Chapter 6 Integral solutions of Maxwells equations 6. 1 Vector Kirchoff solution: method of stratton and chu One of the most powerful tools for the analysis of electromagnetics problems is the integral solution to Maxwells equations formulated by Stratton and Chu [187, 188 These authors used the vector Green's theorem to solve for e and h in much the same way as is done in static fields with the scalar Green's theorem. An alternative approach is to use the Lorentz reciprocity theorem of$ 4. 10. 2, as done by Fradin[ 74. The reciprocity approach allows the identification of terms arising from surface discontinuities, which must be added to the result obtained from the other approach [187 6.1.1 The Stratton-Chu formula Consider an isotropic, homogeneous medium occupying a bounded region V in space The medium is described by permeability i(o), permittivity E(@), and conductivity o() The region V is bounded by a surface S, which can be multiply-connected so that S is the union of several surfaces S1,..., SN as shown in Figure 6.1: these are used to exclude unknown sources and to formulate the vector Huygens principle. Impressed electric and magnetic sources may thus reside both inside and outside V We wish to solve for the electric and magnetic fields at a point r within V. To do this we employ the Lorentz reciprocity theorem(4. 173), written here using the frequency-domain fields as an integral over primed coordinates ea(r,o)x Hb(r,a)-Eb(r,o)x Ha(r, o) [Eb(r,o)·Ja(r,a)-Ea(r,a)·J6(r,o)- (6.1) Hr,o)·Jma(r,o)+H2(r,o)·Jmb(r,o)]dv (6.2) Note that the negative sign on the left arises from the definition of f as the inward normal to V as shown in Figure 6. 1. We place an electric Hertzian dipole at the point r= rp where we wish to compute the field, and set E,= En and H,= H, in the reciprocity theorem,where Ep and H, are the fields produced by the dipole(5. 88)-(5.89) IpG(rrp: o), Ep(r, o)==Vx(V x IpG(rrp; o))) 4 ②2001 by CRC Press LLC
Chapter 6 Integral solutions of Maxwell’s equations 6.1 Vector Kirchoff solution: method of Stratton and Chu One of the most powerful tools for the analysis of electromagnetics problems is the integral solution to Maxwell’s equations formulated by Stratton and Chu [187, 188]. These authors used the vector Green’s theorem to solve for E˜ and H˜ in much the same way as is done in static fields with the scalar Green’s theorem. An alternative approach is to use the Lorentz reciprocity theorem of § 4.10.2, as done by Fradin [74]. The reciprocity approach allows the identification of terms arising from surface discontinuities, which must be added to the result obtained from the other approach [187]. 6.1.1 The Stratton–Chu formula Consider an isotropic, homogeneous medium occupying a bounded region V in space. The medium is described by permeability µ(ω) ˜ , permittivity (ω) ˜ , and conductivity σ(ω) ˜ . The region V is bounded by a surface S, which can be multiply-connected so that S is the union of several surfaces S1,... , SN as shown in Figure 6.1; these are used to exclude unknown sources and to formulate the vector Huygens principle. Impressed electric and magnetic sources may thus reside both inside and outside V. We wish to solve for the electric and magnetic fields at a point r within V. To do this we employ the Lorentz reciprocity theorem (4.173), written here using the frequency-domain fields as an integral over primed coordinates: − S E˜ a(r ,ω) × H˜ b(r ,ω) − E˜ b(r ,ω) × H˜ a(r ,ω) · nˆ d S = V E˜ b(r ,ω) · J˜a(r ,ω) − E˜ a(r ,ω) · J˜b(r ,ω)− (6.1) H˜ b(r ,ω) · J˜ma(r ,ω) + H˜ a(r ,ω) · J˜mb(r ,ω) dV . (6.2) Note that the negative sign on the left arises from the definition of nˆ as the inward normal to V as shown in Figure 6.1. We place an electric Hertzian dipole at the point r = rp where we wish to compute the field, and set E˜ b = E˜ p and H˜ b = H˜ p in the reciprocity theorem, where E˜ p and H˜ p are the fields produced by the dipole (5.88)–(5.89): H˜ p(r,ω) = jω∇ × [p˜G(r|rp; ω)], (6.3) E˜ p(r,ω) = 1 ˜ c ∇ × ∇ × [p˜G(r|rp; ω)] . (6.4)
