*Article* **An Inverse Design Framework for Isotropic Metasurfaces Based on Representation Learning**

**Jian Zhang, Jin Yuan \*, Chuanzhen Li and Bin Li**

State Key Laboratory of Media Convergence and Communication, Communication University of China, Beijing 100024, China; zzjj1818@cuc.edu.cn (J.Z.); lichuanzhen@cuc.edu.cn (C.L.); libin08@cuc.edu.cn (B.L.) **\*** Correspondence: yuanjin@cuc.edu.cn

**Abstract:** A hybrid framework for solving the non-uniqueness problem in the inverse design of isomorphic metasurfaces is proposed. The framework consists of a representation learning (RL) module and a variational autoencoder-particle swarm optimization (VAE-PSO) algorithm module. The RL module is used to reduce the complex high-dimensional space into a low-dimensional space with obvious features, with the purpose of eliminating the many-to-one relationship between the original design space and response space. The VAE-PSO algorithm first encodes all meta-atoms into a continuous latent space through VAE and then applies PSO to search for an optimized latent vector whose corresponding metasurface fulfills the target response. This framework gives the solution paradigm of the ideal non-uniqueness situation, simplifies the complexity of the network, improves the running speed of the PSO algorithm, and obtains the global optimal solution with 94% accuracy on the test set.

**Keywords:** metasurfaces; representation learning; variational autoencoder; inverse design

## **1. Introduction**

Metasurfaces are synthetic composites composed of subwavelength structures arranged in different geometric shapes and distribution functions, which have been successfully applied in spectrum filtering, focusing, holographic imaging, polarization conversion, and other fields in recent years [1]. The subwavelength scatters that make up the metasurfaces are called meta-atoms, which resemble atoms or molecules of natural materials. When a beam of electromagnetic (EM) waves hits a metasurface, it is the strong resonances or spatial orientations of meta-atoms that cause the phase mutation, so metasurfaces have the ability to control the phase, amplitude, and polarization of reflected/transmitted waves in space [2].

Cui et al. proposed the concept of digital metasurface in 2014, in which the meta-atom pattern is discretized and encoded, greatly expanding the design space of metasurfaces [3]. Different coding sequences are proposed on this basis to achieve various EM responses, while the difficulty of inverse design has also increased dramatically with the number of codes for a digital metasurface increasing. Although some evolutionary algorithm (EA) has advantages in terms of fast random search without a problem domain [4], plenty of primary parameter settings of EAs have a dramatic impact on the evolution procedure and convergence result, which might lead to a fall in locally optimal solutions.

Fortunately, the development of deep learning, which is based on artificial neural networks to create computer reasoning by simulating human learning patterns in massive data, has brought new solutions [5]. Deep learning is a completely data-driven approach that mines implicit rules and relationships by learning from enough data sets, which has shown great advantages in computer vision, natural language processing, knowledge graph, and other fields. The basic idea is to design an algorithm to find rules between input data and output data according to a set of given data. The mined rules are stored in the deep learning model as the parameters given. If the rules between input and output

**Citation:** Zhang, J.; Yuan, J.; Li, C.; Li, B. An Inverse Design Framework for Isotropic Metasurfaces Based on Representation Learning. *Electronics* **2022**, *11*, 1844. https://doi.org/ 10.3390/electronics11121844

Academic Editors: Naser Ojaroudi Parchin, Mohammad Ojaroudi and Raed A. Abd-Alhameed

Received: 13 May 2022 Accepted: 9 June 2022 Published: 10 June 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

remain unchanged, the model that has been trained can automatically and quickly predict its output as soon as another input is given. Since deep learning has its natural advantages in automatically mining undefined rules, we associate deep learning theory with metasurface inverse design to mine the relationship between the massive metasurface geometric parameters and response parameters. This could lead to a disruptive breakthrough in the field of metasurface design, skipping the physical interpretation and being faster and more efficient. Thus, there will be no professional theoretical requirements on designers so engineers are only required to pay attention to their practical demands instead of to the complicated design process and obscure physical theory [6]. The most difficult aspect of employing deep learning in the inverse design of metasurfaces is the non-uniqueness challenge [7]. This means that the expected EM response may correspond to multiple design patterns. Another challenge in using deep learning to design a complex meta-atom is the large size of the response and design spaces resulting in the need to train large neural networks [8], which means more parameters are required and networks are difficult to train. Adequate sampling points are required to achieve the desired EM response over a wide band, which typically results in thousands of data points in the response space. How to deal with the large design space and response space is of great importance.

In this article, we propose a new framework for meta-atom structure inverse design based on representation learning by addressing both the non-uniqueness issue and network-size issue. This framework consists of a representation learning (RL) module and a variational autoencoder-particle swarm optimization (VAE-PSO) algorithm module, which can achieve a high accuracy on the test data set. We use a sufficient number of data sets to train two different kinds of autoencoder, embedding high-dimensional space into low-dimensional space with a low loss rate, and the trained final network can output reduced design vectors (RDV) according to the target response. Then VAE-PSO algorithm is used to search for the optimal solution according to the RDV output by the RL module in the continuous space generated by VAE. Transforming the non-uniqueness issue into an optimal solution can greatly reduce the verification time and design cost. In contrast to earlier work, the most notable technique in this work is that the complex high-dimensional space is transformed into the characteristic low-dimensional space by the RL module, which gives the optimal solution to the non-uniqueness problem, greatly reduces the network complexity, and improves the running speed of the optimization algorithm enormously.

## **2. Theory and Design**

As different metasurfaces can achieve the same EM response, when the framework inputs the target response, which metasurface will be returned is the non-unique problem. In practical application, though different metasurfaces can generally produce a similar response, it is almost impossible to produce an identical response at each of the sampling points. Therefore, in the global design space, there must be a metasurface that could realize the response closest to the target response, which means a framework is expected to find the optimal solution. The overall framework flow chart is shown in Figure 1. The task of the RL module is to translate the target EM response into RDV through the representation learning network. Any target EM response corresponds to a unique RDV, while one RDV might correspond to more than one metasurface. To solve this many-to-one problem, the VAE-PSO module generates latent design space, where the PSO algorithm is applied to search globally for the optimal latent vector. The VAE decoder is used to decode the latent vector into the optimal metasurface.

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**Figure 1.** Flowchart of the hybrid framework. **Figure 1.** Flowchart of the hybrid framework.

#### *2.1. Metasurface Design 2.1. Metasurface Design 2.1. Metasurface Design*

Figure 2 illustrates the detailed structure of a meta-atom. The top pattern layer (Layer 1), with a side length *L*1, is evenly divided into 16 × 16 discrete cells, where '1' means copper and '0' means vacuum in digital coding. Thus, the digital metasurface can be modeled as a 16 × 16 binary design matrix, which has 28×8 possible patterns. Each discrete copper piece is a square patch with a thickness of *h*1 and a side length of *L*'1. The second layer is a dielectric layer (Layer 2) with a dielectric constant of = 2.65, and a loss tangent of tan = 0.001. The back layer (Layer 3) is the backing copper sheet. The thickness and the side length of Layer 2 and Layer 3 are *h*2, *h*3, *L*2, and *L*3, respectively. To reduce the influence of polarization, the object of our study is an isotropic metasurface composed of an 8 × 8 coding sequence as a subblock, which forms a 16 × 16 matrix through four-fold symmetry [9]. Figure 2 illustrates the detailed structure of a meta-atom. The top pattern layer (Layer 1), with a side length *L*1, is evenly divided into 16 × 16 discrete cells, where '1' means copper and '0' means vacuum in digital coding. Thus, the digital metasurface can be modeled as a 16 <sup>×</sup> 16 binary design matrix, which has 28×<sup>8</sup> possible patterns. Each discrete copper piece is a square patch with a thickness of *h*<sup>1</sup> and a side length of *L*'1. The second layer is a dielectric layer (Layer 2) with a dielectric constant of *ε<sup>r</sup>* = 2.65, and a loss tangent of tan *δ* = 0.001. The back layer (Layer 3) is the backing copper sheet. The thickness and the side length of Layer 2 and Layer 3 are *h*2, *h*3, *L*2, and *L*3, respectively. To reduce the influence of polarization, the object of our study is an isotropic metasurface composed of an 8 × 8 coding sequence as a subblock, which forms a 16 × 16 matrix through four-fold symmetry [9]. Figure 2 illustrates the detailed structure of a meta-atom. The top pattern layer (Layer 1), with a side length *L*1, is evenly divided into 16 × 16 discrete cells, where '1' means copper and '0' means vacuum in digital coding. Thus, the digital metasurface can be modeled as a 16 × 16 binary design matrix, which has 28×8 possible patterns. Each discrete copper piece is a square patch with a thickness of *h*1 and a side length of *L*'1. The second layer is a dielectric layer (Layer 2) with a dielectric constant of = 2.65, and a loss tangent of tan = 0.001. The back layer (Layer 3) is the backing copper sheet. The thickness and the side length of Layer 2 and Layer 3 are *h*2, *h*3, *L*2, and *L*3, respectively. To reduce the influence of polarization, the object of our study is an isotropic metasurface composed of an 8 × 8 coding sequence as a subblock, which forms a 16 × 16 matrix through four-fold symmetry [9].

In the simulation, boundary conditions and excitations are added to the metasurface **Figure 2.** Schematic illustration of a meta-atom structure. **Figure 2.** Schematic illustration of a meta-atom structure.

model. The calculation principle of the software is solved by Maxwell**'**s equations based on meshing, whose calculation time surges with the increase of model complexity. The design matrices of meta-atoms are selected as the input features and the S-parameter as the EM responses, in which data sets are randomly collected to train deep learning models. MATLAB to control CST STUDIO is used to generate digital metasurfaces, calculate S parameters, and save data sets automatically. *2.2. Representation Learning Module*  In the simulation, boundary conditions and excitations are added to the metasurface model. The calculation principle of the software is solved by Maxwell**'**s equations based on meshing, whose calculation time surges with the increase of model complexity. The design matrices of meta-atoms are selected as the input features and the S-parameter as the EM responses, in which data sets are randomly collected to train deep learning models. MATLAB to control CST STUDIO is used to generate digital metasurfaces, calculate S In the simulation, boundary conditions and excitations are added to the metasurface model. The calculation principle of the software is solved by Maxwell's equations based on meshing, whose calculation time surges with the increase of model complexity. The design matrices of meta-atoms are selected as the input features and the S-parameter as the EM responses, in which data sets are randomly collected to train deep learning models. MATLAB to control CST STUDIO is used to generate digital metasurfaces, calculate S parameters, and save data sets automatically.

#### The key to our framework is transforming design space and response space into low parameters, and save data sets automatically. *2.2. Representation Learning Module*

dimensional space by the RL module to solve the non-uniqueness problem and reduce network complexity [10]. First, the different spaces and their corresponding vectors need to be clearly defined: The original space includes the original design space (ODS) and *2.2. Representation Learning Module*  The key to our framework is transforming design space and response space into low dimensional space by the RL module to solve the non-uniqueness problem and reduce The key to our framework is transforming design space and response space into low dimensional space by the RL module to solve the non-uniqueness problem and reduce network complexity [10]. First, the different spaces and their corresponding vectors need to be clearly defined: The original space includes the original design space (ODS) and original

network complexity [10]. First, the different spaces and their corresponding vectors need to be clearly defined: The original space includes the original design space (ODS) and

response space (ORS), and the corresponding vectors are the original design vector (ODV) and original response vector (ORV). The reduced space includes reduced design space (RDS) and reduced response space (RRS), and the corresponding vectors are reduced design vector (RDV) and reduced response vector (RRV). As multiple sets of meta-atoms can result in the same EM response, it is desired to map to the same vectors in the reduced space so that RDS and RRS can form a one-to-one mapping controlled by a nonlinear function, and this process is invertible. In this way, we transfer the many-to-one relationship between ODS and ORS into the many-to-one relationship between ODS and RDS, which could be solved by the VAE-PSO module. Figure 3 illustrates the mapping relationship between different spaces, with the red line representing one-to-one mapping and the blue line representing many-to-one mapping. tor (ODV) and original response vector (ORV). The reduced space includes reduced design space (RDS) and reduced response space (RRS), and the corresponding vectors are reduced design vector (RDV) and reduced response vector (RRV). As multiple sets of meta-atoms can result in the same EM response, it is desired to map to the same vectors in the reduced space so that RDS and RRS can form a one-to-one mapping controlled by a nonlinear function, and this process is invertible. In this way, we transfer the many-toone relationship between ODS and ORS into the many-to-one relationship between ODS and RDS, which could be solved by the VAE-PSO module. Figure 3 illustrates the mapping relationship between different spaces, with the red line representing one-to-one mapping and the blue line representing many-to-one mapping.

original response space (ORS), and the corresponding vectors are the original design vec-

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**Figure 3.** The relationship between different spaces. The red line represents one-to-one mapping and the blue line represents many-to-one mapping. **Figure 3.** The relationship between different spaces. The red line represents one-to-one mapping and the blue line represents many-to-one mapping.

Representation learning is a machine learning technology that extracts effective features from raw data. Its essence is a dimensionality reduction technology, which can be realized by random forest, principal component analysis, autoencoder (AE), etc. In this article, the RL module uses AE, which is composed of an encoder and a decoder that can be used separately. The back propagation algorithm is used to continually train AE in order to minimize the cost function during the training process. Equations (1)–(3) illus-Representation learning is a machine learning technology that extracts effective features from raw data. Its essence is a dimensionality reduction technology, which can be realized by random forest, principal component analysis, autoencoder (AE), etc. In this article, the RL module uses AE, which is composed of an encoder and a decoder that can be used separately. The back propagation algorithm is used to continually train AE in order to minimize the cost function during the training process. Equations (1)–(3) illustrate features *<sup>h</sup>*, reconstruct data *<sup>x</sup>*<sup>e</sup> and cost function *<sup>L</sup>*.

$$h = \sigma(\mathcal{W}\_{\text{enc}}\mathfrak{x} + b\_{\text{enc}}) \tag{1}$$

$$
\widetilde{\mathfrak{X}} = \sigma(\mathcal{W}\_{\text{dec}}h + b\_{\text{dec}}) \tag{2}
$$

(1)

(2)

(3)

$$L = \frac{1}{N} \sum\_{i} \left\| \mathbf{x}\_{i} - \widetilde{\mathbf{x}}\_{i} \right\|\_{2}^{2} \tag{3}$$

where *x* represents the input data, *W*, *b* represents the transformation matrix and bias of the encoder and decoder, and *N* represents the amount of training data. *σ* is the active function. The dimension of *h* should be less than *x* in order to achieve feature dimension reduction.

where *x* represents the input data, *W*, *b* represents the transformation matrix and bias of the encoder and decoder, and *N* represents the amount of training data. *σ i*s the active function. The dimension of *h* should be less than *x* in order to achieve feature dimension reduction. Figure 4 illustrates the design process of AEs. The dimension of the ORS is reduced by training the autoencoder shown in Figure 4a. S11 between 0–20 GHz is selected as the Figure 4 illustrates the design process of AEs. The dimension of the ORS is reduced by training the autoencoder shown in Figure 4a. S11 between 0–20 GHz is selected as the EM response of this study, with a sampling frequency of 0.02 GHz and a length of 1000 dimensions. Subsequently, a pseudo-autoencoder (the input space is different from the output space) in Figure 4b is used to train the mapping between ODS and RDS, RDS and RRS together. The mapping of the former is many-to-one, and that of the latter is one-to-one. The one-to-one relationship between RDS and RRS is trained by a multi-layer perceptron (MLP). The decoder uses the "decoder1" trained in Figure 4a.

EM response of this study, with a sampling frequency of 0.02 GHz and a length of 1000 dimensions. Subsequently, a pseudo-autoencoder (the input space is different from the output space) in Figure 4b is used to train the mapping between ODS and RDS, RDS and RRS together. The mapping of the former is many-to-one, and that of the latter is one-to-

ceptron (MLP). The decoder uses the "decoder1" trained in Figure 4a.

**Figure 4.** (**a**) Network 1, reduce the dimension of the ORS. (**b**) Network 2, used to forward prediction. (**c**) Network 3, RL module's ultimate network. **Figure 4.** (**a**) Network 1, reduce the dimension of the ORS. (**b**) Network 2, used to forward prediction. (**c**) Network 3, RL module's ultimate network.

When the two reduced spaces are completed by a pseudo-autoencoder, an effective forward prediction model can be formed. It maps different design matrices that can achieve the same EM response to the same RDV, then maps them to the RRS one-to-one through MLP, and finally decodes ORV of 1000 dimensions. Subsequently, the inverse design model is built, and the inverse of the MLP should be found in Figure 4b. As demonstrated in Figure 4c, an inverse design network composed of the encoder1 in Figure 4a When the two reduced spaces are completed by a pseudo-autoencoder, an effective forward prediction model can be formed. It maps different design matrices that can achieve the same EM response to the same RDV, then maps them to the RRS one-to-one through MLP, and finally decodes ORV of 1000 dimensions. Subsequently, the inverse design model is built, and the inverse of the MLP should be found in Figure 4b. As demonstrated in Figure 4c, an inverse design network composed of the encoder1 in Figure 4a and the inverse structure of the MLP in Figure 4b is established. It can generate unique RDV by inputting a 1000-dimensional ORV.

and the inverse structure of the MLP in Figure 4b is established. It can generate unique RDV by inputting a 1000-dimensional ORV. The internal structures of encoders and decoders are various. A convolutional neural network (CNN) and a fully connected layer (FCL) are utilized in this work. Network 1 in Figure 4a is an autoencoder that reduces ORV of 1000 dimensions. If all FCL is adopted, the network scale and network parameters will be greatly increased, which is difficult to train. Thus, a hybrid network of one-dimensional convolution layer (Conv1D), pooling layer, and FCL are adopted in encoder1, and FCL is used in decoder1 in Figure 5a. Network 2 in Figure 4b is a pseudo-autoencoder, which transforms the design matrix into a reduced design vector and established a one-to-one mapping with the reduced response vector using MLP. The aim of network 2 is to train an encoder that can effectively compress an original design matrix to the RDV and train a MLP that connects two reduced spaces. Well-trained network 2 is a forward prediction network. CNN shows good feature extraction effect when processing 2D raster data [11], so our encoder2 uses a two-dimen-The internal structures of encoders and decoders are various. A convolutional neural network (CNN) and a fully connected layer (FCL) are utilized in this work. Network 1 in Figure 4a is an autoencoder that reduces ORV of 1000 dimensions. If all FCL is adopted, the network scale and network parameters will be greatly increased, which is difficult to train. Thus, a hybrid network of one-dimensional convolution layer (Conv1D), pooling layer, and FCL are adopted in encoder1, and FCL is used in decoder1 in Figure 5a. Network 2 in Figure 4b is a pseudo-autoencoder, which transforms the design matrix into a reduced design vector and established a one-to-one mapping with the reduced response vector using MLP. The aim of network 2 is to train an encoder that can effectively compress an original design matrix to the RDV and train a MLP that connects two reduced spaces. Well-trained network 2 is a forward prediction network. CNN shows good feature extraction effect when processing 2D raster data [11], so our encoder2 uses a two-dimensional convolution layer (Conv2D) as the main network. The specific network structure is shown in Figure 5b. In this article, we collected 30,480 samples, among them, 21,366 were used for the training set, 6096 for the validation set, and 3018 for the test set. The training of these networks needs to strictly follow the order shown in Figure 5.

of these networks needs to strictly follow the order shown in Figure 5.

sional convolution layer (Conv2D) as the main network. The specific network structure is shown in Figure 5b. In this article, we collected 30,480 samples, among them, 21,366 were used for the training set, 6096 for the validation set, and 3018 for the test set. The training

**Figure 5.** (**a**) The internal structure of encoder1, decoder1. (**b**) The internal structure of encoder2. **Figure 5.** (**a**) The internal structure of encoder1, decoder1. (**b**) The internal structure of encoder2.

