The Method of Moments in Electromagnetics

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A Brief Review of Electromagnetics 19E 1,H 1E 1,H 1R 1R 2 SJ 1 M 1(a) Original ProblemnÊ 1,H 1E,HR 1R 2 SM s= -n ^ x (E 1- E)J s= n^x (H 1- H)(b) Equivalent ProblemFigure 2.2: Surface equivalence theorem.
20 The Method of Moments in ElectromagneticsM × n ¢ É ½ Eµ (2.109)where the fields E and H inside Ë remain undefined for now. Since Ê ½ and Ê ¾ havethe same dielectric properties, J × and M × radiate in a homogeneous environment and(2.65) and (2.71) may be used to obtain the radiated fields everywhere in Ê ¾ .Thisiscalled the surface equivalent and is depicted in Figure 2.2b. It should be noted thatin this case only E ½ or H ½ needs to be known on Ë, since the other can be obtainedby the Maxwell Equations (2.1) or (2.2).Because the fields E and H inside Ë are totally arbitrary, let us furthergeneralize the surface equivalence theorem. Consider first the problem of Figure2.3a, where external currents J ½ and M ½ and internal currents J ¾ and M ¾ create thefields E and H everywhere. Next, consider the similar problem of Figure 2.3b,where external currents J and M and internal currents J ¿ and M ¿ create the fieldsE and H everywhere. Let us now create a problem that is externally equivalent toFigure 2.3a and internally equivalent to Figure 2.3b. To do so, the fields and sourcesoutside Ë should remain the same as in Figure 2.3a and those inside Ë the same as inFigure 2.3b. The surface currents on Ë must satisfy the boundary conditions, henceJ × n ¢ ´H H µ (2.110)M × n ¢ É E µ (2.111)This is illustrated in Figure 2.4a. By similar arguments, we can create an equivalentthat is internally equivalent to Figure 2.3a and externally equivalent to Figure 2.3b.In this case, the currents satisfy the boundary conditionsJ × n ¢ ´H H µ (2.112)M × n ¢ É E µ (2.113)This is illustrated in Figure 2.4b.2.6.2 Physical EquivalentLet us consider a concept very important in the area of scattering and radiationproblems called the physical equivalent. Suppose that we have currents J ½ and M ½that generate the fields E ½ and H ½ in a homogeneous region with parameters ¯½and ½ . If we now introduce a conducting object into the region, the reflected (orscattered) fieldsE × and H × are created, as illustrated in Figure 2.5a. Our goal is toobtain these scattered fields, so we will create an equivalent problem that allows usto remove the conducting object and replace it with equivalent surface currents. Letus apply the boundary conditions on the electric and magnetic fields at the surfaceof the scatterer. Over the boundary, the total tangential electric field must be zero,hencen ¢ É E Ø µ n ¢ É ½ · E × µM × ¼ (2.114)and the total tangential magnetic field must equal the induced electric current density:n ¢ ´H H Ø µn ¢ ´H ½ · H × µJ × (2.115)
Page 2 and 3: The Methodof Moments inElectromagne
Page 4 and 5: ContentsPrefaceAcknowledgmentsAbout
Page 6 and 7: Contentsv4.6.5 Matching a Yagi Ante
Page 8: Contentsvii8.5.2 Matrix-Vector Prod
Page 11 and 12: xPreface3. A focus on radiation and
Page 14: AcknowledgmentsThis book is a resul
Page 18 and 19: Chapter 1Computational Electromagne
Page 20 and 21: Computational Electromagnetics 3sec
Page 22 and 23: Chapter 2A Brief Review of Electrom
Page 24 and 25: A Brief Review of Electromagnetics
Page 44: A Brief Review of Electromagnetics
Page 47 and 48: 30 The Method of Moments in Electro
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70 The Method of Moments in Electro
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Chapter 7Three-Dimensional Problems
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Three-Dimensional Problems 163(a) S
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Three-Dimensional Problems 165to, a
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Three-Dimensional Problems 167packa
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Three-Dimensional Problems 169where
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Three-Dimensional Problems 171that
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Three-Dimensional Problems 173can b
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Three-Dimensional Problems 175inacc
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Three-Dimensional Problems 177Edge
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Three-Dimensional Problems 1797.3.4
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Three-Dimensional Problems 181OrR n
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Three-Dimensional Problems 183Ampli
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Three-Dimensional Problems 185Objec
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Three-Dimensional Problems 1877.6 C
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Three-Dimensional Problems 1897.7.1
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Three-Dimensional Problems 1913020R
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Three-Dimensional Problems 1932.510
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Three-Dimensional Problems 1950−5
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Three-Dimensional Problems 19750−
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Three-Dimensional Problems 199ABowt
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Three-Dimensional Problems 201terms
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Three-Dimensional Problems 203Input
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Three-Dimensional Problems 205is th
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Three-Dimensional Problems 207[25]
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Chapter 9IntegrationThroughout this
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Integration 257Summing (9.4) across
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Integration 259x 1=a x 2x 3x 4x 5x
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Integration 261(0,1,0)v 3A 1A 3A 2v
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Integration 263or ½ ½ «Á ¾ ×
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Integration 265withÁ Ì ´r×·r
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Integration 267When s is normal to
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Integration 269Table 9.2: Four-Poin
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IndexAntennasArchimedean spiral, 53
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The Method of Moments in Electromagnetics

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