天线原理第三章.ppt

13 Radiation Integrals and Auxiliary Potential Functions（辐射积分和辅助势函数）（辐射积分和辅助势函数）1Antenna Theory and D13.1 IntroductionAntenna analysis and synthesis-Analysis problem to specify the sources and then require the fields radiated by the sources.-synthesis problem where the radiated fields are specified,and we are required to determine the sources.2Vector potentials auxiliary functions introduced to aid in the solution of the problems.-A(magnetic vector potential)and F(electric vector potential)Although the electric and magnetic field intensities(E and H)represent physically measurable quantities,among most engineers the potentials are strictly mathematical tools.3Computing fields radiated by electric and magnetic sources.-The one-step procedure relates the E and H fields to J and M by integral relations,requiring only integration.-The two-step procedure relates the A and F potentials to J and M by integral relations.The E and H are then determined simply by differentiating A and F.The integrands in the two-step procedure are much simpler.Review of Electromagnetic TheoryReview of Electromagnetic Theory4-MaxwellsMaxwells EquationsEquationsQuantitySymbolUnitElectric field intensityEV/m(volt per meter)Electric flux densityDC/m2 Magnetic field intensityHA/m(ampere per meter)Magnetic flux densityBT(tesla)Fields due to electric charge and currentFields due to magnetic charge and current Time-harmonic Time-harmonic(ejt)Fields)Fields5-Constitutive RelationsConstitutive RelationsIn an isotropic and linear medium,we know is the permittivity of the medium is the permeability of the mediumIn free space or air,6-Boundary ConditionsBoundary ConditionsOn the interface between two dielectrics,On the surface of a perfect conductor,Medium 1Medium 27 Elementary Radiating ElementsElementary Radiating ElementszxyI0dlHertzian Dipole(for wire antennas)Huygens Element(for aperture antennas)Equivalent magnetic current:1-89Coordinate systems for computing fields radiated by sources-Bounds of the sources The integration required to determine A and F or E and H is restricted over the bounds of the sources J and M.-Observation point coordinates Integration for A and F,and differentiation to determine E and H must be done in terms of the observation point coordinates.Source point(x,y,z)(r,)Observation point(x,y,z),(r,)R=r-r Far-Field ApproximationFar-Field Approximation0RField pointSource pointField point:(x,y,z)Source point:(x,y,z)1-103.2 The Vector Potential A for anElectric Current Source J11Magnetic vector potential AConsider electric source(J,)Because and using the vector identity The magnetic field intensity:(3.2a)Using the first Maxwells equation,we knowScalar Potential e From the vector identityUsing the Lorenz condition,(3.7a)(3.15)(3.13)12Helmholtz EquationThe curl of A is defined as We use the vector identityFor a homogeneous medium,Equating Maxwells equation,andUsing the Lorenz condition,(3.10)(3.14)133.3 The Vector Potential F for aMagnetic Current Source M14Electric vector potential FConsider electric source(M,m)BecauseUsingIntroducing an arbitrary magnetic scalar potential(3.16)(3.19)15Taking the curl of(3-16)and equating it to Maxwells equationand(3.26)Using the Lorenz condition,(3.25)3.4 Computing E and H from(J,M)16Summary1.Specify J and M(electric and magnetic current density sources).2.a.Find A(due to J)using-They are solutions of the inhomogeneous vector wave equation of(3-14)and(3-25),respectively.-k2=2 and R is the distance from any point in the source to the observation point.173.a.Find HA and EAb.Find EF and HFwith J=0with M=0oror4.The total fields areororwith J=0with M=03.5 Solution of the Inhomogeneous Vector Wave Equation18We verify that the solution of the inhomogeneous vector wave equation is-an infinitesimal source with current density Jz,is placed at the origin.-At points removed from the source Jz=0,19-Az is not a function of direction(and),Az=Az(r)-two independent solutions-In the static case(=0,k=0),the solution simplifies asthe time-varying solution can be obtained by multiplying the static solution by ejkr.20Poissons equation-In the presence of the source(Jz 0)and k=0,-The time-varying solution can be obtained by multiplying the static solution by ejkr.21The wave equation for the current densities in the x-and y-directions(Jx and Jy),the solution to the vector wave equation of(3-14)asIf the source is removed from the origin and placed at(x,y,z),22For linear densities J and M,For electric and magnetic currents Ie and Im,In a similar fashion,we can show that the solution ofis3.6 Far-field Radiation23A general solution to the vector wave equation of(3-14)in spherical components-The amplitude variations of r in each component are of the form 1/rn.Neglecting higher order termsThe r variations are separable from those of and.