The antenna performance advantages of using a ferrite loading material are shown analytically by the following spherical model with infinitesimally small Hertzian excitation sources. The spherical volume of material has the following relative permeability, relative permittivity and loss tangents:

    (1)

The sphere of radius a is excited at the origin O by either a magnetic (TE) or an electric (TM) multipole source having exp[-jωt] time dependence. The fields are expressed as exp[jmφ] P_n^m (cosθ)R(r), where n=1,2,3…∞, m=0,1,2,3…n, (r, θ,ψ) are spherical coordinates, P_n^m (cosθ) are Associated Legendre functions and R(r) is either a spherical Bessel function of the first kind j_n(r) or of the third kind h_n⁽¹⁾(r). On invoking continuity of the tangential fields at r=a for each (nm)th eigenfunction, the fields for r≥a have an amplitude governed by

   (2)

where [f(ρ)]’ denotes differentiation of f(ρ) with respect to ρ, A_nm is the (nm)th source amplitude, ρ₁=k₁a, ρ₂=k₂a, k₁=k₀(μ_Tε_T) ^1/2, k₂=2π/λ₀, λ₀=freespace wavelength and

   (3)

When the sphere is electrically small the low order TE_nmv and TE_nmv resonator modes are narrow band and the dielectric resonator handset antenna (DRA) fields can be represented by the mode alone where relates to the sphere electrical size. In this present paper we are interested in the dipole-like far-fields of the TE₁₀₁ and TM₁₀₁ modes corresponding to magnetic and electric dipole excitation sources respectively, as realized in practice by a small wire loop or dipole wire probe. In the presence of a conductive ground plane the TM₁₀₁ mode is excited by a monopole probe but the TM₁₀₁ mode is not compatible with the ground plane boundary conditions. A small slot in the ground plane will however excite the TE₁₁₁ mode which is the TE₁₀₁ mode with a 90^° rotation of axis.

The complex roots of the denominator of (2) for the TE₁₀₁ and TM₁₀₁ modes are respectively P_M= P’_M+jP’’_M and P_E= P’_E+jP’’_E. The handset antenna Q factors are then given by

  (4)

and relate to the energy lost to radiation. The reciprocal of Q is a useful measure [10] of the antenna impedance bandwidth when a finite size loop or probe is the excitation source. Losses in the material, lower the Q values to Q_M^L and Q_E^Land the antenna radiation efficiencies and are respectively given by

  (5)

The antenna resonant frequencies f_o^M and f_o^E are derived from the real part of the complex roots as

  (6)

 

Fig. 1. Theoretical radiation performance of spherical material coated antenna, radius 20 mm, with central Hertzian electric or magnetic dipole excitation showing variation with K and losses; tan δ_ε= tanδ_μ = tand. (a) Q factor and efficiency η and (b) resonant frequency f_o^M and f_o^E.

 

Examination of (1) to (6) indicates that for moderate material losses the radiation is enhanced when μ’ is of comparable value to ε’. This important property is illustrated in Fig. 1(a) for both the TM₁₀₁ and TE₁₀₁ modes, with a=20 and constant μ’ε’ product of 36. According to the definition of K in (3), Fig. 1 represent either the TM₁₀₁ mode when K=ε_r or the TE₁₀₁ when K=μ_T, corresponding to Hertzian electric and magnetic dipole excitation sources respectively. Note that the small losses allow K to be taken as ε’ or μ’ for ease of presentation. For both modes, the minimum Q and hence maximum BW occurs when μ’=ε’=6; this holds over a wide range of material loss values. In the lossless case both modes have the same Q values. It should be noted that (3) has symmetry in tanδ_μ and tan δ_ε so each loss affects the antenna performance to the same extent.

The ratio μ’/ε’ has an influence on the resonant frequency but the latter is relatively insensitive to the level of material loss as shown in Fig. 1(b).

A similar resonant frequency insensitivity is observed when the sphere is immersed in a lossy media thus supporting the claims, noted above, that material loaded antennas are less affected by their environment. Fig. 1(a) gives the η and Q plots, showing that maximum radiation is released when μ’=ε’=6. From a physical standpoint it would seem for the spherical geometry at least, that equality of μ’ and ε’ provides the best impedance match at the air/material interface. The greatest electrical size reduction however does not correspond to the latter condition and from the resonant frequency plot in Fig. 1(b) is seen to occur for the TM₁₀₁ mode, when K=ε’=1 and μ’=36; for this condition the spherical surface is approximately an electric wall and the nature of the antenna external near field is predominantly electric [13]. The boundary conditions on the sphere are thus compatible with the electric excitation source [10]. Conversely for the mode, minimum electrical size occurs when K=μ’=1 and ε’=36; for this condition the spherical surface is approximately a magnetic wall, the nature of the external antenna near field is predominantly magnetic [13] and the boundary conditions on the sphere are thus compatible with the magnetic excitation source [10]. The boundary conditions are incompatible for other combinations of excitation sources and material composition and no clear distinction can be made regarding the nature of the antenna near fields.

Another interesting property is that the resonant frequencies of the TM₁₀₁ and TE₁₀₁ modes [Fig. 1(b)] converge to the same frequency at μ’=ε’=6. This could be of practical use in the design of circularly polarized material loaded antennas [14], given the appropriate combined electric and magnetic dipole excitation with quadrature phase. From the examination of a few higher modes it would appear that the above properties hold for all higher modes in this spherical geometry.