TCM

In the TCM, a field integral equation is discretized into a matrix form by the MoM in a frequency domain; consequently, its corresponding complex impedance matrix Z at any frequency f is generated to construct a GEE as [Reference Chen and Wang17]:

(1)\begin{equation}{\mathbf{X}}\left(\, f \right){{\mathbf{J}}_n}\left(\, f \right) = {\lambda _n}\left(\, f \right){\mathbf{R}}\left(\, f \right){{\mathbf{J}}_n}\left(\, f \right)\end{equation}

where matrices R and X are real and imaginary parts of Z, respectively. By solving equation (1), a set of eigenvalues λ_n and their corresponding eigenvectors J_n at f are obtained. The subscript n indicates the index of CMs.

As the most important CM quantity, λ_n is utilized to characterize the total stored field energy within a radiation or scattering problem. Since the value range $\left( { - \infty , + \infty } \right)$ of λ_n is too wide to observe, researchers often substitute it with the modal significance (MS) and the characteristic angle (CA) as:

(2)\begin{equation}{\text{MS}}\left(\, f \right) = \frac{1}{{\left| {1 + j{\lambda _n}\left(\, f \right)} \right|}}\end{equation}

(3)\begin{equation}{\text{CA}}\left(\, f \right) = {180^ \circ } - {\tan ^{ - 1}}{\lambda _n}\left(\, f \right)\end{equation}

The MS and the CA are derived quantities of λ_n, and their variation ranges are $[0,1]$ and $[{90^ \circ },{270^ \circ }]$, respectively. The case of ${\lambda _n} = 0$, which can be substituted with either ${\text{MS}} = 1$ or ${\text{CA}} = 180^\circ $, indicates that the associated CM is resonating.

In addition, the solved eigenvectors with distinct mode indexes satisfy the orthogonality conditions as:

(4)\begin{equation}{\mathbf{J}}_m^T\left(\, f \right){\mathbf{R}}\left(\, f \right){{\mathbf{J}}_n}\left(\, f \right) = {0}\end{equation}

(5)\begin{equation}{\mathbf{J}}_m^T\left(\, f \right){\mathbf{X}}\left(\, f \right){{\mathbf{J}}_n}\left(\, f \right) = {0}\end{equation}

where $m \ne n$. Also, they are normalized by the following equation:

(6)\begin{equation}{\mathbf{J}}_n^T\left(\, f \right){\mathbf{R}}\left(\, f \right){{\mathbf{J}}_n}\left(\, f \right) = {1}\end{equation}

The normalization is necessary in the CM tracking, ensuring the unit radiation power of CMs.

Wideband modal tracking problems

At the beginning of this subsection, we provide a brief overview of the classical ECBA in paper [Reference Raines and Rojas11]. Its core tracking method is to construct a correlation matrix C for CMs at consecutive frequency samples f_i and f _i+1 as

(7)\begin{equation}{\mathbf{C}} = {\left[ {{c_{mn}}} \right]_{{\text{ }}N \times N}}\end{equation}

where elements c_mn of C are the correlations between corresponding two sets of eigenvectors ${{\mathbf{J}}_1}\left( {{\,f_i}} \right)\sim{{\mathbf{J}}_N}\left( {{\,f_i}} \right)$ and ${{\mathbf{J}}_1}\left( {{\,f_{i + 1}}} \right)\sim{{\mathbf{J}}_N}\left( {{\,f_{i + 1}}} \right)$ as

(8)\begin{equation}{c_{mn}} = \left| {\rho \left( {{{\mathbf{J}}_m}\left( {{\,f_i}} \right),{{\mathbf{J}}_n}\left( {{\,f_{i + 1}}} \right)} \right)} \right|\end{equation}

where operator $\rho ( \cdot )$ is the Pearson correlation coefficient (PCC). If there are two vectors ${\mathbf{X}} = {\left[ {{x_1},{x_2},\ldots,{x_N}} \right]^T}$ and ${\mathbf{Y}} = {\left[ {{y_1},{y_2},\ldots,{y_N}} \right]^T}$, their PCC is calculated as

