Electromagnetic Theory as an Aid to Understanding Electromagnetic Field Analysis

Transcription

1 PECAL Electomagnetic Theoy as an Aid to Undestanding Electomagnetic Field Analysis Tomohio KE Electomagnetic field analysis is an indispensable tool in the design and development of electical devices and appliances. oftwae development has now made it possible to analyze electomagnetic fields with ease; howeve, in ode to model poblems and evaluate analysis esults, it is still impotant to have a good undestanding of electomagnetic theoy. t is difficult fo beginnes to undestand electomagnetics, but gaining such knowledge should help them to undestand the wokings of electical devices and appliances. Keywods: electomagnetics, electomagnetic field analysis, electic field analysis, magnetic field analysis 1. ntoduction Advances in numeical analysis techniques and the advent of high-speed, high-capacity analytical hadwae, such as the pesonal compute, have made it possible to pefom electomagnetic field analysis using the finite element method and othe numeical analysis techniques, thus making investigation of even the most complex electomagnetic phenomena possible. As such, electomagnetic field analysis has become an indispensable tool in the design and development of electical devices and appliances (1). As electomagnetic field analysis softwae has become inceasingly poweful in ecent yeas, analysis can now be pefomed vey simply by just inputting the shape and setting conditions. When pefoming analysis, howeve, it is necessay to constuct an electomagnetic field analysis model and evaluate analysis esults appopiately. n ode to do this, an undestanding of electomagnetic phenomena based on electomagnetic theoy is useful. n an ideal situation, such electomagnetic field analysis should be pefomed by those well vesed in electomagnetics, but in an inceasing numbe of cases such analysis is pefomed by those without any expetise. Futhemoe, it would take a geat deal of time and effot fo novices in electomagnetics to lean fom the beginning the fundamental pinciples of thei discipline. This pape is intended to explain how knowledge of electomagnetics can lead to an undestanding of the mechanisms of devices and appliances, so that those pesons who ae studying electomagnetics may develop an inteest in this subject. At the end of this pape ae bief desciptions of the bae essentials of what students of electomagnetics should know, and which the autho hopes may seve as a efeence. By definition, electical devices and appliances ae devices that, though the use of electic cuents * 1 and magnetic fluxes * 2, geneate electic enegy out of mechanical enegy (geneatos), convet electic enegy into mechanical enegy (motos), o convet voltage to facilitate electic enegy tansmission (tansfomes). These ae oughly divided into two types: tansfomes and otay machines. Figue 1 shows a typical example of a tansfome. Conducto 1 is wound aound a magnetic mateial and connected to a powe souce with voltage 1 so that an electic cuent 1 flows though the tansfome s coil, geneating magnetic flux ø. To concentate the magnetic field that passes though it, Conducto 2 is wound aound a magnetic mateial. f powe-supply voltage 1 is an altenating voltage, the voltage changes sinusoidally ove time. Theefoe, the magnetic flux also changes. This then geneates electomotive foce * 3 2 in Conducto 2, and electic cuent 2 flows if Conducto 2 is connected to a load (see Appendix 1 fo details). Powe ouce 1 Conducto 1 1 ø Magnetic mateial 2 Conducto 2 Load (1) Apply time-vaying (altenating) electic cuent to Conducto 1 (2) aying magnetic flux is geneated and passes though the magnetic mateial and Conducto 2. (Conducto 2 and the magnetic flux ae intelinked.") (3) Electomotive foce is geneated in Conducto 2. (4) Electic cuent flows if Conducto 2 is connected to a load. 2 Fig. 1. Example of a Tansfome 2. Electomagnetic Phenomena of Electical Devices and Appliances Conductos ae used to cay electic cuent, while magnetic mateials ae used to encouage the magnetic flux to pass though. When the conducto path is alteed, the electic cuent flows along it. Likewise, changing the magnetic mateial path can make the magnetic flux to flow along it. E TECCAL REEW UMBER 72 APRL

