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by: Rodger Reichel
Rodger Reichel
GPA 3.65


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This 52 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 28 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
Vector and Dyadic Analysis I am greatly astonished when I consider the weakness of my mind and its proness to error iDescartes This appendix summarizes a number of useful relationships and transformations from vector calculus and dyadic analysis that are especially relevant to electromagnetic theory A1 COORDINATE SYSTE MS A11 Rectangular Coordinates xyz Z A1iAyyAz d dydzidxdzzjdxdy d2 id17jdy2dz dV dandde Z T zz1pane y Y1 plane 2 VECTOR AND DYADIC ANALYSIS A12 Cylindrical Coordinates 12 A z fly3 AM Az d dp 3pd 2dz dV pd dzpdpdz 3pdpd 2 pdpd dz I Cbl 2 21 plane p p1 y cylinder 1 71 m plane X A13 Spherical Coordinates rpA Z AT 4g Agb 3 d rzsinadederrsinedrdwwdrdw z fdr6rd6 rsin6d dV rzsin drd cw R dsr25in 0 d0 d4 r r1 sphere 4 1 plane COORDINATE TRANSFORMATIONS 3 A2 COORDINATE TRANSFORMATIONS Note that relations between the unit vectors in the different coordinate systems are obtained from the following by replacing the components of A with the corresponding unit vector for example AT i f A21 Rectangular lt gt Spherical transformation These coordinates are related by It rsin cos rsin sin Al 2 roost Q and conversion between vectors is given by AI AT sin cos Ag cos cos 7 A Sln Spherical gt Rectangular Ay AT sin 6 sin Ag cos 6 sin 14 cos A AT 00567149 sing A2 AT AI sin 6cos Ay sin 65in A c056 Rectangular gt Spherical Ag AI C056 cos Ay cos 95in 7 A sin 6 Avg iAzsin Ay 005 A3 A22 Rectangular lt gt Cylindrical transformation These coordinates are related by It pcos y psin A4 2 2 and conversion between vectors is given by AI AP cos 7 A Sln Cylindrical gt Rectangular Ay AP sin Avg cos A5 A z AP AI cos Ay sin Rectangular gt Cylindrical Avg 7A1 sin Ay cos A6 A A A23 Cylindrical lt gt Spherical transformation These coordinates are related by p rsin 4 VECTOR AND DYADIC ANALYSIS 2 rcos the azimuthal angle is common to both coordinate systems vectors is given by A sin Az C056 AT Cylindrical gt Spherical Ag AP C056 7 A sing Avb Avb AP ATsin9A cos Spherical gt Cylindrical Aqb A v5 A Arcos iAgsin A7 Conversion between A8 A9 A3 ELEMENTS OF VECTOR CALCULUS A31 Flux and Circulation Maxwell s equations are expressed in terms citwo important vector field concepts flux and circulation The ux 1b ofa vector field A through some surface S is defined as uxofAthroughS 1p 2 SZd and the circulation of A around some path C is defined as circulation ofA around C f0 d7 These concepm are expressed in differential form as the divergence and curl XE Divergence V AE liIn Curl39 V X A 11111 where we have defined the del operator which in rectangular coordinates is v7 BA 8A 8 x 2 81 Byy 82 szZ S I C VaO dV SaO d V S A 10 An A 12 A13 The V operator takes on different forms in other coordinate systems Section A4 lism explicit divergence and curl operations in the three most common coordinate systems Note that the concepm of fluxdivergence are also related through the Divergence theorem A 54 and the concepts of circulationcurl are also related through the Stokes theorem A 59 ELEMENTS OF VECTOR CALCLLUS 5 A32 The Gradient Another important operation is the gradientofa vector field which is the vector equivalent of a derivative operation The gradient only operates on scalar fields and is written as V Explicit forms for the gradient operation in the three common coordinate systems is given in section A4 The gradient produces a vector which points in the direction of greatest change of the scalar field This property is useful in a geometric sense for determining tangent planes and normal directions to an arbitme surface 2 In three dimensions an arbitrary surface can be described by the functional relation C A14 where C is a constant and f7 is shorthand for a function of the three coordinate variables for example in rectangular coordinates fac y 2 A plane tangent to this surface at the point 739 is described by 77 739 Vf739 0 tangent plane A15 The gradient points in the direction normal to the surface so a unit normal to the surface described by Al 4 at the point 7 can be found from unit normal A16 A33 Vector Taylor Expansion The multi dimensional Taylor series expansion of a function fT 3 around the point T can be represented in vector form as fltrmZ ltEvquotfltrgt Am A34 Change of Variables Inthree dimensions a change ofvariables from the coordinates at y 2 to new coordinates u v w is given by 2 8 fxy2 d1dyd2guvw Lylz dudvdw Al8 Buvw where BapBu BatBU Bataw 8 M ByBu ByBv ByBw Buvw BzBu 8281 828741 is called the Jacobian of the transformation and gu v w f 1u v w yu v w 2u v 6 VECTOR AND DYADIC ANALYSIS and it has been assumed that 1y2 can be expressed functionally in terms of u v w or vicaversa A similar result applies to transformations in twodimensions A4 EXPLICIT DIFFERENTIAL OPERATIONS A41 Rectangular Coordinates x y z Aaltrgt A A8 vltrgt x y Z A19 A20 8A 8A 8A 8A 8A 8A v A e y If 2 ye 1 A21 X 1a 82gty82 81gtZ81 6ygt v2ltrgt 7 8 8 8 A 22 7 812 By2 822 I v iva1 gvay NZAZ A23 A42 Cylindrical Coordinates 12 A z vltrgt 2 A24 10pA 1 BAG 8A v A A25 p 02 p Bab 82 V X 1824 7 8A2 p p B 82 2 0A 8A A1 apA 0A 7 7 A26 lien apl pi 6p ad 2 v2ltrgtli pa CD Lag 821 A27 pap 8p p28 2 822 2 72 2 1 VA pVA 228 22 2 2 l gt A 2 ltv14 p2 W p2 2V Az A28 VECTOR RELA TIONS 7 A43 Spherical Coordinates r p A 01gt A1alt1gt A 1 01gt vlt1gt 9 A29 TBr rB9 rsin9 B 1 00214 1 0Ag sin 9 1 0A v A A30 r2 Br rsin9 B9 rsin9 B 7 i BA sin 9 8A9 VXA rsm9 09 73 1 0A 00A 3 0149 0A 7 7 A 31 r sin9 B r Br B9 I v2lt1gt 7 i3 Ha D 1 3 5111982 1 8 A 32 7 r2 Br Br r2 sin 9 B9 B9 r2 sin2 9 B 2 I 1 02 1 a alt1gt 1 0ch 7 39 9 A33 raw T 5111989 51 B9gtrzsinz9B 2 2 BA BA 2 7 A 2 vb 9 VA7rV AT T2 ATA9COt6CC6 a 86 A 1 BA BA 2 7 2 7 T 4 9 V Ag T2 Ag csc 9 2 86 2cot9csc9 a A 1 8A 0A V2A 7 T z Av csc29 7 2csc9 a 7 2cot 9c9 85 A34 A5 VECTOR RELATIONS A51 Dgt And czross Product Identities A A 2 A35 2332 A36 ZxE 7gtltZ A37 Zx ExZEx A38 A X B X ZE i XE A39 AxBx xExZExxo A40 AC A41 x36 A42 8 VECTOR AND DYADIC ANALYSIS A52 Vector Differential operations v v v2 A43 WW wwww A44 vf Zv vZ A45 V X W xiizxw A46 VZX VX izVX A47 vXZx ZV 7 VZEVZ7ZV A48 WEE Xx V X EHEX v xZ vZZVEA49 v xw 0 A50 vv gtlt 0 A51 VXVXZ VVZ7V2Z A52 The last identity essentially defines the vector Laplacian V2 which reduces to three scalar Laplacians in rectangular coordinates only A53 Integral relations From the Fundamental Theorem ofCalculuS b b a V 112 Edi b 7 a A53 In the following V is a volume bounded by a closed surface S with the direction of d taken as pointing outward from the enclosed volume by convention Divergence theorem V dV d A54 V S Vv dV ME A55 Vv x2 dV d x A56 Note that A54 combined with A51 gives 5 v gtltA d 0 A57 S VECTOR RELA TIONS 9 and that A50 and A56 give A58 37 d xv 0 S In the following 5 is anopen surface bounded by a contour C described by line element 12 The direction of di is tangent to C The direction of dS is normal to the surface following the righthand rule with the fingers curled in the direction of C Sk th v ZE Edi A59 to es eorem gtlt f0 d xv f M A60 S C Note that A50 and A59 give W d 0 A61 C Green s identities and theorems provide additional relations between surface and volume integrals These are o en useful in proving orthogonality of eigenfunctions of the scalar and vector wave equations and also for boundaryvalue problems using Green s functions For two scalar functions and 1b which are continuous through the second derivatives in the volume V we have Green s rst identity V Vw V2w dV Vw dS A62 V S Interchanging and 1b and subtracting gives Green s theorem V2w 7 wvzw dV Vw wWJ 39 d V S A63 The vector forms of Green s identity and Green s theorem are VXZngizVXVXB dVZgtltVgtlt d V S A64 EvaxzizVXVXB dVZgtltVX 7 xVXd A65 V S which also require that Z and E are continuous through the second derivatives 10 VECTOR AND DYADIC ANALYSIS A54 Distance Vector Identities Let Ebe the position vector de ned by the two points 7 and 739 as shown below Also define E Eli where R is the distance between the points and R is the unit vector in the direction of R VB R A66 v1R iRRZ i RS A67 v E 3 A68 v R 2 R from A67 and A68 A69 V X R 0 from A66 A70 V X E 0 from A66 and A70 A71 v2 1 R 7476 A72 v ERS 4776 from A67 and A72 A73 In the following Eis any constant vector v ER EV1R 7 RZ A74 V2ER Wz R 74736 A75 V X Ex it122 477366 7 v Ram122 A76 EVR if fR A77 3 WE E from A66 and A77 A78 VECTOR RELATIONS 11 A55 The Helmholtz theorem The Helmholtz theorem 3 states that a vector function can be expressed as the sum of two vector functions one which has zero divergence the solenoidal or rotational part and one with zero curl the lamellar or irrotational part that is 27 v XEW A79 To show that such a decomposition is possible take the divergence and curl of A79 which gives v25 v E A80a vaXEvXZ ASOb Since A is assumed known these two differential equations age uncoupled and can in principle be solved independently for the pair of function s so This essentially proves the theorem Note that in order to uniquely determine A both its divergence and curl must be specified this is an alternative statement of the Belmholtz theorem Theriare an infinite number of possible functions g which can be used to uniquely determine A since the gradient of an arbitrary scalar function V canalways be added to g without changing A80 that is if is a solution of A80 sois 6 V We can pick any function that is convenient if is chosen such that V f 0 then V X W XE VV ivzg 7v and A80 become v25 v E A81a v22 7v x ASIb From electrostatics we know these have the solution for unbounded regions v Em v gtltAT39 7 dV dV 5W Vim 67 Vim and so A79 can be written as v 17 v XE A 7V dV V dV A82 T 477577 X trim If the field is to be represented in a bounded region then the solutions to A8l must be modified accordingly and it can be shown that the representation is more generally 17 7v v V 3937 dV 7 S Vim VX v AdV ATXi5 A83 V 47T 7 77 S 47139l77rl where