Method of Moments Solution of Hallen's Equation for a Center-Fed Dipole Antenna, Assignments of Classical Philology

Download Method of Moments Solution of Hallen's Equation for a Center-Fed Dipole Antenna and more Assignments Classical Philology in PDF only on Docsity! HOMEWORK #3 3 METHOD OF MOMENTS SOLUTION OF HALLEN’S EQUATION Hallen’s equation was derived in class as: L 2 −L2 I(z ) e−jkR R dz = −j 2π η0 Vg sin k0|z|− C cos k0z (1) where R = a2 + (z − z )2. Hallen’s equation applies to the center-fed dipole. From the symmetry of the problem, we expect the current on each half of the dipole to be the same I(z) = I(−z) and so the integral can be rewritten as L 2 −L2 I(z ) e−jkR R dz = L 2 0 I(z ) e−jk0R1 R1 + e−jk0R2 R2 G(z,z ) dz (2) where R1 = a2 + (z − z )2 R2 = a2 + (z − z )2 Following the Method of Moments procedure, we expand the current in a set of basis func- tions I(z) = N n=1 InΦn(z) (3) so that N n=1 In L 2 0 Φn(z )G(z, z ) dz + C cos k0z = −jVg 60 sin k0|z| (4) Note that there are N+1 unknowns in this equations. Multiply both sides by a set of N+1 weighting functions wm(z) and integrate to give N n=1 InZmn + CYm = Vm (5) 1 2 where Zmn = L 2 0 L 2 0 wm(z)G(z, z )Φn(z ) dz dz Ym = L 2 0 wm(z) cos k0(z) dz Vm = −jVg 60 L 2 0 wm(z) sin k0|z| dz For notational convenience let C = IN+1, so that N+1 n=1 InAmn = Vm (6) where Amn = Zmn for n ≤ N Ym for n = N + 1 The solution of this matrix equation is I = A −1 V (7) Far-Fields In our discussion of wire antennas we found that a filamentary current along the z-axis gives rise to an electric field with a θ̂ component given by Eθ ∝ sin θ L/2 −L/2 I(z )ek0z cos θdz (8) Substituting (3) and exploiting the symmetry of the current distribution gives Eθ ∝ 2 sin θ N n=1 In L/2 0 Φ(z ) cos (k0z cos θ) dz (9) For simple basis functions, the integrals can usually be carried out analytically. Pulse Basis and Point Matching For point matching we enforce the equation at N + 1 discrete points on the wire given by zn = (n− 1)∆ ∆ = L 2N n = 1 . . . N + 1 (10) The weighting functions are then Dirac delta functions, wm(z) = δ(z − zm) (11)