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by: Rodger Reichel
Rodger Reichel
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This 31 page Class Notes was uploaded by Rodger Reichel on Thursday October 22, 2015. The Class Notes belongs to CLASS 201 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/226983/class-201-university-of-california-santa-barbara in Classical Studies at University of California Santa Barbara.

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Date Created: 10/22/15
Elements of Electromagnetic Theory Ring the bells that still can ring Forgetyour perfect o ering There is a crack in everything That is how the light gets in iLeonard Cohen Essential elements of engineering electrodynamics and relevant terminology for antenna analysis are presented here for convenient reference The reader should already have been ex posed to electromagnetic theory at the intermediate level such that much of the material to follow should be familiar However even wellprepared readers should consider reviewing the follow ing source concepts including fictitious magnetic sources field energy and power flow concepts Poynting s theorem the reciprocity theorem and consequences and the field equivalence princi ples Although field equivalence principles are not strictly necessary to treat radiation problems they have proved to be an invaluable analytic tool for the antenna engineer and are widely used in the engineering literature i e master them The chapter concludes with a discussion of the behavior of fields at material edges which has ramifications in the numerical evaluation of fields and currents discussed in later chapters 11 BASIC LAWS AND FIELD QUANTITIES 111 Maxwell s equations The system of units used in this book is the rationalized MKS system In these units Maxwell s equations in differential timedependent form are Gauss law 1 la 0 llb 4 ml 6 3 2 ELEMENTS OF ELECTROMAGNETIC THEORY V X E 788 2 Faraday s law 11c 85 V X 9 7 5 MaxwellAmpere law 11d where E E Electric field intensity Vm g E Magnetic field intensity Hm D E Electric flux density Cmz B E Magnetic flux density Wmz pe E Electric charge density Cm3 7 E Electric current density AJmZ By convention time varying quantities are written with script letters reserving roman letters for phasor quantities The equations are consistent with the conservation of charge expressed by the continuity equation for current 3 815 which follows by taking the divergence of 11d and inserting 11a As commonly discussed in texts equation 11b follows from the apparent absence of magnetic charges or monopoles However even in the absence of physically real magnetic charges it is often convenient to introduce magnetic charges and currents into equations 11 which will be appreciated in our later discussion of field equivalence principles This is done by analogy with 11a39 if magnetic charges did exist 11b would become V E pm where pm is the magnetic charge density Assuming such magnetic charges would also be conserved leads to a continuity equation for magnetic current v7 0 12 30m V 13 M at where H is the magnetic current density In addition 1 1c must also be augmented by a magnetic displacement current in order to satisfy 13 giving a modified set of Maxwell s equations v5 pe 14a VE pm 14b VXE 7M5 14c a VX9JE 14a The symmetry of 14 with respect to electric and magnetic quantities leads directly to the duality and complementarity principles of C pter 3 The linearity of 11 can be exploited using Fourier transform theory for both time and space dependences this will be done throughout the book Using the Fourier transform pair in time with an assumed e3 field dependence ZFw Xte3 dt H ZFt mime iw dw and operating on Maxwell s equations gives the timeharmonic form V pe 15a v3 pm 15b VXE i iw 15c VXF 7gw 15d BASIC LAWS AND FIELD QUANTITIES 3 and similarly the continuity equations become V J ijwpe 16a VM am 16b where the factor of 97 is dropped so that field quantities are now complex phasor functions of position only The connection between the complex phasor fields and the physically meaningful timevarying quantities is given by E Re Eem 17 and similarly for the other field quantities Some books define the phasors to be rootmeansquare rms quantities so that factors of do not appear after timeaveraging operations In this book the phasor represents peak field quantities and so we retain the factors of 5 112 Charges and Currents Electromagnetic radiation is fundamentally a result of accelerating charges or equivalently a time varying current The current 7 is related to the motion of electric charges according to 7 pg convection current density 18 where F is the velocity of the charges and pe is the volume density associated with charges that are actually moving and thus contributing to the current flow The mechanical motion of charges is in turn related to the fields through the Lorentz force law Tq rx 19 For electric charge distributions we can also define a Lorentz force density as fep 7xe 110 and similarly for magnetic charges and currents we would have m mg 7 H X 5 In the presence of applied fields a charged particle will be accelerated in the direction of the Lorentz force If the charge is able to move through matter in response to an applied field as in a conductor it may experience a variety of scattering events which collectively impede this motion The net result is that the charges acquire an average drift velocity which is directly proportional to the applied electric field so 18 becomes 7 09E Ohm s law 111 where the proportionality constant Te is the electrical conductivity of the medium with units of Sm Such currents are called conduction currents For magnetic sources we can similarly postulate a magnetic conductivity am and corresponding magnetic Ohm s law W 0mg this has proved useful in the development of fictitious absorbers for numerical solution of radiation problems If the charges are not free to move in the material but rather are bound closely to a constituent atom as in an insulating material they can still give rise to an AC current since an applied harmonic field will induce some oscillatory motion of the charge about an equilibrium point This polarization current is usually accounted for indirectly using the dielectric permittivity relating 5 to E but can also be useful explicitly in the formulation of integral equations for radiation or scattering problems involving fields in matter This will be discussed again in connection with the volume equivalence theorem 4 ELEMENTS OF ELECTROMAGNETIC THEORY 113 Constitutive relationships When the sources are known 11 represents a system of six independent scalar equations the two vector curl equations in twelve unknowns four field vectors with three scalar components each This is true because 11a and 11b are not independent of 11c and 11d39 they can be derived from these equations using the conservation laws 12 and 13 Therefore we need six more scalar equations to completely determine the fields These could be obtained by expressing two of the field vectors as functions of the other two suc 5 mm E mm 112 in which case Maxwell s equations can be expressed entirely in terms of the vectors 3 and From a purely mathematical point of view the choice of which electric and magnetic field variables to eliminate is arbitrary Physically it can be argued that the set 33 are most fundamental 5 being derived quantities that incorporate macroscopic polarization effects in materials However most engineering texts choose to eliminate 5 and E as above since the set 3 are more directly related to important circuit quantities in the MKS system We will stick with that convention The relations 112 are called the constitutive relationships and are completely determined by the material medium occupied