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Electromagnetic Theory As an Aid to Understanding Electromagnetic Field Analysis

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Electromagnetic Theory As an Aid to Understanding Electromagnetic Field Analysis SPECIAL Electromagnetic Theory as an Aid to Understanding Electromagnetic Field Analysis Tomohiro KEISHI Electromagnetic field analysis is an indispensable tool in the design and development of electrical devices and appliances. Software development has now made it possible to analyze electromagnetic fields with ease; however, in order to model problems and evaluate analysis results, it is still important to have a good understanding of electromagnetic theory. It is difficult for beginners to understand electromagnetics, but gaining such knowledge should help them to understand the workings of electrical devices and appliances. Keywords: electromagnetics, electromagnetic field analysis, electric field analysis, magnetic field analysis 1. Introduction ical energy (motors), or convert voltage to facilitate electric energy transmission (transformers). These are roughly di- Advances in numerical analysis techniques and the ad- vided into two types: transformers and rotary machines. vent of high-speed, high-capacity analytical hardware, such Figure 1 shows a typical example of a transformer. as the personal computer, have made it possible to perform Conductor 1 is wound around a magnetic material and electromagnetic field analysis using the finite element connected to a power source with voltage V1 so that an elec- method and other numerical analysis techniques, thus mak- tric current I1 flows through the transformer’s coil, gener- ing investigation of even the most complex electromagnetic ating magnetic flux ø. To concentrate the magnetic field phenomena possible. As such, electromagnetic field analysis that passes through it, Conductor 2 is wound around a has become an indispensable tool in the design and devel- magnetic material. If power-supply voltage V1 is an alter- opment of electrical devices and appliances(1). nating voltage, the voltage changes sinusoidally over time. As electromagnetic field analysis software has become Therefore, the magnetic flux also changes. This then gen- 3 increasingly powerful in recent years, analysis can now be erates electromotive force * V2 in Conductor 2, and electric performed very simply by just inputting the shape and set- current I2 flows if Conductor 2 is connected to a load (see ting conditions. When performing analysis, however, it is Appendix 1 for details). necessary to construct an electromagnetic field analysis model and evaluate analysis results appropriately. In order to do this, an understanding of electromagnetic phenom- ena based on electromagnetic theory is useful. In an ideal situation, such electromagnetic field analysis should be per- ① ④ Power I1 I2 formed by those well versed in electromagnetics, but in an Source ③ increasing number of cases such analysis is performed by V1 V2 Load those without any expertise. Furthermore, it would take a great deal of time and effort for novices in electromagnet- ② ø ics to learn from the beginning the fundamental principles Conductor 1 Conductor 2 of their discipline. This paper is intended to explain how knowledge of electromagnetics can lead to an understand- Magnetic material ing of the mechanisms of devices and appliances, so that those persons who are studying electromagnetics may de- (1) Apply time-varying (alternating) electric current to Conductor 1 (2) Varying magnetic flux is generated and passes through the velop an interest in this subject. At the end of this paper magnetic material and Conductor 2. (Conductor 2 and the are brief descriptions of the bare essentials of what students magnetic flux are “interlinked.") of electromagnetics should know, and which the author (3) Electromotive force is generated in Conductor 2. hopes may serve as a reference. (4) Electric current flows if Conductor 2 is connected to a load. Fig. 1. Example of a Transformer 2. Electromagnetic Phenomena of Electrical Devices and Appliances Conductors are used to carry electric current, while magnetic materials are used to encourage the magnetic By definition, electrical devices and appliances are de- flux to pass through. When the conductor path is altered, vices that, through the use of electric currents *1 and mag- the electric current flows along it. Likewise, changing the netic fluxes *2, generate electric energy out of mechanical magnetic material path can make the magnetic flux to flow energy (generators), convert electric energy into mechan- along it. SEI TECHNICAL REVIEW · NUMBER 72 · APRIL 2011 · 17 (1) The current sensor shown in Fig. 2 is also a type of Conductor transformer. The sensor’s wire harness conductor corre- Magnetic material sponds to Conductor 1 of Fig. 1, but this sensor does not include Conductor 2, and a part of the magnetic material (magnetic core) is left open to create a gap. N ø S S S N N Permanent magnet Rotations Flowing current I N S S Wire harness conductor N Magnetic material (copper) N N S S Magnetic core S (PC Permalloy) N Gap (Hall element at the center) Fig. 4. Interior permanent magnet motor Fig. 2. Current sensor magnets are also