As the range of frequency bands available for radio frequency (RF) devices becomes increasingly diverse, the dielectrics utilized in these devices are beginning to significantly influence their performance. The Nicolson–Ross–Weir (NRW) algorithm [1, 2] is a widely utilized method for extracting the complex permittivity of dielectrics used in waveguides or coaxial cables. Notably, the measurement setup for this algorithm inherently requires precise machining and insertion of the dielectric material into the waveguide or coaxial line [3, 4]. Any dimensional inaccuracy between the machined dielectric and the waveguide aperture could potentially yield erroneous complex permittivity extraction results [5, 6]. Moreover, given that waveguide aperture sizes vary across frequency bands, researchers must prepare dielectrics that precisely match the waveguide aperture size corresponding to each measurement frequency band. Consequently, each modification in the measurement frequency band necessitates the preparation of a new dielectric sample.

To address the aforementioned issues, researchers developed a methodology featuring a measurement setup [7, 8] that placed the dielectric material between two waveguides, thus eliminating the need to insert a fabricated dielectric into the waveguide. Furthermore, to address electromagnetic (EM) phenomena arising from the dielectric’s exposure to air, such as scattering and refraction, a large flange structure was added to a commercial waveguide and solved using a time-gating method [9–11]. Notably, the scattering parameters (S-parameters) employed in algorithms for measuring complex permittivity are determined based on the first reflected wave at the interface of the dielectric material, as observed in the time domain. The time-gating technique, commonly employed to address multiple reflections caused by the exposure of dielectric materials to air, is implemented to eliminate secondary reflected waves in the time domain. Distinguishing between the first and second reflected waves in the time domain requires flanges of sufficient electrical length and varying sizes tailored to different frequency bands. Alternatively, research has also been conducted on the extraction of complex permittivity through the use of non-destructive dielectrics and commercial waveguides, without the need to introduce additional flange structures [12]. However, the above methods also present challenges in fabricating the dielectric to be measured since variations in the waveguide aperture become significant for measurements across disparate frequency bands. As a simple example, the WR-770 waveguide, operating at 1 GHz, has an aperture size of 195.5 mm × 97.79 mm, while the WR-28 waveguide, operating at 28 GHz, has an aperture size of 7.11 mm × 3.56 mm, indicating a size difference of approximately 27 times along the waveguide width.

In this study, we extracted complex permittivity by employing the same non-destructive dielectric material across frequency bands. As proof of concept, we considered the 0.95–1.23 GHz and 26.5–40 GHz frequency ranges. To enable the utilization of a single dielectric material across these bands, we conducted waveguide miniaturization [13–16] targeting the 0.95–1.23 GHz band. In addition, we integrated a high-permittivity material to address the inherent challenges associated with a large waveguide aperture in the lower frequency band. We designed and fabricated an alumina-inserted waveguide measuring 65 mm × 32.5 mm—approximately one-third the size of a standard WR-770 waveguide (195.5 mm × 97.79 mm)—operating at around 1 GHz. We employed the miniaturized waveguide in the 0.95–1.23 GHz band and the commercial WR-28 waveguide in the 26.5–40 GHz band, using identical dielectric samples across both frequency ranges. Notably, the extraction of complex permittivity necessitates the measurement of the S-parameters at the dielectric interface, which was conducted using thru-reflect-line (TRL) calibration [17]. Furthermore, to mitigate potential EM scattering and refraction effects arising from our non-destructive measurement setup, we applied a Gaussian-weighted moving average (WMA) [18, 19] to the S-parameters. Finally, we implemented a machine-learning-based complex permittivity extraction algorithm to enhance extraction accuracy and address multimode phenomena within the dielectric.

In conventional approaches, WR-770 and WR-28 waveguides are required to extract complex permittivity in the 0.95–1.23 GHz and 26.5–40 GHz frequency bands, respectively. This study aimed to simplify the measurement process by allowing the same dielectric material to be used across different frequency bands without the need for its direct insertion into the waveguide or additional machining during band switching. However, since the aperture size of WR-770 is 195.5 mm × 97.79 mm and that of WR-28 is 7.11 mm × 3.56 mm—an approximately 27-fold difference in size—the dielectric sample must exceed the dimensions of 195.5 mm × 97.79 mm to extract the complex permittivity in the 0.95–1.23 GHz band, which leads to significant challenges in specimen preparation.

To address this issue, we designed a miniaturized waveguide with a reduced aperture size that operates in a frequency band similar to that of WR-770 by incorporating a high-permittivity dielectric. Considering the waveguide aperture size and operating frequency, alumina ( ɛr′=9.8) was selected as the high-permittivity dielectric to be inserted into the waveguide. The cutoff frequency for the TE_mn mode can be expressed as [20]:

where c₀ is the speed of light in vacuum, ɛ_r denotes the relative permittivity of the waveguide, μ_r refers to the relative permeability of the waveguide, W signifies the waveguide width, and H is the waveguide height. The dimensions of the designed miniature waveguide equipped with the alumina insertion, as shown in Fig. 1(a), are 62 mm × 32.5 mm × 60 mm.

Detailed values pertaining to the proposed waveguide are provided in Table 1. These parameters, selected using the ANSYS HFSS simulator with a parameter sweep, achieved good matching performance in the operating frequency band. To validate the performance of the designed waveguide, we compared the measured and simulated results obtained using thru configuration of the two waveguides, as depicted in Fig. 1(b). Furthermore, Fig. 2 shows the S₂₁ magnitude and phase of the fabricated waveguide compared to those of the simulated waveguide, confirming their similarity.

