Section 4 4 TRƯỜNG ĐIỆN TỪ

4 4 Wave Equation and Solution for Material Medium Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by N[.]

Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by Nannapaneni Narayana Rao Edward C Jordan Professor of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, Urbana, Illinois, USA Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India 4.4 Wave Equation and Solution for Material Medium 4.4-3 Waves in Material Media H y  z , t  Ex  z , t    z t H y  z , t  Ex  z , t    Ex  z , t    z t Ex  j H y z H y   Ex  j Ex    j  Ex z 4.4-4 Combining, we get  Ex  j   j  Ex z Define    j   j   j  Then 2 Ex Wave equation   E x z 4.4-5 Solution: Ex  z   Ae  z z  Be Ex  z , t  Re  Ex  z   e jt   z z jt  Re   Ae  Be  e  Re  Ae e  j    z  j  z  z  Ae e e jt  Be e e j  z e jt   cos t   z   j   z    Be z cos t   z     4.4-6 Ae  z cos t   z             attenuation   wave B e z cos t   z              attenuation   wave  = attenuation constant, Np/m  = phase constant, rad/m  = propagation constant, m 4.4-7 f  z , t  e  z cos t   z  f -1   t 4 t 0 2  t 2 z 4.4-8 g  z , t  e z cos t   z  g t 2 - z t 4 t 0 2    -1 4.4-9 Ex  j H y z Ex H y  j z   Ae  z  Be  z   j z   Ae  z  Be  z   j where    intrinsic impedance of the medium   j 4.4-10 Summarizing,    j   j   j  j   e    j j conversely,    j   Re     Im   4.4-11 Example: For dry earth,  10 s/m,  50 , and  0 Let us compute  ,  , vp ,  , and  for f 100 kHz Solution:   j   j      j  j    j     j   j  2 f    10   j 0.36 j 108 4.4-12  j 0.004683 1.0628  19.8  j 0.004683 1.0309  9.9  j 0.004683 1.0155  j 0.1772  0.00083  j 0.004756  0.00083 Np/m  0.004756 rad/m 4.4-13    10 vp   1.32110 m/s  0.004756   2  2 1321.05 m  0.004756 j    j j   j   j 4.4-14      j  j 120 168.6 1  j 0.36 1.0309  9.9 163.559.9 161.1  j 28.1 