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					       ELEG 648
       Lecture #2

   Mark Mirotznik, Ph.D.
    Associate Professor
 The University of Delaware
    Tel: (302)831-4221
Email: mirotzni@ece.udel.edu
  Maxwell’s Equations in Differential Form

             
         B 
 E         M         Faraday’s Law
           t
           
      D  
 H          J c  J i Ampere’s Law
         t
    
D                      Gauss’s Law
    
  B  m                 Gauss’s Magnetic Law
        Faraday’s Law
         
        B
                               
                            B
        t        E  
                             t
                                    
                 c E  dl   t s B  ds
C

    S        
             E
         Ampere’s Law

               
              J
D
t
                                    
                          D
                    H  J 
                                  t
                                               

                   c H  dl  t s D  ds  s J  ds
J          
           H
     
     H
       Gauss’s Law


                  
             D  
Qtot              
             s D  ds v  dv  Qtot


 
 D
Gauss’s Magnetic Law


                     
                B  0
                     
                s B  ds 0
             “all the flow of B entering the
             volume V must leave the volume”



   
   B
Current Continuity Equation


                           q
                    J  
                            t

                 “Charge must be conserved”




      
      J
CONSTITUTIVE RELATIONS

         r o=permittivity (F/m)
D  E
           o=8.854 x 10-12 (F/m)
         r o=permeability (H/m)
BH       o=4p x 10-7 (H/m)

      
Jc   E   =conductivity (S/m)
                            POWER and ENERGY
                      
                  H
(eq1)   E            M d
                    t                                                                              n
                    
                 E                    
(eq 2)   H           E  Ji  J d  Jc  Ji
                  t
                      
     take H  (eq1)  E  (eq 2)                                                             E, H
                                                                                 Ji                 S
                                                
(eq3) H    E  E    H   H  M d  E  ( J d  J c  J i )                        V
Using the vector identity   ( A  B)  B  (  A)  A  (  B)
                                                                               , , 
(eq 4)   ( E  H )  H  M d  E  ( J d  J c  J i )  0
Integrate eq4 over the volume V in the figure
                                                           
(eq5)    v   ( E  H ) dv  v [ H  M d  E  ( J d  J c  J i )] dv

   Applying the divergence theorem
                                                  
                                    H       E         
 (eq6) s ( E  H )  ds  v [  H      E      E  E  E  J i )] dv  0
                                         t       t
              POWER and ENERGY (continued)
                                                  
                                   H        E          
(eq6) s ( E  H )  ds  v [  H       E      E  E  E  J i )] dv  0
                                        t        t
                                              
    H   1            2                E   1          2                 
H         H   wm ,  E                      E   we ,  E  E   E
                                                                                       2
      t t  2            t                t t  2           t
                                w     w             
(eq7) s ( E  H )  ds  v [ m  e ] dv  v [ E  J i ] dv  v  E dv  0
                                                                               2
                                   t     t
                          
                                                       
(eq8) s ( E  H )  ds  v [ we  wm ] dv  v [ E  J i ] dv  v  E dv  0
                            t
                                                                                2


                            
                Ps  s ( E  H )  ds
                             1                         1
                 Wm  v [  H ] dv , We v [  E ] dv
                                        2                     2

                             2                         2
                              
                 Pi  v [ E  J i ] dv  0, Pd  v  E dv  0
                                                            2


               Stored magnetic power (W)       Supplied power (W)


                                         
                            Ps       Wm  We  Pi  Pd
      What is this term?           t     t
                                                                Dissipated power (W)
                                Stored electric power (W)
     POWER and ENERGY (continued)
                        
            Ps  s ( E  H )  ds
                       1                         1
           Wm  v [  H ] dv , We v [  E ] dv
                                  2                     2

                       2                         2
                        
           Pi  v [ E  J i ] dv  0, Pd  v  E dv  0
                                                      2


         Stored magnetic power (W)       Supplied power (W)


                                       
                          Ps       Wm  We  Pi  Pd
What is this term?               t     t
                                                          Dissipated power (W)
                          Stored electric power (W)


