电磁讲义_长庚电子 Ch6.pdf

電磁學 2 Chapter 6 Time Varying Fields and Maxwell s Equation 國生副教授 長庚大學電子系 國生副教授 長庚大學電子系 2 Time varying Fields and Maxwell s Equations Electrostatic Model For linear and isotropic media Constitutive relation 0 static densityflux Electric law s Gauss intensity field Electric 0 t B t B E D D E E v E D 3 Magnetostatic model intensity field Magnetic law s Amperl densityflux Magentic law s Gauss 0 HJH BB H B Constitutive relation Time varying Field H BD E torelated are E and D in the electrostatic model are not related to B and H in the magnetostatic model Static Field 4 Faraday s Law of Electromagnetic Induction The experiment of Michael Faraday 1831 The quantitative relationship between the induced emf electromotive force and the rate of change of flux linkage Faraday s Law form integratedor formregion entire theoremsStokes form aldifferentior form point sd t B dE sd t B sdE t B E cs ss 5 Faraday s Law of Electromagnetic Induction c sd t B dE cs S電 與磁場關係有以下三種可能 1 電 靜止 磁場時變 2 電 移動 磁場 變 3 電 移動 磁場時變 B V 6 V unitcircuit thelinkingflux magnetic changed of rate theminus iscircuit closed stationary ain emf induced The 12 6 Eq induction neticelectromag of law sFaraday source response flux magnetic induced emf field magnetic varying timeain circuit stationary aFor 1 dt d v v sdB dt d dE sc 7 Sol emf find sin 2 cos given plane in wireconducting of turns of loopcircular A t b r BaB xyN Oz dr b r rtB rdrat b r Ba rdrasdsdB b o z b oz z s 0 0 2 cos sin2 2 sin 2 cos 2 y xz b Example 6 1 電 動 磁場時變電 動 磁場時變 8 with phase timeofout 90 V cos1 2 8 turns sin 1 2 8 1 2 4 2 cos 2 2 sin 2 2 os 0 0 2 0 2 2 2 0 2 0 tB bN dt d Nv NN tB b b b r b b r b r dr b r rc b b b rb b rb b r r 2 cos 2 0 2 sin 2 1 2 cos 2 分部積分法分部積分法 9 Transformers A transformer is an alternating current device that transforms voltages currents and impedances k kk j jjI N Primary circuit Secondary circuit Ferromagnetic core Around a closed path in a magnetic circuit the algebraic sum of ampere turns is equal to the algebraic sum of the products of the reluctances and fluxes 2211 iNiN reluctance 10 Transformers 1 2 2 1 N N i i 2 1 2 1 N N v v Leff R N N R 2 2 1 1 0 ers transformidealFor dt d Nv Load resistance Effective load resistance seen by source Leff Z N N Z 2 2 1 1 Load impedance Effective load impedance seen by source Sinusoidal source tv law sFaraday 11 Eddy Currents When time varying magnetic flux flows in the ferromagnetic core an induced emf will result in accordance with Faraday s law This induced emf will produce currents in the conducting core normal to the magnetic flux Ohmic power loss and local heating If the eddy current is undesirable we can use core materials that have high and low or laminated cores to reduce eddy current power loss that is to make transformer cores out of stacked ferromagnetic sheets insulated 12 charged positively endother theleave andconductor theof end one rddrift towa oelectron t thecause willforce a field magnetic static ain velocity a with movesconductor aWhen B u field magnetic static ain conductor movingA 2 BuqFm Magnetic force 13 磁場靜止而導体移動所引生之emf稱為motional emf PS 積分 徑 之方向與 之面積方向有關 需以右手定則判斷 c circuit closed a around