S is the surface enclosing the volume V This is the formal statement of the Helmholtz theorem 1 2 VECTOR AND DYADIC ANALYSIS A56 Useful Vector Relations in TwoDimensions Situations arise where one dimension usually taken as 2 can be factored out of the analysis Let the subscript t represent vector components that are transverse to 2 so that 8 VVi AAi Az 82 Transverse and longitudinal componenm of other common operations can then be similarly decomposed Xx Az X iiBz xZi 1 x3 A84 w transverse longitudinal i 8 A v X A 72 x ViAz 2 X At Vi X A A85 82 W longitudinal tr ansverse We use the earlier vector relations in three dimensions to prove the following iden tities vi Wm vf A86 vi 2 X vm 0 A87 vi gtlt vm 0 A88 5 X 2 X Vm evid A89 vi X 2 X vm 2vf A90 2 X 2 XE J A91 mmME 1E A92 Z X 2 x3 22 E A93 vi X 2 XE 2vi E A94 2 xi X 2 XE Z XE A95 vi 2 x2 72vi x3 A96 Xi 2 XE 722 XE A97 In the following 5 is an open surface bounded by a contour C described by line element 12 The direction of di is tangent to C while the normal to C is described by 1 2D Divergence theorem Vi d5 f X 7 d2 A98 S C VECTOR RELATIONS 13 Green s identity A60 and Green s theorem A61 generalize to two dimensions as follows 2 7 aw 2D Green s identity Svi w Viwd5 7 fob dz A99 2 2 7 aw 7 ad 2D Green s theorem S Viw 7 wViwdS 70 771 w an dZA100 A57 Solid Angle An element of surface area for a sphere of radius a centered at the origin of a spherical coordinate system is given by dA a2 sin 6 d6 d It is sometimes convenient to view this element of surface area as subtending a solid angle d so that the angular integration in 6 and is replaced by an integration over the range of solid angles subtended by the surface That is we write dA a2 d and integrating over the surface ofthe sphere gives dAa2 d 47139a2 which is interpreted as meaning that the entire closed surface of the sphere subtends a total solid angle of 47139 The solidangle is a unitless concept but it is conventionally given the dimensionless units of Steradians This concept can be extended to any arbitrary surface S by forming the projection of each surface element d5 onto a sphere In the figure above CIA is the projection of the surface element d5 along the radial direction onto a sphere of radius a centered at the origin In doing so both CIA and d5 subtend the same solid angle d which from the 14 VECTOR AND DYADIC ANALYSIS discussion above is defined as d dAaz The projection CIA is found by taking the dot product of 15 dS with the radial unit vector and scaling the result by a factor of azrz where r is the distance to the surface element 2 dA dsa 72 and therefor e d d d T 2 and 61 T 2 477 A101 7 It is important to note that the result f d 47139 is critically dependent on having chosen the surface enclose the origin ofthe coordinate system Clearly if the origin were outside of the surface S then the surface no longer subtends a total solid angle of 47139 Mathematically this can be seen as follows From A 67 note that d 7Vlr d From the divergence theorem d 7 V2lrdV47T 6dV where the last equality follows from A72 The last integral is zero unless the volume bounded by 5 contains the point r 0 Shifting the coordinate system by To this result takes the more general form R39d 74 39 70insideS A102 5 R2 7 0 To outsideS where E 77 To This is essentially Gauss law A6 DIRAC DELTA FUNCTIONS Dirac delta functions are a convenient mathematical shorthand that are used to help us out of difficult situations In the context of Maxwell s equations such difficulties can arise from our description of charge and current distributions as density functions p and 7 respectively For example consider the charge density of a single electronihow do we represent such a thing From a macroscopic point of view the actual size of the electron is neglible and acounting for it would unnecessarily complicate the mathematics For an electron located at 7 0 with charge q a mathematical description of the charge density must have the properties pm0 for77 0 and p7dVq DIRAC DELTA FUNCTIONS 15 where the integral is taken over the region contining the charge A delta function in one dimension written as 61 is defined to have similar properties ie 1 agaco b A103 0 otherwise b 617100 forxy 1o and 6x7xod1 The most important property of the delta function follows from the above definition and involves its appearance in an integrand with another ordinary function As long as f is continuous at the location of the delta function singularity then the only contribution to the integral will come from this point and we get in one dimension b f151 7 atom 5 b A104 where the range of integration is taken over all values of It This is called the sifting property of the delta function The extension to three dimensions is straightforward at least in rectangular coordi nates We define 677 739 by the properties 67770 for73 7 and 6T7TdV 1 7 13quot A105 V 0 otherw1se which in turn lead to the sifting property 7 r 7 f39 if in V V Irma Tdvi0 otherwise AA106 In rectangular coordinates dV d1 dy dz and therefore 6T7 739 can be represented as a product of three one dimensional delta functions rectangular 6777 61 7 16y 7 y62 7 2 A107 Returning to our original example we find that the charge density function associated with a point charge at 7 can now be represented concisely as p0 165 As another example consider a current 0 flowing along a thin wire colinear with the zaxis Using the delta function we can represent the corresponding current density as 7069 10615y5 Although the current density so defined is singular at It y 0 the integral over the cross section of the wire will remain finite and provide the correct answer I 7d 06x6y d1 dy 10 These examples also illustrate that the delta function must have units If It represents a physical length dimension then 61 has the unis of inverse length Examining the 1 6 VECTOR AND DYADIC ANALYSIS expressions for the charge and current density above we see that the correct units of Crns and Arnz are obtained respectively with this association of units From the sifting property of the three dimensional delta function we see that it has the units of inverse volume ldV In three dimensions the differential element of volume takes different forms in different coordinate systems and so the delta function must be represented somewhat differently in each case To transform from the representation in rectangular coordinates A 107 to some other set of coordinates u v m we use the change of variable theorem of the previous section and note that the volume element in the new coordinate system is given by lJldudv dw where lJl is the Jacobian of the transformation Therefore a representation for the delta function is 6777 6u7u6v7v6w7w A108 Using this we nd for cylindrical coordinates l cylindrical 6777 6p 7 p6 7 62 7 2 A109 p and for spherical coordinates so 7 mm 7 66 7 W AllO rzsin 9 spherical 677739 There are situations where this approach breaks down however corresponding to the singularities of the Jacobian This occurs when the delta function peak is located such that one of the variables u v w is irrelevant in the transformation For example in cylindrical coordinates if the delta function is located on the zaxis the azimuthal angle does not appear in the transformation and the representation is instead 4 M77 lp p6z72 A111 One can always check the validity of a delta function representation using the integral properties de ned above Similarly in spherical coordinates points on the zaxis corre sponding to 9 0 or 9 7139 are represented by l 6 7d T T 27T7 25ln9 6r7r66 76 A112 For points at the origin both 6 and are irrelevant and 6 r 5 7quot A113 7 T 4W2 Having shown how the threedimensional delta function can be represented by producm of onedimensional delta functions we now list some additional properties of the latter that are useful in electromagnetic analysis 1 6a1 7 b a 6x7ba All4 DYADC ANALYSIS 17 6127amp2617a61a AllS 6xia6xibd16aib AUG fx61 7 ad1 ifa All7 where in the last relation the prime denotes a derivative with respect to the argument Another useful transformation is given by 61 7 11 5 f 1 ANS lt ldfmdavl where 11 are the zeroes of fac ie 0 and the summation is over all the possible zeroes A more exhaustive collection of delta function properties relevant to electromagnetic theory is found in 4 A7 DYAD Ic ANALYSIS In elementary vector analysis we frequently encounter scalar relationships between two vectors such as in Ohm s law J 0E where a is a scalar quantity In matrix form J I Eat J y 0 E3 J Ez This is a very simple relationship which takes the Ector quantity f and scales each component by the number a to give a new vector J which consequently retains the original direction of E A more general linear transformationwould allow each component of f to influence each component of 7 so that the transformation changes the direction as well as the magnitude ie involves a rotation in addition to a scaling We could write this in matrix form as Jr a 013 01 E1 J31 0311 03131 w E31 J z an 0 y 0 E The matrix a is referred to as a secondrank tensor Each component of the tensor describes the in uence of one field quantity on another for example 01 describes the icomponent of current ow due to the component of the electric field Such tensor relationships arise in many physical contexts such as current flow in an anisotropic crys tal or wave propagation in a plasma Naturally the mathematics becomes more coran cated which is why tensor relationships are rarely covered in elementary electromagnetim courses 18 VECTOR AND DYADIC ANALYSIS The tensor relationship can be written in a different way using vector notation as 7 7 f A 1 19 where 7is defined as 011323 013557 01552 031793 7ny ayzyj azz i azy zj mg The only new feature is the appearance of products of unit vectors This definition gives the correct result using the normal rules for the vector dot product provided we Strictly obey the order of the unit vectors and the dot product For example i E iEy but interchanging the order of the unit vectors clearly gives a different result 7932 QEr Similarly we can see that integha