Bài giảng Engineering electromagnetic - Chapter XII: The Uniform Plane Wave - Nguyễn Công Phương

Nguyễn Công Phương Engineering Electromagnetics The Uniform Plane Wave Contents I. Introduction II. Vector Analysis III. Coulomb’s Law & Electric Field Intensity IV. Electric Flux Density, Gauss’ Law & Divergence V. Energy & Potential VI. Current & Conductors VII. Dielectrics & Capacitance VIII.Poisson’s & Laplace’s Equations IX. The Steady Magnetic Field X. Magnetic Forces & Inductance XI. Time – Varying Fields & Maxwell’s Equations XII. The Uniform Plane Wave XIII.Plane Wave Reflection & Dispersion XIV.Guided Waves & Radiation Uniform plane wave - sites.google.com/site/ncpdhbkhn 2 The Uniform Plane Wave 1. Wave Propagation in Free Space 2. Wave Propagation in Dielectrics 3. The Poynting Vector 4. Skin Effect 5. Wave Polarization Uniform plane wave - sites.google.com/site/ncpdhbkhn 3 Wave Propagation in Free Space (1) E H   0 t H E   0 t .E  0 .H  0 Uniform plane wave - sites.google.com/site/ncpdhbkhn 4 Wave Propagation in Free Space (2) E Ex a x Ex  E( x , y , z )cos( t   ) ej t cos t  j sin  t j() t    j  j  t  Ex Re Exyze (,,)   Re  Exyzee (,,)  j Exs  E(,,) x y z e Es E xs a x E Re  E e j t  x xs  Uniform plane wave - sites.google.com/site/ncpdhbkhn 5 Ex. Wave Propagation in Free Space (3) Find the time – varying function of the vector field: o o o Es 100 30ax  20 50a y  40 210az V/ m If f = 1 MHz j30o j 50 o j 210 o Es100e a x  20 e a y  40 e a z V/ m j30o j 50 o j 210 o j 2 10 6 t Es(t )  100 e a x  20 e a y  40 e a z  e j(2 106 t 30 o ) j (2  10 6 t  50 o ) j (2  10 6 t  210 o ) 100eax  20 e a y  40 e a z 6 o E(t )  100cos(2 10 t  30 ) ax  6 o 6 o 20cos(2 10t  50 )ay  40cos(2  10 t  210 ) a z Uniform plane wave - sites.google.com/site/ncpdhbkhn 6 Wave Propagation in Free Space (3) Ex  E( x , y , z )cos( t   ) E  x E( x , y , z )cos( t   )    E ( x , y , z )sin(  t   ) t  t j t   j  t  j  Rej Exs e  Re j   E ( x , y , z ) e  e  j() t   Rej E ( x , y , z ) e  ReE ( x , y , z ) j cos(  t   )  j sin(  t   ) ReE ( x , y , z ) j cos(  t   )  sin(  t   )  E( x , y , z )sin(  t   ) E x  Re j E e j t  t xs  Uniform plane wave - sites.google.com/site/ncpdhbkhn 7 Wave Propagation in Free Space (4) Ex  E( x , y , z )cos( t   ) E x  Re j E e j t   j E t xs  xs E H   HE  j 0 t s0 s H EH   j E   s0 s 0 t .E  0 .E  0 s .H  0 .Hs  0 Uniform plane wave - sites.google.com/site/ncpdhbkhn 8 Wave Propagation in Free Space (5) EHs   j0 s EHHs  j0 s  j  0  s HEs  j 0 s 2  EEs   0  0 s 2 Es  ()  .E s   E s .Es  0  ( .Es )  0 2 2  EEs  k0 s k0   0  0 (wavenumber) 2 2 Exs   k0 E xs 2 2 2 EEExs  xs  xs 2 2 2  2  2  k0 Exs d Exs 2 x  y  z   k0 Exs dz2 Suppose Exs does not vary with x or y Uniform plane wave - sites.google.com/site/ncpdhbkhn 9 Wave Propagation in Free Space (6) 2 d