A) Computation of the equivalent antenna current sources

First let us consider an antenna immersed in ES satisfying the resonance conditions. The magnetic energy at resonance can be computed in a defined volume V _n−f , which includes the whole near-field generated by the antenna, thus the energy stored satisfies the following equality:

(1) $$\int_{{V_{n - f}}} {{\mu _0}\mathop {\left\vert {{\bf H}(r,\theta ,\varphi )} \right\vert}\nolimits^2 \;dV} = \int_{{V_{n - f}}}^{} {{\varepsilon _0}\mathop {\left\vert {{\bf E}(r,\theta ,\varphi )} \right\vert}\nolimits^2 \;dV}.$$

It is noteworthy that the integration volume should be carefully selected to be large enough to include the entire near-field region, but limited to avoid divergence in (1) due to the far-field contribution. Its exact definition is dependent on the particular circumstances, theoretical or practical, but it is not an issue and is stated case by case in the following. Let us now substitute the ES with another isotropic medium: in this case we adopt an OM material with permeability μ _r μ ₀ and permittivity ε ₀. We can reasonably suppose that the volume containing the OM material-antenna near-field ( $V_{n - f}^{(\mu )} $ ) is reduced, with respect to the ES case, according to the rule relating the guided wavelength to the relative electromagnetic constants. Accordingly, we assume that the same energy balance computed by (1) can be obtained in the reduced volume:

(2) $$V_{n - f}^{(\mu )} = \displaystyle{{{V_{n - f}}} \over {\sqrt {\mu _r^3}}},$$

where it will be:

(3) $$\eqalign{& \int_{V_{n - f}^{(\mu )}} {{\mu _0}{\mu _r}\mathop {\left\vert {{{\bf H}_\mu} ({r_\mu}, \theta, \varphi )} \right\vert}\nolimits^2 \;d{V^\mu}} \cr & = \int_{V_{n - f}^{(\mu )}} {{\varepsilon _0}\mathop {\left\vert {{{\bf E}_\mu} ({r_\mu}, \theta, \varphi )} \right\vert}\nolimits^2 \;d{V^\mu}}}.$$

Now, as a hypothesis, we can adopt the following relationship between the corresponding electric field in the ES (E) and in the OM (E _μ ):

(4) $$\left\vert{\bf E}(r,\theta, \varphi )b \right\vert = \mu _r^{ - {{1 \over 2}}} \left\vert{{\bf E}_\mu} ({r_\mu}, \theta, \varphi )\right\vert,$$

where

(5) $${r_\mu} = r\mu _r^{ - {{1 \over 2}}}.$$

These relationships rely on the condition of obtaining the same radiated power in the far field with a reduced wavelength, due to the adopted substrate, for a patch antenna without losses (4). These results are also validated in the next section by the numerical simulation for two patch antennas. Notice μ _r should be thought as an effective value.

Let us now adopt the variable substitutions (2), for dV ^μ , and (5) for r _μ , and the hypothesis (4) and (1) to compute equation (3) which becomes:

(6) $$\int_{V_{n - f}^{(\mu )}} {{\varepsilon _0}\mathop {\left\vert{{\bf E}_\mu} ({r_\mu}, \theta, \varphi )\right\vert}\nolimits^2 \;d{V^\mu} = \mu _r^{ - {{1 \over 2}}} \int_{{V_{n - f}}} {{\mu _0}\left\vert{\bf H}(r,\theta, \varphi )\right\vert\;dV}}.$$

The left-hand side (LHS) of (3) by substitutions can also be cast:

(7) $$\eqalign{& \int_{V_{n - f}^{(\mu )} } {{\mu _0}{\mu _r}{{\left\vert {{{\bf H}_\mu }({r_\mu },\theta ,\varphi )} \right\vert}^2}\;d{V^\mu } = \mu _r^{ - {1 \over 2}} } \cr & \quad \int_{{V_{n - f}}} {{\mu _0}\left\vert {{{\bf H}_\mu }({r_\mu },\theta ,\varphi )} \right\vert\;dV} .}$$

Thus we can write:

(8) $$\int_{{V_{n - f}}} {\mathop {\left\vert{\bf H}(r,\theta, \varphi )\right\vert}\nolimits^2 \;dV =} \int_{{V_{n - f}}} {\mathop {\left\vert{{\bf H}_\mu} ({r_\mu}, \theta, \varphi )\right\vert}\nolimits^2 \;dV},$$

where H _μ is the magnetic field in the OM case.

