电介质电容器LC传输线色散关系波速上课讲义.ppt

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1、电介质电容器LC传输线色散关系波速Established in 1873 by James Clerk MaxwellGeneralized Ampere circuit lawFaradays magnetic induction lawGauss law for electric fieldsGauss law for magnetic fieldsE(r,t)=electric field strength(volts/m)B(r,t)=magnetic flux density(webers/m2)H(r,t)=magnetic field strength(amperes/m)D(r

2、,t)=electric displacement(coulombs/m2)J(r,t)=electric current density(amperes/m2)(r,t)=electric charge density(coulombs/m3)Maxwell EquationsThe Continuity LawThe electric current and charge densities at r are conserved.The divergence of current J from an infinitesimal volume surrounding r is equal t

3、o the decreasing of electric charge density with time t.连续性方程连续性方程Constitutive RelationsThere are a total of 12 scalar unknowns for the four field vectors E,H,B and D.The two Gauss law equations in Maxwell Equations are not independent ones.Only two of them are independent.Thus we need six more scal

4、ar equations.They are the constitutive relations.For an isotropic mediumD=E where =permittivity(=0 in free space)B=H where =permeability(=0 in free space)By isotropy we mean that E is parallel to D and H is parallel to B本构关系本构关系（各向同性介质）（各向同性介质）Constitutive Relations for Free SpaceD=0EB=0Hwhere 0=(1/

5、36)10-9 8.85 10-12 farad/meter0=4 10-7 henry/meter真空中的本构关系真空中的本构关系Wave EquationHelmholtz wave equationThe simplest solution takes the formThe dispersion relation provides an important connection between the spatial frequency k and the temporal frequency.k2=200波动方程波动方程Transverse EM Wave横波横波Boundary C

6、onditionsBoundary Conditions for Moving Boundaries 边界条件边界条件Polarization Vector PA dielectric material can be described by a free-space part and a part that is due to the material alone:D=E=0E+PThe polarization P symbolizes the electric dipole moment per unit volume of the dielectric material.极化矢量极化矢

7、量Magnetization Vector MA magnetic material can also be described by a free-space part and a part characterized by a magnetization vector M such that:B=H=0H+0MA medium is diamagnetic(反磁性)if 0.磁化矢量磁化矢量Ohms LawJc=Ewhich relates conduction current Jc to the electric field E by the conductivity.J=(+j)E (

8、include displacement current)欧姆定律欧姆定律复数介电常数当介质中存在有传导电流（有耗介质）时，通常用电导率（张量）来描述。全电流包含传导电流Jc和位移电流Jd在谐变情况下可以写成式中复介电张量 相当于把传导电流等效为位移电流时的介电常数Polarization(极化)The polarization of a wave is defined by the time variation of the tip of the electric field vector E at a fixed point in space.where h,v,and the direc

9、tion of propagation are mutually perpendicular and thus form an orthogonal system.We assume the amplitudes Ah and Av are both positive.Linear Polarizationv-h=2m v-h=(2m+1)EhoAyAxoAyAxEvEhEvCircular Polarizationv-h=/2 v-h=-/2Right hand Left handEhoAx=AyEvEhoAx=AyEvElliptical Polarizationv-h=/2 v-h=-/

10、2 general caseRight hand Left handEhoEvEhoEvEhoEvPolarization VariationsLet A=Av/Ah and =v-h A0.512LC传输线上的波动方程耦合的耦合的LC振荡器振荡器基本关系基本关系电压满足的波动方程电压满足的波动方程电流满足的波动方程电流满足的波动方程电压电压vs电流电流阻抗阻抗电抗电抗阻抗阻抗阻抗匹配阻抗匹配传播速度传播速度讨论：信号在平行双线上的传播速度讨论：信号在平行双线上的传播速度平行双线平行双线感应系数的计算感应系数的计算感应系数的计算感应系数的计算电容的计算电容的计算电容的计算电容的计算平行双线上的

11、阻抗与波速平行双线上的阻抗与波速光速光速多普勒频移多普勒频移波源移动波源移动观测者移动观测者移动波源和观测者同时移动波源和观测者同时移动运动方向不平行运动方向不平行介质运动介质运动电磁波电磁波相对论意义下的多普勒频移相对论意义下的多普勒频移相对论意义下的多普勒频移相对论意义下的多普勒频移膨胀的宇宙膨胀的宇宙冲击波冲击波突破音障突破音障马赫锥马赫锥色散（Dispersion）The dependence of wave velocity on frequency or wavelength.Dielectric dispersion,the dependence of the permittiv

12、ity of a dielectric material on the frequency of an applied electric field.脉冲传播过程中色散的影响脉冲传播过程中色散的影响群速度（Group Velocity）波包波包波包的传播波包的传播波包的传播波包的传播群速度群速度已知电磁波在等离子体中的色散关系为2(k)=p2+c2k2其中 p2=Ne2/(0m)某一个脉冲星向空间发出短而窄的电磁脉冲，测得其中两个脉冲的频率分别为：1=2563MHz,2=3833MHz假定星际电离氢气体的电子密度 N=105m-3，引起这种现象的原因是该气体的色散，问：(a)哪个频率的脉冲先到

