6不同媒质中的波数和波阻抗

1、7. Intrinsic ConstantsIntrinsic constants in general media As mentioned previously, the wave number k (the intrinsic wave number) and the intrinsic impedance 门 (the intrinsic wave impedance) are defined byk = (-Z0)宁0) and 门=wherez (w)= jtofi 0)y()=s ()+ j我 0)and0) = + j咬一2 +|10)= R + jR _2R +cy 0) =

2、 b + jb 2b + To solve k = J-Zy and 门=JZ y for Z and y, one obtains the compleximpedance Z and the complex admittance y asProofkn = % - yZ = = - Z 2Z 2 = -k2门 2andk2 y2 = 门2 Therefore a knowledge of the wave number k and the intrinsic impedance 门 is equivalent to the knowledge of the complex impedanc

3、e Z and thecomplex admittance y, or to the knowledge of , Q, 6, s, in other wordsthe knowledge of k and 门 specifies the characteristics of the medium. Both the wave number k and the intrinsic impedance 门 are complex, andmay be written ask =厂 zy = k- jk y=r+jx=”, En=正=yin terms of which, the +z-propa

4、gated traveling plane wave becomes,E = e E e - jkzx 0=e El e-j(k-jk“)zx 0ri.=e E e-jk ze-k zx0ande- jkz.=e o e - j(kk - jk )ze- jy间-E=eo e - j Ik z+Oe-k%y间where k is termed the phase constant, which gives rise to a variation of thewave phase,。-狄 2.k is termed the attenuation constant, which causes a

5、n exponential attenuation of the wave amplitude, e-k%.In addition to the wave number k = k 一 jk”, another parameter defined by y = jk =以 + j P and called the propagation constant is often utilized in some textbooks.It is easy to show that a = k is the attenuation constant, and P = k is the phase con

6、stant. In facty = jk=a + j p= j（k，- jk） =k + jk kNamely, the wave number k and the propagation constant y are related byk = -jy = k - jk = p - jaandy = jk =a + j p = k+ jkIt is physically understood that for a z-propagated and attenuated plane wave,e - jk = e - k%e - jk z = e -a ze - jP z = e -y zbo

7、th the attenuation constant a = k and the phase constant P = k should be positive, and this is why the wave number is denoted by k = k - jk rather than by k = k + jk , and the propagation constant is denoted by y =a + jP rather than by y = a - jP .Intrinsic constants in lossy media For lossy media,

8、the material parameters are assumed to be independent of frequency, TOC o 1-5 h z () = + j咬一2 += |10)= R + jR _2R += Rcy 0) = b + jb 2b += bThe complex impedance /(3 ) and the complex admittance (w )areZ ()= jR (3)= j3Ry(3) = C (3)+ j30) = b + j3The wave number isk = k - jkJ f=寸一 yz=- (c + j38 )(j3R

9、)=J-( j3 )(j3R)= 3.rwhere =8r-j8r = 8 , = C3The phase constant k and the attenuation constant k are determined as follows.k 2 = (k，- jk )2=(k 2 - k 2)- j(2kk )=3 2 r8=3 2 R(8，- j8 )thenk2 -k 2 =32R8 2k k = 3 2 R8 thenk =（2k，）then妙2 CD2|LIF（2A） 2 =C02 时4”4 一4（2四%2 Co2|H） = 0then4（02 m2JL18/ +16 原8CD2

10、|Ll,2thenthenthenThe intrinsic impedance isThe intrinsic resistance R and the intrinsic reactance X are determined as follows.，1 )2-2 R 庞2+时 2m2 + 2n = r+jx = 一-一V-庞thenn 2 = (r+jx )=(r 2 X 2)+ j 2 RX= 曰-j=日(+ j ) 2 + 2陀 . 卜 2 + 2 + j 2 +, 2then网R 2 - X 2 = 2 + 2c日2 RX = 2 + 2then(1 A UX = 2R 兴 2 +2

11、thenthenphLe2 + ”2 JpL e2 +2 j、2+16V8 2 +8 2 J、2=03 j2 V8 2 +8 2 J、82 +81 ) 四 W + 8 2 +82 )then2Q 2 +8 2 )R =.8R 2 - X 2 = 8 2 +8 2then四(+8 2 +8 2)日82(8 2 + 8 2 )8 , 2 + 8 2pX + F8 2 +8 2 )2(8,2 +82 )then2 +8 2 -8r) 2 V8 2 +8 2 )P=工= 281 +(8 8)2 The magnitude and the phase 匚 of the intrinsic impedan

12、ce 门 aredetermined as follows.hl =R 2 + X 2Ji+G +1 + Ji+Gw)2 -11 + (e” 8P)1(1 +(88少1 +(8 8)+1=tan-1=tan-1and(/2： +(88少- +(88少28 The intrinsic constants in a simple lossy medium are summarized as follows.k = k jk = P ja =侦|H8k，=p22，8、21 +1WJk =a=wandJ】+(8宾)2 28，,(时：(1 +(8 8小28回=侦1 +(8 8匚=tan-1(8 8少wh

13、ere8=8Intrinsic constants in perfect dielectrics (loss-free) For perfect dielectrics, the complex permittivity isb 8，8 = 0, 8 = 8 j8 = 8 , namely tan 5 =08 8， The wave number isk = k jk = P jawherePU8, =W - 2=叭，四8,=0The intrinsic impedance iswhereR =：1 +(1+； 1)Ji Izx:1 +()旦匚=tan-i、/1+(.* 少-1、顼 +(，+1

14、D.Intrinsic constants in good dielectrics (low loss)For good dielectrics, the complex permittivity isb 时 = j , , namely tan8 =一I8,：1 +(8 8人匚=tan-i机 tan-i机 tan-iE2sJ=tan-iE. Intrinsic constants in good conductors (high loss). The complex permittivity for good conductors isb&=-j& , & & , namely tan8 =

15、一 1 . The wave number iswhere1V；8 2 + 2 +8k =以The intrinsic impedance for good conductors isn = R+jX = |n|e力whereR=1戛丁_ m “：#2+8% +8=?82 +82口： p82 +8气*二n工28曰铲V2c匚=tan-i=tan-i=tan-i机 tan-i：（b 时+1 -（8r 时） p （? +1 +（，）（Eg） + 1一（矿时） 2 2（8七+1+（8七）,/1）tan-1 -1）兀=4Problem 2-6A plane wave propagates in a seawater medium of = = 8 x10-s F：m,日=日 =4兀 x 10-7 Him , and b= 144兀 S/m. The working frequencies are assumed to be f1= 10kHz and / = 10GHz, respectively..5.Find the prop