#### *2.3. Variational Autoencoder-Particle Swarm Optimization Module*

*2.3. Variational Autoencoder-Particle Swarm Optimization Module*  Although the final inverse design network can generate the required RDV, we still cannot find all the design matrices from the RDV due to the one-to-many relationship. Theoretically, a RDV may correspond to a multiple design matrix due to the non-uniqueness problem, and there is no suitable deep learning model that can carry out one-to-many mapping. In recent years, optimization algorithms have been widely used in the design of metasurfaces, but their running speed is slow due to the huge computational cost of original space [12]. Since the RL module has mapped the inverse design problem to a re-Although the final inverse design network can generate the required RDV, we still cannot find all the design matrices from the RDV due to the one-to-many relationship. Theoretically, a RDV may correspond to a multiple design matrix due to the non-uniqueness problem, and there is no suitable deep learning model that can carry out one-to-many mapping. In recent years, optimization algorithms have been widely used in the design of metasurfaces, but their running speed is slow due to the huge computational cost of original space [12]. Since the RL module has mapped the inverse design problem to a reduced space, a Particle Swarm Optimization (PSO) is applied in this low dimensional space with the purpose to improve the speed of the algorithm.

duced space, a Particle Swarm Optimization (PSO) is applied in this low dimensional space with the purpose to improve the speed of the algorithm. First, the global vector space of the 16 × 16 meta-atom space is generated to identify the target vector effectively and expeditiously in a global manner. A generative model can well complete this task [13], for which VAE is chosen to complete the generation of global vector space. A VAE can convert high-dimensional discrete data to a low-dimensional continuous space known as the latent space, where any sampling point can be encoded as a meaningful output. The VAE training process requires continuous learning of conditional probability distributions of inputs given latent variables based on the input data. When VAE is trained with the available binary matrix of 30,000 different isotropic metasurfaces, a continuous low-dimensional design space can be obtained and the re-First, the global vector space of the 16 × 16 meta-atom space is generated to identify the target vector effectively and expeditiously in a global manner. A generative model can well complete this task [13], for which VAE is chosen to complete the generation of global vector space. A VAE can convert high-dimensional discrete data to a low-dimensional continuous space known as the latent space, where any sampling point can be encoded as a meaningful output. The VAE training process requires continuous learning of conditional probability distributions of inputs given latent variables based on the input data. When VAE is trained with the available binary matrix of 30,000 different isotropic metasurfaces, a continuous lowdimensional design space can be obtained and the reduced dimension is K [14]. Figure 6a illustrates an architecture of a VAE. Since the design matrix of the metasurface is a twodimensional grid shape, the V-encoder and V-decoder of VAE are implemented by a convolutional neural network, which is illustrated in Figure 6b,c respectively.

duced dimension is K [14]. Figure 6a illustrates an architecture of a VAE. Since the design matrix of the metasurface is a two-dimensional grid shape, the V-encoder and V-decoder of VAE are implemented by a convolutional neural network, which is illustrated in Figure

6b,c respectively.

**Figure 6.** (**a**) The structure of variational autoencoder. (**b**) The internal structure of V-encoder. (**c**) The internal structure of V-decoder. **Figure 6.** (**a**) The structure of variational autoencoder. (**b**) The internal structure of V-encoder. (**c**) The internal structure of V-decoder.

The encoder first converts the input binary matrix into mean vectors μ and standard deviation vectors σ. The latent vector *Z* is sampled from the Gaussian distribution ~ሺ, ଶሻ during the training process. After that, the decoder *G* reconstructs *Z* into the design matrix. Pattern topologies of meta-atom with comparable features are mapped to the same region of latent space in this process, and similar decoding meta-atoms are constantly modified by disrupting latent vectors. A well-trained decoder can recover any hidden vector sampled in the *Z* space into a binary matrix like the training set, which accomplishes the purpose of generating new samples. Equations (4)–(6) illustrate the loss func-The encoder first converts the input binary matrix into mean vectors µ and standard deviation vectors σ. The latent vector *Z* is sampled from the Gaussian distribution *Z* ∼ *N µ*, *σ* 2 during the training process. After that, the decoder *G* reconstructs *Z* into the design matrix. Pattern topologies of meta-atom with comparable features are mapped to the same region of latent space in this process, and similar decoding meta-atoms are constantly modified by disrupting latent vectors. A well-trained decoder can recover any hidden vector sampled in the *Z* space into a binary matrix like the training set, which accomplishes the purpose of generating new samples. Equations (4)–(6) illustrate the loss function of a VAE, denoted by *L*VAE.

$$L\_{VAE} = L\_{\rm rec} + L\_{KL} \tag{4}$$

$$L\_{\rm rec} = |\mathfrak{x} - \widetilde{\mathfrak{x}}|^2 \tag{5}$$

(4)

(5)

(6)

(7)

$$\begin{split} L\_{KL} &= \sum\_{k=1}^{K} KL[N(\mu\_k, \sigma\_k)\_\prime N(0, 1)] \\ &= \frac{1}{2} \sum\_{k=1}^{K} \left( \sigma\_k^2 + \mu\_k^2 - \ln \sigma\_k^2 - 1 \right) \end{split} \tag{6}$$

where *x*, represent input data and reconstruct data. *KL* refers to Kullback-Leibler diverwhere *<sup>x</sup>*, *<sup>x</sup>*<sup>e</sup> represent input data and reconstruct data. *KL* refers to Kullback-Leibler divergence. After training VAE, a new network *ϕ* can be trained to find many-to-one mappings between continuous design space *V* and RDV *D* in Figure 7a. As shown in Figure 7b, the many-to-one relation can be converted into a multi-solution problem in functions. This can be expressed in Equation (7):

7b, the many-to-one relation can be converted into a multi-solution problem in functions.

$$D = \varrho(Z)Z = \varrho^{-1}(D) \tag{7}$$

This can be expressed in Equation (7):

tion of a VAE, denoted by *L*VAE.

Therefore, if the objective RDV and the function expression of the inverse network in Figure 5a are known, the non-uniqueness problem can be solved. Practically, it is impossible to express the function fitted with the neural network. Simultaneously, it is difficult for different meta-atoms to achieve the same response curve in every sample point. Therefore, it is necessary to introduce an optimization algorithm to find the optimal latent vectors with the smallest distance from the target RDV *D*tar. Searching for the optimal solution to achieve the target response in the huge ODS will save a lot of verification work and greatly improve the inverse design efficiency of the metasurface. In this work, PSO was chosen as the optimization algorithm, which is implemented in Python using the

**Figure 7.** (**a**) A network for the VAE generated continuous latent vector into RDV. (**b**) The functional relationship between latent continuous vectors and RD. **Figure 7.** (**a**) A network for the VAE generated continuous latent vector into RDV. (**b**) The functional relationship between latent continuous vectors and RD.

Figure 8 illustrates the optimization process. The continuous latent vector space is mapped to the RDS through the network . The fitness function is obtained by calculating the mapped vector and the target vector in RDS, which can be written in Equation (8). The fitness function is the objective function of optimization. The condition at the end Therefore, if the objective RDV and the function expression of the inverse network in Figure 5a are known, the non-uniqueness problem can be solved. Practically, it is impossible to express the function fitted with the neural network. Simultaneously, it is difficult for different meta-atoms to achieve the same response curve in every sample point. Therefore, it is necessary to introduce an optimization algorithm to find the optimal latent vectors with the smallest distance from the target RDV *D*tar. Searching for the optimal solution to achieve the target response in the huge ODS will save a lot of verification work and greatly improve the inverse design efficiency of the metasurface. In this work, PSO was chosen as the optimization algorithm, which is implemented in Python using the PySwarms toolkit [15].

of the optimization iteration is , where is an arbitrarily small value based on training data. After optimization, the optimal latent design vector is returned and the binary matrix can be generated by V-decoder. To evaluate the performance of our framework Figure 8 illustrates the optimization process. The continuous latent vector space is mapped to the RDS through the network *ϕ*. The fitness function is obtained by calculating the mapped vector *D<sup>ϕ</sup>* and the target vector *Dtar* in RDS, which can be written in Equation (8). *Electronics* **2022**, *11*, x FOR PEER REVIEW 9 of 13

$$F(z) = \left| \Delta D \right| = \left| D\_{\text{tar}} - D\_{\varphi} \right|^2 \tag{8}$$

(8)

(9)

(10)

framework is calculated. By transforming the non-uniqueness problem into a global optimal solution problem in lower-dimensional space, a high average accuracy of 94% on test **Figure 8.** Flowchart of the VAE-PSO algorithm. **Figure 8.** Flowchart of the VAE-PSO algorithm.

PySwarms toolkit [15].

sets can be achieved in our framework. **3. Design Result and Analysis**  Initially, it is necessary to discuss whether network 1 learns the low-dimensional representation of ORS effectively. Network 1 is a typical AE for squeezing 1000 dimensional ORV down to 10 dimensions. Equation (10) illustrates the relative response error (RRE) to The fitness function is the objective function of optimization. The condition at the end of the optimization iteration is |∆*D*| < *ε*, where *ε* is an arbitrarily small value based on training data. After optimization, the optimal latent design vector is returned and the binary matrix can be generated by V-decoder. To evaluate the performance of our framework quantitatively [16], the accuracy of each target EM response is defined in Equation (9).

$$a = 1 - \frac{\int\_{f\_1}^{f\_2} \left| \mathcal{R}\_{\text{tar}} - \mathcal{R}\_{\text{gen}} \right| df}{\int\_{f\_1}^{f\_2} \left| \mathcal{R}\_{\text{tar}} \right| df} \tag{9}$$

where *f* <sup>1</sup> and *f* <sup>2</sup> are the frequency bounds of the input spectra, *Rtar* is the target EM response, *Rgen* is the generative response calculated by the generated design matrix from the VAE-PSO framework, which can be obtained directly using the trained network 2 in Figure 4b. The

**Figure 9.** (**a**,**b**) Diagram of input response and output response of network 1.

Then, the performance of network 2 should be discussed. The essence of network 2 is a forward prediction network, which will input the design matrix of the metasurface and output the predicted ORV. Its main role in the overall framework is to quickly predict the response curve such as S11 of the inverse-designed metasurface instead of using a traditional physical simulator, so as to calculate the accuracy of this framework rapidly. Prediction accuracy is usually used to describe the performance of a forward prediction

where is the discretized value of ORV, is the discretized value of the reconstruct

mensions of the S11 curve into network 1 and the average RRE is 3.72% in the test set. The results of the two examples randomly selected are shown in Figure 9a,b. These results prove that a perfect consistency between the input response and the output response has been established and it is feasible to use RRV extracted by AE to represent the high-di-

mensional ORV.

accuracy measures the matching degree between *Rtar* and *Rgen*. Applying Equation (9) to all test sets and then averaging them, the average accuracy of the entire framework is calculated. By transforming the non-uniqueness problem into a global optimal solution problem in lower-dimensional space, a high average accuracy of 94% on test sets can be achieved in our framework. **Figure 8.** Flowchart of the VAE-PSO algorithm. **3. Design Result and Analysis**  Initially, it is necessary to discuss whether network 1 learns the low-dimensional rep-

*Electronics* **2022**, *11*, x FOR PEER REVIEW 9 of 13

#### **3. Design Result and Analysis** resentation of ORS effectively. Network 1 is a typical AE for squeezing 1000 dimensional

Initially, it is necessary to discuss whether network 1 learns the low-dimensional representation of ORS effectively. Network 1 is a typical AE for squeezing 1000 dimensional ORV down to 10 dimensions. Equation (10) illustrates the relative response error (RRE) to measure the quality of the autoencoder in the test set. ORV down to 10 dimensions. Equation (10) illustrates the relative response error (RRE) to measure the quality of the autoencoder in the test set.

$$\text{RRE} = \left(\frac{\sqrt{\sum\_{i=1}^{n} (r\_i - g\_i)^2}}{\sqrt{\sum\_{i=1}^{n} r\_i^2}}\right) \times 100\% \tag{10}$$

where *r<sup>i</sup>* is the discretized value of ORV, *g<sup>i</sup>* is the discretized value of the reconstruct EM response, and n is the number of discrete points of the spectrum. We input 1000 dimensions of the S11 curve into network 1 and the average RRE is 3.72% in the test set. The results of the two examples randomly selected are shown in Figure 9a,b. These results prove that a perfect consistency between the input response and the output response has been established and it is feasible to use RRV extracted by AE to represent the high-dimensional ORV. mensions of the S11 curve into network 1 and the average RRE is 3.72% in the test set. The results of the two examples randomly selected are shown in Figure 9a,b. These results prove that a perfect consistency between the input response and the output response has been established and it is feasible to use RRV extracted by AE to represent the high-dimensional ORV.

**Figure 9.** (**a**,**b**) Diagram of input response and output response of network 1. **Figure 9.** (**a**,**b**) Diagram of input response and output response of network 1.

Then, the performance of network 2 should be discussed. The essence of network 2 is a forward prediction network, which will input the design matrix of the metasurface and output the predicted ORV. Its main role in the overall framework is to quickly predict the response curve such as S11 of the inverse-designed metasurface instead of using a traditional physical simulator, so as to calculate the accuracy of this framework rapidly. Prediction accuracy is usually used to describe the performance of a forward prediction Then, the performance of network 2 should be discussed. The essence of network 2 is a forward prediction network, which will input the design matrix of the metasurface and output the predicted ORV. Its main role in the overall framework is to quickly predict the response curve such as S11 of the inverse-designed metasurface instead of using a traditional physical simulator, so as to calculate the accuracy of this framework rapidly. Prediction accuracy is usually used to describe the performance of a forward prediction network and it can be expressed as 1-REE. Figure 10a,b illustrate the S11 curve respectively calculated by the physical simulator and the forward prediction network of two randomly selected metasurfaces. The average prediction accuracy of network 2 on the test set is up to 96%.

set is up to 96%.

network and it can be expressed as 1-REE. Figure 10a,b illustrate the S11 curve respectively calculated by the physical simulator and the forward prediction network of two randomly selected metasurfaces. The average prediction accuracy of network 2 on the test

train. To deal with the overfitting problem, *L2* regularization is necessary.

changed in subsequent iterations, which proves the algorithm has converged.

The training process of network 1 and network 2 is based on the back propagation algorithm, and the loss function value decreases rapidly with the number of epochs during the training process. Figure 10c shows the curve of the loss function for the two networks on the validation set. After about 25 epochs, the loss function values of the two networks can be reduced to less than 0.5, which achieved a good degree of training for an autoencoder network. The overall loss of network 1 is greater than that of network 2 because network 1 has high dimensions of input and output, which will be more difficult to

The VAE-PSO module realizes the application of the PSO algorithm in VAE generated latent space and the optimal latent vector is selected. The algorithm initializes 20 particles and sets the upper limit of iteration to 100. If the number of particles is too small, PSO will fall into the local optimal solution, while if the number of particles is too large, the running time will be greatly increased. It should be noted that the number of iterations should be set large enough to ensure the convergence of the PSO algorithm with the possibility of early stopping in case the fitness function does not change after about 15 iterations. Figure 10d illustrates how the fitness function changes during iteration. When iterating fifteen times, the fitness function reaches the minimum value and is almost un-

**Figure 10.** (**a**,**b**) The response curve calculated by the physical simulator and the prediction curve of the forward prediction network. (**c**) Validation loss of network 1 and network 2. (**d**) The process of VAE-PSO. **Figure 10.** (**a**,**b**) The response curve calculated by the physical simulator and the prediction curve of the forward prediction network. (**c**) Validation loss of network 1 and network 2. (**d**) The process of VAE-PSO.

Next, we introduce how to use the framework. Firstly, the target response is applied to network 3 to obtain the RDV, and then transmitted to the VAE-PSO algorithm frame-The training process of network 1 and network 2 is based on the back propagation algorithm, and the loss function value decreases rapidly with the number of epochs during the training process. Figure 10c shows the curve of the loss function for the two networks on the validation set. After about 25 epochs, the loss function values of the two networks can be reduced to less than 0.5, which achieved a good degree of training for an autoencoder network. The overall loss of network 1 is greater than that of network 2 because network 1 has high dimensions of input and output, which will be more difficult to train. To deal with the overfitting problem, *L<sup>2</sup>* regularization is necessary.

The VAE-PSO module realizes the application of the PSO algorithm in VAE generated latent space and the optimal latent vector is selected. The algorithm initializes 20 particles and sets the upper limit of iteration to 100. If the number of particles is too small, PSO will fall into the local optimal solution, while if the number of particles is too large, the running time will be greatly increased. It should be noted that the number of iterations should be set large enough to ensure the convergence of the PSO algorithm with the possibility of early stopping in case the fitness function does not change after about 15 iterations. Figure 10d illustrates how the fitness function changes during iteration. When iterating fifteen times, the fitness function reaches the minimum value and is almost unchanged in subsequent iterations, which proves the algorithm has converged.