13、？(b)如果接收到两种脉冲的时间间隔是0.367s，估计脉冲星到地球的距离。研习问题：估计脉冲星到地球的距离电介质电介质电容器电容器LC传输线传输线色散关系色散关系波速波速课后作业1.（上述脉冲星到地球距离问题）2.3.63.3.164.3.17附1：Anisotropic Media(各向异性介质)The constitutive relations for anisotropic media are usually written as D=E where =permittivity tensor B=H where =permeability tensorE is no longer p

14、arallel to D,and H is no longer parallel to B.electrically anisotropic:tensor and scaler magnetically anisotropic:scaler and tensor Both electrically and magnetically anisotropic:both of and are tensors群传播与相传播方向的夹角(P8687)基波前子波前OMNCMN波包 nrnpAB2CrystalsCrystals are described in general by symmetric pe

15、rmittivity tensors which can be transformed into a diagonal matrix.The three coordinate axes are referred to as the principal axes of the crystal.x=y=z -isotropicUniaxial crystal:z is the optical axisx y z -biaxial crystalBianisotropic MediaIsotropic or anisotropic media become polarized when placed

16、 in an electric field and become magnetized when placed in a magnetic field.A bianisotropic medium provides the cross-coupling between the electric and magnetic fields:D=E +H where and are tensors B=E+H where and are tensorsWhen placed in an electric or a magnetic field,a bianisotropic medium become

17、s both polarized and magnetized.Almost any medium becomes bianisotropic when it is in motion.The bianisotropic description of material has fundamental importance from the point of view of relativity.The constitutive relations are form-invariant when they are written in bianisotropic form.(The princi

18、ple of relativity postulates that all physical laws of nature must be characterized by mathematical equations that are form-invariant from one observer to the other.)Biisotropic MediaThe gyrator element(微波回转元件)possesses constitutive relations of the form D=E +H B=E+H where,and are scalersand 2/is ne

19、arly equal to 1.Chiral media(手征介质)have the following constitutive relations:where is the chiral parameter.附2：宏观介质类型按介质的宏观电磁特性划分，传播信道大致可分为7种类型1 均匀各向同性无耗介质 介质的电磁参数、为实常数，电磁波以恒速 沿直线传播；由点源辐射的能量随距离r沿球面扩散，则观测点在t时刻的瞬时电场为 即场的幅度反比于r，而相位延迟正比于r。2 均匀有耗介质 介质的、为复常数，传播常数 ，沿波矢量方向 r 处的瞬时电场为 即仍然以恒速 v=/沿直线传播，但由于损耗而产生幅度

20、沿路径的指数衰减。损耗一般源于介质分子（例如对流层中的氧气与水汽分子及电离层中的带电粒子）对电子运动能量的阻尼吸收，并消耗于焦耳热和再辐射。3 均匀色散介质 介质效应表现为在电磁场作用下的介质极化和磁化，当场量频率超过一定数值时，由于带电粒子的质量有限而可能使效应建立的速度跟不上场的变化，因而介质电磁参数、与频率有关，传播常数 与不为线性关系，则介质称为时间色散的。当电磁场在介质中的波长很短，即介质的传播常数k很大时，极化和磁化效应同外加电磁场不能视为局域对应，还与附近空间的场量有关，则介质称为空间色散的 4 均匀各向异性介质从介质中的一点沿不同方向所测的介质特性不同，称为各向异性。各向异性介

21、质有其特征方向，例如，重力或地球磁场方向。因而均匀各向异性介质中单色（单频）波的等相面不为球面，波矢量方向k与能量传播（射线）方向S不一致，相速是k与特征方向夹角的函数。在此种介质中，物质的极化和磁化矢量与外加电磁场矢量不一定同向，即介质电磁参数（除铁磁物质外，一般只是介电常数）为张量，因此，特定方向的介质效应，不仅取决于该方向的场分量，还与其他方向的场分量有关，从而发生波模间的耦合 5 均匀非线性介质 当介质电磁参数、是场强的函数时，本构关系则具有非线性特性。电离层在强电波加热的情况下就表现出这种非线性特性。一般情况下都设为线性介质 6 非均匀介质 非均匀介质的电磁参数、一般为空间点的函数，

22、因而沿射线路径s,传播常数 。对于慢变介质，沿波矢量方向r处的瞬时电场为波的空间相位与路径长度不仅是简单的线性关系，还存在波的折射即射线弯曲现象。当电磁参数不满足慢变条件而具有任意的空间分布时，还可能出现反射、散射等效应，波的传播路径和场特性是非常复杂的，一般难于从场方程获得解析解通常只能针对相对简单的介质特性分布模式进行求解，例如，对于平面分层和球面分层以及球形和圆柱形不均匀体等，可以求得一些优势波型的解析解7 非稳定和随机时变介质 一般情况下介质电磁参数是时间和空间坐标的函数，包含着不同空间尺度的非均匀性和不同时间周期的非稳定性以及随机的时空变化。有耗非均匀时变介质是最普遍的情况，其电磁参数为 。对于电离层还需考虑色散、各向异性以及非线性特性。要同时考虑所有效应，信道特性是很复杂的 此课件下载可自行编辑修改，仅供参考！此课件下载可自行编辑修改，仅供参考！感谢您的支持，我们努力做得更好！谢谢感谢您的支持，我们努力做得更好！谢谢