2 The cuent senso (1) shown in Fig. 2 is also a type of tansfome. The senso s wie haness conducto coesponds to Conducto 1 of Fig. 1, but this senso does not include Conducto 2, and a pat of the magnetic mateial (magnetic coe) is left open to ceate a gap. Conducto Magnetic mateial ø Pemanent magnet Flowing cuent Wie haness conducto (coppe) Rotations Magnetic mateial Magnetic coe (PC Pemalloy) Gap (all element at the cente) Fig. 2. Cuent senso Fig. 4. nteio pemanent magnet moto Figue 3 is a epesentational model of a geneato, an example of a otay machine. Pemanent magnets located above and below geneate magnetic flux ø. A conducto is wound aound a magnetic mateial, and they otate togethe. Consequently, the magnetic flux passing though the conducto changes, theeby geneating electomotive foce in the conducto. f the conducto is connected to a load, electic cuent is applied (see Appendix 2 fo details). Figue 4 shows a coss section of an inteio pemanent magnet synchonous moto, which is an example of a otay machine. Embedded inside the oute magnetic mateials, conductos ae oganized into goups, to which thee-phase altenating cuent is then applied. Pemanent magnets ae also embedded in the inne magnetic mateials on the inside. The thee-phase cuent applied to the outside conductos geneates otating magnetic fluxes in the magnetic mateials and pemanent magnets on the inside, which then poduce tuning foce (toque) in the magnetic mateials and pemanent magnets on the inside (see Appendix 3 fo details about motos). 3. Basic Theoies of Electomagnetics 3-1 Electic cuent * 1 As shown in Fig. 5, electic cuent [A] flows if diect voltage [] is applied at both ends of the conducto. The esistance R [Ω] of this conducto is given by Equation (1), if the conducto has the same coss-section pofile lengthwise and its coss section aea, length, and esistivity ae [m 2 ],l[m], and ρ[ωm], espectively. Pemanent magnet Load ø Pemanent magnet Rotation Magnetic mateial Conducto (1) Pemanent magnets geneate magnetic flux. (2) The magnetic mateial in which the conducto is embedded otates. (n eality, moe than one conducto is used, but only one is shown in the figue above.) (3) As the conducto otates, the magnetic flux passes though (inteacts with) the conducto. As the magnetic flux changes, electomotive foce is geneated in the conducto. (4) Electic cuent flows when the conducto is connected to a load. Electic potential 0 Distance Fig. 3. Rotay machine Fig. 5. Electic field by the cuent flowing though a conducto 18 Electomagnetic Theoy as an Aid to Undestanding Electomagnetic Field Analysis

3 Rρ...(1) The elationship called Ohm s law, given by Equation (2), exists among voltage, electic cuent, and esistance R. =R...(2) E...(6) 2 1 The capacitance of the insulato can be calculated by using Equation (7) (see Appendix 4). C 2πε (7) An electic field exists in the conducto, and an electic cuent flows though the conducto. 3-2 Electic field * 4 n Fig. 5, due to the battey s voltage, the electic potential at the left end of the conducto is and that at the ight end is 0. ince voltage is applied ove the conducto s entie lengthl, electic field E[/m] can be calculated using Equation (3). Conducto nsulato 0 [] 2 [] 1 nsulato [mm] Pemittivity ε E...(3) Fig. 7. Electic field of a concentic cylindical electode As shown in Fig. 6, the electic field geneated when voltage is applied to an insulato placed between paallel plate electodes can also be obtained though Equation (3). Electic chage Q[C] that is accumulated in the electodes is given by Equation (4). Q=C...(4) ee capacitance C[F] can be expessed by Equation (5). Cε εε 0...(5) Whee ε is the insulato s pemittivity [F/m], ε is elative pemittivity, and ε 0 is pemittivity in a vacuum ( F/m). 3-3 Magnetic field * 5 A magnetic field is poduced aound a conducto though which electic cuent passes (Fig. 8). When a ight-handed scew is otated, a magnetic field is geneated in the diection of the otation, if an electic cuent is applied in the diection of the scew taveling fowad (the ight-hand ule). The stength of magnetic field [A/m] at a adial distance fom the cente of the conducto can be obtained by Equation (8) (see Appendix 5)....(8) 2π Magnetic flux density B[T] in the atmosphee is given by Equation (9), if pemeability in a vacuum is µ0=4π 10-7 [/m]. Bµ 0 µ 0 2π...(9) Electode nsulato Right-handed scew Electic potential Electode Diection of Diection of Fig. 8. Magnetic field ceated by electic cuent unning though a conducto 0 Distance Fig. 6. Electic field of an insulato between paallel plate electodes As shown in Fig. 9, a magnetic field is poduced aound a pemanent magnet. Electic field E geneated when voltage is applied to an insulato placed between concentic cylindical electodes as shown in Fig. 7 can be obtained though Equation (6) (see Appendix 4). Fig. 9. Magnetic field ceated by a pemanent magnet E TECCAL REEW UMBER 72 APRL