     Ps = power exiting the volume through radiation
       
     S  E  H W/m2 Poynting vector
                 TIME HARMONIC EM FIELDS
  Assume all sources have a sinusoidal time dependence and all materials
  properties are linear. Since Maxwell’s equations are linear all electric
  and magnetic fields must also have the same sinusoidal time dependence.
  They can be written for the electric field as:
                                                         
         Euler’s Formula              E ( x, y, z , t )  Eo ( x, y, z ) cos( t   ( x, y, z ))
                                       
 e jt  cos(t )  j sin(t )                                 ~
                                      E ( x, y, z , t )  Re[ E ( x, y, z ) e jt ]
  ~
  E ( x, y, z ) is a complex function of space (phasor) called the time-harmonic electric
field. All field values and sources can be represented by their time-harmonic form.
                                                    ~
                           E ( x, y, z , t )  Re[ E ( x, y, z ) e jt ]
                                                     ~
                           D( x, y, z , t )  Re[ D( x, y, z ) e jt ]
                                                     ~
                           H ( x, y, z , t )  Re[ H ( x, y, z ) e jt ]
                                                    ~
                           B ( x, y, z , t )  Re[ B ( x, y, z ) e jt ]
                                                   ~
                           J ( x, y, z , t )  Re[ J ( x, y, z ) e jt ]
                           
                            ( x, y, z , t )  Re[  ( x, y, z ) e jt ]
                                                     ~
PROPERTIES OF TIME HARMONIC FIELDS


                           ~                               ~
Time derivative:       [Re[ E ( x, y, z ) e jt ]]  j[Re[ E ( x, y, z ) e jt ]
                    t




                            ~                          1      ~
Time integration:     [Re[ E ( x, y, z ) e jt ]dt     [Re[ E ( x, y, z ) e jt ]
                                                      j
TIME HARMONIC MAXWELL’S EQUATIONS
             
     
 E  
           B
                                       
                                       ~
                                                  
                                                  t
                                                          ~
                                 Re E e jt   Re B e jt          
           t
           
      D 
 H         J                       
                                       ~ j t    
                                                  
                                                 t
                                                        ~
                                                            ~
                                 Re H e  Re D e jt  Re J e jt          
         t
    
D                                        
                                 Re D e jt  Re e jt 
                                      ~             ~
    
  B  m                                   
                                 Re B e jt  Re e jt 
                                      ~             ~
                                                       m


Employing the derivative property results in the following set of equations:
                             ~       ~
                          E   j B
                             ~      ~ ~
                          H  j D  J
                            ~ ~
                        D  
                            ~ ~
                          B  m
             TIME HARMONIC EM FIELDS
    BOUNDARY CONDITIONS AND CONSTITUTIVE PROPERTIES



The constitutive properties and boundary conditions are very similar
for the time harmonic form:
                                        General Boundary Conditions
    Constitutive Properties                      ~ ~
         ~     ~                          n  ( E2  E1 )  0
                                          ˆ
         D  E                                  ~     ~      ~
         ~      ~                         n  ( H 2  H1 )  J s
                                          ˆ
         BH                                   ~     ~
         ~      ~                         n  ( D2  D1 )   s
                                          ˆ                  ~
         Jc   E                               ~ ~
                                          n  ( B2  B1 )  0
                                          ˆ
                                        PEC Boundary Conditions
                                                   ~
                                              n  E2  0
                                              ˆ
                                                   ~     ~
                                              n  H2  Js
                                              ˆ
                                                  ~
                                              n  D2   s
                                              ˆ         ~
                                                  ~
                                              n  B2  0
                                              ˆ
         TIME HARMONIC EM FIELDS
               IMPEDANCE BOUNDARY CONDITIONS


If one of the material at an interface is a good conductor but of finite
conductivity it is useful to define an impedance boundary condition:

         2,2,2

                                                  
    1,1,1          Z s  Rs  jX s  (1  j )
                                                  2
  1>> 2             ~        ~             ~                ~
                      Et  Z s J s  Z s n  H  (1  j )
                                         ˆ                   n H
                                                             ˆ
                                                          2
  POWER and ENERGY: TIME HARMONIC