by the fields The functional form of 112 can be deduced from the microscopic physics of the material The material is then classified according to this functional dependence For example an isotropic material is defined by the timeharmonic constitutive relations 56E p 113 where e is the permittivity p the permeability and both are scalar quantities These materials are called isotropic because they respond uniformly to the fields in all directions that is e material parameters 6 and y do not depend on the direction of the fields Materials which behave according to 113 are often loosely referred to as simple media A special case is a vacuum or free space for whic e 2 60 885 X 1012 Fm p 2 p0 477 X 107 Hm In practice isotropic material properties are specified relative to the freespace values using 6 E 673960 M 3 WHO where 67 is called the relative permittivity or dielectric constant and in is called the relative per meability Materials which obey 113 but for which the material parameters vary with frequency or time are called temporally dispersive 1f the material parameters depend on position as would be the case in a layered or stratified medium like the ionosphere or a printed circuit board the medium is termed inhomogeneous or spatially dispersive There are few truly isotropic media in nature Polycrystalline ceramic or amorphous materialsithose for which the material constituents are only partially or randomly ordered through out the mediumiare approximately isotropic and hence many dielectrics and substrates used in commercial antenna work are often made from ceramics or other disordered matter or mixtures containing such materials Other materials and especially crystalline matter interact with fields in ways that depend to some extent on the orientation of the fields this is most easily appreciated when one considers the atomic structure of the material Such materials are called anisotropic BASIC LAWS AND FIELD QUANTITIES 5 and can often be described by the relationships EEE fi 114 where E and i are now dyadic quantities In such matgials the direction of 5 for example will in general be different than 5 and each component of D will depend on all of the components of 3 We can similarly extend Ohm s law for anisotropic materials by writing 73 115 In writing 113 we have also made an assumption of linearity Generally the material parameters could also depend on the strength of the applied field in which case the material is classified as nonlinear and Maxwell s equations will subsequently include nonlinear terms All materials exhibit some type of nonlinearity a common and generally unpleasant example being dielectric breakdown which occurs at large field strengths typically on the order of 106 Vcm On the other hand many materials also exhibit approximately linear field dependence over a significant range of applied field strength When this is true Maxwell s equations are linear differential equations and the principle afsuperpasiiian can be used The superposition principle implies that the fields satisfying 15 can be expressed as a summation of terms each of which is a perfectly valid solution to Maxwell s equations This is very useful and is often invoked implicit y Many interesting phenomena involving the interaction of fields and matterisuch as Faraday rotation optical amplification in lasers birefringence etcican only be described by complicated constitutive relations However the associated mathematics is often quite cumbersome and would tend to obscure our treatment of fundamental radiation concepts In our study of radiation we will mostly confine attention to simple isotropic media and assume the constitutive equations 113 Where a derivation is critically dependent on this assumption it will be noted For lossy media the fields can expend energy due to the interaction of moving charges For free electrons in conductors scattering processes leading to energy dissipation are accounted for using a finite conductivity which through Ohm s law allows us to write VXE jiaeEjweE 116 7i JweCE 117 where 6C is a complex permittivity defined by Te 6C6 118 Since the loss associated with freeelectrons can be equivalently represented in the form of a complex permittivity ie as loss associated with bound charges the physical origin of the loss is rarely important In a practical sense it is impossible to distinguish between the two as far as macroscopic effects are concerned The complex permittivity is also commonly written in the forms 6C66H61t31 16 119 where tan is called the loss tangent of the material given by II tand E 6 2 e we In practice losses are commonly specified using either an effective conductivity or loss tangent Phenomenologically we can represent magnetic loss in a similar way 6 ELEMENTS OF ELECTROMAGNETIC THEORY 114 Boundary conditions Using the divergence theorem A54 and Stokes theorem A59 Maxwell s equations can be expressed in integral form as 5 d pedV 120a ds pde 120b jigdz 7ltM d 120c iiidz lt7D d 120d These can be used to determine the behavior of fields at the boundary between two dissimilar materials Consider first a tiny fictitious pillbox of height Ah and cross section AA surrounding a point at the interface between two regions as shown in figure 11 Evaluating the integrals in 120a by letting Ah and AA become infinitesimal gives Aligo D dS m n D1 7D2AA Agile ing dV Qenclosed pseAA where pse is a possible surface charge density with units of Cmz which exists in an infinitesimal layer along the interface any volume charge density contributes nothing to the integral as Ah a 0 The integrals in 120b can be similarly evaluated which leaves us with the boundary conditions for the normal components of the fields 51 1 B 3 732 p59 121a fl AEZ psm 121b Boundary conditions for the tangential field components can be derived from 120b c by con structing a closed rectangular path of length AZ and height Ah as shown in figure 11 Again letting Ah and AZ shrink to zero gives AliiLrgojiHd 3X 7H2AZ lim 78 D d 75 AZ Ahgt0 315 where 75 is a possible surface current density which has the units of NIH The integrals in 120c can be similarly evaluated which leaves f X E 7E J5 122a f X E 7E2 7M5 122b Any set of fields which simultaneously satisfy Maxwell s equations and 122 will automatically satisfy 121 Boundaryvalue problems in radiation theory are most frequently formulated in ELECTROMAGNETIC ENERGY AND POWER FLOW 7 reg ionl 7 Figure 11 Volume and surface elements for de termining boundary conditions reg ion2 terms of currents so 122 will be of most use Conducting bodies are often idealized as perfect electric conductors PEC characterized by vanishing tangential electric field at the conducting surface and zero total electric field inside In setting up equivalent field 3problems the concept of a perfect magnetic conductor PMC is useful characterized by a vanishing tangential magnetic field at the surface Both are described by an infinite conductivity and the vanishing fields can be argued on the physical grounds that the current density inside remain finite as Te a 00 From Faraday s law zero electric field in a PEC implies that the timevarying component of the magnetic field also vanish Similar arguments also hold for a PMC Thus for radiation problems we assume all fields to be zero everywhere inside of both the PEC and PMC The boundary conditions 122 at the surface of such materials then become PEC a X E 0 123a a X F 75 123b PMC a X E 7M5 124a a X F 0 124b 12 ELECTROMAGNETIC ENERGY AND POWER FLOW We know from practical experience that energy delivered to a transmitting antenna can be faith fully recovered at a distant receiver This transfer of energy is attributed to fields generated at the transmitter which propagate to the receiver Our classical description of electromagnetic radia tion