, to be fully convinced of their existence" (see "The Land for the People": The Land Question in Independent Ireland, Dublin, University College Dublin Press, 2004, p. 17). This observation is also consistent with the fact that such subversive experiments as the 1918-1923 Irish Soviets have hitherto been examined somewhat on the fringe of mainstream academic research. As part of a project devoted to the latter, the present article will therefore seek to reappraise the relevance of Hart's analysis by assessing the extent to which "class struggle" was an integral part of the Irish Revolution.

Résumé

Dans le sillage de pensée de la plupart des historiens de ces dernières décennies, l'ouvrage de Peter Hart, The IRA at War, 1916-1923 (2003), tend à minimiser grandement l'importance des conflits sociaux qui eurent lieu au cours de la Révolution irlandaise de 1916-1923. Hart justifie cette approche par le fait que ces derniers n'ouvrirent pas la voie à de profonds bouleversements sociaux au sein du nouvel Etat libre d'Irlande. Toute respectable que puisse être cette affirmation, elle n'en demeure pas moins discutable, étant donné que « hormis le fait de reconnaître aux problèmes agraires quelque rôle dans la révolution, les historiens n'ont pas pris la peine de les étudier en profondeur, voire se sont montrés des plus dubitatifs quant à leur existence », pour citer Terence Dooley (voir « The Land for the People »: The Land Question in Independent Ireland, Dublin, University College Dublin Press, 2004, p. 17). Cette remarque vaut également pour les expériences subversives, telles que les soviets irlandais de 1918-1923, que la recherche universitaire traditionnelle a jusqu'ici quelque peu ignorées. S'inscrivant dans le cadre d'un projet consacré à ces dernières, cet article se veut donc un réexamen de l'analyse de Hart, à travers lequel est évalué l'ampleur de la "lutte des classes" dans la Révolution irlandaise.

Mots clés : Irlande, révolution, lutte des classes, conflits sociaux, syndicats, soviets, conservatisme.

Up until the 1970s most of the studies exploring the Irish revolution or general Irish history had focused exclusively on its political and military aspects. This unique approach may be explained by the fact that the revolutionary activity had been primarily devoted to military and political purposes, so as to federate the cross-class nationalist community around the supreme goal of Irish independence. A marked turning point in Irish historiography, therefore, occurred only after F. S. L. Lyons' 1971 famous essay Ireland Since the Famine had paid a particular attention to social and economic issues, thereby paving the way for other themes being dealt with, including gender, sectarianism, the social background of revolutionaries [START_REF] Lyons | Ireland Since the Famine[END_REF] … But while the socioeconomic dimension of the Irish revolution has since been recognized, on the other hand, analyses have tended to downplay its historical significance [START_REF]Reflective of this trend is notably Richard English's research work on republican socialism. See, for example, Richard English, "Socialism: Socialist Intellectuals and the Irish Revolution[END_REF] . Hence, except for a few monographs (e.g. Diarmaid Ferriter's most recent A Nation and not a Rabble, the Irish Revolution 1913-23 and Gavin M. Foster's The Irish Civil War and Society: Politics, Class and Conflict, also published in 2015 [START_REF] Diarmaid Ferriter | The Irish Civil War and Society: Politics, Class and Conflict, Basingstoke[END_REF] ) and those specifically dedicated to labour and land disputes in the revolutionary era (e.g. Emmet O'Connor's Syndicalism in Ireland, 1917-1923, and Conor Kostick's Revolution in Ireland: Popular Militancy 1917 to 1923 [START_REF] Emmet | Syndicalism in Ireland, 1917-23[END_REF] ), socioeconomic issues have hitherto been pushed into the background or diluted in political and military studies that have remained dominant in mainstream academic research -one particularly relevant example is the otherwise compelling monograph by Charles Townshend entitled The Republic: The fight for Irish Independence, 1918-1923 [START_REF] Townshend | The Republic: The fight for Irish Independence, 1918-1923[END_REF] .

• [START_REF] Dooley | The Land for the People[END_REF] According to Charles Townshend and Terence Dooley, this traditional approach can be traced back to Patrick Lynch's 1966 article "The social revolution that never was" in which, to explain why the Irish Revolution was never a social revolution, the author contends that, "by 1916 most agricultural holdings had been purchased by their occupiers, subject to land annuities, which were not then a matter of contention. The tenant had become a proprietor, the owner of his land; and little land remained, to which the system of voluntary purchase could be applied 6 ". Lynch nonetheless acknowledges that "sporadically, advantage was taken of the unsettled conditions in the country to show that an aggressive spirit of social unrest still existed 7 ". Likewise, John Regan would much later concur with this finding:

Any chance of real social revolution had been substantially undermined by land reform and the creation of an increasingly conservative peasant proprietorship in Ireland sponsored by various British Governments in the four decades before independence 8 .

In line with this view, which has been held by other historians over the past decades 9 , Peter Hart argues in his 2003 The IRA at War, 1916-1923 that "not only did the Irish revolution not bring social transformation, there was no socially revolutionary situation in Ireland, even in prospect". This not least because "most farmers owned their farms by 1922", although "this is not to say that nothing happened in terms of social unrest, or that nothing important happened, but rather that nothing revolutionary happened 10 ". Hence the substantial amount of social unrest, whether industrial or agrarian, that spread across revolutionary Ireland is not included in this otherwise convincing three-dimensional process consisting of three interwoven struggles: a war against the British State; an ethnic and communal conflict; a fratricidal struggle in the nationalist community 11 .

Even though Hart's study undoubtedly challenges the prevailing historiographical tradition in many respects, it remains nonetheless disputable. The present article will therefore seek to reappraise its relevance by assessing the extent to which "class struggle" was an integral part of the Irish Revolution, as much as the other overlapping struggles suggested by Hart 12 . In so doing, it will, first, define the concept of "class struggle" and then depict its various manifestations in revolutionary Ireland.

Although the concept of class struggle is central to Marxist theory of history, many authors before Karl Marx had analysed social classes and their development and antagonism throughout history. Writing on March 5, 1852, to his friend Joseph Weydemeyer Marx himself acknowledged that he did not originally draw up the concept:

And now as for myself, I do not claim to have discovered either the existence of classes in modern society or the struggle between them. Long before me, bourgeois historians had described the historical development of this struggle between the classes, as had bourgeois economists their economic anatomy 13 .

The "bourgeois" historians and economists whom Marx alluded to actually included liberal thinkers such as François Guizot, Augustin Thierry, François Mignet, John Wade and David Ricardo 14 . Further down, in the same letter, Marx summarized his own contribution to class theory in these words: • 27

My own contribution was 1. to show that the existence of classes is merely bound up with certain historical phases in the development of production; 2. that the class struggle necessarily leads to the dictatorship of the proletariat; 3. that this dictatorship itself constitutes no more than a transition to the abolition of all classes and to a classless society.

But how, in Marx's view, were the different classes dialectically connected? What were the conditions and mechanisms whereby the process of class struggle was triggered? Elaborating on his thoughts on this issue in other writings 15 , Marx thus posited that a class really comes into existence when its members not only share common socioeconomic features, but also become aware of the inherent antagonism between their common interests and those of other social groups, as a result of conflicting relationships to the means of production. Class consciousness and identity, therefore, acts as a catalyst for social conflict and competition that will inevitably lead to revolutionary change and, in the case of industrial capitalist society, to "the dictatorship of the proletariat".

Among the proponents of the liberal class-conflict theory, some dismiss Marx's conception of class struggle as totally irrelevant 16 , while others recognize its merits, at least to some extent 17 . Thus, for example, the French philosopher and sociologist, Raymond Aron, contends that Marx's empirical definition of class makes it acceptable for non-Marxists, albeit with reservations. Economic conflicts between employees and business owners for the distribution of national income undeniably exist in industrial society. On the other hand, Aron disputes Marx's assertion whereby each of the antagonistic classes has its own conception of society and, as such, aspires to power. In Marx's class theory, the growing pauperization of the workers arising from the development of capitalist production will thus inevitably incite them to overthrow the capitalist system to establish a new mode of production. The fact is, however, that as aggregate income increased over time in the most developed countries, the rivalry between classes grew less intense and violent, thereby weakening the revolutionary movement. Class struggle in the Marxist sense of the term gradually gave way to what Aron calls "quarrelsome satisfaction" (« La satisfaction querelleuse »), in the form of peaceful resolutions of conflicts designed to ensure more equitable income distribution within the existing order. Such a trend, of course, is at variance with Marx's prediction. But while capitalism does not necessarily create the conditions condu- The failure of the Dublin general strike of 1913, together with the outbreak of the First World War, somewhat curbed industrial unrest in Ireland, which had intensified since the creation of the Irish Transport and General Workers' Union (ITGWU) by James Larkin in 1908. The significant growing demand for food products and raw materials engendered by the war effort actually brought prosperity to the country, which the industrial workers and labourers hardly benefited from, unlike many farmers, employers and shopkeepers. Thus, the postwar economic boom provided an incentive for the former to claim their share of the general growth in prosperity. They did so in an unprecedented wave of wage strikes that coincided with the upsurge in trade union militancy in the later war years 19 . At that time, the Irish Labour movement was embodied by the Irish Trade Union Congress (ITUC), which had dramatically increased the number of its affiliates since the 1914 annual meeting, rising from below 110,000 to 230,000 20 in 1920. This dramatic growth in union membership was mainly the result of the diverse campaigns led by the most powerful Irish trade union, the ITGWU, which could count on 100,000 members by 1920 -as compared with 30,000 before the 1913 Dublin Lockout -, about 40 % of them working in agriculture 21 • 29