(9)\begin{align}& \rho ({\mathbf{X}},{\mathbf{Y}})\nonumber \\ & \quad = \frac{{N\sum\limits_{k = {1}}^N {{x_k}} {y_k} - \sum\limits_{k = {1}}^N {{x_k}} \cdot \sum\limits_{k = {1}}^N {{y_k}} }}{{\sqrt {N\sum\limits_{k = {1}}^N {x_k^2} - {{\left( {\sum\limits_{k = {1}}^N {{x_k}} } \right)}^2}} \cdot \sqrt {N\sum\limits_{k = {1}}^N {y_k^2} - {{\left( {\sum\limits_{k = {1}}^N {{y_k}} } \right)}^2}} }}\end{align}

Drawing on a widely acknowledged interpretation of PCC in Table 1 [Reference Hinkle, Wiersma and Jurs18], judgements of the eigenvector mapping are delineated as follows: (1) If c_mn is the maximum element in the column n with a value above 0.9, then ${{\mathbf{J}}_n}\left( {{\,f_{i + 1}}} \right)$ is mapped to ${{\mathbf{J}}_m}\left( {{\,f_i}} \right)$; (2) If multiple eigenvectors at f _i+1 are mapped to a particular eigenvector at f_i, only the one with maximum |ρ| is selected. The first judgement ensures that mapped eigenvectors exhibit very high correlations, while the second one aims to alleviate correlation ambiguities.

Table 1. Rule of thumb for interpreting PCC

Moreover, a classical adaptive strategy for frequency adjustment is also presented in paper [Reference Raines and Rojas11]. Specifically, if there are unmapped eigenvectors at f_i or f _i+1, a new sample is added between f_i and f _i+1. This strategy has been widely used in other reported tracking methods [Reference Safin and Manteuffel9, Reference Chen, Pan and Su12], which can further improve the accuracy. The traces of tracked CMs can be determined by iteratively applying the above procedure across all sampling frequencies. Although the above classical ECBA is effective and significantly mitigates the mode swapping, it still suffers from the following problems that need to be addressed:

CRA

The CRA, also referred to as “mode degeneracy” in other literature [Reference Akrou and Silva7, Reference Chen, Pan and Su12], is exemplified in Fig. 1. The CRA may occur at a specific frequency when a pair of modes possess approaching eigenvalues. This phenomenon is well documented in the field of quantum physics, which is elucidated as a perturbation for the Hermitian matrix [Reference Neumann and Wigner19]. In electromagnetics, this perturbation is recognized as a coupling effect, indicating that the occurrence of the CRA is more likely to be found in an energy-coupled system [Reference Haus and Huang20]. The CRA frequency f _c featuring the strongest coupling energy between the investigated mode pair exhibits the closest eigenvalue difference ∆λ_mn, defined as

(10)\begin{equation}{\,f_{\text{c}}} = \arg \min \left\{ {\left| {\Delta {\lambda _{mn}}\left(\, f \right)} \right|} \right\}\end{equation}

Figure 1. Sketches of CRA (incorrect) and normal intersection (correct). Dots and solid lines represent the raw unsorted data and the tracking traces, respectively.

where

(11)\begin{equation}\Delta {\lambda _{mn}}\left(\, f \right) = {\lambda _m}\left(\, f \right) - {\lambda _n}\left(\, f \right)\end{equation}

The closest eigenvalue difference is termed as G, given by

(12)\begin{equation}G = \min \left\{ {\left| {\Delta {\lambda _{mn}}\left(\, f \right)} \right|} \right\}\end{equation}

At frequency f _c, eigenvalue curves are split; the surface current of CM is strongly deformed; the current distribution can transition from one situation to another, which is why the CRA should be corrected. Three significant challenges must be tackled to effectively eliminate the CRA: recognizing all CRA mode pairs, determining their CRA frequencies, and correcting eigenvalues and eigenvectors.