The magnetic fields relation satisfying (8) can be obtained by resorting to a modified Wilcox expansion [Reference Wilcox12]:

(9) $$\left\vert{{\bf H}_\mu} ({r_\mu}, \theta, \varphi )\right\vert = \left\vert {\sum\limits_{n = 0}^\infty \,{c_{n,\mu}} \,\displaystyle{{{{\bf B}_n}(\theta, \varphi )} \over {r_\mu ^{n + 1}}}} \right\vert,$$

(10) $$\left\vert{\bf H}(r,\theta, \varphi )\right\vert = \left\vert {\sum\limits_{n = 0}^\infty \,{c_n}\,\displaystyle{{{{\bf B}_n}(\theta, \varphi )} \over {{r^{n + 1}}}}} \right\vert.$$

This representation is addressed as modified due to the presence of the c _n,μ and c _n terms, which are dependent only on the substrate material; it is noteworthy that the vector function B _n (θ, ϕ), representing the dependence on the elevation and azimuth variables, is the same for both expansions, i.e. when n = 0 the two antennas share the same shape of the radiation functions.

Now it is possible to represent (8) using (8); its right-hand side (RHS), considering (5) and (9), becomes:

(11) $$\eqalign{& \int_{V_{n - f}} {\left\vert {\bf H}_{\mu} \left(r_{\mu}, \theta, \varphi \right)\right\vert} ^2 \;dV \cr & \quad = \mu _r^{{{{m + n + 2} \over 2}}} \displaystyle \int\limits_{{V_{n - f}}} \sum\limits_{m = 0}^\infty \,\sum\limits_{n = 0}^\infty \,{c_{m,\mu}} c_{n,\mu} ^{\ast} {{{{\bf B}_m}(\theta, \varphi ) \cdot {\bf B}_n^{\ast} (\theta, \varphi )} \over {{r^{m + n + 2}}}}\;dV.}$$

Similarly, using (10) in the LHS of (8), (8) becomes:

(12) $$\eqalign{& \int\limits_{{V_{n - f}}} \,\;\sum\limits_{m = 0}^\infty \,\sum\limits_{n = 0}^\infty \,\left( {\displaystyle{{{{\bf B}_m}(\theta, \varphi ) \cdot {\bf B}_n^{\ast} (\theta, \varphi )} \over {{r^{m + n + 2}}}}} \right) \cr & \quad \times \left( {\displaystyle{{{c_{m,\mu}} c_{n,\mu} ^{\ast}} \over {\mu _r^{ - \{ [m + n + 2]/2\}}}} - {c_m}c_n^{\ast}} \right)\;dV = 0}.$$

The LHS and RHS integrals in (1), and thus in (8), are singularly divergent due to the radiated power in the far field [Reference Mikki and Antar8]. Since a difference is computed inside (12) and the far-field radiated powers were forced to be the same for both materials, the integral in (12) is not divergent anymore. Mathematically, given the above consideration, the volume can be thought as going to infinite. This removes the ambiguity in its definition for this theoretical case, but for completeness and easiness of notation, in what follows is still specified as V _n−f .

