Meander Structure Analysis Techniques Using Artificial Neural Networks

Abstract

Typically, analyses of meander structures (MSs) for transfer characteristics are conducted using specialized commercial software based on numerical methods. However, these methods can be time-consuming, particularly when a researcher is seeking to perform a preliminary study of the designed structures. This study aims to explore the application of neural networks in the design and analysis of meander structures. Three different feedforward neural network (FFNN), time delay neural network (TDNN), and convolutional neural network (CNN) techniques were investigated for the analysis and design of the meander structures in this article. The geometric dimensions or top-view images of 369 different meander structures were used for training an FFNN, TDNN, and CNN. The investigated networks were designed to predict such electrodynamic parameters as the delay time ($t_{\mathrm{d}}$), reflection coefficient ($S_{11}$), and transmission coefficient ($S_{21}$) in the 0–10 GHz frequency band. A sufficiently low mean absolute error (MAE) was achieved with all three methods for the analysis of MSs. Using an FFNN, the characteristic $t_{\mathrm{d}}$ was predicted with a $3.3$ ps average MAE. The characteristic $S_{21}$ was predicted with a $0.64$ dB average MAE, and $S_{11}$ was predicted with a $2.47$ dB average MAE. The TDNN allowed the average MAEs to be reduced to $0.9$ ps, $0.11$ dB, and $1.63$ dB, respectively. Using a CNN, the average MAEs were $27.5$ ps, $0.44$ dB, and $1.36$ dB, respectively. The use of neural networks has allowed accelerating the analysis procedure from approximately 120 min on average to less than 5 min.

1. Introduction

Microwave devices have experienced a remarkable increase in demand, especially over the past decade, driven by rapid advances in wireless communications. This rise is driven by the increasing need for compact microwave devices that provide high performance and seamless integration into modern systems. Traditionally, microwave devices’ design methodology is based on designers’ experience and creativity, often involving iterative cycles of electromagnetic simulations and experimental testing until the desired specifications are met [1,2,3]. However, as circuit complexity increases and the need for faster development cycles based on innovative intelligent methods increases, intelligent methods are tasked with address complex circuit design and reducing computational resources and processing times, among other challenges [4,5,6]. In this context, the evolution of microwave devices represents not only technological progress but also a paradigm shift toward intelligent adaptive design methods, which are promising for revolutionizing the field.

Artificial neural networks (ANNs) have been extensively used in the field of microwave devices in the past decade due to their ability to autonomously learn complex patterns and relationships from data [7,8,9]. ANNs excel at capturing nonlinearities and handling high-dimensional data, which are often encountered in real-world problems [10]. The application of neural networks is widely distributed for the analysis and parametric synthesis of microwave devices [11,12,13] and also for optimization techniques in computer-based modeling of nonlinear microwave devices [14,15]. ANNs are widely used for modeling antennas [16], antenna arrays [17], filters [18], phase shifters [19], delay lines [20], and other devices [21,22]. ANNs are also used in microwave-based systems to automate the local tasks of nonlinear functions, which may speed up and simplify the design process [23,24].

Multilayer perceptron (MLP) networks with backpropagation are commonly employed for the analysis and synthesis of microwave devices in order to map between the related geometric dimensions of microwave devices and their electrodynamical parameters [25,26]. For example, the optimal geometrical parameters of custom-designed resonators were determined with a feedforward MLP network in order to reduce undesirable coupling in an antenna [27]. A similar MLP network was used to obtain the desired operating frequency, bandwidth, and insertion loss parameters for interdigital filters and a compact dual-band diplexer in [28].

Advanced types and structures such as deep neural networks (DNNs), recurrent neural networks (RNNs), convolution neural networks (CNNs), and Transfer Learning are used in order to approach the modeling problems of microwave devices in which electrodynamical characteristics change dynamically [29,30,31,32]. The parameters of the matching circuit for the matching of antenna impedance were determined using a DNN [33]. The $S_{11}$ parameter was used in the input of the DNN, while the DNN predicted the values of the series and parallel capacitors of the impedance matching circuit in the output. A nonlinear microwave device modeling technique, which is based on the time delay neural network (TDNN), was used to analyze the DC, small-signal, and large-signal data from different models of transistors in [34]. According to the authors, the TDNN approach has advantages in accuracy over the static modeling technique of MLP networks when the number of delay buffers increases.

The most recent and scientifically uncertain direction of research is to study the possible mapping of the geometric dimensions and electrodynamic parameters of microwave devices by learning the image content [35]. For example, a feature mapping between images and the values of nonlinear phase shifts in optical fiber communication systems with polarization division multiplexing is presented in [36]. The initial signals were converted into future maps which were fed into the CNN input. The authors of [37] trained a CNN to estimate the direction of arrival of interference with scanty snapshots and, as a result, to have an undistorted beamformer response to the wideband signal of interest. The CNN helped solve the synthesis task by mapping the stepped-impedance planar filter geometry and its frequency response using top-view images of the planar filter [38].