This means that, in addition to the industrial sector, the ITGWU had also made a significant breakthrough in rural areas since the agrarian outbreak of 1917, thereby organising tens of thousands of labourers, landless farmers and smallholders within four years [START_REF] Emmet | [END_REF] . Thus, from 1918 to 1921, what was known as the wages movement translated into 782 industrial strikes -as against 307 in the years 1914-1918 -, most of them being successful [START_REF]the Irish workers' achievements should be somewhat qualified given that the significant wage increases won as[END_REF] . But social turmoil also included boycott, cattle driving and land seizures perpetrated by all those left behind the land reforms implemented since 1903, including not only the landless tenants and labourers but also the significant portion of smallholders who had not purchased sufficient land for the economic viability of their holdings [START_REF] Dooley | 867 people, of whom 761,791 were employed in fields and pastures, comprising 613,021 farmers ([END_REF] . Knowing that between two thirds and three quarters of the Irish farmers became landowners by 1914 and that the land purchase and emigration processes were suspended for the duration of the War, such a phenomenon can by no means be ignored and tends to indicate that agrarian unrest was more intense and radical than suggested by Lynch, Regan, Cronin and Hart [START_REF] Dooley | The Land for the People[END_REF] . Two trends of rural agitation actually prevailed in the revolutionary period: one based on strike actions for better wages and working conditions led by labourers belonging to the ITGWU -and this mostly in the east; and the other on land seizures carried out by the smaller farmers and labourers for the breakup of the larger estates and their redistribution -and this mostly in the west 26 . Concurrent with these traditional patterns of conflict, however, emerged an alternative organizational method, named "soviet" after the council movement that developed as part of the 1917 Russian Revolution. Initially established as auxiliaries within the wages movement, the one hundred or so Irish soviets nonetheless differed from the latter in two aspects: their action did not rest upon work stoppage but on the continuity of production or management conducted exclusively by the workers themselves, not without having first ousted or ignored the official owners, managers or rulers.

Anyhow, the wages movement that widely spread throughout the country in the thriving period 1917-21 essentially aimed at achieving a fairer distribution of the wealth created during the war years. The conflicts were most certainly fought between antagonistic classes aware of themselves as groups sharing common socioeconomic interests adverse to those of other groups (employers and shopkeepers vs workers and employees; farmers vs labourers), each being specifically represented by one or several organizations: the various trade unions for the workers, labourers and employees; the Irish Farmers' Union for the farmers; the Irish Association of Employers for the employers. But the aim for each class was not to claim or seize power in the name of a specific conception of society. Class struggle, therefore, was not used as a political tool designed to maintain or overthrow the current system, although many striking workers were imbued with the radical spirit of syndicalism 27 instilled by the most active trade union, the ITGWU 28 . And such radicalism often expressed itself symbolically through the "red flag" that was conspicuously hoisted or displayed during rallies and demonstrations, all along what Emmet O'Connor calls "the red flag times 29 " that also saw the ITUC -re-named the Irish Labour Party and Trade Union Congress (ILPTUC) -endorsing a socialist constitution and manifesto at a special conference in November 1918 30 .

As for the agrarian conflicts for land widely carried out by small farmers and landless labourers against landlords and graziers in the west of Ireland, here too class struggle aimed less to subvert the existing system based on private property than to conquer land or redistribute it more fairly -except for those attempts at establishing collectively-run farms in County Galway in 1920 31 . 27. Syndicalism can be defined as a doctrine aimed at organising the workers as a whole into "one big union" to achieve working class control of all industries by direct means, with a view to establishing a socialist society. 28. Emmet • 31

The same finding also partly applies to the "soviet" occupations that some workers resorted to, when work stoppage or negotiation proved insufficient to get the expected results. As Donal Ó Drisceoil points out: "In all cases of 'soviets' up to 1921, the employers' property was returned once demands had been met 32 ". However, regarding the fact that "the employers' property was returned once demands had been met", it must be reminded that, in some cases, the workers did it reluctantly and at times following negotiations brought about under the threat of the Sinn Féin Government, as in the Castleconnell soviet and the Bruree soviet where Constance Markievicz, the then Minister for Labour, would have threatened to dispatch republican soldiers to force the workers to leave the premises peacefully [START_REF]See the offical organ of the ITGWU[END_REF] . Furthermore, the practice of workers' self-management that underpinned the soviet occupations made them appear as genuine subversive experiments -comparable to those Russian factory committees which took over their plants and ran them independently of the owners and managers throughout 1917 [START_REF] Anweiler | On the Russian factory committees[END_REF] . This is well illustrated by the poster displayed at the entrance door of the Bruree mill (Co. Limerick), which read: "Bruree Mills and Bakery are now the property of the workers. The mill and shop are open for the sale of bread, flour and meal. It is hoped to reduce prices and do away with profiteering within a day. By order of the workers [START_REF][END_REF] ". Not to mention the subversive symbols that accompanied these workers' action: starting with the name "soviet" used to identify them, but also through the "red flag" raised over some of the occupied factories or slogans such as, "Bruree Workers Soviet Mills, We make Bread not Profits", that was printed over the door of entrance to the Bruree mill. Here class struggle in the Marxist sense of the term was arguably still in its infancy. But fairly soon, the economic conditions would allow it to mature.

Thus, the post-war economic boom gradually gave way to a slump as of late 1920. With it, social unrest in Ireland gradually turned into struggles against wage cuts imposed by the employers and farmers. The tide was now turning in favour of the latter, to the extent that, to quote Emmet O'Connor, "labour was […] coming close to conceding what had been won since 1914 [START_REF] Emmet | Syndicalism in Ireland[END_REF] ". Such circumstances arguably fostered further hardline stances among labourers and farmers, as well as the resurgence of soviets which, from tactical tools used to complement the wages [START_REF] Donal Ó Drisceoil | [END_REF] • movement, were henceforth increasingly viewed as genuine alternatives to traditional private property rights 37 . Emblematic of this trend were the Cleeves soviets.

Cleeve is actually the name of a family who, at the time of the Irish Revolution, ran a network of some 100 creameries, separation stations, condensed milk factories and mills located in Counties Limerick, Tipperary and Cork. About 3,000 people worked for this business empire, which also processed the milk of some 5,000 farmers. From 1918 onwards, Cleeves, like many other Irish companies, underwent major social disputes over wages, working hours and conditions, some of which were to take the form of factory seizures in which the workers kept their plant running under their control. These workers' self-managed occupations were naturally dubbed "soviets", starting with the Knocklong creamery (Co. Limerick) in May 1920, followed by the Bruree mills and bakery (Co. Limerick) in August 1921, and 39 creameries, along with mills and other workshops (Co. Limerick, Tipperary and Cork), for several months in 1922. In the latter case, the plants were taken over in response to a lockout resulting from an unsettled dispute about pay and staff cutbacks. Here the workers' decision to resume production was justified on the ground that such closing down would "[imperil] the means of livelihood of 5,000 farmers, [risk] the destruction of national produce to the extent of thousands of pounds a week, and [throw] 3,000 workers and their family out of work, to beg and starve 38 ". It was therefore "in the interest of the community, and to preserve the industry for the nation" that the workers were instructed to carry on work. These designs, both communal and national, were encapsulated in their motto, "the Sovereign People 39 ". Yet this revolutionary impulse, embodied in the need to create a new mode of production for themselves and the whole nation, proved short-lived due to the joint effect of the farmers' boycott and the intervention of the Free State Army 40 , both being in line with what Emmet O'Connor calls "the conservative response" to social disorder 41 . Not to mention the official Labour leaders whose refusal to use subversive methods to carry out their revolutionary goals 42 naturally led them to ignore the soviets 42. The report of the 1922 annual meeting of the ILPTUC thus specifies: "It is the duty of the Labour Party to make use of whatever instrument and power the political struggle has placed in its hands. As we have been willing to make use of the opportunities which Town and Urban Councils and Poor Law Boards provide to further the cause and protect the interests of the workers, though neither of them accords with our ideals, so we ought to work the new Governmental machine if it is established, even though it is not built according to our design,

• [START_REF]See the offical organ of the ITGWU[END_REF] showing the slightest sign of subversion 43 . This also means that the ITGWU's involvement in virtually all the soviets set up in the revolutionary era was mostly initiated by the rank and file -or by the union's communist officials from 1921 44 .

Of course, such a subversive trend in the period 1921-23 did not spare the rural areas, which witnessed several attempts at implementing collective ownership of land, as in the village of Broadford (Co. Clare) in 1922 where the Going estate was managed as a soviet for a few months by tenant farmers merely demanding "a reduction in first and second term rents and […] distribution of grass lands among the small tenants". To this end, a "Committee of farmers, tenants, workers and Transport Union workers on the Going estates" was created, with one of its members elected as its secretary. Paradoxical as it might seem, however, the Going estate became self-managed and autonomous, while remaining officially the landlord's private property. This ambiguous situation was characterized by the Broadford Committee's decision to pay what it reckoned to be a fair rent for six months, namely £110 which was much less than the payment expected by the landlord's agent. In addition to this, the Committee or "soviet" endeavoured to support the local community in two ways: first, by converting part of the estate into common fields for meadowing; and second, by letting lands for tillage to landless men in Broadford 45 . Other similar "soviet" experiments were also conducted in County Clare in 1922-23 -in Toovahera, Kilfenora, Ballyvaughan and other areas in the neighbourhood of Crab Island -, here too, by tenant farmers left on the sidelines of the 1903 and 1909 land reforms, and yearning for a fairer distribution of land and a drastic reduction of rents that they imposed by taking over the estates they worked on 46 .