Preprocessing EF stage

After presetting the tracking range (from f _start to f _end) and the initial frequency step (STP), the proposed procedure starts by solving the GEE at two consecutive frequency samples (f_i and f _i+1). Subsequently, their corresponding eigenvalues λ _1∼max and eigenvectors J_1∼max can be obtained. The acquired data then proceed to the preprocessing EF stage. In this stage, a filter is introduced to remove insignificant CMs by constraining the range of |λ_n| to be less than 100. Consequently, the number of mapped eigenvectors can be greatly decreased. Similar strategies and the threshold of 100 have been commonly adopted in other literature [Reference Capek, Hazdra, Hamouz and Eichler8, Reference Safin and Manteuffel9], aiming to alleviate the computational complexity.

CTS

After the EF, the data are transmitted to the CTS, which is a modified version of the classical ECBA discussed in Section “Theoretical foundations”. Compared with the classical ECBA, our modification focuses on adjusting the execution condition for the AFA. Due to the utilization of the EF, the number of eigenvectors in the CTS may not be equal at f_i and f _i+1, i.e., $K\left( {{\,f_i}} \right) \ne K\left( {{\,f_{i + 1}}} \right)$, and their matrix C may not be a square matrix. When this happens, according to the classical ECBA, a new frequency sample should be added between f_i and f _i+1 to avoid the presence of unmapped eigenvectors at either f_i or f _i+1. However, such processing is inappropriate and time-consuming since the issue of different K persists in the tracking range no matter how many samples are added.

We illustrate our adaptive strategy for $K\left( {{\,f_i}} \right) \lt K\left( {{\,f_{i + 1}}} \right)$ and $K\left( {{\,f_i}} \right) \gt K\left( {{\,f_{i + 1}}} \right)$ through two special cases (Case I and Case II) depicted in Fig. 3(a) and (b), respectively. In Case I, no unmapped eigenvectors exist at f_i; the remaining ${{\mathbf{J}}_4}\left( {{\,f_{i + 1}}} \right)$ indicates the emergence of a new CM index M_a at f _i+1. Conversely, in Case II, all the eigenvectors at f _i+1 are mapped; the remaining ${{\mathbf{J}}_4}\left( {{\,f_i}} \right)$ indicates the disappearance of the previous CM index M_b. In these cases, we only need to consider the appropriateness of the start frequency f _i+1 for M_a and the end frequency f_i for M_b. For a complete CM trace, its |λ_n| should feature a large value at the start and end frequencies. Therefore, if $\left| {{\lambda _4}\left( {{\,f_{i + 1}}} \right)} \right|$ of M_a or $\left| {{\lambda _4}\left( {{\,f_i}} \right)} \right|$ of M_b is smaller than a certain threshold, then the AFA is executed to ensure the integrity of tracking results. After numerous experiments, we set the threshold to 10 in this paper.

Figure 3. C matrixes of two special cases. (a) Case I: $K\left( {{\,f_i}} \right) \lt K\left( {{\,f_{i + 1}}} \right)$.

(b) Case II: $K\left( {{\,f_i}} \right) \gt K\left( {{\,f_{i + 1}}} \right)$.

