Analysis of the attenuation constant of electromagnetic waves in TEM mode under different loss regimesin the X-band

Abstract

This work proposes a unified formulation of the electromagnetic wave propagation constant for media exhibiting both electric and magnetic losses, from which a general expression for the attenuation coefficient $\alpha \left(\omega \right)$ is derived. The approach is applied comparatively to five classes of materials (low-loss and resonant dielectrics, metallic conductors and conductive polymers, composites, and magnetic resonant materials), revealing distinct physical signatures and practical criteria for the selection of dispersive models in the X-band (8–12 GHz). The results enable the identification of frequency regions with peak attenuation and the comparative assessment of the isolated and combined impact of each loss mechanism on wave decay. Furthermore, this study provides a comparative synthesis serving as a guide for selecting dispersive models and addressing common limitations (e.g., the indiscriminate use of constant $\tan \left({\delta }_e\right)$), while also identifying gaps and perspectives for future experimental validations.

Keywords:
Attenuation constant; dielectric and magnetic losses; X-band; dispersive models

1. Introduction

The attenuation of electromagnetic waves in lossy media has been a subject of continuous investigation due to its direct impact on communication systems, radars, electromagnetic shielding and the design of absorbing materials[¹, ²]. The complex wave propagation constant ($\gamma =\alpha +i\beta$) is defined in terms of the phase constant $\beta$ and the attenuation constant $\alpha$, which encapsulate the conductive, dielectric, and magnetic contributions of the materials[³]. These constants are fundamental tools for the comparative analysis of different classes of materials in the microwave range [⁴, ⁵].

Several dispersive models have been proposed to describe the frequency-dependent electromagnetic response of real materials. The classical Lorentz model describes the response of bound electrons modeled as damped harmonic oscillators, leading to resonant dielectric dispersion and providing an accurate representation of polarization mechanisms in dielectrics and insulating materials [⁶, ¹⁸]. In contrast, the Drude model assumes free electrons undergoing collisions with the crystal lattice, successfully describing the frequency-dependent electrical conductivity of metals and other conductive media, where attenuation is predominantly governed by conduction losses [¹⁸].

For materials in which free-carriers conduction and resonant polarization mechanisms coexist, such as conductive polymers, composites, and advanced functional materials [¹²], the combined Drude–Lorentz model provides a more realistic description by superimposing the free-electron contribution of the Drude model with the resonant response of Lorentz oscillators. This hybrid formulation enables the simultaneous representation of conductive and dielectric losses over the same frequency range, making it particularly suitable for describing complex electromagnetic materials [¹⁰, ¹⁸].

An analogous Lorentz formulation can also be applied to the complex magnetic permeability, allowing the description of frequency-dependent magnetic resonances associated with the processional dynamics of magnetic moments. This approach provides an accurate representation of magnetic losses in ferrites and other magnetic materials employed in microwave absorbers and electromagnetic interference (EMI) suppression [⁶, ¹⁹].

Understanding the physical assumptions, limitations, and applicability of these dispersive models is essential for correctly interpreting wave propagation and attenuation over different frequency ranges and constitutes the theoretical basis of the unified formulation developed in this work.

Figure 1 presents a conceptual map synthesizing the relationships between electromagnetic loss mechanisms and the dispersive models that describe them (Drude, Lorentz, magnetic Lorentz, and Drude–Lorentz) [⁶]. Each model describes distinct contributions to the attenuation constant $\alpha \left(\omega \right)$, allowing for the identification of transitions between dominant loss regimes: from conductive to magnetic.

Figure 1
Conceptual structure of the electromagnetic loss regimes and respective dispersive models applied to the calculation of the attenuation constant $\alpha \left(\omega \right)$.

This graphical representation allows for the visualization of the different physical regimes considered in the unified formulation of the propagation constant $\gamma \left(\omega \right)$, in addition to describing how each type of loss affects the attenuation $\alpha \left(\omega \right)$ and the phase $\beta \left(\omega \right)$.

Recent studies have experimentally characterized typical microwave materials, such as PTFE (low-loss) [⁷], BaTiO3 (strong resonant response) [⁸], metals such as copper, where attenuation is dominated by the skin effect[⁹, ¹⁰], conductive polymers such as doped polyaniline [¹¹], PPy/CNT composites as tunable hybrids [¹²], and Ni-Zn ferrites [¹³] as selective microwave absorbers.

Although studies specific to material classes exist, there is a notable lack of a unified comparative analysis bridging conductive, dielectric, and magnetic losses within a single formulation. This work formalizes such unification, derives a general expression for $\alpha \left(\omega \right)$ and contrasts theoretical predictions with behaviors reported in the literature for key materials in the X-band.

In this context, the central objective is to highlight how each dispersive model describes distinct regimes and how such results align with experimental data reported in the literature. Based on the conceptual synthesis presented in Fig. 1, a unified mathematical formulation of the propagation constant in media exhibiting electric and magnetic losses is developed. The study also offers a critical analysis of the manifestation of each dispersive mechanism in $\alpha \left(\omega \right)$, highlighting design implications, limitations of simplified models, and opportunities for experimental validation in advanced materials, such as broadband absorbers.

2. Theoretical Background

Consider the propagation of electromagnetic waves in linear, isotropic, and homogeneous media under TEM mode, characterized by complex constitutive parameters: electric permittivity $\epsilon ={\epsilon }^{\prime }-i{\epsilon }^{\prime \prime }$, magnetic permeability $\mu ={\mu }^{\prime }-i{\mu }^{\prime \prime }$ and conductivity $\sigma$, represented here as a purely real number. Electric and magnetic losses are accounted for by the imaginary parts of $\epsilon$ and $\mu$, associated with dissipation due to conductive losses, polarization processes, and magnetic mechanisms (hysteresis and eddy currents). In the frequency domain, Maxwell’s equations are expressed as [³]:

(1) $$\begin{array}{c}\nabla \times \overrightarrow{H}=\left(\sigma +i\omega \epsilon \right)\overrightarrow{E},\hfill \\ \nabla \times \overrightarrow{E}=-i\omega \mu \overrightarrow{H},\hfill \end{array}$$

for a Fourier Transform definition of

(2) $$X\left(\omega \right)={\int }_{-\infty }^{\infty }f\left(t\right)e^{-i\omega t}\mathit{d}t,$$

commonly used in Engineering.

Combining these equations yields the complex wave equation, whose propagation constant $\gamma$ is given by:

(3) $$\gamma =\alpha +i\beta =i\omega \sqrt{\mu \epsilon }$$

expressing $\epsilon ={\epsilon }^{\prime }\left(1-i\tan {\delta }_e\right)$ and $\mu ={\mu }^{\prime }\left(1-i\tan {\delta }_m\right)$ in terms of the electric $\tan {\delta }_e={\epsilon }^{\prime \prime }/{\epsilon }^{\prime }$ and magnetic ($\tan {\delta }_m={\mu }^{\prime \prime }/{\mu }^{\prime }$) loss tangents, the general formula for propagation constant is

(4) $$\gamma =i\omega \sqrt{{\mu }^{\prime }{\epsilon }^{\prime }}\cdot \sqrt{\left(1-i\tan {\delta }_e\right)\cdot \left(1-i\tan {\delta }_m\right)}$$

This formulation forms the basis for the general expressions of the attenuation constant $\alpha$ and the phase constant $\beta$, which, after some algebraic manipulation (fully explained in the Appendix), can be written as:

(5) $$\begin{array}{cc}\hfill \alpha & =\omega \sqrt{{\mu }^{\prime }{\epsilon }^{\prime }}\sqrt{\sqrt{{\left(1-\tan {\delta }_e\cdot \tan {\delta }_m\right)}^2+{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}}\hfill \\ & \cdot {\left[\frac{\sqrt{1+\frac{{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}{{\left(1-\tan {\delta }_e\cdot \tan {\delta }_m\right)}^2}}-1}{2\sqrt{1+\frac{{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}{{\left(1-\tan {\delta }_e\cdot \tan {\delta }_m\right)}^2}}}\right]}^{\frac{1}{2}}\hfill \end{array}$$

(6) $$\begin{array}{cc}\hfill \beta & =\omega \sqrt{{\mu }^{\prime }{\epsilon }^{\prime }}\sqrt{\sqrt{{\left(1-\tan {\delta }_e\cdot \tan {\delta }_m\right)}^2+{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}}\hfill \\ & \cdot {\left[\frac{\sqrt{1+\frac{{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}{{\left(1-\tan {\delta }_e\cdot \tan {\delta }_m\right)}^2}}+1}{2\sqrt{1+\frac{{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}{{\left(1-\tan {\delta }_e\cdot \tan {\delta }_m\right)}^2}}}\right]}^{\frac{1}{2}}\hfill \end{array}$$

