* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Full Article in PDF - Advances in Radio Science
Three-phase electric power wikipedia , lookup
Power engineering wikipedia , lookup
Stray voltage wikipedia , lookup
Wireless power transfer wikipedia , lookup
Opto-isolator wikipedia , lookup
Aluminium-conductor steel-reinforced cable wikipedia , lookup
Nominal impedance wikipedia , lookup
Resonant inductive coupling wikipedia , lookup
Amtrak's 25 Hz traction power system wikipedia , lookup
Skin effect wikipedia , lookup
Electrical substation wikipedia , lookup
Loading coil wikipedia , lookup
Transmission tower wikipedia , lookup
Electromagnetic compatibility wikipedia , lookup
Electric power transmission wikipedia , lookup
Alternating current wikipedia , lookup
Two-port network wikipedia , lookup
Telecommunications engineering wikipedia , lookup
Transmission line loudspeaker wikipedia , lookup
Network analysis (electrical circuits) wikipedia , lookup
Adv. Radio Sci., 10, 221–225, 2012 www.adv-radio-sci.net/10/221/2012/ doi:10.5194/ars-10-221-2012 © Author(s) 2012. CC Attribution 3.0 License. Advances in Radio Science Simplified susceptibility analysis of shielded multiconductor transmission lines B. Schetelig1 , S. Parr1 , R. Rambousky2 , and S. Dickmann1 1 Faculty of Electrical Engineering, Helmut-Schmidt-University/University of the Federal Armed Forces Hamburg, Germany Research Institute for Protective Technologies and NBC Protection (WIS) Munster, Germany 2 Bundeswehr Correspondence to: B. Schetelig ([email protected]) Abstract. In this contribution, a new kind of coupling parameters for shielded multiconductor transmission lines is presented. These parameters are easy to calculate and simplify the vulnerability analysis of shielded cables with multiple inner conductors. In this paper, the underlying theory is derived in detail. The gained insight is demonstrated with a sample measurement. 1 Introduction If the susceptibility of complex structures in large volumes is analyzed, the attention has to be drawn to a smart integration of the transmission lines into the entire model of the structure. A numerical field analysis can be accelerated by separating the analysis into two parts. The field analysis is carried out numerically and afterwards, the cables are added using multiconductor transmission line theory. This approach is particularly favourable for shielded transmission lines. Cable shields can be integrated into transmission line theory by introducing the so-called transfer impedance and transfer admittance. These quantities are well-defined for single conductor transmission lines. Transfer parameters for multiconductor transmission lines are determined in several ways in literature (e.g. Weitze, 1991; Kley, 1991; Degauque and Hamelin, 1993; Kasdepke, 1997). In this paper, a new definition of multiconductor coupling parameters is introduced that uses some approximations to simplify the description of the coupling process. 2 Theory of the transfer parameters In this section, the theory of the transfer parameters is briefly reviewed to the extent as it is applied in the proposed procedure. To describe the voltages and currents on shielded cables, the well-known equivalent circuits from transmission line theory are used. Because of the shield, which separates the inner conductors from the outer world, one circuit is used for each of these areas (Fig. 1). The inner system consists of the inner conductor and the inside of the shield (reference conductor). The outer system is made up by the outside of the shield and the remaining structure around the transmission line. The shield is assumed to be a good one, which allows the application of the “good shielding approximation” and the concept of electromagnetic topology (Baum, 1980). This permits to consider the influence of external electromagnetic fields on the inner conductor primarily as an indirect coupling via the cable shield. In this concept, the field couples to the shield and the resulting voltages and currents of the shield then cause interferences inside. The outer and the inner systems are linked via the transfer impedance and the transfer admittance. They can be defined, using a short section of a cable (Fig. 2), as (Vance, 1978): 1 dUi 0 ZT = · ; Ii = 0 , IS dz 1 dIi Y 0T = − · ; Ui = 0 . US dz (1) (2) These two parameters can be found in the definitions of the distributed sources of the inner system (Fig. 1): U 0 s,i = Z 0 T · IS , (3) Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V. ckmann1 Fig. 2. Very small section of transmission line to define the transfer University / University of the Federal Armed Forces Hamburg, Germany impedance (left) and transfer admittance (right) ologies and NBC Munster, Germany 222 Protection (WIS) B. Schetelig et al.: Simplified susceptibility analysis of shielded multiconductor transmission lines s briefly d proce- 0 Us,i = ZT0 · IS L0i Ii Ci0 G0i inner system Ui 0 Is,i = −YT0 · US ∆z Ra0 2 Fig. 3. Principle of using an equivalent cable description to define 0 Us,a Fig.susceptibility 3. Principle of using an equivalent cable description alternative transfer Schetelig et al.: Simplified analysis of parameter. shielded multiconductor transmission lines IS L0a to define alternative transfer parameter 0 l << λ outer system l G0a dz Ca0 0 l << λ l 0 10 US wire A wire B wire C wire D dz 0 Is,a I impedance and the transfer admittance. They can be defined, 10 U Fig. 1. Equivalent circuit of a shielded transmission line a shortanalysis section of a cable (Fig. 2), as (Vance, Equivalent circuit of a shielded transmission line. using 2 Schetelig et al.: Simplified susceptibility of shielded multiconductor transmission lines 1978): Ui i −1 IS 0 cable shield l << λ S inner conductor l 0 dz l << λ l dz Fig. dure.2. Very small section of transmission line to define the transfer impedance (left) and transfer admittance (right) To describe the voltages and currents on shielded cables, Ui U2/U0 umes is egration e strucy sepacarried d using oach is s. ine thence and ned for ters for several gauque , a new s introdescrip- Ri0 0 10 −2 10 Ii 1 dUi ; Ii = 0 , · IS dz 1 dIi YT0 = − · ; Ui = 0 . US dz ZT0 = wire A wire B wire C wire D (1) the well-known equivalent circuits from transmission line −1 IS 10 −3 US (2) theory are used. Because of the shield, which separates the 10 inner conductors from the world, one circuit is used for cable shield innerouter conductor each of these areas (Fig. 1). The inner system consists of the −2 These two1010 parameters can be found in the definitions of the Very small section transmission line Fig. smalland section of transmission linetotodefine definethe thetransfer transfer inner2.conductor theof inside of the shield (reference con−4 0 0.5 1 1.5 impedanceThe (left) and (left) and transfer transferadmittance admittance (right). ductor). outer system is made up(right) by the outside ofdistributed the sources of the innerf system (Fig. 1): x 1092 / Hz shield and the remaining structure around the transmission −3 line. The shield is assumed to be a good one, which allows 10 0 3. Principle 0 0 Fig. of using an equivalent cable description to define Ithe = −Y · U . (4) Us,i =inner ZT0conductors · IS , of a shielded LiYCY (3) application of the “good shielding approximation” and T s,i S Fig. 4. Coupling to the four LiYCY alternative transfer parameter the concept of electromagnetic topology (Baum, 1980). This 0 transmission line line. This description of cable shields can be expanded to multiIs,i = −YT0 · US . (4) permits to consider the influence of external electromagnetic −4 conductor transmission lines. While the equivalent circuit of 10 0 0.5 1 1.5 2 fields on theand inner conductor primarily They as an can indirect cou9 f / Hz impedance theremains transfer admittance. defined, the outer system the same, the model ofbethe inner x 10 pling via the cable shield. In this concept, the field couples cable be expanded the border of aof bundle of shields conductorscan is rather small. The to mulusing sectionby of the a cable (Fig. 2),circuit as (Vance, 1978):This description systema short is replaced equivalent of a multiconsimilar, if the cable is not mechanically damaged. This is to the shield and the resulting voltages and currents of the difference between the less important transfer admittances is ductor transmission line. In this model, each inner conducFig. 