rests entirely on this on this field interpretation so it is appropriate to carefully review the relationships between electromagnetic fields and energy Figure 12 Poynting s theorem is an expression of conservation of energy Any change in the total energy inside a volume must be accompanied by a ow of energy into or out of the volume As a starting point we simply assume that conservation of energy holds for electromagnetic fields Therefore if the total energy in any region of space is observed to increase in time then there must be a corresponding flow of power into that region to account for the change Using the 8 ELEMENTS OF ELECTROMAGNETIC THEORY notation of fig 12 the total energy in the volume V can be expressed as MMTdV where MT is the total energy density JmS Defining 5 as a power density vector WmZ then the net power flow into the volume is 7 5 d s The conservation of energy can then be written as fd 7MT11V 125 Using the divergence theorem A54 125 can be written as a continuity equation auT VTPW70 126 We must now try to relate the power and energy densities to the field variables 1f the volume V contains matter charges then the total energy MT will include both the energy stored in the fields and the field energy that is converted to mechanical energy or viceversa through the charge motion Writing MT HEM Md the conservation law 126 can be written as 31117 BMEM iwivwT 127 where HEM is the field energy density and Md is the energy lost or gained by the interaction of fields with matter The latter can be expressed using the Lorentz force laws If we describe the matter in the volume using charge density functions pe and pm then the fields act to displace the charges and hence mechanical energy is consumed Recall that the incremental energy dW required to move a charge q moved through a distance d by a force is dW 7 d and therefore the rate of change of energy power is f i where F dFdt Similarly the rate of change of energy density in the volume V due to motion of the charge distributions pe and pm is described by 57apgp 7zm 128 The currents can in turn be related to the electric and magnetic fields through Maxwell s equations 14cd giving 821d 813 8D at 7V5x998t53t 129 where use has been made of vector identity A47 So far this equation is generally applicable to any material Specializing to simple media via 113 allows us to write 337 mi 1 80191 9 5 E WT and similarly for the last term in 129 Therefore 129 becomes after substituting 128 821d E 1 1 i ivExa u eEE 130 ELECTROMAGNETIC ENERGY AND POWER FLOW 9 Comparing 130 to 127 we can tentatively identify the electromagnetic power density as f E X R this is known as Payn ng s vector and 130 is a statement of Poynting s theorem The term in brackets in 130 is the stored field energy Integrating 130 over a volume V bounded by a surface S and using 128 and the diver gence theorem A54 gives the integral form of the Poynting s theorem i x dS EHm dvEp f dV 131 The lefthand side of 131 is interpreted as the total power owing into the volume through the surface S which is equal to the total power absorbed in the volume first term on the right plus the rate of change of field energy in the volume second term on right Although 130 is an exact statement of energy conservation for simple media the iden tification of E X F with the power density is open to question since the curl of any arbitrary vector field can be added to 5 without changing 130 This mathematical ambiguity is resolved by appealing to experiment Poynting s vector E X F does correctly predict the magnitude and direction of power flow measured in the lab and so it is accepted as fact If this makes the reader uncomfortable remember that Maxwell s equations are also just postulates based on experimental evidence Further discussions of the ambiguity of the Poynting theorem can be found in 2 Specializing to harmonic time variations is complicated by the products of field quantities appearing in 130 According to the prescription 17 the timedependent Poynting s vector should be expressed in terms of phasor fields as f Re Eewt gtlt Re Few HE gtlt F E gtlt F 3 gtlt new my Few 5R9 E gtlt F 5R9 E gtlt new since Re The instantaneous power density thus consists of a timeindependent contribution average power and a periodic uctuation at frequency 2w Considering just the average power density gives 1 Pave Re E x H 132 In view of 132 we define a complex Poynting vector as R E X if Taking the divergence of this expression using A47 and substituting Maxwell s equations leads to the phasor form of the Poynting theorem analogous to 130 1 1 1 7 VExH EJHM2WZHBiED 133 Integrating over a volume V bounded by a surface S and using the divergence theorem A54 gives the integral form of the phasor Poynting s theorem 7 lt x dS 134 E7FM WWW a fgj W The lefthand side of 134 is interpreted as the average total complex power flowing into the 10 ELEMENTS OF ELECTROMAGNETIC THEORY volume through the surface S which is equal to the power absorbed in the volume plus the rate of change of field energy in the volume In simple media where the volume integrals in 134 are all real the first term on the right represents the real timeaveraged power and the last term represents the net reactive power in the volume This interpretation of 134 may be more acceptable when gure 13 RLC circuit for interpreting Poynting s theorem compared to a corresponding problem in circuit theory Consider the RLC circuit in figure 13a The complex power delivered to this circuit Pin can be written as 1 1 1 P VI II R L 135 m 2 2 w Um We know from circuit theory that the real power delivered to the circuit is Ploss RI I and the energy stored in the inductor and capacitor is given by U iLIF and Ue iCVV respectively which enables us to write 135 as Pin Ploss 27m Um 7 Ue 136 which is exactly the same form as 134 Furthermore this suggests the following expressions for stored magnetic energy and stored electric energy in terms of the fields Um iRe FEW Ue iRe E rv 137 It should be noted that these expressions for energy density are not valid for dispersive media but can be suitably modified see p94 of 13 SOURCES AND GENERATORS 1n formulating electromagnetic problems we may postulate a set of charges or currents as known sources of fields and subsequently attempt to formulate more direct solutions for the field quantities in terms of these sources However we must remember that the fields so produced are also capable of inducing surface charges and currents in neighboring matter These induced currents and charges will then give rise to another set of fields which are superimposed on the first and so on This phenomenon is called scattering and the secondary fields produced by the induced currents are the scattered elds Although the induced charges and currents also act as sources of the scattered fields they are clearly different than the original set of charges and currents which were assumed to exist independent of the presence of any fields Using the superposition principle we therefore express the total currents 7 and M in Maxwell s equations as 77i7f MMiMf 138 where jiMi are the impressed currents and 7fMf are the currents induced by the fields Impressed currents are assumed to be fixed in some way that is not affected by the fields these SOURCES AND GENERATORS 11 are analogous to the ideal current generators used in circuit theory Induced currents are those that flow only in response to the fields and arise physically from motion of either free or bound electrons The motion of free electrons is described classically through Ohm s law and is called a conduction current whereas the oscillatory motion of bound charges is called a polarization current The linearity of Maxwell s equations assuming linear media means that the fields can be similarly