Based on the above analysis, we propose an NEC for the AFA, involving three comparison statements as

1. (1) If $K\left( {{\,f_i}} \right) = K\left( {{\,f_{i + 1}}} \right)$, the AFA executes when unmapped eigenvectors exist;

2. (2) If $K\left( {{\,f_i}} \right) \lt K\left( {{\,f_{i + 1}}} \right)$, the AFA executes when unmapped eigenvectors exist at f_i, and |λ_n| of emerging CMs is smaller than 10;

3. (3) If $K\left( {{\,f_i}} \right) \gt K\left( {{\,f_{i + 1}}} \right)$, the AFA executes when unmapped eigenvectors exist at f_{i +1}, and |λ_n| of disappearing CMs is smaller than 10;

Among them, the first statement is identical to the one in the classical EBCA. The second and third statements enable us to obtain complete traces of emerging and disappearing CMs while also reducing the probability of executing the AFA when $K\left( {{\,f_i}} \right) \ne K\left( {{\,f_{i + 1}}} \right)$. This approach is time-saving and enhances the overall efficiency.

In the proposed AFA, the interval between f_i and f_{i+ 1} (∆f) will be repeatedly divided into subintervals, and the process will stop when the NEC is not satisfied, or ∆f is smaller than the preset threshold ∆f _min. The CTS commences at f _start and concludes when the mapping at f _end is complete.

Postprocessing

Finally, the preliminary tracking data acquired through the CTS flows to the postprocessing stage, designed to tackle the CRA issue. The fundamental concept of this stage is to integrate the coupled-mode theory into the TCM. The theory and the implementation of the CRA elimination are presented in the following subsections.

Theory of CRA elimination

The theory employed for CRA elimination is grounded in coupled-mode theory, as documented in paper [Reference Schab, Outwater, Young and Bernhard21]. If M₁ (with J₁ and λ ₁) and M₂ (with J₂ and λ ₂) are a pair of CRA modes, their CRA issues at any frequency f can be resolved by replacing their eigenvalues and eigenvectors with the qualities in an auxiliary system, defined as

(13)\begin{equation}{\mathbf{X}}\left(\, f \right){{\mathbf{J}}_a}\left(\, f \right) = {\lambda _a}\left(\, f \right){\mathbf{R}}\left(\, f \right){{\mathbf{J}}_a}\left(\, f \right) + {\mathbf{H}}{{\mathbf{J}}_b}\left(\, f \right)\end{equation}

(14)\begin{equation}{\mathbf{X}}\left(\, f \right){{\mathbf{J}}_b}\left(\, f \right) = {\lambda _b}\left(\, f \right){\mathbf{R}}\left(\, f \right){{\mathbf{J}}_b}\left(\, f \right) + {\mathbf{H}}{{\mathbf{J}}_a}\left(\, f \right)\end{equation}

where J_a and J_b are two auxiliary vectors with orthonormality, and H is a coupling matrix. Notably, the values of λ_a and λ_b are set to be equal at f_c for the normal intersection. Equations (13) and (14) establish a new system that effectively integrates the coupled-mode theory with the TCM. Subsequently, the eigenvectors of M₁ and M₂ are decomposed into the linear combination of J_a and J_b, guided by their orthonormality, expressed as

(15)\begin{equation}{{\mathbf{J}}_1}\left(\, f \right) = \cos \alpha \left(\, f \right){{\mathbf{J}}_a}\left(\, f \right) + \sin \alpha \left(\, f \right){{\mathbf{J}}_b}\left(\, f \right)\end{equation}

(16)\begin{equation}{{\mathbf{J}}_2}\left(\, f \right) = - \sin \alpha \left(\, f \right){{\mathbf{J}}_a}\left(\, f \right) + \cos \alpha \left(\, f \right){{\mathbf{J}}_b}\left(\, f \right)\end{equation}

Consequently, a relationship between λ _1,2 and λ_a,b is obtained as

(17)\begin{equation}{\lambda _{1,2}}\left(\, f \right) = \frac{{{\lambda _a}\left(\, f \right) + {\lambda _b}\left(\, f \right)}}{2} \pm \sqrt {{{\left( {\frac{{{\lambda _a}\left(\, f \right) - {\lambda _b}\left(\, f \right)}}{2}} \right)}^2} + {{\left( {\frac{G}{2}} \right)}^2}} \end{equation}

Moreover, the expression of the coefficient α is derived as

(18)\begin{equation}\alpha \left(\, f \right) = {\tan ^{ - 1}}\frac{{2\left( {{\lambda _1}\left(\, f \right) - {\lambda _a}\left(\, f \right)} \right)}}{G}\end{equation}

The expressions of eigenvalues and α in paper [Reference Schab, Outwater, Young and Bernhard21] are slightly different from the proposed ones, stemming from variations in the definition of decomposition coefficients and G.

