THE RECIPROCITY THEOREM APPLIED TO FIND THE BEST COUPLING INCIDENCE

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1 THE RECIPROCITY THEOREM APPLIED TO FIND THE BEST COUPLING INCIDENCE RECHERCHE DE LA DIRECTION D INCIDENCE DE PLUS FORT COUPLAGE A L AIDE DU THEOREME DE RECIPROCITE Jean-Pierre ADAM 1, Jean-Chrisophe JOLY 2, Bernard PECQUEUX 2, Didier ASFAUX 1 1 IEEA, 13 promenade Paul Doumer, 924 Courbevoie, Frane 2 Cenre d Eudes de Grama, DGA/DET/CEG, 465 Grama, Frane jean-pierre.adam@lub-inerne.fr Absra This doumen presens an appliaion of he eleromagneism reiproiy heorem in he ime domain. The far field radiaed by a sysem in a given direion is used o evaluae he effe of an iniden plane wave oming from ha direion. The useful formulas are derived for a urren soure and a volage soure. These resuls are applied wih he FDTD sofware Gorf3D. A single run wih a sysem radiaing in N direions enables o predi he resul of N runs wih a sysem illuminaed by a plane wave. Appliaion examples are used o esimae he ompuaion ime gain. Key words : Reiproiy heorem, ime domain, FDTD Résumé Ce doumen présene une appliaion du héorème de réiproié en éleromagnéisme dans le domaine emporel. La forme emporelle du hamp rayonné par un sysème dans une direion es uilisée pour déerminer l effe d une onde plane provenan de ee direion. Les relaions uiles son monrées pour une soure de ouran e une soure de ension. Ces résulas son mis en œuvre à l aide du ode DFDT Gorf3D. Un seul as de alul déermine le hamp rayonné dans N direions e perme de remplaer N as de alul ave une onde plane inidene. Des appliaions illusren la méhode e permeen d esimer le gain de emps de alul. Mos lés : Théorème de réiproié, domaine emporel, DFDT 1) INTRODUCTION The reiproiy heorem is a powerful simplifiaion ool in eleromagneism. I is ofen applied in he frequeny domain. On he onrary, he appliaion of his priniple in he ime domain is no so ommon. This paper deals wih his problem and illusraes i wih appliaions using he FDTD (Finie Differenes in Time Domain) Gorf3D ode. In hese examples, he parameer under sudy is he peak value of a wire urren or load volage when a sysem is illuminaed by an iniden plane wave. This sae will be alled reepion. In he seond sae alled ransmission, he load is replaed by a generaor. The reiproiy heorem is he link beween boh saes. Here, i is used o deermine he influene of he inidene direion and of he polarizaion of he plane wave. 2) TIME DOMAIN RECIPROCITY THEOREM A full proof of a general ime domain reiproiy heorem is no he aim of his work. A saemen and proof of suh a heorem may be found in [1] for example. The presen paper is appliaion oriened. However, i may be useful o reall some formulas and inerpreaions. The saring poin is a onsequene of he reiproiy heorem applied o an anenna in he frequeny domain : he behavior of a reeiving anenna only depends on he iniden

2 plane wave and he behavior of he anenna when i is ransmiing. For he load, a reeiving anenna an be represened as a Thevenin equivalen soure. The impedane of his soure is he same as he impedane of he anenna as a ransmier. When he anenna is illuminaed by an iniden eleri field E r oming from he direion u r, he open irui volage V o of his soure is given by (f. [2]) : V r r F(. E = (1) I ( ) o ( ω where F r is dedued from : r exp( ikr) r E = F( (2) r E r is he ransmied eleri far field a a disane r in he direion u r, when he anenna is fed by a urren soure I. is he impedane of he surrounding medium. A dire ransposiion of V o expression in he ime domain would imply onvoluions. In order o avoid hese CPU (Cenral Proessing Uni) ime onsuming operaions, he feeding urren I is hosen o simplify he expression. A simple hoie onsiss in seing he same ime dependeny for he urren I as for he iniden field E r. In ha ase he formula (1) is simplified as following : V o = r E (3) equivalen in he ime domain o : V o ( ) = r E ( τ ) dτ (4) The sep beween (3) and (4) is obvious, bu i shows he major differene beween he frequeny domain and he ime domain. In (3), for a given frequeny, he magniude of V o ( is proporional o he magniude of E (. This is he familiar onsequene of he reiproiy heorem in he frequeny domain : he reeiving paern is he ransmiing paern. The formula (4) shows ha hey are no exaly he same in he ime domain. Indeed, V o () is proporional o he ime inegral of E (). This saemen is alled derivaive relaion for ransmission and reepion in [3]. For simplifiaion purposes, here are omissions in (3) and (4). The phase is no aken ino aoun beause i simply adds a delay in he ime domain. In addiion, he salar produ is omied beause he omponens E θ and E φ (spherial oordinaes) of he radiaed field are separaely reaed. This is illusraed wih an arbirary linear polarizaion. The polarizaion angle is noed α. Formulas (1) and (2) are hen replaed by : E( V o = r( Eθ os( α) + Eϕ sin( α)) (5) I ( ) ω Anoher remark is ha he relaion ould be even simpler if he urren I is he inegraed form of he iniden field E r. Bu suh a urren waveform is no always easily available in he onsidered sofware. An analog resul is obained when he anenna is fed by a volage soure. In ha ase, he ransmied eleri far field is relaed o he shor irui urren I s observed when he anenna is illuminaed by a plane wave. This relaionship is : I s ( ) = r E ( τ ) dτ (6)