J/JA S Figure 6. 1: Geometry used to derive the Stratton-Chu formula We also let Ea=E and Ha=h, where E and h are the fields produced by the impressed sources Ja=J and Jma=Jm within V that we wish to find at r=rp. Since the dipole fields are singular at r=rp, we must exclude the point rp with a small spherical surface Ss surrounding the volume Va as shown in Figure 6. 1. Substituting these fields into(6.2) we obtain ,、匡B一Eד=⊥,国了-B,:(6 A useful identity involves the spatially-constant vector p and the Greens function G(rrp) V××(cp=vv·(Cp]-v2(Gp) =v·(Gp)}-pv2G =V师p·VG)+pk2G (6.6) where we have used V G=-k-G for r*rp We begin by computing the terms on the left side of(6.5). We suppress the rde- pendence of the fields and also the dependencies of G(rrp). Substituting from(6.3)we E x Hp]. nds xVx(p]·的 Using f.xVx(Gp=mx(VG×p)=('xE)·(VG×p) we can write E X Hp]- nds= ja ×E]×vGdS'. ②2001 by CRC Press LLC
Figure 6.1: Geometry used to derive the Stratton–Chu formula. We also let E˜ a = E˜ and H˜ a = H˜ , where E˜ and H˜ are the fields produced by the impressed sources J˜a = J˜i and J˜ma = J˜i m within V that we wish to find at r = rp. Since the dipole fields are singular at r = rp, we must exclude the point rp with a small spherical surface Sδ surrounding the volume Vδ as shown in Figure 6.1. Substituting these fields into (6.2) we obtain − S+Sδ E˜ × H˜ p − E˜ p × H˜ · nˆ d S = V−Vδ E˜ p · J˜i − H˜ p · J˜i m dV . (6.5) A useful identity involves the spatially-constant vector p˜ and the Green’s function G(r |rp): ∇ × ∇ × (Gp˜) = ∇ [∇ · (Gp˜)] − ∇2 (Gp˜) = ∇ [∇ · (Gp˜)] − p˜∇2G = ∇ (p˜ · ∇ G) + p˜ k2G, (6.6) where we have used ∇2G = −k2G for r = rp. We begin by computing the terms on the left side of (6.5). We suppress the r dependence of the fields and also the dependencies of G(r |rp). Substituting from (6.3) we have S+Sδ [E˜ × H˜ p] · nˆ d S = jω S+Sδ E˜ × ∇ × (Gp˜) · nˆ d S . Using nˆ · [E˜ × ∇ × (Gp˜)] = nˆ · [E˜ × (∇ G × p˜)] = (nˆ × E˜) · (∇ G × p˜) we can write S+Sδ [E˜ × H˜ p] · nˆ d S = jωp˜ · S+Sδ [nˆ × E˜ ] × ∇ GdS .
Figure 6.2: Decomposition of surface S, to isolate surface field discontinuit examine Ep×田ndS H× V'xVX(Gp)]·fdS Use of (6.6) along with the identity(B 43) gives 中叵×前d I(H X p)kG S+Sa EC [(·vGm+(pVG 0} We would like to use Stokes's theorem on the second term of the right-hand side. Since the theorem is not valid for surfaces on which h has discontinuities, we break the closed rfaces in Figure 6.1 into open surfaces whose boundary contours isolate the disconti- nuities as shown in Figure 6.2. Then we may write nVx[(p VG)H]ds'= dr.A(pVG) For surfaces not containing discontinuities of h the two contour provide equ and opposite contributions and this term vanishes. Thus the left-hand side of (6.5)is 只、国xB,一ExS= [joe(ixE)xvG+k(×mG+·(J+ joE E)VG]dS where we have substituted J+ joe E for V’ x H and used(×p)·=p·(’×H Now consider the right-hand side of (6.5). Substituting from(6.4 )we have E,.J Using(6.6) and(B 42), we have ②2001 by CRC Press LLC
Figure 6.2: Decomposition of surface Sn to isolate surface field discontinuity. Next we examine S+Sδ [E˜ p × H˜ ] · nˆ d S = − 1 ˜ c S+Sδ H˜ × ∇ × ∇ × (Gp˜) · nˆ d S . Use of (6.6) along with the identity (B.43) gives S+Sδ [E˜ p × H˜ ] · nˆ d S = − 1 ˜ c S+Sδ (H˜ × p˜)k2G − − ∇ × (p˜ · ∇ G)H˜ + (p˜ · ∇ G)(∇ × H˜ ) · nˆ d S . We would like to use Stokes’s theorem on the second term of the right-hand side. Since the theorem is not valid for surfaces on which H˜ has discontinuities, we break the closed surfaces in Figure 6.1 into open surfaces whose boundary contours isolate the discontinuities as shown in Figure 6.2. Then we may write Sn=Sna+Snb nˆ · ∇ × (p˜ · ∇ G)H˜ d S = na+ nb dl · H˜ (p˜ · ∇ G). For surfaces not containing discontinuities of H˜ the two contour integrals provide equal and opposite contributions and this term vanishes. Thus the left-hand side of (6.5) is − S+Sδ E˜ × H˜ p − E˜ p × H˜ · nˆ d S = − 1 ˜ c p˜ · S+Sδ jω˜ c (nˆ × E˜) × ∇ G + k2 (nˆ × H˜ )G + nˆ · (J˜i + jω˜ c E˜)∇ G d S where we have substituted J˜i + jω˜ cE˜ for ∇ × H˜ and used (H˜ × p˜) · nˆ = p˜ · (nˆ × H˜ ). Now consider the right-hand side of (6.5). Substituting from (6.4) we have V−Vδ E˜ p · J˜i dV = 1 ˜ c V−Vδ J˜i · ∇ × ∇ × (p˜G) dV . Using (6.6) and (B.42), we have V−Vδ E˜ p · J˜i dV = 1 ˜ c V−Vδ k2 (p˜ · J˜i )G + ∇ · [J˜i (p˜ · ∇ G)] − (p˜ · ∇ G)∇ · J˜i dV .
