Equivalent Circuit Model of Electromagnetic Behaviour of Wire Objects by the Matrix Pencil Method

Transcription

1 ERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol. 5, No., May 008, -0 Equvalent Crcut Model of Electromagnetc Behavour of Wre Objects by the Matrx Pencl Method Vesna Arnautovsk-Toseva, Khall El Khamlch Drss, Leond Grcev Abstract: The electromagnetc behavour of thn-wre objects s an mportant engneerng problem. A general analyss of such objects s done accordng to the moment method approach wthn an antenna-based electromagnetc model. Ths paper concerns possble extracton of parameters that would enable determnaton of an equvalent crcut model of such wre-structure objects. As an example, the authors focused on the electromagnetc behavour of hgh frequency power converters and groundng conductors. Keywords: Electromagnetc model, Numercal modellng, Matrx pencl method. Introducton Analyss of groundng systems at hgh frequences s very mportant n EMC studes related to lghtnng. On the other hand, wth the ever-ncreasng swtchng frequency power converters are also the subject of EMC/EMI analyss as sources of radaton emsson n the surroundngs. In both cases, n order to predct the transent behavour, these objects are consdered as thn-wre conductors, whch are treated at hgh frequences by electromagnetc models based on the antenna theory. In ths paper, the authors are nterested n the possblty of extractng the parameters of equvalent crcut models of such wre-objects. More specfcally, our analyss s related to transent analyss of:. Rectangular loop model of power converter;. Vertcal groundng rod. In both cases, the analyss s frst performed n the frequency doman usng developed electromagnetc antenna-based numercal models. In the second step, after transformaton of the functons of nterest to the tme doman, the objectve s to extract the poles and resdues of functons usng the matrx pencl method. s. Cyrl and Methodus Unversty, Faculty of Electrcal Engneerng and Informaton Technology, Karpos II, 000 kopje, Macedona, E-mal: atvesna@fet.ukm.edu.mk Unversty Blase Pascal, LAMEA, Aubère, France, E-mal: drss@ubpmes.unv-bpclermont.fr

2 V. Arnautovsk-Toseva, K. El Khamlch Drss, L. Grcev Usng these parameters, t may be possble to later determne the equvalent crcut models. Ths analyss also enables observaton of the nfluence of the wre-object s geometry on ther transent behavour that cannot be analyzed usng crcut theory approaches. Electromagnetc Analyss of Wre Objects at Hgh Frequences. Electromagnetc analyss of the converter crcut The power converter crcut s constructon s based on several elementary cells of commutaton that may be represented by Fg.. The PCB trace prnted over a delectrc layer crcut behaves lke a transmttng antenna that radates drectly nto the mmedate surroundngs. The geometry of such a converter crcut s consdered to be a rectangular loop havng length L and wdth W. PCB traces are modeled usng the thn-wre approxmaton. Here, V s the source voltage and R L represents the load. It s assumed that the swtch s n poston, so that the nfluence of PCB wthout supply s not taken nto account. Fg. Rectangular crcut loop model of the power converter. In our analyss, the swtch and the DC source together are replaced by an equvalent voltage source of V step functon wth a rse tme of the order of ns, whch s placed on the poston of the swtch, v () t as shown on Fg.. + V + V R L W v (t) R L L Fg.. Equvalent voltage source functon v () t. Our mathematcal model for electromagnetc analyss of the power converter crcut gven on Fg. s based on rgorous formulatons derved from the full set of Maxwell s equatons, based on the theoretcal background of mcrostrp antenna analyss. The model s developed n the frequency doman and takes nto account all electromagnetc effects related to the geometry of the

3 Equvalent Crcut Model of Electromagnetc Behavour of Wre Objects... converter crcut and the nfluence of the delectrc board nterfaces. More detals about the mathematcal model and the numercal results are gven n []. In ths paper the authors are more focused on the analyss of transent responses of the converter crcut when excted by voltage source exctaton of the V step functon wth a hgh rse tme of ns. Our objectve was to nvestgate the nfluence of the crcut layout on the transent behavour of the system. These computatons were carred n the frequency doman n conjuncton wth the nverse Fourer transform method. Frstly, usng the developed electromagnetc model we obtaned the transfer functon of the converter s rectangular loop-crcut H ( f) = VRL( f)/ V, wth V = V and R = 0.Ω n the frequency doman of nterest from 0MHz up to several GHz. In our analyss the load R L was defned as 0kΩ to smulate an open crcut n order to obtan only the response of the rectangular crcut-loop tself. Our objectve was to nvestgate the nfluence of the crcut layout on the transent behavor of the system. 0cm + V, R RL V, R + 6cm (c) (a) 0cm R L + V, R R L 4cm 6cm Fg. 3 Three confguraton of the converter crcut. To cover the above mentoned hgh frequency range, t s assumed that the frequency of the voltage harmonc s n range from 0MHz to GHz. Three loop geometres are consdered: (a) square loop (0 0cm), (b) rectangle loop (4 6cm) and (c) rectangle loop (6 4cm), as shown on Fg. 3. The length of the leads (permeter of the crcut) s fxed to 40cm, whle the equvalent thnwre radus of the lead s 0.5mm. The delectrc board s defned by relatve permttvty 4.7 and board thckness of.5mm. Two dstnct postons of the equvalent source are assumed n the analyss: Case - opposte to the load, Case - n the mddle of length L, as shown on Fg. 4. V v (t)v v (t) R L Fg. 4 Two postons of the source wth respect to the load. 3