Furthermore, since the Poynting vector is not divergent, it is also possible to swap the integral and sums [Reference Mikki and Antar8] to get:

(13) $$\eqalign{&\sum\limits_{m = 0}^\infty \,\sum\limits_{n = 0}^\infty \,\left( {\int\limits_{{V_{n - f}}} \displaystyle{{{{\bf B}_m}(\theta, \varphi ) \cdot {\bf B}_n^{\ast} (\theta, \varphi )} \over {{r^{m + n + 2}}}}\;dV} \right) \cr & \quad \times\left( {\displaystyle{{{c_{m,\mu}} c_{n,\mu} ^{\ast}} \over {\mu _r^{ - \{ [m + n + 2]/2\}}}} - {c_m}c_n^{\ast}} \right) = 0}.$$

Considering the same boundary conditions, equation (13) has multiple solutions. There are different relations regarding the various c coefficients and thus the field within the OM. Using the uniqueness theorem the field has to be unique. Here we consider the solution by zeroing all the terms of the series:

(14) $${c_{m,\mu}} c_{n,\mu} ^{\ast} = \mu _r^{ - \{ [m + n + 2]/2\}} \;{c_m}c_n^{\ast}.$$

Using (5) and (14) to (9), the relationship between H _μ and H becomes:

(15) $$\eqalign{\left\vert {\bf H}_{\mu} \left(r_{\mu},\theta, \varphi \right) \right\vert ^2 & = \left\vert {\bf H} \left(r,\theta,\varphi \right) \right\vert ^2 \Rightarrow \left\vert {\bf H}_{\mu} \left(r_{\mu},\theta,\varphi \right) \right\vert \cr & = \left\vert {\bf H} \left(r,\theta,\varphi \right) \right\vert}.$$

The solution for the OM material is validated by solving the homogeneous Helmholtz equation in the domain (r _μ ):

(16) $$\nabla _\mu ^2 \,{{\bf H}_\mu} ({r_\mu}, \theta, \varphi ) = \mu {\varepsilon _0}\displaystyle{{{\partial ^2}} \over {\partial {t^2}}}{{\bf H}_\mu} ({r_\mu}, \theta, \varphi ),$$

which by substituting (5) results in:

(17) $${\nabla ^2}{\bf H}(r,\theta, \varphi ) = {\mu _0}{\varepsilon _0}\displaystyle{{{\partial ^2}} \over {\partial {t^2}}}{\bf H}(r,\theta, \varphi ),$$

Hence, if H(r, θ, ϕ) is a valid solution to the homogeneous Maxwell equations, H _μ (r _μ , θ, ϕ) is also valid, in particular it is the same solution in the corresponding spatial domain. Relation (15) will be numerically validated later.

It is known that the far field has transverse electromagnetic (TEM) properties, it is thus easy to express the relation in (4) and (15) as the components tangent to the contour sphere of the volume (V _∞ − V _n−f ):

(18) $$\left\vert{{\bf E}_\tau} \right\vert = \mu _r^{ - {{1 \over 2}}} \left\vert{\bf E}_\tau ^{(\mu )} \right\vert,\quad \left\vert{{\bf H}_\tau} \right\vert = \left\vert{\bf H}_\tau ^{(\mu )} \right\vert.$$

Note that the relations (4) and (15) have been extended to be valid outside the V _n−f , since the term in the Wilcox expansion (8) with n = 0 is present even in the near field and the near-field volume could be theoretically extended to r → ∞, as also said in [Reference Mikki and Antar8].

We can now make use of the Love's field equivalence principle [Reference Love10] to get the corresponding electric and magnetic magnitudes of surface equivalent currents J _S , ${\bf J}_S^{(\mu )} $ and M _S , ${\bf M}_S^{(\mu )} $ :

(19) $$\eqalign{ & \left\vert{\bf J}_S^{(\mu )} \right\vert = \left\vert{{\bf J}_S}\right\vert = \left\vert\hat n \times {{\bf H}_\tau} \right\vert, \cr & \quad \left\vert{\bf M}_S^{(\mu )} \right\vert = \sqrt {{\mu _r}} \left\vert{{\bf M}_S}\right\vert = \left\vert{{\bf E}_\tau} \times \hat n \right\vert},$$

where ${\hat n}$ is the versor normal to the surface.

From (19) is noteworthy that the equivalent electric sources are not affected by the introduction of a magnetic material, while the equivalent magnetic sources are increased by a factor $\sqrt {{\mu _r}} $ , in the respective domain.

