RADIATED EM FIELDS FROM A ROTATING CURRENT- CARRYING CIRCULAR CYLINDER: 2-DIMENSIONAL NUMERICAL SIMULATION

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1 Prgress In Electrmagnetics Research M, Vl. 18, , 2011 RADIATED EM FIELDS FROM A ROTATING CURRENT- CARRYING CIRCULAR CYLINDER: 2-DIMENSIONAL NUMERICAL SIMULATION M. H Department f Electrnic Engineering WuFeng University, Taiwan F.-S. Lai Department f Security Technlgy and Management WuFeng University, Taiwan Y.-H. Chen and W.-C. Lin Department f Fire Science, WuFeng University, Taiwan Abstract The radiated electrmagnetic (EM) fields frm a rtating current-carrying circular cylinder were numerically simulated in tw dimensins using the methd f characteristics (MOC), and the numerical results were presented in this paper. T vercme the difficulty f the grid cell distrtin caused by the rtating cylinder, the passing center swing back grids (PCSBG) technique is emplyed in cllabratin f MOC in a mdified O-type grid system. In rder t have clear demnstratin f radiated EM fields, the circular cylinder is set t be evenly divided in radial directin int an even number f slices that are made f perfect electric cnductr (PEC) and nnelectric nnmagnetic material, alternatively. The surface current is assumed t have a Gaussian prfile and t flw unifrmly alng the axial directin n the PEC surface. The radiated electric and magnetic fields arund the cylinder were recrded as functins f time and reprted alng with the crrespnding spectra which were btained thrugh prper Furier transfrmatin. Several field distributins ver the whle cmputatinal space are als given. Received 9 March 2011, Accepted 4 May 2011, Scheduled 12 May 2011 Crrespnding authr: Mingtsu H (hmt@wfu.edu.tw).

2 104 H et al. 1. INTRODUCTION The analysis f electrmagnetic radiatin and scattering prblems have been active tpics f research because f theretical interest and practical imprtance in bth physics and engineering. Varius techniques were designed t slve the reflected and scattered EM waves frm slightly rugh surfaces dated back t as early as 1951 [1] and 1963 [2]. In the presence f multiple bjects the theretical derivatin turns ut t be s cmplex that varius appraches, thugh equipped with simplified assumptins and apprximatin techniques, can hardly prgress any further. The scattering EM prblems becme even mre cmplicated when the bject f interest is nt statinary. The difficulty is nt nly theretical but als numerical. If researchers can cme up with any analytical slutins fr such prblems, the resultant equatins wuld be nt nly nnlinear but in cmplicated frms. Mst f the time, such effrts typically end up in vain. Since the 1970s, many researchers have been making effrts in the study f scattered EM waves frm unifrmly mving r vibrating mirrrs; sme develped varius theries while thers wrked n numerical slutins [3 9]. It is bvius that the rle f numerical techniques becmes mre and mre imprtant because many nnlinear and intricate prblems can be numerically slved as well as because f the tremendus advance f cmputing pwer in bth hardware and algrithm. The advance in cmputing pwer prmises researchers t tackle difficult prblems with cmplex structure. The cmputatinal results can hence prvide researchers better understanding f detailed phenmena. In numerical simulatin, due t the mvement f the targeted bject, the shapes f grid cells adjacent t the mving bject becme time dependent. Grid cells can be changed in size and frm and/r eliminated frm the grid system; fr sme cases new cells can als be intrduced int the grid system simultaneusly. In [6], Harfush et al. numerically simulated the EM scattering prblem arund the fundamental frequency using the finite-difference timedmain (FDTD) technique where the incident electric field was linearly interplated whenever the mving bundary mves away frm the grid pint. It is reprted that the eliminatin f already-existing cells and the additin f new cells can be taken care f by the MOC apprach. One f the basic differences between FDTD and MOC is the psitining f field cmpnents. FDTD defines the electric and magnetic field cmpnents at the cell crners in an interlacing fashin [10]; MOC places all field cmpnents at the center f grid cell. When cells are gradually eliminated r added, the change des nt affect the fact that all field variables are lcated at each cell s centrid. MOC evaluates