4 V. Arnautovsk-Toseva, K. El Khamlch Drss, L. Grcev. Electromagnetc analyss of groundng rod Consder a thn vertcal groundng rod of radus a whch s placed n homogeneous sol. It s assumed that the rod extends from depth d to ( d + L) as shown on Fg. 5. The sol s characterzed by permttvty ε=ε0ε r, permeablty μ 0 and conductvty σ. μ 0 Ar '' ε 0 d V 0 z=0 L a Ground '' ε,μ 0,σ Fg. 5 Vertcal groundng rod n homogeneous sol. The hgh frequency response of the groundng rod s analyzed by a complete electromagnetc model. It s based on ntegral equatons derved from the Maxwell's equatons and the nfluence of the earth s taken nto account by an exact ommerfeld formulaton []. The current dstrbuton s computed as a response to njected current I. Ths leads to matrx equaton [ Z ][ I] = [ ZI]. () where, the column matrx [I] represents the unknown currents to be determned, [Z] s the generalzed mpedance matrx, and [ Z I ] represents the energzaton matrx. The developed computer model s used to compute the mpedance to ground Z g( f) = V0 ( f)/ I, where V 0 ( f ) s the voltage between the feed pont and remote ground, and I = A s the njecton current. The calculatons are performed wthn frequency range from khz to 00MHz. Our objectve was to nvestgate the nfluence of the conductor length L and the sol conductvty σ on ts transent behavour. These computatons were carred n the frequency doman n conjuncton wth the nverse Fourer transform method. 3 Introducton After frequency-dependent functons H( f ) and Z g ( f ) were once computed, the results were converted to the tme doman usng the nverse fast Fourer transform algorthm. Thus we obtaned hkt ( ) and zg( kt ), k = 0,, N ; wthn N samples and T samplng perod respectvely. 4 I

5 Equvalent Crcut Model of Electromagnetc Behavour of Wre Objects... Afterwards, usng the matrx pencl method [3] we determned ther approxmate representatons: hkt ( ) M k = yk+ Rz for k 0,, N zg( kt) =. () = Here, R and z are resdues and poles of the Matrx Pencl method basc st ( j ) T functon respectvely. Each pole z may be represented by z = e = e α + ω, where = 0,, M. Here, R represents resdues, α represents dampng factors and ω are angular frequences. It s possble to defne matrx Y of dmensons ( L+ ) ( N L) by regroupng the terms y k y y yl+ y y3 y L+ Y = = [ Y Y YL YL+ ] (3) yn L yn L+ yn and two sub-matrces [ Y Y Y ] [ ] Y = (4) L Y = Y Y3 Y L +. (5) It s shown n [3] that Y Y s dagnosable and permts precse determnaton of the z poles. Fnally, n order to optmze the performances of the method, t s chosen that L s close to N /, so that the choce of M s what determnes the number of domnant poles n the dentfcaton process of the functon hkt ( ) and zg( kt ). The above matrces are not generally square matrces, and the nverse matrx of Y s practcally pseudo-nverse. The correspondng tme-doman responses hkt ( ) and zg( kt ) were subjects of the matrx pencl method analyss. Thus we obtaned ther z poles whch were compared wth the frequency dependent functons H( f ) and Z g ( f ). The results are presented n the next secton. 4 Numercal Results 4. Converter crcut Fgs. 6a and 6b, shows the transfer functon H( f ) n the frequency range from 0MHz to GHz obtaned for all three geometres wth respect to the 5