If credence is to be given to Raymond Aron's proposition whereby hope and despair are necessary prerequisites for class struggle to develop into revolution, as provided it can be adapted for turning out the products we require. […] The object of the Irish Labour Party is, and must be, to establish in Ireland a Co-operative Workers' Republic. In the course of the struggle to attain this ultimate social, political, and economic freedom, Labour must demand and work for many ameliorative reforms". Marx saw it, a large part of the lower strata of Irish society was therefore imbued with revolutionary sentiment in 1921-23. From industrial workers to agricultural labourers to small farmers, many of them undoudtedly felt extreme dissatisfaction with existing conditions and, in this case, not only with their employers' attempt at cutting their salaries and the slow pace of land distribution, but also with their failure to win concessions through strike action, negotiations and land seizures. This feeling of despair towards the system in place led some of them to practically express hope for a new and better organisational structure, particularly in the form of self-managed soviets, as an alternative to the current private property rights, and this outside the country's most industrialized areas. This revolutionary impulse, embodied in the need to create a better life for themselves and the local community or national community, was all the more conspicuous in one of the 1922-23 Cleeve soviets that, according to the Voice of Labour:

These men were straining every nerve to secure perfection in the product they were manufacturing; that no possible slur should be cast upon the Workers' Factory, (in which), the minutest detail failed to escape the keen observation of the Works manager (who was only a worker and) whose fervent enthusiasm and love of his work was a constant urge to the best in every man and woman 47 .

Class struggle most assuredly permeated the revolutionary era in a two-stage process, each corresponding to a specific approach: the liberal or nonexclusively Marxist approach prevailing in the period 1917-21; and the Marxist approach prevailing in the period 1921-23. Class struggle, therefore, can hardly be considered of secondary importance compared with the political and military conflicts confronting Ireland at that time, not least because of the high number of strikes and soviets that swept across the country between 1917 and 1923: in the industrial sector alone, there were 1199 strikes -including 28 involving over 1,000 workers each -and more than 100 self-managed soviets as of 1920, most of which were creameries 48 . So much so that, as Emmet O'Connor puts it: "Never since has the working class confronted the rationale of capitalism so profoundly or so extensively 49 ." As for the land conflicts in the west of Ireland, Fergus Campbell points out that "far from being resolved, serious grievances remained unaddressed and provided the basis for renewed land agitation on a scale not seen since the days of the Land War (1871-81) 50 ". This, Campbell adds, notably translated into • [START_REF][END_REF] the forcible seizure and redistribution of hundreds of grazing farms throughout Connacht, at times in the form of experiments in collective land ownership, thereby infringing private property rights 51 . Furthermore, all these manifestations of class struggle were not without causing great concern in the Irish revolutionary movement, which responded accordingly. The strategy of cross-class unity adopted by the republican leadership thus justified using every possible means to eradicate anything that could sow the seeds of division within the nationalist community, and in particular class struggle 52 . As a result, some of the self-managed soviets were compelled to cease operations under the threat of military intervention uttered by the Sinn Féin government, and several significant measures were taken to resolve the agrarian crisis and take control of the land agitation -a National Land Bank was created in 1919 to accelerate the process of land purchasing through loans made available for the tenants anxious to purchase their own farms; the following year saw the establishment of the Land Settlement Commission and land courts designed to implement a redistribution of grazing land; then, during the Civil War, the Free State government decided upon using strong-arm tactics too, through the Special Infantry Corps specifically created to suppress agrarian turmoil, strikes and any infringement of the law altogether 53 . Thus, while there is no denying that, in Peter Hart's words, "nothing [socially] revolutionary happened" during the Irish Revolution, at least on a large scale, that "there was no socially revolutionary situation in Ireland, even in prospect", is however more than debatable 54 . For there was clearly a socially radical and revolutionary potential, whether industrial or agrarian, in Ireland during the 1916-23

And despite the support and involvement of some sections of the republican and trade union rank-and-file members in the social conflicts, such a potential was discouraged and hampered by the main leaders of both movements: in the name of a moderate and conservative vision of Irish political independence, which admitted of no social hindrance, for the former; and, in the name of a 51. Ibid. 52. Referring to the agrarian unrest that had swept across the country before the setting up of the Dáil courts in 1920, a Dáil report pointed out: "All this was a grave menace to the Republic. The mind of the people was being diverted from the struggle for freedom by a class war, and there was every likelihood that this class war might be carried into the ranks of the republican army itself which was drawn in the main from the agricultural population and was largely officered by farmer's sons". peaceful and gradualist vision of social revolution, which eschewed subversive and illegal action, for the latter.

Hence one is entitled to believe that the Irish Revolution was actually a melting pot of "motivations, expections and opportunities 55 ", to quote Diarmaid Ferriter's own words, in which class struggle featured as prominently as the other ingredients -even though social radicalism was eventually defeated and was not to give rise to progressive forces sufficiently powerful to counterbalance independent Ireland's conservatism established by "the reactionary elements among the elite of Sinn Féin 56 ".

  Toronto, University of Toronto Press, 1965, p. 83-110. 8. John M. Regan, The Irish Counter-Revolution, 1912-36, Dublin, Gill and Macmillan, 1999, p. 377. Regan's position on this issue is confirmed in the introduction to the collection of chapters he edited with Mike Cronin in 2000: "There is little evidence of a social component within the Irish revolution and less again in its settlement. Such potential as there wasn for social upheaval had to agreat extent been defused by the transfer of land back to native ownership under a series of reforming land acts at the end of the nineteenth, and the beginning of the new century". Mike Cronin and John Regan, "Introduction: Ireland and the Politics of Independence, New Perspectives and Re-consideration", Mike Cronin and John Regan (eds.),

	Ireland: The Politics of Independence,
	1922-49, Basingstoke, Palgrave, 2000, p. 1-2.
	9. See, for example, Michael Laffan, The Resurrection of Ireland: the Sinn Féin Party, 1916-1923, Cambridge,
	Cambridge University Press, 1999, p. 315; John A. Murphy, Ireland in the Twentieth Century, Dublin, Gill and
	Macmillan, 1975, p. 10; Francis Costello, "Labour, Irish Republicanism and the social Order during the Anglo-
	Irish War", The Canadian Journal of Irish Studies, vol. 17, n o 2, 1991, p. 16.

6. Charles Townshend, "Historiography: Telling the Irish Revolution", in Joost Augusteijn (dir.), From Public Defiance to Guerrilla Warfare: The Experience of Ordinary Volunteers in the Irish War of Independence, 1916-1921, op. cit., p. 5; Terence Dooley, "The Land for the People": The Land Question in Independent Ireland, Dublin, University College Dublin Press, 2004, p. 17. 7. Patrick Lynch, "The Social Revolution that Never Was", Desmond Williams (dir.), The Irish Struggle, 1916-26, London, Routledge and Kegan Paul, 1966, p. 41, p. 49. However, one year before Lynch, Nicholas Mansergh had pointed out that although the 1903 Wyndham Act had virtually settled the land question, it did not prevent Irish farmers' patriotic feelings from stirring during the War of Independence. Hence the so-called Irish question was at that time essentially a political question as "national sentiment was shown to be the fundamental force". See Nicholas Mansergh, The Irish Question, 1840-1921: A Commentary on Anglo-Irish Relations and on Social and Political Forces in Ireland in the Age of Reform and Revolution,

  15. See, for example, Chapter VII in The Eighteenth Brumaire of Louis Bonaparte, first published in 1852. [https:// www.marxists.org/archive/marx/works/1852/18th-brumaire/ch07.htm], last accessed 6 April 2016. to revolution, this does not mean that those conditions do not exist. For class struggle to develop into revolution, Aron argues, two seemingly contradictory sentiments must prevail: hope and despair -that is, hope for a new society resulting from profound dissatisfaction with existing conditions. Two sentiments, Aron points out, that are mostly felt in backward capitalist and predominantly agrarian countries, such as Russia in 1917 18 . It remains to be seen to what extent the industrial and agrarian conflicts that occurred in the 1916-23 Irish revolution fit with the Marxist and liberal or nonexclusively Marxist perspectives on class struggle.

16. See, for instance, Ludwig Von Mises, Socialism: An Economic and Sociological Analysis, New Haven, Yale University Press, 1962 [orig. 1951], p. 314-358. 17. See, for instance, Joseph Shumpeter, Capitalism, Socialism and Democracy, New York/London, Routledge, 2003 [orig. 1942], p. 1-58.

cive

  . 18. Raymond Aron, La Lutte de classes : nouvelles leçons sur les sociétés industrielles, Paris, Gallimard, 1964, p. 21-127, p. 197-309. 19. Emmet O'Connor, op. cit., p. 8-13, p. 20-25. 20. If one refers to the 1911 census, this figure accounts for approximately 37 % of all workers -including agricultural labourers -, themselves representing roughly one third of the working population, out of a total population of 4,390,219 inhabitants. See W. E. Vaughan and A. J. Fitzpatrick (eds.), Irish Historical Statistics, Dublin, Royal Irish Academy, 1978, p. 3; Census of Ireland, 1911, General report, Occupations of the people, 1912-13, Cd.6663, CXVIII, 1, p. xxviii-xxx. 21. Irish Labour Party and Trade Union Congress, Report of the Twenty-Sixth Annual Meeting, 1920, Published by Authority of the National Executive, p. 145-154; Irish Labour Party and Trade Union Congress, Report of the Twenty-Sixth Annual Meeting, 1921, Published by Authority of the National Executive, p. 75, [http:// centenaries-ituc.nationalarchives.ie/annual-reports/], last accessed 20 December 2016. For the proportion of ITGWU members in agriculture in 1920 and ITGWU membership in 1913, see Desmond Greaves, The Irish

Transport and General Workers's Union: The Formative Years, 1909-1923, Dublin, Gill and Macmillan, 1982, p. 91, p. 259. It is important to note that, except for Congress membership for 1914, which is only mentioned in the 1921 annual report, these figures do not take account of such affiliated organisations as the trades' councils, but only include the members of each affiliated trade union.

