51. IWK Internationales Wissenschaftliches Kolloquium International Scientific Colloquium

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1 51. IWK Intenationales Wissenschaftliches Kolloquium Intenational Scientific Colloquium PROCEEDINGS Septembe 26 FACULTY OF ELECTRICAL ENGINEERING AND INFORMATION SCIENCE INFORMATION TECHNOLOGY AND ELECTRICAL ENGINEERING - DEVICES AND SYSTEMS, MATERIALS AND TECHNOLOGIES FOR THE FUTURE Statseite / Index:

2 Impessum Heausgebe: Redaktion: De Rekto de Technischen Univesität llmenau Univ.-Pof. D. e. nat. habil. Pete Schaff Refeat Maketing und Studentische Angelegenheiten Andea Schneide Redaktionsschluss: 7. Juli 26 Fakultät fü Elektotechnik und Infomationstechnik Susanne Jakob Dipl.-Ing. Helge Dumm Technische Realisieung (CD-Rom-Ausgabe): Institut fü Medientechnik an de TU Ilmenau Dipl.-Ing. Chistian Weigel Dipl.-Ing. Maco Albecht Dipl.-Ing. Helge Dumm Technische Realisieung (Online-Ausgabe): Univesitätsbibliothek Ilmenau Postfach Ilmenau Velag: Velag ISLE, Betiebsstätte des ISLE e.v. Wene-von-Siemens-St llnenau Technische Univesität llmenau (Thü.) 26 Diese Publikationen und alle in ih enthaltenen Beitäge und Abbildungen sind uhebeechtlich geschützt. Mit Ausnahme de gesetzlich zugelassenen Fälle ist eine Vewetung ohne Einwilligung de Redaktion stafba. ISBN (Duckausgabe): ISBN (CD-Rom-Ausgabe): Statseite / Index:

3 K. Niayesh, H. Ghobani, A. Hayati-Soloot 51 st Intenationales Wissenschaftliches Kolloquium Technische Univesität Ilmenau Septembe 11 15, 26 A Maximum Q inducto design fo pulsed powe applications APPLIED ELECTROMAGNETICS AND CIRCUIT THEORY 1 Intoduction Ai coe inductos find divese applications in high cuent pulse technology. Because of vey high amplitudes of cuents and vey fast time vaiations, it is not efficient to use inductos with feomagnetic coes. The high cuent inductos can be used as enegy stoage components in inductive enegy stoage systems o as pulse shaping elements in capacitive enegy stoage systems. In both applications, it is impotant to decease the losses of inductos to be able to achieve the highest possible efficiencies o altenatively to ealize the necessay inductance using the minimum cable length. High-Q inductos ae also vey impotant fo othe applications like in micowave cicuits and in on-chip designs of the RF integated cicuits. These inductos ae ealized in solenoid o spial shape. In this pape, a method to maximize the inductance of an ai coe inducto constucted fom a given length high voltage cable is poposed. Fo this pupose, an expeimentally veified exact calculation method based on Neumann equation fo magnetic vecto potential is applied. In this method the inducto is assumed to be a combination of a numbe of windings located in diffeent positions. This genealized fomulation makes it possible to conside solenoid, spial and any abitay combination of solenoid and spial shapes. The self and mutual inductances between diffeent windings ae calculated and a mathematical fomulation to find the maximum inductance of the inducto is deived. The esults indicate inductance inceases of many folds can be achieved using the optimum dimensions fo the ai coe inducto with the same cable length. The method poposed in this pape can also be applied to othe applications like inductos fo RF and micowave cicuits. 2 Theoetical appoach In geneal, any ai coe inducto can be consideed as a combination of a numbe of one-winding units located somewhee in the space. The total inductance ( Ltotal ) of such a combination can be simply expessed as: L total = L ij n n i= 1 j= 1 Whee L ii is the self inductance of the ith unit, L ij the mutual inductance between ith and jth units and n the numbe of windings. In this way, the calculation of the inductance of any abitay inducto with a numbe of windings can be educed to find a geneal fomulation fo the self-inductance of a one-winding unit and the mutual inductance between two such units. (1)

4 2.1 Mutual-inductance between two windings To fomulate the mutual inductance between two windings, the magnetic vecto potential ( A ) poduced by a cuent souce distibution is taken into consideation. Applying the Coloumb s gauge [1], the magnetic vecto potential ( A ) obtained solving the following equation: 2 A = µ j (2) With the answe as: µ j A = ( ) dv 4π v (3) Unde the simplifying assumption that all cuents ae concentated at the mid of the cable, the answe can be simplified to: µ ( ) I dl A = (4) 4π c whee is the vecto which its end points to cuent component I. dl, a vecto that A is calculated at its end and c is the path which cuent flows in it (see Figue 1). Figue 1: Two windings of an inducto Fom the figue 2, the following geometical elationships can be deived: = z + 2 R 2R cos ( θ ) d l = R d θ cos ( sin ( θ ) i + ( θ ) j ) (5) Substituting (5) in (4) esults in: µ 2π I R ( sin ( θ ) i + cos ( θ ) j ) dθ A = 4π z + 2 R 2 R cos θ ( ) (6) Consideing the definition of the magnetic vecto potential, the magnetic flux though the second winding geneated by the cuent of fist winding can be easily calculated using:

5 φ = A dl (7) S ( A) nds ˆ = c So fo mutual inductance between two windings, the following expession can be deived: L 12 = ( θ ) 2 φ µ π R cos dθ = = I z + 2R 2R cos ψ 2 I ( θ ) (8) 2.2 Self-inductance of a winding The magnetic flux though one winding poduced by its cuent can be calculated in a simila way. The only diffeence is that the magnetic flux though the conducto itself can not be calculated accuately because of the simplifying assumption that the cuent is concentated at the middle point of the conducto. Theefoe, the fomulation used to calculate the mutual inductance between two windings is applied only to calculate a potion of the self inductance elated to the magnetic flux, which is not though the conducto. Fo the othe potion (inne inductance), it is assumed that magnetic field of cuent in ing cable makes the same inductance as that in a long diect cable, because the adius of the inducto R is at least ten times bigge than the adius of conducto a. A long diect cable has.5 µh in one mete [2]. This value is simply added to the inductance calculated using equation 1. Accoding to the figue 2, end of vecto points to a ing with adius ( R a ). Figue 2: one ing of the inducto In this case, the following expession fo magnetic vecto potential and self-inductance ae deived: A = π I R 4π 2 2 a + a µ 2 ( sin( θ ) i + cos( θ ) j ) dθ 2R 2R a 2R( R ) cos( θ ) and (9) L 11 = ψ 2 ( R a) π cos( θ ) φ µ R dθ = = I I a + 2R 2Ra 2R a ( R ) cos( θ ) (1) 3 Results 3.1 Expeimental veification To veify the calculation method, simulation and measuement of a numbe of diffeent inductos

6 constucted using a wie, which has 1 mm conducto diamete, 1.35 mm entie diamete, have been compaed. The constucted inductos have been solenoid type with a diamete of 25 cm. As it can be seen in figue 3, thee is almost no diffeence between simulation and measuement esults fo inductos with lowe numbe of tuns. In case of vey high numbe of tuns, the diffeences ae still smalle than 2 pecent. Figue 3: Compaison between expeimental esults and simulation esults fo an inducto with a=.5 mm, b=.675 mm, R=12.5 cm 3.2 Maximum inductance of one-laye solenoid inductos In pactical applications, it is desied to find out the optimum shape of an inducto constucted fom a given cable to maximize its inductance. In those applications the minimum distance between two adjacent windings is given by the chaacteistics of the cable like the total diamete and the conducto diamete. Figue 4 shows the simulation esults fo one-laye solenoid type inductos constucted fom cables with diffeent length and total diamete of.388 cm and conducto diamete of.138 cm. As it can be seen, thee is a maximum achievable inductance of 531 µh using the 5 m cable. In this case, the solenoid type inducto has 3 tuns and a adius of 26.5 cm. It is also inteesting to note that the optimum numbe of tuns fo a given inducto shape is dependent on the cable length used to constuct it.

7 Figue 4: inductance of 1 mm 2 6 cable in diffeent length and ings 3.3 Maximum inductance of solenoid inductos with seveal layes A moe geneal inducto would be a solenoid type inducto with seveal layes. Fo such inductos, thee ae diffeent paametes, i.e. numbe of layes and numbe of tuns pe laye, which could be optimized. In figue 5, a typical simulation esult is shown. In this case, a 1 m cable with total diamete of.4 cm and conducto diamete of.2 cm is used. The maximum inductance is about L=71 µh fo the inducto with 5 layes of winding and 5 tuns pe laye. Figue 5: Inductance of inductos with same cable and seveal layes of winding

8 So using the poposed fomulation fo the inductance of an inducto constucted in the fom of any abitay combination of diffeent ound windings, it is possible to find the optimum shape and dimensions of the inducto to achieve the highest inductance and maximum quality facto. 4 Conclusion In this pape a method has been pesented to fomulate the inductance of any abitay inducto constucted in the fom of a combination of a numbe of windings, based on the self inductance of the windings and the mutual inductances between diffeent windings. The simulation esults have been veified by measuements. It has been shown that application of the poposed fomulation makes it possible to find out the optimum shape and dimension of the inducto to achieve the maximum inductance level. Especially in cases of solenoid type inductos with one o seveal layes, the simulation esults fo inductos constucted fom a given cable length ae pesented. It has been shown that thee ae maximum values of inductances depending on the numbe of layes and the numbe of tuns pe laye. Refeences: [1] Field and Wave Electomagnetic, David k. Cheng [2] Powe System Analysis John Gainge, William Stevenson Authos: D. Kaveh Niayesh Hossein Ghobani Ami Hayati-Soloot School of Electical and Compute Engineein Faculty of Engineeing, Univesity of Tehan Noth Kaga Ave., P.O.B /515 IR-14395, Tehan, Ian Phone: Fax: kniayesh@ut.ac.i

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