Replacing V.J with -jop from the continuity equation and using the divergence theorem on the second term on the right-hand side. we then have 1 E. dV==p (2J'G+jop'v'g)dv (的·Jv"GdS Lastly we examine Jn·Vx(Gp) Use of J.V×(p)=Jn(VG×p)=p(Jn×VG)gi JmV=jp·/Jmxv We now substitute all terms into(6.5)and note that each term involves a dot product with p Since p is arbitrary we have axE×VG+(,EG-(x(dS+ dr.HV'G Jm x V'G+=V'G-jojJ' The electric field may be extracted from the above expression by letting the radius of the excluding volume Vs recede to zero. We first consider the surface integral over Ss Examining Figure 6.3 we see that R=rp-r=8,n=-R R VG(rrp) d VRER Assuming E is continuous at r=rp we can write [(×E)×VG+('EVG-joj(xmG]dS R R 40xE)×+(R.E-xm|82aB 1切~(,E+食RE+(RER]d2=E(r ②2001 by CRC Press LLC
Figure 6.3: Geometry of surface integral used to extract E at rp. Replacing ∇ · J˜i with − jωρ˜i from the continuity equation and using the divergence theorem on the second term on the right-hand side, we then have V−Vδ E˜ p · J˜i dV = 1 ˜ c p˜ · V−Vδ (k2 J˜i G + jωρ˜i ∇ G) dV − S+Sδ (nˆ · J˜i )∇ GdS . Lastly we examine V−Vδ H˜ p · J˜i m dV = jω V−Vδ J˜i m · ∇ × (Gp˜) dV . Use of J˜i m · ∇ × (Gp˜) = J˜i m · (∇ G × p˜) = p˜ · (J˜i m × ∇ G) gives V−Vδ H˜ p · J˜i m dV = jωp˜ · V−Vδ J˜i m × ∇ G dV . We now substitute all terms into (6.5) and note that each term involves a dot product with p˜. Since p˜ is arbitrary we have − S+Sδ (nˆ × E˜) × ∇ G + (nˆ · E˜)∇ G − jωµ(˜ nˆ × H˜ )G d S + + 1 jω˜ c a+ b (dl · H˜ )∇ G = V−Vδ −J˜i m × ∇ G + ρ˜i ˜ c ∇ G − jωµ˜ J˜i G dV . The electric field may be extracted from the above expression by letting the radius of the excluding volume Vδ recede to zero. We first consider the surface integral over Sδ . Examining Figure 6.3 we see that R = |rp − r | = δ, nˆ = −Rˆ , and ∇ G(r |rp) = d d R e− jkR 4π R ∇ R = Rˆ 1 + jkδ 4πδ2 e− jkδ ≈ Rˆ δ2 as δ → 0. Assuming E˜ is continuous at r = rp we can write − lim δ→0 Sδ (nˆ × E˜) × ∇ G + (nˆ · E˜)∇ G − jωµ(˜ nˆ × H˜ )G d S = lim δ→0 1 4π (Rˆ × E˜) × Rˆ δ2 + (Rˆ · E˜) Rˆ δ2 − jωµ(˜ Rˆ × H˜ ) 1 δ δ2 d = lim δ→0 1 4π −(Rˆ · E˜)Rˆ + (Rˆ · Rˆ )E˜ + (Rˆ · E˜)Rˆ d = E˜(rp).