  O'Connor, A labour History of Ireland, 1824-1960, Dublin, Gill and Macmillan, 1992, p. 67-116. 29. Emmet O'Connor, "Syndicalism, Industrial Unionism, and Nationalism in Ireland", Steven Hirsch and Lucien Van Derr Walt (eds.), Anarchism and Syndicalism in the Colonial and Postcolonial World, 1870-1940, Brill, Brill Academic Publishers, 2010, p. 210. For accounts of struggles or gatherings of workers or labourers displaying the red flag, see for instance The Kerryman, 10 May 1919, p. 4; Leitrim Observer, 12 April 1919 p. 2;

	The Nenagh Guardian, 4 September 1920, p. 4, 26 November 1921, p. 3; The Freeman's Journal, 22 November
	1921, p. 3.
	30. The 1918 manifesto reveals that the ILPITUC's ultimate aims were notably "to win for the workers of Ire-
	land, collectively, the ownership and control of the whole produce of their labour; to secure the democratic
	management and control of all industries and services by the whole body of workers". See Irish Labour Party
	and Trade Union Congress, Report of a Special Congress, Nov. 1 st and 2 nd , 1918, Published by Authority of
	the National Executive, p. 122-123, p. 165-169, [http://centenaries-ituc.nationalarchives.ie/wp-content/

uploads/2014/10/24th-annual-report-1918.pdf ], last accessed 22 January 2017. 31. Fergus Campbell, Land and Revolution, op. cit., p. 280-282, "Land and Revolution Revisited", op. cit., p. 151-152. For a retrospective account of this collective experiment, see Kevin O'Shiel, "The Dail Land Courts", The Irish Times, 14 novembre 1966, p. 10.

  37. Ibid., p. 103, p. 114-116. 38. Cited in The Irish Times, 15 May 1922, p. 5. 39. Ibid. 40. For a detailed account of the Cleeve soviets, see David Lee, "The Munster Soviets and the Fall of the House of Cleeve", David Lee and Debbie Jacobs (eds.), Made in Limerick, vol. 1, History of Industries, trade and com-Syndicalism in Ireland, op. cit., p. 154.

	merce, Limerick Civic Trust, 2003, p. 287-306, [http://www.limerickcity.ie/media/limerick%20soviet%2015.
	pdf ], last accessed 8 January 2016; D. R. O'Connor Lysaght, "The Munster Soviet Creameries", Irish History
	Workshop, vol. 1, 1981, p. 36-49.
	41. Emmet O'Connor,

  See Irish Labour Party and Trade Union Congress, Report of the Twenty-Seventh Annual Meeting, 1922, Published by Authority of the National Executive, p. 62, p. 64, [http://centenaries-ituc.nationalarchives. ie/wp-content/uploads/2014/10/28th-annual-report-1922.pdf ], last accessed 23 January 2017. 43. For example, while the 1920 Knocklong and the 1921 Arigna soviet are referred to in the report of the 1921 annual meeting of the ILPTUC, no mention is made of the 1922-23 Cleeve soviets in the 1922 and 1923 reports. See Irish Labour Party and Trade Union Congress, Report of the Twenty-Sixth Annual Meeting, 1921, op. cit., p. 13; Irish Labour Party and Trade Union Congress, Report of the Twenty-Seventh Annual Meeting, 1922, op. cit.; Irish Labour Party and Trade Union Congress, Report of the Twenty-Eighth Annual Meeting, 1923, Published by Authority of the National Executive, [http://centenaries-ituc.nationalarchives.ie/annualreports/], last accessed 23 January 2017. 44. Emmet O'Connor, Syndicalism in Ireland, op. cit., p. 131. 45. On the Broadford soviet, see Michael McCarthy, "The Broadford Soviet", The Old Limerick Journal, n o 4, 1980, p. 37-40, [http://www.limerickcity.ie/media/Media,3937,en.pdf ], last accessed 23 January 2017. 46. On these land soviets, see The Freeman's Journal, 3 May 1923, p. 4, p. 5, 9 May 1923, p. 5, 17 May 1923, p. 6, 25 May 1923, p. 6.

  47. The Voice of Labour, 27 May 1922, p. 8. 48. Public Record Office London, Strikes and lockouts, 1914-21, Lab 34/14-20, Lab 34/32-39; Public Record Office London, Strikes and lockouts, 1921-23, Lab 34/39-41, figures cited in Emmet O'Connor, Syndicalism in Ireland, op. cit., p. 25, p. 100. 49. Emmet O'Connor, Syndicalism in Ireland, ibid., p. 110. 50. Fergus Campbell, Land and Revolution, op. cit., p. 280.

  Ministry for Home Affairs (compiled and edited by Erskine Childers), The constructive Work of Dáil Éireann, n° 1, The National Police and Courts of Justice, Dublin, The Talbot Press, 1921, p. 12. 53. Terence Dooley, "The Land for the People": The Land Question in Independent Ireland, op. cit., p. 46-49, p. 82; Fergus Campbell, Land and Revolution, op. cit., p. 255-257; Emmet O'Connor, Syndicalism in Ireland, op. cit., p. 162-163. 54. Peter Hart, The IRA at War, op. cit., p. 21.

  55. Diarmaid Ferriter, A Nation and not a Rabble, op.cit., p. 231. 56. Fergus Campbell, Land and Revolution, op. cit., p. 282.


Lazhar Roubache
email: roubache.lazhar@gmail.com

Kamel Boughrara
email: kamel.boughrara@g.enp.edu.dz

Frédéric Dubas
email: fdubas@gmail.com

Rachid Ibtiouen
email: rachid.ibtiouen@gmail.com

Technique en Sous-Domaines Élémentaires dans les Machines Asynchrones à Cage d'Écureuil : Saturation Magnétique Locale & Courants de Foucault dans les Barres

Keywords: Technique en sous-domaines élémentaires, Saturation magnétique locale, Courants de Foucault, Machine asynchrone à cage d

RESUME

-Dans cet article, les auteurs présentent un modèle semi-analytique d'une machine asynchrone à cage d'écureuil en considérant la saturation magnétique locale et les courants de Foucault induits dans les barres rotoriques. Les régions du rotor et du stator sont divisées en sous-domaines élémentaires (SDs-E) caractérisés par des solutions générales au premier harmonique des équations de Maxwell en magnéto-harmonique. Ces SDs-E sont connectés dans les deux directions (i.e., selon les bords r et θ). Le calcul du champ magnétique a été validé pour plusieurs valeurs de la perméabilité relative du fer et du glissement. Toutes les grandeurs électromagnétiques ont été comparées à ceux obtenues par une méthode numérique en deux dimensions (2-D), viz., FEMM. Les résultats semi-analytiques sont satisfaisants par rapport aux numériques, en considérant l'amplitude et la forme d'onde.

INTRODUCTION

Les modèles (semi-)analytiques sont rapides, précis et principalement adaptés aux premières étapes de conception optimale. La technique en sous-domaines (SDs) est l'une des méthodes semi-analytiques les plus récentes pour prédire le champ électromagnétique et pour calculer les performances électromagnétiques. Plusieurs états de l'art sur cette technique, dans les machines électriques, peuvent être trouvés dans [START_REF] Dubas | Analytical solution of the magnetic field in permanent-magnet motors taking into account slotting effect: No-load vector potential and flux density calculation[END_REF]- [START_REF] Pfister | Slotted permanent-magnet machines: General analytical model of magnetic fields, torque, eddy currents, and permanent-magnet power losses including the Diffusion effect[END_REF]. L'application de cette modélisation aux machines asynchrones à cage d'écureuil peut être trouvée dans [START_REF] Lubin | Analytic calculation of eddy currents in the slots of electrical machines: Application to cage rotor induction motors[END_REF]- [START_REF] Devillers | An improved 2-D subdomain model of squirrel-cage induction machine including winding and slotting harmonics at steady state[END_REF], mais malheureusement en négligeant la saturation magnétique globale ou locale. Jusqu'en 2015, la prise en compte du fer dans la technique en SDs était effectuée uniquement dans des régions : i) périodiques (e.g., dans les machines asynchrones à rotor solide [START_REF] Boughrara | 2-D analytical prediction of eddy currents, circuit model parameters, and steady-state performances in solid rotor induction motors[END_REF]), ou ii) avec des conditions aux limites (CLs) homogènes (e.g., dans les ponts magnétiques des machines synchrones à concentration de flux [START_REF] Liang | Analytical prediction of magnetic field distribution in spoke-type permanent-magnet synchronous machines accounting for bridge saturation and magnet shape[END_REF]). En 2017, Dubas et al. ont proposé une nouvelle contribution scientifique dans la technique en SDs en insérant des régions ferromagnétiques [START_REF] Dubas | New scientific contribution on the 2-D subdomain technique in Cartesian coordinates: Taking into account of iron parts[END_REF] et [START_REF] Dubas | New scientific contribution on the 2-D subdomain technique in polar coordinates: Taking into account of iron parts[END_REF]. Cette méthode dite exacte en 2-D a été appliquée à une bobine à noyau d'air ou de fer (en coordonnée cartésienne [START_REF] Dubas | New scientific contribution on the 2-D subdomain technique in Cartesian coordinates: Taking into account of iron parts[END_REF] et polaire [START_REF] Dubas | New scientific contribution on the 2-D subdomain technique in polar coordinates: Taking into account of iron parts[END_REF]) alimenté par un courant continu. La solution générale des équations de Maxwell en magnétostatique 2-D est décomposée en série de Fourier à partir deux solutions dans les deux directions et respecte les CLs sur les différents bords en appliquant le principe de superposition. Roubache et al. ont modélisé semi-analytiquement une machine synchrone à concentration de flux avec les parties ferromagnétiques en utilisant la technique en SDs ainsi que les polynômes de Taylor pour connecter les régions sur les bords [START_REF] Roubache | Semi-analytical modeling of spoke-type permanent-magnet machines considering the iron core relative permeability: Subdomain technique and Taylor polynomial[END_REF]- [START_REF] Roubache | Semi-analytical modeling of spoke-type permanent-magnet machines considering nonlinear magnetic saturation: Subdomain technique and Taylor polynomial[END_REF]. Dans [START_REF] Roubache | New subdomain technique for electromagnetic performances calculation in radial-flux electrical machines considering finite soft-magnetic material permeability[END_REF], la méthode exacte de Dubas et al. [START_REF] Dubas | New scientific contribution on the 2-D subdomain technique in polar coordinates: Taking into account of iron parts[END_REF] a été implémentée sur les machines électriques à flux radial et à aimants permanents. Dans [START_REF] Roubache | Elementary subdomain technique for magnetic field calculation in rotating electrical machines with local saturation effect[END_REF], Roubache et al. ont développé un modèle semianalytique avec la saturation magnétique locale en utilisant l'algorithme itératif de Newton Raphson. L'idée principale de ce nouveau type de modèle est de mailler les régions du rotor et du stator en SDs-E. La solution générale est identique à celle de [START_REF] Dubas | New scientific contribution on the 2-D subdomain technique in polar coordinates: Taking into account of iron parts[END_REF]. Néanmoins, les SDs-E sont supposés suffisamment petits de telle sorte que la variation du potentiel vecteur magnétique z A peut être considérée comme non importante, conduisant ainsi de ne considérer que les premiers harmoniques selon la direction r et  dans la solution générale.