embedded in the inner magnetic materi- Figure 3 is a representational model of a generator, als on the inside. The three-phase current applied to the an example of a rotary machine. Permanent magnets lo- outside conductors generates rotating magnetic fluxes in cated above and below generate magnetic flux ø. A con- the magnetic materials and permanent magnets on the in- ductor is wound around a magnetic material, and they side, which then produce turning force (torque) in the rotate together. Consequently, the magnetic flux passing magnetic materials and permanent magnets on the inside through the conductor changes, thereby generating elec- (see Appendix 3 for details about motors). tromotive force V in the conductor. If the conductor is connected to a load, electric current I is applied (see Ap- pendix 2 for details). Figure 4 shows a cross section of an interior perma- 3. Basic Theories of Electromagnetics nent magnet synchronous motor, which is an example of a rotary machine. Embedded inside the outer magnetic 3-1 Electric current *1 materials, conductors are organized into groups, to which As shown in Fig. 5, electric current I[A] flows if direct three-phase alternating current is then applied. Permanent voltage V[V] is applied at both ends of the conductor. The resistance R [Ω] of this conductor is given by Equation (1), if the conductor has the same cross-section profile length- wise and its cross section area, length, and resistivity are S[m2],ℓ[m], and ρ[Ωm], respectively. Permanent N magnet ② ④ ① V I ø Rotation ③ V Load Magnetic Permanent material I magnet S Conductor S Electric ℓ potential (1) Permanent magnets generate magnetic flux . (2) The magnetic material in which the conductor is embedded rotates. V (In reality, more than one conductor is used, but only one is shown in the figure above.) (3) As the conductor rotates, the magnetic flux passes through (interacts with) the conductor. As the magnetic flux changes, electromotive force is generated in the conductor. 0 Distance (4) Electric current flows when the conductor is connected to a load. ℓ Fig. 3. Rotary machine Fig. 5. Electric field by the current flowing through a conductor 18 · Electromagnetic Theory as an Aid to Understanding Electromagnetic Field Analysis V ℓ ................................................................... E＝ ............................................................... R＝ρ (1) r2 (6) S rln r1 The relationship called Ohm’s law, given by Equa- The capacitance of the insulator can be calculated by tion (2), exists among voltage V, electric current I, and using Equation (7) (see Appendix 4). resistance R. 2πε ＝ ...............................................................(7) C r2 ln V＝IR ......................................................................(2) r1 An electric field exists in the conductor, and an elec- tric current flows through the conductor. *4 Conductor 3-2 Electric field 0 [V] r2 In Fig. 5, due to the battery’s voltage V, the electric po- V [V] tential at the left end of the conductor is V and that at the r [mm] right end is 0. Since voltage V is applied over the conduc- r1 E Insulator Permittivity ε tor’s entire lengthℓ, electric field [V/m] can be calcu- Insulator lated using Equation (3). V Fig. 7. Electric field of a concentric cylindrical electrode E＝ ......................................................................(3) ℓ As shown in Fig. 6, the electric field generated when voltage V is applied to an insulator placed between parallel 3-3 Magnetic field *5 plate electrodes can also be obtained through Equation (3). A magnetic field is produced around a conductor Electric charge Q[C] that is accumulated in the elec- through which electric current I passes (Fig. 8). When a trodes is given by Equation (4). right-handed screw is rotated, a magnetic field is generated in the direction of the rotation, if an electric current is ap- Q＝CV .....................................................................(4) plied in the direction of the screw traveling forward (the right-hand rule). The strength of magnetic field H[A/m] Here capacitance C[F] can be expressed by Equation (5). at a radial distance r from the center of the conductor can be obtained by Equation (8) (see Appendix 5). S S .................................................... C＝ε ＝εrε (5) ℓ 0 ℓ I H＝ ...................................................................(8) 2πr Where ε is the insulator’s permittivity [F/m], ε r is rel- ative permittivity, and ε 0 is permittivity in a vacuum Magnetic flux density B[T] in the atmosphere is (8.854×10-12 F/m). given by Equation (9), if permeability in a vacuum is -7 µ0=4π×10 [H/m]. µ I B＝µ H＝ 0 ..........................................................(9) 0 2πr V Electrode H Right-handed S Insulator r screw I ⇒ Electrode Direction of I Electric potential ℓ S Direction of H V Fig. 8. Magnetic field created by electric current running through a conductor 0 Distance ℓ As shown in Fig. 9, a magnetic field is produced around a permanent magnet. Fig. 6. Electric field of an insulator between parallel plate electrodes Electric field E generated when voltage V is applied to NS an insulator placed between concentric cylindrical elec- trodes as shown in Fig.
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