Exs 2  k0 Exs dz2  jk0 z Exs  E x0 e Ex( z , t )  E x0 cos( t  k 0 z ) Ex( z , t ) E x 0 cos( t  k 0 z ) k0   0  0  1 k0  2.998  108  3  10 8 m/s c 0  0 Ex( z , t ) E x0 cos[ ( t  z / c )]   Ex( z , t ) E x cos[ ( t  z / c )] Uniform plane wave - sites.google.com/site/ncpdhbkhn 10 Wave Propagation in Free Space (7) Ex(,) z t E x0 cos[( t  z /)] c  Ex(,) z t E x 0 cos[( t  z /)] c dE EH  j xs   j  H s0 sdz 0 ys  jk0 z Exs E x0 e 1  jk0 z0  jk 0 z Hys  ()  jk0 E x 0 e  E x 0 e j0  0  H( z , t )  E0 cos( t  k z ) y x0 0 E  0 x  0 H y 0 Ex(,) z t E x0 cos[( t  z /)] c Uniform plane wave - sites.google.com/site/ncpdhbkhn 11 The Uniform Plane Wave 1. Wave Propagation in Free Space 2. Wave Propagation in Dielectrics 3. The Poynting Vector 4. Skin Effect 5. Wave Polarization Uniform plane wave - sites.google.com/site/ncpdhbkhn 12 Wave Propagation in Dielectrics (1) 2 2 EEs  k s k   k0 r  r  0 , 0   377  120    0 2 d Exs 2  k Exs dz2 jk  j  jkz  z  j  z Exs E x0 e  E x 0 e e  z Ex  E x0 ecos( t   z ) Uniform plane wave - sites.google.com/site/ncpdhbkhn 13 Wave Propagation in Dielectrics (2)   j    0() r   j  r k   k0 r  r  k  (   j   )     1  j    j    0() r   j  r 1/ 2 2      Re[jk ]    1    1  2      1/ 2 2      Im[jk ]    1    1  2      Uniform plane wave - sites.google.com/site/ncpdhbkhn 14 Wave Propagation in Dielectrics (3)  E E e z cos( t   z ) v  x x0 p  2 2      EEx x0 z  j  z   Hys  e e H y   1/ 2 2    0     Re[jk ]    1    1   2              1 c 1/ 2  vp    2     r  r      Im[jk ]   1    1  2 1  2        0      f  r  r   0 Uniform plane wave - sites.google.com/site/ncpdhbkhn 15 Wave Propagation in Dielectrics (4)   0 Ex E x0 cos( t   z )    E Ex  Hx0 cos( t   z )    y  H y   Uniform plane wave - sites.google.com/site/ncpdhbkhn 16 Wave Propagation in Dielectrics (5) Ex. Find the attenuation of a 2.5 GHz wave propagating in fresh water, given ε’r = 78, ε’’r = 7, μr = 1. 1/ 2 2        1    1  2        k0 r  r k0   / c 1/ 2 9 2  2  2.5  10 78 7  1  1    1   21 Np/m   4.8 cm 3 108 2 78      Uniform plane wave - sites.google.com/site/ncpdhbkhn 17 Wave Propagation in Dielectrics (6) JE  HEs  j s    j     HEEEs j()   j   s    s  j   s HJEs  s  j s   HEJJs () j  s  s  ds JEJE,  j    s s ds s        Uniform plane wave - sites.google.com/site/ncpdhbkhn 18 Wave Propagation in Dielectrics (7) JEds j s JE()  j   s s JE s  s 1/ 2 2        1    1    2    E tan     s               J s   JEJE s s, ds j  s    Jds j j   Uniform plane wave - sites.google.com/site/ncpdhbkhn 19 Wave Propagation in Dielectrics (8)  Good dielectrics :  1   Conductor:      jk  j  1  j  j   ' 1  j    n( n 1) n ( n  1)( n  2) (1x )n  1  nx  x2  x 3  ... 2! 3! 2 1     jk  j  1  j   ...     