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A simple design method for electromagnetic wave absorbers under normal incidence is presented. For the fabrication of the microwave absorbing material a novel design chart is used, including curves of constant ratio in a plane. The practical use of the chart is explained and, as example, a microwave absorber fabricated with magnetite-impregnated plastisol is designed and tested at X-band frequencies.

The single-layer homogeneous absorber (SLHA) backed by a perfectly conducting plate, known also as the Dallenbach layer, has been the subject of several investigations in the last decades due to its practical importance [

All design theories for SLHA compute the reflection coefficient at the interface, usually in terms of the input impedance at the absorbing layer. The general solution for the perfect SLHA was found by defining the electric loss angle δ e = arctan ( ε ″ / ε ′ ) and the magnetic loss angle δ m = arctan ( μ ″ / μ ′ ) as independent variables to present the zero reflection condition in the (d_{e}, d_{m}) plane

[

Using the same theoretical approach employed to obtain general solutions [_{m}, d_{e}), curves of constant d_{e}/d_{m} ratio in the ( δ e + δ m , ε ′ / μ ′ ) plane are presented.

Absorbers are usually fabricated by impregnating a dielectric binder with a lossy filler; the main contribution of this work is to facilitate the design of the SLHA, by providing a graphical aid that shows when the optimum filler to binder ratio that gives perfect absorption has been achieved. Another contribution is the successful use and measurements of natural magnetite-impregnated plastisol as microwave absorbing material since the electromagnetic characteristics and the use of this composite at X-band frequencies seemingly have not been reported in the literature.

This paper is organized as follows: Section 2 provides the theoretical approach for developing a universal design chart for the SLHA. Section 3 explains the practical use of the chart for designing a microwave absorber. As a practical demonstration of this novel design procedure, an absorber fabricated with a composite material made of magnetite-impregnated plastisol is designed and tested at X-band frequencies; the main results are presented and discussed. Section 4 summarizes the main conclusions of this work.

The material of the SLHA is characterized by a thickness d and complex relative electromagnetic parameters ε = ε ′ − j ε ″ = ε ′ ( 1 − j tan δ e ) and µ = μ ′ − j μ ″ = μ ′ ( 1 − j tan δ m ) . When imposing the zero reflection condition, the following equations are obtained [

ε ′ μ ′ = cos δ e ( cosh θ s − cos θ ) cos δ m ( cosh θ s + cos θ ) (1)

sin θ = r sinh θ s (2)

where θ = 4 π d / λ is the two-way electric length of the absorber, λ is the wavelength inside the material, r = tan 1 / 2 ( δ m − δ e ) , s = tan 1 / 2 ( δ m + δ e ) .

We depart from the charts presented in previous works [_{m},d_{e}) plane, we generate universal design curves of constant d_{e}/d_{m} ratio in the ( δ e + δ m , ε ′ / μ ′ ) plane. The main advantage of this new approach is that in the later plane the electromagnetic properties of any type of material (binder and filler composite) can be plotted as a single point, whereas the former plane is suitable only for representing ideal absorbing conditions.

Using Equations (1) and (2), the universal design chart of _{e}/d_{m} = 0) and zero magnetic loss (d_{e}/d_{m} = ∞). No solutions can be found outside the band defined by the two extreme cases, the range of total loss (d_{e} + d_{m}) for perfect absorption is relatively narrow, especially for high ε ′ / μ ′ ratio. The intermediate condition of equal electric and magnetic loss angles (d_{e}/d_{m} = 1) is also presented for reference purposes.

These design curves are exact provided that the conditions of _{e}/d_{m}. However, this is not necessary when ε ′ ≫ μ ′ since all solutions (d_{e} + d_{m}) for total absorption will lie in a narrow band.