Next, we introduce how to use the framework. Firstly, the target response is applied to network 3 to obtain the RDV, and then transmitted to the VAE-PSO algorithm framework and optimized iteratively. Finally, the design matrix that can produce the most similar EM response is output. As a verification, we select S11 curves as target responses from the test set to feed into the framework and get an average accuracy of up to 94%. Figure 11a illustrates a concrete example selected from the test set, with the target response in the blue line, and the generative response in the red line. For practical applications, perfectly ideal response curves made up of straight lines are usually input. We expect to design a

metasurface capable of implementing a −2 dB S11 parameter at 12 GHz, so an ideal EM response without transition bands is fed into the framework and returns the most accurate design matrix in the global latent space in Figure 11b. An artificial neural network (ANN) can learn complex nonlinear input-output relationships through the training process and data adaptation, which are their main characteristics [17,18]. FCL is the most common type of ANN, which can directly train the corresponding rules between ORV and ODV. For performance comparison, the responses generated based on the FCL method, whose matching degree with the target response is far less than that of the framework, are also illustrated in Figure 11a,b. The phase response of the metasurface is also a significant property, which is illustrated in Figure 11c,d. Abrupt phase changes can be obtained in the planar metasurface structures over the subwavelength scale, which provides a new avenue to enable a variety of applications, including large-scale planar imaging, electromagnetic virtual shaping, and holographic display with a large field of view [19]. response without transition bands is fed into the framework and returns the most accurate design matrix in the global latent space in Figure 11b. An artificial neural network (ANN) can learn complex nonlinear input-output relationships through the training process and data adaptation, which are their main characteristics [17*,*18]. FCL is the most common type of ANN, which can directly train the corresponding rules between ORV and ODV. For performance comparison, the responses generated based on the FCL method, whose matching degree with the target response is far less than that of the framework, are also illustrated in Figure 11a,b. The phase response of the metasurface is also a significant property, which is illustrated in Figure 11c,d. Abrupt phase changes can be obtained in the planar metasurface structures over the subwavelength scale, which provides a new avenue to enable a variety of applications, including large-scale planar imaging, electromagnetic virtual shaping, and holographic display with a large field of view [19].

work and optimized iteratively. Finally, the design matrix that can produce the most similar EM response is output. As a verification, we select S11 curves as target responses from the test set to feed into the framework and get an average accuracy of up to 94%. Figure 11a illustrates a concrete example selected from the test set, with the target response in the blue line, and the generative response in the red line. For practical applications, perfectly ideal response curves made up of straight lines are usually input. We expect to design a metasurface capable of implementing a −2dB S11 parameter at 12 GHz, so an ideal EM

*Electronics* **2022**, *11*, x FOR PEER REVIEW 11 of 13

**Figure 11.** (**a**,**b**) Input the target EM response into the framework. (**c**,**d**) Phase curves corresponding to the metasurfaces of (**a**,**b**). **Figure 11.** (**a**,**b**) Input the target EM response into the framework. (**c**,**d**) Phase curves corresponding to the metasurfaces of (**a**,**b**).

Finally, this framework is named "STARRY", and the performance difference among STARRY, the conventional design method, and the FCL design method is compared in Figure 12. Traditional design methods take genetic algorithms as an example. FCL design method directly trains the mapping of response curve to design matrix with full connection layer. The design time and area of deep learning methods are much smaller than that of the conventional design method. The STARRY framework consumes more time and Finally, this framework is named "STARRY", and the performance difference among STARRY, the conventional design method, and the FCL design method is compared in Figure 12. Traditional design methods take genetic algorithms as an example. FCL design method directly trains the mapping of response curve to design matrix with full connection layer. The design time and area of deep learning methods are much smaller than that of the conventional design method. The STARRY framework consumes more time and space resources than FCL but shows the highest accuracy. At the same time, STARRY solves the problem of non-uniqueness and network complexity.

space resources than FCL but shows the highest accuracy. At the same time, STARRY

**Figure 12.** (**a**) Accuracy comparison (**b**) Other performance comparison. **Figure 12.** (**a**) Accuracy comparison (**b**) Other performance comparison.

time compared with the traditional inverse design methods.

solves the problem of non-uniqueness and network complexity.

#### **4. Conclusions 4. Conclusions**

In conclusion, an isomorphic metasurfaces inverse design framework based on representation learning has been proposed and demonstrated. By using AEs of various architectures, the high-dimensional original space is mapped to the low-dimensional space with little loss, and the non-uniqueness problem is changed into a multi-solution problem in the low-dimensional space. The VAE-PSO algorithm is used to swiftly identify the best global search space value and deliver the binary matrix with the best response accuracy. This hybrid framework, which gives the optimal solution to the non-uniqueness problem and reduces the network complexity, achieves a high average accuracy of 94% on test sets, and it can give the design matrix in just a few seconds, which saves a lot of resources and In conclusion, an isomorphic metasurfaces inverse design framework based on representation learning has been proposed and demonstrated. By using AEs of various architectures, the high-dimensional original space is mapped to the low-dimensional space with little loss, and the non-uniqueness problem is changed into a multi-solution problem in the low-dimensional space. The VAE-PSO algorithm is used to swiftly identify the best global search space value and deliver the binary matrix with the best response accuracy. This hybrid framework, which gives the optimal solution to the non-uniqueness problem and reduces the network complexity, achieves a high average accuracy of 94% on test sets, and it can give the design matrix in just a few seconds, which saves a lot of resources and time compared with the traditional inverse design methods.

**Author Contributions:** Conceptualization, J.Z.; methodology, J.Z. and J.Y.; software, J.Z.; validation, J.Z. and J.Y.; formal analysis, C.L.; investigation, B.L.; data curation, J.Z.; writing—original draft preparation, J.Z.; writing—review and editing, J.Y.; supervision, J.Y.; **Author Contributions:** Conceptualization, J.Z.; methodology, J.Z. and J.Y.; software, J.Z.; validation, J.Z. and J.Y.; formal analysis, C.L.; investigation, B.L.; data curation, J.Z.; writing—original draft preparation, J.Z.; writing—review and editing, J.Y.; supervision, J.Y.; project administration, J.Y.; funding acquisition, J.Y. All authors have read and agreed to the published version of the manuscript.

project administration, J.Y.; funding acquisition, J.Y. All authors have read and agreed to the published version of the manuscript. **Funding:** National Natural Science Foundation of China (NSFC) (62105306). National un-**Funding:** National Natural Science Foundation of China (NSFC) (62105306). National undergraduate innovation and entrepreneurship training program of Communication University of China (202210033007). Fundamental Research Funds for the Central Universities (CUC220B004).

dergraduate innovation and entrepreneurship training program of Communication University of China (202210033007). Fundamental Research Funds for the Central Universities (CUC220B004). **Acknowledgments:** This work is supported by the National Natural Science Foundation of **Acknowledgments:** This work is supported by the National Natural Science Foundation of China (NSFC) (62105306), the National undergraduate innovation and entrepreneurship training program of Communication University of China (202210033007) and the Fundamental Research Funds for the Central Universities (CUC220B004). Jian Zhang and Jin Yuan thank the National Science Foundation for assistance in identifying collaborators for this work.

training program of Communication University of China (202210033007) and the Fundamental Research Funds for the Central Universities (CUC220B004). Jian Zhang and Jin Yuan thank

China (NSFC) (62105306), the National undergraduate innovation and entrepreneurship **Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


## *Article* **High-Performance Magnetoinductive Directional Filters**

**Artem Voronov \* , Richard R. A. Syms and Oleksiy Sydoruk**

Optical and Semiconductor Devices Group, EEE Department, Imperial College London, Exhibition Road, London SW7 2AZ, UK; r.syms@imperial.ac.uk (R.R.A.S.); o.sydoruk@imperial.ac.uk (O.S.) **\*** Correspondence: artem.voronov14@imperial.ac.uk

**Abstract:** Multiport magnetoinductive (MI) devices with directional filter properties are presented. Design equations are developed and solved using wave analysis and dispersion theory, and it is shown that high-performance directional filters can be realised for use both in MI systems with complex, frequency-dependent impedance and in conventional systems with real impedance. Wave analysis is used to reduce the complexity of circuit equations. High-performance MI structures combining directional and infinite rejection filtering are demonstrated, as well as multiple-passband high-rejection filtering. A new method for improving filtering performance through multipath loss compensation is described. Methods for constructing tuneable devices using toroidal ferrite-cored transformers are proposed and demonstrated, and experimental results for tuneable MI directional filters are shown to agree with theoretical models. Limitations are explored, and power handling sufficient for HF RFID applications is demonstrated, despite the use of ferrite materials.

**Keywords:** directional filter; infinite rejection; magnetoinductive waveguide; metamaterial

## **1. Introduction**

Directional filters (DFs) are four-port directional couplers with frequency filtering capability [1–4]. They can remove band-limited noise or unwanted signals at specific frequencies or be arranged in cascade as multiplexers for frequency-division multiplexing [5–7]. Lumped element and waveguide formats have been developed for frequencies ranging from VHF to millimetre wave [8–15]. Two-port filters with high attenuation have also been developed and use multipath cancellation, reflection, or absorption to achieve infinite rejection [16–23]. Applications include transmit/receive modules, multiplexers in ultra-wide-band antennas, and superheterodyne receivers [24–28]. Apart from rare examples [29], development has involved systems with real impedance. However, work on RF and microwave metamaterials showed that (in addition to other novel properties) periodic arrays of coupled metallic resonators allow the propagation of lattice waves including electroinductive (EI) [30–33] and magnetoinductive (MI) [34,35] waves. The latter have attracted considerably more attention for applications that depend on magnetic rather than electric fields. These include near-field communication in weakly conductive media [36–38], inductive power transfer [39–41], and magnetic field sensing [42–46]. MI waveguides consist of arrays of magnetically coupled LC resonators. They are band-limited and dispersive and have complex frequency-dependent impedance. Although quasi-optical devices, such as matching networks, mirrors, resonators splitters, and couplers, can be synthesised [47–49], MI systems are still embryonic, and it is difficult to integrate other functionality, such as filtering.

Here we show that a pair of coupled right- and left-handed MI waveguides can form a directional filter and investigate its potential for signal blocking in MI-based high-frequency radio frequency identification (HF RFID), where MI antennas may also find application [46]. The filters are designed to effectively operate at high RF power, since operation is based on complementary outputs rather than absorption, and to ensure effective blocking, frequency selectivity is tuneable. Furthermore, a new method for realising four-port directional

**Citation:** Voronov, A.; Syms, R.R.A.; Sydoruk, O. High-Performance Magnetoinductive Directional Filters. *Electronics* **2022**, *11*, 845. https:// doi.org/10.3390/electronics11060845

Academic Editors: Naser Ojaroudi Parchin, Mohammad Ojaroudi and Raed A. Abd-Alhameed

Received: 28 January 2022 Accepted: 4 March 2022 Published: 8 March 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

filters with tuneable infinite-rejection responses is presented. Wave theory is developed to explain the principle of multipath loss compensation, and high attenuation is confirmed by experiment. Designs are extended to allow multiple stopbands with improved rejection. Three-port directional filters are also shown to be feasible and can be made using a section of MI waveguide with the addition of as few as three resonators. In each case, wave theory is used to reduce the complexity of the circuit equations. Simple multiplexers can be implemented by cascading directional filters. Together, these features can introduce additional functionality and complexity in magnetoinductive systems, since the filter components can be integrated directly with MI waveguides. Additionally, such devices can be successfully used in conventional systems. presented. Wave theory is developed to explain the principle of multipath loss compensation, and high attenuation is confirmed by experiment. Designs are extended to allow multiple stopbands with improved rejection. Three-port directional filters are also shown to be feasible and can be made using a section of MI waveguide with the addition of as few as three resonators. In each case, wave theory is used to reduce the complexity of the circuit equations. Simple multiplexers can be implemented by cascading directional filters. Together, these features can introduce additional functionality and complexity in magnetoinductive systems, since the filter components can be integrated directly with MI waveguides. Additionally, such devices can be successfully used in conventional systems. Essential background is first reviewed in Sections 2.1 and 2.2. An analytic model for

realising four-port directional filters with tuneable infinite-rejection responses is

Essential background is first reviewed in Sections 2.1 and 2.2. An analytic model for MI directional filters is presented in Section 2.3, the circuit equations are solved using discrete travelling wave analysis, and methods for tuning are proposed. Conditions for infinite rejection are identified in Section 2.4, and the theory is extended to multiple passbands. Methods for constructing tuneable devices with high rejection and multiple passbands are described. Theoretical predictions are shown to be in good agreement with experimental results in the HF band in Section 3. Performance parameters including power handling and frequency scaling are also considered, with conclusions in Section 4. MI directional filters is presented in Section 2.3, the circuit equations are solved using discrete travelling wave analysis, and methods for tuning are proposed. Conditions for infinite rejection are identified in Section 2.4, and the theory is extended to multiple passbands. Methods for constructing tuneable devices with high rejection and multiple passbands are described. Theoretical predictions are shown to be in good agreement with experimental results in the HF band in Section 3. Performance parameters including power handling and frequency scaling are also considered, with conclusions in Section 4.

#### **2. Materials and Methods 2. Materials and Methods**

( +

elements are:

#### *2.1. Magnetoinductive Waveguides 2.1. Magnetoinductive Waveguides*

1 

*Electronics* **2022**, *11*, x FOR PEER REVIEW 2 of 16

Figure 1a is a schematic of a magnetoinductive waveguide, which consists of magnetically coupled resonators with inductance *L*, capacitance *C*, resistance *R*, and nearestneighbour mutual inductance *M*. Although non-nearest-neighbour coupling is a common problem, we assume that steps have been taken to avoid it. Assuming that the loop current in resonator *n* is *In*, the circuit equations relating the currents in adjacent elements are: Figure 1a is a schematic of a magnetoinductive waveguide, which consists of magnetically coupled resonators with inductance , capacitance , resistance , and nearest-neighbour mutual inductance . Although non-nearest-neighbour coupling is a common problem, we assume that steps have been taken to avoid it. Assuming that the loop current in resonator is , the circuit equations relating the currents in adjacent

$$\left(j\omega L + \frac{1}{j\omega \mathbb{C}} + R\right)I\_{\mathbb{R}} + j\omega M(I\_{n-1} + I\_{n+1}) = 0. \tag{1}$$

**Figure 1.** Magnetoinductive (MI) waveguide: (**a**) equivalent circuit; (**b**) frequency dependence of ' and '' for κ = {±0.6, ±0.3} and Q = 200. Crosses show measured points for = 0.3. **Figure 1.** Magnetoinductive (MI) waveguide: (**a**) equivalent circuit; (**b**) frequency dependence of *k* 0 *a* and *k* <sup>00</sup> *a* for *κ* = {±0.6, ±0.3} and *Q* = 200. Crosses show measured points for *κ* = 0.3.

For convenience, we define the self-impedance = + 1 + , mutual impedance = , and magnetic coupling coefficient = 2 . Assumption of a travelling current wave in the form = 0 <sup>−</sup> , where = ′ − ′′ is the complex For convenience, we define the self-impedance *Z* = *jωL* + <sup>1</sup> *<sup>j</sup>ω<sup>C</sup>* + *R*, mutual impedance *Z<sup>m</sup>* = *jωM*, and magnetic coupling coefficient *κ* = <sup>2</sup>*<sup>M</sup> L* . Assumption of a travelling current wave in the form *I<sup>n</sup>* = *I*0*e* <sup>−</sup>*jkna*, where *k* = *k* <sup>0</sup> − *jk*<sup>00</sup> is the complex propagation constant and a is the element spacing, yields the well-known dispersion relation [34]:

$$1 - \frac{\omega\_0^2}{\omega^2} - \frac{j}{Q} \frac{\omega\_0}{\omega} + \kappa \cos(ka) = 0. \tag{2}$$

Here *ω*<sup>0</sup> = <sup>√</sup> 1 *LC* is the angular resonant frequency and *<sup>Q</sup>* <sup>=</sup> *ω*0*L R* is the quality factor. For lossless waveguides, the Q-factor is infinite, and the left-hand side of Equation (2) is real with *k* <sup>00</sup> = 0. In real systems, *Q* must be finite. If losses are low (*k* <sup>00</sup> *k* 0 ), dispersion and loss may be approximated using:

$$1 - \frac{\omega\_0^2}{\omega^2} + \kappa \cos(k'a) = 0\tag{3}$$

$$k''a = \frac{\omega\_0}{\kappa \mathcal{Q} \omega \sin(k'a)}.\tag{4}$$

Equation (3) is the dispersion relation for lossless guides. It implies that MI waveguides are band limited, with propagation in the range √ 1 1+|*κ*| <sup>≤</sup> *<sup>ω</sup> ω*0 < √ 1 1−|*κ*| . Guides with positive *M* are right-handed and have positive phase velocity, while negative *M* yields left-handed guides with negative phase velocity. At the resonant frequency, *k* 0 *a* = *<sup>π</sup>* <sup>2</sup> when *<sup>M</sup>* <sup>&</sup>gt; <sup>0</sup> and <sup>−</sup>*<sup>π</sup>* <sup>2</sup> when *M* < 0. Equation (4) implies that losses are inversely proportional to |*κQ*|, are lowest at the resonant frequency, and rise rapidly at the band edges. Figure 1b shows the frequency dependence of *k* 0 *a* and *k* <sup>00</sup> *a* for *Q* = 200 and two different values of |*κ*|, which clearly show the band-limited nature of propagation. Crosses show three experimentally measured values along the dispersion relation for *κ* = 0.3 and are clearly in agreement with Equation (3).