3. Principle of using an equivalent cable description to define ticonductor lines.andWhile the Other equivalent valid transmission for the absolute values the phases. inves- circuit shield then cause interferences 1 dUinside. i Fig. 4. Coupling to the four inner conductors of a shielded LiYCY slightly larger (Kasdepke, 1997). 0individual alternative transferby parameter tor is stimulated distributed voltage and current ZT = · ; Ii = 0 , tigations show that even thethe difference between the transfer of (1) the outer system same, the model of the inner The outer and thebe inner linked via the transfer transmission line remains IS systems dz asare These observations are of course only for cables sources. They can described follows: impedances of a conductor in the centre andvalid a conductor at 1 dIi without mechanical defects. If shielded multiconductor casystem is replaced by the equivalent circuit of a multicon0 0 0 the border of a bundle of conductors is rather small. The Y = − · ; U = 0 . (2) i They can be defined, T (5) U · ISthe , transfer impedance s,i = Z Tand bles are bent by force, the coupling parameters of the indiUadmittance. S dz difference between the less transfereach admittances ductor transmission line. Inimportant this model, inneris conduc0 0 section of a cable (Fig. 2), as (Vance, 1978): short vidual ifwires can become different from each other. the cable is not mechanically damaged. ThisAs is it =a −Y · US . Iusing (6) similar, T s,i slightly larger (Kasdepke, 1997). These two parameters can be found in the definitions tor of the is stimulated by individual distributed voltage and is usually not known throughout a susceptibility analysis, if current valid for the absolute values and the phases. Other inves1 inner dUparameter i system (Fig. distributed sources the 1): 0of These formulas use transfer vectors of dimension These observations are of course only valid for cables ZT = · ; Ii = 0 , (1) there areshow mechanically stressed cables, these differences tigations that the difference between the transferbesources. They can be even described as follows: IS dz N to describe the coupling to 0a shielded N conductor trans- impedances without mechanical defects. If shielded multiconductor tween the various wires cannot be taken into consideration, of a conductor in the centre and a conductor atca0 Us,i 1= ZdI (3) T ·i IS , mission line. bles are bent by force, the coupling parameters of the indi0 anyhow. the border of a bundle0 of conductors is rather small. The YT = −0 · 0 ; Ui = 0 . (2) dzT · US . Ufollowing Z 0Tit ·isISassumed , each (5) Is,iU= (4) vidual wires can become different from As S −Y s,i difference between the less=important transfer admittances is it Therefore, in the thatother. an external is usually not known throughout a susceptibility analysis, if 0 0 slightly larger (Kasdepke, 1997). electromagnetic field quite ·similar on two parameters can beshields foundcoupling in the of multhe 3These Concept of simple equivalent parameters I s,icauses = −Y . interferences (6) This description of cable can be definitions expanded to T USthese there are conductors. mechanically stressed cables, differences be4, the shield of a four-wire-cable the inner In Fig. distributed sources of the inner system (Fig. 1): These observations are of course only valid for cables for multiconductor ticonductor transmissiontransmission lines. While lines the equivalent circuit tween the at various wires cannot be intoUconsideration, is excited one end of the cable bytaken a voltage 0 . The resultwithout mechanical defects. If shielded multiconductor caof the outer system remains the inner formulas 0 0 same, the model of theThese anyhow. use transfer parameter vectors of are dimension Us,i = Z (3) between each inner wire and the shield ing voltages U T · IS , industrial multiconducPrevious work (Jung, 2003) analyzed bles are bent by2,i force, the coupling parameters of the indisystem is replaced by the equivalent circuit of a multicon0 0 Therefore, in the following assumed external measured atcan the other enddifferent (setup analogous Fig.an 5). Is,iwires. = US .shown tor cables with stranded ItT ·was that the coupling N(4)to describe the coupling to ait isshielded N conductor transvidual wires become from eachtothat other. As