decomposed into a component due only to the impressed currents the applied or incident field and a component produced by the induced currents the scattered field E Einc Escatt H Fine scatt 139 As a example of these ideas consider the re ection of an incident field from a perfectly conducting plane as illustrated in figure 14 Mathematically Maxwell s equations describe a selfconsistent relationship between the total currents and total fields for the problem but physically it is more appealing to consider the situation as a chain of events the incident field is produced by an impressed source distribution 71 this incident field induces a conduction current on the surface of the conductor39 the induced current then acts as a source radiating fields which must exactly cancel the incident fields on and within the conductor as required by the boundary conditions In later chapters we will develop quite general methods for attacking such scattering problems based on this causal viewpoint which can be applied to almost any problem at least in principle incident eld W W Figure 14 Re ection from a mirror as mduced a simple example of scattering processes Jf surface described by impressed and induced cur cunth rents scattered eld mirror The distinction between impressed and induced currents is therefore a natural breakdown in terms of cause and effect but it can sometimes lead to confusion in analysis The trouble starts when making statements such as currents are induced which in turn radiate In order to calculate the scattered fields we may temporarily view the induced currents as fixed generators that is like impressed currents In this way the same mathematical relationship between incident fields and impressed currents can be used to relate the scattered fields to the induced currents But it must be remembered that the currents in question are in fact induced currents when making such calculations They produce only part of the field and hence must be related back to the impressed currents in such a way that all boundary conditions are satisfied This discussion may seem rather pedantic but a clear understanding of the differences is especially helpful in our application of field equivalence principles Impressed currents can be used to represent circuit generators as shown in fig 15 A current source is modeled as a short filament of impressed current 1 in series with a perfectly conducting wire as shown in fig 15a Assuming the dimensions of the circuit are small enough so that Kirchoffs circuit laws apply then the impressed current will induce a current in the external circuit of the same magnitude irrespective of the load impedance If we compute the complex 12 ELEMENTS OF ELECTROMAGNETIC THEORY a Figure 15 Electromagnetic representation of independent circuit sources a Current generator im pressed electric current lament b Voltage generator impressed magnetic current loop power flow out of a volume surrounding the generator the dashed box in fig 15 we find 1 FEW 711 Edi lFV 140 2 2 W 2 which is in accordance with our expectations of a current generator Note that the internal im pedance of the source is infinite since removal of the impressed current leaves an open circuit in the gap Similarly a voltage source in circuit theory can be represented as in fig 15b using a fil amentary loop of magnetic current around a perfectly conducting wire From Maxwell s equations and neglecting the magnetic flux linked by the circuit jiEdi Md C If the path 0 is coincident with the wire leads and closes across the terminals then we find the magnitude of the magnetic current filament is just 7V and therefore the complex power owing out of the generator through the dashed box in fig 15b is 1 1 1 7 HMdV V HdZ VI 141 2 2 2 The internal impedance in this case is zero since removal of the current loop leaves a short circuit 14 RECIPROCITY THEOREMS RUMSEY S REACTION Suppose there are two separate source distributions 71M1 and 72M2 in a certain localized region defined by volume V as shown in figure 16 Physically this situation is representative of a general twoport electrical network such as an antenna link Characterization of this electrical network involves examining the interaction of fields and sources between the ports We assume the volume is filled with a simple isotropic media described by 113 and 111 These sources produce the fields E1731 and E2F2 respectively in accordance with Maxwell s equations V X E1 ijwp l 7M1 V X F1 JweE1 71 V X E2 ijw z 7 M2 and V X F2 JWEEz 72 142 Using the vector identity A4D VZXEVgtltZBiVgtlt Z RECIPROCITY THEOREMSRUMSEY S REACTION 13 wefindthat v E1 x r z x l 71E2772E1M2317M1 2 143 Integrating 143 over the volume V and using the divergence theorem A54 gives gure 16 Two source distributions and corresponding elds Within a volume V mlx rmx ld S pyrrjzi mnlinymdv 144 V Note that the only currents on the right hand side which contribute to the integral are the impressed sources induced current terms cancel by virtue of 111 This result is usually applied and more easily interpreted for certain special cases Where either the surface integral or volume integral vanishes For example if the surface is chosen to exclude any impressed sources so that 71 72 M1 M2 0 then 144 reduces to mlx r zxm d o s which is called the Lorentz reciprocity theorem In this case the fields are due to sources external to S We Will later use this result to establish the reciprocal properties of an antenna li Alternatively if the surface S coincides with a PEC or PMC boundary then the surface integral vanishes since using A38 E1 XF27 L7E2XF1fl XE1F2 XE2F1 7mmE1m 1EZ and either ft X E 0 for a PEC boundary or ft X F 0 on a PMC boundary Then 144 reduces to Eljzi lMZdVME27li ZM1dV This is a more familiar statement of reciprocity for those knowledgeable in circuit theory and is often simply referred to as the reciprocity theorem This last result can also be obtained if S is 145 Note conditions of validity 146 Note conditions i ity 14 ELEMENTS OF ELECTROMAGNETIC THEORY taken as a sphere at infinity Then the Sommerfeld radiation condition see Chapter 2 insures that the fields produced by localized currents in V will be spherical outward waves at infinity so that 72 gtlt E 77 Interestingly many of the physical observables important in applied electromagneticsithat is quantities that can be measured directlyican be expressed in terms of integrals like those in 146 Rumsey 2 has argued for the physical significance of these integrals which he called reaction integrals The left hand side of 146 is then thought of as the reaction of source 2 on the fields from source 1 This refers to the fact that in order to keep owing the sources must react to the fields in their vicinity by supplyingabsorbing energy Reaction integrals are commonly abbreviated as F XE1F27 XE2 10 lt2j gt Eig r j dV 147 The reciprocity theorem 146 can then be represented concisely as lt 239j gtlt j239 gt 148 An important problem for later work that can be described in terms of reaction integrals is the determination of equivalent circuit parameters representing a multiport electromagnetic network as shown in figure 17 Using fig 17a the impedance matrix is defined by N VZf or 142271 149 j1 Each term in the summation ZijIj gives the contribution to the terminal voltageithe induced EMFiat port 239 due to currents impressed at port j with all other ports opencircuited Assuming that the independent current sources of fig 17a are implemented in the sense of fig 15a we can compute the reaction lt j239 gt as lt 13239 gt iii1V I E 212 7142711 150 port 239 porti where the last equality follows since the path integral or EJ over port 239 is just the voltage induced at port 239 due to the current source at port j or ZijIj Therefore lt j239 gt 7 1 E J dV 151 1in 1in J 1 