Here we have used a ds2= 4T for the total solid angle subtending the sphere Ss. Finally, assuming that the volume sources are continuous, the volume integral over Vs vanishes E(r,)= JmxV'G+=V'G-joij'Gdv'+ G-jo(×mG]ds (dI·mVG (6.7) A similar formula for H can be derived by placing a magnetic dipole of moment pm at r=rp and proceeding as above. This leads to H(r,)=(J'xVG+mV'G-joe Ji) dv'+ 点x+(B下O+“+ (dI·E)VG (6.8) =i Jop We can also obtain this expression by substituting(6.7)into Faraday's law 6.1.2 The Sommerfeld radiation condition In 8 5.2.2 we found that if the potentials are not to be influenced by effects that are infinitely removed, then they must obey a radiation condition. We can make the same argument about the fields from(6.7) and(6.8). Let us allow one of the excluding surfaces, say Sw, to recede to infinity(enclosing all of the sources as it expands). AS Sw-00 any contributions from the fields on this surface to the fields at r should vanish Letting Sw be a sphere centered at the origin, we note that ft= -f and that as 4r VG(rr; w)=R jki Substituting these expressions into(6.7)we find that xVG+(·EvG [(×E)×P+(f·E)门] jop(fx [rakE E ②2001 by CRC Press LLC
Here we have used d = 4π for the total solid angle subtending the sphere Sδ . Finally, assuming that the volume sources are continuous, the volume integral over Vδ vanishes and we have E˜(r,ω) = V −J˜i m × ∇ G + ρ˜i ˜ c ∇ G − jωµ˜ J˜i G dV + + N n=1 Sn (nˆ × E˜) × ∇ G + (nˆ · E˜)∇ G − jωµ(˜ nˆ × H˜ )G d S − − N n=1 1 jω˜ c na+ nb (dl · H˜ )∇ G. (6.7) A similar formula for H˜ can be derived by placing a magnetic dipole of moment p˜ m at r = rp and proceeding as above. This leads to H˜ (r,ω) = V J˜i × ∇ G + ρ˜i m µ˜ ∇ G − jω˜ c J˜i m G dV + + N n=1 Sn (nˆ × H˜ ) × ∇ G + (nˆ · H˜ )∇ G + jω˜ c (nˆ × E˜)G d S + + N n=1 1 jωµ˜ na+ nb (dl · E˜)∇ G. (6.8) We can also obtain this expression by substituting (6.7) into Faraday’s law. 6.1.2 The Sommerfeld radiation condition In § 5.2.2 we found that if the potentials are not to be influenced by effects that are infinitely removed, then they must obey a radiation condition. We can make the same argument about the fields from (6.7) and (6.8). Let us allow one of the excluding surfaces, say SN , to recede to infinity (enclosing all of the sources as it expands). As SN → ∞ any contributions from the fields on this surface to the fields at r should vanish. Letting SN be a sphere centered at the origin, we note that nˆ = −rˆ and that as r → ∞ G(r|r ; ω) = e− jk|r−r | 4π|r − r | ≈ e− jkr 4πr , ∇ G(r|r ; ω) = Rˆ 1 + jkR 4π R2 e− jkR ≈ −rˆ 1 + jkr r e− jkr 4πr . Substituting these expressions into (6.7) we find that lim SN→S∞ SN (nˆ × E˜) × ∇ G + (nˆ · E˜)∇ G − jωµ(˜ nˆ × H˜ )G d S ≈ lim r →∞ 2π 0 π 0 (rˆ × E˜) × rˆ + (rˆ · E˜)rˆ 1 + jkr r + jωµ(˜ rˆ × H˜ ) e− jkr 4πr r2 sin θ dθ dφ ≈ lim r →∞ 2π 0 π 0 r jkE˜ + jωµ(˜ rˆ × H˜ ) + E˜ e− jkr 4π sin θ dθ dφ .