6 V. Arnautovsk-Toseva, K. El Khamlch Drss, L. Grcev poston of the source: and (opposte to the load and n the mddle of length L, as gven on Fg. 4). The curves n red, green and blue correspond to the three crcut-loop geometres consdered: (a) rectangle loop (4 6cm); (b) square loop (0 0cm), and (c) rectangle loop (6 4cm). (a) Case : ource opposte to the load. (b) Case : ource n the mddle of length L. Fg. 6 H ( f ) n frequency range from 0MHz to GHz. After transformng H( f ) n tme we obtaned samples of hkt ( ), k = 0,, N, whch were subject to the pencl method analyss. The results obtaned whch represent complex values of z are gven n Tables -a and -b. Table -a Values of α and f n case of converter, source opposte to the load. Case α n Np/ns f = ω/( π ) n GHz Cr. a Cr. b Cr. c Table -b Values of α and f n case of converter, source n the mddle of length L. Case α n Np/ns f = ω/( π ) n GHz Cr. a Cr. b Cr. c Fgs. 7a and 7b, gves a vsual presentaton of the results obtaned usng the pencl method. Comparng Fgs. 6 and 7 t may be observed that magnary values of z n GHz (on the ordnate) correspond to pck frequences of H( f ). Respectvely, real values of z (on the abscssa) correspond to the dampng factors.

7 Equvalent Crcut Model of Electromagnetc Behavour of Wre Objects (a) poles z of ht (): Case - ource opposte to the load (b) poles z of ht (): Case : ource n the mddle of length L. Fg. 7 z poles of ht (): values on the ordnate frequency n GHz, values on the abscssa attenuaton n Np/ns. It s shown that not only the geometry of the crcut but also the poston of the source (swtch wth respect to the load) nfluence the transent behavour of the converter crcut. 7

8 V. Arnautovsk-Toseva, K. El Khamlch Drss, L. Grcev 4. Groundng rod Fgs. 8a and 8b, represents the groundng mpedance Z g ( f ) n the frequency range from khz to 00MHz. On Fg. 8a, the curves n red, green and blue correspond to the rod lengths consdered: L = 4m, L = 3m and L = m. The depth s d = 0.5m. Fg. 8-b shows the mpedance Z g ( f ) n the case of a groundng rod of L = 4m when placed n homogeneous sol wth varous conductvty: σ = 0.00/m (red), σ= 0.005/m (magenta) and σ = 0.0/m (orange). (a) Rod of L = 4m, L = 3m and m L = n sol wth σ = 0.00 ( / m). (b) Rod of L = 4m n sol wth σ = 0.00, and 0.0 ( / m). Fg. 8 Z ( f ) n the frequency range from khz to 00MHz. g 8

9 Equvalent Crcut Model of Electromagnetc Behavour of Wre Objects (a) poles z of zg () t : Varous lengths of the groundng rod (b) poles z of zg () t : Varous conductvty of the sol. Fg. 9 z poles of zg () t : values on the ordnate frequency n MHz, values on the abscssa attenuaton n Np/μs. 9

10 V. Arnautovsk-Toseva, K. El Khamlch Drss, L. Grcev The correspondng values of z poles obtaned usng the pencl method are gven n Tables -a and -b. The results are vsualzed on Fgs. 9a and 9b. The nfluence of the rod length as well as the nfluence of the sol conductvty may also be observed. Real values of z (on the abscssa) correspond to attenuaton factors. Table -a Values of α and f n the case of a converter, where the source s opposte to the load. Rod L (m) α n Np/ μ s f = ω/( π ) n MHz Table -b Values of α and f n the case of a converter, the source s n the mddle of length L. σ (/m) Rod L = 4m Rod L = 4m Concluson In ths paper, we have appled the matrx pencl method to determne the poles of specfc frequency-doman functons that represent the transent behavour of two dstnct thn-wre objects. The results obtaned n ths way were compared wth those obtaned n the frequency doman usng a complete electromagnetc approach. The man objectve of ths work was to present the possblty of parameter extracton that would enable determnaton of an equvalent crcut model of such wre-structure objects. 6 References [] L. Grcev, V. Arnautovsk-Toseva: Groundng ystems Modelng for Hgh Frequences and Transents: ome Fundamental Consderatons, n Proc. IEEE Bologna Power Tech., Bologna, Italy, June 3-6, 003. [] V. Arnautovsk-Toseva, Y. Rousset, K. El Khamlch Drss, L. Grcev: Radated EMI from Power Converters, erban Journal of Electrcal Engneerng, Vol., No., Nov. 005, pp [3] T.K. arkar, O. Perera: Usng the Matrx Pencl Method to Estmate the Parameters of um of Complex Exponentals, IEEE Antennas and Propagaton Magazne, Vol. 37, No., Feb. 995, pp