4 3-4 Electomagnetic induction When placing a conducto aound anothe conducto though which time-vaying electic cuent passes and aound which vaying magnetic field has been geneated, electomotive foce out is geneated as shown in Fig. 10. f the conducto has a closed loop stuctue as shown in Fig. 11, electic cuent out flows. foce out is geneated. f the conducto has a closed loop stuctue as shown in Fig. 13, electic cuent out flows. OUT Motion Fig. 13. Electic cuent geneated by a vaying magnetic field OUT Fig. 10. Electomotive foce geneated by a vaying magnetic field As Fig. 12 shows, when alteing a magnetic flux that is linked with a conducto by moving anothe conducto within an aea whee magnetic field exists, electomotive OUT Fig. 11. Electic cuent geneated by a vaying magnetic field 4. Electomagnetic Consideation of Electomagnetic Field Analysis Results 4-1 Electic field (1) nsulatos between paallel plate electodes When voltage was applied to the insulato between paallel plate electodes shown in Fig. 14 and the electic field was analyzed using the finite element method (FEM), the electic field was found to be 5 [/mm] (E= [/m]), which matched the theoetical value calculated using Equation (3). Based on the FEM electic field analysis esults, capacitance is obtained as follows: Q εe C F 100 ts theoetical value is calculated using Equation (5): Cε ε ε F As demonstated above, the FEM electic field analysis esults matched the theoetical value. = 1000mm 2 = m 2 OUT 100 Ai (eative pemittivity ε = 1 = 20mm 10mm Motion 100mm Fig. 12. Electomotive foce geneated by a vaying magnetic field Fig. 14. nsulato betweem paallel plate electodes 20 Electomagnetic Theoy as an Aid to Undestanding Electomagnetic Field Analysis

5 (2) Concentic cylindical insulato When voltage was applied to the concentic cylindical insulato (XLPE cable insulato) shown in Fig. 15 and the electic field was analyzed using FEM, the electic field matched the theoetical value calculated using Equation (6) as shown in Fig. 16. Based on the FEM electic field analysis esults, capacitance was calculated using electic field values of elements on the suface of the inne semiconducto laye: Q εe C π F/m was analyzed using FEM, magnetic flux density was calculated as shown in Fig. 18. The esults obtained with FEM analysis matched the theoetical value calculated using Equation (9). Radius a z a = 5mm = 1m = 100A Based on Equation (7), the theoetical value is obtained as follows: C 2πε 2 1 2πεε π F/m 33.1 The FEM electic field analysis esults matched the theoetical value (1.4% magin of eo). Magnetic flux density B (T) Fig. 17. Linea cicula conducto FEM Theoetical value mm Conducto nsulato oute semiconducto inne semiconducto k = k (RM value) 0 = k 1 = 33.1 nsulato 2-dimensional model 2 = 60.1 [mm] Relative pemittivity ε = 2.3 Radius (m) Fig. 18. Magnetic field aound a linea cicula conducto Electic Field E (k/mm) Fig. 15. Concentic cylindical insulato Radius (mm) Theoetical value FEM Fig. 16. Electic field of a concentic cylindical insulato 4-2 Magnetic field When electic cuent was passed though the linea cicula conducto shown in Fig. 17 and the magnetic field 5. Conclusion Electical devices and appliances achieve thei intended functions though the inteaction of an electic cuent and a magnetic flux. Electic cuent passes though the conducto, and a magnetic flux uns though the magnetic mateial. Electic cuent that passes though the conducto foms a magnetic flux, which in tun uns though the magnetic cicuit made of magnetic mateials. When the magnetic flux that uns though the inside of the looped conducto is alteed, electomotive foce is geneated in the conducto. Electomotive foce causes the electic cuent to flow, and electic cuent flows once a conducto is connected to a load, etc. to fom a closed cicuit. Pemanent magnets can also be used to ceate a magnetic flux. Electomagnetic theoy explains such electomagnetic phenomena. ince physical quantities such as electic fields and magnetic fields ae vectos, equations using vecto analysis ae employed, and these have been compiled into Maxwell s Equations (1). These equations can be esolved using numeical tools such as FEM, whee a set of conditions ae given, including the shapes of the objects to be analyzed, pemittivity, pemeability, and othe physical popeties, E TECCAL REEW UMBER 72 APRL