it ductor transmission line. In−Y this model, each inner conducelectromagnetic field causes quite similar interferences on parameters of the wires within such a cable usually are quite This property of approximately identical voltages and if curis usually not known throughout a susceptibility analysis, tor is stimulated by individual distributed voltage and current line. This description of cable can be expanded tomission multhe inner conductors. fig. 4, cables, thethe shield of adifferences four-wire-cable similar, ifThey the can cable not shields mechanically rents shall be used toIn simplify calculation of thebecouare mechanically stressed these sources. be isdescribed as follows:damaged. This is there ticonductor transmission lines. While the equivalent circuit is excited at one end of cannot the cable atransmission voltage U0 . The the various wires beby taken into consideration, valid for the absolute values and the phases. Other inves- tween pling parameters of multiconductor lines.resultThat of the outer system 0 same, the model of the inner ing U2,imulticonductor between each inner wire andline the will shield anyhow. for, voltages the shielded transmission be are retigations show that remains even between the transfer U 0s,ithe =the Zdifference (5) T · IS , system is replaced by the equivalent circuit ofa aconductor multicon-at measured at the other end (setup analogous to fig. 5). 0 0 centre impedances of a conductor in the and placed by an equivalent model, which is made up by only one 3 (6)Concept of simple equivalent coupling parameters for Therefore, in the following it is assumed that an external I s,i In = −Y · US . each inner conducductor transmission line. this Tmodel, This property of approximately identical voltages andon curelectromagnetic field causes quite similar interferences tor is stimulated by individual distributed voltage and currentmulticonductor transmission lines These formulas use transfer parameter vectors of dimension the inner conductors. In fig. 4, the shield of a four-wire-cable rents shall be used to simplify the calculation of the couAdv. Radio 221–225, as 2012 www.adv-radio-sci.net/10/221/2012/ sources. TheySci., can10, be described follows: N to describe the coupling to a shielded N conductor transispling excited at one endofofmulticonductor the cable by a voltage U0 . The resultparameters transmission lines. That Previous work (Jung, 2003)each analyzed mission line. ing voltages U2,i between inner wireindustrial and the arerefor, the shielded multiconductor transmission lineshield willmulticonducbe U 0s,i = Z 0T · IS , (5) measured theequivalent other end (setup analogous fig. 5). placed an model,It which istomade up by the onlycoupling tor(6)cables withbyatstranded wires. was shown that I 0 = −Y 0 · U . U2/U0 ng paras is preimplify tiple inderived sample s,i T S 4 multiconductor transmission Schetelig lded multiconductor transmission lines susceptibility analysis of shielded 3 B. Schetelig et al.: Simplified lines et al.: Simplified 223susceptibility cable under test Zein,a (9) (10) the (11) (12) and (13) (14) (15) (16) inner conductor Zi inner conductor and the shield (Fig. 3). Nevertheless, that This means that coupling to theconductors inner system a multicondoes not mean that the inner are of simply shortcircuited or that allline wires one are neglected. Instead, ductor transmission canexcept be described by scalar coupling the observation of almost identical parameters. The consequence is that currents one can and nowvoltages use the from above is applied the following way: simple procedure for a coaxial line to describe the multiconFollowing ductor coupling(Andrieu, process. 