By means of the duality principle an OD material can be described following the same steps, thus providing results similar to (19), with the superscript (ε) indicating the vectors in the OD case:

(20) $$\eqalign{\left\vert{{\bf E}_\tau} \right\vert = & \left\vert{\bf E}_\tau ^{(\varepsilon )} \right\vert,\quad \left\vert{{\bf H}_\tau} \right\vert = \varepsilon _r^{ - {{1 \over 2}}} \left\vert{\bf H}_\tau ^{(\varepsilon )} \right\vert, \cr \left\vert{\bf J}_S^{(\varepsilon )} \right\vert = & \sqrt {{\varepsilon _r}} \left\vert{{\bf J}_S} \right\vert = \left\vert\hat n \times {{\bf H}_\tau} \right\vert,\quad \left\vert{\bf M}_S^{(\varepsilon )} \right\vert = \left\vert{{\bf M}_S}\right\vert = \left\vert{{\bf E}_\tau} \times \hat n\right\vert.}$$

It is noteworthy that, differently from the OM case, the dual hypothesis on H in (20) does not provide a constant radiated power in the far field for a patch antenna, but it is a purely mathematical assumption derived from the duality theorem.

B) Computation of the antenna radiation characteristics

The near-field properties provided by (4), (15), and (19) can also be adopted for the far-field vectors, thus we can evaluate the active radiated power, the energy stored and the losses of antennas exploiting an OM material and ES as substrates; it is then possible to obtain the relations for BW and efficiency.

By referring to the Poynting vector dependence in the far field, the domain dependence can be removed. The absolute value of the Poynting vector for ES and OM in the domain r results as:

(21) $$\eqalign{S_{{J_S}}^{(\mu )} (r,\theta, \varphi ) = & \mu _r^{ - 1} \;{S_{{J_S}}}(r,\theta, \varphi ), \cr S_{{M_S}}^{(\mu )} (r,\theta, \varphi ) = & {S_{{M_S}}}(r,\theta, \varphi ).}$$

According to the results described in the previous subsection, it is clear that the miniaturization by means of an OM material is not convenient in case of an antenna exploiting electrical equivalent currents J _S . Conversely, for antennas exploiting magnetic equivalent currents M _S , an OM substrate allows a suitable antenna dimensions reduction since it preserves the radiation performances without affecting the active radiated power density with respect to ES.

The power density (21) allows us to straightforwardly calculate the radiated power (P):

(22) $$P_{{J_S}}^{(\mu )} = \mu _r^{ - 1} {P_{{J_S}}},\quad P_{{M_S}}^{(\mu )} = {P_{{M_S}}},$$

while using (4) for the stored energy (W) we can write:

(23) $$W_E^{(\mu )} = \mu _r^{ - {{1 \over 2}}\,} {W_E} \Rightarrow W_{{\rm TOT}}^{(\mu )} = \mu _r^{ - {{1 \over 2}}} \,{W_{{\rm TOT}}}.$$

Note that (21)–(23) are strictly dependent on the initial hypothesis (4).

It is now immediate to get the quality factor and thus approximately the BW:

(24) $$\eqalign{Q_{{J_S}}^{(\mu )} = 2\pi {f_r}\;\displaystyle{{\mu _r^{ - {{1 \over 2}}} \,{W_{{\rm TOT}}}} \over {{P_{{J_S}}}\,\mu _r^{ - 1}}} = \mu _r^{{{1 \over 2}}} \,{Q_{{J_S}}} & \Rightarrow BW_{{J_S}}^{(\mu )} \sim \displaystyle{{B{W_{{J_S}}}} \over {\mu _r^{(1/2)}}}, \cr Q_{{M_S}}^{(\mu )} = 2\pi {f_r}\;\displaystyle{{\mu _r^{ - (1/2)} \,{W_{{\rm TOT}}}} \over {{P_{{M_S}}}}} = \mu _r^{ - {{1 \over 2}}} \,{Q_{{M_S}}} & \Rightarrow BW_{{M_S}}^{(\mu )} \sim \mu _r^{{{1 \over 2}}} \,B{W_{{M_S}}}.}$$