## *2.2. Impedance and Power Flow*

It is simple to show that the characteristic impedance of an MI waveguide is [47]:

$$Z\_0 = j\omega M e^{-jka}.\tag{5}$$

If the waveguide is lossless, the characteristic impedance *Z*0*<sup>M</sup>* at resonance is purely real, with *Z*0*<sup>M</sup>* = *ω*0*M*. In general, the impedance is complex valued and frequency dependent, which implies that care must be taken when matching to real-valued loads. An important consequence is that conventional scattering parameters cannot be used; instead, the formulation of 'power waves', originally developed by Kurokawa [50] and recently adapted to MI waveguides [51], is required. For MI lines extending to the left, type 1 power waves *A*1 and *B*1 are used at element n, as shown in Figure 1a, defined as:

$$A1 = \frac{-Z\_m I\_{n-1} + Z\_0 I\_n}{2\sqrt{\text{Re}(Z\_0)}} \qquad B1 = \frac{-Z\_m I\_{n-1} - Z\_0^\* I\_n}{2\sqrt{\text{Re}(Z\_0)}} \tag{6}$$

Here *Z* ∗ 0 represents the complex conjugate of *Z*0, and *Re*(*Z*0) its real part. For MI lines extending to the right, type 2 power waves *A*2 and *B*2 must be used, defined as:

$$A2 = \frac{Z\_m I\_{n+1} - Z\_0 I\_n}{2\sqrt{\text{Re}(Z\_0)}}, \qquad B2 = \frac{Z\_m I\_{n+1} + Z\_0^\* I\_n}{2\sqrt{\text{Re}(Z\_0)}}.\tag{7}$$

The squared moduli of these coefficients provide the correct definition of power.

#### *2.3. Magnetoinductive Directional Filters*

A directional filter has four ports connected to lines of characteristic impedance *Z*0. A voltage source connected to port 1 then gives rise to incident and reflected waves *A*1<sup>1</sup> and *B*1<sup>1</sup> and transmitted waves *B*22, *B*23, *B*1<sup>4</sup> from ports 2, 3, and 4. Port 2 has the response of a band-reject filter, and Port 4 that of a bandpass filter. In a matched and lossless device, *S*<sup>21</sup> and *S*<sup>41</sup> are complementary, *S*<sup>31</sup> = 0 for all *ω*, and *S*<sup>11</sup> = 0 at resonance. Equivalent properties apply no matter which port is used as input. We now develop the analysis for MI directional filters and consider their frequency response, bandwidth, and tuneability.

Figure 2a shows the equivalent circuit of a filter based on a pair of infinite horizontal MI waveguides consisting of identical resonators, but with opposite coupling polarity so

that the lower guide is right-handed (blue) and the upper guide left-handed (red). The mutual inductances are equal in magnitude, with *M* > 0, so the coupling coefficients are *κ <sup>f</sup>* = <sup>2</sup>*<sup>M</sup> L* and *<sup>κ</sup><sup>b</sup>* <sup>=</sup> <sup>−</sup>2*<sup>M</sup> L* . The waveguides are connected vertically via two resonators, which couple the guides via mutual inductances *M<sup>C</sup>* and coupling coefficients *κ<sup>c</sup>* = 2*Mc L* , such that *κ<sup>c</sup>* < *κ <sup>f</sup>* . The network has four ports, 1–4, formed by sections of an MI waveguide. The lower line supports currents *In*, and the top line supports *Jn*, while the two coupling resonators have currents *K* and *P*. Note that guide terminations can also be represented by lumped impedances, as shown in Figure 2b. There are three main possibilities: an infinite MI waveguide, represented by *Z*<sup>0</sup> (1); a resistive termination, represented by *Z*0*<sup>M</sup>* (2); and a broadband termination to *Z*0*M*, achieved by halving the inductance and doubling the capacitance in the resonant element (3). Termination 2 provides an impedance match at *ω*0, while termination 3 provides matching at two frequencies, *ω*<sup>0</sup> and <sup>√</sup> *ω*0 1−*κ* 2 , with low reflection in between [49]. that the lower guide is right-handed (blue) and the upper guide left-handed (red). The mutual inductances are equal in magnitude, with > 0, so the coupling coefficients are 2 and = − 2 . The waveguides are connected vertically via two resonators, which couple the guides via mutual inductances and coupling coefficients = 2 , such that < . The network has four ports, 1–4, formed by sections of an MI waveguide. The lower line supports currents , and the top line supports , while the two coupling resonators have currents and . Note that guide terminations can also be represented by lumped impedances, as shown in Figure 2b. There are three main possibilities: an infinite MI waveguide, represented by <sup>0</sup> (1); a resistive termination, represented by 0 (2); and a broadband termination to 0 , achieved by halving the inductance and doubling the capacitance in the resonant element (3). Termination 2 provides an impedance match at <sup>0</sup> , while termination 3 provides matching at two frequencies, <sup>0</sup> and <sup>0</sup> √1 − 2 , with low reflection in between [49].

develop the analysis for MI directional filters and consider their frequency response,

Figure 2a shows the equivalent circuit of a filter based on a pair of infinite horizontal MI waveguides consisting of identical resonators, but with opposite coupling polarity so

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bandwidth, and tuneability.

=

2 , <sup>3</sup>

resonators:

**Figure 2.** (**a**) MI directional filter equivalent circuit; (**b**) lumped element terminations: (1) magnetoinductive, (2) resistive, and (3) broadband resistive. **Figure 2.** (**a**) MI directional filter equivalent circuit; (**b**) lumped element terminations: (1) magnetoinductive, (2) resistive, and (3) broadband resistive.

For now, we assume type 1 terminations and that an MI wave is incident on port 1 from the left. Experience suggests that this wave will be partially reflected at = 0 due to the discontinuity caused by the coupling resonator, and that the remaining power, less any loss, will appear as MI waves from ports 2 to 4. We write expressions for the currents and in terms of incident, reflected, and transmitted MI waves as: For now, we assume type 1 terminations and that an MI wave is incident on port 1 from the left. Experience suggests that this wave will be partially reflected at*n* = 0 due to the discontinuity caused by the coupling resonator, and that the remaining power, less any loss, will appear as MI waves from ports 2 to 4. We write expressions for the currents *I<sup>n</sup>* and *J<sup>n</sup>* in terms of incident, reflected, and transmitted MI waves as:

$$\begin{array}{c} I\_{\mathbb{M}} = I e^{-jk\_{f}na} + \mathrm{Re}^{jk\_{f}na}, \ n \le 0\\\ I\_{\mathbb{M}} = T\_{2} e^{-jk\_{f}na}, \ n \ge 1\\\ I\_{\mathbb{M}} = T\_{3} e^{-jk\_{b}na}, \ n \ge 1\\\ I\_{\mathbb{M}} = T\_{4} e^{jk\_{b}na}, \ n \le 0. \end{array} \tag{8}$$

 = 4 , ≤ 0. Here and are the amplitudes of the incident and reflected waves at port 1; , <sup>4</sup> are the wave amplitudes from ports 2, 3, and 4; = ′ − ′′ is the propagation constant of the forward waveguide; and = ′ − ′′ is the propagation constant of the backward one. From Figure 1b and Equation (2), we can assume: Here *I* and *R* are the amplitudes of the incident and reflected waves at port 1; *T*2, *T*3, *T*<sup>4</sup> are the wave amplitudes from ports 2, 3, and 4; *k <sup>f</sup>* = *k* 0 *<sup>f</sup>* − *jk*<sup>00</sup> *f* is the propagation constant of the forward waveguide; and *k<sup>b</sup>* = *k* 0 *<sup>b</sup>* − *jk*<sup>00</sup> *b* is the propagation constant of the backward one. From Figure 1b and Equation (2), we can assume:

$$k\_b a = k\_f a - \pi. \tag{9}$$

Combining Equation (1) with circuit equations for the currents and in the coupling resonators, a set of coupled equations may be obtained at the six central Combining Equation (1) with circuit equations for the currents *K* and *P* in the coupling resonators, a set of coupled equations may be obtained at the six central resonators:

$$\begin{array}{ll} ZI\_0 + Z\_m(I\_{-1} + I\_1) + Z\_{\rm mc}K = 0, & ZI\_1 + Z\_m(I\_0 + I\_2) + Z\_{\rm mc}P = 0\\ ZK + Z\_{\rm mc}(I\_0 + I\_0) = 0, & ZP + Z\_{\rm mc}(I\_1 + I\_1) = 0\\ ZJ\_0 - Z\_m(I\_{-1} + I\_1) + Z\_{\rm mc}K = 0, & ZJ\_1 - Z\_m(I\_0 + I\_2) + Z\_{\rm mc}P = 0. \end{array} \tag{10}$$

=

Substituting the assumed solutions and using the dispersion equation, *K* and *P* may be eliminated, allowing the reflection and transmission coefficients to be calculated:

$$\begin{split} \frac{R}{I} &= \frac{-\left(e^{j2k\_f a} + 1\right)\mu^2}{1 + 2\mu^2 e^{j2k\_f a} - e^{j4k\_f a}}, \quad \frac{T\_4}{I} = \frac{\left(1 - e^{j2k\_f a}\right)\mu^2}{1 + 2\mu^2 e^{j2k\_f a} - e^{j4k\_f a}}\\ \frac{T\_2}{I} &= \frac{1 - e^{j4k\_f a}}{1 + 2\mu^2 e^{j2k\_f a} - e^{j4k\_f a}}, \quad \frac{T\_3}{I} = 0. \end{split} \tag{11}$$

Here *µ* = *Mc <sup>M</sup>* is the mutual inductance ratio. Note that the coefficients above are complex valued and frequency dependent. They are expressed only in terms of *µ* and *k <sup>f</sup>* , which may be found from Equation (2). However, care must be taken to ensure that *k* 00 *<sup>f</sup>* ≥ 0 and 0 ≤ *k* 0 *<sup>f</sup>* ≤ *π*. The results imply that no signal leaves port 2 at resonance, provided the guides are lossless. Similarly, the transmission from port 4 is maximum at resonance, while the reflectance from port 1 is zero. Port 3 is isolated at all frequencies, both within and outside the MI passbands. This behaviour is inherent in the way signals are combined. The phase shift at resonance along path P1 in Figure 2a is *<sup>π</sup>* 2 , while the phase shift along path P2 is <sup>3</sup>*<sup>π</sup>* 2 . If the device is lossless, the signal from port 2 is then zeroed by cancellation. If Q-factors are finite, losses are different along P1 and P2 and cancellation is partial. Two signal paths, P3 and P4, to port 3 are in antiphase for all frequencies.

Although the results above provide valid amplitude reflection and transmission coefficients, they offer no insight on power flow in devices that are either lossy or operating out of band. We therefore define three sets of discrete power wave pairs, {*A*11, *B*11}, {*A*22, *B*22}, and {*A*14, *B*14}, for ports 1, 2, and 4, as shown in Figure 2a, as follows:

$$\begin{array}{ll} A1\_1 = \frac{-Z\_m \mathcal{I}\_{-1} + Z\_0 \mathcal{I}\_0}{2\sqrt{\mathcal{R}\mathcal{E}\left(\mathcal{Z}\_{0f}\right)}}, & A2\_2 = \frac{Z\_m \mathcal{I}\_2 - Z\_0 \mathcal{I}\_1}{2\sqrt{\mathcal{R}\mathcal{E}\left(\mathcal{Z}\_{0f}\right)}}, & A1\_4 = \frac{Z\_m \mathcal{I}\_{-1} + Z\_0 \mathcal{I}\_0}{2\sqrt{\mathcal{R}\mathcal{E}\left(\mathcal{Z}\_{0b}\right)}},\\ B1\_1 = \frac{-Z\_m \mathcal{I}\_{-1} - Z\_0^\* \mathcal{I}\_0}{2\sqrt{\mathcal{R}\mathcal{E}\left(\mathcal{Z}\_{0f}\right)}}, & B2\_2 = \frac{Z\_m \mathcal{I}\_2 + Z\_0^\* \mathcal{I}\_1}{2\sqrt{\mathcal{R}\mathcal{E}\left(\mathcal{Z}\_{0f}\right)}}, & B1\_4 = \frac{Z\_m \mathcal{I}\_{-1} - Z\_0^\* \mathcal{I}\_0}{2\sqrt{\mathcal{R}\mathcal{E}\left(\mathcal{Z}\_{0b}\right)}}. \end{array} \tag{12}$$

Here *Z*<sup>0</sup> *<sup>f</sup>* = *jωMe*−*jk <sup>f</sup> a* and *<sup>Z</sup>*0*<sup>b</sup>* <sup>=</sup> <sup>−</sup>*jωMe*−*jk<sup>b</sup> <sup>a</sup>* are the characteristic impedances of the forward and backward guides. In fact, *Z*<sup>0</sup> *<sup>f</sup>* = *Z*0*<sup>b</sup>* whenever Equation (9) applies. Power wave scattering parameters with respect to port 1 may then be found as:

$$\mathcal{S}\_{11} = \frac{B1\_1}{A1\_1}, \mathcal{S}\_{21} = \frac{B2\_2}{A1\_1}, \mathcal{S}\_{41} = \frac{B1\_4}{A1\_1}.\tag{13}$$

Using the transmission and reflection coefficients, we may obtain analytic expressions for the scattering parameters of a directional filter with MI line terminations (1) as:

$$\begin{split} S\_{11} &= \left( \left( 1 - e^{2\binom{k''}{f}a} \right) \left\{ e^{2\underline{k'}\_f a} + e^{2(\underline{k''}\_f + j2\underline{k'}\_f)a} \right\} + \mu^2 \left( 1 + 2e^{2(\underline{k''}\_f + j\underline{k'}\_f)a} - e^{j2\underline{k'}\_f a} \right) \right) / d \\ S\_{21} &= \left( e^{(\underline{k''}\_f + j\underline{k'}\_f)a} - e^{(\underline{k''}\_f + j3\underline{k'}\_f)a} + e^{-(\underline{k''}\_f + j\underline{k'}\_f)a} - e^{(\underline{k''}\_f + j\underline{k'}\_f)a} \right) / d \\ S\_{41} &= \mu^2 \left( e^{j2\underline{k'}\_f a} - 1 \right) / d, \\ d &= 1 + 2\mu^2 e^{2(\underline{k'}\_f + j\underline{k'}\_f)a} - e^{4(\underline{k''}\_f + j\underline{k'}\_f)a}. \end{split} \tag{14}$$

Moduli squared of Equations (11) and (14) are equal whenever *k* 00 *<sup>f</sup>* = *k* 00 *<sup>b</sup>* = 0.

#### 2.3.1. Frequency Response

To illustrate the above, Figure 3a shows the frequency dependence of magnitude S-parameters for different terminations, with *L* = 1 *µH*, *κ* = 0.6, and *κ<sup>c</sup>* = 0.3 (so that *µ* = 1/2). Results for lossless and lossy elements with magnetoinductive terminations are

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2.3.1. Frequency Response

obtained using Equation (14), and results for lossy elements with narrowband resistive terminations using numerical solution of circuit equations. obtained using Equation (14), and results for lossy elements with narrowband resistive terminations using numerical solution of circuit equations.

To illustrate the above, Figure 3a shows the frequency dependence of magnitude Sparameters for different terminations, with = 1 , = 0.6, and = 0.3 (so that = 1⁄2). Results for lossless and lossy elements with magnetoinductive terminations are

′′ =

′′ = 0.

(16)

Moduli squared of Equations (11) and (14) are equal whenever

**Figure 3.** MI directional filter performance for κ = 0.6 and κ<sup>c</sup> = 0.3: (**a**) scattering parameters for lossless and lossy (Q = 200) resonators with MI terminations (1) and real narrowband terminations (2); (**b**) variation of with and for a lossless filter with MI terminations (1); (**c**) scattering parameters for lossless resonators with MI terminations (1) for ω<sup>0</sup> ′ /<sup>0</sup> = 0.9, 1.0, and 1.1. **Figure 3.** MI directional filter performance for *κ* = 0.6 and *κ<sup>c</sup>* = 0.3: (**a**) scattering parameters for lossless and lossy (*Q* = 200 ) resonators with MI terminations (1) and real narrowband terminations (2); (**b**) variation of *Qbp* with *κ* and *κ<sup>c</sup>* for a lossless filter with MI terminations (1); (**c**) scattering parameters for lossless resonators with MI terminations (1) for *ω*0 0 /*ω*<sup>0</sup> = 0.9, 1.0, and 1.1.

In each case, frequencies near <sup>0</sup> are diverted into port 4, while the remaining power is sent to port 2. Port 4 is therefore bandpass, while port 2 is a bandstop. The MI passband is as estimated from previous arguments. The isolation of port 3 is unaffected by loss; however, power emerging from port 4 is reduced and port 2 rejection diminished. Since losses are inversely proportional to and , high coupling coefficients and Q-factors are required to minimise loss. The effect of a narrowband resistive termination is to reduce the bandwidth. However, this can be partly restored by using a broadband termination. In each case, frequencies near *ω*<sup>0</sup> are diverted into port 4, while the remaining power is sent to port 2. Port 4 is therefore bandpass, while port 2 is a bandstop. The MI passband is as estimated from previous arguments. The isolation of port 3 is unaffected by loss; however, power emerging from port 4 is reduced and port 2 rejection diminished. Since losses are inversely proportional to *κQ* and *κcQ*, high coupling coefficients and Q-factors are required to minimise loss. The effect of a narrowband resistive termination is to reduce the bandwidth. However, this can be partly restored by using a broadband termination.

#### 2.3.2. Bandwidth 2.3.2. Bandwidth

An analytic expression for the 3-dB bandwidth of |41| <sup>2</sup> may be obtained in the lossless case. The half-power frequencies can be found as follows: An analytic expression for the 3-dB bandwidth of |*S*41| <sup>2</sup> may be obtained in the lossless case. The half-power frequencies can be found as follows:

$$\left| S\_{41} \right|\_{k''=0}^2 = \frac{\mu^4 e^{j2k\_f'a} \left( e^{j2k\_f'a} - 1 \right)^2}{1 - e^{j4k\_f'a} \left( 4\mu^4 + 2 - e^{j4k\_f'a} \right)} = \frac{1}{2}. \tag{15}$$

2

, , and <sup>0</sup> , the half-power bandwidth is given by: 1 1 √4 − √8+4+2 Within the passband, there are two solutions for *k* 0 *f* . If these are expressed in terms of *κ*, *µ*, and *ω*0, the half-power bandwidth *B* is given by:

$$B = \omega\_0 \left( \sqrt{\frac{1}{1 - \kappa p}} - \sqrt{\frac{1}{1 + \kappa p}} \right), \quad p = \frac{\sqrt{\mu^4 - \sqrt{\mu^8 + 4} + 2}}{2}. \tag{16}$$

map of the variation of with and thus obtained. High-filter Q-factors require high values of and low values of ; however, the achievable Q-factor is limited since the use of real resonators with finite Q-factors will result in high losses if is too low. Thus, the width of the bandpass channel increases with *µ* and *ω*0. It is simple to calculate the Q-factor of the bandpass response, as *Qbp* = *ω*0 *B* . Figure 3b shows a contour map of the variation of *Qbp* with *κ* and *κ<sup>c</sup>* thus obtained. High-filter Q-factors require high values of *κ* and low values of *κc*; however, the achievable Q-factor is limited since the use of real resonators with finite Q-factors will result in high losses if *κ<sup>c</sup>* is too low.