A measurement setup in which the dielectric is exposed to air may lead to EM wave scattering at dielectric boundaries and the excitation of multiple modes within the dielectric material. To address these issues, researchers have employed the method of moments (MoM) [21–23] for EM wave numerical analysis and optimization. However, since this approach was developed for commercial waveguides, extensive modifications must be implemented to the existing algorithm to adapt it to small waveguides with alumina inserts, which is a laborious task.

Therefore, in this study, a machine learning technique [24] was employed to extract the complex permittivity by accounting for various EM phenomena that may occur in the proposed measurement setup. Notably, machine learning techniques have previously been employed to extract the complex permittivity in the resonator material measurement method [25] and the microstrip transmission line material measurement method [26, 27].

In the present work, to generate training data for the machine learning model, the alumina-inserted miniaturized waveguide designed in Section II was considered for the 0.95–1.23 GHz band, while a commercial WR-28 waveguide was employed for the 26.5–40 GHz band. Training data were generated using full-wave simulations, similar to those conducted in previous studies [26, 27]. Notably, simulations were performed for the two frequency bands—0.95–1.23 GHz and 26.5–40 GHz—considering the same range of complex permittivity values using a PC equipped with an Intel Xeon Gold 6248R processor at 3.00 GHz and 768 GB RAM. The configuration settings for generating the training data are detailed in Table 2 under “Training data.” The generated training data consisted of 24,969 samples for the 0.95–1.23 GHz band and 117,096 samples for the 26.5–40 GHz band.

Notably, the S-parameters of the cross-section of the test material were required to extract complex permittivity. The S-parameters generated from the simulations included the effects of the waveguide and the SMA connector. To calculate the S-parameters at the cross-section of the material and mitigate the scattering phenomena at the material’s boundaries, TRL calibration was applied, and a Gaussian-WMA was used to produce the final input data for the training set.

With regard to machine-learning modeling for complex permittivity extraction, a regression model [28–30] was employed to predict continuous values and model the relationships between the input and output variables. The input variables were frequency and S-parameters, while the output variables were the real and imaginary parts of the complex permittivity. Detailed settings of the machine learning model employed in this study are provided in Table 2 under the section “Machine learning modeling.” Notably, since this study aimed to extract the complex permittivity using a simplified measurement setup, the research was conducted with an emphasis on minimizing the complexity of the machine learning model.

The implemented machine learning model was validated using scatter plots comparing the actual and the predicted values. For this validation process, 80% of the data were used as training data, and 20% were used as test data. Figs. 3 and 4 present the scatter plots of the real and imaginary parts of the complex permittivity for the two selected frequency bands.

The mean squared error (MSE) [31] for the real and imaginary parts of the complex permittivity at the two frequency bands is provided in Table 3. The formula for calculating the MSE is as follows:

where n represents the number of samples, y_actual denotes the actual values, and y_predict refers to the predicted results calculated using the machine learning algorithm. Notably, although the predicted results for the imaginary part may appear to differ from its corresponding actual values on the graph, the calculated error rate was low, as shown in Table 3, indicating a high level of accuracy.

The process of extracting complex permittivity using the machine learning technique proposed in this study is illustrated in Fig. 5. The detailed steps are as follows:

To verify the results of the proposed complex permittivity extraction algorithm constructed using machine learning techniques, an RO3003 (ɛr′=3.005, ɛr′′=0.0011) and RO4003C (ɛr′=3.345, ɛr′′=0.0035) were utilized while maintaining the settings noted in Table 2. Fig. 6 shows the measurement setup comprising a VNA (Anritsu MS4647B), two miniature waveguides, and a nondestructive dielectric material. Figs. 7–10 compare the machine-learning-based complex permittivity results obtained for RO3003 and RO4003C across the two frequency bands against the data sheet values.

Figs. 7 and 8 show the complex permittivity extraction results obtained using the proposed machine learning algorithm in the 0.95–1.23 GHz band upon employing the alumina-inserted miniaturized waveguide designed in this study. To calculate the error rate, the root mean square relative error (RMSRE) was computed, achieving values of 2.09% and 0.97% for RO3003 and RO4003C, respectively. Figs. 9 and 10 present graphs comparing the datasheet values of RO3003 and RO4003C with the results obtained using the proposed machine learning algorithm in the 26.5–40 GHz frequency band, showing that the RMSRE for the proposed machine-learning algorithm is 3.84% for RO3003 and 6.47% for RO4003C. These results confirm the high accuracy and low deviation of the machine-learning-based algorithm in the face of measurement material changes.

This study proposes an efficient technique for measuring complex permittivity that involves using a dielectric with consistent dimensions across frequency bands. The proposed measurement setup determines complex permittivity by positioning the dielectric material between two waveguides rather than inserting it into the waveguide. To use a consistent dielectric, a compact waveguide design with an alumina insert operating in the 0.95–1.23 GHz frequency band was fabricated. Furthermore, machine learning techniques were employed to determine the complex permittivity. First, the S-parameters and complex permittivity across the selected frequency bands were estimated through simulation. Subsequently, TRL calibration was applied to the S-parameters. Additionally, Gaussian-WMA filtering was implemented to mitigate the EM scattering effects caused by the air-exposed structure. The machine learning model was then implemented and trained. Finally, S-parameters across the frequency bands were measured using a VNA, and the measured data were used as input for the machine learning model to calculate the complex permittivity as the predicted result, thereby demonstrating the validity of the proposed approach.