(3-55)Using24Neglecting higher order terms of 1/rn,the radiated E-and H-fields have only and components.They can be expressed asRadial field components exist only for higher order terms of 1/rn.25The far-zone fields due to a magnetic source M(potential F)can be written asSimply stated,the corresponding far-zone E-and H-field components are orthogonal to each other and form TEM(to r)mode fields.3.7 Duality Theorem（对偶定理）（对偶定理）26Duality theorem-When two equations that describe the behavior of two different variables are of the same mathematical form,their solutions will also be identical.-The variables in the two equations that occupy identical positions are known as dual quantities and-a solution of one can be formed by a systematic interchange of symbols to the other.2728Notify:Duality only serves as a guide to form mathematical solutions.There are no magnetic charges or currents in nature.3.8 Reciprocity Theorem（互易互易定理定理）29Reciprocity theorem applied to EM theory-a linear and isotropic medium,but not necessarily homogeneous”,-two sets of sources J1,M1,and J2,M2 which are allowed to radiate simultaneously or individually inside the same medium at the same frequency and-produce fields E1,H1 and E2,H2.The sources and fields satisfyLorentz Reciprocity Theorem in differential formLorentz Reciprocity Theorem in integral form30Lorentz Reciprocity Theorem for source-free regionsConsider that the fields(E1,H1,E2,H2)and the sources(J1,M1,J2,M2)are within a medium that is enclosed by a sphere of infinite radius.-Assume that the sources are positioned within a finite region and that the fields are observed in the far field(ideally at infinity).(3.66)(3.62)(3.63)31Reaction:Each of the integrals in(3-66)can be interpreted as a coupling between a set of fields and a set of sources,which produce another set of fields.(3.66)Coupling,not powerFor reciprocity to hold,32a linear and isotropic(but not necessarily homogeneous)medium 3.8.1 Reciprocity for Two Antennas3.8.1 Reciprocity for Two AntennasConjugate matching33The power delivered by the generator to antenna#1If the transfer admittance of the combined network is Y21,the current through the load is Vg Y21 and the power delivered to the load iswhen antenna#2 is transmitting and#1 is receivingwhen antenna#1 is transmitting and#2 is receivingUnder conditions of reciprocity(Y12=Y21),the power delivered in either direction is the same.343.8.2 Reciprocity for Antenna Radiation Patterns3.8.2 Reciprocity for Antenna Radiation PatternsPattern in the receiving mode is identical,because of reciprocity,to that of the transmitting mode.-Reciprocity for antenna patterns is general provided the materials used for the antennas and feeds,and the media of wave propagation are linear.-there is a distinct single propagating mode at each port.-the antennas in the transmit and receive modes are polarization matched,including the sense of rotation.A special case:Measurement system using a linear polarized probe to measure the circularly polarized pattern of a transmitting AUT.-If the probe antenna is used twice to measure the-component and the-component,then the sum of the two components can represent the pattern of the circularly polarized antenna in either the transmit or receive modes.35Procedure and foundation of pattern measurements and reciprocityThe voltages and currents at terminals 11 and 22 are related byMutual impedanceIf the medium between the two antennas is linear,passive,isotropic,and the waves monochromatic,then because of reciprocityIf in addition I1=I2,then36Figure 3.5 Antenna arrangement for pattern measurements and reciprocity theorem.-The three-dimensional plots of V2oc and V1oc,as a function of and,have been defined in Section 2.2 as field patterns.-The most convenient mode of operation is that of Figure 3.5(b)with the test antenna used as a receiver.Thank you for your attention!37