3 A serial impedane an be added beside he volage generaor in ransmiing mode. In reeiving mode, he generaor is replaed by a shor irui and he urren a his poin is dedued from formula (6). If his poin is near enough he serial impedane o mee he elerosai ondiion, hen he observed urren is he same as he one flowing hrough he impedane. In oher words, he mehod ould easily be exended o ompue a load urren. In addiion, he volage indued in a small dipole is proporional o he eleri near field omponen whih is parallel o he dipole. As a onsequene, he reiproiy heorem may be applied o he far field radiaed by his small dipole fed by a urren soure in order o evaluae he emporal variaion of he near field a he loaion of he dipole when he sysem is illuminaed by a plane wave. 3) APPLICATIONS The ime domain appliaions of he reiproiy heorem are usually found in he field of he UWB (Ulra- Wideband) anennas (f. [3]). In his work, i is used for EMC (Eleromagnei Compaibiliy) appliaions : he influene of he inidene direion of an EMP (Eleromagnei Pulse) is sudied. The simulaions are performed wih he FDTD sofware Gorf3D developed by he DGA (Frenh Defense Adminisraion). A desripion of he FDTD mehod may be found in [4] and some deails of is sofware implemenaion are presened in [5]. The ool inludes a near field o far field ransformaion rouine, allowing onsequenly ompuing he field radiaed ouside he FDTD domain. This ransformaion is performed in he ime domain. The firs example is a verial monopole anenna plaed over a perfe ground and fed by a urren soure. The reiproiy heorem is applied o ompue he maximum value of he open irui volage ha would be implied by an iniden plane wave. This iniden plane wave is verially polarized. In his work, he maximum value of he volage is sudied. However, his is no a limiaion : any oher ime domain reamen is appliable, like for example derivaion, inegraion, filering, FFT Figure 1 presens he resuls obained for differen monopole lenghs. This figure also shows he frequeny sperum of he pulse used o feed he anenna. This sperum drops down afer 3 MHz. This frequeny orresponds o a wavelengh λ of 1 m. For small monopoles (lengh below λ/4 =.25 m), he maximum value is ahieved wih an iniden plane wave perpendiular o he wire. This resul is lassial in he frequeny domain. For longer monopoles, he maximum oupling is reahed wih a non-zero elevaion angle. E L Wire Iniden plane wave elev Vo() Perfe ground (a) (b) Figure 1: (a) upper par : monopole anenna in reeiving mode; lower par : frequeny sperum of he iniden plane wave (reeiving mode) and of he feeding urren (ransmiing mode). (b) maximum value of he open irui volage (dedued by reiproiy) as a funion of inidene direion, for differen monopole lenghs.

The erm anenna inludes he radiaing elemen and he whole arrier, as presened in figure 2. This shielded ruk arries a verial monopole anenna. The sruure is plaed over a perfe ground.

This polarizaion is hosen o maximize he oupling wih he wire anenna. The waveform is presened in figure 2a. I orresponds o a ypial EMP waveform.

(a) (b) () (d) Figure 2: (a) waveform of he iniden field (reeiving mode) and of he feeding urren (ransmiing mode).