2006), the differential equations of the inner multiconductor system, U1 I1 d U 2 = (R 0 + j ωL0 ) · I2 , (7) 4 −Measurement setup to determine the coupling para... ... dz metersUN IN I1 U1 The application coupling para of the conceptof equivalent d I2 = (G0 + j ωC 0 ) · U2 , − (8) meters from the measurement procedure used ... ... dzisindependent to determine approach is the IN the parameters. A well-known UN “triaxial This measurement setup be simplified can be setup“. simplified. This is done by using thecan assumptions as shown in fig. 5. The measurement principle is, to excite N P the shield via the source U0 and to Ui measure the voltage U2 n=1the shield. Using transmisbetween the inner conductor and , (9) Ueq = U1 = U2 = ... = UN = sion line theory, these quantities N can be used to calculate the N X transfer impedance and the transfer admittance (Tiedemann, I = In = N · I1 . (10) eq 2001). n=1 There is no difference between the calculation of the transThen, the Eqs. (7) and (8) reduce from dimension N to the ferdimension parameters of a physical coaxial line and of the substitute 1: transmission line from above. The equivalent cable is ded 0 · Ieq j ωL0 eq )of − Uineqthe = (R scribed usual a single conductor with(11) the eq +notation dz 0 0 0 0 and the equiva, C , L , G per unit length parameters R eq eq eq eq d 0 0 Ieq =U(G j ωCequivalent lent−voltage If + these are eq ) · Ueq . substitute quantities(12) eq . eq dz used0 instead0 of 0physical ones, any measurement setup can be 0 Req results of parameters merging theof rows eq , Geq and eq are thecoupling used to, L determine the C equivalent eq. and columns: 17 and eq. 18. N P N P the hey (17) (18) R 0 eq = R 0 ij i=1 j =1 N2 , (13) 5 Susceptibility analysis with equivalent coupling paraN P N P meters L0 ij L0 eq = R0 ,L0 , G0 ,C 0 10 U2 Simplifiedsetup setuptotodetermine determinethe the transfer transfer parameters parameters of Fig.Fig. 5. 5.Simplified ofaa shielded single conductortransmission transmissionline line. shielded single conductor ling anstute paetry U0 Z2 10 i=1 j =1 N2 , (14) The overview of the procedure (Fig. 6) shows that the first N X N three0 stepsX are completed so far. The fourth step is to do the G eq = G0 ij , (15) susceptibility analysis of the structure. This includes calcui=1 j =1 lating the distributed sources of the inner system using the N X N X 0 0 0 0 quantities Z and Y C eq = T,eq C ij ,T,eq . Common single conductor trans(16) mission line can then be used to calculate the interi=1theory j =1 ferences Ueq , Ieq at the end of the lines (Tesche et al., 1997). After that, eq. 9 and eq. 10 can be used to derive the interwww.adv-radio-sci.net/10/221/2012/ ferences U , I, of the original transmission line. 10 calculation of the p.u.l. parameters of the substitute transmission line Determination of the transfer parameters of the substitute transmission line Vulnerability analysis of the substitute transmission line Derivation of the vulnerability of the original transmission line 0 ,L0eq , Req 0 0 Geq ,Ceq Z’T / Ω/m (8) Z1 one equivalent conductor (7) calculation of the p.u.l. parameters of the original transmission line 10 10 0 ZT,eq , 0 YT,eq 10 Ueq ,Ieq 10 U,I Fig. 6. Fig. 6. Summary Summary of of the theconcept conceptofofalternative alternativecoupling couplingparameters. parameters with ij : indexation of the matrix elements of the prevailing 6original Conversion of the parameters. equivalent coupling parameters to per unit length the conventional ones Equations (11) and (12) describe a shielded equivalent transmission line with only one inner conductor. This substitute system iscoupling solely defined by its equivalent per unit The equivalent parameters in combination with the length parameters. The corresponding and exact substitute equivalent single conductor notation of the inner system can geometry known nor the of interest for this application. be used tois neither determine easily interferences on the inner This substitute cable can now be used conductors. Nevertheless, it is possible to to calculate transformthe the equivalent transfer admittance.descripThey equivalent transferimpedance parametersand to transfer the conventional are defined as: be useful, if there is already a multi conduction. This might Ueq line implementation for the inner system. tor0 transmission Z T,eq = , (17) IS The relationship can be drawn from the definitions of the parameters: IeqUsing the assumption of approximately identiY 0 T,eq = . on the inner conductors, the conventional (18) cal interferences US transfer impedance can be written as: This means that coupling to the inner system of a multiconductor transmission can be described by scalar couline U1 pling parameters. The consequence is that one can now IS use the simple procedure line