Datasets of the parameters and characteristics of microwave devices for ANN training are usually quite small due to the complex and time-consuming procedure of data collection using commercial software packages like CST Studio Suite^©, HFSS^©, or Sonnet^© [39]. CST Studio Suite^© works on basis of the finite-difference time domain (FDTD) method, HFSS^© uses the finite element method (FEM), and Sonnet^© uses the method of moment (MoM). All of these methods belong to a family of techniques which use a differential form of Maxwell’s equations. The authors consider such datasets reliable, since they are obtained using traditional full-wave methods. For example, only 64 training and 49 testing samples were collected using HFSS^© software for mapping the geometry and corresponding electromagnetic parameters of the antenna using cross-sectional binary images [40]. An ANN’s training and prediction accuracy are usually evaluated with measures such as the mean squared error (MSE), mean absolute error (MAE), and root mean squared error (RMSE) [41].

In this article, significant attention is paid to meander structures (MSs), which are more commonly applied in microwave devices for the purpose of miniaturization [42]. MSs allow miniaturization of microwave devices while maintaining the operational parameters of the device [43].

Three scientific problems are investigated in this article. Firstly, the design of one unique meander structure using traditional full-wave methods is time-consuming and takes approximately two hours. This is especially relevant when simulations need to be repeated many times until an MS with the desired characteristics is achieved. Secondly, gathering a correctly structured training dataset is complicated and has a significant impact on the accuracy of neural network training. Thirdly, selection of the type and structure of the ANN is complex and depends on the training dataset. Therefore, the dataset and ANN must be selected individually each time in order to maintain a high prediction accuracy.

ANNs are rarely applied in the design and analysis of meander structures. Therefore, the applied studies on the possible usage of ANNs in order to predict preliminary characteristics, such as $S_{11}$, $S_{21}$, and the group delay $t_{\mathrm{d}}$, hold the potential to accelerate the meander structure design process by an order of magnitude while maintaining the desired prediction accuracy.

2. Materials and Methods

The investigated meander structure (MS), collected dataset, and used methods, which are based on artificial neural networks, will be discussed in detail in this section.

2.1. Investigated Meander Structure

The applied research on the possible usage of ANNs for the prediction of the parameters of microwave devices was conducted using a meander structure (MS) (Figure 1) from our previous article [44]. The introduced meander structure was designed for signal synchronization by delaying signals. Meander structures are used to create longer electrical paths within a confined space, effectively increasing the signal’s travel time without significantly enlarging the physical size of the circuit. These structures are commonly found in microwave circuits, delay lines, phase shifters, and antennas.

The main electrodynamical parameters for the analysis of the meander structures were $S_{11}$ (reflection coefficient), $S_{21}$ (transmission coefficient), $\Delta f$ (operating frequency range), $t_{\mathrm{d}}$ (delay time), and $Z_{\mathrm{c}}$ (characteristic impedance) [45,46]. The meander structure (Figure 1) was designed for a delay system, and therefore the most relevant $S_{11}$, $S_{21}$ and $t_{\mathrm{d}}$ characteristics will be predicted. The $\Delta f$ could be seen from the $S_{21}$ characteristic, and the $Z_{\mathrm{c}}$ characteristic’s impedance matching in the line can be seen from the $S_{11}$ characteristic. Therefore, the list of $S_{11}$, $S_{21}$, and $t_{\mathrm{d}}$ was sufficient to analyze the meander structure.

The electrodynamical parameters of the presented MS depended on its geometric dimensions and the materials used in it. The width of the meander strip $L_{\mathrm{c}}$, the number of central meander strips N, the gap between adjacent meander strips l, the length of the central meander strip $2a$, and the length c of the meander conductors near the connectors have a direct influence on the electrodynamical parameters of the meander structure. The characteristic impedance $Z_{\mathrm{c}}$ is mainly dependent on the ratio of the width $L_{\mathrm{c}}$ of the meander strip and the height of the substrate. The operating frequency range $\Delta f$ and thus $S_{11}$ and $S_{21}$ are mainly dependent on the width l of the gap between the meander strips.

The provided meander structure has limitations. The gap between adjacent meander conductors was selected to allow the MS to work in the 0–10 GHz frequency range. The number of meander strips was selected in the range from 10 to 20 to have approximately 1 ns for its delay time.

The exact dimensions of the initial model of the meander structure (Figure 1) from which the data gathering was started were the following: the total length of the meander structure $L=14.8$ mm; the width of the central part of the meander structure $2a=10$ mm; the length of the additional conductors near the connectors $c=6$ mm; the width of the meander conductors $L_{\mathrm{c}}=0.4$ mm; the gap between adjacent conductors $l=0.4$ mm; the thickness of the substrate $h_{\mathrm{s}}=0.221$ mm; and the thickness of the conductors $h_{\mathrm{c}}=0.035$ mm. An FR4 material was used for the substrate, where ${\epsilon }_{\mathrm{r}}=4.3$ and $tan\delta =0.02$. Copper was used for the conductor, whose inductance was equal to 5.8107 S/m. The general dimensions of the meander structure were $16.24\times 17.35\times 0.291$ mm.

Data for the investigation were collected using the full-wave Sonnet^© commercial software package (Sonnet 13.52, Sonnet Software, Syracuse, NY, USA). Sonnet^© is based on the method of moments (MoM). The MoM belongs to a family of techniques which use a differential form of Maxwell’s equations. The calculations when using the MoM are time-consuming. Calculating the particular model (Figure 1) takes approximately two hours. It is important to search for new methods that will allow real-time analysis of MSs. Therefore, a sufficient dataset for training three different ANNs was collected.

