The effective permittivity (ε_eff) of a chaff cloud with a random distribution is as follows:

where ε₀ denotes the permittivity in free space, η₀ denotes the characteristic impedance in free space, ρ denotes the density of the chaff cloud, λ denotes the wavelength, l denotes the length of the chaff, and s denotes nondimensional parameter of effective permittivity. Furthermore, I¯ denotes the effective average current of the chaff per unit wavelength and is expressed as follows:

where I.̇ (θ,ϕ) denotes the scattered field component in the -x direction when a z-polarized plane wave is incident on the chaff as shown in Figure 1. This can be calculated as follows:

Figure 1: 
Geometry of the chaff

where l^(θ,ϕ) denotes the orientation vector of chaff in the spherical coordinate system. Equation (2) assumes that chaff is randomly oriented in all directions.

However, if there are many distributions in the specific orientation of the chaffs, the probability density function (pdf) of the orientation distribution is set as the weighting function W and weight is applied to the specific orientation by using the weighted average [11]. The weighted average is as follows:

Based on these procedures, the chaff cloud can be replaced by an effective medium with the effective permittivity of the chaff cloud considering the orientation distribution of the chaffs.

The reason for calculating the number of chaffs by replacing them with an effective medium with effective permittivity involves obtaining a coherent RCS. A coherent RCS is the average RCS. Specifically, the coherent RCS is determined by the overall shape of the effective medium and can be calculated from the reflection coefficient at the effective medium interface as shown in Figure 2.

Figure 2: 
Geometry of the effective medium

The input reflection coefficient of the slab structure shown in Figure 2 is as follows:

where Γ₁₂ and Γ₂₃ denote the interference reflection coefficients between Regions 1 and 2 and between Regions 2 and 3, respectively, and γ₂ denotes the complex propagation constant in Region 2.

The field scattered by the effective medium is a coherent scattered field, and the coherent RCS can be obtained as ollows:

where A denotes the area of the effective medium.

When a large number of scatterers exist, there are not only coherent scattered fields but also incoherent scattered fields, and the incoherent scattered fields are changed by a spatial change in the scatterers. Therefore, the incoherent field represents fluctuations in the scattered field. Specifically, as the number of scatterers increase and similarity between the scatterers decrease, the incoherent scattered field component increases. Therefore, even in the field scattered by the chaff with the same density, as the thickness of the effective medium increases, the coherent component decreases and incoherent component increases. To calculate the incoherent RCS of the effective medium, the incoherent scattered field can be obtained by ignoring the mutual coupling between the chaffs, calculating only the scattered field by the chaffs, and considering only the location of the scatterers. The incoherent scattered field of the effective medium is as follows:

where μ₀ denotes the permeability in free space, J denotes the volume current density, R denotes the distance between the target and observation point, k₀ and k₁ denote the wavenumber of the free space and effective medium, respectively, and A denotes the cross-section of the effective medium. Using this scattered field, the incoherent RCS can be obtained as follows:

Finally, the RCS of the effective medium is obtained by adding the coherent RCS and incoherent RCS as follows:

The length of the chaff, composing the chaff cloud, was set to 0.5λ. When simulating the RCS of 900 chaffs using MoM, the calculation time required for one simulation was approximately 95 min. When using the EMM, a total of 1.7 s were required. Additionally, the MoM requires a large amount of calculation time by repeating the analysis 20 times for each case using the Monte Carlo method.

As shown in Figure 4, there is a small difference of 2 dB between the results of EMM and MoM. If the number of iterations in the Monte Carlo method increases, then the difference is expected to decrease.

Figure 4: 
RCS of the chaff cloud composed of 0.5λ-long chaffs

The RCS obtained by EMM can be separated into coherent and incoherent RCS as shown in Figure 5. The level of coherent RCS increases as the density increases. Even at a density of 2.3 N/λ³, there is a 5-dB difference between the two components, and the coherent RCS is relatively negligible. Additionally, the incoherent RCS level increases as the density increases, but it can be shown that it gradually becomes saturated as the density increases. At a density of 2.3 N/λ³, the difference between RCS_ϕϕ and incoherent RCS is very small (0.5 dB), and accordingly, it can be confirmed that the effect of coherent RCS is very small.