Microwave absorbing materials are usually prepared by blending a lossy filler in powder (e.g. carbon black, aluminum, iron, magnetite, ferrites) with a dielectric binder (e.g. epoxy resin, silicon rubber, plastics). These materials are characterized by a complex permittivity and permeability. When increasing the filler/dielectric ratio (maintaining constant frequency), the ε ′ , d_{e} and d_{m} values increase [

d λ 0 = θ 4 π λ λ 0 = [ π − r ( sinh π s − π s cosh π s ) 1 + r s cosh π s ] 1 4 π cos δ m cos δ m cos δ e μ ′ ε ′ cos 1 2 ( δ m + δ e ) (3)

This approximation has an error smaller than 1% for d_{e} ≤ 0.6 and d_{m} ≤ 0.7 [

When the loss angles are small, Equation (3) simplifies to the well-known quarter wavelength approximation d = λ / 4 .

The design method will be illustrated using a composite made of magnetite-impregnated plastisol. The binder is a PVC emulsion and plasticizer compound. The filler is natural magnetite in powder sieved under 325 mesh (44 μm). The components are blended according to a previously specified magnetite/plastisol weight ratio p. The homogeneous paste is then heated in a 180˚C oven for about 20 minutes, allowing the solidification of plastisol. The resulting rubber-like material is flexible and molded samples can be easily cut and thinned as necessary.

Initially, two composite materials were fabricated with p = 1 and p = 3. The corresponding electromagnetic parameters were measured at 9.8 GHz using microwave reflection techniques in a rectangular waveguide, giving the results included in _{e}/d_{m} = 0 of

p | μ ′ | µ ″ | ε ′ | ε ″ | ε ′ / µ ′ | d_{e} + d_{m} |
---|---|---|---|---|---|---|

1.0 | 0.98 | 0.25 | 7.50 | 0.50 | 7.65 | 0.32 |

1.7 | 0.98 | 0.40 | 11.54 | 0.61 | 11.77 | 0.44 |

2.0 | 0.96 | 0.43 | 13.50 | 1.11 | 14.06 | 0.50 |

3.0 | 0.90 | 0.52 | 18.40 | 1.75 | 20.44 | 0.62 |

parameters of this composite are included in

The SLHA was fabricated with a layer of magnetite-impregnated plastisol of weight ratio p = 1.7 and thickness d = 2.1 mm (according to the design value predicted by Equation (3)), glued to a 20 × 20 cm aluminum plate. The reflection loss measured as a function of frequency in the X-band is shown in

In order to predict the absorption characteristics of the SLHA, the electromagnetic parameters of the magnetite-plastisol composite with p = 1.7 were measured as a function of frequency at X-band. The results for μ and ε are shown in

From the data included in

µ ′ = − 0.045 ⋅ f + 1.425 (4)

µ ″ = − 0.055 ⋅ f + 0.955 (5)

ε ′ = − 0.004 ⋅ f + 11.407 (6)

ε ″ = − 0.008 ⋅ f + 0.615 (7)

In Equations (4)-(7) f is the frequency in GHz.

Using these expressions and considering now an adhesive layer of thickness g = 0.1 mm and relative permittivity ε_{d} = 2, a prediction of the absorption

characteristics of the two-layer structure was obtained (considering normal incidence of plane waves). As shown in

In

A universal design method for designing single-layer electromagnetic wave absorbers under normal incidence has been presented. The method is based on a novel design chart which includes curves of constant d_{e}/d_{m} ratio in a ( δ e + δ m , ε ′ / μ ′ ) plane. When representing the electromagnetic characteristics of composite absorbing materials in the design chart, it is observed that the corresponding points are very sensitive to the lossy filler concentration. Therefore, the optimum concentration at a fixed frequency is easily found after representing the measured parameters of a few trial materials with different concentrations in the design chart. Once the optimum absorbing material has been determined, the optimum thickness of the SLHA is readily calculated using a simple expression. The method has been successfully applied for designing a high performance microwave absorber using a low cost composite material made of magnetite-impregnated plastisol.

The key difference with related studies [

The key difference with [

This work has been funded by the University of Chile and FONDECYT Project N˚1940439.

The authors declare no conflicts of interest regarding the publication of this paper.

Jacard, B., Valenzuela, A. and Gonzalez, M. (2018) Practical Universal Method for Designing Single-Layer Electromagnetic Wave Absorbers. Open Journal of Antennas and Propagation, 6, 84-92. https://doi.org/10.4236/ojapr.2018.64008