2.2. Dataset

In total, 369 different combinations of the presented MS (Figure 1) were collected with different sets of $L_{\mathrm{c}}$, N, l, $2a$, and c geometric dimensions. The gap l between the adjacent conductors varied between 0.2 mm and 1.8 mm with a step of 0.05 mm (33 different combinations). The width $L_{\mathrm{c}}$ of the meander conductors varied from 0.2 mm to 1 mm with a step of 0.05 mm, except for samples where $L_{\mathrm{c}}$ was equal to 0.25, 0.35, 0.45, 0.75, 0.85, and 0.95 mm (11 different combinations). Six additional samples were collected when $l=0.2$ mm and $L_{\mathrm{c}}$ equaled 0.25, 0.35, 0.45, 0.75, 0.85, and 0.95 mm. Therefore, the overall number of samples was equal to $11\times 33+6=369$.

All other N, $2a$, and c parameters were kept constant but were added to the network training in order to maintain the same structure of the dataset for future plans. Such a decision was made due to the long data collection procedure and the fact that variations in several parameters were sufficient to research our specified scientific uncertainties for the potential usage of ANNs to predict an MS’s characteristics. The $S_{11}$ and $S_{21}$ characteristics were obtained during a computer-based simulation in Sonnet^©. The $t_{\mathrm{d}}$ values were found during the additional calculations. A 120 min simulation was required on average in order to obtain the EM characteristics of a single MS. A cumulative simulation duration exceeding 738 h was required in order to collect this dataset of 369 unique MSs.

The input matrix for each individual MS consisted of $L_{\mathrm{c}}$, N, l, $2a$, and c geometric dimensions and a frequency f. The simulations in Sonnet^© were carried out in the 0.1–10 GHz frequency range with a frequency step of 25 MHz (397 frequency samples overall). Therefore, the size of the prepared input matrix for each individual MS with a unique combination of parameters was equal to $[6\times 397]$. The target matrix of one particular model of the meander structure was equal to $[3\times 397]$. The target matrix consisted of the $S_{21}$, $S_{11}$, and $t_{\mathrm{d}}$ parameters. The overall size (146,493) of the input and target matrices was obtained by multiplying the 369 unique MSs by the number of samples of each unique MS (397). The overall dimensions of the collected input and target matrices were equal to 6 × 146,493 and 3 × 146,493, respectively. The collected dataset was divided into training, validation, and testing data in the following approximate percentages, depending on the neural network used: 70%, 10%, and 20%.

In addition, the top view of every particular unique meander structure was saved for CNN training. In general, 369 top view images of unique MSs were collected. The original $2298\times 1318$, 24 bit depth top view image of every unique MS was converted to a $1024\times 1024$, 8 bit depth image to increase contrast and facilitate the CNN learning process. Examples of black and white top view images of the meander structures with a different set of $L_{\mathrm{c}}$ and l geometric dimensions are presented in (Figure 2).

In all, 70% of the data from the collected dataset were used for training, 10% were used for validation, and 20% were used for testing. The next three subchapters will present the artificial methods used for analysis of the presented meander structure.

It should also be emphasized that the collected dataset imposed limitations of the research. These limitations were independent of the chosen neural network structures. The prediction of frequency characteristics would only be possible to carry out in the 0–10 GHz frequency range, while l varied in the 0.2–1.8 mm range and $L_{\mathrm{c}}$ varied in the 0.2–1 mm range, and all other parameters were constant ($2a$ = 10 mm, c = 6 mm and N = 14). The parameters of the cross-sectional structures and the parameters of the used materials were not included in the training dataset. Therefore, in order to expand the prediction limits of the electrodynamical characteristics of the MS, additional data for training should be collected.

2.3. Structure of the Feedforward Neural Network

A multilayer feedforward neural network (FFNN) was used for the prediction of $t_{\mathrm{d}}$, $S_{21}$, and $S_{11}$. The general FFNN network description equation is as shown in Equation (1):

$$z\left(x\right)=(\sum _{i=1}^m{w}_i\times x_i)+b,$$

(1)

where w is a random weight, x is the input value, b is a random bias, m is the number of neurons in the previous layer, and i is an index representing the individual input connections to the neuron from the previous layer. In our case, the input vector consisted of $x=[L_{\mathrm{c}},N,l,2a,c,f]$ for the parameters.

The sigmoid activation function was used for all hidden layers (Equation (2)). The linear activation function was used in the output layer. The sigmoid function is described by the following equation:

$$g\left(z\right)=(1/(1+e^{-z})),$$

(2)

where z is a real-valued number in the input to the sigmoid function which can be positive, negative, or zero, and $g\left(z\right)$ is the output of the sigmoid function, which is also a real-valued number which shrinks the input z into a range between 0 and 1.

