Lecture 2 Review of Maxwell s Equations, EM Energy and Power

Transcription

1 Lecture 2 Review of Maxwell Equation, EM Energy and Power Optional Reading: Steer Appendix D, or Pozar Section 1.2,1.6, or any text on Engineering Electromagnetic (e.g., Hayt/Buck)

2 ime-domain Maxwell Equation: Faraday Law Faraday law in integral form V electromotive force (EMF) d E dt d E d l d dt B C S C Are the two expreion below equivalent? If not, can you tate what the relationhip i between the two? d? d B Hayt/Buck, ection 9.1 B d dt t SC SC Hint: Conider both the tranformer EMF and the motional EMF. Faraday law in differential (point-wie) form ( E) d B B d E t t S C S C ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 2

3 ime-domain Maxwell Equation: Ampère Law Ampère law in integral form D H d l J d I I I t C S C Ampère law in differential (point-wie) form J D H J E J t Show that Ampère law of magnetotatic doe not hold for timevarying EM field (it i inconitent with the conervation of charge).? H J J t Show that Ampère law with Maxwell correction oberve the conervation of charge for time-varying field. J ource D D conduction diplacement Maxwell correction: diplacement current ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 3

4 All Four ime-domain Maxwell Equation the curl ME Faraday Law Ampère Law Gau Law of Electricity integral form E d l B d t C S C D H d l J d t C S C D d vdvq S V S free differential form E B t D i H EJ t D v J Gau Law of Magnetim B S d 0 B 0 Gau law follow from the conervation of charge and the curl ME ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 4

5 ime-domain Maxwell Equation and Charge Conervation Prove that Gau law of electricity follow from Ampère law and the conervation of electrical charge. Prove that Gau law of magnetim follow from Faraday law. ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 5

6 Contitutive Relation Maxwell equation are 4 but only 2 of them are independent (the two curl equation) there are 4 unknown vector in the 2 curl Maxwell equation we need two more vector equation for a complete olution the contitutive EM equation are eential in decribing the EM field interaction with matter in vacuum (SI): D 0E, J 0, B0H H/m F/m , c m/ c 0 in matter: D F ( E, H), J F ( E, H), B F ( E, H) p c m ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 6

7 Contitutive Relation (2) in microwave engineering we often aume that material are iotropic, linear and diperion-free: D0EP 0 re, r 1e J E B HM H, r r m the above aumption doe not hold for many material, for example water and living tiue plama magnetized plama ferrite piezoelectric crytal ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 7

8 Contitutive Relation (3) Decribe how the contitutive relation of the following type of material are decribed mathematically heterogeneou nonlinear aniotropic bi-aniotropic diperive Each of the material lited a example in the previou lide i either heterogeneou, or nonlinear, or aniotropic, or diperive. Which property applie to which material? ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 8

9 ime-harmonic EM Analyi: Field Phaor time-domain field vector E( xyzt,,, ) Re E ( xyze,, ) H( xyzt,,, ) Re H ( xyze,, ) jt jt E( x, yzt,, ) xˆ E( xyz,, )co[ t ( xyz,, )] x yˆ E ( x, y, z)co[ t ( x, y, z)] y zˆ E ( x, y, z)co[ t ( x, y, z)] z vector-field phaor (frequency-domain field vector) j x ( x, y, z) E xˆ E ( x, y, z) e E xˆ E ( x, y, z) x yˆ E ( x, y, z) e y zˆ E ( x, y, z) e z j ( x, y, z) y j ( x, y, z) z x z y ( xyz,, ) ye x y ( x, y, z) zˆ E ( x, y, z) z ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 9

10 Maxwell Equation in Phaor Form phaor of the time-derivative of a function f f( x, y, z, t) F( x, y, z) t ( xyzt,,, ) f F patial derivative, x, yz, jf( x, y, z) time domain B E t D H EJ t D v B 0 frequency domain E jb H jd E J uually given a jd only D v B 0 ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 10

11 he Continuity Relation in Phaor Form Write the continuity equation in phaor form. J t Prove that the equation H j E E J D where the permittivity i a complex number uch that j, (εꞌ and εꞌꞌ being poitive real) can alo be written a H j E J where j j ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 11

12 Contitutive Relation in the Frequency Domain dielectric polarization and complex permittivity D E, j or ( j) (1 j tan ), tan / E( x, yzt,, ) eˆ Exyz (,, )co( t) D( xyzt,,, ) dˆ Exyz (,, )co( t ) magnetization and complex permeability B H, j or 0( r jr) (1 j tan ), tan / Why are the imaginary part of ε and μ negative? d m 0 d r m r d Im E Im H E H d m j E D 0 Re j H B 0 Hint: Conider a capacitor. he flux denity D relate to the charge Q on the plate while E relate to the voltage V applied to the plate. ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 12 Re