3 Prgress In Electrmagnetics Research M, Vl. 18, the flux change within each cell including the fractinal nes all the time. Meanwhile, MOC adjusts the time step sizes f thse fractinal cells since the numerical time step fr each cell is dependent n the grid size. It is shwn that MOC yields results which are cmpatible with data generated by the FDTD technique [11] and in gd agreement with the theretical values when EM fields reflected frm a traveling and/r vibrating perfect surface [12]. Als, MOC prduces reasnable trends in the fllwing cases: the effects f medium cnductivity n the prpagatin f EM pulse nt cnducting dielectric half space [13] and the prpagatin f EM pulse thrugh lssless nn-unifrm dielectric slab [14]. In the case where bject is rtating grid cells arund bject are twisted s badly that the numerical scheme is n lnger applicable. T vercme this difficulty, thanks t the nature f MOC, the PCSBG technique was prpsed in calitin with MOC [15, 16]. MOC with the aid f PCSBG was demnstrated t generate reasnable cmputatinal results in the scattered EM fields frm rtating circular cylinder under the illuminatin f plane Gaussian EM pulse [17]. The main purpse f this paper is t bserve, thrugh the applicatin f MOC cmbined with PCSBG t the slutins f Maxwell s equatins, the EM fields radiated frm a circular rtating cylinder n whse PEC prtins surface currents are numerically set t flw. Bth electric and magnetic fields arund the cylinder are recrded as functins f time. The crrespnding frequencydmain infrmatin becmes available via the use f Furier transfrm. This paper is rganized as fllws: the numerical frmulatin and the bundary cnditin treatment f the prblem are presented in Sectin 2; Sectin 3 illustrates the idea f the PCSBG technique; Sectin 4 defines the prblem with the lcatins f the sampling pints and the mdified O-type grid system; the descriptin f hw the divided circular cylinder was arranged and the specificatins f surface current are als given; Sectin 5 demnstrates the numerical results and is fllwed by the cnclusin sectin. 2. GOVERNING EQUATIONS, SURFACE CURRENT AND BOUNDARY CONDITIONS In the present prblem, the EM fields are radiated frm a rtating circular cylinder f infinite length and prpagate in free space. Therefre, it can be fully described by the time-dmain Maxwell s equatins. MOC is an implicit apprach numerically slving Maxwell curl equatins in a curvilinear crdinate system. In free space, the

4 106 H et al. Maxwell s equatins are B t + E = 0 (1) D t H = 0 (2) where E and H are the electric and magnetic field intensities, and D and B are the electric and magnetic flux densities. In the numerical mdel, the surface current n the rtating circular cylinder f infinite length has a Gaussian prfile and is assumed t flw alng the axial directin, namely the psitive-z directin. Cnsequently, the electric cmpnents f the radiated EM fields are plarized in the z-directin, and the tw magnetic field cmpnents are n the x-y plane. Fr the current task, a tw-dimensinal numerical frmulatin is adequate. Because MOC directly apprximates Maxwell s equatins in a curvilinear crdinate system, the gverning equatins have t be rearranged in the frm f Euler equatin prir t the transfrm frm the Cartesian system (t, x, y) t a curvilinear system (τ, ξ, η). After all these have been dne, Maxwell s equatins becme where Q τ + F ξ + G η = 0 (3) q = [B x B y D z ] T (4) f = [0 E z H y ] T (5) g = [E z 0 H x ] T (6) Q = Jq, F = Jf, G = Jg. (7) In the abve equatin, the symbl J is the Jacbian f the inverse transfrmatin and defined as J = det (x, y) (ξ, η). (8) With the definitin f the central difference scheme in mind, δ k (φ) = (φ) k+ 1 2 (φ) k 1, (9) 2 the numerical frmulatin starts with the applicatin f the central difference peratr t (3). In (9), the symbl (φ) stands fr the variable vectr F r G crrespnding t the subscript (k) representing the cell index alng either ξ- r η-directin in the curvilinear crdinate system. Mrever, that the cell index (k) designates the centrid f