Three separated FFNN networks were selected in order to predict $t_{\mathrm{d}}$, $S_{21}$, and $S_{11}$ separately according to our collected dataset of geometrical dimensions and electrodynamical characteristics of MSs. The reason for this is that the FFNN network cannot learn correctly how to predict 3 quite different $t_{\mathrm{d}}$, $S_{21}$, and $S_{11}$ characteristics at the same time.

The input layer of each selected FFNN had 6 neurons to provide $L_{\mathrm{c}}$, N, l, $2a$, c, and f for the FFNN network. Each FFNN consisted of one neuron in the output layer in order to separately predict $t_{\mathrm{d}}$, $S_{21}$, or $S_{11}$. The prediction of $t_{\mathrm{d}}$ was performed with a (6-9-11-1) structure for the FFNN network, where there were two hidden layers with approximately 9 and 11 neurons. The (6-9-11-1) structure of the FFNN network for the prediction of $t_{\mathrm{d}}$ is provided in Figure 3.

The prediction of $S_{21}$ was performed with a (6-9-16-1) structure due to the completely different dataset used in comparison with $t_{\mathrm{d}}$. The same (6-9-16-1) FFNN structure was used to predict $S_{11}$.

The optimal structure of the FFNN network was discovered by evaluating the smallest MSE of the validation data. The selected structure for the FFNN for the prediction of $t_{\mathrm{d}}$ yielded $2.23\times {10}^{-5}$ for the MSE value. The selected structures for the FFNN for the prediction of $S_{21}$ and $S_{11}$ yielded MSEs of $0.086$ and $8.618$, respectively.

The structure of each FFNN were selected by taking into account the recommendation to use systematic experimentation in order to discover the appropriate structure for the neural network for each dataset. The determination of the number of hidden layers and the number of neurons in each hidden layer was conducted empirically by changing the number of hidden layers from 1 to 2, with the number of neurons in each hidden layer ranging from 0 to 21.

During the empirical investigation of different training methods, the Levenberg–Marquardt training method was chosen for further research. The Levenberg–Marquardt method is especially effective for training feedforward neural networks with a few hidden layers and is commonly used for regression problems. The MSE was employed as the loss function. Evaluation of the training methods for the FFNN network was performed with test data.

2.4. Structure of the Time Delay Neural Network

Due to the generalization and model flexibility features, the time delay neural network (TDNN) is capable of generalizing from historical data, allowing it to make predictions even when the exact input–output relationships are complex and nonlinear. The buffer size, which is the size of the input sequence window, can have an impact on the accuracy of the TDNN’s predictions. The size of the buffer determines the amount of historical data that the TDNN can take into account when making predictions. It is especially important when characteristics vary dynamically.

The general TDNN equation is presented in Equation (3):

$$Y\left(t\right)=f(x\left(t\right),x(t-\tau ),\cdots ,x(t-N_d\times \tau ),w),$$

(3)

where $x=[x_1,x_2,\cdots x_m]$ represents the vector of the input signals, $Y=[Y_1{Y}_2\cdots Y_n]$ represents the output vector of the output signals, $N_{\mathrm{d}}$ represents the total number of the delay steps, $\tau$ represents the time delay parameter, and w represents the internal weight of the neural network. Here, m is the number of input signals, and n is the number of output signals.

The base of the TDNN network consists of the previously discussed FFNN network. The same suitable FFNN structures were used in the TDNN network. The (6-9-11-1) structure was used for the prediction of $t_{\mathrm{d}}$. The $S_{11}$ and $S_{21}$ characteristics were predicted with the (6-9-16-1) structure. The presented TDNN predicted the $t_{\mathrm{d}}$ characteristic (Figure 4), where x represents the vector of input parameters $x=[L_{\mathrm{c}},N,l,2a,c,f]$ and Y represents the vector of $Y=\left[t_{\mathrm{d}}\right]$ in the output. In this particular case, $m=6$ and $n=1$.

The only difference in this case is that the TDNN had time buffers, and the input vector with the parameters $x=\left[L_{\mathrm{c}}Nl2acf\right]$ would be provided to the network repeatedly in different time steps. The size of the time delay buffer $N_{\mathrm{d}}$ would vary during the investigation in order to obtain more accurate predictions of $S_{21}$, $S_{11}$, and $t_{\mathrm{d}}$. The same dataset would be used for the FFNN was used here.

During training, the MSE would be calculated in order to analyze the influence of the time delay buffers. Verification would be performed with the test dataset. The range of the test data was within the range of the training data. The predictions of $S_{21}$, $S_{11}$, and $t_{\mathrm{d}}$ using the TDNN and FFNN would be compared.

2.5. Structure of the Convolutional Neural Network

In this subsection, the convolutional neural network (CNN) model is investigated for the analysis and synthesis of meander structures. The presented CNN model was used to predict the electromagnetic (EM) responses of the meander structure and speed up the design process. The layering architecture enables CNNs to learn the underlying nonlinear relationships effectively, making these models particularly adept at handling the complexities in microwave device data for more accurate predictions. A prediction was made by mapping the relationship between the geometric dimensions of the meander structure and its EM responses. In total, three separate CNN models were trained for each EM response. All three models shared the same structural architecture and training settings. This consistency in structure and settings provided a uniform framework which allowed for comparative analysis and validation.