#### 2.3.3. Frequency Tuning

So far, the filter centre frequency has been determined by *ω*0, which also determines the operation of the MI waveguides. It would clearly be beneficial to introduce tuning within the MI passband. Figure 3c shows the result of tuning the resonance of the coupling resonators to different values, namely, (a) *ω*0 <sup>0</sup> = 0.9*ω*0, (b) *ω*<sup>0</sup> <sup>0</sup> = 0.95*ω*0, and (c) *ω*<sup>0</sup> <sup>0</sup> = 1.1*ω*0, assuming lossless resonators and lossless MI terminations. Although the best performance is obtained when *ω*0 <sup>0</sup> = *ω*0, tuning around *ω*<sup>0</sup> is feasible for the variations in *ω*<sup>0</sup> 0 shown.

From the above, we conclude that the network shown in Figure 2a with lossless terminations possesses all the properties of a directional filter except that it has null reflectivity only within the MI passband. *S*<sup>11</sup> is zero at the resonant frequency and very small in its vicinity. Lossy systems and systems with narrowband terminations have similar properties, the main difference being a reduction in the magnetoinductive bandwidth.

## *2.4. Advanced Filters*

In this section, we consider the properties of more advanced filters, which offer either infinite rejection or multiple bandstop frequencies.

#### 2.4.1. Infinite Rejection

In the absence of loss, complete rejection occurs at *ω*<sup>0</sup> at port 2 when port 1 is used as an input. However, for finite Q-factors, the two signal paths from port 1 to port 2 in Figure 2a incur unequal losses, and some leakage into port 2 then occurs even at resonance. To equalise signal amplitudes along the two paths, the magnetic coupling strength between resonators 0 and 1 in the forward waveguide may be adjusted. We define a new positive coefficient, *κ* 0 *<sup>f</sup>* = *λκ <sup>f</sup>* , between these elements and reconsider the port reflection and transmission coefficients. The circuit equations must be modified to:

$$\begin{array}{c} ZI\_0 + Z\_m(I\_{-1} + \lambda I\_1) + Z\_{\rm mc} \mathcal{K} = 0, \; ZI\_1 + Z\_m(\lambda I\_0 + I\_2) + Z\_{\rm mc}P = 0\\ ZK + Z\_{\rm mc}(I\_0 + I\_0) = 0, \; ZP + Z\_{\rm mc}(I\_1 + I\_1) = 0\\ ZJ\_0 - Z\_{\rm m}(I\_{-1} + I\_1) + Z\_{\rm mc}\mathcal{K} = 0, \; Zj\_1 - Z\_{\rm m}(I\_0 + I\_2) + Z\_{\rm mc}P = 0. \end{array} \tag{17}$$

As before, we may substitute the assumed solutions and use the dispersion relation to obtain modified reflection and transmission coefficients. The transmission from port 2 may then be expressed as a rational polynomial in *λ* as follows:

$$\begin{split} \frac{T\_2}{\dot{\Gamma}} &= \frac{p\_0 + \lambda p\_1}{\mu\_0 + 2\mu^2 e^{i2\tilde{k}\_f^a \mu\_A} + q\_2 \lambda^2} \\ p\_0 &= \mu^4 e^{j2k\_f a} \left( 1 - e^{j2k\_f a} \right) \end{split}$$

$$\begin{split} p\_1 &= \left( e^{j4k\_f a} - 1 \right)^2 + \mu^2 e^{j2k\_f a} \left( 2 - 2e^{j4k\_f a} + \mu^2 \left( e^{j2k\_f a} - 1 \right) \right) \\ q\_0 &= e^{j2k\_f a} \left( e^{j6k\_f a} + e^{j4k\_f a} - e^{j2k\_f a} - 1 \right) + \mu^2 e^{j2k\_f a} \left( 2 - 2e^{j2k\_f a} - 4e^{j4k\_f a} + 4\mu^2 e^{j2k\_f a} - \mu^2 \right) \\ q\_2 &= 1 + e^{j2k\_f a} - e^{j4k\_f a} - e^{j6k\_f a} + \mu^2 e^{j2k\_f a} \left( 2 + 2e^{j2k\_f a} - \mu^2 \right) \end{split} \tag{18}$$

For infinite rejection at midband, we require the numerator of *T*2/*I* to equal zero:

$$
\lambda = -\frac{p\_0}{p\_1}.\tag{19}
$$

At resonance, the propagation constant can be rewritten in terms of *κ* and *Q* alone:

$$k\_f a \Big|\_{\omega = \omega\_0} = \frac{\pi}{2} + j \ln \left( \sqrt{1 + \frac{1}{\kappa^2 Q^2}} - \frac{1}{\kappa Q} \right). \tag{20}$$

Substituting into Equation (19), we obtain the condition for infinite rejection as:

$$\lambda = \frac{\mu^4 \rho^4}{\mu^4 \rho^4 - 2\mu^2 \rho^4 + 2\mu^2 \rho^2 + \rho^6 - \rho^4 - \rho^2 + 1}, \quad \rho = \sqrt{1 + \frac{1}{\kappa^2 Q^2}} - \frac{1}{\kappa Q}. \tag{21}$$

Figure 4a shows the variation of *λ* with *µ* for a range of Q-factors. Lower values of *λ* are required for higher losses, as well as for lower values of *µ*. In each case, *λ* → 1 as *µ* increases, with sharper increases at lower losses. When *λ* is set as above, port 2 is isolated at resonance. However, there is an impact on filter performance; namely, port 3 is no longer fully isolated, the passband of port 2 reduces, and reflectance at port 1 is improved.

Figure 4b shows the frequency dependence of scattering parameters for filters with lossy MI (1) and resistive narrowband (2) terminations for *κ* = 0.6, *κ<sup>c</sup>* = 0.3, *Q* = 200, and *λ* = 0.8746 from Equation (21). Both cases demonstrate excellent directional filtering with reflectance mostly <−30 dB within the passband, as well as infinite rejection at resonance. If isolation is important, *λ* may be adjusted to balance rejection with leakage into port 3. The mutual inductance discontinuity generated by tuning *λ* implies that the complementary waveguide pair is no longer symmetrical, meaning the ports cannot be used interchangeably. Only the two ports immediately adjacent to the *λ*-tuned resonator pair would yield an infiniterejection response. The remaining two ports would suffer from reduced performance if used as inputs. Figure 4b shows the frequency dependence of scattering parameters for filters with lossy MI (1) and resistive narrowband (2) terminations for = 0.6, = 0.3, = 200, and = 0.8746 from Equation (21). Both cases demonstrate excellent directional filtering with reflectance mostly <−30 dB within the passband, as well as infinite rejection at resonance. If isolation is important, may be adjusted to balance rejection with leakage into port 3. The mutual inductance discontinuity generated by tuning implies that the complementary waveguide pair is no longer symmetrical, meaning the ports cannot be used interchangeably. Only the two ports immediately adjacent to the -tuned resonator pair would yield an infinite-rejection response. The remaining two ports would suffer from reduced performance if used as inputs.

Figure 4a shows the variation of with for a range of Q-factors. Lower values of are required for higher losses, as well as for lower values of . In each case, → 1 as increases, with sharper increases at lower losses. When is set as above, port 2 is isolated at resonance. However, there is an impact on filter performance; namely, port 3 is no longer fully isolated, the passband of port 2 reduces, and reflectance at port 1 is improved.

,

= √1 +

1 <sup>2</sup><sup>2</sup> <sup>−</sup>

1 

. (21)

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 4 4 44 − 224+222+6 − 4 − 2+1

=

**Figure 4.** (**a**) Variation of λ with μ for different unloaded resonator Q-factors with = 0.6. (**b**) Frequency dependence of scattering parameters for infinite rejection filter with lossy MI and resistive narrowband terminations with κ = 0.6, κ<sup>c</sup> = 0.3, and Q = 200. **Figure 4.** (**a**) Variation of *λ* with *µ* for different unloaded resonator Q-factors with *κ* = 0.6. (**b**) Frequency dependence of scattering parameters for infinite rejection filter with lossy MI and resistive narrowband terminations with *κ* = 0.6, *κ<sup>c</sup>* = 0.3, and *Q* = 200.

#### 2.4.2. Multiple Bandstop Frequencies 2.4.2. Multiple Bandstop Frequencies

A directional filter with complementary bandstop and bandpass responses can be realised by increasing the number of coupling resonators from 1 to , as shown in Figure 5a. Here MI waveguides with coupling coefficients ± are connected using vertical lines containing weakly coupled resonators with coupling . The resonator rows are labelled from 0 (bottom) to top ( − 1) and have the currents and flowing in the left and right columns, respectively. The lines of coupling resonators can be considered as two identical uniform MI waveguides with a passband of <sup>1</sup> √1+| | ≤ 0 ≤ 1 √1 − | | and dispersion relation + ( − + ) = 0, where is the propagation constant. The A directional filter with *N* complementary bandstop and bandpass responses can be realised by increasing the number of coupling resonators from 1 to *N*, as shown in Figure 5a. Here MI waveguides with coupling coefficients ±*κ* are connected using vertical lines containing *N* weakly coupled resonators with coupling *κc*. The resonator rows are labelled from 0 (bottom) to top (*N* − 1) and have the currents *K<sup>m</sup>* and *P<sup>m</sup>* flowing in the left and right columns, respectively. The lines of coupling resonators can be considered as two identical uniform MI waveguides with a passband of √ 1 1+|*κ<sup>c</sup>* | <sup>≤</sup> *<sup>ω</sup> ω*0 ≤ √ 1 1−|*κ<sup>c</sup>* | and dispersion relation *Z* + *Zmc*(*e* <sup>−</sup>*jkc<sup>a</sup>* + *e jkca* ) = 0, where *k<sup>c</sup>* is the propagation constant. The currents *P<sup>m</sup>* and *K<sup>m</sup>* may also be expressed in terms of reflected and transmitted waves as:

currents and may also be expressed in terms of reflected and transmitted waves as:

$$P\_{\mathfrak{m}} = P\_T e^{-j\mathbf{k}\_\ell \mathfrak{m} a} + P\_R e^{j\mathbf{k}\_\ell \mathfrak{m} a}, \quad \mathcal{K}\_{\mathfrak{m}} = \mathcal{K}\_T e^{-j\mathbf{k}\_\ell \mathfrak{m} a} + \mathcal{K}\_R e^{j\mathbf{k}\_\ell \mathfrak{m} a}. \tag{22}$$

**Figure 5.** (**a**) Multiple bandstop MI directional filter; (**b**) frequency dependence of S-parameters for multiple passband filter with lossless MI terminations (1) for κ = 0.6, κ<sup>c</sup> = 0.3, = 2, and = 5. **Figure 5.** (**a**) Multiple bandstop MI directional filter; (**b**) frequency dependence of S-parameters for multiple passband filter with lossless MI terminations (1) for *κ* = 0.6, *κ<sup>c</sup>* = 0.3, *N* = 2, and *N* = 5.

The solutions for currents and remain unchanged from Equation (8). Once

) = 0, <sup>0</sup> + (<sup>1</sup> + <sup>1</sup>

Equation (23) can again be solved for the reflection coefficient at port 1 and transmission coefficients at ports 2–4 by substituting the dispersion relations and eliminating the terms , , and , . When = 1 , Equation (24) is identical to

(1 +

2

(3+)

The coefficients can be converted to scattering parameters as before. Figure 5b shows the frequency dependence of S-parameters for = 2 and = 5. The additional coupling resonators generate multiple bandpass responses in <sup>41</sup> and complementary notches in

2(+2)

frequencies, with a larger separation between peaks for higher values. The quality factors of the peaks are higher for increasing , and the effect of loss is to reduce rejection in <sup>21</sup>

or the minima of <sup>21</sup> within the MI passband. The central frequencies simplify to:

1 1+ cos(

The notch spacing is nonuniform and can be adjusted by varying <sup>0</sup> or

 +1 )

at the resonant frequency. As previously shown, infinite rejection can be achieved at resonance for = 1 . However, since path losses cannot be equalised at several

= 0√

for are independent of , provided that >

The frequencies of peaks and notches can be found by computing the maxima of <sup>41</sup>


) − 2 

)+ <sup>−</sup> <sup>1</sup> = 0, <sup>1</sup> − (<sup>0</sup> +<sup>2</sup>

(+)

) −

(1 −

) = 0, <sup>−</sup> <sup>1</sup> + ( <sup>−</sup> <sup>2</sup> + <sup>1</sup>

2

(1 +

(1 −

)+ 2 

2

4

)+ <sup>0</sup> = 0

) = 0

(23)

(24)

)+ <sup>−</sup> <sup>1</sup> = 0.

, ∈ [1, ]. (25)

. When is odd, there is always a peak

. The values

) = 0

2(+)

2(1 −

)} /

)} /

} /

2

(+3) )

<sup>0</sup> +(<sup>0</sup> +<sup>1</sup>

<sup>−</sup> <sup>1</sup> + ( <sup>−</sup> <sup>2</sup> +<sup>0</sup>

2

) −

2(+) −

(+)

2

(1 −

<sup>0</sup> + (−1 + <sup>1</sup>

<sup>0</sup> − (−1 + <sup>1</sup>

2

2(+1)

{2

2(+1)

(1 +

) {2

(+)

) +2

Equation (11).

2(+1) − 1){

= (1 −

= (+1)

(1 −

2 

4 

2

2 − 1)

 = (

= (

21. The value of |

and increase insertion loss in 41.

again, we may write down the equations at the junctions as:

*Electronics* **2022**, *11*, x FOR PEER REVIEW 9 of 16

The solutions for currents *I<sup>n</sup>* and *J<sup>n</sup>* remain unchanged from Equation (8). Once again, we may write down the equations at the junctions as:

$$\begin{array}{c} \text{ZI}\_{0} + Z\_{\text{m}}(I\_{-1} + I\_{1}) + Z\_{\text{m}c}K\_{0} = 0, \,\text{Z}I\_{1} + Z\_{\text{m}}(I\_{0} + I\_{2}) + Z\_{\text{m}c}P\_{0} = 0\\ \text{Z}K\_{0} + Z\_{\text{m}c}(I\_{0} + K\_{1}) = 0, \,\text{Z}P\_{0} + Z\_{\text{m}c}(I\_{1} + P\_{1}) = 0\\ \text{Z}K\_{N-1} + Z\_{\text{m}c}(K\_{N-2} + I\_{0}) = 0, \,\text{Z}P\_{N-1} + Z\_{\text{m}c}(P\_{N-2} + I\_{1}) = 0\\ \text{Z}[\_{0} - Z\_{\text{m}}(I\_{-1} + I\_{1}) + Z\_{\text{m}c}K\_{N-1} = 0, \,\text{Z}I\_{1} - Z\_{\text{m}c}(I\_{0} + I\_{2}) + Z\_{\text{m}c}P\_{N-1} = 0. \end{array} \tag{23}$$

Equation (23) can again be solved for the reflection coefficient at port 1 and transmission coefficients at ports 2–4 by substituting the dispersion relations and eliminating the terms *PT*, *PR*, and *KT*, *KR*. When *N* = 1, Equation (24) is identical to Equation (11).

$$\begin{split} \frac{R}{\overline{\tau}} &= \left(e^{\overline{\nu}2k\_{c}a(N+1)} - 1\right) \left\{ e^{\overline{\nu}2k\_{c}a} \left(1 + e^{\overline{\nu}2k\_{f}a}\right) - \mu e^{i(k\_{f}+k\_{c})a} \left(1 + e^{\overline{\nu}2k\_{c}a}\right) + \mu^{2}e^{i(2k\_{f}+k\_{c})a} \right\}/D \\ &\frac{T\_{2}}{\overline{\tau}} = \left(1 - e^{\overline{\nu}(N+1)k\_{c}a}\right) \left\{ 2e^{\overline{\nu}(k\_{f}+k\_{c})a} - \mu e^{i(3k\_{f}+k\_{c})a} \left(1 + e^{\overline{\nu}2k\_{c}a}\right) \right\}/D \\ &\frac{T\_{4}}{\overline{\tau}} = \mu e^{i(N+1)k\_{c}a} \left\{ 2e^{i(k\_{f}+k\_{c})a} \left(1 - e^{2k\_{c}a}\right) - \mu e^{2k\_{f}a} \left(1 - e^{i\overline{\nu}k\_{c}a}\right) \right\}/D \\ &D = \left(e^{\overline{\nu}2k\_{c}a} - 1\right) e^{\overline{\nu}2k\_{c}a} \left(1 - e^{\overline{\nu}2k\_{c}a(N+1)}\right) + 2\mu e^{i(k\_{f}+k\_{c})a} \left(1 - e^{\overline{\nu}2k\_{c}a(N+2)}\right) - \mu^{2}e^{i2k\_{c}\left(1 - e^{\overline{\nu}2k\_{c}\left(N+3\right)a}\right)} \end{split} \tag{24}$$

The coefficients can be converted to scattering parameters as before. Figure 5b shows the frequency dependence of S-parameters for *N* = 2 and *N* = 5. The additional coupling resonators generate multiple bandpass responses in *S*<sup>41</sup> and complementary notches in *S*21. The value of |*κc*| determines the sharpness of the peaks in *S*41, as well as the notch frequencies, with a larger separation between peaks for higher values. The quality factors of the peaks are higher for increasing *N*, and the effect of loss is to reduce rejection in *S*<sup>21</sup> and increase insertion loss in *S*41.

The frequencies of peaks and notches can be found by computing the maxima of *S*<sup>41</sup> or the minima of *S*<sup>21</sup> within the MI passband. The central frequencies *ω<sup>k</sup>* simplify to:

$$
\omega\_k = \omega\_0 \sqrt{\frac{1}{1 + \kappa\_\varepsilon \cos\left(\frac{\nu \pi}{N+1}\right)}}, \ \nu \in [1, N]. \tag{25}
$$

The notch spacing is nonuniform and can be adjusted by varying *ω*<sup>0</sup> or *κc*. The values for *ω<sup>k</sup>* are independent of *κ*, provided that *κ* > *κc*. When *N* is odd, there is always a peak at the resonant frequency. As previously shown, infinite rejection can be achieved at resonance for *N* = 1. However, since path losses cannot be equalised at several frequencies, infinite rejection cannot be obtained for all notches simultaneously when *N* > 1. However, simulations show that it is simple to improve performance by adjusting *λ*.