4 The erm anenna inludes he radiaing elemen and he whole arrier, as presened in figure 2. This shielded ruk arries a verial monopole anenna. The sruure is plaed over a perfe ground. Like in he previous example, he maximum value of he open irui volage indued a he monopole feed por when illuminaed by a verially polarized plane wave is analyzed. This polarizaion is hosen o maximize he oupling wih he wire anenna. The waveform is presened in figure 2a. I orresponds o a ypial EMP waveform. Again, he reiproiy heorem is used o redue he ase in only one simulaion run in ransmiing mode. The influene of he inidene direion has been invesigaed. (a) (b) () (d) Figure 2: (a) waveform of he iniden field (reeiving mode) and of he feeding urren (ransmiing mode). (b) eleri field (log sale) for a given ime during he simulaion of he monopole fed by a urren soure. () and (d) maximum value of he open irui volage indued in he anenna por as a funion of he direion of inidene. () and (d) are dedued from he simulaion shown in (b) wih he use of he reiproiy heorem. This omplex example is a good ase for esimaing he CPU ime reduion. The simulaions were performed on a sandard ompuer wih an AMD Ahlon proessor. The FDTD domain onains 1.5 millions ells. A dire ompuaion of he reeiving mode requires 855 runs of 1 minues, oaling 142 hours. The ransmiing mode ombined wih he reiproiy requires 1 simulaion of 16 hours wih defaul onfiguraion. The ime onsuming near o far field ransformaion is he ause of he CPU ime inrease. This ompuaion ime an be dropped o 3 minues by uning he under sampling opions of he near o far field ransformaion. Despie his drawbak, he use of he proposed mehod provides a signifian speed up of 284 imes. This observaion is pariularly rue if he number of onsidered inidenes is imporan. The nex example is based on he same sruure and implies he same EMP. However, wo oher eleromagnei weak poins are onsidered : a horizonal slo a he rear of he ruk and a longer verial wire going hrough he roof of he ruk. The observed parameer is he maximum value of he eleri field a he ener of he ruk when an EMP illuminaes i. In order o apply he reiproiy heorem as i was presened previously, he observaion poin is replaed by a small dipole (a one FDTD ell size long wire). This verial dipole is a sensor able o measure he verial omponen of he eleri field. The figure 3 presens he resuls for hree differen onfiguraions : wire alone, slo alone, wire and slo. These differen resuls may be used o invesigae he privileged peneraion way for a given direion of inidene. For example, he slo is he main oupling way for a wave oming from he rear of he ruk. On he onrary, he oupling hrough he wire seems o be raher omnidireional.

(a) (b) wire () slo (d) wire and slo Figure 3: (a) eleri field (log sale) for a given ime during he simulaion of a small wire fed by a urren soure. There is a horizonal slo in he rear fae.

(d) resuls are dedued by reiproiy from he ompuaion shown in (a). (b) and () resuls are also dedued by reiproiy, bu he slo and he wire are respeively absen.

5 (a) (b) wire () slo (d) wire and slo Figure 3: (a) eleri field (log sale) for a given ime during he simulaion of a small wire fed by a urren soure. There is a horizonal slo in he rear fae. A verial wire goes hrough he roof of he ruk. (b), () and (d) maximum value of he verial omponen of he eleri field a he small wire loaion for differen direions of inidene. (d) resuls are dedued by reiproiy from he ompuaion shown in (a). (b) and () resuls are also dedued by reiproiy, bu he slo and he wire are respeively absen. 4) CONCLUSION The reiproiy heorem is applied in he ime domain wih a FDTD sofware. This mehod provides a fas evaluaion of he bes oupling direion of inidene. However, he resuls are available for only one observaion poin in he sruure. This poin is he soure poin in he ransmiing mode. For example, he mehod is unable o predi a near field map. This seems o be he main limiaion. This mehod is perfely suied o sudy he influene of he inidene direion on a volage or a urren a a pariular loaion, for example he inpu por of a vulnerable devie. REFERENCES [1] B. Ru-Shao Cheo, A Reiproiy Theorem for Eleromagnei Fields wih General Time Dependene, IEEE Trans. Anennas Propaga., vol. AP-13, pp , Marh [2] S.W. Lee, Theorems and Formulas, in Anenna Handbook, Anenna fundamenals and mahemaial ehniques, vol. 1, hap. 2, Edied by Lo and Lee, Chapman & Hall, [3] Glenn S. Smih, A Dire Derivaion of a Single-Anenna Reiproiy Relaion for he Time Domain, IEEE Trans. Anennas Propaga., vol. 52, no. 6, pp , June 24. [4] Allen Taflove, Compuaional Elerodynamis, The Finie-Differene Time-Domain Mehod, Areh House, [5] L. Morel, B. Pequeux, R. Vézine, Noie d insallaion e d uilisaion de la version exporable du ode GORF3D, Noe inerne CEG I94-28, Oober 1994.

B Signals and Systems I Solutions to Midterm Test 2. xt ()

B Signals and Systems I Solutions to Midterm Test 2. xt () 34-33B Signals and Sysems I Soluions o Miderm es 34-33B Signals and Sysems I Soluions o Miderm es ednesday Marh 7, 7:PM-9:PM Examiner: Prof. Benoi Boule Deparmen of Elerial and Compuer Engineering MGill