The designed CNN consisted of a total of 18 layers, including four convolutional layers, each followed by a batch normalization layer and an activation layer of rectified linear units (ReLUs). The first two ReLU layers were further linked to the average pooling layers with a size of 2 and a stride of 2, reducing the spatial dimensions of the feature maps while improving the network’s ability to extract and generalize nonlinear EM responses as input data. The final ReLU layer led to a dropout layer with a probability of 10%, which helped prevent overfitting by randomly omitting some network connections during training. This architecture also included a fully connected layer which integrated the learned features, culminating in a regression output layer designed to predict 397 response values. The network had a total of 1.6 million learning parameters, allowing it to effectively capture and predict the characteristics of nonlinear microwave devices. The architecture of the investigated model is shown in Figure 5.

The model designed for this application began with an input layer which accepts images $64\times 64$ pixels in size. The first convolutional layer had 4 filters, the second had 8 filters, and the third and fourth layers each had 16 filters. These filters were $3\times 3$ in size, ideal for capturing the fine details in an image without losing spatial resolution. To maintain the output size, padding was applied in each convolutional layer, ensuring that the dimension of the output feature map remained unchanged. The stride, which defines the step size of the filter as it moves through the image, was set to 1. The dilation factor was also set to $1\times 1$, indicating that there was no dilation in the convolution process.

The size of the mini-batch was set to 128. The network was trained over 200 epochs. A key aspect of training was the initial learning rate, established to be $0.0001$. A piecewise learning rate schedule was implemented. Furthermore, the learning rate drop factor was set to $0.1$, and the learning rate drop period was in all 20 epochs, helping to gradually accelerate the convergence of the network to optimal performance. The MSE was employed as the loss function.

In this study, the number of filters used in the investigated CNN architecture was notably lower compared with typical CNN structures. This reduction was primarily attributed to the nature of the input data, consisting of grayscale images, which inherently possess less complexity than their colored counterparts as they contain only intensity information, rather than the additional color channel data present in RGB images. Consequently, a lower number of filters was sufficient to capture the essential features and patterns in the grayscale images, as there was a reduced requirement to process and distinguish between multiple color channels [47]. This simplification allowed us to design a more streamlined network with a reduced number of filters while maintaining effective feature extraction and learning capabilities.

3. Results

In this section, the prediction results for $t_{\mathrm{d}}$, $S_{11}$, and $S_{21}$ will be presented with three different solutions using artificial neural networks. The prediction results will be compared with the results obtained with the Sonnet© commercial software package, which is based on the MoM method. The neural network methods are also compared with the well-known ensemble learning technique of least-squares boosting (LSBoost), which comprises an ensemble of boosted regression trees. The experimentation and evaluation were conducted utilizing the implementation available in MATLAB.

The quality of the prediction was evaluated based on the mean absolute error (MAE) with a standard deviation (SD). Training and prediction with ANNs was performed using a stationary computer with an NVIDIA GeForce RTX 4090 graphics card and an Intel Core i7-7700 CPU operating at 3.60 GHz supported by 16 GB of RAM.

The dataset of 369 structures was collected for the investigation, where 20% of the overall dataset was left for testing, constituting 74 unique structures. For visual purposes, and to be easy to read, the results for $t_{\mathrm{d}}$, $S_{11}$, and $S_{21}$ for four unique meander structures, whose geometrical dimensions are in

Table 1. Configurations of meander structures which were used for result visualization.

Title of Structure	${\mathit{L}}_{\mathbf{c}}$ (mm)	l (mm)	$2\mathit{a}$ (mm)	c (mm)	N
MS1	0.4	0.2	10	6	14
MS2	0.55	0.45	10	6	14
MS3	0.9	0.7	10	6	14
MS4	0.6	1.45	10	6	14

, are presented graphically. The average prediction results, represented as the MAE with the SD for all 74 unique meander structures, are summarized in

Table 2. Prediction accuracy of $t_{\mathrm{d}}$, $S_{21}$, and $S_{11}$, represented by MAE with SD values.

Method	${\mathit{t}}_{\mathbf{d}}$ (ps)	${\mathit{S}}_{21}$ (dB)	${\mathit{S}}_{11}$ (dB)
LSBoost	5.1 ± 7.2	0.38 ± 0.44	1.94 ± 1.89
FFNN	3.3 ± 1.5	0.64 ± 0.42	2.47 ± 1.55
TDNN when $N_{\mathrm{d}}$ = 8	0.9 ± 0.64	0.11 ± 0.12	1.63 ± 1.71
CNN	27.5 ± 3.5	0.44 ± 0.46	1.36 ± 0.72 dB

The text in bold represents the best neural network option for prediction of $t_{\mathrm{d}}$, $S_{11}$ and $S_{21}$ characteristics.

.

These visual representations contribute to a nuanced understanding of how changes in a meander structure impact a system’s frequency-dependent response, aiding in the interpretation and optimization of the temporal characteristics in the studied phenomena.

3.1. Results Obtained with Feedforward Neural Network

The results for $t_{\mathrm{d}}$, $S_{11}$, and $S_{21}$ are presented in Figure 6. The prediction results for the $t_{\mathrm{d}}$ characteristic in four meander structures, labeled as MS1, MS2, MS3, and MS4, were obtained with the (6-9-11-1) structure for the FFNN network, and they are presented in Figure 6a. The prediction results for the $t_{\mathrm{d}}$ characteristic were compared with the calculation results when using the MoM method in the Sonnet© software package Version 18.