ginithiorder of the dot product in A119 also markedly affects the result ie 3 E E E This is not surprising given the obvious similarity between this new quantity 7md an ordinary matrix Since the components of are characterized by pairs of unit vectors it is called a dyad or a dyadic quantity the word dyad means pair Clearly there is a close relationship between dyads and secondrank tensors A dyad or dyadic operator is expressable as the algebraic product of two vectors or vector operators much like a matrix can be formed from the product of two vectors Y7 A120 To the extent that the vector fields represent or can be related to physically meaningful quantities a dyad only has meaning when it acts upon another vector However we can often ascribe an independent physical signi cance to dyads such as E in this case the conductivity dyadic As noted above dyadvector multiplications do not obey the familiar vector commutation rules A36 A37 but obey instead the matrixlike commutative laws 2 ZT T P X A 7 A X PT where the superscript T suggesm a matrixlike transpose operation For example T 011555 03195 0121 013327 7ny7 azy zj 011322 ayzyj azz Consequently one must resist the temptation to use dyads in place of vectors in the vector identities of section A 5 which are derived assuming the simpler vector commutation laws A36 A37 where ordering of the vectors is not as significant DYADIC ANALYSIS 19 It is frequently useful to employ a unit dyad I defined such that tel 77ZZ A121 In rectangular coordinates 7 33yg22 A122 This is analogous to the identity matrix in linear algebra ln electromagnetic theory dyadic notation is frequently used for brevity Once the reader becomes familiar with the notation we find it can be employed in many situations formerly handled by vector manipulations A simple example given in the text is the function VV A Ordinarily expression is understood to mean VV A but it can also be represented as WV A where VV is a dyadic operator Similarly the function 77 5 which appears in the radiation integrals can be represented by Y7 M A71 Dyadic Dot and Cross Product Identities M U H H UJI a A123 ZXET7XZ A124 Z ZE Z gt A125 ZF FTZ A126 xEZ x gt7 Zxgt A127 ZxE Fx gtZxZgt E A128 Xx Ex EQ ia E A129 Xx Zx gtZx A130 2 x54 RE A g A131 Zx x 1xx gt2xx A132 5gtT3T r A133 K 32 3 Z3 A134 6 Z3Zgt3Z A135 1x6 xj 5 A136 20 VECTOR AND DYADIC ANALYSIS A72 Differential Operations Involving Dyads v ME v Z vZ v v v VX gtV x vX vZEvZEXv v A 7 A VXEXZ VXZBVgtltA 72XV Vijv14 gtlt 7VB x2 VZXEVXZgtizVX VXVZ0 VVx 0 VXltVXVVgt7V2 A73 Properties of the Unit Dyad T 1771 7xZx 2X7 32 Raga 7x 3379 42 2x3 Zx v 7gtv v 7XZgtVgtltZ A137 A138 A139 A140 A141 A142 A143 A144 A145 A146 A147 A148 A149 A150 A151 A152 A153 A154 A155 A156 A157 A158 DYADIC ANALYSIS 21 A159 A74 Integral relations We can generalize the earlier vector integral theorems in a straightforward manner to accomodate dyadic functions In the following V is a volume bounded by a closed surface S with the direction of 15 taken as pointing outward from the enclosed volume by convention39 v dv 5 61 A160 V S V dev 5652 A161 VvXd1 s x A162 In the following 5 is an open surface bounded by a contour C described by line element dz The direction of d is tangent to C while the normal to C is described by fr The direction of d is normal to the surface following the righthand rule with the fingers curled in the direction of C Sd VXE 612 A163 Sd x v2 f0 6122 A164 The vectordyadic form of Green s identity A64 and Green s theorem A65 are see 5 for a derivation VVXZVXZWW61V5 6zxvxds A165 vaXmizvaX 61v 6zxvxwmx ds A166 where 65 51615 The 6661666616 16666 61166 66666 666 VW6TWimx5f 6v vXT6X 66 6167 Fundamental Properties of Antennas The beginning of wisdom is the de nition of terms iSocrates An incredible diversity of radiating structures can be found in modern communication and radar systems Ignoring superficial geometrical or mechanical differences what distinguishes these structures from each other In later theoretical work we will find it convenient to classify antennas according to the analytical methods used andor the physical basis for the radiation But in a practical sense the most important distinguishing characteristics are those that describe the performance of the antenna in a real system Our goal in this chapter will be to define several important parameters representing the radiation characteristics in transmission and reception as well as the circuit properties of antennas and show how these can be computed from a field analysis This blackbox description of the antenna will provide welldefined goals for later theoretical work as well as providing the necessary tools for using antennas in practical systems 31 TRANSMITTING PROPERTIES In a communications system the antenna is a transducer that couples energy from one electronic system to another at some distant location Two important aspects of the system are immediately apparent l the antennafield interaction and 2 the antennacircuit interaction In the first case we are concerned with the directive properties of the antenna and the polarization that is we must determine in which direction and how effectively the antenna interacts with electromagnetic waves In the second case we require the circuit properties of the antenna when used as both a transmitter or receiver39 for example we must know the equivalent impedance of an antenna as perceived by the generator or the Thevenin equivalent circuit when used as a receiver We will begin by considering the directive properties of the antenna 2 FUNDAMENTAL PROPERTIES OF ANTENNAS 311 Radiation Patterns and Directivity A useful idealization in antenna theory is the concept of an isotropic radiator which is a fictitious point source that radiates uniformly in all directions For this case the Poynting s vector P N34 Figure 31 Isotropic source should have only a radial component and depend only on the radial distance from the source ie 5 PAT The total average power radiated Prad is found by integrating the power flux through a sphere of radius r around the source giving Prad f d 5 f r2 d9 4773150 31 P rad 47772 50 f isotropic source 32 Since the Poynting vector is related to the fields through 1 P ReEgtltH 33 then the radiation fields those that are responsible for carrying real power away from the antenna must vary as eijm E F X 34 which is the form of an outward spherical propagating wave This behavior was discussed in Chapter 2 in connection with the Sommerfeld radiation conditions but here we see it as a simple consequence of energy conservation The Sommerfeld radiation conditions also tell us that far enough away from the source the radiated fields are transverse to the direction of propagation TEM waves so in spherical coordinates we expect E E9 E q3 35 A A 7E A E A H H96H 6 9 n n The fields close to the antennaithe nearfields 4can be quite complicated and not described well by the simple spherical wave 34 but the radiation condition assures us that in the farfield we always have a transverse wave with 1r dependence In reality no physically realizable source can radiate uniformly in all directionsithere is always some angular dependence to the radiation We will prove this later So for real antennas we must modify 34 and write the farfields as eijm tljl gl 67 lt15 T farfield 36 f gtlt E El 1 77 TRANSMITTING PROPERTIES 3 where Rio is the farfield radiation pattern or scattering function The vector nature of T is necessary to describe the polarization properties of the fields The farfield Poynting s vector for a real antenna then takes the form m lf6 l2 37 where l f 6 l2 is referred to as the power or intensity pattern Unless otherwise stated radiation pattern plots are usually of the powerpatterns and not the field patterns But remember on a dB scale the two are exactly the same since 10log if M2 20 log if Furthermore the pattern functions are often normalized so that the maximum value of l f 6 M is 1 in which case U0 is the maximum radiation intensity A plot of the pattern function conveys much information about the directive properties of the antenna As an example fig 32 shows part of a radiation pattern for an electrically small dipole antenna oriented along the zaxis Such plots also provide a nice visual method for comparing igv 0quot 41 r m 1 In39O n I I 7 gure 32 3D radiation pattem for a dipole an tenna f6 o sin cut to show cross section antennas Comparisons of the magnitude of power density in various directions are most meaningful when the two antennas radiate the same total power For example a plot of the relative power density patterns not normalized in the 902 plane for a idirected point dipole antenna alongside that of an isotropic radiator for the same total radiated power is shown in figure 33 In this Di ole Isotropic re 33 Comparison of isotropic source and point dipole power pattems case we see that in some directions the radiation intensity in the farfield is greater if a dipole is used than if an isotropic source were used and in some directions it is smaller This increase or 4 FUNDAMENTAL PROPERTIES OF ANTENNAS decrease in power over that of an isotropic source is called the directive gain or directivity of an antenna It is not a power gain in the sense of an active amplifier The directive gain is only a function of the angular direction so it can be represented as D67 lt15 and is defined quantitatively so that the power density is given by 72 m Dams 38 An isotropic source therefore has a directivity of D67 lt15 1 Rear ranging this expression gives L DM am or in terms of the pattern function from 37 6 2 D67 lt15 47TM 310 MN m In practice the maximum value of the directivity function is most often used and in fact this maxrmum value is often casually referred to as THE directivity If the pattern function has been normalized the maximum direcivily D0 is D 7 1 0 41 MN m As we will see the directivity is an extremely important parameter in practice However a graphical representation of the pattern function f 6 lt15 is also quite useful and can be drawn in a number of ways to enhance or suppress various features