5 Prgress In Electrmagnetics Research M, Vl. 18, grid cell is being understd. The ne-half index reveals the fact that MOC slves Maxwell s equatins by evaluating flux acrss the interface between cells. By applying the central difference scheme (9) t (3), ne btains Q n+1 Q n τ + δ if ξ + δ jg = 0. (10) η Nte that the finite difference symbl ( ) has been applied t time and space. On vectr Q the superscripts (n + 1) and (n) represent the tw successive time levels; the first term therefre designates the apprximatin f the tempral derivative f vectr Q. MOC slves the system f linear equatins thrugh the incrpratin with the flux vectr splitting technique and the lwer-upper apprximate factrizatin scheme. The z-directed currents n the infinite, circular PEC cylinder surface are set t have a Gaussian prfile with a peak value f ne as in Figure 1(a). The Gaussian envelp has a width f T = 0.2 ns measured frm the center t the pint having the value f e 0.5 and a cut-ff level at 100 db fr practical reasns. The crrespnding spectrum was btained thrugh Furier transfrm and pltted in Figure 1(b) where the highest frequency cntent lcates at abut 3.8 GHz. (a) (b) Figure 1. (a) Definitin f the surface current (nt t scale) and (b) its spectrum. In the numerical mdel, there are tw types f bundary cnditins: ne is n the cylinder PEC surface, and the ther is n the truncated cmputatinal bundary. Given the surface currents (J z ), the bundary cnditins at any pint n the PEC surface can be

6 108 H et al. derived frm the fllwing ˆn H = 2J z (11) ˆn B = 0 (12) ˆn E = 0 (13) where ˆn is the exterir nrmal vectr f the grid n the cylinder surface. It is nted that (12) ensures n EM fields with the magnetic field being perpendicular t the surface that can be radiated frm the cylinder and that the electric field intensity must be vanished n the PEC cylinder surface accrding t (13). The radiated EM fields are then in the transverse electric mde. Additinally, n the surface f the nnelectric and nnmagnetic prtin f the rtating cylinder bth electric and magnetic fields are set t be null in magnitude. At the mst uter numerical bundary the prper bundary cnditins are t guarantee that the utging EM fields keep prpagating utward. 3. THE PASSING CENTER SWING BACK GRIDS TECHNIQUE The idea f the PCSBG technique was prpsed with the purpse f slving the cell distrtin difficulty. It is based n the nature f MOC which defines all field cmpnents at the cell centrid. A sequence f plts is given in Figure 2 t explain hw the PCSBG (a) (b) (c) (d) Figure 2. Schematic presentatin f PCSBG: (a) Grids align at certain time instance; (b) grid lines are tilted; (c) grid lines pass the center pint and are set t swing; (d) after the swing rtatin cntinues.