Figure 5: 
Comparison of incoherent and coherent RCSs of the chaff cloud composed of 0.5λ-long chaffs

A chaff cloud composed of 0.75λ-long chaff fibers is considered. When the RCS of 900 chaffs is simulated using the MoM, the calculation time required for a single iteration is approximately 185 min, and when the EMM is used, a total of 2.9 s are required.

The results of the EMM and MoM are plotted in Figure 6, and the difference between the results of the two methods is small within 1.8 dB. Given that a chaff fiber with a length of 0.7λ does not resonate at the corresponding frequency, the average RCS at a density of 2.3 N/λ³ is about 5.6 dB, which is 10 dB lower than that of a chaff cloud composed of 0.5λ-long chaffs where resonance occurs.

Figure 6: 
RCS of the chaff cloud composed of 0.75λ-long chaffs

As shown in Figure 7, the difference between coherent RCS and incoherent RCS even at a density of 2.3 N/λ³ is 14 dB. Given that resonance does not occur in 0.75λ-long chaff, the RCS level is 9 dB higher than that of 0.5λ-long chaffs. Moreover, the difference between RCS_ϕϕ and incoherent RCS at the density of 2.3 N/λ³ is very small, i.e., 0.1 dB. Hence, coherent RCS is negligible.

Figure 7: 
Comparison of incoherent and coherent RCSs of the chaff cloud composed of 0.75λ-long chaffs

The length of the chaff composing the chaff cloud was set to 1λ. In the case of simulating the RCS of 900 chaffs using the MoM, the calculation time required for a single iteration was approximately 266 min; in the case of using the EMM, a total of 3.8 s were required. As shown in Figure 8, there is a small difference of 1.2 dB between the RCSs of the MoM and EMM. At a density of 0.25 N/λ³, the difference from the chaff cloud with a 0.5λ-long chaff was 0.1 dB, which was small. However, as the density increased, the difference with the RCS_ϕϕ of the chaff cloud, composed of 0.5λ-long chaffs, increased, resulting in a difference of 4.7 dB at a density of 2.3 N/λ³.

Figure 8: 
Comparison of incoherent and coherent RCSs of the chaff cloud composed of 1λ-long chaffs

Given that resonance occurs in 1λ-long chaff, as shown in Figure 9, the difference between the coherent RCS and incoherent RCS at a density of 2.3 N/λ³ is 11.9 dB, which is 6.9 dB higher than that of the 0.5λ-long chaffs. When comparing the coherent RCS with the incoherent RCS, the coherent RCS is at a similar level within 1.2 dB when compared to the chaff cloud composed of 0.5λ-long chaffs. However, the incoherent RCS is within 5.6 dB, it is less than 5.6 dB, and the effect of the coherent RCS is smaller and effect of incoherent RCS is higher. Additionally, at a density of 2.3 N/λ³, the difference between RCS_ϕϕ and incoherent RCS is very small and corresponds to 0.2 dB, and the coherent RCS is negligible.

Figure 9: 
Comparison of incoherent and coherent RCSs of the chaff cloud composed of 1λ-long chaffs

Based on these results, as the number of chaffs increases and the chaffs resonate with an odd multiple of the half-wavelength of the radar operating frequency, the effect of incoherent RCS becomes greater than that of coherent RCS. As the number of scatterers increases, the incoherent scattered fields owing to various orientations and locations increase. Additionally, it can be shown that this occurs because individual chaffs generate more scattered fields near the resonance frequency. Furthermore, the larger deviation of the scattered field between scatterers owing to the orientation distribution of the chaff is due to the similarity between the scatterers.