Lecture 11 Maxwell's Equations

1. Maxwell's Equations 2. Power and Energy in Electromagnetic Fields 3. Poynting's Theorem 4. Impedance and Admittance 5. Wave Equation 6. Planewave 7. Coding Example

1 1. Maxwell's Equations

. Maxwell's equations (ME) . Materials electrical properties

 D  (Gauss's law) DE  (dielectric property) B E  (Faraday's law) BH  (magnetic property) t JE  (conductive property) B 0 (Gauss's law)

D HJ   (Ampere's law) t

. ME in terms of E and H  E  /

H E   (Faraday's law) t H 0 (Gauss's law)

E HE   (Ampere's law) t 2 . Sinusoidal field

E  E0 cos(t  e )  H H0 cos(t h )   1 jdt and tj

je  EE 0e : electric field phasor

j h HH 0e : magnetic field phasor

. ME in phasor form

E  /

EH j  H 0 HE ()  j

3 2. Power and Energy in Electromagnetic Fields . Power in terms of voltage and current

DC: 11Q2 WVQ (dc capacitor stored energy) 22C dW12 Q dQ PVI  (dc capacitor power) dt2 C dt AC:

vt() V00 cos( t vi ),() it I cos( t ) jjvi VVeIIe00, 1 pt() vi VI cos(  )  cos(2 t  ) 2 00 vi vi 1 PVI dc2 0 0 1 PVI cos(2t  ) ac2 0 0 vi 11T Ppav  pdVI()  0 0 cos(vi  ) T 0  2  11 PVIVIRe(** ) Re( ) av 22 11 PVIVI** (complex power) 22 4 . Power loss in dielectric materials 11q2 wvq (capacitor stored energy) 22C dv 11 1 pviiC,()()||  P VI** VjCV  jCV 2 dt 22 2  S Cj() d 11SS PVjV( )||22 (  )|| 22dd 1|V |2 1 1 PjCVYV||22  || 222R 1 YjC R   dd R  (resistance in parallel with capacitance) () SS S C   (capacitance) d 5 . Complex permittivity and electric loss tangent

    j  (complex permittivity)  tan  (loss tangent)  

=0r : accounts for dielectric polarization  : accounts for dielectric loss 8.854 1012 F/m (permittivity of vacuum) 0  r : relative permittivity = dielectric constant 

, : materials with dielectric polarization and dielectric loss

,  : materials with dielectric polarization and conduction loss

6 . Power loss in magnetic materials

11 wi Li2 (magnetic energy stored in an inductor) 22 di 11 1 pvivL,()()|| P IV** IjLI  jLI 2 dt 22 2 S Lj( ) (inductance of a torroidal inductor) d 11SS PIjI( ) ||22 (  ) || 22 dd 111 PRIjLI||22 || ZV||2 222 ZRjL  S R   (resistor in series with a lossless inductor) d S L    (inductance) d 7 . Complex permeability and magnetic loss tangent

    j  (complex permeability)  tan (magnetic loss tangent) m  

=0r : accounts for magnetization of a material  : accounts for magnetic loss of a material 4 107 H/m (permeability of vacuum) 0  r : relative permeability

8 . Power density of electromagnetic fields - Use an ideal parallel-plate capacitor to understand the concept

VEd IWH 11 PIVVI** 22 11 PHEWdEHWd**() () 22 11 SHEEH** (complex power density of electromagnetic field) 22 11 SEHEH** (Poynting vector = electromagnetic power density) 22 S: complex power density vector (in the direction of wave propagation)

9 . Stored energy in electric field

11S 1 wCv222( EdESd )  ( ) : parallel-plate capacitor 22d 2 11 wDEDE (J/m3 , electric field energy density) 22 11 1 WDEE**= ( jE ) ( jE ) | | 2 (phasor for electric energy density) 44 4   1 1 PjW(2 ) ( )| E |2  jE ( ) | |2 (phasor for electric power density) 2 2 111 PEEJE( ) | |22 | | (electric loss power density) r 222 1 PE ( ) | |2 (stored electric power density) i 2

10 . Stored energy in magnetic field

11S 1 wLi222( HdESd )  ( ) : torroidal inductor 22d 2 11 wBHBH (J/m3 , magnectic field energy density) 22 11 1 WBEH**= ( jE ) ( jH ) | | 2 (phasor for magnetic energy density) 44 4   11 PjW(2 ) ( )| H |2  j( ) |H |2 (phasor for magnetic power density) 22 1 PH ( ) | |2 (magnetic loss power density) r 2 1 PH ( ) | |2 (stored magnetic power density) i 2

11 3. Poynting's Theorem . Power balance equation in electromagnetic field

PPPjsrl 2( WW m  e )

1 *** Psss(EJ   H  M )dV : source power in V 2 V

1 * Pdr EH S: power radiated through S 2 S

1 22 Pl [( )EHdVV   ] : power loss in 2 V

1 2 WHdVm    : stored magnetic energy 4 V

1 2 WEdVe   : stored electric energy 4 V 

12 4. Impedance and Admittance . Impedance and admittance in terms of terminal current and terminal voltage

2Pin PPjrl2( W m  W e ) Zin  : input impedance ||II22 ||/2 I : terminal current

2Pin PPjrl2( W m  W e ) Yin  : input admittance ||VV22 ||/2 V : terminal current

13 5. Wave Equation . Electromagnetic field in charge-free region

  0 (in charge-free region) EEH 0,  j

HHE 0,   j 

EEE ()  22 j  HE   (22 )E  0: wave equation for   electric field Similarly,  (22 )H  0: wave equation for magnetic field

14 6. Planewave . Planewave: one-dimensional wave

 0, 0, 0 xyz

E E E E x y z 00 E  xyz z

d 2 2 dz2

d 2 ()22 EE   2  0 2 dz

EEeejkz E  jkz

Eezjkz : wave propagating in + direction

Eezjkz : wave propagating in  direction 2 k  ( , real) : propagation constant    : wavelength  15 . Intrinsic impedance

 jk z Ex Eex ˆ

kzˆ  ˆ : direction of propagation

11 HEzE ˆ  jjz 

1  jk z yˆ() jk Ex e j 

k jk z1 jk z HyEexxˆˆ Ee y  kEˆ  H     : intrinsic impedance k     0 0 376.73  0

EHk ˆ

16 7. Coding Example

Planewave

Input: f, ur, εr Calculate: k, λ, η

  2 f   7 00r ,410 H/m  12 00r , 8.854 10 F/m k    2/k

   

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