Dans ce papier, les auteurs se proposent de modéliser une machine asynchrone à cage d'écureuil en utilisant la technique en SDs-E avec l'effet de saturation magnétique locale en prenant en compte les courants de Foucault induits dans les barres rotoriques. Les simulations ont été effectuées pour plusieurs valeurs de la perméabilité relative du fer et du glissement. Les résultats semi-analytiques en 2-D ont été comparés avec ceux obtenus par FEMM [START_REF] Meeker | Finite Element Method Magnetics ver[END_REF].

FORMULATION MATHÉMATIQUE

      , , ,
j rm t zz A r t e A r e       dans le rotor (1)       , , , j t zz A r t e A r e       dans le stator (2) avec 1 j  et rm r sp        où rm
 représente la pulsation électrique au rotor, s le glissement,  la pulsation électrique au stator (ou de synchronisme), p le nombre de paires de pôles, et r  la pulsation mécanique au bout de l'arbre du rotor. La formulation utilisée dans ce document nécessite la définition des éléments de maillage représentés sur la Fig. 2 :  

0 i z sz AJ     pour i Se    (4) 0 z A  pour Rf Sf Si Ri I      (3)
1 1 Lr l Lr         (7)   1 1 Lr l Lr a a a a     (8)   1 1 Ls l Ls         (9)   1 1 Ls l Ls b b b b     (10)   ( 1) 1 int 1 Kr k Kr Rr R Rr Rr      (11)   ( 1) 1 1 Ks s k Ks Rs R Rs Rs      (12)   ,     k l Kr Lr II II  (13)   ,     k l Kr Lr II II  (14)   ,     k l Ks Ls III III  (15)   ,     k l Ks Ls z III III JJ (16) avec , 0 , , 0 1 if 1 if                kl Rb Ri kl II kl Rf r    (17) , , , 0 if if              kl Rf Ri kl II kl Rb   ( 18
) , 0 , , 0 1 if 1 if                kl Sf Si kl III kl Sf r    (19) , , , 0 if if              kl Sf Si kl III i k l i sz Se J J (20)
La solution générale de (4) ~ ( 6) est donnée par [START_REF] Dubas | New scientific contribution on the 2-D subdomain technique in polar coordinates: Taking into account of iron parts[END_REF] :

  , rp z z z z A r A A A      (21)
  , , , , 2 1 2 0 , , 3 4 , 5 , 6 1
= ln( ) 4 cos 2 sinh 2 sin ln sinh 2                                                                          sl sl k l k l k l k l z III III k l k l l sl l kl l sk l sk k kl l sk l A C C r J r b C r C r b C r Rs b C              (23) avec   1 ln   sk kk Rs Rs   & sl l b    (24) où   ,, 16
k l k l
CC sont les constantes d'intégrations dans le stator.

 dans le rotor (Région II) :

Selon [START_REF] Sprangers | Magnetic saturation in semi-analytical harmonic modeling for electric machine analysis[END_REF], la solution générale dans le rotor peut être définie par :

  , , , , 2 1 2 0 , , 3 4 , 5 , 6 1
= ln( ) 4 cos 2 sinh 2 sin ln sinh 2                                                                         rl rl k l k l k l k l z II z II k l k l l rl l kl l rk l rk k kl l rk l A B B r J r a B r B r a B r Rr b B              (25) avec   1 ln rk kk Rr Rr     & rl l a    (26) mais , , ,    k l k l rm zI I I kl I II z A J  j , ce qui donne :   , , , , , , 1 2 2 , 5 , 5 6
, ,

          k l k l k l k k l k l k l z II II rm l k l z l I kl k I B B J B J B     j (27) avec   2 2 , 1 0 2    k l k k Rr Rr   (28)     , 2 2 1 1 , 2 ln ln 1 2 2           kl l k k k k kl a Rr Rr Rr Rr S  (29)       1 , 2 ln , 2 5 2 cosh 1 1 4             
      , , , , , , , , , 2 1 2 1
2 2 5 5 6 , , 3 4 , 5 , 6
= ln( )

cos 2 sinh 2 sin l sinh 2                                                                  rl rl k l k l k l k l k l k l k l k l k l k l z II k l k l l rl l kl l rk l rk kl l rk l A B B r B B B B r a B r B r a B a B                n            k r
         Y (36)     4 2 1 0 0 1 1 a 1 1 2 2 3 3 4 4    N air N N N N A A A A A A A A (37)   6 1 , , 1,1 1,1 1,2 1,2 1 6 1 6 1 6         rr r r r r KL K L K L rotor b B B B B B B ( 38
)   6 1 , , 1,1 1,1 1,2 1,2 1 6 1 6 1 6         ss s s s s KL K L K L stator c C C C C C C (39)
Il est à noter que   H , dépendent de la réluctivité des matériaux (hormis 41  ). Le développement des CLs ainsi que les sous-matrices peuvent être trouvés dans [START_REF] Roubache | Elementary subdomain technique for magnetic field calculation in rotating electrical machines with local saturation effect[END_REF]. Dans cet article, le cas d'une perméabilité constante a été traité. Néanmoins, il possible d'introduire l'algorithme itératif de Newton Raphson pour l'analyse non-linéaire du champ magnétique [START_REF] Roubache | Elementary subdomain technique for magnetic field calculation in rotating electrical machines with local saturation effect[END_REF] en utilisant la courbe B(H) effective [START_REF] Paoli | Complex representation in nonlinear time harmonic eddy current problems[END_REF].

RESULTATS ET VALIDATIONS

Performances électromagnétiques

La méthode semi-analytique développée est utilisée pour déterminer le champ magnétique dans toutes les régions de la machine, la densité surfacique de courant de Foucault induits dans les barres rotoriques et le couple électromagnétique. Les paramètres géométriques et physiques de la machine sont donnés dans le Tableau 1.

Les Figs. 3 ~ Pour l'alimentation en tension, le fondamental du courant statorique (i.e., l'amplitude et le déphasage) peut être calculé en utilisant un circuit électrique équivalent [START_REF] Boughrara | Analytical analysis of cage rotor induction motors in healthy, defective, and broken bars conditions[END_REF]- [START_REF] Roubache | Analytical electromagnetic analysis of multi-phases cage rotor induction motors in healthy, broken bars and open phases conditions[END_REF]. Fig. 10 représente les caractéristiques électromagnétiques de la machine en fonction de glissement. On remarque que pour des valeurs de glissement inferieures à 0,1 les résultats sont en accord avec ceux obtenus par FEMM. Néanmoins, pour des valeurs élevées de glissement le modèle donne une erreur de 5% entre les courants obtenus analytiquement et numériquement. Bien entendu, cette erreur est également obtenue pour les résultats du couple. Cette erreur est due à la prise en compte des courants de Foucault induits dans les barres du rotor en tant que constantes.

Limite du modèle -Taille, précision et temps de calcul

Un point très important devrait être traité concernant la limite d'utilisation du modèle semi-analytique par rapport au modèle numérique d'un point de vue précision et temps de calcul. Il est clair que la considération des harmoniques de courant de Foucault induits dans les barres rotoriques [START_REF] Devillers | An improved 2-D subdomain model of squirrel-cage induction machine including winding and slotting harmonics at steady state[END_REF], pour l'étude de régime permanent pour glissement donné, donne un avantage très important par rapport au modèle numérique d'un point de vue temps de calcul, à cause du temps de calcul de régime transitoire consommé par le modèle numérique. Néanmoins, une comparaison entre le modèle proposé et le modèle numérique magnéto-harmonique a été faite.