Since this gives the contribution to the field in V from the fields on the surface receding to infinity, we expect that this term should be zero. If the medium has loss, then the exponential term decays and drives the contribution to zero. For a lossless medium the contributions are zero if lim rE(r,o)<∞, (69) limr[nf x H(r,o)+E(r,o]=0 To accompany(6. 8)we al lim rh(r,o)<∞, (6.11) lim r[nH(r,o)-fxE(r,o)=0 We refer to (6.9) and(611)as the finiteness conditions, and to(6.10) and(6. 12)as the Sommerfeld radiation condition, for the electromagnetic field. They show that far from the sources the fields must behave as a wave tem to the r-direction We shall se 86.2 that the waves are in fact spherical TEM waves 6.1.3 Fields in the excluded region: the extinction theorem The Stratton-Chu formula provides a solution for the field within the region V, external to the excluded regions. An interesting consequence of this formula, and one that helps us identify the equivalence principle, is that it gives the null resultH=E=0when evaluated at points within the excluded regions We can show this by considering two cases. In the first case we do not exclude the particular region Vm, but do exclude the remaining regions Vn, n# Then the electric field everywhere outside the remaining excluded regions(including at points within Vm) is,by(6.7), E(r,o)= JmxVG+2V'G-joij'Gdv+ G+(·E)VG-jd h)G]ds' drHVG In the second case we apply the Stratton-Chu formula only to Vn, and exclude all other regions. We incur a sign change on the surface and line integrals compared to the first case because the normal is now directed oppositely. By(6.7) we have E(r, o)= Jm xVG+=)dV E)V'G-joa(n'xhGdS'+ ②2001 by CRC Press LLC
Since this gives the contribution to the field in V from the fields on the surface receding to infinity, we expect that this term should be zero. If the medium has loss, then the exponential term decays and drives the contribution to zero. For a lossless medium the contributions are zero if lim r→∞ rE˜(r,ω) < ∞, (6.9) lim r→∞ r ηrˆ × H˜ (r,ω) + E˜(r,ω) = 0. (6.10) To accompany (6.8) we also have lim r→∞ rH˜ (r,ω) < ∞, (6.11) lim r→∞ r ηH˜ (r,ω) − rˆ × E˜(r,ω) = 0. (6.12) We refer to (6.9) and (6.11) as the finiteness conditions, and to (6.10) and (6.12) as the Sommerfeld radiation condition, for the electromagnetic field. They show that far from the sources the fields must behave as a wave TEM to the r-direction. We shall see in § 6.2 that the waves are in fact spherical TEM waves. 6.1.3 Fields in the excluded region: the extinction theorem The Stratton–Chu formula provides a solution for the field within the region V, external to the excluded regions. An interesting consequence of this formula, and one that helps us identify the equivalence principle, is that it gives the null result H˜ = E˜ = 0 when evaluated at points within the excluded regions. We can show this by considering two cases. In the first case we do not exclude the particular region Vm, but do exclude the remaining regions Vn, n = m. Then the electric field everywhere outside the remaining excluded regions (including at points within Vm) is, by (6.7), E˜(r,ω) = V+Vm −J˜i m × ∇ G + ρ˜i ˜ c ∇ G − jωµ˜ J˜i G dV + + n =m Sn (nˆ × E˜) × ∇ G + (nˆ · E˜)∇ G − jωµ(˜ nˆ × H˜ )G d S − − n =m 1 jω˜ c na+ nb (dl · H˜ )∇ G, r ∈ V + Vm. In the second case we apply the Stratton–Chu formula only to Vm, and exclude all other regions. We incur a sign change on the surface and line integrals compared to the first case because the normal is now directed oppositely. By (6.7) we have E˜(r,ω) = Vm −J˜i m × ∇ G + ρ˜i ˜ c ∇ G − jωµ˜ J˜i G dV − − Sm (nˆ × E˜) × ∇ G + (nˆ · E˜)∇ G − jωµ(˜ nˆ × H˜ )G d S + + 1 jω˜ c na+ nb (dl · H˜ )∇ G, r ∈ Vm.