#### 2.4.3. Three-Port Filters and Multiplexers

In the absence of infinite or improved rejection (*λ* = 1), the current in port 3 is identically zero at all frequencies with port 1 as an input. This implies that port 3 can be physically removed from either the single- or multiple-notch device, along with a resonator at *n*=1 in the backward-wave guide in Figure 2a or Figure 5a. Hence, a threeport directional filter can be generated with the addition of just three weakly coupled resonators, and a backward-wave resonator at *n* = 0 loaded with one of the terminations from Figure 2b. Loss compensation, however, is then impossible due to the removal of signal paths. Nevertheless, filter performance would be indistinguishable from Figure 3a or Figure 5b, as long as *λ* is unity. Multiplexers can also be implemented by cascading filters, with the notch frequencies of the constituent filters defining the frequencies of extracted signals.

#### **3. Results and Discussion** 13.56 MHz to remove residual carrier from tag responses in HF RFID. A tuneable design

**3. Results and Discussion**

signals.

*Electronics* **2022**, *11*, x FOR PEER REVIEW 10 of 16

2.4.3. Three-Port Filters and Multiplexers

In this section, we present experimental results for filters constructed for *f*<sup>0</sup> = 13.56 MHz to remove residual carrier from tag responses in HF RFID. A tuneable design was developed to allow the demonstration of filter operation and extension to complex filters. was developed to allow the demonstration of filter operation and extension to complex filters. *3.1. Single-Notch Filter*

In this section, we present experimental results for filters constructed for <sup>0</sup> =

frequencies, infinite rejection cannot be obtained for all notches simultaneously when > 1. However, simulations show that it is simple to improve performance by adjusting .

In the absence of infinite or improved rejection ( = 1), the current in port 3 is identically zero at all frequencies with port 1 as an input. This implies that port 3 can be physically removed from either the single- or multiple-notch device, along with a resonator at *n*=1 in the backward-wave guide in Figure 2a or Figure 5a. Hence, a threeport directional filter can be generated with the addition of just three weakly coupled resonators, and a backward-wave resonator at = 0 loaded with one of the terminations from Figure 2b. Loss compensation, however, is then impossible due to the removal of signal paths. Nevertheless, filter performance would be indistinguishable from Figure 3a or 5b, as long as is unity. Multiplexers can also be implemented by cascading filters, with the notch frequencies of the constituent filters defining the frequencies of extracted

#### *3.1. Single-Notch Filter* The overall design was as shown in Figure 2a; however, resistive terminations were

The overall design was as shown in Figure 2a; however, resistive terminations were used, and ports were floated using paired capacitors. To allow frequency tuneability, all capacitors included varicaps. To confine magnetic fields and ensure nearest-neighbour coupling, all inductors used low-loss ferrimagnetic toroidal cores with inner and outer diameters of 13.2 and 21 mm (Fair-Rite 5967000601, based on "67" Ni–Zn ferrite material with cutoff at 50 MHz). The initial relative permeability was ≈40, implying that the magnetic flux was almost completely contained in the core. Two windings (with inductances *L<sup>C</sup>* and *LT*, such that *L<sup>C</sup>* + *L<sup>T</sup>* = *L*) were used for the waveguide elements, and three (with inductances *LC*, *LC*, and *LS*, such that 2*L<sup>C</sup>* + *L<sup>S</sup>* = *L*) for the coupling resonators. Adjacent windings were reversed to obtain negative coupling coefficients. A method was used to allow the tuning of mutual inductance using components constructed by 3D printing with a polycarbonate filament. Figure 6a shows this mechanism, allowing control over the relative position of two copper windings. The coupling coefficient was a nonlinear function of the winding separation with minimum separation yielding maximum coupling. used, and ports were floated using paired capacitors. To allow frequency tuneability, all capacitors included varicaps. To confine magnetic fields and ensure nearest-neighbour coupling, all inductors used low-loss ferrimagnetic toroidal cores with inner and outer diameters of 13.2 and 21 mm (Fair-Rite 5967000601, based on "67" Ni–Zn ferrite material with cutoff at 50 MHz). The initial relative permeability was ≈40, implying that the magnetic flux was almost completely contained in the core. Two windings (with inductances and , such that + = ) were used for the waveguide elements, and three (with inductances , , and , such that 2 + = ) for the coupling resonators. Adjacent windings were reversed to obtain negative coupling coefficients. A method was used to allow the tuning of mutual inductance using components constructed by 3D printing with a polycarbonate filament. Figure 6a shows this mechanism, allowing control over the relative position of two copper windings. The coupling coefficient was a nonlinear function of the winding separation with minimum separation yielding maximum coupling.

**Figure 6.** MI directional filter: (**a**) mechanism for tuning of magnetic coupling (LH–maximum coupling; RH–minimum coupling); (**b**) complete PCB with labelled port numbers. **Figure 6.** MI directional filter: (**a**) mechanism for tuning of magnetic coupling (LH–maximum coupling; RH–minimum coupling); (**b**) complete PCB with labelled port numbers.

Figure 6b shows a photograph of the completed filter constructed as a copper-clad printed circuit board using low-loss mica capacitors. Mechanical tuning was provided for all mutual inductances. The value of *M* needed for impedance matching to a 50 Ω load at resonance was first identified. Inductors *L<sup>s</sup>* were then chosen to equalise the self-inductance of all resonators, and capacitances were tuned to achieve resonance at 13.56 MHz. Table 1 shows the values of the most important parameters after tuning.



Four-port scattering parameters were measured using an electronic network analyser (Agilent E5061B). To begin with, *λ* was set to unity so that responses were independent of the port used as input. Figure 7a compares the frequency dependence of S-parameters with the predictions of the theoretical model for *κ<sup>c</sup>* = 0.28. Excellent agreement with experimental results can be seen. Larger values of *µ* were investigated and found to reduce losses and increase the passband of the bandpass channel and the notch depth. (21). For parameters in Table 1, the value of required for infinite rejection is 0.79, well within range for the coupling tuning mechanism. Figure 7b compares the simulated and measured scattering parameters obtained in this case. A performance improvement of over 50 dB is achieved compared with that in Figure 7a, with 70 dB rejection at port 2. This improvement is at the expense of reduced isolation of port 3 with a minimum attenuation of 18 dB at resonance between ports 1 and 3. However, losses in <sup>21</sup> are still extremely low.

Stopband rejection can be greatly improved by fine-tuning according to Equation

**Figure 7.** Experimental (solid) and theoretical (dashed) frequency dependence of scattering parameters for MI filter with resistive narrowband terminations (2) with ω<sup>0</sup> = 13.56 MHz, Q = 200, and 0 = 50Ω. (**a**) κ<sup>c</sup> = 0.28 and = 1; (**b**) infinite rejection with κ<sup>c</sup> = 0.25 and = 0.79. **Figure 7.** Experimental (solid) and theoretical (dashed) frequency dependence of scattering parameters for MI filter with resistive narrowband terminations (2) with *ω*<sup>0</sup> = 13.56 MHz, *Q* = 200, and *Z*0*<sup>M</sup>* = 50 Ω. (**a**) *κ<sup>c</sup>* = 0.28 and *λ* = 1; (**b**) infinite rejection with *κ<sup>c</sup>* = 0.25 and *λ* = 0.79.

#### *3.3. Double-Notch Filters 3.2. Single-Notch Filter with Infinite Rejection*

*Electronics* **2022**, *11*, x FOR PEER REVIEW 11 of 16

**Table 1.** Experimental parameter values.

*3.2. Single-Notch Filter with Infinite Rejection*

Figure 6b shows a photograph of the completed filter constructed as a copper-clad printed circuit board using low-loss mica capacitors. Mechanical tuning was provided for all mutual inductances. The value of needed for impedance matching to a 50Ω load at resonance was first identified. Inductors were then chosen to equalise the selfinductance of all resonators, and capacitances were tuned to achieve resonance at 13.56

Four-port scattering parameters were measured using an electronic network analyser (Agilent E5061B). To begin with, was set to unity so that responses were independent of the port used as input. Figure 7a compares the frequency dependence of S-parameters with the predictions of the theoretical model for = 0.28. Excellent agreement with experimental results can be seen. Larger values of were investigated and found to reduce losses and increase the passband of the bandpass channel and the notch depth.

> **Parameter Value Range of Adjustment** 1.16 μH N/A (not applicable) 460 nH N/A 710 nH N/A 0.725 0.51–0.92 0.28 0.19–0.34 <sup>0</sup> 13.56 MHz DC to 50 MHz 200 ± 10% N/A

MHz. Table 1 shows the values of the most important parameters after tuning.

A two-notch directional filter with parameter values shown in Table 1 was implemented in the same way by increasing the number of coupling resonators in each vertical branch from one to two. The magnitude of determines the frequency locations of the notches in <sup>21</sup> according to Equation (25). However, smaller values of also Stopband rejection can be greatly improved by fine-tuning *λ* according to Equation (21). For parameters in Table 1, the value of *λ* required for infinite rejection is 0.79, well within range for the coupling tuning mechanism. Figure 7b compares the simulated and measured scattering parameters obtained in this case. A performance improvement of over 50 dB is achieved compared with that in Figure 7a, with 70 dB rejection at port 2. This improvement is at the expense of reduced isolation of port 3 with a minimum attenuation of 18 dB at resonance between ports 1 and 3. However, losses in *S*<sup>21</sup> are still extremely low.

#### *3.3. Double-Notch Filters*

A two-notch directional filter with parameter values shown in Table 1 was implemented in the same way by increasing the number of coupling resonators in each vertical branch from one to two. The magnitude of *κ<sup>c</sup>* determines the frequency locations of the notches in *S*<sup>21</sup> according to Equation (25). However, smaller values of *κ<sup>c</sup>* also increase propagation losses through the coupling resonators, reducing notch depth and increasing losses in the bandpass channel. Figure 8a compares theoretical and experimental results for *κ<sup>c</sup>* = 0.3. Agreement between simulation and experimental results is again excellent. Attenuation in *S*<sup>21</sup> is the same as for the single-notch filter, implying that two signals can be extracted simultaneously without hindering performance. *Electronics* **2022**, *11*, x FOR PEER REVIEW 12 of 16 increase propagation losses through the coupling resonators, reducing notch depth and increasing losses in the bandpass channel. Figure 8a compares theoretical and experimental results for = 0.3 . Agreement between simulation and experimental results is again excellent. Attenuation in <sup>21</sup> is the same as for the single-notch filter, implying that two signals can be extracted simultaneously without hindering performance.

**Figure 8.** Experimental (solid) and theoretical (dashed) frequency dependence of S-parameters for double passband MI filter with resistive narrowband terminations with ω<sup>0</sup> = 13.56 MHz, Q = 180, 0 = 50Ω, and κ<sup>c</sup> = 0.3. (**a**) No improved rejection; (**b**) improved rejection. **Figure 8.** Experimental (solid) and theoretical (dashed) frequency dependence of S-parameters for double passband MI filter with resistive narrowband terminations with *ω*<sup>0</sup> = 13.56 MHz, *Q* = 180, *Z*0*<sup>M</sup>* = 50 Ω, and *κ<sup>c</sup>* = 0.3. (**a**) No improved rejection; (**b**) improved rejection.

Although infinite rejection appears impossible in multinotch filters, rejection can be

Toroid transformers based on soft ferrites allow magnetoinductive designs to be realised with low non-nearest-neighbour coupling. However, core materials must have low loss and high linearity. Some applications also require high power handling. In HF RFID, a modulated AC magnetic field delivers power and data to tags, which couple to the field using inductive loops. For the ISO/IEC 14443 type A standard, the carrier frequency is = 13.56 MHz, while tags use an internally generated subcarrier at =

to return data, generating sideband spectra with central frequencies, ± . Since the carrier power can be orders of magnitude greater than that of the sidebands, receiver protection is required. The single-notch filter presented here can be integrated into existing MI systems via direct connection to the waveguides, has high rejection at resonance, and may be used to transfer residual carrier to a load. However, the filter must be effective at the representative carrier powers, despite potential nonlinearities introduced by ferrite cores. To emulate generic HF RFID signals, we modulated a highpower carrier at using a simple cosine message waveform at . The waveform of the

Here and are the carrier and message amplitudes. The signal was derived from a first signal generator (Agilent E4433B) modulated by a second signal modulator (Agilent N5181A) with amplification by a 27 dB power amplifier (Minicircuits ZFDWHA-1–20+).

() = [ + (2)] (2). (26)

 16

significant performance improvement in a rejection of over 15 dB is obtained, once again at the expense of port 3 isolation. As regards single-notch filters, lower values of are needed for higher values of . Low insertion loss is observed around the resonance frequency but increases elsewhere as matching between the guides and load deteriorates.

*3.4. Double-Notch Filter with Improved Rejection*

*3.5. Applications*

input signal () is:

#### *3.4. Double-Notch Filter with Improved Rejection*

Although infinite rejection appears impossible in multinotch filters, rejection can be improved by adjusting *λ*. Figure 8b compares theoretical and measured S-parameters for a two-notch filter with *κ<sup>c</sup>* = 0.3 with *λ* chosen to equalise rejection at the two notches. A significant performance improvement in a rejection of over 15 dB is obtained, once again at the expense of port 3 isolation. As regards single-notch filters, lower values of *λ* are needed for higher values of *µ*. Low insertion loss is observed around the resonance frequency but increases elsewhere as matching between the guides and load deteriorates.

### *3.5. Applications*

Toroid transformers based on soft ferrites allow magnetoinductive designs to be realised with low non-nearest-neighbour coupling. However, core materials must have low loss and high linearity. Some applications also require high power handling. In HF RFID, a modulated AC magnetic field delivers power and data to tags, which couple to the field using inductive loops. For the ISO/IEC 14443 type A standard, the carrier frequency is *f<sup>c</sup>* = 13.56 MHz, while tags use an internally generated subcarrier at *f<sup>m</sup>* = *fc* <sup>16</sup> to return data, generating sideband spectra with central frequencies, *f<sup>c</sup>* ± *fm*. Since the carrier power can be orders of magnitude greater than that of the sidebands, receiver protection is required. The single-notch filter presented here can be integrated into existing MI systems via direct connection to the waveguides, has high rejection at resonance, and may be used to transfer residual carrier to a load. However, the filter must be effective at the representative carrier powers, despite potential nonlinearities introduced by ferrite cores. To emulate generic HF RFID signals, we modulated a high-power carrier at *f<sup>c</sup>* using a simple cosine message waveform at *fm*. The waveform of the input signal *s*(*t*) is:

$$s(t) = \left[A\_{\mathfrak{c}} + A\_{\mathfrak{m}} \cos(2\pi f\_{\mathfrak{m}} t)\right] \cos(2\pi f\_{\mathfrak{c}} t). \tag{26}$$

Here *A<sup>c</sup>* and *A<sup>m</sup>* are the carrier and message amplitudes. The signal was derived from a first signal generator (Agilent E4433B) modulated by a second signal modulator (Agilent N5181A) with amplification by a 27 dB power amplifier (Minicircuits ZFDWHA-1–20+). *A<sup>c</sup>* was adjusted to deliver a carrier power, *Pc*, into a 50 Ω load, while the modulation index *Am*/*A<sup>c</sup>* was chosen to generate typically small sidebands. Detection was performed using a digital storage oscilloscope (Keysight InfiniiVision DSOX3024T, Santa Rosa, CA, USA) using a 30 dB attenuator for unfiltered signals. Figure 9a shows the power spectral density (PSD) of the signal input to port 1 of the single-notch filter for *P<sup>c</sup>* = 1 *W*. Amplifier nonlinearities have resulted in minor harmonics of the AM spectrum, which have been partially suppressed at high frequencies with a low-pass filter. Figure 9b shows the PSD at port 2, the stopband channel. The output shows a carrier attenuation of 73 dB and a sideband attenuation of 1.2 dB, implying an output signal with almost intact message and negligible carrier. No significant additional harmonics are introduced. Figure 9c shows the attenuation of the two sidebands and the carrier over a power range between 0.1 and 1 W. Attenuation is almost independent of *Pc*, implying effective power handling despite the use of ferrites.

National

**Figure 9.** Power spectral densities of (**a**) input signal to port 1 of MI single-notch directional filter; (**b**) output signal at port 2 normalised by the input carrier power . Dashed lines: filter bandwidth; red: harmonics at multiples of carrier frequencies; green: harmonics at message sidebands. (**c**) Attenuation of sidebands and carrier. **Figure 9.** Power spectral densities of (**a**) input signal to port 1 of MI single-notch directional filter; (**b**) output signal at port 2 normalised by the input carrier power *Pc*. Dashed lines: filter bandwidth; red: harmonics at multiples of carrier frequencies; green: harmonics at message sidebands. (**c**) Attenuation of sidebands and carrier.

 was adjusted to deliver a carrier power, , into a 50 Ω load, while the modulation index / was chosen to generate typically small sidebands. Detection was performed using a digital storage oscilloscope (Keysight InfiniiVision DSOX3024T, Santa Rosa, CA, USA) using a 30 dB attenuator for unfiltered signals. Figure 9a shows the power spectral density (PSD) of the signal input to port 1 of the single-notch filter for = 1 . Amplifier nonlinearities have resulted in minor harmonics of the AM spectrum, which have been partially suppressed at high frequencies with a low-pass filter. Figure 9b shows the PSD at port 2, the stopband channel. The output shows a carrier attenuation of 73 dB and a sideband attenuation of 1.2 dB, implying an output signal with almost intact message and negligible carrier. No significant additional harmonics are introduced. Figure 9c shows the attenuation of the two sidebands and the carrier over a power range between 0.1 and 1 W. Attenuation is almost independent of , implying effective power handling despite

#### *3.6. Frequency Scaling 3.6. Frequency Scaling*

reduce performance.

the use of ferrites.