From these general expressions, distinct loss scenarios in materials of practical interest can be analyzed, as discussed in the following sections. It is observed that when applying these expressions to non-magnetic media ($\tan {\delta }_m=0$) with dielectric losses represented by $\tan {\delta }_e=\sigma /\omega {\epsilon }^{\prime }$, Eqs. 5 and 6 reduce to:

(7) $$\alpha =\omega \sqrt{\frac{{\mu }^{\prime }{\epsilon }^{\prime }}{2}}\cdot {\left[\sqrt{1+{\left(\frac{\sigma }{\omega {\epsilon }^{\prime }}\right)}^2}-1\right]}^{1/2}$$

and

(8) $$\beta =\omega \sqrt{\frac{\mu \prime \epsilon \prime }{2}}\cdot {\left[\sqrt{1+{\left(\frac{\sigma }{\omega \epsilon \prime }\right)}^2}+1\right]}^{1/2}$$

Equations 5 and 6 are valid for linear isotropic media and simultaneously incorporate the effects of conductivity, complex permittivity, and effective permeability. In the case of conductors or dielectric materials without magnetic losses, these expressions reduce to the well-known expressions for the attenuation constant $\alpha$ and phase constant $\beta$ as a function of $\epsilon$, $\mu$, and $\sigma$ given by Eqs. 7 and 8.

Limiting case for a good conductor ($\sigma \gg \omega \epsilon$):

(9) $$\alpha \approx \beta \approx \sqrt{\pi f\mu \sigma }$$

Limiting case for a dielectric ($\sigma <<\omega \epsilon$):

(10) $$\alpha \approx \frac{\sigma }{2}\sqrt{\frac{\mu }{\epsilon }};\beta \approx \omega \sqrt{\mu \epsilon }$$

This ensures consistency with well-established limiting cases [³, ⁴]. Such canonical forms are widely employed in electromagnetics literature and serve as verification cases for the generalized models proposed herein. It should be noted that Eqs. (8) and (9) are limiting approximations obtained from Eqs. (6) and (7). Therefore, the constitutive parameters appearing in these expressions correspond to the real quantities ${\epsilon }^{{}^{\prime }}$ and ${\mu }^{{}^{\prime }}$, whereas the dissipative effects are already represented by the conductivity term.

From Eq. 5, it is observed that the attenuation constant $\alpha \left(\omega \right)$ results from the combined contribution of conduction currents and dielectric losses associated with the imaginary part of the complex permittivity ${\epsilon }^{\prime \prime }$. To highlight the physical nature of these mechanisms, Fig. 2 presents three basic regimes: (i) dominant conduction ($\sigma >0$; $\tan {\delta }_e\approx 0$), (ii) predominant dielectric losses ($\sigma =0$; $\tan {\delta }_e>0$), and (iii) ideal medium ($\sigma =0$; $\tan {\delta }_e=0$).

Figure 2
Comparison of the attenuation constant $\alpha \left(\omega \right)$ in three distinct regimes: dominant conduction, predominant dielectric losses, and ideal medium.

Using Eq. 5 as a unified analysis framework, it is observed in Fig. 2 that, in the dominant conduction regime (solid line), $\alpha \left(\omega \right)$ is significant even at low frequencies and tends to stabilize as the conductive term becomes dominant in the imaginary part of $\epsilon$. In the dielectric loss regime (dashed line), $\alpha \left(\omega \right)$ increases quasi-linearly with frequency (assuming constant $\tan {\delta }_e$), characterizing dipolar relaxation and internal absorption mechanisms. In the ideal medium (dotted line), $\alpha \left(\omega \right)$ remains zero across the entire range, exhibiting no conduction, no ${\epsilon }^{\prime \prime }$ and thus no energy dissipation.

Based on these theoretical considerations, it is possible to establish a direct correlation between the analytically predicted loss regimes and the experimental behavior of the complex quantities ${\epsilon }_r\left(\omega \right)$, the relative electric permittivity, and ${\mu }_r\left(\omega \right)$, the relative magnetic permeability. The spectral analysis of these functions enables the identification of regions dominated by conductive, dielectric, or magnetic mechanisms, in addition to demonstrating the physical consistency of Eq. 5 with respect to the measurements performed. This comparison between theoretical models and experimental data is presented below.

In the following sections, microscopic mechanisms (dielectric/magnetic response) are analyzed through the application of dispersive models (Lorentz, Drude, and Drude–Lorentz), linking them to observable macroscopic effects, such as the attenuation constant, across different classes of materials (conductors, conductive polymers, hybrid composites, and ferrites).

3. Analysis of Loss Scenarious

In this section, distinct combinations of dielectric, conductive and magnetic losses are analyzed within the X-band frequency range regarding the attenuation constant. It is observed that in electromagnetic wave propagation calculations at high frequencies, the material permittivity $\epsilon$ and permeability $\mu$ are not constant.

For lossy media, permittivity and permeability are frequency-dependent functions, $\epsilon \left(\omega \right)$ and $\mu \left(\omega \right)$, modeled by dispersive expressions such as the Drude, dielectric Lorentz and magnetic Lorentz models [³, ⁶, ¹⁰]. These models capture the physical and dissipative responses of materials across the frequency spectrum, leading to a frequency dependent propagation constant $\gamma \left(\omega \right)=\alpha \left(\omega \right)+i\beta \left(\omega \right)$.

In this work, five distinct electromagnetic wave propagation scenarios are analyzed for materials with specific characteristics using appropriate dispersive models. It should be emphasized that the dispersive models used for permittivity and permeability in the graphs presented in this section are intended for didactic and illustrative purposes and do not correspond to experimental values of specific materials. That means the curves in this section employ synthetic parameters designed to illustrate the typical signatures of each loss regime within the X-band. The descriptive analysis of attenuation for real materials is presented in Section IV – Material Analysis and Results.

3.1. Case 1: Ideal medium (lossless) –($\sigma =0$, ${\epsilon }^{\prime \prime }=0$, ${\mu }^{\prime \prime }=$)

In this theoretical scenario, an ideal medium free from any loss mechanisms is considered. This idealized case should be regarded solely as a mathematical limiting configuration adopted for comparison purposes, rather than as a physically realizable material. In the material analysis presented in Section IV, this limiting behavior is approximated by PTFE, a real low-loss dielectric that remains physically consistent with causality and the Kramers–Kronig relations [¹⁰].

Within this idealized framework, permittivity and permeability are assumed to be purely real and frequency-independent. Substituting these values into Eq. 5 directly yields $\alpha =0$, indicating the absence of attenuation and purely conservative electromagnetic wave propagation. This limiting case serves as a reference for comparison with the subsequent dispersive scenarios.

3.2. Case 2: Resonant dielectric medium(Lorentz Model) – ($\sigma =0$, ${\epsilon }^{\prime \prime }\ne 0$, ${\mu }^{\prime \prime }=0$)

In this case, losses arising exclusively from the non-ideal behavior of the dielectric response are considered. The complex permittivity introduces an imaginary term associated with polarization losses, typical of real dielectric materials subjected to high-frequency fields. The attenuation constant becomes directly dependent on frequency through the term $\omega {\epsilon }^{\prime \prime }\left(\omega \right)$ and although there is no electric conduction, dissipation due to molecular hysteresis is sufficient to produce measurable losses in the X-band range. Thus, the dispersive Lorentz model is employed, describing the oscillations of bound dipoles, which results in a dispersive response with a loss peak at resonance.

In resonant dielectrics (Lorentz), electrons are bound to atoms via a restoring force, governed by the equation of motion of a damped harmonic oscillator, such that the attenuation $\alpha \left(\omega \right)$ exhibits a peak around ${\omega }_0$, and the permittivity can be written in the form:

(11) $$\epsilon \left(\omega \right)={\epsilon }_{\infty }+\frac{\Delta \epsilon \cdot {\omega }_0^2}{{\omega }_0^2-{\omega }^2-\left(i\cdot {\gamma }_L\cdot \omega \right)}$$

where ${\epsilon }_{\infty }$ is the material permittivity at high frequencies ($\omega \to \infty$), where dipole-related resonances no longer respond to the oscillating field; $\Delta \epsilon ={\epsilon }_s-{\epsilon }_{\infty }$ denotes the dispersion amplitude, i.e., the difference between the static permittivity, ${\epsilon }_s=\epsilon \left(\omega \to 0\right)$ and ${\epsilon }_{\infty }$. Here, ${\omega }_0$ denotes the resonance frequency, and ${\gamma }_L$ denotes the damping coefficient associated with the oscillation of bound charges, determining the linewidth of the dielectric resonance.