7 Prgress In Electrmagnetics Research M, Vl. 18, design wrks. Each plt is cmpsed f tw layers f grids whse inner circle represents the cylinder. The layer f grids immediately next t cylinder is designated as the passing center swing back grids. They will suffer frm defrming nce the cylinder rtates since ne side f these grids is attached t the cylinder. Frm the PCSBG layer utward are free space and grids are statinary thrughut the prcess. Figures 2(a) and 2(b) shw the grid lines are first at their alignment psitin at certain time instance and then twisted tward left due t the rtatin f the cylinder (as indicated by arrw). Displayed in Figure 2(c), at the mment the grid lines are abut t pass r at the cell center as depicted by the skewed black lines, and the PCSBG technique maneuvers them t swing back as depicted by the dtted red lines. The cylinder cntinues rtating after thse grid lines has been reset by ne grid backward as in Figure 2(d). Nte that during the prcedure the field variables are always defined at the cell centrid. Thugh MOC vercmes the difficulty caused by the rtating bject, MOC cautiusly bk-keeps every change in the metrics terms f the swing back grids alng with the exterir nrmal vectr and the cell index f each grid n the cylinder surface. Hwever, there is ne exemptin; the numerical time step f these swing back grids, in spite f the fact that it is grid size sensitive, remains cnstant even when the grid is twisted. It is because the grid size stays the same the whle time. 4. THE MODIFIED O-TYPE GRID AND THE PROBLEM SET UP The present prblem is characterized by a rtating circular cylinder carrying surface current, which results in the radiatin f EM fields. The surface current is assumed t have a tempral Gaussian prfile and t flw axially as described in the previus sectin. T numerically mdel the prblem, the O-type grid system is a suitable candidate when a circular bject is invlved. Yet, regular O-type grid system is made f cncentric circles. Even it is unifrmly divided radially, grids turn int lng thin trapezids as radius increases and turn ut t be unacceptable in mdeling EM scattering related prblems. T imprve, the mdified O-type grid is then prpsed in which grid cells are dubled in number as radius grws t a certain size s that each grid remains within specific measurements in bth dimensins. Fr clear illustratin, a simplified versin f the mdified O-type grid is demnstrated in Figure 3, which is cmpsed f three znes f grids. The cell numbers in each ring frm zne t zne are 32 (black), 64 (red) and 128 (blue), respectively. The surface currents flwing n the rtating cylinder have the

8 110 H et al. Figure 3. Mdified O-type grid. Figure 4. Cylinder has fur sectins. previusly defined Gaussian prfile with a tempral span f abut T span = 1.92 ns measured frm end t end (cut-ff pints). Any pint utside this range is set t be zer in magnitude. The angular frequency f the rtating cylinder is defined as the number f the revlved cycle within ne T span. Fr example, if the cylinder makes ne cmplete rtatin (360 ) in ne T span, the angular frequency wuld be a little bit greater than ne π per nansecnd. Under such arrangement, it is equivalent that the instantaneus velcity f any pint n the cylinder surface wuld be abut 1.1 times as fast as the speed f light. Thugh an impractical figure this wuld be, the chief bjective f the current task stays intact. Fr simplificatin and with the previus definitin at hand, hencefrth the term cycle will be used t define the angular frequency f the rtating cylinder. Fr similar reasns, anther term sectin-cycle number stands fr the prduct f the sectin number and the cycle number. The sectin-cycle number is eight, fr instance, if a fur-sectin cylinder rtates at 2 cycles. Figure 4 illustrates a cylinder divided int fur sectins. Fr the purpse f clear demnstrating the effects f radiated EM fields frm the rtating circular cylinder, these sectins are assumed t be made f PEC and nnelectric nnmagnetic material, alternatively. In additin, there are eight sampling pints lcated arund at abut cm away frm the cylinder surface. And they are equally spaced by 45 started frm the x-axis (0 ). 5. NUMERICAL RESULTS The mdified O-type grid system emplyed in the numerical mdel is cmpsed f tw prtins: a cylinder in the center with a radius