First of all, it should be mentioned that the $t_{\mathrm{d}}$ characteristic is an important parameter for meander structures in the field of signal synchronization indicating the possible signal delay in the operating frequency range. The delay time characteristic depends on the frequency. Therefore, it is important to perform the analysis across the entire selected operating frequency range.

The comparison showed good agreement between the results calculated with Sonnet© and predicted with an FFNN network. The average MAE between the predicted and simulated results of the 74 tested MSs did not exceed $3.3\pm 1.5$ ps (Table 2), or approximately 0.3% when taking into account that the desired delay time was to the order of 1 ns. The average MAE of $t_{\mathrm{d}}$ was even smaller if only the bandwidth from 0 until approximately 3 GHz was analyzed, being equal to $3\pm 1.5$ ps. Meanwhile, the $t_{\mathrm{d}}$ values for four MSs are presented in Figure 6a for visual analysis. For example, the maximum peak error value of MS1 was equal to 20 ps at 5.4 GHz.

The $S_{21}$ parameter represents information about a meander structure’s transmission efficiency, frequency, and phase responses. It is extremely important to understand the frequency-dependent behavior of meander structures in order to ensure optimal performance of microwave devices. The prediction results for $S_{21}$ were obtained with the (6-9-16-1) structure for the FFNN network. The average MAE of $S_{21}$ did not exceed $0.64\pm 0.42$ dB in the overall investigated 0–10 GHz frequency range (Table 2). The cutoff frequency of the bandwidth varied depending on the meander structure (MS1 = 3.78 GHz, MS2 = 3.65 GHz, MS3 = 3.05 GHz, and MS4 = 2.9 GHz; see Figure 6b). For example, the maximum peak error value of MS1 was equal to 5.36 dB at 9.15 GHz.

The largest average MAE was obtained by predicting the $S_{11}$ results. The $S_{11}$ parameter represents the impedance matching in a meander structure, which affects reflections from the meander structure’s ports and periodical inhomogeneities. The FFNN structure for the prediction of $S_{11}$ was the same as that for $S_{21}$ and was equal to (6-9-16-1). For clarity in the results, in order to avoid a huge overlap for the curves, the $S_{11}$ results for MS1 and MS2 (Figure 6c) and for MS3 and MS4 (Figure 6d) are presented in separate figures.

The average MAE of $S_{11}$ for all 74 MSs did not exceed $2.47\pm 1.55$ dB (Table 2). The largest MAE was obtained in the passband due to the greater variation in $S_{11}$, influenced by the reflection of the signal from the output of the meander structure. The maximum peak error value of MS1 was equal to 14.55 dB at 4.03 GHz (Figure 6c).

3.2. Results Obtained with Time Delay Neural Network

The prediction results for $t_{\mathrm{d}}$, $S_{11}$, and $S_{21}$ were carried out with all 74 structures’ test datasets. The average MAE with SD values for the predicted $t_{\mathrm{d}}$, $S_{11}$, and $S_{21}$ characteristics when the size of the delay buffer $N_{\mathrm{d}}$ = 8 are presented in Table 2. For clarity of the results, to avoid a huge overlap in the curves, only the prediction results for MS2 with different numbers of delay buffers are provided graphically (Figure 7). The chosen size of the delay buffer $N_{\mathrm{d}}$ was 1, 2, 4, or 8.

The prediction results for $t_{\mathrm{d}}$ for MS2 which were obtained with the (6-9-16-1) structure for the TDNN network are presented in Figure 7a. The MAE varied according to the chosen size $N_{\mathrm{d}}$ of the delay buffer. The TDNN worked as the FFNN network did with the size $N_{\mathrm{d}}$ = 1, and the MAE was equal to $2.64\pm 2.35$ ps. The delay buffer $N_{\mathrm{d}}$ = 2 allowed reducing the MAE to $1.91\pm 1.73$ ps. Here, $N_{\mathrm{d}}$ = 4 until $1.4\pm 1.25$ ps for the MAE, and $N_{\mathrm{d}}$ = 8 until $0.9\pm 0.64$ ps for the MAE. For example, the maximum peak error value of $t_{\mathrm{d}}$ changed from 12.4 ps to 3.8 ps when the delay buffer size $N_{\mathrm{d}}$ increased from 1 to 8, respectively.

The same effect was observed when predicting the $S_{21}$ characteristic for MS2 (Figure 7b). The MAE of $S_{21}$ with a delay buffer $N_{\mathrm{d}}$ = 1 was equal to $0.59\pm 0.68$ dB. The larger delay buffer allowed reducing the MAE for $N_{\mathrm{d}}$ = 2–to $0.22\pm 0.31$ dB, $N_{\mathrm{d}}$ = 4–to $0.16\pm 0.19$ dB, and $N_{\mathrm{d}}$ = 8–to $0.11\pm 0.12$ dB. The maximum peak error value of $S_{21}$ decreased from 4.8 dB to 1 dB, with $N_{\mathrm{d}}$ increasing from 1 to 8.