to describe the fora coaxial ... U1 N ×1 multiconductor coupling ZT0 = process. ·1 . (19) = IS U N Measurement setup ItoS determine the coupling parameters Using the definition of the equivalent transfer impedance 0 application of the concept of equivalent coupling paThe Z T,eq (Eq. 17) and eq. 9, the demanded link can be estabrameters lished: is independent from the measurement procedure used to determine the parameters. A well-known approach is the “triaxial setup“. This measurement setup can be sim0 N ×1 ZT0 =5.ZT,eq . The· 1measurement principle (20) is, plified as shown in Fig. to excite the shield via the source U0 and to measure the 4 Under the same conditions, the conventional transfer admitAdv. Radio Sci., 10, 221–225, 2012 10 Fig. 7 LiYC tance For th The s 7 S The p shield ture w equiv age U scalar admit If t bility at the lated the m Schetelig et al.: of Simplified susceptibility of shielded transmission lines fied susceptibility shielded multiconductor transmission lines multiconductor 224 analysis B. Schetelig et al.:analysis Simplified susceptibility analysis of shielded multiconductor transmission lines5 04 1 10 10 L0 , C0 10 calculation with ZT and YT of cable 2 3 10 −1 10 measurement cable 1, wire A measurement cable 1, wire B measurement cable 1, wire C measurement cable 1, wire D 0 10 2 10 −2 T U2 / V 10 1 10 Y’T // Ω/m S/m Z’ L0eq , 0 Ceq 0 10−3 −1 10 10 −1 10 , −2 10 −4 10 −2 10 −5 −3 10 10 4 4 10 10 Ieq −3 6 6 10 10 10 8 f f/ /Hz Hz 8 10 10 4 6 10 8 10 10 f / Hz Fig. 9. Calculation Fig.8.7. Absolute Absolutevalue valueof of the equivalent transfer impedance Fig. Calculation of of the the induced inducedvoltage voltageatatan aninner innerconductor conductorofof Fig. ofthe theequivalent equivalenttransfer transferadmittance impedancefor foraa aa shielded cable. Schetelig et al.: Simplified susceptibility analysis transmission lines 5 LiYCYmulticonductor transmission line (4 inner wires). LiYCY transmission line wires) shielded LiYCY LiYCY cable LiYCY multiconductor transmission line(4 (4inner inner wires)of shielded multiconductor ith the em can e inner m the escriponducem. of the identintional (19) edance estab- (20) admit- Y’T / S/m ters to 1 10 istance applied is: on a susceptibility analysis of the ”cable piece no. 1”. I −1 1 10 Because of eq. 9, the values of the induced voltages of the U Sas original cable are the same the voltage of the sub induced I1 N ×1 further calculation 0 ... stitute. −2That means, no has YT = = ·1 . to be carried (21) 10 US out. I N −3 10 US 8ForConclusions the equivalent transfer admittance, the following is valid: −4 N The 10 procedure, which is presented here, is a considerable P In coupling parameters of simplification when calculating the I I1 eq n=1 0 YT,eq = = = Nlines · .with stranded (22) industrial multiconductor transmission −5 US US US 10 4 6 8 wires. The 10 deviations are quite 10 acceptable. The10application f / Hz toThe thesecond susceptibility analysis of the cables under test results in demanded link then becomes: a reduced complexity, since only a simple single conductor 0 Yconsidered. T,eq 0 be N ×1 transmission line has to procedure Fig. 8. Absolute value of the equivalent admittance for aa · 1 transfer .The total (23) Absolute valueYof the equivalent transfer admittance for T = N isLiYCY summarized in fig. 6. LiYCY multiconductor transmission line (4 inner wires). multiconductor transmission line (4 inner wires) voltage U2on between the inner analysis conductor the shield. References 7is Sample measurements applied a susceptibility of and the ”cable pieceUsno. ing transmission line theory, these quantities can be used to 1”. Andrieu, G.: et presented application d’une méthode de faiscalculate theElaboration transfer impedance and the transfer admittance The procedure which was was applied Because of eq. 9, the values of thebefore induced voltages ofon thea ceau équivalent pour l’étude des couplages électromagnetiques (Tiedemann, 2001). shielded LiYCY cable with four inner conductors. The strucoriginal cable are the same as the induced voltage of the subsur réseaux câblages automobiles, Ph.D. thesis, Université de There is nodedifference the calculation of the transture wasThat condensed tonoa between