ANALYSIS OF ELECTROMAGNETIC FIELD USING THE CONSTRAINED INTERPOLATION PROFILE METHOD PHÂN TÍCH TRƯỜNG ĐIỆN TỪ SỬ DỤNG PHƯƠNG PHÁP CIP

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1 ANALYSIS OF ELECTROMAGNETIC FIELD USING THE CONSTRAINED INTERPOLATION PROFILE METHOD PHÂN TÍCH TRƯỜNG ĐIỆN TỪ SỬ DỤNG PHƯƠNG PHÁP CIP LÊ VŨ HƯNG Cao đẳng kỹ thuật quốc ga Kushro, Nhật Bản Kushro Natonal College of Technolog, Japan ABSTRACT A new tme doman method, CIP (Constraned Interpolaton Profle) method has been developed as a numercal solver for mult-phase problems. The method s based on the upwnd scheme for the fnte dfference method, but the varables to be calculated are not onl the values of the quantt, but also the spatal dervatves. Ths stud uses CIP method to anale one-dmensonal electromagnetc feld and compare wth the FDTD (Fnte Dfference Tme Doman) method. TÓM TẮT Một phương pháp mền thờ gan mớ. Phương pháp CIP (Constraned Interpolaton Profle) được phát trển để tính toán số trị các vấn đề đa pha. Phương pháp CIP không chỉ tính toán gá trị chuển động của trường mà còn tính cả v phân không gan. Trong nghên cứu nà sử dụng phương pháp CIP để phân tích trường đện từ một chều và tến hành so sánh vớ phương pháp Sa phân mền thờ gan (FDTD: Fnte Dfference Tme Doman). Ⅰ. INTRODUCTION Recentl, tme doman numercal analss of electromagnetc (EM) felds has come to be nvestgated wdel as a result of computer development. Development of accurate schemes s an mportant techncal ssue related to EM feld calculaton. Although man methods have been proposed, the fnte-dfference tme-doman (FDTD) method s the most wdel used method for tme doman numercal analss. However, the FDTD method solves Maxwell s equaton usng fnte dfference approxmaton. Usng ths approxmaton certanl causes error because of numercal dsperson. For that reason, EM feld propagaton cannot be analed exactl. CIP method was proposed b Yabe.et.al. A notceable feature of the CIP method s advantageous that t allows analss not onl of EM felds on grd ponts, but also analss of ther spatal dervatves on grd ponts. In ths stud, CIP method s appled to one-dmensonal EM feld equaton (Maxwell s equaton) and compare analtcal results wth the FDTD method. Ⅱ. CALCULATION A. The Constraned nterpolaton profle method Let us consder the lnear wave propagaton n non-unform mesh sstem wth an advecton equaton: Fg. : The prncple of the CIP method: (a) sold lne s ntal profle and dashed lne s an exact soluton after advecton. (b) Spatal dervatve also propagates and the profle nsde a grd cell s retreved.

2 f f v () Then, let us dfferentate Eq. () wth spatal varable x, we get: g g v v g () f Where g stands for the spatal dervatve of f ( g ). In the smplest case where the x veloct v s constant, Eq. () concdes wth Eq. () and represents the propagaton of spatal dervatve wth a veloct v. If two values of f and g are gven at two grd ponts, the profle between these ponts can be nterpolated b cubc polnomal: 3 F ( x) a ( x x ) b ( x x ) c ( x x ) d (3) Here F ( x ) d f (4) From the condtons n the lefthand sde the coeffcents a and b can df be descrbed such as: ( x ) c g (5) g g ( f f ) dx a (8) 3 3 F D D ( x ) a b c d f (6) 3( f f ) g g df b (9) ( x ) 3a b c g (7) D D dx Thus, the profle at the n+ step and x x, D can be obtaned b shftng the profle b vso that: n 3 n n f n n a b g f f F( x v), g df( x v t)/ dx and n n g 3a b g () Where v t. B. Electromagnetc Feld Numercal Analss Usng the CIP Method Farada's and Ampere's laws are shown respectvel as () and (): B c E () B J c () Here, usng CGS-Gauss sstem of unt and assumng E, E,), B (,, B ) and J= n Fg. : Propagaton of EM. Propagaton of B E wth veloct v s constant. B c (3); ( order to anale one-dmensonal EM feld propagaton of the x-drecton n the vacuum, we can obtan the followng equatons from Eq. () and (). B addton and subtracton of Eq. () and (), we obton: In addton and subtracton of Eq. (3) and Eq. (4) for the dervatve are also gven as: B c (4)

