Magnetic field
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(2) 7.1 Introduction 1º.Static fields for any medium:. Electrostatic field. Magnetic field. 2º. Static charge → Electrostatic field Moving charge → Steady-state current → quasi-static magnetic field or magnetostatic field. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(3) 3º.. ⇒ Maxwell's equations ⇒. Space. Wave equation Time source Gauss's law of E.F. Gauss's law of M.F. (Divergenceless field) Faraday's law. Ampere's circuital law. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(4) 7.2 Faraday's law 1º. Fundamental postulation of eletromagnetic inductions. By Stokes's theorem:. 2º. For a time invariant field. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(5) 7.2 Faraday's law 1º. Fundamental postulation of eletromagnetic inductions. By Stokes's theorem:. 2º. For a time invariant field. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(6) 7-2.1 A stationary circuit in a time-varying magnetic field 1º. For a stationary surfaces and c. 2º.. The EMF in circuit contour C (V) Total flux in S (Wb). 3º.. Faraday's law (1839) ↑Lenz's law(1834). 4º. Transformer emf Rotational/motional emf. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(7) EX7-1. A circular of N turns of conducting wire lines in the xy-plane with its center at the origin of a magnetic field specified by B = azb0 cos(πr/2b) sin ωt, where b is the radius of the loop and ω is the angular frequency. Find the emf induced in the loop. Sol:. 1º.. 2º.. 3º.. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(8) 7-2.2 A moving conductor in static magnetic field Lorentz's law. 1º.. 2º.. 3º.. 4º. The induced emf at a close loop: ↑flux-cutting emf or motional emf. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(9) EX7-2 A metal bar slide over a pair of conducting rails in a uniform magnetic field B = acB0 with a constant velocity u, as shown in Fig. 7-3. a) Determine the open-circuit voltage V0 that appears across terminals 1 and 2. b) Assuming that a resistance R is connected between the terminals, find the electric power dissipated in R. c) Show that this electric power is equal to the mechanical power required to move the sliding bra with a velocity u. Neglect the electric resistance of the metal bar and of the conducting rails. Neglect also the mechanical friction at the contact points. Sol: a). b). Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(10) c). (mechanical power). Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(11) EX.7-3 The Faraday disk generator consists of a circular metal rotating with a constant angular velocity ω in a uniform and constant magnetic field of flux density B= azB0 that is parallel to the axis of rotation. Brush contacts are provided at the axis and on the rim of the disk, as depicted in Fig. 7-4 Determine the open-circuit voltage of the generator if the radius of the disk is b. Sol:. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(12) 7-2.3 A moving circuit in a time-varying magnetic field 1°. By Lorentz's force equation. 2°.. ......(*). Proof:(*). Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(13) 3°. The times current rate of flux is. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(14) 通過s1 s2 s3 所圍之體積之通量可用之散度表示. (*)故得証. Note:封閉路徑之感應電動勢等於交鏈該回路之磁通之負值遞增率. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(15) For Ex 7.2. For Ex 7.3. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(16) EX7-4 An h by w rectangular conducting loop is situated in a changing magnetic field B = ay B0 sin ωt. The normal of the loop initially makes an angle α with ay, as shown in Fig. 7-6. Find the induced emf in the loop:(a)when the loop is at rest, and (b) when the loop rotates with an angular velocity ω about the x-axis Sol: (a) stationary : (Transformer EMF). (b) Rotate about x-axis with the angular speed ω. (Motional EMF + Transformer emf). .. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(17) ...motional emf. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(18) 7-3 Maxwell's Equations Electrostatic field. Magnetic field. 1°. Static fields. Faraday's law Times varying fields postulations:(假設). Ampere's circuital law field (static m field) Gauss's law Gauss's law. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(19) Continuous equation of a conservation field: 滿足a b c d. The Divergence of (b) is. compare with. ∴The continuous equation could not satisfy the equation (*). Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(20) 2°. Modify the equation a b c d such that satisfy the equation * :. .....satisfy conservation theorem (1831-1879 maxwell). Given. or/and Given. magnetic field. :displacement current density (A/m2). (convervction current) and. (conduction current). Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(21) 3°. The Maxwell's equation (1831-1879). Differential form of Maxwell's equs.. 4X3 = 12 unknown parameters ⇒12 scalar equations continuous equation:. Lorentz's force equation:. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(22) 7-3.1 Integral form of Maxwell's equations. 1°.. 2°.. .....Faraday's law. .....Ampere's circuital law. Gauss's law of M.F.. 3°.. 4°.. .....Gauss's law of E.F.. .....通過任意封閉面的線磁通為零 Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(23) Diff. form. Maxwell's equation Integral form 法拉第. 安培迴路. 高斯定律 of E.F. 高斯定律 of M.F.. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(24) EX7-5 An a-c voltage source of amplitude V0 and angular frequency ω , vc = V0 sin ωt, is connected across a parallel-plate capacitor C1, as shown in Fig. 7-7. (a)Verify that the displacement current in the capacitor is the same as the conduction current in the wires. (b) Determine the magnetic field intensity at a distance r from the wire. Sol: a). iP:current displacement. proof:. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(25) b) at distance r away the conductor : from Amperes circuital law:. where HФ 為導線C上之HФ ,對導線對稱. ∴HФ 為定值. ∵The stored charge on conductor is zero⇒D=0. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(26) 7-4.Potential function 1°.. is solenoidal. ,the vector magnetic potential. 2°.. 3°.For a static electro field (與前述靜電場相同). Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(27) 4°.. 5°.To determine ρ and. :. (the solution of. (the solution of. for a very small frequency and R<<λ ⇒ Quasi-static field for a high frequency and R>>λ ⇒ 上述解不成立,必須包含延時效應(Delay effects) Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(28) 6°.. Left:. Right:. .....for a homogeneous medium. 7°.. i.e. we get. .....Lorentz conduction of vec A .....(Passion equ. of Time-Varying field) Nonhomogeneous wave-equ. of vector potential. ⇒ 速度. 之波動波. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(29) 8°.. ,. .....Nonhomogeneous wave equs. of scalar potential V 9°. The Poisson equs. of time vary field (Ordinary nonhomogeneous wave equs.) ...... 的波動方程式. .....V的波動方程式. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(30) 7-5 Electromagnetic Boundary Conditions 1°.From. and. determine the boundary conduction of tangential component of S=ΔhΔw→0 (i). (ii). ∵ d s = Δ h ΔW→0. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(31) 2°.From. and. (i). (ii). .....E之切線分量為連續. 3°.Notation:. .....if Js ≠ 0 ∴H之切線分量不連續 .....if ρs≠ 0 ∴D之法線分量不連續 .....B之法線分量為連續. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(32) 4°.For following two special case 1.Two lossless linear media 2.Dielectric and a perfect conductor 7-5-1:Interface between two 1°.The lossless media define as ε ,μ and σ = 0. 2°.If ρs=0 then. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(33) 7-5-2 Inter face between a dielectric and a perfect conductor 1°.The ideal conductor that is σ→∞ ⇒ supper conductor and in the inner of conductor ρ exist at the surface of conductor under time varying field －導體內之. If B is a time varying field. 2°.Media 1 (dielectric) Media 2 (ideal conductor). Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(34) Media 1 (介質) Media 2 (理想導體). Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(35) EX7-6 A rectangular wave guide (a x b) , and. passing through the cross area:. where. H0, ω, μ, β are const. the inner wall is ideal conductor. (a). Find the surface charge density (b). Surface current density. Sol: The outer normal of the wall x=0 , x=a , y=0 , y=b are respectly.. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(36) (a). The surface charge density :. ,. (b). The surface current density :. ※Maxwell's equs. ⇒ solution + boundary ⇒ engineering solution. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(37) 7-6 Wave Equations and Their Solutions. 1°.If ρ and. 2°.Then. are well know const. solved. and. and V by. could be find from. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(38) 7-6.1 Solution of wave equation for potentials 1°.Set the point charge (ρ(t)Δv') in origin of the conductions system. Choose the sphere coordinate system for analysis: If the coordinate system is symmetric ∴V = V(R,t) independent of θ and φ and ρ =0 (The origin is except). 2°.We let. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(39) Let. Let. are two solution of. we choose the solution of ＊ is .....波以. 之速度沿+R方向移動. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(40) 3°.If R = R+ΔR , t= t+Δt ,then. where. ..... propagation speed. the Quasi-static point charge ρΔV' in the origin. 4°. static point charge ρΔV' time-varying charge ρ(t)ΔV'. ∴The solution of eq. Is .....retarded scalar potential. .....