                       ~ ~
            Ps  s ( E  H * )  ds
                       1 ~2                     1 ~2
            Wm  v [  H ] dv , We v [  E ] dv
                       4                        4
                       1~ ~                         1 ~2
            Pi  v [ E  J i* ] dv  0, Pd  v  E dv  0
                       2                            2


                  Time average magneticenergy (J)     Supplied complex power (W)



               Ps  j 2 (Wm  We )  Pi  Pd
                                  Time average electric energy (J)   Dissipated real power (W)
Time average exiting power
      CONTINUITY OF CURRENT LAW


                                            
        B                                D                    
 E                   (  H )    [     J ]  [  D]    J
         t                                 t        t
                                                 
     D               vector identity   (  A)  0
 H       J
        t                                 
                      0  [  D]    J
D                       t
  
B  0                                   
                         0  []    J
                              t
                                     
                            J  
                                      t

                              
            time harmonic   J   j
                                        SUMMARY
                  Time Domain                                       Frequency Domain
                                       
             B                  D                  ~        ~ ~                  ~     ~ ~
       E    M            H        J           E   j B  M             H  j D  J
              t                      t                ~ ~                           ~ ~
                                                   D                          B  m
      D                    B  m

                                                       ~ ~                  ~     ~      ~
     n  ( E2  E1 )  0 n  ( H 2  H1 )  J s
     ˆ                     ˆ                         n  ( E2  E1 )  0 n  ( H 2  H1 )  J s
                                                     ˆ                     ˆ
                                                       ~    ~      ~ ˆ ~ ~
     n  ( D2  D1 )   s n  ( B2  B1 )  0
     ˆ                       ˆ                       n  ( D2  D1 )   s n  ( B2  B1 )  0
                                                     ˆ
                                                                                  
                                                     Z s  Rs  jX s  (1  j )
                                                                                  2
                                                                   ~     ~
                     D  E                                          D  E
                                                                   ~      ~
                     BH                                            BH
                                                                   ~      ~
                     Jc   E                                        Jc   E
                                                              ~ ~
Ps  s ( E  H )  ds                              Ps  s ( E  H * )  ds

                                                              1 ~2                     1 ~2
             1         2                1
Wm  v [  H ] dv , We v [  E ] dv
                                             2       Wm  v [  H ] dv , We v [  E ] dv
            2                         2                       4                        4
                                                              1~ ~                         1 ~2
Pi  v [ E  J i ] dv  0, Pd  v  E dv  0
                                           2
                                                     Pi  v [ E  J i* ] dv  0, Pd  v  E dv  0
                                                                2                            2
   Electromagnetic Properties of Materials

Primary Material Properties   Secondary Material Properties

     Electrical Properties         Electrical Properties
r o=permittivity (F/m)    n  r          Index of refraction
o=8.854 x 10-12 (F/m)
                                 e           Electric susceptibility
 =conductivity (S/m)

  Magnetic Properties                 Magnetic Properties
r o=permeability (H/m)       m          Magnetic susceptibility
o=4p x 10-7 (H/m)
            Electric Properties of Materials

                       Qi Eext
                        +                                 ++++
                                                 Eext     - - - -
      +                             li                    ++++        Eext
                        -                                 - - - -
No external field                                         ++++
                             Eext                       - - - -
                                                Bulk material (N molecules)
                Applied external field


                                        Net dipole moment or    N       N   
    Electric dipole moment   
                             pi  Qili   polarization vector:   P   pi   Qili
    of individual atom or                                           i 1     i 1
    molecule:
Electric Properties of Materials (continued)

           ++++
  Eext                          N       N   
           - - - -
           ++++      Eext      P   pi   Qili
                                    i 1     i 1
           - - - -
           ++++                         
         - - - -               D  oE  P
                                                  
                                  o E  o eE
 Bulk material (N molecules)

                                                  
         ?                       o (1   e ) E
         P oeE                         
                                   o r E
                                  r 1   e
  What are the assumptions                      Static permittivity
  here?                                         or relative permittivity
Electric Properties of Materials (continued)
               Conductivity
              E
                                                 J=current density=qvvz where
                                                 qv=volume charge density and
                                                 vz= charge drift velocity
         y             J
                                                 When subjected to an external
                                                 electric field E the charge
                                                 velocity is increased and is
                                                 given by
z            x              E
                                                 ev    e vq