By inverting equation (17) and ensuring the consistency of eigenvalue traces, we derive the expressions of independent λ_a and λ_b as

(19)\begin{equation}{\lambda _a}\left(\, f \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{{\lambda _1}\left(\, f \right) + {\lambda _2}\left(\, f \right)}}{2} + \sqrt {{{\left( {\frac{{{\lambda _1}\left(\, f \right) - {\lambda _2}\left(\, f \right)}}{2}} \right)}^2} - {{\left( {\frac{G}{2}} \right)}^2}} }&{\,f \lt {\,f_c}} \\ {\frac{{{\lambda _1}\left(\, f \right) + {\lambda _2}\left(\, f \right)}}{2} - \sqrt {{{\left( {\frac{{{\lambda _1}\left(\, f \right) - {\lambda _2}\left(\, f \right)}}{2}} \right)}^2} - {{\left( {\frac{G}{2}} \right)}^2}} }&{\,f \gt {\,f_c}} \end{array}} \right.\end{equation}

(20)\begin{equation}{\lambda _b}\left(\, f \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{{\lambda _1}\left(\, f \right) + {\lambda _2}\left(\, f \right)}}{2} - \sqrt {{{\left( {\frac{{{\lambda _1}\left(\, f \right) - {\lambda _2}\left(\, f \right)}}{2}} \right)}^2} - {{\left( {\frac{G}{2}} \right)}^2}} }&{\,f \lt {\,f_c}} \\ {\frac{{{\lambda _1}\left(\, f \right) + {\lambda _2}\left(\, f \right)}}{2} + \sqrt {{{\left( {\frac{{{\lambda _1}\left(\, f \right) - {\lambda _2}\left(\, f \right)}}{2}} \right)}^2} - {{\left( {\frac{G}{2}} \right)}^2}} }&{\,f \gt {\,f_c}} \end{array}} \right.\end{equation}

In addition, J_a and J_b can be calculated by inverting equations (15) and (16) as

(21)\begin{equation}{{\mathbf{J}}_a}\left(\, f \right) = \cos \alpha \left(\, f \right){{\mathbf{J}}_1}\left(\, f \right) - \sin \alpha \left(\, f \right){{\mathbf{J}}_2}\left(\, f \right)\end{equation}

(22)\begin{equation}{{\mathbf{J}}_b}\left(\, f \right) = \sin \alpha \left(\, f \right){{\mathbf{J}}_1}\left(\, f \right) + \cos \alpha \left(\, f \right){{\mathbf{J}}_2}\left(\, f \right)\end{equation}

For M₁ and M₂, the CRA can be eliminated by replacing the quantities λ ₁, λ ₂, J₁, and J₂ with their respective auxiliary counterparts λ_a, λ_b, J_a, and J_b.

Implementation of CRA elimination

In this subsection, we discuss the detailed information of the postprocessing stage. This stage is based on the CRA elimination theory mentioned in the previous subsection and comprises three key steps.