The effect of the size of the delay buffer $N_{\mathrm{d}}$ was more noticeable when predicting $S_{11}$ for MS2 (Figure 7c). The MAE varied as follows: $N_{\mathrm{d}}$ = 1–$2.47\pm 2.13$ dB; $N_{\mathrm{d}}$ = 2–$2.24\pm 2.09$ dB; $N_{\mathrm{d}}$ = 4–$2.10\pm 2.01$ dB; and $N_{\mathrm{d}}$ = 8–$1.63\pm 1.71$ dB. The maximum peak error value of $S_{11}$ decreased from 10.97 dB to 9.07 dB, with $N_{\mathrm{d}}$ increasing from 1 to 8.

The summarized prediction results of the TDNN for all 74 unique structures are presented in Table 2 for when the size of the delay buffer was equal to $N_{\mathrm{d}}$ = 8.

3.3. Results Obtained with Convolutional Neural Network

A visual representation of the predictions of $t_{\mathrm{d}}$, $S_{21}$, and $S_{11}$ using a CNN is presented in Figure 8. The dashed lines represent the predicted characteristics, while the straight lines represent the target characteristics. The prediction results were obtained by training three individual CNN models. For each figure, four prediction results from the test set are visualized. The meander structures and their indexes in the figures were the same as in the previous subsections and are given in Table 1. CNN training depends mainly on the batch size and maximum epoch number, and for this research, the training of a CNN for a single characteristic lasted approximately 90 s, and the prediction time was approximately $0.13$ s.

Figure 8a provides an overview of the frequency-dependent behavior in four configurations of meander structures, detailing the characteristic $t_{\mathrm{d}}$ for each setting. Visually, it can be observed that the predicted characteristics differed to the order of hundreds of nanoseconds when working within the 1 ns range. The MAE in the prediction of the characteristic $t_{\mathrm{d}}$ in the entire investigated frequency band when using the test set of the 74 unique testing structures was $27.5\pm 3.5$ ps (Table 2) and $27.7\pm 9$ ps for the displayed characteristics in Figure 8a.

Figure 8b provides an overview of the frequency-dependent behavior, detailing the characteristic $S_{21}$ for each setting of the meander structure in the 0–10 GHz frequency range. Visually, it can be observed that there were minimal differences between the actual and predicted signals in the bandwidth, ranging from 0 to approximately 3 GHz. The MAE in the prediction of the characteristic $S_{21}$ for all investigated frequencies when using the test set was $0.44\pm 0.46$ dB (Table 2), and it was $0.4\pm 0.31$ dB for the displayed characteristics in Figure 8b.

Figure 8c provides an overview of the frequency-dependent behavior, detailing the characteristic $S_{11}$ for each setting of the meander structure. In the figure, a dynamically changing region is highlighted. When comparing the predicted and actual values of the characteristic $S_{11}$, particularly when $S_{11}$ was close to 0 dB, the differences between the two sets of values were minimal. The MAE in the prediction of the characteristic $S_{11}$ throughout the investigated frequency band when using the test set was $1.36\pm 0.72$ dB (Table 2), and it was $1.2\pm 1.18$ dB for the displayed characteristics in Figure 8c,d.

The summarized prediction accuracy results in terms of MAE with an SD for $t_{\mathrm{d}}$, $S_{21}$, and $S_{11}$ in the overall 74 MS training set are presented in Table 2.

The MAEs with the SDs for $t_{\mathrm{d}}$, $S_{21}$, and $S_{11}$ were also compared with those for the baseline LSBoost method. The best $t_{\mathrm{d}}$ and $S_{21}$ prediction results were achieved with the TDNN network when $N_{\mathrm{d}}$ = 8. The best $S_{11}$ prediction results were achieved with the CNN network.

4. Discussion

Meander structures can be used in many different microwave devices to delay the signal or to increase the signal path while keeping or reducing the occupied area. The main parameters characterizing meander structures are $S_{21}$ (transmission coefficient), $S_{11}$ (reflection coefficient), and $t_{\mathrm{d}}$ (delay time). Usually, meander structures are analyzed using commercial software based on numerical methods. On the other hand, numerical methods are time-consuming, and calculations using numerical methods can last hours or even days. Therefore, three different intelligent methods were used to replace traditional numerical methods with methods based on neural networks. The obtained prediction results were compared with numerical results from Sonnet© and also the baseline machine learning LSBoost technique.

The FFNN is already widely used in the synthesis and analysis of microwave devices. However, each new dataset for every different MS is unique and requires an individual FFNN network structure to achieve the best characteristic prediction results. Quite different dynamics for $t_{\mathrm{d}}$, $S_{21}$, and $S_{11}$ resulted in the requirement of predicting each parameter with a separately trained FFNN network.

The average MAE with its SD did not exceeded $3.3\pm 1.5$ ps for $t_{\mathrm{d}}$ in the 0–10 GHz frequency range. If only the passband were considered in the 0–3 GHz frequency range, then the size of the MAE would decrease even more to $3\pm 1.5$ ps. The MAEs of the individual characteristics without averaging did not exceed 20 ps.