Each of the expressions for E is equally valid for points within Vm. Upon subtraction we xV'G+ opj'gdv'+ 点人 (AXE)XV'G+Bwo-0mxm(4y (d·HVG,r∈V This expression is exactly the Stratton-Chu formula(6.7)evaluated at points within the excluded region Vm. The treatment of H is analogous and is left as an exercise. Since we may repeat this for any excluded region, we find that the Stratton-Chu formula returns the null field when evaluated at points outside V. This is sometimes referred to as the vector Ewald-Oseen extinction theorem 90. We must emphasize that the fields within the excluded regions are not generally equal to zero; the Stratton-Chu formula merely returns this result when evaluated there 6.2 Fields in an unbounded medium Two special cases of the Stratton-Chu formula are important because of their applica- tion to antenna theory. The first is that of sources radiating into an unbor region The second involves a bounded region with all sources excluded. We shall consider the former here and the latter in 86.3 Assuming that there are no bounding surfaces in(6.7)and( 6.8), except for one surface that has been allowed to recede to infinity and therefore provides no surface contribution ve find that the electromagnetic fields in unbounded space are given by JixV G+P G)dv n=(①xv+vo-/)w We can view the right-hand sides as superpositions of the fields present in the cases There(1)electric sources are present exclusively, and(2)magnetic sources are present clusively. With Pm =0 and Jm=0 we find that ⅴG-jojG)d (6.13) H=/J×vGdv. 6.14) Using V'G=-VG we can write (r ②2001 by CRC Press LLC
Each of the expressions for E˜ is equally valid for points within Vm. Upon subtraction we get 0 = V −J˜i m × ∇ G + ρ˜i ˜ c ∇ G − jωµ˜ J˜i G dV + + N n=1 Sn (nˆ × E˜) × ∇ G + (nˆ · E˜)∇ G − jωµ(˜ nˆ × H˜ )G d S − − N n=1 1 jω˜ c na+ nb (dl · H˜ )∇ G, r ∈ Vm. This expression is exactly the Stratton–Chu formula (6.7) evaluated at points within the excluded region Vm. The treatment of H˜ is analogous and is left as an exercise. Since we may repeat this for any excluded region, we find that the Stratton–Chu formula returns the null field when evaluated at points outside V. This is sometimes referred to as the vector Ewald–Oseen extinction theorem [90]. We must emphasize that the fields within the excluded regions are not generally equal to zero; the Stratton–Chu formula merely returns this result when evaluated there. 6.2 Fields in an unbounded medium Two special cases of the Stratton–Chu formula are important because of their application to antenna theory. The first is that of sources radiating into an unbounded region. The second involves a bounded region with all sources excluded. We shall consider the former here and the latter in § 6.3. Assuming that there are no bounding surfaces in (6.7) and (6.8), except for one surface that has been allowed to recede to infinity and therefore provides no surface contribution, we find that the electromagnetic fields in unbounded space are given by E˜ = V −J˜i m × ∇ G + ρ˜i ˜ c ∇ G − jωµ˜ J˜i G dV , H˜ = V J˜i × ∇ G + ρ˜i m µ˜ ∇ G − jω˜ c J˜i m G dV . We can view the right-hand sides as superpositions of the fields present in the cases where (1) electric sources are present exclusively, and (2) magnetic sources are present exclusively. With ρ˜i m = 0 and J˜i m = 0 we find that E˜ = V ρ˜i ˜ c ∇ G − jωµ˜ J˜i G dV , (6.13) H˜ = V J˜i × ∇ G dV . (6.14) Using ∇ G = −∇G we can write E˜(r,ω) = −∇ V ρ˜i (r ,ω) ˜ c(ω) G(r|r ; ω) dV − jω V µ(ω) ˜ J˜i (r ,ω)G(r|r ; ω) dV = −∇φ˜ e(r,ω) − jωA˜ e(r, ω),
p(r, o) E@)G(rr; o)dv (r,a=/A(o)(r,o)G(rr;a)dV are the electric scalar and vector potential functions introduced in$5.2. UsingJ'xV'G G=V×(JjG) we have H(r, o) vxa(oJ(r, o)G(rr; o)dv V×A(r,a). (6.16) These expressions for the fields are identical to those of (5.56)and(5.57), and thus the integral formula for the electromagnetic fields produces a result identical to that obtained using potential relations. Similarly, with p=0,J=0 we have E I xV'GdV H V'G-joe G dv E(r,) V×A H(r, o)=-Vph(r, o) φh(r,o) G(rr dv 正(au) Ah (r, o)= 2(o)J (r,o)G(rr;o)dv are the magnetic scalar and vector potentials introduced in 8 5.2 6.2.1 The far-zone fields produced by sources in unbounded space Many antennas may be analyzed in terms of electric currents and charges radiating in unbounded space. Since antennas are used to transmit information over great distances the fields far from the sources are often of most interest assume that the sources are contained within a sphere of radius rs centered at the origin. We define the far zone of the sources to consist of all observation points satisfying both r >>rs(and thus r>r) and kr >1. For points in the far zone we may approximate the unit vector R directed from the sources to the observation point by the unit vector f directed from the origin to the observation point. We may also approximate dR(4R VR=R(I+jkr)e-jkR =PikE R 4丌R 6.17) ②2001 by CRC Press LLC