shieldedcalculation single conductor line. The stitute. means, further has to be carried Lille, 2006. fer parameters of a physical coaxial lineand andthe of the substitute equivalent unit length parameters induced voltout. C. E.:per Baum, Electromagnetic topology: A formal approach the transmission line fromThis above. The equivalent cable isto deage U were derived. data then is used to calculate the eq and design of complex electronic systems, Tech. rep., Inanalysis scribed in the usual notation of a single conductor with the scalars equivalent transfer impedance (Fig. 7) 0and transfer teraction Notes - Note 400, http://socorro.eece.unm.edu/summa/ 0 0 0 per unit length admittance (Fig.parameters 8).1980. R eq , L eq , G eq , C eq and the 8 notes/In/0400.pdf, Conclusions equivalent Ueq . If these equivalent substitute quantiIf theseP.voltage coupling parameters are applied to the susceptiDegauque, and Hamelin, J.: Electromagnetic Compatibility, Oxties are used instead of physical ones, any measurement setup fordprocedure, Science bility analysisPublications, ofwhich the cable under test, theisinduced voltage The is 1993. presented here, a considerable can be used to determine the equivalent coupling parameters Jung, Einfluß von Schirminhomogenitäten bei Mehrleiterkabeln at theL.:inner conductor of the equivalent cable can be calcusimplification when calculating the coupling parameters of ofauf Eqs. (17) and (18). die komplexe Ph.D.lines thesis, Technische lated (Fig. 9). TheTransferimpedanz, transfer parameters are calculated with industrial multiconductor transmission with stranded Universität Hamburg-Harburg, 2003. the measured data of the no. 2“.The This data then wires. The deviations are”cable quite piece acceptable. application to the susceptibility analysis of the cables under test results in Adv. Radio Sci., 10, 221–225, 2012 a reduced complexity, since only a simple single conductor transmission line has to be considered. The total procedure is summarized in fig. 6. 10 5Kasdepke, Susceptibility analysis with equivalent coupling T.: Simulation von Störströmen auf geschirmten calculation with ZTthesis, and YT of cable 2 mehradrigen Ph.D. Technische Universität parametersKabeln, Hamburg-Harburg, 1997. measurement cable 1, wire A 0 measurement cable 1, wire Kley,overview T.: Optimierte Kabelschirme Theorie und Messung, 10 The of the procedure (Fig. 6)Bshows that the Ph.D. first measurement cable 1, wire C thesis, Eidgenössische Technische Hochschule Zürich, three steps are completed so far. The fourth step is to1991. do the measurement cableKarlsson, 1, wire D Tesche, F. M., analysis Ianoz, M. EMC Analysis susceptibility of V., the and structure. ThisT.:includes calcuMethods and Computational Methods, Wiley, 1997. −1 lating10the distributed sources of the inner system using the Tiedemann, R.: Schirmwirkung koaxialer Geflechtsstrukturen, quantities Z 0 T,eq and Y 0 T,eq . Common single conductor Ph.D. thesis, Technische Universität Dresden, 2001. transmission line theory can then be used to calculate the inVance, E. F.: Coupling to shielded cables, John Wiley & Sons, Inc., terferences Ueq , Ieq at the end of the lines (Tesche et al., −2 1978. 10 After that, Eqs. (9) and (10) can be used to derive the 1997). Weitze, F.: Ausbreitung eingekoppelter Störströme auf interferences U , Leitungen, I , of the original line. Univerabgeschirmte Ph.D. transmission thesis, Technische U2 / V 0 meters sität Hamburg-Harburg, 1991. −3 10 6 4 6 10 10 8 10 Conversion of the equivalent coupling parameters f / Hz to the conventional ones Fig. equivalent 9. Calculation of the induced voltageinatcombination an inner conductor The coupling parameters with of a shielded LiYCY cableconductor notation of the inner systhe equivalent single tem can be used to determine easily the interferences on the inner conductors. Nevertheless, it is possible to transKasdepke, T.: Simulation von Störströmen auf geschirmten form the equivalent transfer parameters to the conventional mehradrigen Kabeln, Ph.D. thesis, Technische Universität description. This might Hamburg-Harburg, 1997. be useful, if there is already a multi conductor transmission lineTheorie implementation for the Kley, T.: Optimierte Kabelschirme