9 Prgress In Electrmagnetics Research M, Vl. 18, f 10 cm and anther 60 cm fr the free space. There are three znes f grid utside the cylinder and 176 layers f grid in ttal. The grid number in each layer is dubling up frm 336 (inner zne) t 1344 (uter zne). The grid density, in the units f grids per meter, ranges frm 275 t 550 circumferentially and frm 250 t 530 radially. The surface current is set t start ut frm zer in magnitude and takes 1 ns t reach the peak value. Remember that the time measured frm the peak t the cut-ff is 0.96 ns. The cylinder may be divided int tw, fur, r eight sectins and may rtate at ne, tw, r fur cycles. Figure 5 shws tw sets f field distributin taken at 1.6 ns where cylinders were severed int tw halves and fur quarters and may be at rest r rtate. Figures 5(a) and 5(e) are the radiated z-directed electric fields (E z ) frm tw statinary PEC cylinders. With clse bservatins f these tw radiated E z distributin patterns, it is pinted ut that E z are wrapped arund the cylinder in bth cases and that E z are super-psitined in the latter case. Figures 5(b) thrugh 5(d) are respectively E z, the x- and y-cmpnents f the magnetic field (H x and H y ) when the tw-sectin cylinder rtating at 4 cycles. And Figures 5(f) thrugh 5(h) are fr the fur-sectin cylinder at 2 cycles. The fllwings are clearly shwn: E z frms a single-whirl pattern in (a) E z (b) E z (c) H x (d) H y (e) E z (f) E z (g) H x (h) H y Figure 5. Field distributins ver the whle cmputatinal dmain at 1.6 ns. A tw-sectin cylinder is (a) statinary and (b) (d) rtating at 4 cycles. A fur-sectin cylinder is (e) statinary and (f) (h) rtating at 2 cycles. (backgrund: zer; red: psitive; blue: negative).

10 112 H et al. Figure 5(b) and a duble-whirl pattern in Figure 5(f); there is n H x in the x-directin in Figures 5(c) and 5(g), nr H y in the y-directin in Figures 5(d) and 5(h); H x and H y in these tw grups f plts bear similar patterns but differ in magnitude because they have same sectin-cycle numbers. The radiated electric fields frm varius statinary cylinders, slid r severed, were recrded and displayed. Due t the symmetry, when the electric fields sampled at different pints are alike, nly ne will be shwn with remarks. Fr instance, eight lines cincide in the cases f slid cylinder and eight-sectin cylinder as in Figure 6(a). They are different in strength by a factr f abut tw. Fr a fur-sectin cylinder there are three grups f lines as in Figure 6(b) and fr a twsectin cylinder five grups as in Figure 6(c). It is nticed that the relative strength and the lcatin f primary lbe reveal infrmatin regarding the rientatin f bserver. Nte als that the slid lines in three plts are identical as the result f gemetric symmetry and field superpsitin. The crrespnding nrmalized spectra were given in Figure 7. In Figure 7(a) the spectrum f surface current was included as a reference, and nly the line f slid cylinder was shwn since it is Slid Dash Statinary 8-sectin cylinder Slid cylinder Statinary 4-sectin Cylinder Slid 0, 90, 180, 270 Dash 45, 225 Dt 135, 315 Statinary 2-sectin Cylinder Slid 0, 180 Dash 45, 135 Dt 90 O 225, 315 X 270 (a) (b) (c) Figure 6. Radiated electric fields arund statinary cylinders. (a) Slid and eight-sectin cylinders, (b) fur-sectin cylinder, (c) twsectin cylinder. Slid Dt Statinary Slid Cylinder Gaussian Surface Current Statinary 4-sectin Cylinder Slid 0, 90, 180, 270 Dash 45, 225 Dt 135, 315 Statinary 2-sectin Cylinder Slid 0, 180 Dash 45, 135 Dt 90 O 225, 315 X 270 (a) (b) (c) Figure 7. Spectra crrespnding t the plts in Figure 6.