Using the reciprocity theorem 146 we find that Zij th th which is the familiar result from circuit theory If we had alternatively chosen to express the system in terms of admittance parameters as shown in figure 17b a similar analysis gives ltj239gt 1 y HMdV 152 39J Vth Vth J 39 with a similar consequence of reciprocity 327th RECIPROCITY THEOREMSRUMSEY S REACTION 15 gt 11 V1 V1 11 g 2g 12 V2 E V2 12 e g g Q Q N gt gt N VN VN 1N Figure 17 Multiport circuit representation of an electromagnetic system a Configuration for charac terization in terms of impedance parameters b Con guration for characterization in terms of admittance parameters Closer examination of the derivation of 1 44 shows that it is critically dependent on the assumption of a simple isotropic media in the volume V That is we have only proved reciprocal properties for electrical systems comprised on isotropic media Using more general constitutive relations for anisotropic media 114 and 115 we find Problem 2 that the result 144 is only obtained when the material properties in the volume are described by symmetric dyads T T T 6 27 153 M M mll Such materials are therefore called reciprocal materials An important example of a material that is not reciprocal is a magneticallybiased plasma such as the Earth s ionosphere Antenna links involving propagation through the Earth s ionosphere are therefore not reciprocal Alternatively antennas themselves may be constructed from nonreciprocal media such as magneticallybiased ferrites The resulting nonreciprocal antenna may serve a useful function for example simultane ously transmitting and receiving different polarizations Propagation in nonreciprocal media and analysis of nonreciprocal antennas is however a relatively specialized topic that will not be dealt with in this wor The reciprocity theorem will prove a useful tool in other contexts For example consider the situation in fig 18 where there are two sets of impressed currents denoted as 71 and J2 within a volume V Current 71 is impressed directly adjacent to a PEC object Current 72 is a test source that can be oriented in any arbitrary direction According to 146 these currents and the fields they produce are related by v ljngVE271dV 154 Now the field E2 is the total field produced by the test source 72 which must vanish everywhere along the surface of the PEC object Therefore the integral on the right in 154 is zero and we have E17de 0 V Since 72 can be anything we choose this must mean that E1 0 everywhere inside of V This proves that impressed electric currents on PEC surfaces do not radiate Physically this is because 16 ELEMENTS OF ELECTROMAGNETIC THEORY x t test l source quot I J1 VI J2 I l X l I x x I s Figure 18 Impressed currents above conductors do not radiate as can be shown by aplying the reci procity theorem to this example the induced currents on the object radiate fields which exactly cancel the fields of the impressed current The method just employed is quite powerful Think of what we have just doneiwe ve solved for the fields produced by an arbitrary current distribution 71 radiating in the presence of a conductor an otherwise difficult boundaryvalue problem All that was necessary was a knowledge of the fields produced by our testing source which can be anythign we choose 15 UNIQUENESS OF SOLUTIONS After going to the trouble of finding a solution to Maxwell s equations for a particular problem one may wonder if it is the only possible solution This is guaranteed under certain conditions by the uniqueness theorem To prove the theorem we assume the existence of two possible solutions and derive the Enditionsreqlged to insure they are identical Let E17 H1 and Ez7 H2 be two possible solutions to 15 for a given set of sources VXE17M7WMF1 Vx ljjweE1 155 VXE27M7WMF2 VXF27OJ E2 39 Subtracting these equations and defining the difference fields 6 E1 7 E2 and 6E F1 7 F2 gives V X 5E ijwudH V X 5H JwedE 156 which are just the sourcefree Maxwell equations Therefore the difference fields must satisfy Poynting s theorem 134 swn gtlt a d Jw may 7 armij dV 157 If the solution were indeed unique then this would imply that ii 6F 0 everywhere within the volume of interest so that both sides of 157 vanishes Suppose we now reverse the problem FIELD EQUIVALENCE PRINCIPLES 17 if we can somehow prove that the surface integral in 157 vanishes under what conditions does this imply that the solution is unique Expanding the volume integral in terms of its real and imaginary components a vanishing surface integral would require that v mle 7 SEEP dV 0 158a mle swim dV 0 158b l ln lossy media p and e are always positive As long as there is some finite though perhaps infinitesimal loss in the system the second of 158 can only be satisfied if 6E 6F 0 everywhere in the volume Since there is always some loss in practice uniqueness is therefore guaranteed provided we can make the surface integral vanish Using A38 and d fidS the integrand of the surface integral in 2 can be written as E X 533415 a X E 5 d3 4 X mquot 6EdS 159 If the tangential electric fields are specified on the bounding surfaceifor example if the problem statement fixes the value of ft X E on Sithen this boundary condition must be incorporated into every possible solution hence ft gtlt 6E 0 over S and the surface integral vanishes Similarly if the tangential magnetic fields ft X F are specified on the surface then ft X 6 0 over S and the surface integral vanishes Therefore the elds produced by sources within a lossy region are unique as long as the tangential components satisfy prescribed conditions at the bounding surface To obtain uniqueness in an ideal lossless region we consider the fields to be the limit of a corresponding field in a lossy region as the loss goes to zero 2 16 FIELD EQUIVALENCE PRINCIPLES lt many problems a knowledge of the fields is required not everywhere in space but rather in a certain welldefined region that is separate from the sources of the field for example the radiation fields of an antenna In such cases it may be possible to simplify the problem by replacing the actual sources with fictitious sources that produce the same fields in the region of interest These provide powerful tools for analysis 161 Image Theory The simplest and most familiar equivalence is the method of images This technique is really just a catalog of certain electromagnetic problems that produce identical field distributions These are usually identified by noting that conducting surfaces are surfaces of constant potential and therefore can be placed along equipotential lines in any field distribution without altering the fields For example in fig 19a the fields produced by a positive and negative charge separated by a distance 290 produce an equipotential surface midway between the two charges If we place a conducting object along this equipotential surface as shown in fig l9b then the fields above the surface are unchanged This equivalence is usually applied in reverse Given a situation where there are charges a distance x a conducting plane the conductor can be replaced by a set of image charges that have the opposite sign as the original charge spaced a distance as below the original conducting surface This eliminates the conducting matter leaving only charges in unbounded space a considerably 18 ELEMENTS OF ELECTROMAGNETIC THEORY Cb Figure 19 a Positive and negative charges separated by 21 b Positive charge q a distance as above a ground plane These two situations are identical as far as the fields above the ground plane are concerned easier problem to solve Note however that this equivalence applies only to the fields above the original conductor From this simple example we can derive many other image equivalents involving electric currents and magentic charges and