La précision dans le modèle développé est liée à la taille des SDs-E. Ce paramètre peut être traité en termes de dimension de la matrice M du système linéaire (33). Le Tableau 2 représente l'erreur relative entre le modèle semi-analytique et numérique en fonction de la taille de la matrice M et du temps de calcul. L'erreur relative est représentée en termes de l'induction magnétique au milieu de l'entrefer et du couple électromagnétique. Ces erreurs sont calculées :

      % Num Ana I I Num I RMS E RMS   B BB B (40)   % Num Ana Num E       (41)
Les résultats numériques du champ magnétique sont calculés avec 185341 noeuds et 369262 éléments donnant un temps de calcul égal à 2,26 min. Le Tableau 2 montre que le modèle semi-analytique donne de bons résultats pour une taille de matrice M élevée, mais avec un temps de calcul proche de celui de FEMM. Néanmoins, pour des valeurs faibles de glissement (e.g., 0,1  s

), le vecteur potentiel magnétique n'est pas si déformé. Ceci aboutit à l'approximation de la considération d'une seule harmonique dans les SDs-E sans mailler le rotor et le stator en éléments très fins. L'erreur obtenue est respectivement autour de 6 % et de 4 % pour l'induction magnétique et le couple avec un temps de calcul inférieur à 0,3 min.

CONCLUSION

Dans ce papier, un modèle semi-analytique des machines asynchrones à cage d'écureuil en considérant la saturation magnétique locale et les courants de Foucault induits dans les barres rotoriques est présenté. La modélisation développée est basée sur la technique en SDs-E dont l'hypothèse principale est que la variation du potentiel vecteur magnétique n'est pas importante dans les SDs-E. Ces SDs-E sont connectés dans les deux directions (i.e., selon les bords r et θ). Le modèle semianalytique a été validé pour différentes valeurs de la perméabilité relative du fer et du glissement. Les résultats sont en excellente concordance avec ceux obtenus par FEMM.

Dans le cas où le potentiel vecteur magnétique est très déformé, l'obtention de résultats acceptables nécessite de mailler le rotor et le stator en SDs-E très fins, ce qui augmente le temps de calcul. Par conséquent, une optimisation de l'implémentation de ce modèle est proposée aux chercheurs, soit en focalisant sur la taille optimale des SDs-E ou sur la solution proposée dans les SDs-E afin de minimiser le nombre des constants d'intégrations.

Pour des raisons de programmation, toutes les régions du rotor et du stator sont maillées ce qui conduit à des CLs entre des SDs-E avec les mêmes périodes spatiales. Cependant, les encoches statoriques sont des régions homogènes et le fait de les considérer sans maillage et de les associer par des nombres harmoniques élevés conduit à des résultats plus précis. Il en est de même pour les barres rotoriques, néanmoins, il faudrait déterminer la période spatiale des fonctions Bessel pour connecter les SDs-E selon les bords r. Pour satisfaire ces CLs, il est possible aussi d'utiliser le développement en série de Taylor comme dans [START_REF] Roubache | Semi-analytical modeling of spoke-type permanent-magnet machines considering the iron core relative permeability: Subdomain technique and Taylor polynomial[END_REF].

  EN SDS-E Les Figs. 1 ~ 2 représentent respectivement une machine asynchrone à cage d'écureuil avec l'identification des différentes régions et le maillage en SDs-E du rotor ainsi que du stator. En magnéto-harmonique 2-D, le potentiel vecteur magnétique s'exprime par :

Fig. 1 .Fig. 2 .

 12 Fig. 1. Machine asynchrone à cage d'écureuil.

où i sz J

 sz représente la densité surfacique de courant dans la i ème encoche statorique, σ la conductivité électrique des barres rotoriques, j Rb  & i Se  respectivement la j ème barre rotorique et la i ème encoche statorique, des encoches rotoriques et statoriques, et I  l'entrefer. La résolution analytique de (3) conduit à des fonctions Bessel selon la direction r. Ces fonctions non-périodiques nécessitent la détermination numérique de la période spatiale pour connecter les SDs-E selon les bords r. Étant donné que les SDs-E sont suffisamment petits, la variation de z A peut être considérée comme non importante, ce qui conduit à ne considérer que les premiers harmoniques selon la direction r et  dans z A . De plus, on peut considérer que les courants de Foucault induits dans les barres rotoriques j rz J sont constants. Par conséquent,

  utilisant (27) ~ (30), (25) devient

  5 représentent les différentes composantes de l'induction magnétique au milieu de l'entrefer pour différentes valeurs de la perméabilité relative du fer et du glissement (i.e., résultats semianalytiques sont très satisfaisants par rapport aux résultats numériques, en considérant l'amplitude et la forme d'onde. Le dégradé d'amplitude de l'induction magnétique et du vecteur potentiel magnétique à l'intérieur de la machine asynchrone à cage d'écureuil pour   Figs. 6 ~ 7. Les Figs. 8 ~ 9 représentent les courants de Foucault induits dans les barres rotoriques obtenus analytiquement et numériquement pour       ; 600; 0.1 , 600; 1  r s  . Les résultats semi-analytiques sont très satisfaisants par rapport aux résultats numériques.

Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .

 34567 Fig. 3. Composante r et θ de l'induction magnétique au milieu de l'entrefer pour 100

Fig. 8 .Fig. 9 .

 89 Densité surfacique de courant de Foucault induits dans les barres rotoriques Densité surfacique de courant de Foucault induits dans les barres rotoriques pour 600 r   et 1  s : (a) Analytique et (b) FEMM.

  glissement : (a) Courant statorique et (b) Couple électromagnétique.

  Tableau 2. Temps de calcul, précision et dimension du système.

				M	12522	9882	7378	5746	5170
		r	s	Temps (min)	3,1	1,56	0,62	0,31	0,23
				EB (%)	4,15	4,2	4,18	4,23	4,28
			0,1
				EΓ (%)	0,9	0,93	1,69	2,02	1,98
	100
				EB (%)	5,17	5,94	6,51	9,9	10,04
			1
				EΓ (%)	1,87	2,13	4,02	7,11	7,21
				EB (%)	4,19	4,18	4,5	6,72	6,68
			0,1
				EΓ (%)	1,1	1,17	2,01	4,9	4,77
	1000
				EB (%)	6,2	6,42	7,33	10,9	10,92
			1
				EΓ (%)	0,63	0,69	2,04	6,73	6,64


Soro Gnatiessoro
email: gnatiessoro.soro@femto-st.fr

Eric Lantz

Alexis Mosset

Fabrice Devaux

Imaging Quantum Correlations through a Scattering Medium

We report photon-counting imaging of entangled photon-pairs of high Schmidt number transmitted through a scattering medium lying either in the near-field or in the far-field of the entangled photons source. We demonstrate that spatial momentum or position quantum correlations, measured between twin images recorded onto two separate detectors, exhibit in both cases speckle patterns. Moreover, the total correlation is only slightly lowered by the scattering.

Introduction

Among all the experiments dealing with quantum correlation properties of Spontaneous Parametric Down Conversion (SPDC) [1][2][3][4], some studies adress the propagation of entangled two-photon states through random medium [5][6][7]. While in [6] two single-point detectors are scanned in the image planes, we propose in this paper to directly image, onto two separate electron-multiplying charge coupled device (EMCCD) cameras operating in photon-counting regime [8], the two-photon states transmitted through a scattering medium. Here, far-field and near-field quantum correlations are evidenced depending on the two reported imaging configurations. Finally, we performed stochastic simulations [9] to confirm the experimental results. The experimental set-up for the far-field correlations is illustrated in Fig. 1a. Entangled photon pairs are generated by type-II SPDC in a 0.8 mm long β-barium borate (BBO) crystal pumped by a collimated pulsed laser at 355 nm. The output face of the crystal is imaged onto the diffuser using a 4 -f imaging system. The photon pairs transmitted by the diffuser are separated thanks to walk-off. Far-field detection is performed by two EMCCD cameras lying in the focal plane of lenses L 3 (f 3 =150 mm) and L 4 (f 4 =150 mm). Before detection, SPDC beams are spectrally filtered around degeneracy with narrow-band interferential filters F 1 and F 2 (@710 nm, ∆λ = 4 nm). For the near-field correlations, we only replace the 4 -f imaging system by a 2 -f imaging system (f = 150 mm). In this case, the position of the diffuser lies in the far-field of the crystal and the EMCCD cameras image the near-field. Fig. 1b and Fig. 1e show respectively the far-field and near-field images of the SPDC simply conditioned by the phase matching condition and the pump beam shape, respectively. As shown in [6], the absence of a speckle pattern in the single-beam images of the SPDC results from the incoherent character of the light formed by a single beam of the entangled light. In contrast, and in agreement with [6], the cross correlation of the images exhibits speckle patterns in the far-field (Fig. 1c) as well in the near-field (Fig. 1f). This experimental cross correlation image corresponds to the spatial distribution of signal-idler quantum correlations after traversal of the scattering medium.

Experimental protocol and results

Conclusion

We have experimentally studied at the quantum level spatial correlations of bi-photon states by using two EMCCD cameras. As expected, studies have shown the absence of single-photon speckle patterns when bi-photon states are transmitted through a random medium whereas the near-field and far-field two-photon imaging exhibit speckle patterns. Unlike [6] where only a small part of the photon is used, far-field and near-field correlations have been evidenced on the whole set of photons. Our experiment could be adapted to news technologies using spatial light modulator (SLM) to control quantum state of the light in random medium [10].