As regard the efficiency, firstly we define the losses in the materials. The following expression can be used for the dielectric losses:

(25) $${L_d} = \displaystyle{{\omega \varepsilon ^{\prime\prime}} \over 2}\int_{{V_{sub}}} {\mathop {\left\vert{\bf E}(r,\theta, \varphi )\right\vert}\nolimits^2 \;dV}$$

and similarly for the magnetic losses:

(26) $${L_m} = \displaystyle{{\omega \mu^{\prime\prime}} \over 2} \int_{V_{sub}} \left\vert{\bf H} \left(r,\theta, \varphi \right)\right\vert ^2 dV,$$

where ε″ = σ _e /(ε ₀ ω) and μ″ = σ _m /(μ ₀ ω). The total losses are therefore:

(27) $${L_{{\rm TOT}}} = {L_m} + {L_d}.$$

Considering an OM material with relations (4) and (15), it is immediate to write:

(28) $$L_d^{(\mu )} = \mu _r^{ - {{1 \over 2}}} {L_d},\quad L_m^{(\mu )} = \mu _r^{ - {{3 \over 2}}} {L_m},$$

where is supposed, for the sake of simplicity, the ES and OM to have the same loss-tangent; obviously this represents a “fake” ES with losses. In the losses definition (28), we use the mathematical substitution $V_{sub}^{(\mu )} = {V_{sub}}\;\mu _r^{ - {{3 \over 2}}} $ ; hence outside that volume there are no losses.

The radiation efficiency, defined as δ = P/(P + L _TOT), can be now computed for ES and OM:

(29) $${\delta _{{J_S}}} = \displaystyle{{{P_{{J_S}}}} \over {{P_{{J_S}}} + {L_d} + {L_m}}},\quad {\delta _{{M_S}}} = \displaystyle{{{P_{{M_S}}}} \over {{P_{{M_S}}} + {L_d} + {L_m}}},$$

(30) $$\eqalign{& \delta _{{J_S}}^{(\mu )} = \displaystyle{{{P_{{J_S}}}} \over {{P_{{J_S}}} + \mu _r^{{{1 \over 2}}} {L_d} + \mu _r^{ - {{1 \over 2}}} {L_m}}}, \cr & \quad \delta _{{M_S}}^{(\mu )} = \displaystyle{{{P_{{M_S}}}} \over {{P_{{M_S}}} + \mu _r^{ - {{1 \over 2}}} {L_d} + \mu _r^{ - {{3 \over 2}}} {L_m}}}}.$$

The efficiencies in (29) and (30) are quite different compared with previous works, as [Reference Hansen and Burke15]. Since this is a perturbation theory with a zero-order approximation model of the antenna (i.e. losses introduced by perturbation and no reference to a specific antenna geometry, but only to the stored energy) some differences were expected.

The obtained relations on the radiation properties (24), (29), and (30) allow us to straightforwardly predict the advantage in using an OM material in presence of magnetic equivalent currents and the disadvantage with electric equivalent currents.

Starting from (20) dual relationships hold for antennas on OD (ε) materials:

(31) $$BW_{{J_S}}^{(\varepsilon )} \sim {\varepsilon ^{1/2}}\,B{W_{{J_S}}},\quad BW_{{M_S}}^{(\varepsilon )} \sim \displaystyle{{B{W_{{M_S}}}} \over {{\varepsilon ^{{{1 \over 2}}}}}},$$

(32) $$\eqalign{& \delta _{{J_S}}^{(\varepsilon )} = \displaystyle{{{P_{{J_S}}}} \over {{P_{{J_S}}} + {\varepsilon ^{ - (3/2)}}{L_d} + {\varepsilon ^{ - (1/2)}}{L_m}}}, \cr & \quad \delta _{{M_S}}^{(\varepsilon )} = \displaystyle{{{P_{{M_S}}}} \over {{P_{{M_S}}} + {\varepsilon ^{ - (1/2)}}{L_d} + {\varepsilon ^{(1/2)}}{L_m}}}}.$$