generated emf V emf Motional from motional 2 1 2 1 1221 motional l dBuv l d E l dBu l d EVVV Eq BuqF c m l dsd 14 Example 6 2 velocity abar with sliding themove toreguiredpower mechanical power electric that thisshow c in dissipatedpower find terminals between connected is resistanceA b ltagecircuit vo open Find a locity elocity veconstant v awith uniform ain rails conducting ofpair aover slidesbar metalA 0 0 u R RV u BaB z 15 W 1 to2 terminalfrom flow will current a connected is When the b Using a 2 0 2 0 0 1 2 0 1122 21120 R huB RIP R huB IR huB dlaBaua l dBu l dBuVVVV l dBu v e yzx c c Sol I 16 e x xm c m P R huB huIBuF P hIBaF F hIBaBl dIF BI cB l d IF 2 0 0 0m 0 1 2 force mechanical the in current a carries that circuit closed aon force magnetic the c 電 與dl反向 Conservation of energy 17 ltage circuit vo open theFind field magnetic uniform ain velocity a withrotating figure in theshown isgenerator disk Faraday The 0 BaB z V 22 flux magnetic cut the 34only Since Sol 2 0 0 2 0 0 0 4 3 00 113422 21120 bBr BrdrB draBaraV l dBu l dBuVVVV b b rz c Example 6 3 18 ccs ccc l dBu sd t B l d E l dBul dE l d E BuE E E BuEqF qFBE uq is on rcemagneticfo electro theexist and both reregion whe ain velocity a with moves charge aWhen Transformer emf 由磁場時變所引起 Motional emf 由電 在磁場中移動引起 field magnetic varying timeain circuit movingA 3 Eq 6 32 19 12 Eq 6 1 case the then motion in not iscircuit a If 34 Eq 6 V emf Induced c c vv dt d sdB dt d l dEv sdB dt d l dE l dBu sd t B sdB dt d s s css PS 上式在 time varying B 中同時適用於 stationary or moving circuit 之意義已涵蓋 time varying B 所造成之磁通變化 circuit 移動所造成之磁通改變 dt d 20 V 2 2 3 6 ExampleIn V 2 a 6 ExampleIn 2 0 2 00 0 bB dt d V b tB drrdBsdB huB dt d V hutBsdB o o s bt o o s o 21 hW S tS B dt d v thWB hWatBasdB dt d v a a tBaB Wh a ny x y y ere wh coscos cossin sin 12 6 Eq userest at is loop a Sol about locity angular vean with rotates loop b rest at is loop a loop in the emf induced Find with anglean makesinitially loop of normal The sin changing ain situated is loopr rectangulaA 0 0 0 0 Example 6 5 W 22 tS Btt sS B vv v tttSB tSBh tB W dxatBa W a dxatBa W a l d Bu l d Buv vvvv l d Busd t B l d E aat xyn xyn c a aaat csc 2cos sin co wheresinsin sinsin sinsin 2 2 0 sin 2 0 sin 2 a 326 Eq emf motional emf rtransforme 32 6 Eq use rotates loop when b 0 22 0 0 00 2 3 3 4 0 4 1 1 2 0 21432 得出已由前面 W 23 者之影響及之計算需同時包含但 及可直接用 相同結果 式由 另解 StatB sdB dt d v tSB tSB dt d dt d v tSBttSB tSBStatBt dt d v n t n PS 2cos 2sin 2 1 2sin 2 1 cossin cossin 34 6 s 0 0 00 0 W 24 Maxwell s Equations t J D t B E D E HD t D EB t B v v v 0 varyingtime varyingtime 電 散失 電荷隨時變 之減少 In time varying field Faraday 1831 Maxwell 1864 Static electric field Time varying field 根據 Equation of continuity 修正修正 電荷守恆 25 t D JH t D JD t J t JH HJ B JH v 0 0 0 In static magnetic field A time varying D will give rise to a magnetic field even in the absence of a current flow Ampere s law Not true in a time varying situation 需修正 Ampere s law with Maxwell s correction 位移電 密 t J v 26 Hertzby confirmedally Experinent 4 wave EMan also is light netism electromag called science one into optics magnetism y electricit combined Maxwell With this3 waves field eticeletromagn