j     2 8          Re[jk ]  j     j    2  2     2   1   Im[jk ]    1         8      Uniform plane wave - sites.google.com/site/ncpdhbkhn 20 Wave Propagation in Dielectrics (9)      j     j    2  2    2   1     1         8         1       j     1 j ( /  ') 2 3           1  j   1  j         8   2     2   Uniform plane wave - sites.google.com/site/ncpdhbkhn 21 The Uniform Plane Wave 1. Wave Propagation in Free Space 2. Wave Propagation in Dielectrics 3. The Poynting Vector 4. Skin Effect 5. Wave Polarization Uniform plane wave - sites.google.com/site/ncpdhbkhn 22 The Poynting Vector (1) D HJ   t D E.  H  E.J  E. t .() E  H   E.  H  H.  E D H.  E   .() E  H  J.E  E. t B E   t BD   H.   .() E  H  E.J  E. t  t EH   .() E  H  J.E  E.   H. t  t Uniform plane wave - sites.google.com/site/ncpdhbkhn 23 The Poynting Vector (2) EH  .() E  H  J.E  E.   H. t  t E   D.E  E.    t  t 2  H   B.H  H.    t  t 2  D.E    B.H   .() E  H  J.E      t2   t  2  D.E    B.H    .() E  Hdv  J.E dv   dv    dv VVVV  t2    t  2  DSD..d  dv SV  d1 d 1  ()E  H .d S  J.E dv  D.E dv  B.H dv SVVV dt 2 dt  2 Uniform plane wave - sites.google.com/site/ncpdhbkhn 24 The Poynting Vector (3)  EH2  2  ()E H .d S  J.E dv   dv      SVVt 2 2  SEH  W/m2 EHSxa x y a y  z a z Ex E x0 cos( t   z ) 2 Ex0 2 E Sz cos ( t   z ) Hx0 cos( t   z )  y  2 1 T E2 1 E T Sx0 cos2 ( t   z ) dt x0 [1  cos(2t   z )] dt z, av T 0  2T  0 2T 2 1EEx0 1  1 x 0 2 1  sin( t  2  z )   W/ m 2T  2   0 2  Uniform plane wave - sites.google.com/site/ncpdhbkhn 25 The Poynting Vector (4) E E e z cos( t   z ) x x0 Ex0  z Hy  ecos( t   z   )     E2 S  E H x0 e2 z cos( t   z )cos(  t   z   ) z x y   E2 x0 e2 z[cos(2 t  2  z  2  )  cos  ] 2    1 E2 1 S  x0 e2 z cos ReEH  ˆ  W/m2 z, av 2   2 s s   j z Es E x0 e a x EE j Hˆ x0ej z a  x 0 e e j  z a sˆ y  y Uniform plane wave - sites.google.com/site/ncpdhbkhn 26 The Uniform Plane Wave 1. Wave Propagation in Free Space 2. Wave Propagation in Dielectrics 3. The Poynting Vector 4. Skin Effect 5. Wave Polarization Uniform plane wave - sites.google.com/site/ncpdhbkhn 27 Skin Effect (1)   jk j 1  j  j     j  j  j     j 1  90o 1 1 1 90o  1 45o   j 2 2 1 1  jk  j  j   ( j 1  1)  f     j  2 2      f   z z f  Ex  E x0 ecos( t   z )  E x 0 e cos(  t  z  f  ) Uniform plane wave - sites.google.com/site/ncpdhbkhn 28 Skin Effect (2) E E ez f  cos( t  z  f  ) x x0 Dielectrics z E Ecos t xz0 x0 0 Conductor z f  Jx E x   E x0 ecos(  t  z  f  ) 1 1 1     f    0.066 Cu  f Cu; 50Hz  9.3 mm 4 Cu; 10,000 MHz 6.61  10 mm Uniform plane wave - sites.google.com/site/ncpdhbkhn 29 Skin Effect (3) 1     f            v  p  vp   Uniform plane wave - sites.google.com/site/ncpdhbkhn 30 Skin Effect (4) Ex. Consider an 1 MHz wave propagating in seawater, σ = 4 S/m, ε’r = 81.  