C) BW and efficiency extension to a general domains relationship

Let us re-write the previous resonance energy-based theory with a general relationship between the E-fields in the form:

(33) $$\left\vert{\bf E}(r,\theta, \varphi )\right\vert = \mu _r^{ - x} \left\vert{{\bf E}_\mu} ({r_\mu}, \theta, \varphi )\right\vert$$

the solution for H _μ would have the same relation as (33) but scaled by a factor of $\mu _r^{1/2} $ :

(34) $$\left\vert{\bf H}(r,\theta, \varphi )\right\vert = \mu _r^{ - x + (1/2)} \left\vert{{\bf H}_\mu} ({r_\mu}, \theta, \varphi )\right\vert.$$

Relations (33) and (34) directly affect the absolute value of the Poynting vector:

(35) $$\eqalign{S_{{J_S}}^{(\mu )} (r,\theta, \varphi ) = & \mu _r^{2x - 2} \;{S_{{J_S}}}(r,\theta, \varphi ), \cr S_{{M_S}}^{(\mu )} (r,\theta, \varphi ) = & \mu _r^{2x - 1} \;{S_{{M_S}}}(r,\theta, \varphi ).}$$

By computing the BW and efficiency, as in (24) and (30), the relations are exactly the same. This means that the relations on BW and efficiency (δ) for an OM material compared to the ES, are invariant to an initial hypothesis on the OM material of the kind $\mu _r^{ - x} $ . Dual relations developed for an OD show the same invariance. Therefore it is now possible to extend the theory to a wider set of antennas.

To give an example of its usefulness, we again consider two patch antennas, one with an OD material as substrate, the other in ES. The physical initial hypothesis of a constant radiated power (thus on E) in the far-field without losses (differently from the dual case in (20)) will be:

(36) $$\left\vert{\bf E}(r,\theta, \varphi )\right\vert = \varepsilon _r^{ - (1/2)} \left\vert{{\bf E}_\varepsilon} ({r_\varepsilon}, \theta, \varphi )\right\vert,$$

hence on H:

(37) $$\left\vert {{\bf H}(r,\theta, \varphi )} \right\vert = \varepsilon _r^{ - 1} \left\vert {{{\bf H}_\varepsilon} ({r_\varepsilon}, \theta, \varphi )} \right\vert$$

Using the aforementioned invariance it is possible to say that the relations on BW and efficiency will be again exactly the dual version of (31) and (32).

A) Validation of fields inside a square patch

To validate the field relations, as a first example of application, we analyze a square patch, optimized to be resonant at 2.4 GHz by forcing the imaginary part of the impedance for both structures to be zero at the same frequency.

Two different patch configurations are compared:

1. (i) the space between the patch and the ground plane consists of ES;

2. (ii) the space between the patch and the ground plane consists of an OM material (with ε _r = 1 and μ _r = 4).

The resulting patch lengths were computed to be (Fig. 1): L _a = 54 mm and L _b = 29.5 mm. These dimensions approximately agree with the results obtainable from (5): it is noteworthy that the dimensions scaling is related to the effective permeability and not to the relative one.

Fig. 1. Top views of the analyzed patch antennas: these antenna types are described mainly by magnetic current sources.

First of all, let us discuss the numerical results obtained with the two configurations in terms of the near-field regions of the two antennas. In Fig. 2, the simulated electric and magnetic near-fields of the two patch antennas are shown. The field values are taken on a cut plane in the middle of the substrate brick. These numerical results validate the relationships (4) and (15).

Fig. 2. E and H near-field numerical simulation results of the two patches: the E-field for the antenna on the OM substrate is twice stronger and compressed in size than that on the ES.

The E-field of the patch, exploiting the OM substrate, is approximately twice the one on the ES material, whereas the H-fields are approximately the same, considering for both fields evaluations a size compression by a factor of about two.