Separating the real part, ${\epsilon }^{\prime }\left(\omega \right)$, representing energy storage (effective capacitance of the material), and the imaginary part, ${\epsilon }^{\prime \prime }\left(\omega \right)$, representing dielectric losses (dissipation), we obtain:

(12) $${\epsilon }^{\prime }\left(\omega \right)={\epsilon }_{\infty }+\frac{\Delta \epsilon \cdot \left({\omega }_0^2-{\omega }^2\right)}{{\left({\omega }_0^2-{\omega }^2\right)}^2+{\left({\gamma }_L\cdot \omega \right)}^2}$$

(13) $${\epsilon }^{\prime \prime }\left(\omega \right)=\frac{\Delta \epsilon \cdot {\gamma }_L\omega }{{\left({\omega }_0^2-{\omega }^2\right)}^2+{\left({\gamma }_L\cdot \omega \right)}^2}$$

where ${\epsilon }_{\infty }$ is the material permittivity at high frequencies ($\omega \to \infty$), where dipole-related resonances no longer respond to the oscillating field; $\Delta \epsilon =\left({\epsilon }_S-{\epsilon }_{\infty }\right)$ denotes the dispersion amplitude, i.e., the difference between the static permittivity, ${\epsilon }_s=\epsilon \left(\omega \to 0\right)$ and the high-frequency permittivity ${\epsilon }_{\infty }$. Here, ${\omega }_0$ denotes the resonance frequency, and ${\gamma }_L$ is the damping factor in the Lorentz model.

Figure 3 displays an example of the behavior of the real and imaginary parts of relative permittivity in the Lorentz model, for a resonance frequency $f_0=9,5$ GHz. It is observed that the real part (solid line) approaches the static permittivity ${\epsilon }_s$ in the low-frequency regimes ${\epsilon }_r^{\prime }$($\omega \to 0$), subsequently undergoes anomalous dispersion (rapid drop) near $f_0$ and tends toward ${\epsilon }_{\infty }$ = 1 at high frequencies ($\omega \gg {\omega }_0$). The imaginary part ${\epsilon }^{\prime \prime }\left(\omega \right)$ (dashed line) is virtually zero at low and high frequencies but exhibits an absorption peak at $\omega ={\omega }_0$, associated with energy dissipation in the medium due to hysteresis or relaxation.

Figure 3
Dielectric dispersion in the X-band ($8–12.5$ GHz) for the dielectric Lorentz model. The dotted line indicates the resonant frequency ${\omega }_0$. The peak of ${\epsilon }^{\prime \prime }$ and the sign inversion of ${\epsilon }^{\prime \prime }$ around ${\omega }_0$ are observed, with an asymptotic return to ${\epsilon }_{\infty }$.

While the Lorentz model describes oscillations of bound dipoles, the Drude model represents the opposite regime, characterized by the motion of free electrons under the action of an oscillating electric field. This transition marks the shift from predominantly dielectric materials to conductors, enabling a unified interpretation of the behavior of $\epsilon \left(\omega \right)$ across different frequency ranges [¹¹, ¹²].

3.3. Case 3: Conductivity and absence ofdielectric and magnetic losses(Drude model) – ($\sigma \ne 0$, ${\epsilon }^{\prime \prime }=0$, ${\mu }^{\prime \prime }=0$)

In this case, a conductive medium with $\sigma \left(\omega \right)$ is simulated, assuming no contribution from dielectric or magnetic losses. In this conductive dispersion scenario, the Drude model is adopted to describe the dielectric response of conductive materials, such as metals. It considers electrons as free particles subjected to an external electric field and collisions with the crystal lattice, characterized by a collision rate ${\gamma }_D$. The Drude model is also suitable for describing materials such as certain doped polymers or PCB substrates with minor conductive impurities, where the complex permittivity is given by the expression:

(14) $$\epsilon \left(\omega \right)={\epsilon }_{\infty }-\frac{{\omega }_p^2}{{\omega }^2+i\cdot {\gamma }_D\cdot \omega }$$

(15) $${\epsilon }^{\prime }\left(\omega \right)={\epsilon }_{\infty }-\frac{{\omega }_p^2\cdot {\omega }^2}{{\omega }^4+\left({\gamma }_D\cdot {\omega }^2\right)}$$

(16) $${\epsilon }^{\prime \prime }\left(\omega \right)=\frac{{\omega }_p^2\cdot \left({\gamma }_D\cdot \omega \right)}{{\omega }^4+\left({\gamma }_D\cdot {\omega }^2\right)}$$

where ${\epsilon }_{\infty }$ is the material permittivity at high frequencies, ${\omega }_p=\sqrt{\frac{n{e}^2}{{\epsilon }_0{m}^{\ast }}}$ is the plasma frequency described in terms of the charge carrier density ($n$), the electron charge ($e$), the vacuum permittivity (${\epsilon }_0$), and the effective carrier mass ($m^{\ast }$). Here, ${\gamma }_D$ denotes the free electron collision frequency, which determines the relaxation time of free charge carriers ($\tau =1/{\gamma }_D$).

Considering the low-frequency limit, where $\omega \gg {\omega }_P$, Eqs. (15) and (16) simplify to:

(17) $${\epsilon }^{\prime }\left(\omega \right)\approx -\frac{{\omega }_p^2}{{\omega }^2}<0\text{and}{\epsilon }^{\prime \prime }\left(\omega \right)\approx \frac{{\omega }_p^2}{{\omega }^3}{\gamma }_D$$

This expression shows that ${\epsilon }^{\prime }\left(\omega \right)$ remains negative for $\omega <<{\omega }_P$ (metallic behavior), while ${\epsilon }^{\prime \prime }\left(\omega \right)$ decays smoothly with frequency, reflecting conduction losses [¹⁴], characterizing a conductive plasmonic regime typical of metals and conductive polymers in the low-frequency range.

Figure 4 illustrates the behavior of relative permittivity in the Drude model over a wide frequency range, including values near the plasma frequency (${\omega }_P$ $\approx$ 20 GHz). In the low-frequency regime ($\omega \ll {\omega }_P$), the real component ${\epsilon }_r^{\prime }\left(\omega \right)$ (solid line) assumes negative values, and the material behaves as a metallic reflector. As the frequency approaches ${\omega }_P$, it is observed that ${\epsilon }_r^{\prime }\left(\omega \right)\to 0$, and for $\omega \gg {\omega }_P$, the material becomes transparent, tending toward ${\epsilon }_r^{\prime }\left(\omega \right)\to {\epsilon }_{\infty }$ with positive values.

Figure 4
Electric permittivity dispersion for the Drude model highlighting the regimes: $\omega <<{\omega }_P$ (metallic), the transition near ${\omega }_p$, and $\omega >>{\omega }_P$ (dielectric), without intrinsic resonances.

Regarding the imaginary component ${\epsilon }_r^{\prime \prime }\left(\omega \right)$ (dashed line), it is observed that for frequencies $\omega <<{\omega }_P$, the response exhibits high values, reflecting significant conduction losses. The maximum of ${\epsilon }_r^{\prime \prime }\left(\omega \right)$ occurs well below the plasma frequency, indicating that losses are dominated by the motion of quasi-free electrons. As the frequency approaches ${\omega }_p$ and progresses into the $\omega >>{\omega }_P$ regime, ${\epsilon }_r^{\prime \prime }\left(\omega \right)$ decays progressively, approaching zero. This reveals the transition of the material from a conductor to a quasi-dielectric state, where electromagnetic absorption becomes negligible.

Finally, it is emphasized that in the pure Drude model, there is no significant contribution from slow low frequency polarizations associated with dipole deformation; therefore, the concept of dielectric variation $\Delta \epsilon$, characteristic of the Lorentz model, does not apply in this case.