of the function as desired The accompanying Mathematica file PlottingExamples nb includes a summary of many of these formats and example code for generating radiation patterns from a given f 6 lt15 Two common representations of a highly directive beam are shown in fig 34 While there are many examples of patterns not represented by this figure including the dipole pattern of fig 32 it serves to help us define several common terms that are used to describe radiation patterns in general The central feature of importance is the main beam or main lobe which describes the direction and angular extent of the bulk of the radiated energy Usually there is a single main lobe as shown along with a number of much smaller minor lobes or side lobes which usually account for a small fraction of the total radiated energy The minor lobes immediately adjacent to the main lobe are called the rst sidelobe Of special importance in many applications is the sidelobe level which refers to the peak sidelobe power relative to the peak main lobe power Many radiation patterns also involve deep pattern nulls in one or more directions which are a result of almost perfect destructive interference of the radiation from all parts of the antenna The main lobe is characterized by two commonly used and fairly selfexplanatory beam width parameters a the HalfPower Beam Width HPBVxl which is the angular width between half power 3dB directions around the main beam also sometimes referred to as the fullwidth halfmaximum FWHIVT beamwidth and b the FirstNull Beam Width FN39BVxl which is the angular width between the null locations on either side of the main beam location assuming there are nulls there Clearly either of these definitions are dependent on the cross section of the pattern so That Dw lt1 D0 lf6 l2 311 TRANSWTTWG PROPERTES 5 mm mm r w mm heamwmm FNBW lililrepower heamwidllk uwmw Mmm lobes nm lobr Mme lolm m A R lklmllon mlemily Hullpower hciuuwulllltl ll llW I list null hummuuuFNBM mm 101w side lube Buck lobe lmor lube y on Figure 34 Two VIEWS of a typical highly directive radiation pattan displaying a single main lobe and several walla i bes and illustrating beamrwidth de nitions and sidelobe taminology a Threedimeqmnai View dimmsional cross seotio used For highly directivity antennas with a single narrow lobe or beam sometimes called a pencilbeam i is com t de ne a equivalent beam solid angle as shown in gure The beam is replaced by a cone of solid angle 0A over which all of the radiation is distributed uniformly Therefore the power density at the end ofthe cone at a distance 739 is giv 312 6 FUNDAMENTAL PROPERTIES OF ANTENNAS Figure 35 Beam solid angle Comparing this with 38 we can relate the beamsold angle to the directivity as 47139 47139 Q D 313 A Do or 0 9A The narrower the beam the higher the directivity In terms of the pattern function so l w cm 314 As such we can compute a beam solid angle for any pattern not simply pencilbeam patterns The ratio IA47139 describes the fraction of visible space into which radiation can be foun Show how beamsolidangle can be approximately found by multiplying angles in the or thogonal planes Example Beam solid angle and directivity ofa Herlzian point dipole The normalized field pattern of a Hertzian dipole is M gt sine From 314 we get 9A lf6 lzsin6d6d 27r sin36d6 8 0 And so the maximum directivity is from 313 47139 3 D 15 0 8773 2 1n terrestrial communications links and radar it is common and quite natural to specify the angles 6 and 15 relative to the ground as angles of elevation and azimuth respectively These are described in fig 36 for a typical groundbased re ector antenna system TRANSMITTING PROPERTIES 7 dire ction o f radiation Figure 36 Angles are o en speci ed in terms of azimuth and elevation in ter restrial antennas 312 Applicationspecific radiation patterns There are numerous other terms used to describe radiation patterns in special cases For example radio broadcasting antennas are usually designed for uniform coverage over all angles in azimuth as shown in fig 37a This type of antenna is called omnidirectional Note that omnidirectional antennas are not isotropic radiators they do not and can not radiate uniformly in elevation We may anticipate that omni directional antennas will have azimuthal cylindrical symmetry in their physical construction For example a vertical cylindrical tower or pole can have omni directional characteristics Since the signals are transmitted to groundbased receivers an ideal shortrange broadcasting antenna would in fact have a narrow beam cross section in elevationias narrow as possibleito achieve high gain and hence maximum transmitting range In contrast pointto point communication links have highlydirectional beams as shown in fig 37 0 Figure 37 Omnidirectional directional and sector coverage antennas Surveilance or airtrafficcontrol radars may require the high gain of a directional beam but the full azimuthal coverage of an omni directional antenna this is often achieved by mechanical rotation of the beam which is sometimes referred to as a spotlight beam We will also discuss how to achieve beammotion electronically when we discuss phasedarrays in a later chapter Full 8 FUNDAMENTAL PROPERTIES OF ANTENNAS coverage in azimuth can also be obtained using several sectorcoverage antennas as in fig 37c In this case each antenna has a high gain in elevation and covers a specific sector of space in azimuth In this case the designer must balance the improved gain resulting from limited sector coverage with the complexity and economics of using multiple antennas often 34 antennas prove optimum direction of incomjn aircraft gure 38 Cosecant pattern In some cases the radiation patterns of the antennas are designed for a specific shape in elevation The classic example of a shapedbeam antenna is the socalled cosecant antenna which is used in airtrafficcontrol radar The cosecant pattern maintains over a limited range of angle in elevation a constant power density independent at a fixed height h from the ground independent of the distance d from the antenna This can be easily seen using the geometry of ig WW2 BDO 05026 7 BBQ p 7 PtDO 47139 d2 h2 47Th 477W since csc d2 h2h As we will later see this is particularly useful in radar systems However shaped beam antennas are not confined to radar applications or cosecant patterns For example many modern communications satellites now employ shapedbeams to selectively illumi nate portions of the the globe Show HNS spaceway for shapedbeam directed at various parts of Earth Narrow beams are often used in radar and communications applications and can be realized using large apertures or arrays In these applications narrow highly directive beams with very low sidelobes are required but as we will find later there is an important tradeoff between sidelobe level and beamwidth The optimum balance between the two is very application specific and numerous designs are in use One specific type of radar that deserves special mention is the manapulse radar which requires an antenna that can switch between two narrow beams as shown in figX called sum and difference patterns These beam patterns are used for precise angle tracking The target is first acquired using the sum pattern so that is it known to fall somewhere near the pattern peak The antenna is then switched to the difference mode Unless the target is located exactly in the central null of the difference pattern a signal will be detected in the receiver The signal will have a different sign depending on which central lobe the target falls in The angular position of the antenna is then adjusted until this signal is minimized Difference patterns with very deep nulls are required for precise angle tracking TRANSMITTING PROPERTIES 9 Figure 39 Monopulse a sum and b difference patterns 313 Radiation Efficiency Impedance Mismatch and Gain There are practical complications regarding our definition of directivity that are important in many situations In a real antenna the radiated power comes from a generator connected via some feed network to the antenna terminals as shown in figure 310 Ideally all of the average available power from the generator Pavail VgZSRe 29 V9 is a peak quantity would be radiated by the antenna but this is never the case due to impedance mismatching and ohmic losses in the antenna Let s ignore matching losses for the moment and consider just the effect of antenna losses if an Z g reference plane Figure 310 A transmitting system with Pin Prad the source network replaced by a Thevenin V W M equivalent g amount of power Rn is delivered to the antenna terminals and only some fraction eT is radiated then Had 315 and er is called the radiation e icienqy of the antenna It can be expressed as 6T Prad Prad 316 Pin Prad Plost The lost power could be due to ohmic losses or other mechanisms such as excitation of trapped modes in the dielectric substrate of a planar antenna that prevent energy from being radiated in the desired fashion Substituting 315 into 38 gives Pin PAW 4W2 lerD67 l The quantity in brackets is called the antenna gain denoted by 06 lt15 It is the quantity of most practical interest but unfortunately the term has been abused in the literature so that directivity P 2 06 317 4777 10 FUNDAMENTAL PROPERTIES OF ANTENNAS and gain are now often used interchangeably For efficient antennas where e7 m 1 then little harm is done by this abuse of words but it should always be remembered that there is a significant difference between the gain and directivity of the antenna Matching losses can be treated in much the same way but in this case we do not include the mismatch factor in the antenna gain function since it is not an intrinsic limitation of the antenna but rather the way it is being fed With a mismatch between the source and antenna 315 is modified as Prad 57391 lFal2Pavail 318 where Ta is the re ection coefficient defined by Zant 2 r 9 319 a Zant Zg and Zant is the antenna terminal impedance The farfield power density is then written as Pavail 47772 Pavai 174mb Wu 7 lFal2eTD6 1 7 lFal2G6 320 