FIG. 1 :

 1 FIG. 1: (a) Experimental setup for far-field measurement: (b) average photon number in single far-field images (signal or idler) of SPDC, (c) measured correlation function in dB between 40 000 twin images and (d) correlation function issued from 10 000 stochastic simulations. For the near-field correlations: (e) average photon number in single near-field images (signal or idler) of SPDC, (f) measured correlation function in dB between 70 000 twin images and (g) correlation function issued from 10 000 stochastic simulations


Member, IEEE Nicolas Barbot

Member, IEEE Olivier Rance

Senior Member, IEEE Etienne Perret

Angle Sensor Based on Chipless RFID Tag

Keywords: Chipless RFID, RCS, scatterer, angle sensor

This paper presents theoretical and practical implementation of the chipless RFID technology to measure the orientation of tags attached on a 2D surface. Analytical expressions are used to recover tag orientation from mono-static co-polarization measurements. Simulations are performed to validate the orientation estimation technique. Moreover, measurements in anechoic chamber and real environment are done to evaluate the performance of the proposed solution. Contrary to already implemented techniques, the one introduced in this paper is not sensitive to distance variations which is an advantage for practical applications. Our solution can estimate the orientation of a chipless tag with an error less than 5 • in a range of 90 • .

I. INTRODUCTION

Chipless RFID technology is a new paradigm to realize identification of items at a very low-cost compared to the classical RFID. However, the lack of a chip introduces new challenges in the tag design to store the data and to communicate this information to the reader. A lot of effort has already been made to increase coding capacity, to improve robustness of the reading and to reduce cost [START_REF] Vena | Chipless RFID based on RF encoding particle: realization, coding and reading system[END_REF]. In [START_REF] Vena | Design of chipless RFID tags printed on paper by flexography[END_REF], authors have shown that the price of a fully-printable chipless tag can be reduced to 0.4 euro cent making it a valid alternative or a complementary solution compared to classical RFID.

Like barcodes, chipless tags encode the information within the geometry of the tag so that memory is not required anymore. However to compete with barcodes, chipless RFID needs to propose new functionalities that are difficult or impossible to realize with the barcode technology. These functionalities can be added without altering the main identification function of the chipless tag. Indeed, tag ID can be extracted from the scatterers response, at the same time as other useful physical quantities that can give information about the tag vicinity. This additional information can be used to develop new kind of applications like sensing. Sensors based on chipless tags offer contactless, low-cost, and batteryless solutions which can be deployed in harsh environments. Many researchers have investigated the use of chipless tags to design new type of sensors for temperature [START_REF] Girbau | Passive wireless temperature sensor based on time-coded UWB chipless RFID tags[END_REF]- [START_REF] Amin | Development of a chipless RFID temperature sensor using cascaded spiral resonators[END_REF], humidity [START_REF] Feng | Low-cost printed chipless RFID humidity sensor tag for intelligent packaging[END_REF], gas detection [START_REF] Vena | A fully inkjet-printed wireless and chipless sensor for CO 2 and temperature detection[END_REF], level of fluids [START_REF] Guillet | Design of a chipless RFID sensor for water level detection[END_REF] and positioning [START_REF] Barbot | A chipless RFID method of 2D localization based on phase acquisition[END_REF].

Only few angle sensors using chipless tags are found in the literature. In [START_REF] Vena | A compact chipless rfid tag using polarization diversity for encoding and sensing[END_REF], authors have designed a co-polarization chipless tag which exploits the polarization diversity to detect the tag orientation with a resolution of 20 • . In [START_REF] Genovesi | Chipless RFID tag exploiting cross-polarization for angular rotation sensing[END_REF], authors introduce a periodic structure printed on a grounded dielectric slab. Structure is rotated and authors show the backscattered field variations as a function of the orientation. Different orientations produce the same RCS value which limits the The authors are with Univ. Grenoble Alpes, Grenoble INP, LCIS, F-26000 Valence, France.

M. Perret is also with Institut Universitaire de France, Paris, France.

sensor range. In [START_REF] Genovesi | Chipless RFID sensor for rotation monitoring[END_REF], same authors introduce a structure composed of 2 slightly different dipoles (without ground plane) with a relative orientation of 45 • to overcome the problem of null response. Orientation estimation is however limited to a range of 45 • . Finally, in [START_REF] Genovesi | Chipless radio frequency identification (RFID) sensor for angular rotation monitoring[END_REF] authors consider the same finite periodic surface but with a scatterer composed of 4 different dipoles to determine the orientation. All these methods exploit the magnitude of the backscattered signal at the resonant frequency, and are thus sensitive to the distance between the antenna and the tag.

In this paper, new techniques are introduced to determine the orientation of a tag with an error less than 5 • in a range of 90 • . Contrary to the previous studies the approach is based on an analytical model. The expressions are derived to be independent from the reading distance.

The paper is organized as follows, in Section II, the analytical expressions used to extract the tag orientation from red the measured S-parameters, are introduced. In Section III, two different chipless tags are presented and characterized for orientation sensing in anechoic chamber and real environment. Finally, Section IV concludes the paper.

II. THEORY

This section begins by determining the analytical relation between the measurand θ and the estimated quantity θ obtained by measuring the backscattered S-parameters. It is shown that these relations can be greatly simplified by the use of thin dipoles as scatterers. Then, different methods are proposed to take into account the influence of distance variation.

A. Rotation of a Scatterer

Using the model described in [START_REF] Vena | A depolarizing chipless RFID tag for robust detection and its FCC compliant UWB reading system[END_REF], the S-parameter matrix S m measured at the VNA for a chipless tag can be expressed as:

S m = T • C • R + T • S • R (1)
where T and R represent the transmit path and received path respectively and are function of the distance d between the tag and the antenna , C is the coupling between the emission and reception and S the polarization scattering matrix of the tag. Note that each term is a 2 × 2 matrix of complex and frequency dependent parameters in the vertical and horizontal plane. From (1), assuming T vv T vh and R vv R vh like in [START_REF] Vena | A depolarizing chipless RFID tag for robust detection and its FCC compliant UWB reading system[END_REF], the response in co-polarization of a chipless tag can be extracted:

S 11m = I vv + R vv • S vv • T vv (2)
where I vv = T vv .C vv .R vv and can be estimated by measuring S 11m without the tag. Also, note that the quantity S 11m -I vv is directly proportional to S vv . Moreover, applying a rotation of θ to the scatterer (see Fig. 1) modifies the S matrix as [START_REF] Knott | Radar cross section[END_REF]:

V H θ f 1 V H θ f 2 θ 0 f 1 (a) (b)
S(θ) = Ω T • S • Ω ( 3
)
where T is the transpose operator and Ω is a rotation matrix defined by:

Ω = cos θ -sin θ sin θ cos θ (4)
From ( 3), expression of S vv (θ) can be written as:

S vv (θ) = S vv cos 2 θ + S vh sin 2θ + S hh sin 2 θ (5)
Note that S hh (θ) could also be expressed from [START_REF] Girbau | Passive wireless temperature sensor based on time-coded UWB chipless RFID tags[END_REF]. It is possible to solve (5) analytically to determine θ. However, this estimation depends on S vv , S hh and S vh which have to be known a priori, using at least three reference measurements, at the same distance d than the orientation measurement S vv (θ).

If a single parameter is estimated at a different distance, [START_REF] Amin | Development of a chipless RFID temperature sensor using cascaded spiral resonators[END_REF] can not be used anymore. An efficient way to greatly simplify (3) is to consider a thin dipole instead of a generic scatterer. For θ = 0, the scattering matrix S corresponding to the antenna mode of a dipole with a vertical orientation at the resonant frequency f 1 takes a simple form given by:

S f1 (0) = S f1 vvmax 0 0 0 (6)
where S f1 vvmax depends on the distance between the tag and the antenna. When this thin dipole is rotated by an angle θ [see Fig. 1(a)], (3) simplifies into:

S f1 (θ) = S f1 vvmax • cos 2 θ cos θ sin θ cos θ sin θ sin 2 θ . (7)
If we now consider a tag composed of two scatterers resonating at two different frequencies f 1 and f 2 , separated by an angle θ 0 [see Fig. 1(b)], and by supposing that the coupling between the two resonators can be neglected, [START_REF] Vena | A fully inkjet-printed wireless and chipless sensor for CO 2 and temperature detection[END_REF] can still be expressed separately for the two resonators at their respective frequency simply by replacing θ by, θ -θ 0 /2 for f 1 and θ + θ 0 /2 for f 2 . In all cases, S fi vvmax still depends on the distance between the tag and the antenna.

Fig. 2 presents the simulation of the scattered E-field in vertical co-polarization (VV) as a function of the frequency for a single resonator i.e., the tag configuration corresponding to Fig. 1(a). The resonance is observed at 3.5 GHz. As predicted by [START_REF] Vena | A fully inkjet-printed wireless and chipless sensor for CO 2 and temperature detection[END_REF], the backscattered field is maximum at θ = 0 and null at θ = 90 • .

VV θ = 0 • VV θ = 10 • VV θ = 20 • VV θ = 30 • VV θ = 40 • VV θ = 50 • VV θ = 60 • VV θ = 70 • VV θ = 80 • VV θ = 90 • main resonance parasitic resonance

B. Orientation Determination in Co-polarization

The relations developed in the previous section will be used to perform the extraction of the orientation. Three methods are presented, for which the objective is to determine an estimate of the orientation of the tag, noted θ, while minimizing the number of necessary measurements. A special attention is also given to the distance dependence.

The first one is a direct method which uses a single resonator [see Fig. 1(a)] and two measurements in vertical co-polarization. From ( 5) and ( 6), the vertical co-polarization component can be extracted as: S f1 vv (θ) = S f1 vvmax cos 2 θ. Taking the magnitude and isolating θ leads to:

θ = arccos |S f1 vv (θ)| |S f1 vvmax | = arccos   |S f1 11m -I f1 vv | |S f1 11m max -I f1 vv |   (8)
where the second equality of ( 8) is obtained from (2) and is valid only if T f1 vv (d)R f1 vv (d) are equal for both orientation measurement and reference measurement. This implies that the measurements have to be done at the same distance d.