3.4. Case 4: Conductivity and dielectric losses (Drude – Lorentz model) – ($\sigma \ne 0$, ${\epsilon }^{\prime \prime }\ne 0$ and ${\mu }^{\prime \prime }=0$)

In this case, the effects of electrical conduction and dielectric losses are considered simultaneously, representing a scenario quite common in composite materials or substrates with high dielectric constants. The effective conductivity of the medium is given by ${\sigma }_{\mathrm{eff}}=\left(\sigma +\omega {\epsilon }_d^{\prime \prime }\right)$ and the attenuation constant is obtained from the general equation, Eq. 5. In the high-frequency regime, the contribution of dielectric losses, represented by the term $\omega {\epsilon }^{\prime \prime }$ within the effective conductivity ${\sigma }_{\mathrm{eff}}=\left(\sigma +\omega {\epsilon }_d^{\prime \prime }\right)$, becomes dominant. This effect is implicitly described in Eq. 18 through the imaginary part of the permittivity. Under these conditions, the behavior of $\alpha \left(\omega \right)$ increases markedly with frequency, approaching a quadratic loss regime, depending on the relative values of $\sigma$ and ${\epsilon }_d^{\prime \prime }$.

(18) $$\epsilon \left(\omega \right)={\epsilon }_{\infty }+\frac{\Delta \epsilon \cdot {\omega }_0^2}{{\omega }_0^2-{\omega }^2-\left(i{\gamma }_L\omega \right)}-\frac{{\omega }_p^2}{{\omega }^2+\left(i{\gamma }_D\omega \right)}$$

Figure 5 displays the behavior of the terms ${\epsilon }_r^{\prime }\left(\omega \right)$ and ${\epsilon }_r^{\prime \prime }\left(\omega \right)$ according to the Drude–Lorentz model, where a transition of the material from dielectric to metallic as a function of frequency is observed for ${\epsilon }_r^{\prime }\left(\omega \right)$, in addition to exhibiting, for ${\epsilon }_r^{\prime \prime }\left(\omega \right)$, a resonant behavior typical of systems with oscillatory modes.

Figure 5
Electric permittivity dispersion for the Drude-Lorentz model. Its real part (solid line) crosses the resonant frequency region at ${\omega }_0$ = 9.5 GHz, while the imaginary part (dashed line) exhibits a broadened peak due to the interaction between the conductive (Drude) and resonant (Lorentz) terms.

It is observed that at low frequencies, the real part of the permittivity exhibits a positive and increasing value, typical of dielectric materials. Near ${\omega }_0\approx$ 9.5 GHz, a resonance occurs, triggering a rapid sign reversal typical of the metallic regime; for $\omega \gg {\omega }_0$, it increases again and tends to stabilize, yet remains negative or approaches zero. The imaginary part exhibits an absorption peak centered at ${\omega }_0$, representing maximum energy loss associated with the resonant excitation of an electronic or atomic oscillator.

The Drude–Lorentz model describes the superposition between a conductive background (free carriers) and a resonant response (bound dipoles), producing a well-defined maximum around 9.7 GHz. This hybrid behavior is typical of PPy/CNT and related composites [¹²], and has also been reported in metacomposites designed for X-band shielding [¹², ¹⁵].

3.5. Case 5: Conductivity, dielectric losses, and magnetic losses (Magnetic Lorentz model) – ($\sigma \ne 0$, ${\epsilon }^{\prime \prime }\ne 0$, ${\mu }^{\prime \prime }\ne 0$)

This is the most general scenario, in which the medium exhibits losses associated with all relevant mechanisms (conductive, dielectric, and magnetic). Magnetic materials, such as nickel-zinc ferrite (NiZn), display dispersive and dissipative behavior of the magnetic permeability $\mu \left(\omega \right)$ due to resonances associated with the motion of the electron magnetic dipole moments, which do not respond instantaneously to the action of high-frequency oscillating fields. This characteristic is widely exploited in commercial electromagnetic absorbers, where ferromagnetic materials are frequently combined with conductive particles to maximize the dissipation of incident energy. In this regime, the attenuation constant $\alpha \left(\omega \right)$ also becomes dependent on the magnetic loss tangent, $\tan \left({\delta }_m\right)$, representing the contribution of magnetic absorption to the material’s overall response.

Applying the Lorentz resonance model to the complex permeability $\mu \left(\omega \right)$, we obtain:

(19) $$\mu \left(\omega \right)={\mu }^{\prime }-i{\mu }^{\prime \prime }=1+\frac{\Delta \mu \cdot {\omega }_m^2}{{\omega }_m^2-{\omega }^2-\left(i{\Gamma }_m\omega \right)}$$

where the correction term represents the resonant contribution associated with the magnetic behavior of the material. From Eq. 19, the real and imaginary components of permeability are separated, corresponding, respectively, to the storage and dissipation of magnetic energy:

(20) $${\mu }^{\prime }\left(\omega \right)={\mu }_{\infty }+\frac{\Delta \mu \cdot \left({\omega }_m^2-{\omega }^2\right)}{{\left({\omega }_m^2-{\omega }^2\right)}^2+{\left({\Gamma }_m\omega \right)}^2}$$

(21) $${\mu }^{\prime \prime }\left(\omega \right)=\frac{\Delta \mu \cdot {\Gamma }_m\cdot \omega }{{\left({\omega }_m^2-{\omega }^2\right)}^2+{\left({\Gamma }_m\omega \right)}^2}$$

In these expressions, ${\omega }_m$ represents the magnetic resonance frequency, determined by the internal field and the microscopic structure of the material; $\Delta \mu =\left({\mu }_s-{\mu }_{\infty }\right)$ is the permeability variation between the static value ${\mu }_s$ and the high-frequency value ${\mu }_{\infty }$, defining the maximum amplitude of the resonant response; ${\Gamma }_m$ denotes the damping coefficient of the magnetic Lorentz model. Its physical role is analogous to that of ${\gamma }_L$ in the electric Lorentz model. While ${\gamma }_L$ characterizes the damping of oscillating bound electric charges, ${\Gamma }_m$ characterizes the relaxation of the precessional motion of magnetic moments. In both cases, the damping parameter determines the resonance linewidth and the associated electromagnetic energy dissipation. Consequently, larger values of ${\Gamma }_m$ produce, broader resonance peaks and stronger magnetic losses over a wider frequency range.

Figure 6 depicts the behavior of the relative permeability ${\mu }_r\left(\omega \right)$ for a material described by the magnetic Lorentz model in the X-band range. It is observed that the real part exhibits classic anomalous dispersion behavior, such that below the resonance frequency ($\omega \ll {\omega }_m$), the contribution is positive and elevated. Around the resonance frequency ($\omega \approx {\omega }_m$), a rapid variation occurs, potentially even reaching negative values, a fundamental characteristic of materials with resonant magnetic behavior. Beyond resonance ($\omega \gg {\omega }_m$), the permeability tends toward ${\mu }_{\infty }=1$. On the other hand, the imaginary part exhibits a maximum peak exactly at the resonant frequency, representing maximum magnetic losses at ${\omega }_m$, with the peak width determined by the damping factor ${\Gamma }_m$.

Figure 6
Magnetic permeability dispersion for the Lorentz model. The real part ${\mu }^{\prime }$(solid line) undergoes sign inversion at ${\omega }_m$ = 9 GHz, indicating the transition from inductive to capacitive magnetic behavior, while the imaginary part ${\mu }^{\prime \prime }$(dashed line) exhibits the typical magnetic absorption peak.

This magnetic dispersion profile is widely observed in commercial NiZn ferrites, employed in selective absorbers and electromagnetic interference (EMI) suppressors. The simultaneous presence of high ${\mu }^{\prime \prime }$ and ${\mu }^{\prime }$ near zero in the $9–10$ GHz region maximizes absorption, a characteristic of absorbers based on ferromagnetic resonance [¹³, ¹⁵].

Thus, the five analyzed cases, covering dielectric, conductive and magnetic losses, provide a unified view of electromagnetic absorption mechanisms in the X-band. In all cases, the attenuation constant $\alpha \left(\omega \right)$ arises from the interaction between the complex permittivity and permeability parameters, whose imaginary parts represent the dominant dissipative losses in each regime. For non-magnetic materials (Cases 1 to 4), this contribution can be expressed in terms of a frequency-dependent effective conductivity.

It should be noted that ${\sigma }_{eff}\left(\omega \right)$ represents a dynamic conductivity, explicitly incorporating the frequency dependence of the imaginary part of the permittivity ${\epsilon }^{\prime \prime }\left(\omega \right)$. Thus, the term encompasses both intrinsic dielectric losses and those associated with dispersive electrical conduction, distinguishing it from conventional static conductivity [², ⁷].