Expressions like 320 are frequently used to predict the frequency response of an antenna system in which case it should be remembered that the antenna impedance and gain are both functions of frequency although we have not shown this dependence explicitly in our notation 314 Polarization Properties In the farfield the electric field in general has both a fl and lt13 components as in 35 The relative magnitude and phasing of these components determine the polarization of the antenna which describes the vector orientation of the radiated field and how that orientation varies in time If we take the phase reference as the 6component we can write E iEgi triplequ where Aw is the relative phase diference between the two components The simplest case to consider is a linearly polarized field which is described by either of the following two conditions a one of the two field components is zero such as the fields of the Hertzian dipoles when the dipole moment is coincident with the iaxis or b the relative phase is Aw 0 or 7139 In each of these cases the tip of the field vector traces out a line on the flq surface Another special case is the circularly polarized field when occurs for the very special case of EM EM and Aw lZ7T2 In this case the tip of the field vector rotates around the axis of propagation as depicted in fig 3lla tracing out a circle on the flzib surface The direction of rotation is determined by the sign of Aw and is described in two ways The wave is said to have righthand polarization or RHP if the field rotation is in the direction of the fingers of the right hand when the thumb is pointing in the direction of propagation Otherwise it is called lefthanded circular polarization or LHP Equivalently RHP is sometimes referred to as clockwise or CW polarization since the field is observed to follow a clockwise rotation it when looking along the direction of propagation and LHP is similarly called counterclockwise or CCW polarization When both field components are present with different magnitudes andor a relative phase that is not one of the special cases described above the field vector then traces out an ellipse as shown in fig 3llb We define the axial ratio as the ratio of major to minor axes on the TRANSWTTWG PROPERTES 11 Minor m Minor 1m by Palmmmquot tlllpsc Figure 311 polarized wave b Gmaal polarization ellipse polarization ellipse Therefore a cicularly polarized signal will have an axial ratio of unity and a linearlypolarized signal will have an axial ratio of in nity ideally I eneral we will represent the eld polarization using our pattern function Vector 70 gt which we can write as 7W Mew HM gtgt This nr1ation ernphasi e 39 the antenna quot r quot the direc tion of observation each eld component has its own independent radiation panern Alternatively we can write the pattern Vector in terms of the Vectors long the major and minor axes of the 12 FUNDAMENTAL PROPERTIES OF ANTENNAS polarization ellipse H645 fp67 13fq67 321 in which case the axial ratio is simply fp AR 322 fq For directional antennas we are primarily interesed in the polarizationproperties in the direction of maximum radiation When an antenna is said to transmitreceive a certain polarization it is implied that the radiation is transmittedreceived along this direction However it is rarely the case that the antenna will maintain the same polarization or axial ratio over all possible angles In fig 2 we show examples of measured radiation patterns in the two orthogonal polarizations for circular and linearlypolarized antennas It is clear that the desired characteristics in each case are maintained over a relatively small range of angles around 6 0 Show plots of fp and fq for linear and circular antennas Show a radiation pattern that is measured with a rotating dipole3 Any state of polarization can be decomposed into a combination of two orthogonal polariza tions We have already done the obvious cases of decompsition along the zib or directions but an wave can also be decomposed into a combination of RHP and LHP waves For example we can write 321 as A J B13 J 323 V V LHP RHP where A ft 7 qu2 and B ft qul2 Most communications or radar systems are designed to operate in linear or circular polariza tion However it is difficult to obtain perfect linear or circular polarization When the polarization characteristics differ from the desired characteristics the undesired polarization is called the cross palarizatian Therefore in the antenna of fig b we woudl consider the xcomponent to be the crosspolarized radiation For a circularlypolarized antenna producing a desired rotation the crosspolarization represents a coupling to the opposite sense of rotation but this is hard to measure and hence the axial ratio is specified instead as determined from measurements like that of fig 2 We have already pointed out that threedimensional radiation patterns are often represented by two orthogonal cross sections For linearly polarized antennas these crosssections are commonly chosen based on the orientation of fields produced by the antenna A waveguide horn antenna shown in fig 311 serves as a good example If the horn is fed by a rectangular waveguide operated in the dominant mode the radiated field will have the same linear polarization as the waveguide mode which is the Q sdirection in fig 311 The plane defined by the beam direction 2 and the electric field 92 is then referred to as the Eplane and the cross section of the beam taken along this plane is called the Eplane pattern The H plane pattern is similarly defined Clearly it is only possible to define an E or Hplane pattern for a linearlypolarized antenna 315 Terminal Impedance and Bandwidth From the point of view of the electronic system to which the antenna is connected the antenna is simply another circuit element with a complex impedance or admittance that must be matched to the rest of the network for efficient power transfer Nearly all antennas encountered in practice have a complicated frequencydependence of the input impedance which limits the bandwidth of operation when connected to a generator with a different internal impedance Some traveling wave antennas and selfcomplementary structures that are physically large compared with a wavelength TRANSMTTWG PROPERTES 13 Erl lune 14mm wermre Lllsirmulmn can have a wide range of operating frequencies over which the circuit characteristics are relatively constant but in general smaller antennas support a standing wave of current and consequently structures The frequmcy dependence of the reactance is characterized by alternating poles and W en imaginary part reactance has a positive linear slo e then a series RLC model is appropriate If e admittance displays this variation at conductance positive linear slope for susceptance then a parallel RLC circuit is appropriate e uivalent circuit parameters near resonance can be determined directly from the plots For a series resonator we write 1 ZBwL BjX 324 WC Resonance is de ned as the point where the reactance is zero X O which occurs at the resonant frequency 1 mo 325 FUNDAMENTAL PROPERTIES OF ANTENNAS 1500 w 1000 500 500 Impedance Ohms O 1000 1500 O llll llll llll llll llll llll o 97 o 0010 0005 0000 Admittance mhos 0005 00 o M o 0 o 5 o 01 o 07 0 Frequency GHz Figure 312 Typical impedanceversusfrequency plot top graph for an antenna in this case a center fed dipole and corresponding admittance bottom graph The appropriate circuit models are identified near resonance points Near resonance we can write to we Am and we find 2a m R j2AwL 326 so BXBw m 2L near resonance in other words the slope of the reactance and the zero crossing determine the equivalent reactive elements The real part R is easily read from the graph directly The unloaded Qfactor of a series resonator is 7 MOL y we 8X Q 7 Y 2R 8w 33927 uuo TRANSMITTING PROPERTIES 15 and the impedance is sometimes written in terms of this quantity as A Z N R 1 j2Q w 328 we We can characterize the bandwidth of the antenna in a communications circuit by connecting it directly to a generator with a source resistance of R9 as shown in figure 313 Note that the additional loss in the circuit will lower the overall Qfactor of the system This net Qfactor is called the loaded Qfaclar QL and is given by 7 MOL TRgR QL 329 Assuming the generator is matched to the antenna at resonance Rg R the radiated power is Rg R Vg Figure 313 Series resonant antenna connected to gen erator ZLI 1 V 2 1V lZSR Pm R 9 9 330 d 2 12R1QAww0l 1Q2AwWO2 The halfpower 3 dB bandwidth BW is then found as 7 2010 7 E N 4R BW7 Q 7 QL N aXawLm 331 note this is in unit of rads This simply confirms that a broadband antenna should have the smallest possible variation in reactance with frequenc 1n electronic systems it is comm on to specify bandwidth as a fraction of the carrier or center frequency The fractional bandwidth FBW is defined as BW FBW gtlt 100 332 we A typical communications system may have a fractional bandwidth of 510 which requires an antenna with Q lt 20 At high frequencies the antenna is usually fed from a generator with a 50 2 source impedance and for maximum power transfer over a broad frequency range a matching network fig 314 must be used In principle the matching network can be designed to increase the inherent bandwidth of the antenna as described above but there are both theoretical and practical limitations on such matching networks when the load is reactive The design of impedance matching networks is well beyond the scope of this book but is an important aspect of antenna system design and the interested reader is referred to the encyclopedic work by Matthaei 2 for details 16 FUNDAMENTAL PROPERTIES OF ANTENNAS Matching g Network Figure 314 Impedance matching 32 RECEIVING PROPERTIES OF ANTENNAS 321 Relationship Between Transmitting and Receiving Properties For antennas comprised of linear reciprocal conducting or dielectric materials we can establish a general relationship between the transmitting and receiving properties using the Lorentz reciprocity theorem To do this we consider a typical communications