不可能發生 Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(41) Note (1)V(r,t)←ρ(t – R/u) , given ρ(t – R/u) → V(R,t) (2)t = R/u , V(R,t) ←ρ(0) (3)t + R/u is not solution of ＊because it is impossible to introduce ρ effect the fore ρ exist. 5°.The magnetic vector potential. and. are retarded function of magnetic field (t- R/u). Notation: 延時之時間可提供電磁波移動，可使遠離點能感應時變電荷產生電流效應. (static (忽略延時)→ 立即響應). Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(42) 7.6-2 Source free wave equation 1°.Source free ⇒. and. (σ = 0 , 非導體介質中). 2°.Maxwell equation in simple media. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(43) Similarity. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(44) Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(45) Let. ＊. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(46) If the plane wave has the B.C. as:. we have a solution of the form:. 代入＊. 以速度 面(x,y)上而且整個. 向正Z軸方向運動且 向正Z軸傳播. , 因此可看見整個. 都在一橫截. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(47) 7-7 Time-Harmonic Fields 1°. Source function magnetic field . E.M. Fields 2°.Sinusoidal form of magnetic wave (獨特地位) 3°.Periodic function ⇒ Fourier series Nonperiodic function ⇒ Fourier series Maxwell's equs ⇒ a linear differential equs. 4°.Differential frequency component ⇒ electromagnetic field. Superposition method ⇒ 解全部的場量 7-7.1 The use of phasors 1°.Sinusoidal component ⇒ amplitude , frequency , phase angle. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(48) 2°.L-R-C , e(t) = ρ(t) =E cos ωt. 求I 及. 相當複雜. 3°.Phasor Notation:. where .....參考向位. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(49) 4°.. 5°.. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(50) 7-7.2 Time-Harmonic Electromagnetics 1°. Phasor Transformation:. 2°.For a linear , isotropic , homogeneous media :. with t ⇒ instantaneous field . without t ⇒ phasor (include jω sinusoidal phaser) Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(51) 3°.Scalar potential V and vector potential {vec A newline} , Times Harmonic wave equation :. "Nonhomogeneous Helmholtz's equations". Lorentz's Conduction. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(52) The phasor solution of Helmholtz's equations:. "retarded vector potential". 延時. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(53) The Taylor series expansion of. If. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(54) giving. and. , Then the produce of solving is as following :. 1. From. and. to solve the phasor. 2. Find the phasor. and. and. 3. Total const. as a difference, then find and. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(55) EX.6-9 Gievn nonconducting dielectric medium and the electronmagetic wave is. ,find the. and the volume of β .. Sol: (1). Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(56) (2) For nonconducting ,medium ,. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(57) 7-7.3 Source free fields in simple media 1°.Source free , simple media. 2°.The Maxwell's times harmonic wave equs. became as. 3°.. homogeneous vector Helmholtz's equations 求解將在chap.8 和 chap10 討論. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(58) Ex 7-7 A simple media (ε and μ ) If and are also the solution where. α is arbitrary angle,. and and. are relations of the source free maxwell's equs. then are. :本質組抗. Sol:. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(59) Note 1°.linear transformation ⇐ source free Maxwell's equations 2°. "Principle of duality" 對偶原理 3°. Symmetry of source-free Maxwell's equs.. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(60) 4°.If the simple media is a conductor (σ ≠ 0) and Then. Then Helmholtz's equation became. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(61) 5°.. :loss tangent 用來計算位移電流與導電電流 .....is a measure of the power in the medium. :loss angle good conductivity conductor good insulation material. For example. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(62) EX 7-9 E = 50 V/m , f = 1GHz find P(W/m3) sol:. Excise:7-2 ,7-3 ,7-4 ,7-6,7-7,7-9,7-10,7-12,7-13,7-14,7-17,7-18,7-20,7-22,7-23. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(63) 7-7.4 Spectrum. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(64) Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(65) Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(66) Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(67) 7-2 ,7-3 ,7-4 ,7-6,7-7,7-9,7-10,7-12,7-13,7-14,7-17,7-18,7-20,7-22,7-23. Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
(68) Power Electronic Lab., Institute of Electronic Engineering, Fu Jen Catholic University, Taipei, Taiwan 輔仁大學電子工程學系(所) EMI電力電子研究室.
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