3 ( B E ) ( B E ) c (5); ( B E ) ( B E ) c (6) It s readl apparent that equatons from (3) to (6) are advecton equatons of B E and B E. Therefore, applcaton of the CIP method to these equatons allows a soluton of EM x x feld propagaton n the x-drecton at tme step n. Next, we calculate the felds of the n+ tme step from the felds of n tme step. Here, the calculaton method of the x -drecton propagaton s descrbed below usng Fg.. Eq. (6) shows propagaton of B E to pont wth the veloct +c s constant. In Fg. shows propagaton of B E from A pont to pont s become B E. Eq. (5) shows propagaton of B E to n n pont wth the veloct c s constant. Fg. shows propagaton of B E from B pont to pont s become B E n n. We can obton: B E B E (7); n n B E B E (8) n n In the x-drecton, equatons for the dervatve are gven as: n n n n B B B B (9); () Then, b addton and subtracton of these equatons, we obtan: ( B ) ( ) n E B E ( ) ( ) xb xe n xb xe B (); B () ( B ) ( ) n E B E ( ) ( ) xb xe n xb xe E (3); E (4) Usng the equatons from () to (4), one can obtan EM feld components of tme step n+. From the above equatons, x Ⅲ. RESULTS AND DISCUSSION We show the result of calculatons n Secton Ⅱ. We calculate the EM feld propagaton n the one-dmensonal analss doman n the vacuum. In the orgnal addton: f x E ( x) and B ( x) otherwse Grd se, dx. 5m ; tme step, dt. s ; number of grd ponts, NX=. Fg. 3 and Fg. 4 shows the electrc feld waveform that s calculated usng CIP method and Fg.5 and Fg. 6 shows the magnetc feld waveform that s calculated usng FDTD method at t.s, t. 5s and t. 5s. All parameters of FDTD calculaton are equal to those of the CIP calculaton. Fg. 3, Fg. 4, Fg. 5 and Fg. 6 demonstrates that the CIP results provdes accurac. On the other hand, numercal oscllaton appears n the FDTD result. Next, we calculate the EM feld propagaton wth the orgnal addton usng Gauss s dstrbuton:

4 H (A/m) H (A/m) E (V/m) E (V/m). CIP method (t=.5s) FDTD method (t=.5s). CIP method (t=.5s) FDTD method (t=.5s) Fg. 3: Calculated results of E at t =.5s. Fg. 4: Calculated results of E at t =.5s CIP method (t=.5s) FDTD method (t=.5s) CIP method (t=.5s) FDTD method (t=.5s) Fg. 5: Calculated results of B at t =.5s. Fg. 6: Calculaed results of B at t =.5s - ( ).5exp x 5 E x., B ( x) Parameters of the calculaton are as follows: Grd se, dt. s ; number of grd ponts, NX=.. dx. m ; tme step, Fg. 7, Fg. 8 and Fg. 9 shows the electrc feld waveform that are calculated usng CIP analss and FDTD analss under the condton of = 5., =.5 and =.5 at t = s. When = 5., the results calculated usng FDTD are equal to those of the CIP calculaton. However, =.5 and =.5 numercal oscllaton appears n the FDTD result. Ⅲ. CONCLUSION In ths paper, we have analed EM feld usng the CIP method. We consder the values not onl of the EM feld, but also of ther spatal dervatves. The results obtaned n ths stud have clarfed that EM feld analss usng the CIP method provdes hgher accurac than that obtanable usng the FDTD method. Accordng to these results, we ntend to nvestgate actual one-dmensonal analss ncludng boundares n the near future.

5 E (V/m) E (V/m) E (V/m).3.5 CIP method (sgma=5.) FDTD method (sgma=5.).3.5 CIP method (sgma=.5) FDTD method (sgma=.5) Fg. 7: Calculated results of E at = Fg. 8: Calculated results of E at CIP method (sgma=.5) FDTD method (sgma=.5) Fg. 9: Calculated results of E at REFERENCES [] T. Yabe, T. Utsum, and Y. Ogata, The Constraned Interpolaton Profle Method. Morkta Publshng, Toko, (3). (n Japanese) [] T. Yabe, X. Feng, and T. Utsum, The constraned nterpolaton profle method for multphase analss, J. Comput. Phs., vol. 69, pp ,. [3] H. Takewak, A. Nshguch, and T. Yabe, Cubc nterpolated pseudo-partcle method (CIP) for solvng hperbolc-tpe equatons, J. Comput. Phs., vol. 6, pp.6-68, 985. [4] T. Yabe, Y. Ogata, and K. Taksawa, CG Smulaton b CIP Method and Java, Morkta Publshng, Toko, (7). (n Japanese) [5] K. Okubo, N. Takeuch, Analss of an Electromagnetc Feld Created b Lne Current Usng Constraned Interpolaton Profle Method, IEEE Trans. Antennas Propag., vol. 55, no., pp. -9, Jan. 7. [6] K. Okubo, Y. Yoshda and N. Takeuch, Consderaton of Treatment of the Boundar Between Dfferent Meda n Electromagnetc Feld Analss Usng Constraned Interpolaton Profle Method, IEEE Trans. Antennas Propag., vol. 55, no., pp , Feb. 7.

6 [7] S. Watanabe and O. Hashmoto, An Examnaton of Electromagnetc Feld Usng CIP Method and Characterstc Curve Methods, EuMC 5. (n Japanese) [8] K. Okubo, N. Takeuch, Numercal Analss of Electromagnetc Feld Generated b Lne Currendt Usng the CIP Method, MW 4. (n Japanese) Địa chỉ lên lạc của tác gả: Lê Vũ Hưng, Khoa Kỹ thuật Đện, Cao đẳng kỹ thuật quốc ga Kushro, Nhật Bản. Department of Electrcal Engneerng, Kushro Natonal College of Technolog, Japan Japan, Kushro Ct, Otanoshke-Nsh -3- Emal: levuhung@gmal.com