system shown below in fig 315a The antennas and the relative separation and orientation are completely arbitrary The same physical situation is shown in fig 315b except that the transmitter and receiver circuits have been inter changed We assume that the transmitter and receiver are both connected to the antennas through a length of singlemode transmission line or waveguide In each case we assume that the only Figure 315 Generic pointtopoint link a Antenna 1 is the transmitter b antenna 2 is the transmitter impressed sources in the problem are located in the transmitter We can exclude these sources from the volume V by deforming the surface S to include surfaces 31 and SQ There can still be currents owing inside of V on the antennas or other bodies of matter but they will all be induced currents The fields and currents inside V in each of the two situations of fig 315 are denoted by Em mja and Eb bjb respectively Letting the surface S extend to infinity and applying the Lorentz reciprocity theorem gives Eax i x ald J Z i dv 333 SlSz V Since there are no impressed currents in the volume by design the volume itegral vanishes assuming the medium within the volume is reciprocal Furthermore if we assume the transmitter RECEIVING PROPERTIES OF ANTENNAS 17 and receiver and feed lines are shielded by a perfect conducting surface then the surface integral terms will vanish everywhere along 31 and SQ except over the cross sections of the feed lines at the reference planes Therefore we have Ea gtlt F 7E gtlt F d 0 334 RP1RP2 where RPi is the crosssectional surface defined by the ith reference plane If the reference planes are chosen sufficiently far from the antenna so that higherorder evanescent modes have dies away then the fields in these integrals are therefore described in terms of the dominant mode fields Using normalized mode functions we can write the tangential fields in the transmission lines as superpositions of forward and reverse travelingwaves ET V I e j Vi ej go 335a 1 A HT Z V 7 V e g X 50 335b 0 where 20 is the characteristic impedance 3 is the propagation constant on the feed line represents the axial direction direction of wave propagation on the feed line and E is the normalized transverse mode function for the dominant mode satisfying a aods 1 336 RP In this form and taking the reference planes at g 0 in each feed line the total voltage and current at the reference planes can be written as V V V 337a W V I 2 0 7 70 337b Substituting the fields 335 into 334 and using 336337 gives Vlar V511 V1be Vszg 338 where the superscript a or b denotes the physical situations of fig 315 We have reduced the system to a two port network as shown in fig 316 The voltages and currents at the reference planes can be expressed in terms of the equivalent impedance parameters of the twoport V 21111 21212 339a V2 22111 22212 339b H The Zparameters of this twoantenna system are independent of the transmitter and receiver configuration depending only on the antennas themselves and their separation and the intervening medium Using 339 in 338 gives Z 12 221 which is the familiar statement of reciprocity in circuit theory Parameters 212 and 221 are called the transfer impedances for the system and are responsible for the coupling between the two 18 FUNDAMENTAL PROPERTIES OF ANTENNAS I1 12 gt Tx Rx V1 V2 TX Rx 4 R gt reference reference plane 1 plane 2 Tequivalent Theveninequivalent gure 316 Twoport circuit representation of the generic communications link of g 315 antennas The Thevenin equivalent for the twoport shown in fig 316 gives a nice illustration of the role played by 212 Clearly the transfer impedances are a function of R the distance between the two antennas In the limit of infinite separation we would expect no coupling hence 139 Z 0 152 12 In this limit the driving point impedance of each antenna is then V V 11111 1 m 211 2 211 11111 2 m 222 2 212 R400 1 R400 2 where Zal and Zaz are the antenna impedances measured in isolation Consider the link when antenna 1 is transmitting and antenna 2 is terminated in a receiver with an impedance ZL as in fig 317 In this case V2 iIZZL so from 339 we find Z12 7 222 ZL 1 and hence V 71 Z 2122 3 40 17 1 11 ZZZZL The voltage at terminal 1 has two contributions one from the driving current I1 and the other an induced voltage due to backscatter from antenna 2 sometimes referred to as the contribution from mutual coupling From the perspective of the transmitter the link can be therefore be replaced by an equivalent impedance Zin given by 22 z z i 2n a1 Za2ZL 341 RECEIVING PROPERTIES OF ANTENNAS 19 11 12 lt R gt o Transmitter equivalent circuit ReCelVel qullV alent ClICult gure 317 Equivalent circuits for a typical antenna link When the antennas are close to each other the mutual coupling term can contribute significantly to the impedance On the other hand when the antennas are in the farfield of each other then we expect 212 to be small and then 2m m 21 farfield 342 So as long as the antennas are far apart they are essentially uncoupled as far as the transmitter is concerned and only the intrinsic impedance Zal of antenna 1 is necessary for design of the impedance matching networks Similarly from the perspective of the receiver we can replace the link by a Thevenin equiv alent circuit fig 317 where Veq and Zeq can be easily found as 22 V ZIVi Z 2 ii 343 eq 12 1 ng 21 eq 12 Zal 29 When the antennas are far apart the receiver equivalent circuit parameters become 212 Veq ng Zeq Zaz farfield 344 Notice that in the farfield case the opencircuit voltage Veq does not depend on the load ZL whereas in the general case 343 it does indirectly through changes in Zin due to mutual coupling So in a receiving mode in the farfield an antenna behaves like a source with an intrinsic impedance equal to the driving point impedance when used as a transmitter One must be careful about interpreting this equivalent circuit physicallyiit is tempting to regard power absorbed in the equivalent generator impedance Zeq as the total scattered power but this is not true in general see 2 section 49 The equivalent circuit is physically meaningful only insofar as determining the power delivered to the load receiver impedance We can now prove that the receiving and transmitting properties of an antenna are the same Consider the situation shown in fig 318 We apply a fixed current to antenna 1 and move 20 FUNDAMENTAL PROPERTIES OF ANTENNAS Figure 3 8 Arrangement for demonstrating equivalence of transmitting and receiving properties as discussed in the text antenna 2 along a constant radius circle in the farfield of 1 As long as antenna 2 is always oriented for a polarization match then the measured opencircuit voltage records the transmitting pattern of antenna 1 ie 221 Voc6 I is the normalized field pattern If we reverse the arrangement so that the current generator feeds antenna 2 and antenna 1 is recieving then 212 is the normalized receiving pattern By reciprocity 221 212 and hence we have the desired result for all antenna systems obeying reciprocity the transmitting pattern and receiving pattern are equivalent 322 Receiving Properties Effective Length and Effective Area We have reduced the pointtopoint transmission link to an equivalent circuit without discussing how to calculate the equivalent circuit parameters In fact calculating these parameters is a large part of antenna theory However there are alternative ways to describe energy transfer between antennas Important link calculations can be computed using the concepts of effective receiving area and effective length Antenna 1 Receiver Trans mitter Figure 319 Pointtopoint antenna link RECEIVING PROPERTIES OF ANTENNAS 21 In the system shown in figure 319 the power density indent on receiving antenna is Pin Pine 47TR201617 1 345 The receiving antenna will capture some fraction of this incident field and can deliver a power Prec to a matched load We relate Prec to Pine by defining an e ec ve area Aeg so that Free PincAeff627 Z52 346 The effective area is sometimes referred to as the capture cross section or absorbing crasssection Now if the transmitter and receiver are interchanged then by reciprocity the power received should be the same so Pin Pin MR2 01617 1Aefrz 627 2 47TR202627 2Ae31617 1 347 and hence G 6 G 6 1 1451 2 2452 348 Aeffl617 1 Aeffz627 2 Since we have made no assumptions about the antennas other than constraints associated with the reciprocity theorem then the quantity AeEEG must be a universal constant independent of the antenna and the receiving direction We will also prove this using reciprocity in the next section and again in Chapter 4 using statistical mechanics Effective Area of an Antenna We have alread shown how the reciprocity theorem can link the transmitting and receiving properties of an antenna so it is natural to revisit the reciprocity theorem to relate the antenna gain to the effective area To do this we use a physical situation like that of fig 315 but with antenna 2 replaced by a Hertzian dipole a simple radiator whose fields we have previously computed This situation in shown in fig 320 In fig 320a the generic antenna is radiating and producing the fields Em a and in fig 320b it is receiving in the presence of he impressed current 71 which produces the fields Em b Figure 320 QTransrnitting antenna producing tie fields in in b Same antenna used to receive the elds Eth generated by a testing dipole J5 Also show equivalent circuits 22 FUNDAMENTAL PROPERTIES OF ANTENNAS We assume that the antenna is matched to the transmitter and receiver such that Zg Z Writing Za Ra an and Z9 Rg JXg this implies thath Ba and X 7X11 The reciprocity theorem gives p ax bi bx ad 7M7bEde 349 We have already evaluated the lefthand side in 0 Writing the impressed curret as j Iod x gives 41 V117 Iod K Ea Using the equivalent circuits in fig 320 we can write If Za 29 EaEandZ HP 212 Formign the magnitude squared of both sides we can write lI ZlIi FeRa l6 Ea muf edz 350 W PLF where PLF is the polarization loss factor to be defined later Note that C describes the direction of j and