### I. Introduction

### II. Theory based on an Equivalent Circuit

### 1. Closed-Form Solutions for Thin and Wideband Absorbers

*ɛ*

*. A simple dipole-type structure on a AgNW resistive film is used as a reactive screen. Fig. 2 is the equivalent circuit of Fig. 1 when the EM wave is normally incident. The dipole-shaped structure is modelled as a lumped series RLC resonator characterized by of*

_{r}*R*

_{0},

*L*

_{0}, and

*C*

_{0}. In the equivalent circuit,

*η*

_{0}(= 377 Ω) and

*η*

_{1}are the intrinsic impedances of free space and the substrate, respectively.

*Z*

_{0}is the impedance of the RLC reactive screen alone.

*Y*

_{1}is the input admittance at the input terminal of the shorted transmission line with length

*l*. The characteristic impedance of the transmission line is

*η*

_{1}.

*α*is the attenuation constant, and

*β*is the propagation constant.

*Y*

*is the total input admittance of the RLC screen. If the electrical length*

_{in}*βl*at a design angular frequency

*ω*

_{0}and intrinsic impedance

*η*

_{1}, with the permittivity of the substrate are specified, the RLC values (

*R*

_{0},

*L*

_{0}, and

*C*

_{0}) can be determined and used to obtain the dimensions of the dipole structure through EM simulations, as shown in Section III.

*Y*

_{1}of the shorted transmission line is

*l*is the physical length, and

*γ*=

*α*+

*jβ*is a complex propagation constant of a lossy line. The attenuation constant

*α*can be written in terms of loss tangent (tanδ).

*αl*is given by

*Y*

_{in}as a function of frequency is given by

##### (4)

*A*are given by

*A*= 1) at

*ω*

_{0}is that Eq. (5) is zero at

*ω*

_{0.}This condition results in

##### (7)

##### (8)

*R*

_{0},

*L*

_{0}, and

*C*

_{0}is that the slope of the imaginary part (

*B*

*) of input admittance (4) should be 0 at*

_{in}*ω*=

*ω*

_{0}[24], given by

*B*

*is 0 (9) and satisfies (7) and (8) at*

_{in}*ω*

_{0}, it is guaranteed that the absorption is perfect at

*ω*

_{0}and has the widest bandwidth possible. Note that the real part (G

*) of input admittance (4) is flatter than*

_{in}*B*

*near*

_{in}*ω*

_{0}, especially when

*θ*

_{0}approaches 90°. Eq. (9) is expressed as follows:

##### (10)

##### (11)

##### (12)

*R*

_{0},

*L*

_{0}, and

*C*

_{0}. The resistance

*R*

_{0}in Fig. 2 is given by

##### (13)

*L*

_{0}and capacitance

*C*

_{0}can be obtained from the two equations as follows:

##### (14)

##### (15)

*R*

_{0},

*L*

_{0}, and

*C*

_{0}are all related to the intrinsic impedance (

*η*

_{1}), loss tangent (tanδ), and electrical thickness (

*θ*

_{0}) (at

*ω*

_{0}) of the substrate.

### 2. Verification of Closed-Form Solutions

*R*

_{0}(13) with different electrical thicknesses and loss tangents are shown in Fig. 3. As the electrical thickness

*θ*

_{0}becomes larger, the value of

*R*

_{0}also becomes larger. When the loss tangent is smaller than 0.01, the value of

*R*

_{0}is almost the same as in the case of no loss. The effects of loss on the substrate are shown to be more pronounced when

*θ*

_{0}is greater than 70°. Note that, when the loss tangent is zero and

*θ*

_{0}is 90°,

*R*

_{0}approaches 377 Ω which is same as Salisbury’s. The values of

*L*

_{0}(16) and

*C*

_{0}(17) with different electrical thicknesses and loss tangents are plotted in Fig. 4. The effect of the loss tangent is not significant in

*C*

_{0}but is relatively more significant in

*L*

_{0}, especially when

*θ*

_{0}is greater than 70°.

*θ*

_{0}is 90°, the above solutions are simplified again to those in [5]. Thus, we can see that the solutions presented in this work are general and can be applied to any thickness, permittivity, loss tangent of the substrate, and frequency.

*θ*

_{0}is 70°. The 90% (

*A*= 0.9) bandwidth is approximately 8 GHz, from 8.2 GHz to 16.2 GHz (66% at a center frequency of 12.2 GHz or 80% at a design frequency of 10 GHz).

*R*

_{0},

*L*

_{0}, and

*C*

_{0}using (13) and (16)–(17) are shown in Table 2.

### III. Fabrication and Measurement

### 1. Design and Simulation

*a*is the side length of the square absorber unit, and

*w*and

*h*are the width and height, respectively, of the crossed-dipole pattern on a resistive film.

*R*

_{s}is the resistance per square of the film material, and

*g*is the gap distance. To realize the real part (

*R*

_{0}= 291.86 Ω) of the reactive screen, a AgNW film with 23 Ω/square is used. AgNWs are chosen for fabrication because the sheet resistance of the film can be easily controlled by adjusting the concentration of the AgNW solution. The width (

*w*) and length (

*h*) of the dipole structure affect the resistance

*R*

_{0}, which is given by

*R*

_{s}*h*/

*w*, where

*R*

_{s}is the surface resistance of the resistive sheet (Ω/square). By choosing

*R*

_{s}and the aspect ratio

*h*/

*w*properly, a specifically required

*R*

_{0}can be realized in many ways. Moreover,

*L*

_{0}and

*C*

_{0}are proportional to

*h*/

*w*and

*w*/

*g*, respectively. By using these characteristics, the overall dimensions of the crossed-dipole structure can be determined using an EM simulator.

_{0}is not smaller than the wavelength, a good resemblance between them is observed. The input impedance

*Z*

_{in}, as defined in Fig. 2 and simulated based on Fig. 7(b), is shown in Fig. 9 as a function of the frequency based on the EM and circuit simulations. Without any adjustment to the designed crossed-dipole structure, the EM simulation results reasonably agree with the circuit simulation results. The circuit modelling of the crossed-dipole structure (0.37λ

_{0}in Table 3) by the lumped elements

*R*

_{0},

*L*

_{0}, and

*C*

_{0}is expected to intrinsically lead to some discrepancies between the circuit and EM simulations. Despite these discrepancies, the design guidance by circuit modelling is helpful. Note that the imaginary part is zero (resonant) with a flat slope, and the real part is 377 Ω. This enables perfect absorption (or match) at the design frequency and maximum bandwidth.

*φ*) of 0°–45°. The absorption is almost independent of the polarization angle because of the crossed-dipole structure employed. The degradation effects of microwave absorbers with oblique TE and TM incidence cases are mostly similar to those in [7].

### 2. Fabrication and Measurement

_{0}(15 cm) at 10 GHz. The 3 dB beamwidth (in the E and H planes) of the horn antenna is approximately 45° at 10 GHz. Some variations of this distance (5λ

_{0}) do not affect the measured reflection coefficient and absorption at the absorber. The absorption results are shown to reasonably agree with each other. Some differences between the simulations and the measurement could have resulted from the non-uniformity of the resistivity in the AgNW film and the limited size of the absorber in the measurement. According to the specification, the surface resistance of the AgNW resistive film (Kolon Industries, Seoul, Korea) is 23 Ω/square ±4%.