currents and PMC surfaces These are summarized in fig 110 below Note that the images for current elements depend on the direction of the current element and can be derived from a knowledge of the behavior of the image charges as they are moved relative to the conducting surface For example a horizontal current element 7h above a PEC ground plane corrsponds to a positive charge movement in the direction of the current flow The image charge in this case would move in the same direction but has the opposite sign so that the effective image current direction is reversed Electric Sources Magnetic Sources 0 gt T 7v 0 gt TM 0 lt T j C gt M pe 7h v 3quot Mk v o gt f I 0 TM we Jh V 9quot Mk V Figure 110 Summary of equivalent images for sources near conductors FIELD EQUIVALENCE PRINCIPLES 19 162 LoveShelkunoff Equivalents Readers familiar with circuit theory will remember that a network containing sources which drives a passive load network can be replaced by a Thevenin or Norton equivalent This is illustrated in figure 111 Insofar as the calculation of voltage and current in the load network is concerned the original and equivalent sources behave the same 1 Passive gt Network Source V Passwe Network Network Passive Network a b Figure 111 Thevenin and Norton equivalence principles from circuit theory In the case of the Thevenin and Norton circuits the equivalent sources are expressed in terms of the open circuit voltage VOC and shortcircuit current ISC which are measured at the terminals of the source network The equivalent source impedance in each case Zeq is the impedance of the original source network with all the sources shut off However the Thevenin and Norton sources are not the only possible equivalents In order to relate the concept more directly to field theory it is desirable to express the equivalent sources in terms of the actual terminal voltage and current in the original problem of figure 111 This is accomplished by the equivalent circuits of figure 112 These circuits are equivalent to the original problem because the relationship between V and I is fixed by the passive load network which remains unchanged Simple analysis shows that there is no current flowing in the source impedance of figure 112a so it can be specified arbitrarily39 figures 112b and 112c result from the choice Zeq 0 and Zeq 00 respectively Passive Passive I Passive Network V Network Network a b C Figure 112 Other possible equivalents sources in terms of the actual terminal voltage V and current I from the previous gure The same concepts are extended to field theory by considering the situation depicted in figure 113a Sources within some bounded region possibly containing matter produce the fields EH outside of that region To simplify the calculation of these fields we replace the original sources by impressed surface currents J5 and M5 flowing on the boundary of the source region From 122 we know that the magnitude of the surface currents required to produce the same fields outside of the boundary depends on the difference of the tangential fields across the boundary Since the region within the boundary is of no interest we can arbitrarily specify that the fields are 20 ELEMENTS OF ELECTROMAGNETIC THEORY zero within that region giving the equivalent of figure ll3b This is known as Love s equivalence principle 2 Note that this equivalence is only helpful when the tangential fields at the boundary of the original problem are known or can be approximated Comparing this situation to the circuit model of figure 112a we see that the magnetic current is analogous to the terminal voltage V while the electric current is analogous to the terminal current I E H E H a r u I f I 1 Sources Ze r0 e dy and matter free space I I a E H C F n n 7ix t223232i 25t 3 33333223 Ze ro e Id 32 Zero e Id 333 l I o o PEC l quot PMC 1133 t t quot 323tt221 3 39 39Z111339 N 0 3333 A x M S 0 33 I M S E x n S c d Figure 113 Four possible source configurations which produce the same field configuration external to the boundary S a Original problem b Love s equivalent where the original source region is replaced by freespace and surface currents are impressed on the bounding surface to produce null field within S c and d are Shelkunoff equivalents where the original source region is replaced by a perfect conductor In the latter case the impressed currents induce additional currents on the conductors Since the null field was specified within the original source region the material found within the source region is irrelevant to the calculation of fields external to that region This is analogous to the arbitrariness of the source impedance in the circuit equivalent of figure 112a The most common choices are to fill the volume with freespace as was tacitly assumed in Love s equivalent or to surround the region by a perfect electric conductor or perfect magnetic conductor as shown in figures ll3c and ll3d The latter two choices are due to Shelkunoff 2 and are analogous to the circuit models of figures 112b and 112c In the first case figure ll3c only magnetic currents are required since the impressed current J is shortcircuited by the PEC and does not radiate proved earlier using the reciprocity theorem Similarly the magnetic current M5 is shortcircuited in figure ll3d and only the electric surface current is required From the uniqueness theorem we are assured that the fields calculated in each case will be identical to the original problem as only one of the tangential fields E or F is required Although it may not be immediately obvious the equivalence of the four physical situations depicted in figure 113 can greatly simplify radiation problems A complicated boundaryvalue problem ie sources radiating in the presence of nearby objects such as figures ll3a ll3c FIELD BEHAVIOR NEAR A SHARP EDGE 21 ll3d can be reduced to an equivalent set of currents radiating in a homogeneous unbounded medium figure ll3b Alternatively we will use the equivalence of figures ll3a ll3c and ll3d in our formulation of Huygen s principle in Chapter 2 which reduces a complicated source distribution ll3a to a hopefully simpler surface current on a conductor ll3c or ll3d 163 Volume Equivalence Theorem The volume equivalence theorem is based on the following observation for any material body characterized by a simple scalar permittivity and permeability as in 113 we can write Maxwell s curl equations as V X H 7i jweE 7i 7p jw 0E 160a V X E i i ijwp 7Mi 7 Mp 7704 l60b where we have defined the polarization currents Jp JW 0E Mp JwwtoH 161 In other words we can replace the material by a volume current distribution Jp7 Mp owing in free space The magnitude of the polarization currents are dependent on the total fields and the total fields include some contribution from the polarization currents so this approach is typically used to set up a selfconsistent integral equation for the currents Free space I Free space Free space 0 0 goz o 0 0 4 4 Figure 114 Volume equivalence theorem Simple dielectric or magnetic matter can be replaced by volume polarization currents owing in freespace 17 FIELD BEHAVIOR NEAR A SHARP EDGE At sharp edges of material bodies the charge andor current density may be highly concentrated and in fact can become infinite in the limit of a mathematically perfect edge Consequently some of the field components may also be highly peaked near an edge It is important to understand the mathematical nature of this possible singularity especially in numerical computation where anticipating the correct form of the fields can often greatly speed the convergence to an accurate resu t As a simple and practical example consider the twodimensional conducting wedge in fig 115 which is infinite in the 2 direction A TEtoz excitation as shown will induce idirected currents on this