Under low-loss conditions ($\sigma \ll \omega {\epsilon }^{\prime }$), the effective conductivity can be simply expressed as:

(22) $${\sigma }_{eff}\left(\omega \right)=\omega {\epsilon }^{\prime \prime }\left(\omega \right)$$

This relationship holds for all dispersive models considered in this study, serving as the conceptual foundation for the analysis of Cases 1 to 4, where dielectric and conductive losses described by the Drude, Lorentz, and Drude–Lorentz models predominate. Thus, this concludes the theoretical overview of the different classes of losses, establishing the direct link between the dispersive parameters and the attenuation constant.

4. Materials Analysis and Results

In the first four cases analyzed, the materials do not exhibit magnetic losses; therefore, the reduced forms of Eqs. 7 and 8 are sufficient to describe the behavior of $\alpha \left(\omega \right)$ and $\beta \left(\omega \right)$. In these regimes, the frequency-dependent effective conductivity ${\sigma }_{eff}\left(\omega \right)$ has already been incorporated into the formalism via ${\epsilon }^{\prime \prime }\left(\omega \right)$, simultaneously encompassing the conductive and dielectric losses predicted by the Drude, Lorentz, and Drude–Lorentz models.

Based on the parameters summarized in Tables 1 and 2, five distinct scenarios are investigated, representing regimes of dielectric, conductive, and magnetic losses. In this study, parameterization via $\Delta \epsilon$ was adopted uniformly to allow the direct application of Eq. 5 to different materials. However, it is important to highlight that the physical significance of $\Delta \epsilon$ depends on the adopted model: in purely dielectric media, it is associated with polarization variation; in conductive or mixed materials, with the effective contribution of free carriers and relaxation mechanisms. The formal assignment of $\Delta \epsilon$ for metals aims solely to ensure calculation consistency, without implying physical correspondence to the difference between static and high-frequency permittivities, as occurs in classical dielectrics.

Table 1
Electromagnetic loss scenarios and respective dispersive models applied to the analyzed materials.

Analyzed Cases	Model	Material
Case 1 – lossless	Non-dispersive	Teflon (PTFE)
Case 2 – dielectric losses	Dielectric Lorentz	Barium Titanate ${\mathrm{BaTiO}}_3$)∗∗
Case 3 – conductive losses	Drude $\left(\frac{\sigma }{\omega \epsilon }\gg 1;\frac{\sigma }{\omega \epsilon }\ll 1\right)$	Copper and Polyaniline (PANI)
Case 4 – conductive + dielectric losses	Drude–Lorentz	PPy/CNT Composite
Case 5 – conductive + dielectric + magnetic losses	Magnetic Lorentz	NiZn Ferrite

Table 2
Characteristic parameters of the dispersive models for the five analyzed scenarios, adjusted to be representative for the X-band.

Material	Model	$\Delta \epsilon$	$f_0$; $f_m$ (Ghz)	${\gamma }_L$; ${\gamma }_D$; ${\Gamma }_D$(GHz)	${\omega }_p$ (rad/s)	$\tan \left({\delta }_e\right)$	$\Delta \mu$
Teflon	Low loss	No dispersion	–	–	–	2.00x10-4	–
BaTiO3∗∗	Dielectric Lorentz	3.5	$f_0$ = 10.2	${\gamma }_L$ = 0.80	1.17x1011	–	–
Copper	Drude	Not applicable	–	–	1.60x1016	–	–
PANI	Drude	3.0	–	${\gamma }_D$ = 0.95	2.00x1010	$\approx$ 0.05–0.10	–
PPy/CNT	Drude–Lorentz	2.2	$f_0$ = 9.5	${\gamma }_L$ = 1.20; ${\gamma }_D$ = 0.60	6.50x1010	$\approx$ 0.05	–
NiZn Ferrite	Magnetic Lorentz	–	$f_m$ = 9.0	${\Gamma }_m$ = 3.00	–	$\approx$ 0.02–0.05	9.0

• ** In the case of BaTiO₃, ${\omega }_p=\sqrt{\Delta \epsilon }\omega 0$ was adopted as a parameter representative of the oscillator strength in the Lorentz model.

Table 1 summarizes the main electromagnetic loss regimes considered in this study, associating them with real materials and the equivalent theoretical models that best describe their complex response.

Table 2 presents the parameters for equations 11, 14, 18 and 19. The parameter sets were obtained by fitting experimental permittivity data reported in Refs [⁸, ⁹, ¹¹, ¹², ¹³] and subsequently refined to ensure representativeness within the X-band frequency range (8–12 GHz). The adopted values are consistent, in order of magnitude, with experimental and theoretical results available in the literature, thereby supporting the comparative simulations and ensuring coherence among the dielectric, conductive, and magnetic regimes described in Table 1.

Based on these parameters, the curves for $\alpha \left(\omega \right)$ and $\beta \left(\omega \right)$ were generated for each material class to highlight how the different loss mechanisms (conductive, dielectric, and magnetic) influence the propagation and absorption of electromagnetic waves. The following analyses are presented in increasing order of physical complexity, starting with the behavior of quasi-ideal materials (low-loss dielectrics) and progressing to systems with pronounced dispersion, significant conductivity, and frequency-dependent magnetic response.

4.1. Case 1 – Teflon (PTFE, low-loss dielectric)

The first scenario analyzed corresponds to a quasi-ideal dielectric material, represented by Teflon (PTFE), widely used as a substrate and reference standard in microwave measurements. Although no real material is perfectly lossless over the entire frequency spectrum, PTFE is widely regarded as an excellent approximation to the ideal lossless limit because of its extremely low loss tangent ($\tan \left(\delta \right)$ $\approx$ 2×10^-4) and very weak dispersion within the X-band. Consequently, it provides a physically realistic reference for the theoretical limiting case discussed in Section III while remaining fully consistent with the Kramers–Kronig relations and, therefore, with causality.

In this regime, Eqs. 5 and 6 predict extremely low values for $\alpha \left(\omega \right)$, with a nearly linear variation of $\beta \left(\omega \right)$ as a function of frequency. The numerical results confirm that the attenuation remains close to zero across the entire range from 8.2 to 12.4 GHz, in the X-band, validating the consistency of the proposed formulation for the a low-loss dielectric exhibiting nearly lossless behavior (${\epsilon }^{\prime \prime }\approx 0$), typical of insulating materials employed in waveguides and radomes. This behavior is consistent with the literature for non-conductive polymeric materials used in high-stability microwave devices [¹, ⁷].

Figure 7 confirms that the attenuation constant remains practically zero throughout the X-band, indicating that electromagnetic wave propagation in PTFE is almost lossless over the investigated frequency range. This behavior is consistent with its extremely low dielectric losses and establishes the reference response against which the remaining material classes are compared.

Figure 7
Attenuation constant $\alpha \left(\omega \right)$ for PTFE in the X-band. Nearly zero attenuation confirms the expected behavior of a practical low-loss dielectric that closely approaches the theoretical lossless limit.

4.2. Case 2 – BaTiO₃ (resonant dielectric,Lorentz model)

The second scenario corresponds to resonant dielectric materials, described by the Lorentz model and represented in this work by barium titanate (BaTiO₃), whose physical parameters are presented in Table 2. In this regime, the imaginary part of the permittivity, ${\epsilon }^{\prime \prime }\left(\omega \right)$, exhibits a sharp peak near the resonance frequency $f_0=10.2$ GHz, indicating strong dipolar polarization and dielectric hysteresis, phenomena typical of ferroelectrics with alignable domains under an oscillating field. The profile of ${\epsilon }^{\prime }\left(\omega \right)$ shows a sign inversion of $d{\epsilon }^{\prime }/d\omega$ near $f_0=10.2$ GHz, a characteristic feature of Lorentzian media.

Figure 8 illustrates this pronounced dispersion, in which the maximum of ${\epsilon }^{\prime \prime }$ coincides with the attenuation peak $\alpha \left(\omega \right)$, highlighting the predominance of dielectric relaxation losses. This behavior contrasts strongly with that of non-resonant dielectrics, such as the epoxy and fiberglass composite (FR-4), for which $\tan \left({\delta }_e\right)$ is practically constant and $\alpha \left(\omega \right)$ varies smoothly with frequency. Thus, BaTiO₃ stands out as an example of a typical resonant dielectric in the X-band, whose absorption mechanism is intrinsically frequency-dependent. Such behavior is in agreement with experimental results obtained for BaTiO₃–epoxy nanocomposites, where increasing the ceramic fraction amplifies the dielectric response [⁸].