The first step is to identify all CRA mode pairs within the tracking range. The central challenge in this step is to recognize the CRA phenomenon effectively. It is observed that a CRA is likely to occur between a pair of CMs when their mode behaviors satisfy the following criteria at three consecutive frequency samples (f_{i −1}, f_i, and f _i+1) as

(23)\begin{equation}\left\{ {\begin{array}{*{20}{c}} {\frac{{\Delta {\lambda _{mn}}\left( {{\,f_{i - 1}}} \right)}}{{\Delta {\lambda _{mn}}\left( {{\,f_i}} \right)}} \gt 1} \\ {\frac{{\Delta {\lambda _{mn}}\left( {{\,f_{i + 1}}} \right)}}{{\Delta {\lambda _{mn}}\left( {{\,f_i}} \right)}} \gt 1} \end{array}} \right.\end{equation}

(24)\begin{equation}\left| {\rho \left( {{{\mathbf{J}}_m}\left( {{\,f_i}} \right),{{\mathbf{J}}_n}\left( {{\,f_i}} \right)} \right)} \right| \gt 0.3\end{equation}

The first criterion involves evaluating ∆λ_mn between two modes. As depicted in Fig. 1, when f_c is within the range from f_{i −1} to f_{i +1}, ∆λ_mn exhibits a fixed variation pattern at f_{i −1}, f_i, and f _i+1, i.e., small at f_i and large at f_{i −1} and f _i+1. In addition, the signs of ∆λ_mn at f_{i −1}, f_i, and f _i+1 are identical. The above characteristics of ∆λ_mn can be captured in equation (23). However, mode pairs without CRA issues may also exhibit these features of ∆λ_mn. To address this potential issue, we propose an additional criterion in equation (24) to determine whether the recognized CRA mode pair has a nonnegligible correlation. Consequently, all CRA mode pairs can be identified by iteratively estimating mode behaviors for any two modes across all sampling frequencies. Moreover, the frequency range in which the CRA occurs for each mode pair can also be available.

The second step involves determining the CRA frequency f_c and, consequently, the value of G for a CRA mode pair (Pair i). With knowledge of the frequency range of f_c, adding more frequency samples in that range is conventional to find a frequency closer to f_c [Reference Sun, Li and Liu14]. However, since these newly added samples are unsorted, a significant amount of tracking time will be spent by using this strategy. This paper employs CSI to predict f_c due to its ability to generate smooth curves with a consecutive second derivative [Reference Burden, Faires and Burden22]. For instance, if we have determined Modes m and n in Fig. 1 as a pair of CRA modes with ${f_{i - 1}} \lt {f_c} \lt {f_{i + 1}}$ through the first step, then CSI is performed to obtain the piecewise trace functions for the two CMs in the intervals of ${f_{i - 1}} \lt f \lt {f_i}$ and ${f_i} \lt f \lt {f_{i + 1}}$. Specifically, for Mode n, its piecewise trace functions are calculated using the eigenvalue data of the three knots f_{i −1}, f_i, and f _i+1, and two slope values, m ₁ and m ₂, at f_{i −1} and f _i+1 are also considered as

(25)\begin{equation}{m_1} = \frac{{{\lambda _n}\left( {{\,f_{i - 1}}} \right) - {\lambda _n}\left( {{\,f_{i - 2}}} \right)}}{{{\,f_{i - 1}} - {\,f_{i - 2}}}}\end{equation}

(26)\begin{equation}{m_2} = \frac{{{\lambda _n}\left( {{\,f_{i + 2}}} \right) - {\lambda _n}\left( {{\,f_{i + 1}}} \right)}}{{{\,f_{i + 2}} - {\,f_{i + 1}}}}\end{equation}

where f _i−2 and f_{i +2} are the previous and the subsequent frequencies of f_{i −1} and f _i+1, respectively. Consequently, the piecewise trace functions of the two CRA modes can be determined, and a frequency with the smallest |∆λ_mn| can be found easily. The obtained frequency is the predicted f_c, and its corresponding |∆λ_mn| is considered as G. This strategy is time-saving because no extra unsorted frequency samples are introduced. The prediction error is also acceptable due to the consistency of eigenvalue traces.