6 bounday conditions, and flowing electic cuent. t was confimed that electomagnetic field analysis esults matched the theoetical values of electomagnetics in simple poblems on electic fields and magnetic fields. When designing electical devices and appliances, this numeical solution electomagnetic field analysis is used. Knowledge of electomagnetics plays an impotant ole in the development of analysis models and assessing analysis esults. The autho hopes that the above explanations will help the eade to develop an inteest in and undestanding of electomagnetics. Technical Tem *1 Electic cuent : The flow of electic chage caied by moving electons and othe chaged paticles. t is defined as the ate at which an electic chage tavels though a given suface pe unit of time, and is measued in ampees [A]. Fo an electic cuent of 1 ampee, 1 coulomb [C] pe second of electic chage will pass though any plane (1 A = 1 C/s). Electic chage: A fom of chage found on elementay paticles of matte. Electic chage can be eithe positive o negative and is measued in coulombs [C]. *2 Magnetic flux : Magnetic flux flows along the magnetic line of foce fom the noth to south poles of pemanent magnets. t is the sum of the poducts of nomal diection components multiplied by the coss-sectional aea of the density of the magnetic flux that passes though a given section of a magnetic field. Magnetic flux is measued in [Wb]. Magnetic flux density: Magnetic flux divided by the aea though which it passes. Magnetic flux density is measued in [T]. ([Wb/m 2 ] = [T]) ø=b Whee ø is magnetic flux [Wb], B is magnetic flux density [T], and is the coss-sectional aea though which the magnetic flux passes [m 2 ] *3 Electomotive foce : A foce that pushes chaged paticles such as electons, thus causing a cuent to flow, equivalent to the potential diffeence (voltage) that gives ise to cuent. Electomotive foce is measued in []. *4 Electic field : A egion of space in which foce is exeted on electically chaged objects. Electic fields ae measued in [/m]. *5 Magnetic field : When a pemanent magnet is placed on a piece of pape ove which ion sand has been spinkled, the ion sand will fom a egula patten of lines aound the magnet. This occus in esponse to the magnetic field of the magnet. n addition to pemanent magnets, a magnetic field also occus aound electic cuent. Magnetic fields ae expessed though thei magnetic flux density [T] o intensity [A/m]. B=µ Whee B is magnetic flux density [T], is the magnetic field [A/m], and µ is pemeability [/m]. *6 Magnetic cicuit : A path though which a magnetic flux that is made up of magnetic mateials, etc. tavels. Because Ohm s law applies to a magnetic cicuit, whee magnetic flux coesponds to electic cuent, magnetomotive foce to voltage, and magnetic esistance to electic esistance, a magnetic cicuit can be calculated in the same manne as that fo electic cicuits. Magnetomotive foce: Foce that poduces magnetic flux in a magnetic cicuit. t is the poduct of electic cuent that flows in a coil multiplied by the winding numbe (numbe of tuns) of the coils that ae wound aound an ion coe. Appendix 1. Tansfomes As shown in Appended Fig. 1, in a tansfome, a pimay winding conducto and a seconday winding conducto ae wound aound the ion coe, theeby foming a closed magnetic cicuit*6. As shown in Appended Fig. 2, when altenating cuent is applied to the pimay winding, a time-vaying magnetic flux is poduced and flows though the ion coe, which then induces a time-vaying magnetic flux in the seconday winding s ion coe. 1 1 Pimay winding on coe 2 econday winding Appended Fig. 1. Electomotive foce geneated by a tansfome t (a) Positive cuent t (b) egative cuent Appended Fig. 2. Magnetic flux geneated by a tansfome 22 Electomagnetic Theoy as an Aid to Undestanding Electomagnetic Field Analysis