Figure 2 lets also to address the ambiguity when defining the volume V _n−f for the practical case. This can be done by interpreting V _n−f as the reactive region volume, thus “that portion of the near-field region immediately surrounding the antenna wherein the reactive field predominates”. A common approximation for the boundary is a distance ${d_{n - f}} \simeq 0.62\sqrt {{D^3}/\lambda} $ from the antenna surface, where λ is the guided wavelength [Reference Balanis16]. In Fig. 2, the fields are shown to be well confined inside that region, where d _n−f is approximately 40 and 22 mm for the ES and the OM materials, respectively.

A difference of about 15% in terms of far-field radiated in the broadside direction is observed in Fig. 3. The reasons of this small discrepancy with respect to the previously developed theory can be ascribed to different spacings between the two radiating slots, which the patch antenna radiation mechanism is based on, in the OM and ES situations. From the far-field point of view the reduction of this distance in the OM case is a direct consequence of the corresponding directivity reduction, since the two slots-array has a shorter step. This is also confirmed by the further widening of the radiation surface of a more miniaturized square patch exploiting a denser OM material, as confirmed by simulation.

Fig. 3. E far-field numerical simulation results of the two patches: the $E_{FF}^{(\mu )} $ -field maximum value for the antenna on the OM substrate is different than that on the empty space.

B) Validation of BW relations

As a further test we compare different antenna layouts from the operating BW point of view: the same miniaturized patch antenna (thus exploiting magnetic equivalent surface currents) and a single full-wavelength loop antenna (exploiting electric equivalent surface currents) embedded in the substrate. In these cases two substrates are considered: the same OM as before and the dual OD substrate (ε _r = 4, μ _r = 1), in order to validate the previous theoretical results. The patch antenna keeps the same size for both substrates, since the field is well confined between the conductors; hence the effective parameters have a negligible variation. The loop antennas, whose layouts are shown in Fig. 4, have slightly different sizes, since the fringing of the field in these open structures significantly depends on the electromagnetic characteristics of the substrates: the loop line width is 1 mm, while the inner radius length provides a full-wavelength behavior at 2.4 GHz.

Fig. 4. Top views of the analyzed loop antennas: these antenna types are described mainly by electric current sources.

In Fig. 5 the reflection coefficients for the patch on OM and OD are shown. It is clear the advantage in terms of BW using the OM with respect to the OD: it is increased approximately by a factor of 4, as predicted by our theory.

Fig. 5. S ₁₁ for the patch antennas. It is normalized to the antennas impedance at resonance (250 and 325 Ω for the OM and OD patches, respectively).

Once again in Fig. 6 the reflection coefficients are shown, this time for two loop antennas on the same OD and OM as before. In this case, it is clear the advantage in using the OD instead of the OM: the BWs are again approximately related by a factor of 4.

Fig. 6. S ₁₁ for the loop antennas. It is normalized to the antennas impedance at resonance (60 versus 66 Ω for the OM and OD loop, respectively).

C) Simulations with realistic losses and dispersion

Today's available magneto-dielectric materials have relevant losses and are subject to permeability dispersion.

As an example regarding the losses, in [Reference Aldrigo, Costanzo, Masotti, Baldisserri, Dumitru and Galassi5] the loss tangent is about 0.38 at 900 MHz and it is significantly higher than available OD materials. This obviously results in lower efficiency when using a real MD; in particular the relations (29) and (30) will have heavily different L _m and L _d inside. It is thus still disadvantageous to use present MD materials if the application interest is purely on the efficiency. On the other hand, in this case the BW is obviously further increased due to the increased losses when compared to the energy stored.

An extensive work on frequency dispersion is provided in [Reference Ikonen, Rozanov, Osipov, Alitalo and Tretyakov17]. Most materials containing magnetic inclusions are subject to the ferro-magnetic resonance, which brings a strong dispersion in its vicinity. As a result, there could be a huge difference in terms of radiation properties.