According to the theory of CRA elimination, the CRA issue can be effectively addressed by replacing the original CM quantities with their corresponding auxiliary counterparts. Therefore, in the third step, auxiliary quantities λ_a, λ_b, J_a, and J_b of Pair i are calculated at a certain frequency f_i by equations (19–22) at first. Then, they replace the original eigenvalues and eigenvectors at f_i. While other studies perform the replacement across all frequencies in the tracking range [Reference Sun, Li and Liu14, Reference Schab, Outwater, Young and Bernhard21], in this paper, the replacement is made only at a few frequencies. The decision is supported by the observation that as the investigated frequency moves away from f_c, CMs in Pair i exhibit a weakly coupled effect, causing their eigenvectors and eigenvalues to resemble auxiliary counterparts. Therefore, performing the replacement at all frequencies is deemed unnecessary in most cases, as it would consume more computational time. In this paper, the replacement starts at f_c and proceeds to the higher and lower frequencies. The replacement ends when the original eigenvectors, such as J₁ and J₂, exhibit a very high correlation with their auxiliary vectors, i.e., $\left| {\rho \left( {{{\mathbf{J}}_1}\left( {{\,f_i}} \right),{{\mathbf{J}}_a}\left( {{\,f_i}} \right)} \right)} \right| \gt 0.9$ or $\left| {\rho \left( {{{\mathbf{J}}_2}\left( {{\,f_i}} \right),{{\mathbf{J}}_b}\left( {{\,f_i}} \right)} \right)} \right| \gt 0.9$ (Criterion A). Finally, the entire procedure of the proposed CM tracking method is complete when the replacement of the last CRA mode pair is done.

Example 1: Rectangular conductor plate

As shown in Fig. 4(a), the first example is a rectangular conductor plate with dimensions of 60 × 120 mm², which has been widely used in other studies [Reference Safin and Manteuffel9, Reference Shavakand, Shokouh and Dashti13]. The plate was discretized with 456 triangular elements and subjected to the MoM analysis utilizing 654 Rao–Wilton–Glisson (RWG) basis functions from 1 to 5 GHz with STP of 160 MHz. After applying the EF, there were 50 pairs of consecutive frequency samples with unequal K, and none of them needed to execute the AFA by using the NEC. Consequently, 42 frequency samples were utilized in this tracking example. Following the CTS, 24 valid CMs (M₁–M₂₄) were acquired, and their CA result without the postprocessing is depicted in Fig. 5(a). The result presents no mode sweeping, and all CMs seem to be tracked correctly. However, using the first step of the postprocessing, an unnoticeable CRA error between M₁ and M₉ in the range from 4.2 to 4.36 GHz was found, showing more obviously in Fig. 5(b).

Figure 4. Geometry and mesh views of the demonstration examples. (a) Rectangular conductor plate. (b) Fractal structure.

Figure 5. Tracking results of the conductor plate without the postprocessing. (a) Entire-band view in CAs. (b) Zoomed-in view in eigenvalues.

Then, for M₁ and M₉, CSI was performed with the eigenvalues at 4.2, 4.28, and 4.36 GHz, along with corresponding endpoint slopes. Consequently, f_c for M₁ and M₉ was determined as 4.314 GHz, with a G value of 0.027. The replacement of the original eigenvalues and eigenvectors of M₁ and M₉ with corresponding auxiliary quantities started at 4.314 GHz and stopped at 3.88 and 4.68 GHz, accounting for only 24.5% of the total frequency samples. The final tracking CA results are presented in Fig. 6(a), revealing a normal trace intersection of M₁ and M₉, as observed in Fig. 6(b). Additionally, as shown in Fig. 7(a), there is no erroneous modal current exchange at f_c for M₁ and M₉ after the postprocessing, and their current distribution at f_c is undeformed. For comparison, the original modal current distributions of M₁ and M₉ are provided in Fig. 7(b).

Figure 6. Tracking results of the conductor plate with the postprocessing. (a) Entire-band view in CAs. (b) Zoomed-in view in eigenvalues.