Figure 8
Attenuation constant $\alpha \left(\omega \right)$ for a resonant dielectric (BaTiO₃) in the X-band, with a peak centered near $f_0=10.2$ GHz.

4.3. Case 3 – Copper and Polyaniline(conductors, Drude model)

The third scenario addresses conductive materials, represented by Copper (classical metal) and Polyaniline (PANI), the latter being a doped conducting polymer exhibiting a moderate plasmonic response. Both cases are modeled by the Drude model, in which the permittivity strongly depends on frequency due to the motion of free carriers.

For copper, whose plasma frequency is extremely high (${\omega }_p\approx 1,60\times {10}^{16}$ rad/s), the microwave range clearly satisfies the condition $\omega \ll {\omega }_p$. In this regime, the real part of the permittivity ${\epsilon }^{\prime }\left(\omega \right)$ becomes strongly negative, and the attenuation $\alpha \left(\omega \right)$ assumes very high values (10⁶ dB/m), highlighting the skin effect typical of high-conductivity metals. Thus, electromagnetic waves practically do not penetrate the material, resulting in near-perfect reflective behavior, consistent with the classical theory of good conductors [⁶, ⁹].

Figure 9 illustrates the wave attenuation behavior in metallic materials (e.g., copper) in the X-band, where $\alpha \left(\omega \right)$ increases approximately with the square root of the frequency, as predicted by Eqs. (15) and (16) and by Maxwell’s conduction solutions in the limit of high conductivities.

Figure 9
Typical attenuation dependence $\propto \sqrt{\omega }$ for high-conductivity metallic materials in the Drude regime (e.g., copper), where losses are purely conductive and occur due to the skin effect.

For pure polyaniline (PANI), whose plasma frequency is moderate (${\omega }_p$ $\approx$ $2\times {10}^{10}$ rad/s), the X-band regime satisfies the condition $\omega$ $\lesssim$ ${\omega }_p$, typical of dilute plasma conductors. In this range, the Drude model predicts that the real part of the permittivity ${\epsilon }^{\prime }\left(\omega \right)$ remains slightly negative, while the imaginary part ${\epsilon }^{\prime \prime }\left(\omega \right)$ decreases with frequency, reflecting the gradual reduction of dispersive conductive losses.

Figure 10 shows the attenuation constant $\alpha \left(\omega \right)$ corresponding to this regime, demonstrating an intermediate and smoothly decreasing behavior (on the order of ${10}^3-{10}^4$ dB/m). This profile represents the continuous transition from a conductive to a quasi-dielectric regime, where losses, although significant, decrease progressively with frequency.

Figure 10
Attenuation $\alpha \left(\omega \right)$ in pure conducting polymer (PANI), decaying with frequency in the X-band.

From a physical standpoint, this behavior contrasts with that observed for copper, whose high electrical conductivity ($\sigma \approx 5,8\times {10}^7$ S/m) causes $\alpha \left(\omega \right)$ to increase with the square root of the frequency. In the case of PANI, which has moderate conductivity ($\sigma \approx 1\text{–}20$ S/m), the ratio $\sigma /\left(\omega \epsilon \right)$ becomes of the order of unity, reducing the relative conductive contribution. Thus, both cases are described in a unified manner by (Eq. 5), differing only in the order of magnitude of the effective conductivity, which determines the hierarchy between conductive and dielectric losses.

Although Fig. 10 presents the theoretical behavior of the attenuation constant $\alpha \left(\omega \right)$ for pure polyaniline (PANI), obtained from Eq. (5) and the synthetic parameters in Table 2, it is important to highlight that there are no experimental data widely reported in the literature for pure PANI in the X-band microwave range (8–12 GHz). Therefore, the results presented here should be interpreted as theoretical reference estimates, useful for indicating physical trends and guiding future experimental investigations. This gap in data highlights the need for specific characterizations of pure PANI in this frequency range in order to validate or adjust the adopted model.

Most available studies address composites or filled versions of PANI, frequently combining the conducting polymer with oxides or ferrites. These additives increase the effective conductivity and broaden the loss mechanisms, resulting in higher electromagnetic attenuation. For instance, PANI–Y${}_2{}^{}O_3^{}$ composites exhibit absorption on the order of 33 to 35 dB in the X-band, attributed to the simultaneous contribution of conductive, dielectric, and interfacial losses, intensified by the microstructural heterogeneity of the material [¹¹].

From an application standpoint, PANI stands out as a promising material for electromagnetic absorption and shielding, combining moderate attenuation levels with light weight, flexibility, and easy processability, which are desirable characteristics in microwave devices and practical applications in the X-band.

4.4. Case 4 – PPy/CNT (conductive-dielectriccomposite, Drude–Lorentz model)

The fourth scenario analyzes the hybrid behavior of the conductive-dielectric composite PPy/CNT (polypyrrole with carbon nanotubes), whose electromagnetic response is well described by the Drude–Lorentz model. In this formalism, the Drude term represents the background conductive contribution, associated with the movement of free carriers, while the Lorentz term describes the interfacial polarizations and molecular relaxations responsible for resonant losses.

Figure 11 presents the behavior of the attenuation constant $\alpha \left(\omega \right)$ for this composite, calculated based on the parameters in Table 2. An intermediate profile between purely conductive and purely dielectric regimes is observed, with attenuation growth starting from 8 GHz, a maximum around 9.7 GHz and subsequent gradual decay up to 12 GHz. This peak is a direct consequence of the superposition of conduction and resonance mechanisms, producing a selective and tunable response within the X-band.

Figure 11
Attenuation $\alpha \left(\omega \right)$ for the PPy/CNT composite (Drude–Lorentz model), with a maximum at 9.7 GHz resulting from the combination of conductive and resonant losses.

Such behavior is characteristic of conductive-dielectric metacomposites, in which interfacial coupling and material morphology simultaneously modulate conduction and resonant response [⁹, ¹²]. The result is a controllable dispersive response that combines an adequate background loss level, favoring incident wave coupling, with an efficient resonant mechanism for electromagnetic energy dissipation.

Thus, PPy/CNT presents itself as a promising material for selective absorption and electromagnetic shielding, offering good performance in the microwave range combined with low density, thermal stability and electromagnetic tunability, characteristics typical of conductive-dielectric composites. The observed hybrid profile confirms the coexistence of electrical conduction and dielectric relaxation, consolidating PPy/CNT as a transitional link between the purely conductive and purely dielectric regimes.

4.5. Case 5 – NiZn Ferrite (magnetic Lorentzmodel)

The fifth scenario covers dispersive magnetic materials, represented by nickel-zinc ferrite (NiZn), whose behavior is described by the Lorentz model applied to permeability $\mu \left(\omega \right)$. In these materials, magnetic losses derive from resonances associated with the movement of the magnetic dipole moment of electrons under high-frequency oscillating fields, a phenomenon known as ferromagnetic resonance (FMR).

The real part ${\mu }^{\prime }\left(\omega \right)$ and the imaginary part ${\mu }^{\prime \prime }\left(\omega \right)$ of the permeability exhibit typical magnetic resonance behavior: ${\mu }^{\prime \prime }\left(\omega \right)$ presents a sharp peak around 9 GHz, while ${\mu }^{\prime }\left(\omega \right)$ crosses zero and inverts its sign, marking the transition between magnetic energy storage and dissipation regimes. As a direct consequence, the attenuation constant $\alpha \left(\omega \right)$ presents a pronounced maximum in the same frequency range, indicating that absorption is maximal when ${\mu }^{\prime \prime }$ reaches its peak and ${\mu }^{\prime }\approx 0$, a characteristic condition of FMR.

Figure 12 presents the behavior of $\alpha \left(\omega \right)$ for NiZn ferrite, calculated according to (Eq. 5), simultaneously incorporating dielectric, magnetic, and residual conductive contributions. An intense attenuation peak is observed at $f_m$ $\approx$ 9 GHz, which approximately coincides with the maximum of ${\mu }^{\prime \prime }\left(\omega \right)$ shown in Fig. 6, confirming that the dominant loss mechanism is magnetic. Small differences between the positions of these peaks result from the presence of residual conductive terms (Drude), which broaden the response and introduce a phase shift in $\beta \left(\omega \right)$, without significantly altering the main resonance frequency.