11 Prgress In Electrmagnetics Research M, Vl. 18, indistinguishable frm that f eight-sectin cylinder. It is bserved frm Figures 7(b) and 7(c) that when the bserver faces, fully r partially, the PEC prtin, the spectrum f the radiated electric fields has lwer level at DC and that thse n the hind side have maxima ranging frm DC up t abut 0.3 GHz. T investigate the effects f the rtatin f cylinder n the radiated electric fields and spectra, the fllwing effrts were made. The radiated electric fields frm a tw-sectin cylinder rtating at 4 cycles, whse radiated EM field patterns at 1.6 ns were shwn in Figures 5(b) (d), were recrded and illustrated in Figure 8(a) and their crrespnding spectra in Figure 8(b). Fr clear illustratin nly fur f eight different lines were pltted. A similar arrangement fr a fur-sectin cylinder rtating at 2 cycles, whse radiated EM field distributins at 1.6 ns were shwn in Figures 5(e) (h), was made and summarized in Figures 9(a) and 9(b). Unlike the preceding case, eight lines fall int fur pairs as indicated in Figure 9(a). With clse bservatins n the electric fields, it is shwn, except the twin-lines in the latter case, that tw sets f lines share similar behavir because the tw sectin-cycle numbers are identical. On the spectra, thugh nt very clearly, the separatin between tw successive minima is abut 2 GHz in Figure 8(b), and that in Figure 9(b) is abut 4 GHz. Further effrts were made in the fllwing tw cases with identical sectin-cycle number: a fur-sectin rtates at 4 cycles, and an eightsectin rtates at 2 cycles. The numerical results were summarized in Figures 10 and 11. Frm the synchrnus viewpint, regardless the magnitude, the fur electric fields in Figure 10(a) are almst equally spaced between tw successive peaks while thse in Figure 11(a) fall (a) (b) Figure 8. A tw-sectin cylinder rtating at 4 cycles. (a) Radiated electric fields, (b) spectra. Slid: 180 ; dash: 225 ; dt: 270 ; dashdt: 315.

12 114 H et al. int tw grups separated by a distance twice as much. Thugh s, their spectra are nearly alike as depicted in Figures 10(b) and 11(b). Nte that the secndary envelpes center at abut 4 GHz in bth cases. Finally, a set f radiated EM field patterns was given in Figure 12 fr an eight-sectin cylinder rtating at varius angular frequencies and at different instances f time. Cmpared with thse in Figure 5, it is nticed that the mre slices the cylinder is severed and/r the faster the cylinder rtates the thinner in shape the radiated fields are. (a) (b) Figure 9. A fur-sectin cylinder rtating at 2 cycles. (a) Radiated electric fields, (b) spectra. Slid: (0, 180 ); dash: (45, 225 ); dt: (90, 270 ); dash-dt: (135, 315 ). (a) (b) Figure 10. A fur-sectin cylinder rtating at 4 cycles. (a) Radiated electric fields, (b) spectra. Slid: (0, 180 ); dash: (45, 225 ); dt: (90, 270 ); dash-dt: (135, 315 ).

13 Prgress In Electrmagnetics Research M, Vl. 18, (a) (b) Figure 11. An eight-sectin cylinder rtating at 2 cycles. (a) Radiated electric fields, (b) spectra. Slid: (0, 180 ); dash: (45, 225 ); dt: (90, 270 ); dash-dt: (135, 315 ). (a) (b) (c) (d) Figure 12. Field distributins f an eight-sectin cylinder that is (a) statinary at 1.3 ns, (b) rtating at 1 cycle at 1.6 ns, (c) rtating at 2 cycles at 1.6 ns, (d) rtating at 4 cycles at 1.07 ns. (backgrund: zer; red: psitive; blue: negative). 6. CONCLUSION In this paper, it is shwn that the methd f characteristics incrprating with the passing center swing back grids technique vercmes the difficulty f grid distrtin due t the rtatin f bject f interest while slving the Maxwell curl equatins and prduced feasible trends. The radiated electric fields are recrded and demnstrated alng with the frequency-dmain infrmatin. In the mdel the circular cylinder is divided int an even number f sectins and rtates at varius angular frequencies. Als given are several sets f tw-dimensinal plts f the electric fields and magnetic fields ver the whle cmputatinal space. It is clearly illustrated that the radiated EM field patterns display whirl-like utline due t the

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