7 As the magnetic flux inside the seconday winding changes, electomotive foce is poduced in the conducto. f the voltage of the pimay winding is 1[], the winding numbe (numbe of tuns) of the pimay winding is 1, and that of the seconday winding is 2, electomotive foce 2[] is: (App. 1) 1 1 When a closed cicuit is fomed by having a load connected to the seconday winding o by othe means, electic cuent flows as shown in Appended Fig. 3. B ø...(app. 3) 3. Motos As shown in Appended Fig. 5, by applying electic cuent [A] to a conducto in magnetic flux ø[wb], lengthl[m] whee the conducto cuts acoss the magnetic flux is subjected to foce F[]. Both the magnetic flux and the conducto that is pependicula to it ae subjected to foce F, and since these two foces ae facing in opposite diections, the conducto is subjected to tuning foce (toque). This is how a moto functions. F=Bl...(App. 4) f the aea though which magnetic flux passes is [m 2 ], magnetic flux density B[T] is given by Equation (App. 3). n an actual moto, magnetic flux is geneated and electic cuent is applied in such a way that foce is always geneated in the same diection as the otation, even as the conducto itself otates. Appended Fig. 3. Electic cuent geneated by a tansfome 2. Geneatos As shown in Appended Fig. 4, by otating the conducto in the magnetic flux ø[wb] at a speed of ν[m/s] to cut the magnetic flux, electomotive foce e[] is geneated ove the lengthl[m] whee the conducto cuts acoss the magnetic flux. Electomotive foce e is geneated both in the magnetic flux and the conducto that is pependicula to it. ince these two foces ae facing in opposite diections, electomotive foce = 2e[] is geneated at the ends of the conducto. This is how a geneato functions. e=vbl...(app. 2) f the aea though which magnetic flux passes is [m 2 ], magnetic flux density B[T] is given by Equation (App. 3): F ø Appended Fig. 5. Pinciple of motos 4. Electic field of concentic cylindical insulatos An electic field of an insulato placed between concentic cylindical electodes is calculated as shown in Appended Fig. 6. Fom Gauss s law, F E nd Q ε...(app. 5) v e ø e v Whee E is the electic field [/m], n is the unit nomal vecto of a suface, is aea[m 2 ], Q is the electode s electic chage [C], and ε is pemittivity of the insulato [F/m], and the integal is the suface integal of the planes suounding the electic chage. f E is the electic field at the oute position of adius [m] and the electic chage pe evey 1 m of unit length is Q, then fom the following equation, = 2e Appended Fig. 4. Pinciple of geneatos 2π1E Q ε E Q 1...(App. 6) 2πε E TECCAL REEW UMBER 72 APRL

8 Applied voltage is an integal of the electic field in Equation (App. 6). 2 Ed 2 Q 1 Q 2 2πε d Q [] 2 2πε 2πε Thus Q 2πε 2 1 When substituting this into Equation (App. 6), E (App. 7) Capacitance C[F] is then calculated as follows. C Q 2πε 2...(App. 8) Conducto 1 0 [] 2 [] 1 [mm] Refeences (1) Tomohio Keishi, Electomagnetic Field Analysis and ts Applications to Poduct Development, E Technical Review, o.69, pp.4-12, 2009 (2) Kazuhiko Tamechika, Basic ote of Physics [Electomagnetics] fo the Wok of cience Couse, Chukei Publishing Company, 2007 (3) aohei Yamada, Electomagnetics, The nstitute of the Electical Enginees of Japan (EEJ), 1971 (Refeence (2) and (3) ae witten in Japanese.) Contibuto T. KE enio pecialist D. Engineeing, Pofessional Enginee Japan (Electics & Electonics Engineeing) enio Assistant Geneal Manage, Analysis Technology Reseach Cente Engaged in eseach of compute aided engineeing (CAE) nsulato nsulato Pemittivity ε Appended Fig. 6. Electic field of a concentic cylindical insulato 5. Magnetic field aound a linea cuent A magnetic field aound a conducto though which electic cuent flows is calculated as shown in Appended Fig. 7. Fom Ampee s cicuital integal law, ds=...(app. 9) Whee is the magnetic field [A/m], s is the linea vecto, and is the conducto s electic cuent [A], and the integal is the cuvilinea integal on the contou suounding the electic cuent. Fom Equation (App. 9), 2π =, 2π...(App. 10) f pemittivity in a vacuum is µ0, the magnetic flux density B[T] is B µ 0 2π...(App. 11) Appended Fig. 7. Magnetic field aound a linea cylindical conducto 24 Electomagnetic Theoy as an Aid to Undestanding Electomagnetic Field Analysis