Where e is called the electron            E 
mobility. The current density is           
                                        e  vq   J
                                                            Where  is called the
                                      E                     conductivity. Its units are S/m
thus given by                                     

                                                     Material         Conductivity (S/m)
                  Where  is called the resistivity. Silver            6.1 x 107
     

     1
         m                                           Glass           1.0 x 10-12
                                                       Sea Water       4
 Electric Properties of Materials (continued)
1.    Orientational Polarization: molecules have a slight polarization even in the
absence of an applied field. However each polarization vector is orientated randomly
so the net P vector is zero. Such materials are known as polar; water is a good example.

    rH 2O  81

2.   Ionic Polarization: Evident in materials ionic materials such as NaCl. Positive and
negative ions tend to align with the applied field.




3.    Electronic Polarization: Evident in most materials and exists when an applied
field displaces the electron cloud of an atom relative to the positive nucleus.
              Magnetic Properties of Materials
   Ii   Mi


                                                                 Bext                               Bext
                                    Net magnetic dipole
                                    moment or
                                    magnetization vector:
                                       N         N
                                M   M i   I i dsi ni
                                                      ˆ
                                        i 1      i 1
No magnetic field: random                                         Applied external magnetic field
oriented magnetic dipoles

   Magnetic dipoles randomly oriented                       Magnetic dipoles tend to align with
   resulting in zero net magnetization vector:              external magnetic field resulting in non-zero
                                                            net magnetization vector:
              N                                                    N
             M   I i dsi ni  0
                           ˆ                                       M   I i dsi ni  0
                                                                                 ˆ
                   i 1                                                    i 1
Magnetic Properties of Materials (continued)

                                        N        N
                                       M   M i   I i dsi ni
                                                             ˆ
 Bext                        Bext           i 1      i 1

                                                
                                       B  o H  M
                                                           
                                          o H  o  m H
                                                          
                                           o (1   m ) H
 Applied external magnetic field

         ?                                       
        M  o  m H                      o r H

                                     r 1   m
    What are the assumptions                       Static permeability
    here?                                          or relative permeability
Magnetic Properties of Materials (continued)
1.    Diamagnetic: Net magnetization vector tends to appose the direction of the
applied field resulting in a relative permeability slightly less than 1.0
Examples: silver (r=0.9998)




2.    Paramagnetic: Net magnetization vector tends to align in the direction
of the applied field resulting in a relative permeability slightly greater than 1.0
Examples: Aluminum (r=1.00002)




3     Ferromagnetic: Net magnetization vector tends to align strongly in the direction
of the applied field resulting in a relative permeability much greater than 1.0
Examples: Iron (r=5000)



.
                  Classification of Materials
1.    Homogenous or Inhomogenous: If the material properties are independent of spatial
location then the material is homogenous, otherwise it is called inhomogenous

                         ( x, y, z)  Inhomogenous

2.    Isotropic or Anisotropic: If the material properties are independent of the polarization
of the applied field then the material is isotropic, otherwise it is called anisotropic.
                 Dx   xx    xy    xz   Ex 
                 D                                    
                                yy    yz   E y   D   E     anisotropic
                 y   yx                    
                 Dz   zx
                             zy    zz   E z 
                                            