13 ime-average Quadratic Value of Vector Field time-average energy (quadratic) value of harmonic field av E() t E() t dt [ E 0 0 x()] t [ Ey()] t [ Ez()] t dt E 0 Mxco ( t x) EMyco ( t y) EMzco ( t z) dt ( EMx EMy EMz) E E E Note: EE EE x x EE y y EE z z EMx EMy EMz E root-mean-quare (RMS) value rm 1 E E E 2 2 av EE dt 0 ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 13

14 ime-average Quadratic Quantitie: Stored Energy energy denity a a function of time (ame a in tatic) we() t E() t D() t E() t E(), t J/m electric energy 2 2 in an iotropic medium 1 1 wm() t H() t B() t H() t H(), t J/m 2 2 time-average (tored) energy denity (harmonic field) 1 1 E ( E ) Re{ E D } we,av w 0 e() t dt E() t E() t dt J/m H ( H ) Re{ H B } 3 wm,av w 0 m() t dt J/m magnetic energy 3 ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 14

15 ime-average Quadratic Quantitie: Diipation (Power Lo) diipated power denity a a function of time (Joule law in differential form) p () t J () t E() t E() t E(), t W/m d in an iotropic medium time-average diipated power denity (harmonic field) 1 1 E ( E ) Re{ E J } pd,av p () t dt E() t E() t dt W/m 2 2 d ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 15

16 ime-average Quadratic Quantitie: Power ranfer power-flux denity a a function of time S() t E() t H(), t W/m 2 Poynting vector time-average tranferred power denity (harmonic field) E() t eˆ E0 in( t) 1 1 ˆ Sav S() tdt () t () tdt E 0 H H() t hh 0 0 in( t) 1 Sav ( eˆ hˆ ) EH 0 0 in( t) in( t) dt 0 ˆ 1 ( eˆ h) EH 0 0 co co(2 t) dt 2 0 ( eˆ hˆ ) EH co co(2 t ) dt 2 0 ( ˆ ˆ eh) EH 0 0co Re( E H 0 ) W/m

17 ime-average Quadratic Quantitie: Source Power ource power denity a a function of time p t J t E t M t H t () () () () () time-average ource power denity (harmonic field) E() t eˆ E0 in( t) ˆ J () t j J in( t) 1 av eˆ ˆj ˆ eˆ j EJ 0 0 E J p ( ) EJ in( t) in( t) dt 0 fictitiou ( ) co Re( ) W/m

18 Putting hing ogether: Poynting heorem power-balance equation of an EM ytem (derived from ME): relate quadratic quantitie time domain (all vector quantitie are function of time) o differential form 1 1 E() t H() t () t () t () t () t t D E 2 t B H 2 J () t E() t () t () t () t () t E J H M, W/m o integral form E() t H() t d () t () t () t () t dv S t V D E B H S 2 2 J () t E() t dv () t () t () t () t E J H M dv, W term 1 term 2 term 3 V S S term 4 term 5 V ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 18

19 Putting hing ogether: Poynting heorem 2 Decribe with one entence the phyical meaning of term 1, 2, 3, 4 and 5 in Poynting time-domain theorem in l. 18. ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 19

20 Putting hing in Perpective: Poynting heorem 3 frequency domain o differential form 0.5 ( E H ) 0.5 j( H B E D ) 0.5 E J H M 2 H E explicit active (lo) and reactive (exchange) power term term 1 term 2 term 3 term 4 Re( E H ) E E H H E E Re E J H M Im( E H ) j H H E E Im E J H M term 5 term 6 term 7 Decribe with one entence the phyical meaning of term 1, 2, 3, 4, 5, 6 and 7 in Poynting frequency-domain theorem above. ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 20

21 Summary the 4 Maxwell equation (2 curl and 2 divergence) form the bai of EM and of microwave engineering the 2 div equation (the Gau law) follow from the curl equation and the continuity of charge phaor are ued in harmonic (ingle-frequency) field analyi the imaginary part of the complex permittivity and permeability mut be negative (or zero) indicating lo (no lo) (a poitive imaginary part would indicate a gain material!) quadratic field quantitie uch a energy and power a well the root-mean-quare value are conveniently calculated uing the product of phaor Poynting theorem decribe the power balance in EM ytem ElecEng 4FJ4 Nikolova LECURE 02: MAXWELL'S EQUAIONS: REVIEW 21