hence the polarization of the incident wave We can orient this vector anyway we wish taking 6 E makes PLFl in general Now on the righthand side we can relate the field Ea to the gain of the antenna as lEal2 Pt 1 a 2 GUM G 6 I R 351 277 4m2 7 15 gill a 4mg In addition we can relate the current If to the effective area as W2 Wine1e Men WW 1 Ra Ba 27 The field produced by the Hertizian dipole is e Jk A E I d4 2 MM 0 47 C so we have A 77 I irdz2 352 m Raw m gt lt gt Sustituting O and 0 into 0 and collecting terms gives the result A2 Aeff G67 353 47139 which is the desired result RECEIVING PROPERTIES OF ANTENNAS 23 Effective Length of an Antenna Just as the effective area is defined to give the correct received power given a certain incident power density we can also define an e eclive length of an antenna to relate the induced voltage or current at the terminals to the incident field intensity through Vac Zea Em 354 The effective length also describes the polarization characteristics of the antenna We can use the reciprocity theorem to quantify the effective length In fig 2 is the antenna in question used in both the transmit and receive mode In the first case a transmitter is connected to the antenna producing a field Ea at the observation point F In the second case a Hertzian dipole is placed at F and the antenna is configured to receive the radiation From the reciprocity theorem we have using the same principles as before ap ax bifbx awig7M7bEde 355 where the surface RP is the cross section of the feeding structure at a specified reference plane and V is the volume outside of S We would like to derive an expression for the opencircuit voltage induced at the terminals in terms of an incident field Therefore in Caseb the receiver is configured so that there is an open circuit condition at RP hence F1 0 and g Voc o where EU is the dominant mode field function In Casea the fields on the feed line can similarly be written in terms of mode function so that E V030 m 105 X an 356 where V0 and I0 are the terminal voltage and current in the transmit mode and g is the axial direction on the feed line The surface integral then evaluates to More y a x 5 x 60 ds macro 357 For any antenna we know that when the observation point is in the far field then we can write the fields in the form 16 WM 1 r E 6 6 358 am as 4 T f as where f is the pattern function This equation defines f For the Hertzian testing dipole in Caseb J1 IdZ6 359 we have m I vocIo Z Me WIdIfw gt 360 L Einc where we have identified Einc as the field incident on the antenna from the testing dipole Therefore the effective length of the antenna is just 1 as I fw gt 361 0 where I0 is the current at the antenna terminals Notice that the effective length allows us to write the field produced by an antenna in the form E aozemeeikr 362 47139 24 FUNDAMENTAL PROPERTIES OF ANTENNAS 323 ShortCircuit Current and OpenCircuit Voltage We have just shown how to calculate the opencircuit voltage in a receiving antenna using reci procity concepts The following is alternate derivation see Chapter 13 of that expands on this result and gives us another chance to practice applying the reciprocity theorem to antenna problems We will start by deriving an expression for the shortcircuit current in a receiving antenna We start with the arrangement shown in fig 321 which is physically similar to fig 320 In fig 321a a PEC transmitting antenna driven by a voltage generator V modelled as a magnetic current loop M generates total fields EF and a feed current I In fig 321b an impressed Transmitting Receiving J EH 0 M E0 H a b Figure 321 39 39ansmitting antenna producing Lhe fields b Same antenna used to receive the elds E0Ho generated by a testing dipole Jo current 70 generates the total fields E07 Fe in the presence of the same antenna when configured as a receiver in this case under short circuit conditions This time we enclose the problem by a surface at infinity so the reciprocity theorem 0 becomes MEG77FOM dV Ejodv 363 The first term on the left E0 j is zero since the tangential electric field must vanish on the surface of the conductor The second term can be evaluated as we discussed in Chapter 1 in connection with sources 7 F0MdV7ij FodZVISC loop 15C Ejodv 364 Now in section 2x we found that when the transmitting antenna is replaced by a Love equivalent currents owing in free space the reciprocity theorem relates the fields to those of a testing dipole Ji as EjidV Eidv 365 so that RECEIVING PROPERTIES OF ANTENNAS 25 In this case the field Ei is the field produced by the testing dipole 7 in free space with no other objects around Therefore if we make 70 7 we can also write 364 as 1 1 15CV EJdVV EiJdV 366 324 Reception of Completely Polarized Waves The polarization characteristics of an antenna are usually specified in term of the transmitted wave described by the pattern function In other words the antenna polarization is described in terms of the direction pointing away from the antenna whereas the polarization of a received signal is usually specified in terms of the direction of propagation which is towards the antenna This often leads to confusion in the case of circular or ellipticallypolarized waves If the antenna is designed to transmit CW polarization in a given direction it will receive CCW polarized signals incident from that direction If the polariation of the incident beam is descrbed by the vector then the received signal will be proportional to 1 12 A A 2 m2 m2 if El 367 This is called the polarization loss factor and describes the fraction of incident power that has the correct polarization for reception The complex conjugate of corrects for the opposite sense of polarization when receiving If E 1 the antenna is said to be polarizationmatched to the incoming signal and PLF1 However if the incoming signal is orthogonal to the antenna polarization such that f E 0 then no signal will be recieved For example suppose the antenna is designed to transmit CW polarization in the 2 direction so that f 92 7 J32 The polarization loss factor for various received signal polarizations is 1 92 Linear PLF i g 92J32 CW PLF 0 927ng CCW PLF1 So an antenna that transmits CW radiation receives CCW radiation and vice versa Give example of polarization interleaving in DBS downlinks We have tacitly assumed that the signal to be received in completely polarized meaning that it is a coherent signal in one definite and perpetual polarization In contrast incoherent electromagnetic emission from natural objects such as celestial bodies arrives at an antenna with a randomly changing polarization state If the polarization state varies lowly relative to centroid of frequency this is called a partially polarized wave This is of importance in the field of radio astronomy and also passive radiometric detection for remote sensing applications Howeve it is a relatively specialized topic that will not be considered in this work the interested reader is referred to Chapter 3 of 2 325 Equivalent Circuit Parameters Now that we have eff and Ag we can find Z21 an Z12 26 FUNDAMENTAL PROPERTIES OF ANTENNAS 33 METHODS FOR FINDING THE TERMINAL IMPEDANCE As we have seen the terminal impedance is an important parameter for characterizing the antenna There are some general methods for calculating this impedance which we now consider These methods are often referred to collectively as Induced EMF methods although the concept of an induced EMF electromotive force does not always play an explicit role in the derivation This terminology has its roots in the physical origin of the radiation resistance 331 Induced EMF method The antenna can be represented by a complex terminating impedance or admittance where the real part of the impedance is associated with an average power flow away from the antenna radiation and the reactive part is associated with the nonradiative near fields of the antenna The fie description can be given a mechanical interpretation by considering the motion of electrons on the antenna When the electrons are accelerated by an applied field from the generator they will radiate and hence lose energy This change in energy is equivalent to a force acting against the motion of the electrons This force called the induced EMF or radiation reaction is attributed to the radiated fields acting back on the current Mechanically it can be thought of as the recoil force on the electrons as they eject photons In order to maintain a certain timevarying current distribution on the antenna the generator must therefore expend energy to overcome the radiation reaction This viewpoint allows us to formulate a simple expression for the input impedance of the antenna For simplicity we first consider a simple dipole as shown in fig 322 the concept will be generalized in the next section An applied voltage Vin at the terminals of this antenna will induce a current flow given by the distribution I This current in turn produces a scattered field E563 Note that the scattered field at any point along the antenna includes contributions gure 322 Simple dipole antenna from the induced currents on all parts of the antenna not just the currents at the point in question The induced current must distribute itself such that the induced field plus the applied field satisfy the boundary conditions on the antenna In circuit terms the scattered field is an induced distributed along the antenna The induced EMF over a length dz is therefore dVz ficmz 2 dz This EMF can be related to a corresponding change in terminal current using reciprocity Since an applied voltage Vin at the terminals produces a current I at the point 2 then by reciprocity an EMF dV at point z will produce a current dLn at the terminals W 7 dVz Iltzgt d2 METHODS FOR FINDING THE TERMINAL IMPEDANCE 27 which gives VindIin Emu 72 dz Integrating this expression over the length of the antenna gives VinIin r Esca z 72 dz 368 The input impedance is then Zin Iil ZnEscatdz Tzdz 369 This is the socalled induced EMF formula for the input impedance of a dipole If the true currents and fields for the structure are used