ofn propagatio theexplainingin crucial is termThis 2 charge ofon conservati with theconsistent be To 1 term thisneed wedoWhy 1979 1839 MaxwellClerk James of onscontributimajor theof one asequation w in the onintroducti its and density current nt displaceme called is H t D 27 0 BuEqF t J B D t D JH t B E v v B Faraday s law Ampere s law with Maxwell s correction Gauss s law for field Gauss s law for field Equation of continuity Lorentz s force equation Form the foundation of electromagnetic theory Maxwell equations D 微分型 for point 28 0 s s v v c c sc sdB dv sdD sd t D Jl dH sd t B l dE 電場在封閉線 上引生之電位 通過封閉線 包圍面上 磁通隨時間之變 磁場在封閉線 上引生之電 過此封閉線 上包圍面 之自由電 位移電 通過包住此體積之封閉面之電通 體積內總電荷 無isolated磁荷 往外通過任何封閉面之磁通 0 PS 積分型可由微分型推導而得 藉由 Stokes s theorem Divergence theorem 積分型 for entire region 29 wire thefrom disance aat Determine b in wirescurrent conduction the in current nt displaceme Verify the a capacitor a across connected is sin source voltagec aAn 1 1 0 rH C C tVVc Example 6 6 r 30 C A D c c C itVCtV d A sd t D i t d V d V E D d A C d A tVC dt dV C i cos cos iscurrent nt displaceme The sin ty permittivi of medium dielectric a and separation plate areaan with capacitor plate parallel aFor cos is in wirecurrent conduction The a Sol 010 0 1 011 D i 31 A m 2 cos cos2 RHSLHS or 0surfaceat or cos0surfaceat RHS 2 LHS b 01 01 21 2 011 2 1 r tVC H tVC rH S S iisd t D J S t VCisdJ D S rHdH sd t D JdH CD S C S c sc 皆可選 用安培 r 32 Electromagnetic Boundary Conditions 21 21 212 21 nn snn sn tt BB DD JH Ha EE 由積分型 Maxwell equation 可推得 3 8及5 9節 電場 線 tangential 分 續 V m 磁場 線分 差值造成表面電 電通法線 normal 分 差值造成表面電荷 A m C m2 磁通法線分 續 T 2 na 33 density current Surface s J density charge Surface s areaunit per charge C m charge 2 s da dq s lengthunit per current A m I J s 34 interface an across continous is field a ofcomponent normal The 4 exist charge surface a whereinterfacean across ousdiscontinu is field a ofcomponent normal The 3 existscurrent surface a whereinterfacean across ousdiscontinu is field an ofcomponent l tangentiaThe 2 interface an across continuous is field an ofcomponent l tangentiaThe 1 B D H E Physical meaning 35 nnnn nnnn t t tt t t tt ss HHBB EEDD B B HH D D EE J 221121 221121 2 1 2 1 21 2 1 2 1 21 0 0 0 medialosslessoBetween tw Example 0 0 ss J 36 0 0 0 0 0 0 21 212 212 21 nn nsn tsn tt BB DDa HJHa EE Between a dielectric medium 1 and a perfect conductor medium 2 2 na 37 Potential Functions 0 law sFaraday potential magnetic 0 0 0 t A E A t t B E AB B A B 代入 可表達為 定義一 旋 之散 必定為根據 38 coupled are and m V BEABE t A VE V t A E 有關皆與與 又因為梯 之旋 必定為0 可定義一 scalar electric potential V 由電荷分 佈 v引生 由time varying current J 所引生 39 ngsubstituti and From condition Lorentz 0 Let 2 2 2 2 2 2 t V AJ t A A t A t V J A A t A V t J A t A V E AB t E J B ED B H t D JH 40 If static Poisson s equations in static cases v v v v t V V t V A A t V t A V t A V EE D D J t A A 2 2 2 2 2 2 2 0 from Similarly Non homogenous wave equation for V 全部以V表示 Non homogeneous wave equation for A 全部以 A 表示 41 Solution of Wave Equations 0 1 Let 0 1 on depensnot symmetry spherical and on only depends origin at the chargepoint afor scoordinate sphericalIn 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 