The predictions for $S_{21}$ and $S_{11}$ were slightly worse but insignificant from the perspective of the design of microwave devices. The MAE of $S_{21}$ did not exceed $0.64\pm 0.42$ dB in the overall frequency range. The average MAE of $S_{11}$ did not exceed $2.47\pm 1.55$ dB. The maximum peak errors of MS1 for $S_{21}$ and $S_{11}$ were equal to 5.36 dB and 14.55 dB, respectively. The obviously worse prediction results were obtained due to the nature of the characteristics of $S_{11}$. The reflections from the output port were clearly visible in the bandwidth, and as a result, the fluctuation in $S_{11}$ was quite fast. The worst prediction results for $S_{21}$ and $S_{11}$ were comparable to $t_{\mathrm{d}}$ and could already be expected in the FFNN training stage when the calculated MSE values were equal to $0.086$ and $8.618$, respectively, while the MSE of $t_{\mathrm{d}}$ was equal to $2.23\times {10}^{-5}$.

It could be seen that the FFNN predicted the desired MS characteristics accurately enough to design and analyze an MS (Figure 6). For example, the error margin of 3.3 ps while working in the range of ns is highly desirable. On the other hand, the predictions with the FFNN were worse in the section where the characteristics changed quickly, such as the 7–8 GHz or 8.5–9.5 GHz range of frequencies for $S_{21}$. For these reasons, the time delay neural network was also investigated.

The TDNN is structurally similar to the FFNN, except that the TDNN has delay buffers in the input. These delay buffers allow the TDNN to consider information from previous time steps. The base of the TDNN consists of the previously discussed FFNN. The same FFNN structures were used in the TDNN for predicting $t_{\mathrm{d}}$, $S_{21}$, and $S_{11}$. Regarding the TDNN, the MAEs of the predictions for $t_{\mathrm{d}}$, $S_{21}$, and $S_{11}$ for MS2 are presented, taking into account the different delay buffer size $N_{\mathrm{d}}$ of the TDNN (Figure 7).

The results show (Figure 7a) that by increasing the delay buffer from 1 to 8, the MAE of $t_{\mathrm{d}}$ could be reduced from $2.64\pm 2.35$ ps to $0.9\pm 0.64$ ps. An obviously lower MAE in fast-changing regions can be seen from $S_{21}$ (Figure 7b). The MAE of $S_{21}$ decreased from $0.59\pm 0.68$ dB to $0.11\pm 0.12$ dB. It can also be seen that the TDNN with $N_{\mathrm{d}}$ = 8 already predicted the fast-changing characteristics of $S_{11}$, representing the reflection of the MS’s output (Figure 7c). No larger delay buffer size was attempted, as the current MAE margin was sufficient for MS analysis. In addition, the present results clearly demonstrate the advantages of the TDNN.

The third CNN model was used to investigate the possible mapping between an MS’s images and its electrodynamical characteristics. CNN models are rarely used in the analysis of microwave devices. In this study, we focused on establishing a basic understanding of the application of CNNs in order to predict the electrodynamical characteristics of various meander structures. Using simple CNN models was a deliberate strategy aimed at unraveling the fundamental aspects of the problem, particularly considering the relatively modest size of the dataset. The results obtained using the CNN model architecture showed low MAEs for $S_{21}$ and $S_{11}$ compared with the other neural networks investigated. Furthermore, relatively high standard deviation values were observed, which shows a need for additional post-processing of the signals. In future research, it would be appropriate to explore more advanced CNN architectures, taking into account the specific challenges associated with our task and the modest size of our dataset.

Calculations using the MoM method for a particular MS took approximately 120 min on average. The predictions using an FFNN or TDNN took approximately less than $0.1$ s. The prediction with a CNN took approximately $0.4$ s on average. On the other hand, the training times of all three characteristics using the FFNN, TDNN, and CNN took approximately 3 min, 3.5 min, and 4.5 min, respectively.

5. Conclusions

The scientific novelty of this article is the collected dataset of different sets of geometrical dimensions, top view images, and associated eletrodynamical $t_{\mathrm{d}}$, $S_{21}$ and $S_{11}$ characteristics of meander structures, as well as three artificial neural network models (FFNN, TDNN, and CNN) applied for the prediction of the characteristics of an MS.

The obtained results confirmed that the FFNN, TDNN and CNN artificial neural network models are suitable for predicting the electrodynamical parameters of meander structures. The calculated MAEs with SD margins are sufficient for the analysis of MSs.

The usage of ANNs has allowed the real time analysis of a meander structure since the prediction time is reduced to a margin of 0.4 s, compared with 120 min of calculations using traditional numerical methods.

The selection of parameter variation ranges for the training dataset is extremely important for determining the prediction limits of artificial neural networks. In our particular case, the meander strip width and the width of the gap between adjacent conductors varied. Therefore, the selected artificial neural networks could only predict within the limits of the chosen parameters. Other geometric dimensions were held constant but were added to the network training for future plans. The cross-sectional dimensions of the MS and the properties of the materials used for the design of the MS were not included in this study.

Future plans include expanding the training dataset by gathering all important geometric parameters for the MS while also incorporating the cross-sectional geometrical parameters of the MS and the properties of the used materials.