then this expression is exact insofar as the ideal dipole structure is concerned However if we already knew the true current distribution there would be no reason to use 369 since we wouldjust compute the impedance from Zin VinIn Vin0 II The induced EMF result is therefore only used in situations where the current distribution can only be guessed at or otherwise approximated It is attractive in this respect due to a stationary property which will be discussed in the next section Once we have a trial current distribution for the antenna 369 requires that we compute the fields produced by those currents If we assume that the applied field is concentrated in the feed gap region the applied field can be taken as Einc 2Ving in the gap where g is the gap width and zero elsewhere along the conductors see discussion of sources in Chapter 1 This applied field induces whatever current is necessary to maintain a voltage Vin or equivalently a field iiiing in the gap and a vanishing total field along the conductors Therefore the true current I will produce a scattered field which is E 7 iiiing in the gap sea 7 0 along the conductor When this is inserted in 369 we can see that the integration is just over the gap region However it should be noted that the fields produced by an approximation to the true currents will not necessarily vanish along the conductors and hence the integration must then be carried out over the entire length of the antenna In fact it has been found that relatively small changes in the current distribution can lead to enormous changes in the tangential fields This makes the stationary character of 369 even more remarkable 332 Generalized Impedance From Reciprocity and its Variational Property The derivation above based on circuit principles is limited to thin wire antennas where the current is filamentary A more general expression can be derived from a field statement of reciprocity as follows We start with an arbitrary antenna as shown in fig 323a which is fed by a voltage generator Using fieldequivalence concepts we can replace the conductors and generators by free space as shown in fig 323b The impressed surface currents Ma and Ja in this case can be related to the applied field and induced surface currents respectively from the original problem The combination of these currents produces the same fields outside of S as exist in the original problem and produce a null field within V We now introduce an auxiliary problem in fig 323c This concerns exactly the same volume as in fig 323b but introduces a different current J1 28 FUNDAMENTAL PROPERTIES OF ANTENNAS a b C Figure 323 a A PEC antenna with feed b Using equivalence principles the feed is replaced by an impressed magnetic current around the surface S and the conductors are replaced by freespace c An auxiliary current distribution owing in the same volume V owing in freespace within the volume The reciprocity theorem links the currents and fields of the two cases according to ME7de VEjadVi v bMadV 370 where Em a are the total fields produced by the currents jaMa The left hand side of 370 is therefore zero since the tangential electric field is zero along S in fig 323b the tangential electric field is nonzero just outside of the current distribution Ma given by the negative of the applied field but it is zero on and within S The second integral on the right of 370 can be evaluated as in 2 to give ivarb E 7a dV v where Va is the original generator voltage and I1 is the total current passing through the feed region in fig 323c Therefore the driving point impedance is given by a Z Z 7 7 m Ia IaIbM E 7a dV 371 This is a remarkable result since the auxiliary current 71 can be specified arbitrarily In some cases it may be advantageous to use a simple current distribution in order to simplify the computation of impedance However the formula is only exact when the true currents 7a are used and as we saw in the previous section if the true currents were already known there would be no reason to use this result It is mostly used for situations where the current is approximated and in those cases it is advantageous to use the specific choice of 71 1 E 7 from which follows I1 a E I sot at 1 Zin 7 E JdV 372 2 V where E is now the field produced by the current J This is the desired generalization of the METHODS FOR FINDING THE TERMINAL IMPEDANCE 29 induced EMF result for the input impedance of an antenna Note that this result is essentially identical with 2 for the impedance parameters of a multiport network developed in Chapter 1 The utility of this expression derives from the fact that it is stationary with respect to small errors in the assumed current If the true currents and fields are 70 and E0 respectively then a small error in the assumed current can be represented as j 70 67 and all other derived quantities will be similarly perturbed EaEOJr E Ian6I Zinazinwzm Substituting these perturbed quantities into 372 gives 2 52 7 1 1 2 ME 7 6E 7 5 575E 57pm m m 7 g In 0 0 0 0 From reciprocity we find 6E 70 E0 6739 keeping only first order terms gives 62ini22 M E070dV72 Eo jdV 0 I0 In V0 10 V0 51 which shows that the impedance is unchanged or stationary with respect to small perturbations in the current Expressions which have this property can often be constructed for physically meaningful parameters and are called variational from variational calculus A guess current that has some error 67 will give an impedance accurate to the order of 67 Note that the result 371 does not possess this property in the general case when the two currents 7a and 71 are different In general we are usually concerned with antennas constructed of perfect conductors as in fig a in which case the currents are always surface currents and the impedance is then given by 1 Zin i gE Jsm 13 373 The electric field E is that produced by the impressed current distribution 75 and can be expressed as an yw 75m Emmw 374 3 Using equivalence principles the conductors can usually be replaced by freespace so that the unbounded Green s function can be used The selfimpedance is then given by M zin 7W fs smr GO J5rdS dS 375 333 Power conservation methods Poynting s theorem and power conservation arguments also provide a useful technique for finding the antenna impedance In fact we already used this approach to find the radiation resistance of the Hertzian dipoles Consider the situation of figure 324 where a transmissionline delivers energy to some arbitrary element Z The element is surrounded by a closed surface S with only a 30 FUNDAMENTAL PROPERTIES OF ANTENNAS Figure 324 Illustration for nding the antenna impedance at reference plane 51 via Poynting s the orern and power conservation arguments small aperture Si through which energy is coupled tofrom the transmission line At this aperture plane there exists a voltage Vi and current Ii which are related to the fields by VtI i E X F dS S L where ft is the unit normal to the surface directed out of the volume Using 2 and 2 allows us to define an impedance at the reference plane Si as Zin a lElZdV 4JwF 7E dV9731EgtltF ds 376 where we have substituted 7 0E for the induced currents in the volume The last term accounts for complex power owing out of the volume through the enclosing surface S 7 Si If the medium surrounding the antenna in the volume is lossless then the first volume integral in 376 is just the ohmic loss on the antenna The second integral gives a reactive contribution to the input impedance that is primarily associated with stored energy in the vicinity of the antenna near fields The last integral accounts for complex power radiated out of the surface and in general has both a resistive and reactive component If the surface S extends to infinity as in fig 325a then it can be shown using the Sommer feld radiation conditions that the surface integral term is purely real and constitutes the radiation resistance of the antenna Then 376 reduces to Zn Rios Brad JX0J 377 1 1 1 R EZdV R E2dS Hip aw W snll Xw Re ml 761E dV Using 36 we can write the radiation resistance in terms of the pattern function as 1 Ra 6 2d9 378 d W sw gt1 lt gt where The fields and hence the pattern function lt15 will be proportional to the feed current Ii so this term will cancel in the denominator A different expression for the impedance can be obtained when the surface enclosing the antenna coincides with the surface of the conductors as in fig 325b For simplicity we also METHODS FOR FINDING THE TERMINAL IMPEDANCE 31 1 a b Figure 325 a PEC antenna enclosed by a spherical surface at in nity b PEC antenna enclosed by a surface S which coincides with the antenna surface and also encloses the generator enclose the generator by the surface as shown When the conductors are perfectly conducting the complex power owing through the surface must equal the power supplied by the generator so again we have a power conservation statement 1 1 VI 77 EgtltHds from which we can find the input impedance as V 1 Zin 13st 379 I m2 If 5 where J5 ft X H is the equivalent impressed current flowing on the surface S which includes the induced surface current flowing on the conductors and a contribution from the feed region numerically proportional to the current owing in the generator for small gaps This is quite similar to the induced EMF formula except for the complex conjugate on the current density For real 75 the two are identical In this formulation E is the total field on the antenna surface which of course must vanish on the PEC surfaces so 379 implies that 2quot7L2 Elias lIl feed if the true fields and currents are used This just reinforces the physical expectation that the energy flows out of the gap region and is merely guided by the conductors into space This point is worth emphasizing the antenna structure is simply a device for guiding energy away from the generator and into freespaceiin fact it is possible to treat the antenna as a finite length of transmission line where in general the characteristic impedance varies along its length We will examine this


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