und Messung, Ph.D. inner system. thesis, Eidgenössische Technische Hochschule Zürich, 1991. The relationship canM. be V., drawn the definitions the Tesche, F. M., Ianoz, and from Karlsson, T.: EMC of Analysis parameters: Using the assumption of approximately identiMethods and Computational Methods, Wiley, 1997. Tiedemann, R.: on Schirmwirkung koaxialer the Geflechtsstrukturen, cal interferences the inner conductors, conventional Ph.D.impedance thesis, Technische transfer can be Universität written as: Dresden, 2001. Vance, E. F.: Coupling to shielded cables, John Wiley & Sons, Inc., U1 1978. I Weitze, U1 N×1 eingekoppelter Störströme auf SF.: Ausbreitung Z 0 Tabgeschirmte = ... = Leitungen, ·1 . Ph.D. thesis, Technische Univer(19) IS UN sität Hamburg-Harburg, 1991. I S www.adv-radio-sci.net/10/221/2012/ B. Schetelig et al.: Simplified susceptibility analysis of shielded multiconductor transmission lines Using the definition of the equivalent transfer impedance 0 ZT,eq (Eqs. 17 and 9), the demanded link can be established: N×1 0 0 Z T = Z T,eq · 1 . (20) Under the same conditions, the conventional transfer admittance is: I1 U S I1 N×1 ·1 . (21) Y 0 T = ... = US IN U S For the equivalent transfer admittance, the following is valid: N P In Ieq n=1 I1 0 YT,eq = = =N· . US US US (22) The second demanded link then becomes: Y 0T = 7 Y 0 T,eq N · 1N ×1 . (23) Sample measurements The procedure which was presented before was applied on a shielded LiYCY cable with four inner conductors. The structure was condensed to a shielded single conductor line. The equivalent per unit length parameters and the induced voltage Ueq were derived. This data then is used to calculate the scalars equivalent transfer impedance (Fig. 7) and transfer admittance (Fig. 8). If these coupling parameters are applied to the susceptibility analysis of the cable under test, the induced voltage at the inner conductor of the equivalent cable can be calculated (Fig. 9). The transfer parameters are calculated with the measured data of the ”cable piece no. 2“. This data then is applied on a susceptibility analysis of the ”cable piece no. 1”. Because of Eq. (9), the values of the induced voltages of the original cable are the same as the induced voltage of the substitute. That means, no further calculation has to be carried out. www.adv-radio-sci.net/10/221/2012/ 8 225 Conclusions The procedure, which is presented here, is a considerable simplification when calculating the coupling parameters of industrial multiconductor transmission lines with stranded wires. The deviations are quite acceptable. The application to the susceptibility analysis of the cables under test results in a reduced complexity, since only a simple single conductor transmission line has to be considered. The total procedure is summarized in Fig. 6. References Andrieu, G.: Elaboration et application d’une méthode de faisceau équivalent pour l’étude des couplages électromagnetiques sur réseaux de câblages automobiles, Ph.D. thesis, Université de Lille, 2006. Baum, C. E.: Electromagnetic topology: A formal approach to the analysis and design of complex electronic systems, Tech. Rep., Interaction Notes – Note 400, available at: http://www.ece.unm. edu/summa/notes/In/0400.pdf, 1980. Degauque, P. and Hamelin, J.: Electromagnetic Compatibility, Oxford Science Publications, 1993. Jung, L.: Einfluß von Schirminhomogenitäten bei Mehrleiterkabeln auf die komplexe Transferimpedanz, Ph.D. thesis, Technische Universität Hamburg-Harburg, 2003. Kasdepke, T.: Simulation von Störströmen auf geschirmten mehradrigen Kabeln, Ph.D. thesis, Technische Universität Hamburg-Harburg, 1997. Kley, T.: Optimierte Kabelschirme Theorie und Messung, Ph.D. thesis, Eidgenössische Technische Hochschule Zürich, 1991. Tesche, F. M., Ianoz, M. V., and Karlsson, T.: EMC Analysis Methods and Computational Methods, Wiley, 1997. Tiedemann, R.: Schirmwirkung koaxialer Geflechtsstrukturen, Ph.D. thesis, Technische Universität Dresden, 2001. Vance, E. F.: Coupling to shielded cables, John Wiley & Sons, Inc., 1978. Weitze, F.: Ausbreitung eingekoppelter Störströme auf abgeschirmte Leitungen, Ph.D. thesis, Technische Universität Hamburg-Harburg, 1991. Adv. Radio Sci., 10, 221–225, 2012