Resonant Half-wave Dipole and Its Odd Integral Multiples

Based on the results of several numerical computations of the radiation integrals derived to evaluate the characteristics of the dipole antenna, it is shown that the half-wave dipole precisely resonates at the point (l
_r
, R
_in
) = (0.4889λ,72.38Ω), where l
_r
; and Rin are the resonant electrical length and the input resistance of the dipole, respectively. Use is made of (a= 5 × 10
^-6
λ, f= 550 MHz, N, = 15), where (a, f, N
_s
) are the radius of the wire, operating frequency and the number of segments used in the Method of Moments (MoM) computation, respectively. Several dipole radii were computationally investigated before selecting a = 5 × 10
^-6
λ as the best option for an electrically thin wire, which is consistent with the filamentary dipole requirement of the use of MoM as a numerical tool. A close observation of the design curves describing the impedance variation against the electrical length of the dipole within the range 0 ≤ lλ
^-1
≤ 10, reveals that higher resonances occur at odd integral multiples of that first resonant length symbolized as l
_r
' such that the next five odd integral multiples are 3lr, 5l
_r
, 7l
_r
, 9l
_r
and 11l
_r
, respectively, which are the values of the electrical lengths at the points when the input reactance attenuates to zero. What eventually emerged are doughnut-shaped pattern and asymmetric pattern radiated at first and second resonances, respectively. It is interesting to point out what is observable at the point l == 8A on the design curve. At that point, the input impedance, Zin= (R
_in
, X
_in
) = (1.4 × 10
¹⁷
, 1.48 × 10
¹⁸
) Ω, which is so high that it virtually renders radiation impossible as the input impedance tends to infinity. This justifies why dipoles can only radiate at odd integral multiples of the first resonant length (that is approximately 0.5A). The profiles of the approximate current distribution obtained from MoM also reveal that for x multiples of the first resonant length, l
_r'
of the dipole, the number of complete crests and troughs obtainable is given as 0.5(x - 1).