4 2  8.9  10 1  (2  106 )(81)(8.85  10 12 ) 1 1     0.25 m f  ( 10)(46   10 7 )(4) 2   1.6 m 6 6 vp  (2   10 )(0.25)  1.6  10 m/s Uniform plane wave - sites.google.com/site/ncpdhbkhn 31 Skin Effect (5)  j      j  j     j        ' 1 j 1 45o   f  2 45o 1 1     j    z f  z/ Ex E x0 ecos( t  z  f  )  E x 0 e cos(  t  z /  ) E x  H y  Ex0 z/ z   Hy  ecos t    2  4  Uniform plane wave - sites.google.com/site/ncpdhbkhn 32 Skin Effect (6) x z/ Ex E x0 ecos( t  z /  ) Conductor L  Ex0 z/ z   Hy  ecos t    Dielectrics 2  4  Jx0 1 S Re EH  ˆ  z av2  s s  0 |Jxs| 2 1 Ex0 2z /    Sav  e cos  b 22  4  y δ 1   E2 e 2z / 4 x0 b L 12 2z / 1 2 S S dS Ex0 e dxdy   bLE x 0 L,, avS z av 0  0 4z0 4 Uniform plane wave - sites.google.com/site/ncpdhbkhn 33 Skin Effect (7) 1 x S  bLE2 L, av4 x 0 Conductor JEx0  x 0 L 1 2 Dielectrics S   bLJ Jx0 L, av4 x 0  b z I J dydz 0 0  0 x |Jxs| z / Jx J x0 ecos t  z /   z//  jz  b Jxs  J x0 e e y δ (1  j ) z /  Jx0 e   b  J b I  J e(1  j ) z / dydz  J be(1  j ) z /  x0 s0  0 x0 x0 1 j 0 1 j Jx0 b   I cos t   2 4  Uniform plane wave - sites.google.com/site/ncpdhbkhn 34 Skin Effect (8) x Jx0 b   Icos t   Conductor 2 4  L IJ x0   J  cos t   Dielectrics J b 2 4  x0 1 z S  () J 2 bL 0 L  |Jxs| 2 J x0 2   bLcos   t   2  4  b δ 1 y S  J2 bL L, av4 x 0 (if the current is distributed uniformly throughout 0 < z < δ) 1 S J2 bL L, av4 x 0 (if the total current is distributed throuthout 0 < z < ∞) Uniform plane wave - sites.google.com/site/ncpdhbkhn 35 Skin Effect (9) LL R   S 2  a  103 RCu, 1MHz, a 1mm, l  1km  41.5  (5.8 107 )(2 )(10 3 )(0.066  10  3 ) Uniform plane wave - sites.google.com/site/ncpdhbkhn 36 The Uniform Plane Wave 1. Wave Propagation in Free Space 2. Wave Propagation in Dielectrics 3. The Poynting Vector 4. Skin Effect 5. Wave Polarization Uniform plane wave - sites.google.com/site/ncpdhbkhn 37 Wave Polarization (1) • In the previous sections, E & H are supposed to lie in fix directions • However, the directions of E & H within the plane perpendicular to az may change as functions of time and position • λ, vp, S, • The instantaneous orientation of field vectors • Wave polarization: its electric field vector orientation as a function of time, at a fixed point in space • H can be found from E Uniform plane wave - sites.google.com/site/ncpdhbkhn 38 Wave Polarization (2) y z  j  z Es()E x0 a x  E y 0 a y e e E Ey0 z  j  z Hs()H x0 a x  H y 0 a y e e H Hy0 Ey0 Ex0  z  j  z  ax  a y  e e    x H E 1 x0 x0 S Re[EH  ˆ ] z, av2 s s 1 ReE Hˆ (a  a )  E H ˆ ( a  a )  e2 z 2 x0 y 0 x y y 0 x 0 y x  ˆ ˆ  1 EEx0 x 0 EEy0 y 0 2 z Re    e az 2 ˆ  ˆ  1 1  2 2  2 z 2 Re  Ex0  E y 0  e a z W/m 2 ˆ    Uniform plane wave - sites.google.com/site/ncpdhbkhn 39 Wave Polarization (3)  j z Es()E x0 a x  E y 0 a y e E(z , t )  Ex0 cos( t   z ) a x  E y 0 cos(  t   z   ) a y E(z ,0)  Ex0 cos( z ) a x  E y 0 cos(  z   ) a y E(z, 0)  Ex0  Observer a location Ey0 b z Wave travel Uniform plane wave - sites.google.com/site/ncpdhbkhn 40