3.   Linear or non-Linear: If the material properties are independent on the magnitude
and phase of the electric and magnetic fields, otherwise it is called non-linear
                                     2      3
                  D   o E   o E  1E   3 E  ...
                    Classification of Materials
 4.    Dispersive or non-dispersive: If the material properties are independent of frequency
 then the material is non-dispersive, otherwise it is called dispersive
                     t=t1                    t=t2
                    +++ +                                             t=t3
                                            ++++
                    ----                    ----                      ++++
                    +++ +                   ++++                      ----
                                                                      ++++
                                            -- --                     ----
                                                                      ++++
                    ----                    ++++                      ----
                    +++ +                   ----
                    ----                                                     -
                                                                               -
                                                   -                      -
                                                 --
                                             - - -+                     -
                                                                        - - -+ -
                    ----  -                    -                            +
                    - - -+
                    ++++-                    -+ + +                       +
                    --                      +++                        ++++
                                                                       +
                    ++++
                    +++ -                    -
                                            + - -                       ---
                                            + +++  -
                                                                       + + ++-
                     t=t4                   t=t5
                                                                        t=t6
A material’s atoms or molecules attempt to keep up with a changing electric field. This results
in two things: (1) friction causes energy loss via heat and (2) the dynamic response of the
molecules will be a function of the frequency of the applied field (i.e. frequency dependant
material properties)
       Electric Properties of Materials
 Frequency Behavior (Complex Permittivity)
    ~        ~
  E   j B                     ~            ~
                                   D   * ( ) E         Dielectric
    ~       ~ ~ ~                                         constant                        loss term
  H  j D  J i  J c           ~        ~
   ~ ~                             B  o H
D  
   ~
                                   ~
                                   Jc   s E
                                             ~           * ( )   ( )  j ( )
B  0
                                                      is called the complex permittivity
             ~            ~
          E   j  o H                                ~           ~
              ~          ~ ~         ~                 E   j  o H
          H  j  * E  J i   s E                     ~                  ~ ~           ~
                                                       H  j (   j ) E  J i   s E
                 ~ ~
          ( * E )                                        ~ ~
                                                       ( * E )  
              ~
          (H )  0                                      ~
                                                       (H )  0
               ~            ~
            E   j  o H
                ~               ~ ~
            H  j  eff ( ) E  J i
                                                                                     s
                   ~ ~                         eff ( )   ( )  j ( )  j
            ( * E )                                                             
               ~
            (H )  0
Frequency Behavior of Sea Water
      Electric Properties of Materials
Frequency Behavior (Complex Permittivity)
     ~           ~                               ~            ~
  E   j  o H                            E   j  o H
      ~                  ~ ~           ~          ~          ~ ~                    ~
  H  j (   j ) E  J i   s E      H  j   E  J i  ( s    ) E
         ~ ~                                         ~
  ( * E )                                ( * E )  
                                                           ~
     ~                                           ~
  (H )  0                                  (H )  0

         ~            ~                         ~            ~
      E   j  o H                       E   j  o H
          ~          ~ ~           ~             ~ ~ ~ ~
      H  j   E  J i   eff E         H  J d  J i  J eff
             ~ ~                                    ~ ~
      ( * E )                           ( * E )  
         ~                                        ~
      (H )  0                             (H )  0
     eff ( )   s   a   s       ~             ~
                                            J d  j   E    Displacement current
                                            ~
                                            Ji                Source current
                                            ~             ~
                                            J eff   eff E   Effective electric
                                                              conduction current
      Electric Properties of Materials
Frequency Behavior (Complex Permittivity)
                                                  ~            ~
        ~            ~                          E   j  o H
     E   j  o H
         ~          ~ ~           ~                 ~                    ~ ~
     H  j   E  J i   eff E             H  j   (1  j eff ) E  J i
            ~ ~
                                                                    
     ( * E )                                      ~
                                                ( * E )  
                                                             ~
        ~
     (H )  0                                    ~
                                                (H )  0
    eff ( )   s   a   s   

     ~            ~                                ~             ~
  E   j  o H                              E   j  o H
      ~                             ~ ~             ~ ~        ~
  H  j   (1  j tan(  eff )) E  J i     H  J cd  J i
         ~ ~                                           ~
  ( * E )                                  ( * E )  
                                                             ~
     ~                                             ~
  (H )  0                                    (H )  0
                               eff             ~                       ~
             tan(  eff )                      J cd  j (1  j eff ) E
                                                                
      Electric Properties of Materials
Frequency Behavior (Complex Permittivity)
     ~            ~
  E   j  o H
      ~ ~ ~ ~                                Good Dielectric
  H  J d  J i  J eff                               
                                          ~      ~
         ~ ~                              J d  J eff ( eff  1)
  ( * E )                                           
       ~
  (H )  0
 ~             ~
 J d  j   E    Displacement current
 ~
 Ji                Source current
 ~             ~
 J eff   eff E   Effective electric        Good Conductor
                   conduction current     ~        ~ 
                                          J eff  J d ( eff  1)
    eff ( )   s   a   s                     

				
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