Figure 12
Behavior of the attenuation constant $\alpha \left(\omega \right)$ for NiZn Ferrite described by the magnetic Lorentz model.

Above the peak frequency, the $\alpha \left(\omega \right)$ curve exhibits a sharp drop, reflecting the loss of efficiency of the magnetic absorption mechanism. For frequencies above 10 GHz, the material behaves as a medium with nearly constant effective permeability, and attenuation becomes dominated by dielectric or residual conductive losses.

This profile is widely observed in commercial NiZn ferrites, employed as selective absorbers and electromagnetic interference (EMI) suppressors, wherein the simultaneous presence of high ${\mu }^{\prime \prime }$ and ${\mu }^{\prime }$ near zero in the 9–10 GHz range maximizes absorption [¹⁰, ¹³]. Experimental tests report consistent results, with strong absorption peaks associated with FMR, confirming the high efficiency of these ferrites in the X-band. Thus, Case 5 represents the most comprehensive scenario analyzed, in which Eq. (5) is applied to the general limit that simultaneously includes conductive, dielectric, and magnetic losses, constituting the most complete physical regime within the proposed formulation.

Figures 7 to 12 consolidate the complete overview of the attenuation constant $\alpha \left(\omega \right)$ for the different loss regimes. Dielectric materials (Teflon and BaTiO₃) exhibit contrasting responses: while Teflon presents practically negligible attenuation throughout the X-band, BaTiO₃ manifests a pronounced resonant peak, typical of ferroelectric materials.

Among conductors, distinct behavior is observed: Copper presents attenuation increasing with frequency, following the trend $\alpha \propto \sqrt{f}$, characteristic of good conductors; whereas Polyaniline (PANI), as a plasma conductor with $\omega \ll {\omega }_p$, exhibits decreasing attenuation, reflecting the progressive reduction of conductive losses with increasing frequency. Conductive–dielectric composites (PPy/CNT) reveal a hybrid behavior, with an absorption maximum resulting from the superposition of conduction and interfacial polarization mechanisms.

Finally, NiZn ferrite presents the characteristic magnetic peak around 9 GHz, confirming a predominance of losses associated with magnetic resonance, consistent with the behavior of ${\mu }^{\prime \prime }\left(\omega \right)$ shown in Figure 6.

5. Conclusion

The present work presented a unified formulation of the complex propagation constant $\gamma \left(\omega \right)=\alpha \left(\omega \right)+i\beta \left(\omega \right)$ in media with dielectric and magnetic losses, from which general expressions for the attenuation and phase constants were derived. The main contribution of this work lies in the demonstration that a single analytical expression, (Eq. 5), is capable of continuously describing the dielectric, conductive and magnetic regimes, preserving the classical limiting cases and incorporating standard dispersive models.

This approach allows interpreting, under the same formalism, phenomena traditionally treated separately, providing a bridge between the behavior of insulating, conductive, and magnetic materials throughout the microwave range. Beyond theoretical rigor, the proposed formulation establishes objective criteria for loss comparison and absorber material selection in the X-band, consolidating a unified analysis framework useful for both modeling and experimental validations.

Comparative studies by material class highlighted the importance of adopting realistic dispersive models (Lorentz, Drude, Drude–Lorentz and magnetic Lorentz) in contrast to simplified approximations, such as the indiscriminate use of constant $\tan \left({\delta }_e\right)$ in resonant materials, despite experimental evidence that losses vary significantly with frequency in such materials [⁵, ¹⁶].

The application to five distinct types of materials revealed characteristic behaviors consistent with the literature: low-loss dielectrics (PTFE), with quasi-ideal transparency and $\alpha \approx 0$; resonant dielectrics (BaTiO₃), with peaks near the resonance frequency; metallic conductors (copper), dominated by the skin effect; conductive polymers and hybrid composites (PANI, PPy/CNT), with adjustable intermediate absorption; and magnetic ferrites (NiZn), with selective absorption associated with magnetic resonances.

From an applied point of view, the analytical formulation proved to be a versatile tool for different scenarios: (i) selection of low-loss substrates in antennas and waveguides; (ii) design of narrow-band (BaTiO₃) or broadband (PANI, PPy/CNT) absorbers; (iii) dimensioning of metallic shielding, where extreme attenuation makes propagation unfeasible; and (iv) development of selective ferrite-based absorbers for radars and electromagnetic interference (EMI) suppression. These implications reinforce the utility of the comparative approach in telecommunications projects, automotive radars, and advanced shielding materials.

The global comparison of the $\alpha \left(\omega \right)$ curves showed how different loss mechanisms manifest in the attenuating response, contributing to an integrated understanding of phenomena often treated in isolation in the literature. Nevertheless, relevant gaps persist: (i) dependence on parameters obtained from the literature under limited experimental conditions; (ii) scarcity of experimental data in millimeter ranges, where conductors and polymers present strong dispersion; and (iii) lack of standardization in broadband measurements of $\tan \left({\delta }_e\right)$ and magnetic permeability. Such limitations reinforce the need for more comprehensive experimental studies capable of validating or refining the presented models.

Finally, the formulation presented in this work represents not only a mathematical generalization but a conceptual milestone that connects, under the same formal structure, the various regimes of electromagnetic dissipation. As a perspective, the theoretical understanding of attenuation mechanisms can offer a quantitative reference applicable to new classes of materials, including metamaterials, hybrid composites, and periodic structures [¹⁷], which may exhibit anomalous propagation regimes, such as $\epsilon <0$, $\mu <0$, and broadband absorption, serving as a basis for future correlations between analytical modeling and experimental data.

Acknowledgments

This work is financed in part by the Coordenação de Aperfeiçoamento de Pessoa de Nível Superior (CAPES/Brasil) – finance code 001. D. B. Gavinier would like to thank Centro Paula Souza for the support provided through the Regime de Jornada Integral (RJI).

AI Use

AI platforms ChatGPT and Grammarly have been used to help with the Python codes that generate the figures and with grammar review, respectively.

Appendix

Here is the step-by-step process to obtain (Equations 5) and (6). Remembering Equation (4),

(23) $$\gamma =i\omega \sqrt{{\mu }^{\prime }{\epsilon }^{\prime }}\cdot \sqrt{\left(1-i\tan {\delta }_e\right)\cdot \left(1-i\tan {\delta }_m\right)},$$

we can proceed in the following manner:

I. Developing ${\gamma }^2$ in terms of $\alpha$ and $\beta$

(24) $${\gamma }^2={\left(\alpha +i\beta \right)}^2=\left({\alpha }^2-{\beta }^2+2i\alpha \beta \right)$$

(25) $$\Re \left\{{\gamma }^2\right\}=\left({\alpha }^2-{\beta }^2\right)$$

(26) $$\Im \left\{{\gamma }^2\right\}=2\alpha \beta$$

II. Developing ${\gamma }^2$ in terms of $\omega$, $\tan {\delta }_e$ and $\tan {\delta }_m$

(27) $${\gamma }^2=-{\omega }^2\left({\mu }^{\prime }{ϵ}^{\prime }\right)\left(1-i\tan {\delta }_e\right)\left(1-i\tan {\delta }_m\right)$$

(28) $${\gamma }^2=-{\omega }^2\left({\mu }^{\prime }{ϵ}^{\prime }\right)\left(1-i\left(\tan {\delta }_e+\tan {\delta }_m\right)-\tan {\delta }_e\tan {\delta }_m\right)$$

(29) $$\Re \left\{{\gamma }^2\right\}=-{\omega }^2\left({\mu }^{\prime }{ϵ}^{\prime }\right)\left(1-\tan {\delta }_e\tan {\delta }_m\right)$$

(30) $$\Im \left\{{\gamma }^2\right\}={\omega }^2\left({\mu }^{\prime }{ϵ}^{\prime }\right)\left(\tan {\delta }_e+\tan {\delta }_m\right)$$

III. Now, we make real and imaginary parts equal

(31) $$\left({\alpha }^2-{\beta }^2\right)=-{\omega }^2\left({\mu }^{\prime }{ϵ}^{\prime }\right)\left(1-\tan {\delta }_e\tan {\delta }_m\right)$$

(32) $$2\alpha \beta ={\omega }^2\left({\mu }^{\prime }{ϵ}^{\prime }\right)\left(\tan {\delta }_e+\tan {\delta }_m\right)$$