object and the associated current density function will have a mathematical sin gularity at the edge From the boundary conditions this implies that the 3 and 13 components of the scattered magnetic field will be singular at the edge This is the only solution as guaranteed by the uniqueness theorem Now infinite field quantities may be acceptable as long as the physical 22 ELEMENTS OF ELECTROMAGNETIC THEORY TE excitation Figure 115 Cross section of a PEC wedge with TE excitation observables derived from them remain finite 1f Maxwell s equations are correct the singular solution must therefore behave such that the eld energy remain finite This physical constraint gives us the critical information for predicting the behavior of the fields close to an edge In the present case the electric stored energy per unit length in a cylinder of radius a surrounding the edge is 2 A similar expression describes the magnetic stored energy The fields within the fictitious cylinder can be expressed as a power series in p the dominant term in the series for small p will have the form p where 39y is the exponent to be determined For the electric energy in 162 to remain finite with E X p7 requires that 39y gt 71 From Maxwell s equations the dominant term in the expansion for the magnetic field can be determined as H X V X E X p74 and therefore for the magnetic stored energy per unit length to remain finite requires that 39y gt 0 Clearly then 7 must be positive in order to have a finite total field energy within the cylinder This also agrees with our expectation that E must vanish as p a Near the edge we can write E Ez m M Substituting into Maxwell s equations gives 82 87 72 18 f 0 163 1 2W7Ct a 2 Me 0 lEzl pdpd 162 0 0 Electrically close to the edge where kp ltlt 7 we can neglect the term sz and hence m Asin7 Bcosv 164 The unknown coefficients and allowed values of 39y can be determined by the requirement that E vanish at lt15 0 and lt15 27139 7 a which gives B 0 and 39y TNT2W 7 a where n 17 2 The smallest positive 7 describes the dominant term in the power series for the fields near the edge 139e for small p so we have TE case Ez m 41 sin39yqb where 39y W 165 A 27139 7 Oz LA H7 JKWVX E7 Jam77 pcosvzp s1n7 A similar derivation can be carried out for a TM excitation Problem with the result TM case Hz m 3 cos 15 166 B E V X H L We 061177 7 sinv 7 cosvz l NORMALIZED WAVEGUIDE MODES 1 RECTANGULAR WAVEGUIDE PEC y b a H X 0 Assuming 6quot dependence we have TE modes 6 0 TM modes h 0 hz A22 cosk11coskyy ez A312 siner sinkyy k k 1 er MyAEE zcos z simkyy 61 ymn AEDg Zcosersinkyy TE k1 TM k3 ey ijwyAmn k 2s1nk11coskyy ey 39ymnAmn k 2s1ner coskyy where m7 mr kz k k2k2 W 2 k2 a y b c z y Ymn Note that for TE modes either m or n can be zero but both cannot be zero simultaneously For TM modes neither m or n can ever be zero the lowest order TM mode is TM11 Enforcing the desired normalization condition a b imm 39 im n dy d1 6mm16m 2 o 0 we nd for the TE modes k e e 1 239 TE 7 c m0 710 J Am 7 W1 ab where 6 2 Zj 3 and for the TM modes kc 2 Wm lt4 So we can de ne normalized TE and TM eld vectors for the transverse elds as TM Amn 1 m 71 A A 6321 k 1kg cosk11 s1nkyy 7 ykr s1nk11 coskyy 5a 1 C a 2 TM 1 imn Cam ikr cosk11sinkyy 79kg sink11 coskyy 5b 2 IDEAL PARALLELPLATE WAVEGUIDE f PEC lt 4 b E E PMC 8 W X 5 0 Assuming 77 dependence we have TE modes 6 0 TM modes h 0 h A2 sinkzx coskyy ez A2172 coser sinkyy TE k TM k1 6 er jwyAmn siners1nkyy er 39ymnAmnEsinkzxs1nkyy k1 k 63 jwyAEEk zcos zx coskyy ey ivmnAgbf chos zx coskyy where 77 mr kz ky k k k yfmk2 a b Note that for TE modes m 2 1 and n 2 0 and for TM modes m 2 0 and n 2 1 Enforcing the desired normalization condition a b am am dy dz SW6m 7 o 0 we nd for the TE modes kc l2 7 39 39 ATE E D where eij 1 3 7quotquot jwp ab 2 1 J Enclosed Waveguides and Cavities 19 105 ANALYSIS OF JUNCTIONS AND DISCONTINUITIES For a given waveguide cross section the elds of a closed cylindrical waveguide can be described by an in nite set ofTE and TM modes Only a nite number of these modes can propagate at a given frequency the rest are evanescent Typical problems involving waveguides include the scattering of waves from obstacles placed in the waveguide gure 107 junctions between dissimilar waveguides or excitation of waveguides by sources placed within them The obstacles or sources introduce new boundary conditions so that the original TE and TM modes of the empty waveguide are no longer satisfactory solutions for the elds near the obstacle However since the emptywaveguide modes form a complete orthogonal set any eld can be represented as a superposition of the modes Note that this representation must theoretically include all the modes whether evanescent or propagating These emptywaveguide modes are an especially convenient basis since they already satisfy the boundary conditions on the waveguide wa s After representing the elds near the obstacle or junction in this way the expansion co ef cients are then found by enforcing the boundary conditions impose by the source or obstacle and exploiting the mode orthogonality property The procedure is directly analo gous to Fourier series analysis in the context of waveguide junctions or scattering problems it is usually called modematching reference lanes 39 39 scattered fie Ids InCIdent eld dominant mode transmitted 74 propagating 39 modes re ected lt propagating lt I modes evanescent elds exist in this region Figure 107 Typical waveguide scattering problem The scattered elds can be repre sented as a superposition of propaga ing an evanescent modes as dictated by the bound ary conditions At suit ably chosen reference places far from the obstacle only propagating modes are present and t e system can be modeled as a N port network where N is t e number of propagating modes The goal of analysis for problems such as in gure 107 is usually to determine the effect of the obstacle on the propagating modes of the system By de ning a set of references planes far enough from the obstacle so that all the evanescent elds have decayed to negligible amplitude the obstacle can be characterized entirely by the amplitudes of the propagating modes In many cases only the dominant mode is allowed to propagate in which case the region within the reference planes can be characterized by two complex numbers the 20 Chapter 10 re ection coef cient and transmission coef cient This is similar to an ideal transmission line junction We will nd that in general complicated waveguide problems can often be characterized by equivalent Nport circuits The detailed nature of the elds near the obstacle are only important to the extent that they in uence the amplitudes of the propagating modes We will nd that the expressions for the equivalent circuit elements for obstacles or junctions are often relatively insensitive to errors in the near elds of the junction and hence reasonably accurate results can be obtained with some educated guesswork about the elds 1051 Analysis of Planar Discontinuities by ModeMatching Simple discontinuity or waveguide junction problems that involve planar boundaries dis continuities lying in a 2constant plane can be treated in a straightforward manner by eldmatching techniques As an illustration of this method we will consider the waveg uide junction shown in gure 108 The true elds on either side of the junction can be Waveguide a PEC bomdary Waveguide b incident wave SE do minart mod e 55 Figure 108 Simple waveguide junction problem involving a reduction in ch section from Sm to 3 at z 0 and a PEC boundaries represented by a modal expansion The unknown coef cients are then found by enforcing the continuity of tangential elds across the junction and exploiting mode orthogonality This procedure results in a matrix equation for the unknown coef cients which