However, it is possible to find expressions which do not depend on the distance. The second method also uses a single resonator [see Fig. 1(a)] and is based on the measurement of both vertical and horizontal component of the S matrix. From [START_REF] Vena | A fully inkjet-printed wireless and chipless sensor for CO 2 and temperature detection[END_REF], θ can be obtained directly with the diagonal elements of the matrix S f1 vvmax cos 2 θ and S f1 vvmax sin 2 θ, and the following formula can be derived:

θ = arctan |S f1 hh (θ)| |S f1 vv (θ)| = arctan |S f1 22m -I f1 hh | |S f1 11m -I f1 vv | (9)
Assuming that antenna gain is the same at both access, the second equality remains valid for all distances as T f1 vv (d)R f1 vv (d) and T f1 hh (d)R f1 hh (d) are equal. The third approach consists in using a tag with two resonators [see Fig. 1(b)], in this case, orientation determination can be done with a single measurement and a reference measurement both with a single-port antenna (by measuring S vv or S hh ). From [START_REF] Vena | A fully inkjet-printed wireless and chipless sensor for CO 2 and temperature detection[END_REF], taking the magnitude and considering a relative angle between the resonators of θ 0 = π/2, which involves that cos 2 (θ + θ 0 /2) = sin 2 (θ -θ 0 /2), the tag orientation can simply be extracted and is equal to:

θ = arctan |S f1 vvmax | • |S f2 vv (θ + θ 0 /2)| |S f2 vvmax | • |S f1 vv (θ -θ 0 /2)| + θ 0 /2 (10) = arctan    |S f1 11m max -I f1 vv | • |S f2 11m -I f2 vv | |S f2 11m max -I f 2 vv | • |S f1 11m -I f1 vv |    + θ 0 /2 (11)
where |S f1 vvmax | and |S f2 vvmax | can be obtained with a single reference measurement at a known orientation. Moreover, since both numerator and denominator are scaled by T f1 vv (d 1 )T f2 vv (d 2 ), (11) holds even if orientation measurement and reference measurement are not done at the same distance from the antenna. Finally, for all methods, θ estimation belongs in a 90 • interval due to the positive sign argument of the inverse trigonometric functions.

III. MEASUREMENT BENCH AND RESULTS

In the following, we describe the chipless tags and the measurement bench used to validate ( 9) and [START_REF] Genovesi | Chipless RFID tag exploiting cross-polarization for angular rotation sensing[END_REF]. Measurements are done in anechoic chamber and real environment.

A. Tag Design and Characterization

Two different chipless tags corresponding to the two proposed approaches have been realized for orientation determination (see Fig. 1). Note that the tag scatterers have been first introduced and optimized for identification purposes [START_REF] Vena | A depolarizing chipless RFID tag for robust detection and its FCC compliant UWB reading system[END_REF]. Resonators include a ground plane to increase their selectivities. The first resonator has been designed to resonate at 3.5 GHz and is based on five metallic stripes of dimensions 23 × 2 mm separated by a gap of 0.5 mm which increase the RCS compared to the unitary dipole but introduce a parasitic resonance which is observed at 3.9 GHz in Fig. 2 due to a different mode of resonance. The second resonator is based on the same structure except for the dipole's length which is 17.4 mm (which corresponds to a resonant frequency of 4.5 GHz and a parasitic resonance of 5.15 GHz). Both tags use 0.8 mm thickness Rogers RO4003 as substrate ( r = 3.55).

The measurement bench is presented in Fig. 3 and has been used to characterize the tags (see Fig. 1) in co-polarization (VV and HH) at a distance of 10 cm (and 20 cm) from the antenna. At these distances, the tag is in the radiating near field of the antenna [START_REF] Yaghjian | An overview of near-field antenna measurements[END_REF]. Measurement has been done using the Vector Network Analyser (VNA) N5222A. Dualaccess dual-polarization Satimo QH2000 antenna has been connected to the VNA ports 1 and 2 in the horizontal and vertical polarization respectively. For comparison purposes, full wave simulations using CST Microwaves with transient solver have been performed. Note that time gating has been used to remove the structural mode of the simulations and measurement results [START_REF] Ramos | Temporal separation detection for chipless depolarizing frequencycoded RFID[END_REF]. The impact of the tag orientation on the peak apex magnitude is represented in Fig. 4. The measured backscattered Efield in anechoic chamber and real environment is compared with both theory [START_REF] Vena | A fully inkjet-printed wireless and chipless sensor for CO 2 and temperature detection[END_REF] and simulation. For comparison purposes, all results have been normalized to 1. Simulations are in very good agreement with theory. Measurements present a slight offset compared to theory and simulation due to positioning errors and multi-path interference.

B. Orientation Determination Using a Single Resonator

As described in Section II-B, the tag orientation can be determined considering a single resonator using (8) and [START_REF] Barbot | A chipless RFID method of 2D localization based on phase acquisition[END_REF]. Orientation estimation θ, obtained in real environment is plotted in Fig. 5 as a function of the real value of θ. Orientation can be accurately derived in the interval [0 • ; 90 • ]. In both cases, average value of the absolute orientation error is equal to respectively 2.85 • and 2.98 • for ( 8) and ( 9) over the complete interval. Differences between these two methods will be highlighted in Section III-C and III-D.

C. Orientation Determination Using Two Resonators

As previously described, it is possible to determine the orientation using two resonators with a single port of the antenna using [START_REF] Genovesi | Chipless RFID tag exploiting cross-polarization for angular rotation sensing[END_REF]. Fig. 6 presents the back-scattered field in co-polarization as a function of θ for the tag presented in Fig. 1(b). For comparison purposes, all results have been normalized to 1. We can see that the simulation and measurements are in good agreement. The maximum error between simulation and measurement occurs at θ = ±45 • . Results of the tag orientation extraction using [START_REF] Genovesi | Chipless RFID tag exploiting cross-polarization for angular rotation sensing[END_REF], between -45 • and 45 • , are plotted in Fig. 5. As for ( 8) and ( 9), we can observe that the accuracy of orientation estimation using [START_REF] Genovesi | Chipless RFID tag exploiting cross-polarization for angular rotation sensing[END_REF] increases in the center of the interval. In average, the orientation error is equal to 4.6 • in real environment. As we have seen previously, ( 9) and ( 11) permit to recover the orientation of a chipless tag independently from the distance. However, in real environment, signal to noise ratio decreases when distance increases. This section estimates the performance of the sensor as a function of the distance and noise. Two different distances have been chosen: 10 cm (as previously presented) and 20 cm, both using the same measurement bench. As previously, the reference measurement has been chosen at 10 cm for both distances. Results obtained in real environment are presented in the two first columns of Table I where the average error is estimated for all configurations and for the two distances. We can observe that with (8), accuracy of the estimation depends on the distance i.e., the error becomes important if reference measurement and orientation measurement are not realized at the same distance. On the other side, with ( 9) and ( 11), the error is relatively independent of the distance (the slight increase is due to a lower signal to noise ratio). Since maximal errors appear at 0 • and 90 • (see Fig. 5), performance can be increased by considering a reduced angular range.

Concerning the noise effect, Monte Carlo simulations, presented in Fig. 5 and Table I, have been done to estimate the orientation error against additive white gaussian noise. For Fig. 5, SNR has been set to 25 dB which is consistent with the SNR level seen by the antenna when the tag is placed at 10 cm. Average value (solid line) and standard deviation (vertical bars) of the error show that all the methods are affected by a bias and a higher standard deviation at 0 • and 90 • . From Table I, note that ( 9) and ( 11) are more accurate when SNR is high and are characterized by same average error. Thus the proposed methods can estimate the orientation of a tag in real environment with a good accuracy, are not sensitive to distance variation, and are robust against noise.

IV. CONCLUSION

In this paper, three different approaches have been presented to determine the tag orientation in co-polarization. Analytical formulas have been derived in all cases. Simulations and measurements have been performed to evaluate the performance of the proposed solutions. We have shown that our solution can estimate the orientation of the chipless tag with an error less than 5 • . Moreover, two of the three methods presented in the paper are not sensitive to distance variation between the tag and the antenna. Finally, the same resonators can be used for reading the tag ID and for measuring the orientation so that tag's coding capacity is not reduced. Thus identification and sensing can be combined to address new kind of applications.

Fig. 1 .

 1 Fig. 1. Different tag configurations used: (a) with a single resonator, (b) with two resonators.

  ) Scattered E-field (mV/m)

Fig. 2 .

 2 Fig. 2. Simulated scattered E-field in vertical co-polarization (VV) for the tag presented in Fig. 1(a) for different orientations using time gating.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Measurement bench used for tag orientation determination. A monostatic configuration is used with a dual polarization.

Fig. 5 .θFig. 6 .

 56 Fig.5. Orientation estimation in co-polarization using (8), (9) and (11) obtained in real environment (RE) at 10 cm, and using Monte Carlo (MC) method: average estimated value and standard deviation (vertical bars).

TABLE I AVERAGE

 I ESTIMATION ERROR AS A FUNCTION OF THE DISTANCE (REAL ENVIRONMENT) AND NOISE (MONTE CARLO SIMULATIONS)

	d / SNR	10 cm 20 cm	35 dB 25 dB 15 dB
	Equation (8)	2.85 •	23.6 •	24.0 •	2.87 •	57.4 •
	Equation (9)	2.98 •	4.70 •	1.29 •	3.32 •	8.67 •
	Equation (11)	4.64 •	5.05 •	1.29 •	3.33 •	8.74 •
	D. Evaluation of the Distance and Noise Impact

ACKNOWLEDGMENTS

This research was funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant N • 772539).