IV. we can pick either equation to start with. Let us pick the imaginary equality as an example

(33) $$2\alpha \beta ={\omega }^2\left({\mu }^{\prime }{ϵ}^{\prime }\right)\left(\tan {\delta }_e+\tan {\delta }_m\right)$$

(34) $$\beta =\frac{{\omega }^2}{2\alpha }\left({\mu }^{\prime }{ϵ}^{\prime }\right)\left(\tan {\delta }_e+\tan {\delta }_m\right)$$

V. We substitute that into the real equality equation

(35) $${\alpha }^2=-{\omega }^2\left({\mu }^{\prime }{ϵ}^{\prime }\right)\left(1-\tan {\delta }_e\tan {\delta }_m\right)+\frac{{\omega }^4}{4{\alpha }^2}{\left({\mu }^{\prime }{ϵ}^{\prime }\right)}^2{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2$$

VI. Now, using an auxiliary variable $\Phi$, we solve this equation

(36) $$\Phi ={\alpha }^2$$

(37) $$\begin{array}{cc}& {\Phi }^2+\Phi {\omega }^2\left({\mu }^{\prime }{ϵ}^{\prime }\right)\left(1-\tan {\delta }_e\tan {\delta }_m\right)\hfill \\ & -\frac{{\omega }^4}{4}{\left({\mu }^{\prime }{ϵ}^{\prime }\right)}^2{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2=0\hfill \end{array}$$

(38) $$\Phi =\frac{-\left({\omega }^2\left({\mu }^{\prime }{ϵ}^{\prime }\right)\left(1-\tan {\delta }_e\tan {\delta }_m\right)\pm \sqrt{\Delta }\right)}{2}$$

(39) $$\begin{array}{cc}\hfill \Delta & =({\left({\omega }^4{\left({\mu }^{\prime }{ϵ}^{\prime }\right)}^2(1-\tan {\delta }_e\tan {\delta }_m)\right)}^2\hfill \\ & -4\frac{{\omega }^4}{4}{\left({\mu }^{\prime }{ϵ}^{\prime }\right)}^2{(\tan {\delta }_e+\tan {\delta }_m)}^2)\hfill \end{array}$$

(40) $$\begin{array}{cc}\hfill \Delta & =({\left({\omega }^4{\left({\mu }^{\prime }{ϵ}^{\prime }\right)}^2(1-\tan {\delta }_e\tan {\delta }_m)\right)}^2\hfill \\ & -{\omega }^4{\left({\mu }^{\prime }{ϵ}^{\prime }\right)}^2{(\tan {\delta }_e+\tan {\delta }_m)}^2)\hfill \end{array}$$

(41) $$\Phi =\frac{-{\omega }^2\left({\mu }^{\prime }{ϵ}^{\prime }\right)}{2}((1-\tan {\delta }_e\tan {\delta }_m)$$

(42) $$\pm \sqrt{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2+{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2})$$

VII. And rearrange the variables

(43) $$\Phi =\frac{-{\omega }^2\left({\mu }^{\prime }{ϵ}^{\prime }\right)\sqrt{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2+{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}}{2}\cdot \left(\frac{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}{\sqrt{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2+{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}}\pm 1\right)$$

We notice that

(44) $$\left(\frac{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}{\sqrt{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2+{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}}\right)$$

(45) $$=\frac{1}{\sqrt{\frac{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2+{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2}}}$$

(46) $$=\frac{1}{\sqrt{1+\frac{{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2}}}$$

and

(47) $$1=\frac{1}{1}=\frac{\sqrt{1+\frac{{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2}}}{\sqrt{1+\frac{{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2}}}$$

Therefore,

(48) $$\begin{array}{cc}& \Phi =\frac{-{\omega }^2\left({\mu }^{\prime }{ϵ}^{\prime }\right)\sqrt{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2+{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}}{2}\hfill \\ & \cdot \left(\frac{1}{\sqrt{1+\frac{{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2}}}\pm \frac{\sqrt{1+\frac{{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2}}}{\sqrt{1+\frac{{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2}}}\right)\hfill \end{array}$$

(49) $$\begin{array}{cc}\hfill \Phi & =\frac{-{\omega }^2\left({\mu }^{\prime }{ϵ}^{\prime }\right)\sqrt{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2+{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}}{2}\hfill \\ & \cdot \left(\frac{1\pm \sqrt{1+\frac{{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2}}}{\sqrt{1+\frac{{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2}}}\right)\hfill \end{array}$$

We have two options, a$+$ and a$-$ option. How do we choose which to pick? We have to remember that, for materials with no losses, $\alpha$ returns to zero. Observing the previous equation, we notive that this can only happen if the negative number is chosen. So, we can proceed,

(50) $$\begin{array}{cc}\hfill \alpha & =\omega \sqrt{\left({\mu }^{\prime }{ϵ}^{\prime }\right)}[\frac{\sqrt{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2+{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}}{2}\hfill \\ & \cdot \left(\frac{-1+\sqrt{1+\frac{{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2}}}{\sqrt{1+\frac{{\left(\tan {\delta }_e+\tan {\delta }_m\right)}^2}{{\left(1-\tan {\delta }_e\tan {\delta }_m\right)}^2}}}\right)]{}^{1/2}\hfill \end{array}$$

which is Equation (5) rearranged. The step-by-step process for $\beta$ is basically the same.

References

• ^[1] A.R.V. Hippel and S. Morgan, Journal of The Electrochemical Society 102, 68Ca (1955).
• ^[2] R.C. Portes, B.H. Lopes, M.C. Rezende, G. Amaral-Labat and M.R. Baldan, Advanced Composites and Hybrid Materials 7, 208 (2024).
• ^[3] C.A. Balanis, Advanced engineering electromagnetics (John Wiley & Sons, Hoboken, 2012).
• ^[4] R.E. Collin, Field theory of guided waves (John Wiley & Sons, Hoboken, 1990), 2 ed.
• ^[5] J. Krupka, Materials 14, 5097 (2021).
• ^[6] J.A. Kong, Electromagnetic Wave Theory (MIT Press, New York, 2008).
• ^[7] NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY. <emph font="italic">NIST Measurement Services: Complex Permittivity of Dieletric Materials</emph>, NIST Technical Note 1536 (Gaithersburg, 2012).
• ^[8] H.Y. Yao, Y.W. Lin and T.H. Chang, Polymers 13, 1391 (2021).
• ^[9] S.G. Hassan and E.J. Mohammad, Materials Today: Proceedings 42, 2749 (2021).
• ^[10] M. Dressel and G. Grüner, Electrodynamics of solids: optical properties of electrons in matter (Cambridge university press, Cambridge, 2002).
• ^[11] M. Faisal and S. Khasim, e-Polymers 14, 209 (2014).
• ^[12] R.B. Yang, P.M. Reddy, C.J. Chang, P.A. Chen, J.K. Chen and C.C. Chang, Chemical Engineering Journal 285, 497 (2016).
• ^[13] M. Derakhshani, E. Taheri-Nassaj, M. Jazirehpour and S. Masoudpanah, Scientific reports 11, 9468 (2021).
• ^[14] F. Ruiz-Perez, S. López-Estrada, R. Tolentino-Hernández and F. Caballero-Briones, Journal of Science: Advanced Materials and Devices 7, 100454 (2022).
• ^[15] Q. Jiang, C. Xiang, Y. Luo, L. Wu, F. Qin, Y. Jiao, L. Chen and J.H. Lin, Journal of Materials Research and Technology 9, 4593 (2020).
• ^[16] E. Chan, K. Sivaprasad and K.A. Chamberlin, IEEE transactions on components and packaging technologies 30, 86 (2007).
• ^[17] M.A. Amaral Junior, J.S. Marcuzzo, B.S. Pinheiro, B.H.K. Lopes, A.P.S. Oliveira, J.T. Matsushima and M.R. Baldan, Journal of Materials Research and Technology 8, 4040 (2019).
• ^[18] N.W. Ashcroft and N.D. Mermin, Solid State Physics (Rinehart and Winston, Austin, 1976).
• ^[19] D.M. Pozar, Microwave Engineering (Wiley, Washington, 2012), 4 ed.

*Correspondence email address:

[e-mail]

Editor-in-Chief:

Marcello Ferreira https://orcid.org/0000-0003-4945-3169

Publication Dates

• Publication in this collection
  09 Nov 2026
• Date of issue
  2026

History

• Received
  03 Mar 2026
• Reviewed
  10 July 2026
• Accepted
  12 Aug 2026