where φ˜ e(r,ω) = V ρ˜i (r ,ω) ˜ c(ω) G(r|r ; ω) dV , A˜ e(r,ω) = V µ(ω) ˜ J˜i (r ,ω)G(r|r ; ω) dV , (6.15) are the electric scalar and vector potential functions introduced in § 5.2. Using J˜i ×∇ G = −J˜i × ∇G =∇× (J˜i G) we have H˜ (r,ω) = 1 µ(ω) ˜ ∇ × V µ(ω) ˜ J˜i (r ,ω)G(r|r ; ω) dV = 1 µ(ω) ˜ ∇ × A˜ e(r, ω). (6.16) These expressions for the fields are identical to those of (5.56) and (5.57), and thus the integral formula for the electromagnetic fields produces a result identical to that obtained using potential relations. Similarly, with ρ˜i = 0, J˜i = 0 we have E˜ = − V J˜i m × ∇ G dV , H˜ = V ρ˜i m µ˜ ∇ G − jω˜ c J˜i m G dV , or E˜(r,ω) = − 1 ˜ c(ω) ∇ × A˜ h(r, ω), H˜ (r,ω) = −∇φ˜ h(r,ω) − jωA˜ h(r, ω), where φ˜ h(r,ω) = V ρ˜i m(r ,ω) µ(ω) ˜ G(r|r ; ω) dV , A˜ h(r,ω) = V ˜ c (ω)J˜i m(r ,ω)G(r|r ; ω) dV , are the magnetic scalar and vector potentials introduced in § 5.2. 6.2.1 The far-zone fields produced by sources in unbounded space Many antennas may be analyzed in terms of electric currents and charges radiating in unbounded space. Since antennas are used to transmit information over great distances, the fields far from the sources are often of most interest. Assume that the sources are contained within a sphere of radius rs centered at the origin. We define the far zone of the sources to consist of all observation points satisfying both r rs (and thus r r ) and kr 1. For points in the far zone we may approximate the unit vector Rˆ directed from the sources to the observation point by the unit vector rˆ directed from the origin to the observation point. We may also approximate ∇ G = d d R e− jkR 4π R ∇ R = Rˆ 1 + jkR R e− jkR 4π R ≈ rˆ jk e− jkR 4π R = rˆ jkG. (6.17)
Using this we can obtain expressions for E and H in the far zone of the sources. The approximation(. 17) leads pVG≈ (rjkG)=--fV(GJ)-J'VGI Substituting this into(6.13), again using(6.17) and also using the divergence theorem E(o)≈-,jon[-fe,了]Gd+ oec (nJ)Gds where the surface S surrounds the volume V that contains the impressed sources. If we let this volume slightly exceed that needed to contain the sources, then we do not change the value of the volume integral above; however, the surface integral vanishes cen·J=0 everywhere on the surface. Using f x(×J)=f(·J)- I' we then obtain the far-zone expression E(r,) jof xfx/i(o)j'(r,o)G(rr;)dv jof xf x Ae(r, o) where Ae is the electric vector potential. The far-zone electric field has no r-component and it is often convenient to write E(r, o) (r,a) 18 where Aer is the vector component of Ae transverse to the r-direction fxa=Ae-r(f Ae)=8Ace+oAed We can approximate the magnetic field in a similar fashion. Noting that JxV'G J× gkfG) we hay With this we hav T X (r, E(r,ω)=-nf×H(r,a),H(r,o) in the far zone To simplify the computations involved, we often choose to approximate the vector al in the far zone. noting that R (r·r ②2001 by CRC Press LLC
Using this we can obtain expressions for E˜ and H˜ in the far zone of the sources. The approximation (6.17) leads directly to ρ˜i ∇ G ≈ j ∇ · J˜i ω (rˆ jkG) = − k ωrˆ ∇ · (GJ˜i ) − J˜i · ∇ G . Substituting this into (6.13), again using (6.17) and also using the divergence theorem, we have E˜(r,ω) ≈ − V jωµ˜ J˜i − rˆ(rˆ · J˜i ) G dV + rˆ k ω˜ c S (nˆ · J˜i )GdS , where the surface S surrounds the volume V that contains the impressed sources. If we let this volume slightly exceed that needed to contain the sources, then we do not change the value of the volume integral above; however, the surface integral vanishes since nˆ · J˜i = 0 everywhere on the surface. Using rˆ × (rˆ × J˜i ) = rˆ(rˆ · J˜i ) − J˜i we then obtain the far-zone expression E˜(r,ω) ≈ jωrˆ × rˆ × V µ(ω) ˜ J˜i (r ,ω)G(r|r ; ω) dV = jωrˆ × rˆ × A˜ e(r,ω) , where A˜ e is the electric vector potential. The far-zone electric field has no r-component, and it is often convenient to write E˜(r,ω) ≈ − jωA˜ eT (r,ω) (6.18) where A˜ eT is the vector component of A˜ e transverse to the r-direction: A˜ eT = −rˆ × rˆ × A˜ e = A˜ e − rˆ(rˆ · A˜ e) = θˆ A˜ eθ + φˆ A˜ eφ. We can approximate the magnetic field in a similar fashion. Noting that J˜i × ∇ G = J˜i × (jkrˆG) we have H˜ (r,ω) ≈ − j k µ(ω) ˜ rˆ × V µ(ω) ˜ J˜i (r ,ω)G(r|r ,ω) dV ≈ −1 η jωrˆ × A˜ e(r, ω). With this we have E˜(r,ω) = −ηrˆ × H˜ (r, ω), H˜ (r,ω) = rˆ × E˜(r,ω) η , in the far zone. To simplify the computations involved, we often choose to approximate the vector potential in the far zone. Noting that R = (r − r ) · (r − r ) = r 2 + r2 − 2(r · r )
d remembering that r >>r for r in the far zone, we can use the leading terms of a binomial expansion of the square root to get (6.19) Thus the Green's function may be approximated as Gr;o)≈re Here we have kept the approximation(6.19) intact in the phase of G but have used 1/RN 1/r in the amplitude of G. We must keep a more accurate approximation for the phase since k(f. r)may be an appreciable fraction of a radian. We thus have the far-zone approximation for the vector potential A-(r,o)≈p(a) Let us summarize the expressions for computing the far-zone field E(r, o) (6.21) f×E(r,o) (6.22) Ae(r, o) arri(o)ae(0, a), (6.23) Here ae is called the directional weighting function. This function is independent of r and describes the angular variation, or pattern, of the fields In the far zone e, H, f are mutually orthogonal. Because of this, and because the fields vary as e-jkr /r, the electromagnetic field in the far zone takes the form of a spherical TEM wave which is consistent with the Sommerfeld radiation condition Power radiated by time-harmonic sources in unbounded space. In$5. 2.1 we defined the power radiated by a time-harmonic source in unbounded space as the total time-average power passing through a sphere of very large radius. We found that for a Hertzian dipole the radiated power could be computed from the far-zone fields through P where E×H} is the time-average Poynting vector. By superposition this holds for any localized source. Assuming a lossless medium and using phasor notation to describe the time-harmonic ②2001 by CRC Press LLC
and remembering that r r for r in the far zone, we can use the leading terms of a binomial expansion of the square root to get R = r 1 − 2(rˆ · r ) r + r r 2 ≈ r 1 − 2(rˆ · r ) r ≈ r 1 − rˆ · r r ≈ r − rˆ · r . (6.19) Thus the Green’s function may be approximated as G(r|r ; ω) ≈ e− jkr 4πr e jkrˆ·r . (6.20) Here we have kept the approximation (6.19) intact in the phase of G but have used 1/R ≈ 1/r in the amplitude of G. We must keep a more accurate approximation for the phase since k(rˆ · r ) may be an appreciable fraction of a radian. We thus have the far-zone approximation for the vector potential A˜ e(r,ω) ≈ µ(ω) ˜ e− jkr 4πr V J˜i (r ,ω)e jkrˆ·r dV , which we may use in computing (6.18). Let us summarize the expressions for computing the far-zone fields: E˜(r,ω) = − jω θˆ A˜ eθ (r,ω) + φˆ A˜ eφ(r,ω) , (6.21) H˜ (r,ω) = rˆ × E˜(r,ω) η , (6.22) A˜ e(r,ω) = e− jkr 4πr µ(ω) ˜ a˜e(θ, φ, ω), (6.23) a˜e(θ, φ, ω) = V J˜i (r ,ω)e jkrˆ·r dV . (6.24) Here a˜e is called the directional weighting function. This function is independent of r and describes the angular variation, or pattern, of the fields. In the far zone E˜ , H˜ , rˆ are mutually orthogonal. Because of this, and because the fields vary as e− jkr /r, the electromagnetic field in the far zone takes the form of a spherical TEM wave, which is consistent with the Sommerfeld radiation condition. Power radiated by time-harmonic sources in unbounded space. In § 5.2.1 we defined the power radiated by a time-harmonic source in unbounded space as the total time-average power passing through a sphere of very large radius. We found that for a Hertzian dipole the radiated power could be computed from the far-zone fields through Pav = lim r→∞ 2π 0 π 0 Sav · rˆr 2 sin θ dθ dφ where Sav = 1 2 Re Eˇ × Hˇ ∗ is the time-average Poynting vector. By superposition this holds for any localized source. Assuming a lossless medium and using phasor notation to describe the time-harmonic