Figure 7. Normalized modal current distributions of M₁ and M₉. (a) With the postprocessing. (b) Without the postprocessing.

Example 2: Fractal structure

As shown in Fig. 4(b), the second example features a fractal structure with dimensions identical to those in papers [Reference Capek, Hazdra, Hamouz and Eichler8, Reference Chen, Pan and Su12]. Compared with the plate, the fractal structure presents a more challenging tracking scenario due to an increased number of CRA. The MoM analysis for the fractal structure utilized 700 RWG basis functions over the range from 0.7 to 2.8 GHz with STP of 40 MHz. Following the application of the proposed EF, there were 17 pairs of consecutive frequency samples with unequal K. Among these, only two pairs needed to execute the AFA using the proposed NEC. Consequently, 66 frequency samples were utilized in this tracking example. After the CTS, 11 valid CMs (M₁–M₁₁) were obtained, and their CA result without the postprocessing is depicted in Fig. 8(a). Utilizing the first step of the postprocessing, three CRA mode pairs were identified, i.e., Pair 1 (M₄ and M₉), Pair 2 (M₆ and M₇), and Pair 3 (M₅ and M₁₀), as highlighted in Fig. 8(b).

Figure 8. Tracking results of the fractal structure without the postprocessing. (a) Entire-band view in CAs. (b) Zoomed-in view in eigenvalues.

Like the first example, CRA issues within Pairs 1–3 were resolved by predicting their CRA frequencies f_c using CSI, calculating their corresponding auxiliary quantities, and replacing their original eigenvalues and eigenvectors. The predicted and replacement information for Pairs 1–3 is detailed in Table 2. The final tracking CA results are presented in Fig. 9(a). As illustrated in Fig. 9(b), CMs in Pairs 1–3 exhibit correct intersections after the postprocessing, highlighting the capability of the proposed method to correct multiple CRA mode pairs simultaneously. Moreover, the original incorrect modal current distributions of CMs in Pairs 1–3, as shown in Fig. 10(a), were rectified to a consistent and undeformed state after the postprocessing, as demonstrated in Fig. 10(b).

Figure 9. Tracking results of the fractal structure with the postprocessing. (a) Entire-band view in CAs. (b) Zoomed-in view in eigenvalues.

Figure 10. Normalized modal current distributions of the three CRA mode pairs. (a) With the postprocessing. (b) Without the postprocessing.

Table 2. Detailed information for Pairs 1–3

In addition to its enhanced accuracy, the proposed method provides benefits in terms of reduced tracking time and decreased sensitivity to the initial frequency step size. The computational time for several different tracking methods applied to the two examples is recorded in Table 3. Notably, the time for solving GEE was considered. A comparison of methods i–iii reveals a significant reduction in computational time by incorporating the EF, and an additional reduction is observed with the implementation of the NEC. Moreover, the time difference between methods iii and iv is less than 4.83%, indicating that the extra time cost by the postprocessing is acceptable.

Table 3. Computational time of four different tracking methods (unit: s)

i: Classical ECBA [Reference Raines and Rojas11], ii: Classical ECBA + EF [Reference Safin and Manteuffel9], iii: Classical ECBA + EF + NEC, iv: Classical ECBA + EF + NEC + Postprocessing (Proposed).

To assess the sensitivity to the frequency step, the plate was tested with STP ranging from 10 to 320 MHz, and the fractal structure was tested with STP ranging from 10 to 80 MHz. Their corresponding tracking CA results, along with information on the number of frequency samples, are depicted in Fig. 11. The proposed method here exhibits lower sensitivity to STP variations. As STP increases exponentially, there is no proportional decrease in the number of frequency samples. This behavior can be attributed to the proposed AFA, which ensures that the frequency samples crucial for tracking accuracy are not omitted. Therefore, a larger STP only results in less smooth tracking results rather than introducing errors.

Figure 11. Tracking CA results with different STP. (a) Conductor plate. (b) Fractal structure.