The potential for scaling to higher frequencies is largely determined by ferrite performance. Table 2 shows performance parameters for low-cost NiZn soft ceramic ferrite materials and toroids designed for RF applications, such as EMI suppressors (upper group) and broadband transformers and baluns (lower), including their initial relative permeability , Curie temperature TC, and loss factor. The main difference between groups is the maximum usable frequency Fmax. Following Snoek's limit [52], falls as Fmax is raised. High-frequency performance is therefore accompanied by a reduction in magnetic field confinement and an increase in loss. Small toroids may be used to minimise the volume of magnetic material. However, careful design will be required to optimise performance, power handling will be reduced, and automated coil winding may be required for construction [53]. Despite this, the data suggest that response may be extended to the low UHF band. Above this, air-cored inductors may be used, but nonnearest-neighbour coupling and radiation will inevitably complicate the design and The potential for scaling to higher frequencies is largely determined by ferrite performance. Table 2 shows performance parameters for low-cost NiZn soft ceramic ferrite materials and toroids designed for RF applications, such as EMI suppressors (upper group) and broadband transformers and baluns (lower), including their initial relative permeability *µ<sup>i</sup>* , Curie temperature TC, and loss factor. The main difference between groups is the maximum usable frequency Fmax. Following Snoek's limit [52], *µ<sup>i</sup>* falls as Fmax is raised. High-frequency performance is therefore accompanied by a reduction in magnetic field confinement and an increase in loss. Small toroids may be used to minimise the volume of magnetic material. However, careful design will be required to optimise performance, power handling will be reduced, and automated coil winding may be required for construction [53]. Despite this, the data suggest that response may be extended to the low UHF band. Above this, air-cored inductors may be used, but non-nearest-neighbour coupling and radiation will inevitably complicate the design and reduce performance.


Fair-Rite 5968001801 22.10 13.70 6.35 16 150 >500 68 300 @ 100 MHz

**Table 2.** Properties of a selection of ferrite materials and toroids.

#### *3.7. Future Work*

Magnetics - - - - 7.5 400 >320 M5 <3500 @ 100 MHz MI directional filters can also be designed for use at UHF. Potential research could concentrate on counteracting ferrite losses or, alternatively, minimising non-nearest-neighbour coupling and radiation in open-loop designs. Furthermore, the device wave operating principles can be extended to other metamaterial types, such as elastic or acoustic [54]. In particular, it has already been shown by us that MEMS could be used to generate highperformance two-port notch and comb filters [55]. Once again, wave analysis was used to simplify the coupled dynamical equations. Extension to multiport devices can be a fitting research target. Higher performance is expected due to inherently higher Q-factors of MEMS resonators.

#### **4. Conclusions**

New configurations of a magnetoinductive device with directional filter properties have been introduced based purely on magnetically coupled LC resonators. Design rules have been established, and methods for calculating scattering parameters when filters are used in MI systems (which have complex, frequency-dependent impedance) or conventional systems (with real impedance) have been clarified. The circuit equations have been simplified and solved using wave analysis. Analytic expressions have been developed for S-parameters and bandwidth, a simple method for introducing tuneability has been proposed, and extensions to allow infinite rejection at a chosen frequency or multiple stopbands with high rejection have been described.

Device construction and operation have been verified at HF. Ceramic ferrite cores have been used to ensure nearest-neighbour coupling in a compact layout, and simple mechanical methods of tuning resonant frequency and coupling (essential in blocker applications) have been proposed and demonstrated. Experimental results for tuneable filters, filters with infinite rejection, and filters with multiple bandstop frequencies have all been shown to agree with theoretical models. Power handing capability sufficient for blocker applications in HF RFID has been demonstrated, with little harmonic generation due to ferrite nonlinearity. Further development will involve direct connection to a magnetoinductive antenna, and this work is in progress.

**Author Contributions:** Methodology, A.V., R.R.A.S. and O.S.; software, A.V.; validation, A.V., R.R.A.S. and O.S.; formal analysis, A.V., R.R.A.S. and O.S.; investigation, A.V.; writing—original draft preparation, A.V., R.R.A.S. and O.S.; writing—review and editing, A.V., R.R.A.S. and O.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


**Andre Tavora de Albuquerque Silva 1,2,\*, Claudio Ferreira Dias 1,\*, Eduardo Rodrigues de Lima <sup>1</sup> , Gustavo Fraidenraich 2,\* and Larissa Medeiros de Almeida <sup>3</sup>**


**Abstract:** This work presents a new unit cell electromagnetic bandgap (EBG) design based on HoneyComb geometry (HCPBG). The new HCPBG takes a uniplanar geometry (UCPBG—uniplanar compact PBG) as a reference and follows similar design methods for defining geometric parameters. The new structure's advantages consist of reduced occupied printed circuit board area and flexible rejection band properties. In addition, rotation and slight geometry modification in the HCPBG cell allow changing the profile of the attenuation frequency range. This paper also presents a reconfigurable unit cell HCPBG filter strategy, for which the resonance center frequency is shifted by changing the gap capacitance with the assistance of varactor diodes. The HCPBG filter and reconfiguration behavior is demonstrated through electromagnetic (EM) simulations over the FR1 band of the 5G communication network. Intelligent communication systems can use the reconfiguration feature to select the optimal operating frequency for maximum attenuation of unwanted or interfering signals, such as harmonics or intermodulation products.

**Keywords:** electromagnetic bandgap; photonic bandgap; electromagnetic compatibility; interference; filtering; HCPBG (HoneyComb PBG)

#### **1. Introduction**

Photonic bandgaps (PBGs) are periodic structures that introduce material changes in the waveguide or printed circuit board (PCB), such as holes, patterns, or dielectric rods. They are also known as photonic crystals and are based on Electromagnetic Band Gap (EBG) properties. The geometric modification in the medium produces forbidden frequency bands for propagating waves. In other words, it works as a wave filter with high impedance at desired frequency bands. As a result, the electromagnetic waves in a PBG material are hindered due to the periodic discontinuity, which is equivalent to a photonic crystal in the light domain. They were first reported for structures at optical wavelengths in 1987 [1,2]. Since the first PBG structure publication, designers created many shapes, geometries, and materials for various applications. As an example, we can cite: microwave filters, electromagnetic compatibility (EMC) improvement [3–5], antenna beam steering [6], compact antenna arrays [7], antenna gain, efficiency, and bandwidth improvement [8–10], and beam tilting of 5G antenna arrays [11].

PBG structures have the characteristic of phase control of plane waves enabling suppression of surface waves and higher-order harmonics. It is also an exciting tool for those looking beyond 5G, where the advent of electromagnetic components can shape how they interact with the propagation environment. It is a case for reconfigurable intelligent surface (RIS), a two-dimensional surface of engineered material whose properties are reconfigurable rather than static [8,9,12]. For example, the RIS device can control scattering,

**Citation:** Tavora de Albuquerque Silva, A.; Ferreira Dias, C.; Rodrigues de Lima, E.; Fraidenraich, G.; Medeiros de Almeida, L. A New Reconfigurable Filter Based on a Single Electromagnetic Bandgap Honey Comb Geometry Cell. *Electronics* **2021**, *10*, 2390. https:// doi.org/10.3390/electronics10192390

Academic Editors: Giovanni Andrea Casula and Naser Ojaroudi Parchin

Received: 12 July 2021 Accepted: 20 September 2021 Published: 30 September 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

absorption, reflection, and diffraction properties by software according to how the environment changes over time. In principle, the RIS can form a beam and synthesize the scattering behavior of an arbitrarily shaped surface of the same size. For example, it can create a superposition of multiple beams or act as a diffuse scatterer [13]. Regarding traditional communication systems, the critical difference between a RIS and the common notion of an antenna array in 5G is that a RIS is neither part of the transmitter nor the receiver. However, it is a controllable part of the wireless propagation environment. The system-level role of a RIS is to influence the propagation of the wireless signals sent by other devices without generating its signals [14].

On the scope of the current investigation, we focus on interference filtering, which is a critical problem to be addressed as the number of wireless devices increases every year. It is of great concern that long-term interfering issues could arise due to the increasing number of heterogeneous devices. Aiming on that, it is a necessity to implement such solutions as a filtering option. For example, cable-stayed power line towers monitored by wireless sensors operating at unlicensed spectrum deliver data of utmost importance to the substations. The data help predict the risks of collapse events, and wireless sensor communication may be harmed by adjacent heterogeneous systems interference. In this way, one can use PBG as a tool to contend against interference coming from a crowded environment in several ways. Firstly, the current downscaling of devices makes structure sizes a critical issue. As technologies occupy higher spectrum frequencies, PBG is a viable tool to be adopted since the stopband frequency is proportional to half of the guided wavelength [15]. Secondly, planar PBGs like UCPBG do not need vias or unique materials suitable for standard PCB designs with no extra cost. The UCPBG works as a stopband filter due to the structure of metal etched in slots on the ground plane connected by narrow lines to form a distributed LC network [16,17].

In this paper, we present the HCPBG unit cell, a novel geometry for EBG structures that introduces new features when compared to that of the traditional UCPBG [9]. For instance, it further decreases the consumption area compared to that of the UCPBG equivalent due to the proposed hexagonal geometry. Secondly, the hexagon can keep the same perimeter as the side of a square, where the microstrip line crosses the PBG structure while occupying 65% of the area. Besides, the HCPBG unit cell can be configured with different traces and orientations, adding asymmetry to the design and producing different filtering characteristics, as simulations and measurements demonstrate. As a result of this study, we can highlight the following HCPBG advantages:


As a second novel feature, we also present a reconfiguration method to modify the center frequency of the rejection band by controlling the gap capacitance through a varactor diode. The reverse voltage of the varactor diode changes its capacitance, allowing a control system to shift filtering center frequency and bandwidth range.

Although the technique of using active discrete components to reconfigure different types of PBG structures was studied before, the reconfiguration of the UCPBG type structures, as presented in this work, is not yet reported in the literature.

Different applications of reconfigurable EBG structures were investigated over the years, such as antenna array beam-steering at 6 GHz using metal tape [6], change of patch antenna polarization for navigation systems with the use of varactor diodes [18], modification of antenna frequency and radiation pattern for WiFi/WiMAX using PIN

diode [19], cavity resonator at 10 GHz range with mechanical switching reconfiguration [20], and reconfigurable 3D MEMS filter for optical applications [21].

The HCPBG is a planar structure, and no unique material, vias, or fabrication process is needed to produce such structures. This is an advantage over tunable bandpass filters based on microelectromechanical systems (MEMS) [22]. On the other hand, planar reconfigurable PBG filters, as described in [23–25], have the advantage of large bandwidth and dynamic range. However, to achieve a frequency range of a few GHz, they show an increased area consumption and circuit complexity compared to that of the proposed filter.

This paper organizes as follows: Section 2 refers to the presented HCPBG model. Section 3 presents the simulation results comparing the classic UCPBG to the HCPBG and multiple geometries through transmission loss of a microstrip line over one single cell; it also presents the effect of different orientations. Section 4 shows the measurement setup and transmission loss results for the fabricated samples. Section 5 describes the design of the reconfigurable HCPBG filter and presents the simulation and measurement results. Finally, the discussion is closed in the Section 6.

#### **2. Design of HCPBG**

The method used to evaluate the UCPBG and HCPBG unit cells consists of a PCB having a microstrip line that crosses the entire top layer and one single cell at the bottom layer underneath the microstrip line. Figure 1 illustrates a diagram used in this work for simulation and measurements. By supposing an interfering signal impinging the stripline, the objective here is to calculate a resonance frequency for the structure to filter out the interference.

**Figure 1.** Reference diagram for simulations and measurements.

Figure 2 shows (a) the design reference UCPBG and (b) the new proposed HCPBG unit cells on microstrip substrates. The dark area is the metallic ground layer, while the white area represents the removal of the metal (air gaps). The geometry design parameters for both UCPBG and HCPBG are the gap size *g*1, *g*2, the total structure size *a*, and the trace dimensions where *t<sup>l</sup>* is the length and *t<sup>w</sup>* is the width. The gap sizes define the capacitance, while the trace length and width define the inductance. For example, reducing the gap size *g*1, *g*<sup>2</sup> increases the capacitance and, henceforth, resonance moves to lower frequencies. On the other hand, a longer length *t<sup>l</sup>* will produce a higher inductance, consequently reducing the rejection band frequency. Similarly, the width *t<sup>w</sup>* affects the inductance and, henceforth, the resonance frequency.

**Figure 2.** PBG structure designs—(**a**) Uniplanar Geometry PBG (UCPBG) and (**b**) HoneyComb PBG (HCPBG).

The lumped capacitors and inductors introduced by gaps and traces form a parallel LC network as described in [2]. The determination of a stopband frequency for a single PBG structure is roughly determined by [5]

$$f\_o = \frac{1}{2\pi\sqrt{L\mathcal{C}}},\tag{1}$$

where *L* is the inductance of the trace and *C* is the capacitance of the gaps connecting the PBG structure to the ground. The method for calculating *C* follows the model of two metal sheets coplanar capacitance on a PCB [26], which can be calculated as

$$\mathbf{C} = \begin{cases} \frac{l\varepsilon\_r}{377\pi c} \ln\left[ -\frac{2\left(1 + \sqrt[4]{1 - \frac{s\_1^2}{(2w + g\_1)^2}}\right)}{-1 + \sqrt[4]{1 - \frac{s\_1^2}{(2w + g\_1)^2}}} \right], 0 < \frac{g\_1}{g\_1 + 2w} \ll \frac{1}{\sqrt{2}}, \\\\ \frac{l\varepsilon\_r}{120c\ln\left[ -\frac{2\left(1 + \sqrt{\frac{g\_1}{2w + g\_1}}\right)}{-1 + \sqrt{\frac{g\_1}{2w + g\_1}}} \right]}, \frac{1}{\sqrt{2}} < \frac{g\_1}{g\_1 + 2w} \ll 1, \end{cases} \tag{2}$$

where *w* is the size of the metal plates of the considered capacitor model, *ε<sup>r</sup>* = 4.3 (FR4) is the substrate dielectric constant, *c* = 299,792,458 m/s is the vacuum speed of light, and *l* = *a* − *t<sup>w</sup>* − *g*<sup>1</sup> − *g*<sup>2</sup> is the gap length. It is also convenient to make *g*<sup>2</sup> = *t<sup>w</sup>* to minimize the effect on the inductance *L*.

One can calculate *L* as [27]

$$L = \rho(t\_w) \times t\_{l\prime} \tag{3}$$

where

$$\rho(t\_w) = \begin{cases} 60 \frac{\ln\left[\frac{8h}{t\_W} + \frac{t\_W}{4h}\right]}{c}, & \text{if } \frac{t\_W}{h} \le 1\\ \frac{180h\pi}{c\left(h\left(2.0895 + \ln\left[1.444 + \frac{t\_W}{h}\right]\right) + 1.5t\_W\right)}, & \text{otherwise}' \end{cases} \tag{4}$$

and *h* is the substrate thickness.

The calculated values of *f*0, *C*, and *L* define the dimensions considered in the EM simulations. It is also a starting point where the designer must proceed a fine-tune to achieve a desired resonance center frequency. Also, a second method one can use is scaling based on a predesigned structure. The rejection band moves to lower frequencies as the structure is scaled up and vice-versa. Furthermore, the attenuation further improves if the PBG structure forms a lattice. As the designer works at higher frequency ranges, UCPBG achieves smaller geometric sizes, improving its applicability. In this case, metal slots are etched in the ground plane connected by narrow lines to form a distributed LC network [17]. This method fits well for standard PCB designs because it is planar, countering the need for vias or unique materials, saving costs. Concerning the hexagonal geometry, the reduction

in the PCB-occupied area is the most attractive design feature. It can keep the same gap perimeter on the side of a square while occupying 65% of the area.

The current work investigates six variations of HCPBG geometries. The desired *f*<sup>0</sup> is achieved by simply changing the number of traces, introducing different gap lengths, rotation of PBG structure relative to the microstrip line, and geometric scaling. Table 1 describes and illustrates the variations proposed for simulations and measurements. Each line of the table constitutes the structure case, the number of traces, rotation, and the PCB bottom layer PBG structure relative to the microstrip line represented as dashed contours. The cases of Table 1 (a) and (b) explores the asymmetry in the HCPBG structure by suppressing some of the traces, which changes the LC elements from the basic geometry presented in Figure 2b.

The capacitance of the gap and the trace inductance closer to the point where the transmission line crosses the PBG have a more substantial influence on the resonance frequency. Adding asymmetry changes the values of the LC and the influence over the resonances.

The cases (a) and (b) are the three traces and show asymmetries for both gaps and traces close to the microstrip line. For cases (c) and (d), we can notice that traces and gaps are symmetrically apart from the microstrip line, and in case (e), there are two traces parallel to the microstrip line due to the rotation of 30º, for which it is expected to have similar results to the UCPBG. Finally, the case (f) is the fundamental geometry presented in Figure 2b.


**Table 1.** Six variations of HCPBG geometries considering rotations and suppressed traces.

## **3. Simulation of HCPBG**

The simulation setup consists of a reference plane PCB (without PBG on the ground layer), a UCPBG, and six HCPBG combinations shown in Table 1. Six out of seven simulations on HCPBG structures have dimensions defined in Table 2 case (a). The seventh simulation has dimensions defined in Table 2 case (b), where *g*<sup>2</sup> is the only different parameter. We calculated the PCB's frequency profile using CST Studio® [28], an electromagnetic field simulation software.

In the CST Studio simulation, we set the time domain solver, which employs finite integration technique (FIT) [29]. The time domain solver has the ability to handle large and

complex structures, and it allows for memory efficient computation. We set the accuracy to −40 dB and hexahedral mesh type. For the mesh properties, we defined 12 cells per wavelength near/far from model and 35 cells of fraction minimum cell near model.

The PCB is an FR-4 two-layer of 90 mm length, 60 mm width, and 1.6 mm thickness. The microstrip line is 50 Ω, 3 mm width and 90 mm length on the top layer. The ground plane is placed on the bottom side of the PCB, where the HCPBG and UCPBG structures are located, as shown in Figure 1. The simulations consider the dimensions of Table 2 case (a) where *L* = 3.26 nH, *C* = 1.38 pF and *f*<sup>0</sup> = 2.37 GHz when using Equations (2) and (3). The nomenclature to name the HCPBG structures is defined as HCPBG-xx-yy, where "xx" represents the number of traces and "yy" is the orientation relative to the microstrip line.

