Electromagnetic Field Theory: A Problem Solving Approach Part 71

= sins 0d0 cpn=d21w = I Idlj2 =16d1 t[icos O(sin' 0+2)]1" Ifd1l 2 ='IQ 71k 2 r7 i J,=0 s2 1 676 Radiation As far as the dipole is concerned, this radiated power is lost in the same way as if it were dissipated in a resistance R, (30)

= i l2R where this equivalent resistance is called the radiation resistance: (k d 2 27 dl(31) R = ) , k=--In free space '70-/ Lo/EO0 1207r, the radiation resistance is 2 Ro = 8012() (free space) (32) These results are only true for point dipoles, where dl is much less than a wavelength (dl/A << I). This verifies the validity of the quasi-static approximation for geometries much smaller than a radiated wavelength, as the radiated power is then negligible. If the current on a dipole is not constant but rather varies with z over the length, the only term that varies with z for the vector potential in (5) is I(z): S+d1/2 li(z) A,(r)= Re Sd/2 2 1 e- jkrQp, dz ~-QP e-jkrQ' Re 4 Q rrQP +d1/2 -dU2 (z)dz (33) where, because the dipole is of infinitesimal length, the distance rQp from any point on the dipole to any field point far from the dipole is essentially r, independent of z. Then, all further results for the electric and magnetic fields are the same as in Section 9-2-3 if we replace the actual dipole length dl by its effective length, 1 +dl/2 10 di/2 dleff - I(z) dz (34) where 0ois the terminal current feeding the center of the dipole. Generally the current is zero at the open circuited ends, as for the linear distribution shown in Figure 9-4, I(z) = Io(1-2z/dl), Io(l+ 2z/dl), - z- dl/2 -dl/2-z-0 (35) so that the effective length is half the actual length: dle=ff 2 1 r+d/ - J-I/ 2 10 d/2 I(z) dz = dl 2 (36) 677 Radiation from PointDipoles 7i(z) 'Jo dl1f = d12 -d1/2 d/12 d!.z 7(z) dz x (b) (a) Figure 9-4 (a) If a point electric dipole has a nonuniform current distribution, the solutions are of the same form if we replace the actual dipole length dl by an effective length dl,,. (b) For a triangular current distribution the effective length is half the true length. Because the fields are reduced by half, the radiation resistance is then reduced by 1: JR( : (dleu]'• : 201P\'r ( (37) In free space the relative permeability /A, and relative permittivity e, are unity. Note also that with a spatially dependent current distribution, a line charge distribution is found over the whole length of the dipole and not just on the ends: 1 di I=-- jw dz (38) For the linear current distribution described by (35), we see that: 2I/o j od 9-2-6 0 5 z s dl/2 I-dl/2 <7= 22 12"n'c (42) To approximately compute wo, we use the approximate radius of the electron found in Section 3-8-2 by equating the energy stored in Einstein's relativistic formula relating mass to energy: 2 3Q2 3Q 2 105 mc 2e Ro 20 .IMC x 10L1.69 m (43) Then from (40) o= 3 /5/3 207EImc , 3Q 2 - ~2.3 x 10'" radian/sec is much greater than light frequencies (w becomes approximately lim