t U R U U R t R V R t t V R V R RR tRV t V V t V V v Solution of wave equations for potentials Non homogenous wave equation Homogeneous One dimensional homogeneous wave equation y z x P R 42 R v t RV vt u R t f R t Rf R R tV uR Rtf RRt tf u t R Rt RRt tftR tRU ttRR t RfR tU v p p p 4 origin at the chargepoint static a 29 3 1 1 1 1 or if thus later time aat at funtion heConsider t 式由 速 方向代表波傳播在 相同波形再現 可能解 1 See P 6 11 R Rtf tRV tt t R 43 distance aat felt be to ofeffect for the timeIt takes meearlier tian at density charge theof valueon the depends at time source thefrom distance aat potentialscalar The potentialscalar Retarded 4 1 volumea overon distributi charge a todue potential The RuR ut R tR dv R ut R R tV v vp p v pv 4 vuRt uRtf pv p Wb m 4 is potential vector retarded Thedv R ut RJ R tA v p Similarly y z x t R t R up 44 Time Harmonic Fields In engineering sinusoidal time functions occupy a unique position 1 Easy to generate 2 Periodic time functions Fourier series Non periodic time functions Fourier integrals 3 經過線性微分方程運算還是 sinusoidal 函 Example cos t ej t 45 The use of Phasor 電 中由於電感 電容的作用 使電壓 電 阻抗等電 成為一種有大小及相角的 稱為相 phasor 相 可用複 a jb 表示 根據尤 定 ej cos jsin 相 可用 e 的指 表示及簡化 運算 For a series RLC circuit with a sinusoidal voltage source v t and solvingfor onsmanipulati almathematic dComplicate cos sin 1 cos sin 1 equation Loop cos cos 0 00 00 I tVt C tRtLI tvidt C Ri dt di L tItitVtv 46 where Re Re Re sin cos Re cos 00 0 000 j s tj s tjjtj eIIeIeeIeI tjItItIt i ss tj s tj s tj s VI C LjR tvidt C Ri dt di L e j I dtti eIjeI dt d dt tdi 1 1 equation Loop Re Re Re 好處 It is much simpler to use exponential functions Is and Vs are phasors contain amplitude and phase information but are independent of t Phasor Re Re cos 0 00 tj s tjj eVeeVtVtv Is can be solved easily with cosine reference 47 Re tj se It i j x y zE dttzyxE x y zEj t x y z tE ex y zE x y z tE tj Re phasor vector 好處 The instantaneous current i t can be found by multiplying Is by ej t and taking the real part of the product For time harmonic electromagnetic fields Phasor expression 看 到t Instantaneous expression看到t 0 j s eI I Instantaneous expression Phasor expression 48 Example 6 7 Write the phasor expression Is for the following current functions using a cosine reference 2 0sin b and 30cos a o tIti tIti o o Sol 6 56 30 30 Thus Re 30cos a Re j o j o j os tjj o o o tj s eIeIeII eeItIti eIti o o 3 02 2 02 2 0 Re 2 2 0cos 2 0sin b j o jj os tjjj o oo eIeeIIeeeI tItIti 49 Example 6 8 Write the instantaneous expressions v t for the following phasors using a cosine reference 43 b and a 4 jV eVV s j os Sol 4 cos Re Re a 4 tV eeV eVtv o tjj o tj s 1 53cos 55Re 54343 b 1 53 1 53 3 4 tan22 s 1 otjj jj teetv eejV o o 50 0 0 H E EjJH HjE B D t D JH t B E v v j t tj ex y zEx y z tE Re Write time harmonic Maxwell s equation in terms of phasors with cosine reference 51 The time harmonic wave equation for V and A Non homogeneous Helmholtz s equation in phasor form v v v v VkV VV VjV t V V 22 22 22 2 2 2 r wavenumbecalled 22 pp u f u k JAkAJ t A A VkV t V V vv 22 2 22 2 2 2 2 2 52 dv R J A dv R V e R kR Rk jkRe dv R eJ RAdv R u R tJ tRA dv R e RVdv R u R t tRV v v v j