**Case** *a g***<sup>1</sup>** *g***<sup>2</sup>** *t<sup>w</sup> t<sup>l</sup>* (a) 28.5 mm 1.5 mm 1.5 mm 7.5 mm 1.5 mm 0.23*λ*2.37 GHz 0.012*λ*2.37 GHz 0.012*λ*2.37 GHz 0.06*λ*2.37 GHz 0.012*λ*2.37 GHz (b) 28.5 mm 1.5 mm 0.5 mm 7.5 mm 1.5 mm

**Table 2.** Dimensions used for UCPBG and HCPBG designs.

The most significant characteristic of HCPBG unit cell is that multiple resonance profiles emerge due to rotation and trace suppression. Figure 3 shows the transmission loss profile for UCPBG and HCPBG Table 1 case (e). Both have similar results up to 3.3 GHz, with the first resonance at 2.36 GHz and similar bandwidth (BW); BW = 136 MHz at −10 dB. This case presents a second strong resonance at 4.13 GHz with BW = 100 MHz and the third one at 5.38 GHz, but with an attenuation lower than the −10 dB. This third resonance coincides with the second resonance of the UCPBG structure but with lower attenuation.

0.23*λ*2.37 GHz 0.012*λ*2.37 GHz 0.04*λ*2.37 GHz 0.06*λ*2.37 GHz 0.012*λ*2.37 GHz

**Figure 3.** Simulation—transmission loss for UCPBG and HCPBG-4t-o3.

Figures 4 highlights the effect of HCPBG rotation for Table 1 (c) and (d) cases. HCPBG and UCPBG present a similar filtering profile observed in the previous simulation. Notice that the main differences are due to the rotation of HCPBG-4t-o1 and HCPBG-4t-o2 (Table 1 case (c) and (d), respectively). The HCPBG-4t-o1 has two strong attenuated bands, one at 2.17 GHz (BW = 167 MHz) and the second at 3.61 GHz, with a wide bandwidth (BW = 982 MHz). In HCPBG-4t-o2, the first attenuation band has an extensive bandwidth at 2.25 GHz (BW = 1.4 GHz), and the second resonance locates at 4.14 GHz (BW = 260 MHz).

**Figure 4.** Simulation—transmission loss for HCPBG with 4 traces—orientations o1 and o2.

We also simulated HCPBG with three traces and six traces, keeping the same dimensions to verify the attenuation band profile's behavior. Figure 5 shows the transmission loss for the HCPBG with 3 traces and two different orientations, as indicated in Table 1 by cases (a) and (b). Both have similar filtering profiles up to 4.5 GHz, with a first attenuation around 2 GHz, and with BW of 645 MHz for HCPBG-3t-01 and BW of 380 MHz HCPBG-3t-o2. The second attenuation band is close to 3 GHz for both. The main difference is that HCPBG-3t-o1 produces a third strong wideband attenuation at 5.75 MHz (BW = 1670 MHz). The LC network seen by the microstrip line depends on the HCPBG orientation, and it seems to have more impact when the HCPBG traces are more aligned with the microstrip line.

**Figure 5.** Simulation—transmission loss for HCPBG with 3 traces—orientations o1 and o2.

Figure 6 shows the transmission loss simulation results for the 6-trace HCPBG, Table 1 case (f), with outer gaps of 1.5 and 0.5 mm. A smaller gap results in a greater capacitance, leading to lower rejection band frequencies. The HCPBG-6t with a gap of 1.5 mm has a significant bandwidth response if we use a criterion of −5.8 dB (BW = 3.16 GHz). This structure could be helpful in the suppression of broadband noise.

**Figure 6.** Simulation—transmission loss for HCPBG with 6 traces—gap distances 1.5 and 0.5 mm.

#### **4. Measurements of HCPBG**

We also verified the analytical guidelines and simulation results performing measurements in an actual PCB. An LPKF machine milled the 2-layer FR-4 samples of UCPBG and HCPBG structures. Figure 7 shows the bottom side of two fabricated samples, UCPBG (to the left) and the HCPBG-4t-o3 (to the right). The HCPBG-3t-o1, HCPBG-3t-o2, HCPBG-6t (g = 1.5 mm), and the reference board (ground only) were also fabricated. The sample dimensions and substrates are the same used for the simulations.

**Figure 7.** Fabricated UCPBG (**left**) and HCPBG with 4 traces, orientation 3 (**right**)-PCB bottom view.

The measurement setup, shown in Figure 8, is composed of the Keysight Field Fox RF Analyzer N9914B, 2 RF cables, N-SMA adapters, and the samples. The analyzer is set to network analyzer mode and calibrated from 30 kHz to 6.5 GHz.

The measured logarithmic magnitude of S21 parameter for the UCPBG and HCPBG-4t-o3 can be seen in Figure 9. When comparing the results in Figure 3, we can observe that the simulation curves' profiles are very close to the measurement results. However, the resonance frequencies showed a slight discrepancy, with the UCPBG's case being the more evident one. This discrepancy is probably due to fabrication process variations, such as gap width, depth, and substrate dielectric constant variation. Nonetheless, the simulation results showed good accuracy that can be observed in all measurements. The UCPBG's measured first resonance frequency is at 2.18 GHz (BW = 160 MHz), and the second one at 5.4 GHz. For the HCPBG-4t-o3, the resonance frequencies are 2.4 MHz (BW = 160 MHz), 4.26 GHz (BW = 130 MHz), and 5.56 GHz. The bandwidth and maximum attenuation are

also very similar in both measurement and simulation. The reference line represents the measured S21 for the transmission line with a solid ground plane (no PBG).

**Figure 8.** Measurement Setup for Transmission Loss.

**Figure 9.** Measurement—transmission loss for HCPBG with 4 traces, orientation 3 vs. UCPBG.

The graphics of Figure 10 show the measurement results for the three trace geometries: HCPBG-3t-o1 and HCPBG-3t-o2. The simulation results are in good accordance with the measurements concerning curve profile, rejection band center frequency, and bandwidth. Similarly to the simulation, the HCPBG-3t-o1 first attenuation band is at 2.04 GHz (BW = 650 MHz), and for HCPBG-3t-o2, it is at 2.08 GHz (BW = 390 MHz). The second resonance appears at 2.92 GHz (BW = 260 MHz) and 3.05 GHz (BW = 292 MHz) for orientations 1 and 2, respectively. Also, as predicted by simulation, a third wide resonance occurs at 6.01 GHz (BW = 1.4 GHz) for the case of HCPBG-3t-o1.

**Figure 10.** Measurement—transmission loss for HCPBG with 3 traces, orientations 1 and 2.

Finally, Figure 11 shows the transmission loss measurement results for the six trace HCPBG with a gap of 1.5 mm. Although it shows a significant bandwidth response and a similar profile, when compared to that of the simulated model, the bandwidth is reduced when we use −5.8 dB criteria. To have similar bandwidth, the criteria would need to be around −5 dB. The more substantial attenuation is at 4.55 GHz (−18.8 dB).

The simulation results show an error of less than 2% in the resonance frequencies below 4 GHz, while for values above 4 GHz, the error is less than 4%. The discrepancy between the measurements and simulation results for higher frequencies could be due to a mismatch between the dielectric characteristics of the PCB and the simulation model, over frequency. The variations in the fabrication process of the board would also impact the structure geometry; for example, changing the gap capacitance and displacing the resonance frequency.

**Figure 11.** Measurement—transmission loss for HCPBG with 6 traces—gap distances 1.5 mm.

### **5. Design of Reconfigurable HCPBG**

First, it is essential to remind that the central concept of HCPBG concerns the model of two metal plates separated by gaps of etched copper. Also, the traces of each hexagon face and gaps define the features of an equivalent LC network. Then, it is possible to tune the resonance center frequency of the filter by actively controlling the parameters *L* and *C*. The simplest method to achieve frequency tuning is to switch on/off one or more traces that result in inductance changes. For this purpose, it is possible to use FET transistors or micro-electro-mechanical systems (MEMS) switches. The caveat of this approach is the additional capacitance of the FET transistor or MEMS while in an off state, making the implementation more complex.

On the other hand, it is also possible to change the LC network using a variable capacitor at the gaps close to the transmission line. The technique allows actively reconfigure the first resonance frequency adding parallel capacitance to the overall intrinsic gap capacitance. In this case, a varactor diode connected to a bias tee circuit with a port connected to a DC voltage supply allows controlling the capacitance. The varactor diode method is less complex than active inductor circuits or RF switches and provides fine-tuning of the first resonance frequency. Hence, the cost of implementing switches or the complexity of polarizing FET transistors for each trace of the HCPBG would be higher than using two varactor diodes to change the capacitance.

The selected HCPBG model for the reconfiguration study is the 6-trace type, and it can be observed in Table 1 case (f). The main dimensions used to create the reconfigurable HCPBG structure can be seen in Table 3.

**Table 3.** Reconfigurable HCPBG dimensions.


The model shown in Figure 12 was simulated in the CST studio software. It consists of a 50 Ω microstrip line on a 2-layer FR4 PCB, with a dielectric constant of 4.3, length of 90 mm, a width of 60 mm, and thickness of 1.6 mm. The ground plane is placed on the bottom side of the PCB, where the HCPBG structure is also etched. All pads, traces, and vias are designed to emulate the fabricated board.

On the bottom layer, we have the varactor diode (*D*\_*varicap*) and the DC block capacitor (*C*\_*dcblock*). The inductor (*L*\_*choke*) to isolate the DC from the AC part of the circuit is placed on the top layer. A metal via connects the DC power supply traces from the top layer to the varactor on the bottom layer. At the end of the DC power supply trace, a capacitor (*C*\_*supply*) is connected to the ground to represent the power supply's capacitive coupling to the common ground.

A simplified series RLC varactor model is used in the simulation. It is composed of a capacitor (*CT*) in series with the parasitic inductance (*LS*) and resistance (*RS*). According to the datasheet of SMV1247 from Skyworks, *L<sup>S</sup>* is 0.7 nH for the SC-79 package, and *R<sup>S</sup>* is dependent on the applied reverse voltage (*VR*), having a value that ranges from 2.5 to 9 Ω. The applied reverse voltage controls the varactor's capacitance (*CT*). The rationale behind the choice of the SM1247 is that its capacitance range is in the same order as the gap capacitance of the HCPBG, giving a good dynamic range for frequency shifting. Table 4 shows selected *C<sup>T</sup>* versus *V<sup>R</sup>* used for the simulations.

**Figure 12.** Reconfigurable HCPBG—detailed simulation design view.

**Table 4.** Reverse voltage (*VR*) versus capacitance (*CT*) of SMV1247 varactor diode.


The representation of the varactor's circuit model (*CT*, *RS*, *LS*) and bias circuit electrical connections can be seen in Figure 13. The varactor's anode is connected to a common ground plane (PCB GND) and the cathode to the DC block capacitor (*Cdc*\_*blk*), which is also connected to the HCPBG's inner GND. This configuration allows DC bias isolation from the RF ground, for the *C<sup>T</sup>* works as a second DC block element. Completing the bias tee, the inductor (*L*\_*choke*) isolates the RF signal from the DC power supply line. The DC power supply coupling capacitance (*C*\_*supply*) connects the DC line to the PCB GND.

**Figure 13.** Reconfigurable HCPBG—varactor and bias tee schematic.

The discrete component values used in the simulation can be seen in Table 5.

**Table 5.** Simulation—discrete component values.


To validate the simulation model, we fabricated the testing sample using an LPKF milling machine. In Figure 14a, is the top view of the fabricated sample, showing the microstrip line, SMA connectors, choke inductors, and DC power supply traces and pads. The bottom side of the sample is shown in Figure 14b, containing the GND plane, HCPBG, varactor diodes, and DC block capacitors.

**Figure 14.** Reconfigurable HCPBG-6t-fabricated sample. (**a**) Top view; (**b**) bottom view.

The filtering characteristic of the reconfigurable HCPBG can be observed by the simulated and measured transmission loss (S21) in Figures 15 and 16, respectively. Tables 6 and 7 show the simulated and measured resonance frequencies and bandwidths for different *V<sup>R</sup>* or *C<sup>T</sup>* values.

Both results show similar curve profiles, and reasonable approximation with respect to resonance center frequency and bandwidth. For instance, considering *C<sup>T</sup>* of 0.64 pF (*V<sup>R</sup>* = 7.5 V), the simulated first resonance frequency *f*<sup>1</sup> is 2.49 GHz (*BW*<sup>1</sup> = 331 MHz) and the measured *f*<sup>1</sup> is 2.31 GHz (*BW*<sup>1</sup> = 260 MHz).

Starting the analysis with the simulation results, as we apply the lowest *C<sup>T</sup>* value (0.64 pF) two main resonances occur, the first one at 2.49 GHz (*BW*<sup>1</sup> = 331 MHz) and the second one at 4.63 GHz (*BW*<sup>1</sup> = 647 MHz). As expected, when we increase *CT*, the resonance frequencies are shifted to lower values. For example, if we take *C<sup>T</sup>* values of 0.83 pF and 0.95 pF, *f*<sup>1</sup> is displaced by 100 MHz to 2.32 GHz (*BW*<sup>1</sup> = 244 MHz) and 2.22 GHz (*BW*<sup>1</sup> = 209 MHz), respectively. The same behavior is observed for the measurement results; for example, for *V<sup>R</sup>* of 3.5 V (0.83 pF) and 3.0 V (0.95 pF), the *f*<sup>1</sup> is displaced by 70 MHz, going from 2.15 GHz (*BW*<sup>1</sup> = 160 MHz) to 2.08 GHz (*BW*<sup>1</sup> = 160 MHz).

**Figure 15.** Simulated—transmission loss for voltage values: 7.5, 4.0, 3.5, 3.0, 2.5, 2.0, 0.0 Volts.

**Figure 16.** Measured—transmission loss for voltage values: 7.5, 4.0, 3.5, 3.0, 2.75, 2.5, 2.33, 2.0, 0.0 Volts.


**Table 6.** Simulation—stop band center frequency and bandwidth (BW).


**Table 7.** Measurement—stop band center frequency and bandwidth (BW).

The difference between the simulated and measured resonance frequencies is dependent on the varactor's bias voltage or the selected *CT*. Considering *C<sup>T</sup>* values of 1.22 pF and above, the first resonance frequency showed an error of 15%. On the other hand, for capacitance values of 0.95 pF and bellow, the difference is less than 8%. According to the datasheet, the varactor SMV1247 can show 7% to 11% variation in the capacitance, depending on the applied voltage. Added to that, we have the fabrication process and substrate dielectric variations. For the second resonance, around 4.5 GHz, for which the varactor does not play a strong influence, the error is only 3%.

Another observed characteristic is the bandwidth and maximum attenuation reduction, as the resonance is shifted to lower frequencies. This reduction is strongly influenced by the varactor's *RS*. For example, considering *C<sup>T</sup>* of 0.77 pF, then *R<sup>S</sup>* is 3.9 Ω and simulated S21 equals to −20 dB @ 2.36 GHz. For *C<sup>T</sup>* of 1.22 pF, then *R<sup>S</sup>* = 6.3 Ω and simulated S21 equals to −12.6 dB @ 2.02 GHz. Due to *R<sup>S</sup>* effect, the −10 dB criteria can not be achieved for 1.88 pF (S21 = −7.29 dB @ 1.73 GHz) and 8.86 pF (S21 = −1.35 dB @ 0.98 GHz), limiting the dynamic range of the reconfigurable filter. The measurement results confirm this characteristic. When we set *V<sup>R</sup>* to 2.0 V, or equivalently, *C<sup>T</sup>* to 1.88 pF, the filter attenuation stays above the −10 dB criteria (S21 = −6.97 dB @ 1.5 GHz). If we remove *R<sup>S</sup>* from the simulation, the bandwidth is also reduced as the frequency shifts, however the entire range of *C<sup>T</sup>* achieves the −10 dB attenuation criteria.

*C<sup>T</sup>* has a stronger influence over the first resonance frequency. For example, in Table 6, if we observe the resonance frequencies at 0.64 pF and 8.86 pF, *f*<sup>1</sup> shifts 1.51 GHz, however, the second resonance (*f*2) shifts only 400 MHz.

For a −10 dB criteria, the measurements show a filtering capability that ranges from 1.72 GHz (*V<sup>R</sup>* = 2.33 V) to 2.44 GHz (*V<sup>R</sup>* = 7.5 V, upper band of *f*<sup>1</sup> = 2.31 GHz), that is 720 MHz bandwidth coverage.

#### **6. Conclusions**

In this paper, we presented a new planar PBG geometry based on the UCPBG: the HCPBG. It occupies less PCB area on the reference plane allowing a more compact usage of the space area. The simulations showed good accuracy when compared to that of the measurements. We showed through simulation and measurement that HCPBG unit cell structure can produce similar characteristics to the UCPBG, but it also adds additional resonance bands of interest. Furthermore, the rejection band is reconfigurable depending on the orientation and number of traces used, which results in more flexibility on filter designs for interfering signals. We also presented a reconfigurable HCPBG single-cell structure that can be controlled electronically. It allows for changing the rejection band profile by applying DC voltage at the control port. The control circuit, composed of two varactor diodes and a bias tee circuit, allows changing the gap capacitance of the HCPBG structure. The concept was demonstrated by EM simulation, including the varactor model and bias tee components. The measurement results point out that the resonance center frequency changes proportionally to the varactor's capacitance, thereby agreeing with the results observed in simulations. The simulated model showed less than 4% error in the

resonance center frequency. For higher frequencies, the difference is stronger, probably due to variations in the fabrication process and a model mismatch of the PCB dielectric characteristics. In the case of reconfigurable HCPBG, we observed that the varactor's capacitance plays a strong role in the difference between the simulated and measured S parameters. The first simulated resonances showed 15% differences with respect to the measured ones, however, the second resonance has a maximum of 3% variation.

**Author Contributions:** Conceptualization, A.T.d.A.S.; methodology, validation, formal analysis, investigation, A.T.d.A.S., C.F.D., E.R.d.L. and G.F.; resources, E.R.d.L. and L.M.d.A.; data curation, writing—original draft preparation, A.T.d.A.S.; writing—review and editing, A.T.d.A.S., C.F.D., E.R.d.L., G.F. and L.M.d.A.; visualization, supervision, project administration, funding acquisition, E.R.d.L. and L.M.d.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors are grateful to R&D Project PD-07130-0047, funded by Transmissora Aliança de Energia Elétrica SA (TAESA), with resources from ANEEL R&D ProgramProgram and São Paulo Research Foundation (FAPESP) grant #15/24494-8.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **Abbreviations**

The following abbreviations are used in this manuscript:


## **References**