ý 12 2 Eow o>> 127A mcwo (44) 1015) so that (42) (45) This result was originally derived by Rayleigh to explain the blueness of the sky. Since the scattered power is proportional to w 4 , shorter wavelength light dominates. However, near sunset the light is scattered parallel to the earth rather than towards it. The blue light received by an observer at the earth is diminished so that the longer wavelengths dominate and the sky appears reddish. 9-2-7 Radiation from a Point Magnetic Dipole A closed sinusoidally varying current loop of very small size flowing in the z = 0 plane also generates radiating waves. Because the loop is closed, the current has no divergence so 680 Radiation that there is no charge and the scalar potential is zero. The vector potential phasor amplitude is then 0) A(r) = e--jr,- dl (46) We assume the dipole to be much smaller than a wavelength, k(rQp-r)<< 1, so that the exponential factor in (46) can be linearized to - e ikQp = e - jk r e - lim j (r Pg P- r T) ,e-i er[l - jk( rQp - r)] k(rqp-r) 21i i, d= (-sin i + cos 4i,) di = 0 (49) the integral is again zero as the average value of the unit vector i# around the loop is zero. The remaining integral is the same as for quasi-statics except that it is multiplied by the factor (1+ jkr) e-i. Using the results of Section 5-5-1, the quasi-static vector potential is also multiplied by this quantity: M= sin 0(1 +jkr) e-k'i,, 4 7tr- t = dS (50) 681 Point Dipole Arrays The electric and magnetic fields are then f=lvxA=• jksei, 2cos 1 +i Sjkr x X 1si (jkr) = WE (jkr) (51) 1 (jkr)I 71e-krsin 0 (jkr) 4r 1 + (jkr)' S u + (jkr)2Y The magnetic dipole field solutions are the dual to those of the electric dipole where the electric and magnetic fields reverse roles if we replace the electric dipole moment with the magnetic dipole moment: p 9-3 q dl I dl m (52) POINT DIPOLE ARRAYS The power density for a point electric dipole varies with the broad angular distribution sin 2 0. Often it is desired that the power pattern be highly directive with certain angles carrying most of the power with negligible power density at other angles. It is also necessary that the directions for maximum power flow be controllable with no mechanical motion of the antenna. These requirements can be met by using more dipoles in a periodic array. 9-3-1 A Simple Two Element Array To illustrate the basic principles of antenna arrays we consider the two element electric dipole array shown in Figure 9-6. We assume each element carries uniform currents II and i2 and has lengths dll and dl2, respectively. The elements are a distance 2a apart. The fields at any point P are given by the superposition of fields due to each dipole alone. Since we are only interested in the far field radiation pattern where 01 02 0, we use the solutions of Eq. (16) in Section 9-2-3 to write: + E2 sin 0 e -kZ EI sin Oe- ' jkr, jkr2 where P, dl k• 41r 21 dl2 k"y 4.7 ( 682 Radiation Z 11/2 r r + asinOcoso asinOcos0 a = sin 0 cos 0 Figure 9-6 The field at any point P due to two-point dipoles is just the sum of the fields due to each dipole alone taking into account the difference in distances to each dipole. Remember, we can superpose the fields but we cannot superpose the power flows. From the law of cosines the distances r, and r 2 are related as r 2 = [r2+ a 2 - 2ar cos (7Tr- 6)]j /2 = [r2+ a 2 + 2ar cos rl = [r2 + a2- 2 ] 1/ 2 ar cos]12 (2) where 6 is the angle between the unit radial vector i, and the x axis: = ir, ix = sin 0 cos cos 4 Since we are interested in the far field pattern, we linearize (2) to rf r i/a 2 2 2a r + 2-+--sin s 0 cos r, r-- r + a sin 0 cos • lim a2 2ar sin 0 cos ) r - a sin 0 cos 4 In this far field limit, the correction terms have little effect in the denominators of (1) but can have significant effect in the exponential phase factors if a is comparable to a wavelength so that ka is near or greater than unity. In this spirit we include the first-order correction terms of (3) in the phase I I_·_ 683 Point Dipole Arrays factors of (1), but not anywhere else, so that (1) is rewritten as /E = -/H, jk- sin Oe= 4 rr jkr(l dil ejk si' . • + d12 e - k ' - in "'. ) (4) array factot eltentlt factol The first factor is called the element factor because it is the radiation field per unit current element (Idl) due to a single dipole at the origin. The second factor is called the array factor because it only depends on the geometry and excitations (magnitude and phase) of each dipole element in the array. To examine (4) in greater detail, we assume the two dipoles are identical in length and that the currents have the same magnitude but can differ in phase X: dl = dl2 -dl i, = ie' xe)Xfi i, ý2 = e") ( so that (4) can be written as 0= , = e-i sin ejx/2 cos (ka sin 0 cos jkr 2 (6) Now the far fields also depend on 0. In particular, we focus attention on the 0 = 7r/ 2 plane. Then the power flow, lim S> = 1I 2 1Eol 22 (kr) co s 2 ka cos - (7) depends strongly on the dipole spacing 2a and current phase difference X. (a) Broadside Array Consider the case where the currents are in phase (X= 0) but the dipole spacing is a half wavelength (2a = A/2). Then, as illustrated by the radiation pattern in Figure 9-7a, the field strengths cancel along the x axis while they add along the y axis. This is because along the y axis r, = r2, so the fields due to each dipole add, while along the x axis the distances differ by a half wavelength so that the dipole fields cancel. Wherever the array factor phase (ka cos 0 -X/ 2 ) is an integer multiple of nT, the power density is maximum, while wherever it is an odd integer multiple of 7'/2, the power density is zero. Because this radiation pattern is maximum in the direction perpendicular to the.array, it is called a broadside pattern. 684 Radiation acos2(Tcose), 2 X= 0 cos2( 0 Ecos 2 - 8 '), x = 1! 4 acos2(!cos 02 Broadside (a) = 4 ), X 2 (c) a ( k--x 3 acos2'(2cos0- 1r), X= 32 2 8 4 acos2IIcoS 2 Endfire (d) 2a =X/2 -I 2 ), x= r (e) Figure 9-7 The power radiation pattern due to two-point dipoles depends strongly on the dipole spacing and current phases. With a half wavelength dipole spacing (2a = A/2), the radiation pattern is drawn for various values of current phase difference in the 0 = ir/2 plane. The broadside array in (a) with the currents in phase (X = 0) has the power lobe in the direction perpendicular to the array while the end-fire array in (e) has out-of-phase currents (X = 7r) with the power lobe in the direction along the array. -