is solved numerically Strictly speaking the method is rigorous only if an in nite number ofmodes are considered However excellent results can usually be obtained by truncating the expansions to a nite number of terms This is true because any highorder evanescent modes excited near the junction contribute little to the terms involving the dominant modes if their cutoff frequencies are well above the operating frequency The number of terms required to obtain adequate results is very problemspeci c and usually must be determined empirically Enclosed Waveguides and Cavities 21 We assume that the junction is excited from the left by the dominant mode of unit amplitude in waveguide a Just before the junction on the left 2 0 we write N E2 0 61 Z ai ai 1061a i1 A Na A HM 0 hm 7 2am 1061b and to the right at z 0 we can express the transmitted elds in terms of the modes of waveguide E 2 7 0 Nb 6 i 7 J b A 1062 Hi20i hbj where Na and Nb are the number of modes required in each part to adequately represent the elds For the junction of gure 108 involving a PEC boundary the boundary conditions at the junction 2 0 can be written as 7 7 7 0 on the metal 5a 7 Sb E42 7 0 7 E42 0 in the aperture Sb 106380 F z 0 EQ 0 in the aperture Sb 1063b Substituting the modal expansions for the elds gives N 0 on the metal A a A Nb 6 11 Z aieai Z bj bj in the aperture Sb 106430 11 1 Na Nb Hal 7 Zai ai ZbJ bj in the aperture 1064b 11 j1 The unknown expansion coef cients can now be determined by exploiting the modeorthogonality properties Dotting both sides of 1064a with ak and integrating over the cross section Sa gives Na Nb 6a adeZai ai adeij bj akd5 1065 5a i1 5 j1 55 and subsequently using 1051 allows us to isolate the kth expansion coef cient ak as Nb 6 76161 Eb 6 6 d5 1066 7 Sb J71 Similarly forming the dot product of 1064b with 11 and integrating over the aperture Sb allows us to solve for the expansion coef cients b as N a Z A A b 05 7a Seb eaidS 1067 11 a b 22 Chapter 10 The two equations for a set of Na Nb simultaneous equations for the unknown expansion coef cients For convenience let PJk bj 11de 1068 Sb Substituting 1067 into 1066 to eliminate the bl s gives Na Nb Nb ak Z a Z ijPji 76161 Z gleam 1069 i1 j1 j1 1 This can be written as a matrix equation E E E 1070 where the matrix elements R and ck can be written as Nb Rki Ski 2 j1 Ck Rm 25161 Zh ZfPJkPJi Given a certain setof waveguides and junction parameters the problem then involves com puting the matrix R and inverting it to form the solution vector E E4 c 1071 Once the re ection vector His known the transmission vector 3 can be found using 1067 For computational ef ciency it is desirable to do the overlap integral PJk in 1068 ana lytically if possible For problems involving canonical rectangular or cylindrical waveguide structures this is usually possible There are Nb gtlt Na of these integrals which should only be computed once and stored for later calculation of the RM Note also that the ordering of the indices is very important Many similar problems have been treated by Wexler 1 including boundary enlargement as opposed to the boundary reduction problem above and junctions involving more than two waveguides The procedures involve at most only slight modi cations of the above 1052 Variational Method In the previous problem the expansion coef cients describing the aperture elds were found by enforcing the boundary conditions Equivalent circuits can then be determined using the expansion coef cients for the propagating modes If the equivalent circuits are expressed directly in terms of the aperture elds it is often the case that the resulting expression is relatively insensitive to the exact nature of the aperture eld The expression is said to be variational or stationary with respect to small errors in the elds This is a useful property in practical work since one can often make an educated guess about the eld distribution and consequently deduce a reasonably accurate equivalent circuit for modeling the discontinuity Enclosed Waveguides and Cavities 23 apertLre fields E71 Ii at z 0 PE diaphragm Figure 109 Waveguide with a PEC diaphragm at z 0 Consider a waveguide with a planar metallic diaphragm discontinuity shown in gure 109 Here we assume that the waveguide crosssection is the same on both sides for simplicity If only the dominant mode can propagate in the guide then we can model this discontinuity by an equivalent shunt admittance 72F Yd Y1 1072 where I is the dominantmode re ection coef cient and Y1 is dominant mode wave admit tance Using similar ideas from the mode matching method we expand the elds to the left of the junction at z 0 as Eiz 0 l I 1 3 air 1 1073a i2 F42 0 1 7 1w 7 Zai i 1073b i2 and to the right at z 0 as f z 0 i 00 E l iizwl l l 22 libjl 25 1074 i 1 F2 J where 739 is the dominant mode transmission coef cient and 1111 represent the dominant mode We again enforce continuity of tangential elds at the aperture Denoting the true aperture eld as Ea Ha we have A 77 0 ondiaphragm 7i 7 z X E2 7 7 2 X Ea in aperture 7 z X E2 7 0 1075a f X F0 0 X a 2 X FQ 0 in aperture only 1075b 24 Chapter 10 Using the mode orthogonality and the boundary condition on tangential E eld we can easily nd 1rSEads anSEads ngt1 1070 and TSEads bnSfagnd5 gt1 1077 where the integration is over the aperture 5a not the entire waveguide cross section Note that 1 T 739 as expected from the eiuivalent circuit and that an 1 for n gt 1 Using the boundary condition on tangential H elds we can write 1 7 DE 7 2 an 1 mil 2 Mn 1078 712 712 Substituting for an and 1 and using 1041 gives a a 72F 2 E e d5 1079 Z Zn Salt gt lt gt Forming the dot product of both sides with fa and integrating over the waveguide cross SQE E1ds2221 nUSa aEn lsl2 This can be solved for T Using this result and 1076 we can then express 1072 as Yd 4p Emmy 2 1080 Y1 1I Sa aads This is the variational expressionijr Yd We wish to show that 1080 is stationary with respect to a rst order variation in Ea If the aperture eld is perturbed by a srnallcorrection fa gt fa Afa the adrnittance will change to Yd AYd Making these substitutions in 1080 cross multiplying and keeping only rstorder terrns leaves A7105 EMSF gzllL aamsAfnfaElms iii55EEnds gaAEads 1031 Taking the dot product of both sides of 1079 with Aid and integrating gives leLAE E1dS an aloof EMS section gives Enclosed Waveguides and Cavities 25 and since 1076 allows us to write Yd 7 721 7 721 1 1I EaE1ds Sa then the last two terms in 1081 are identical and AYdEaE1d52 0 1032 and therefore AYd 0 The rstorder arrection to Yd is zero for a small error in Ea In other words an approximate guess at Ea will yield a more accurate solution for Yd 106 INTEGRAL EQUATION FORMALISM FOR WAVEGUIDE SCATTERING The problems treated by model expansion in previous sections dealt exclusively with planar obstacles or discontinuities and consequently only transverse elds were required A more general scattering problem is shown in gure 1010 where an arbitrarily shaped conducting obstacle placed in the waveguide Incident elds on the obstacle will excite currents which will in turn reradiate elds into the waveguide We can treat this problem rigorously using the dyadic Green s function developed in the previous section The approach is typical of most scattering problems E inc W J5 ESL E scat S 71 Figure 1010 Scattering from a PEC obstacle in a waveguide Let the incident eld be fine and the scattered eld be Esau On the surface of the obstacle the total tangential elds must vanish so we write X fine Escaii 0 on S lo83 The scattered elds are computed from the surface currents using the Green s function Em ejw azr 77 dS 1034 One possible representation for G has been found already 1 z gt 2 G f 7 n 1085 G T 2101p ZnEn z lt 2


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