Surrogate‐assisted EM‐driven miniaturization of wideband microwave couplers by means of co‐simulation low‐fidelity models

This article proposes a methodology for rapid design optimization of miniaturized wideband couplers. More specifically, a class of circuits is considered, in which conventional transmission lines are replaced by their abbreviated counterparts referred to as slow‐wave compact cells. Our focus is on explicit reduction of the structure size as well as on reducing the CPU cost of the design process. For the sake of computational feasibility, a surrogate‐based optimization paradigm involving a co‐simulation low‐fidelity model is used. The latter is a fundamental component of the proposed technique. The low‐fidelity model represents cascaded slow‐wave cells replacing the low‐impedance lines of the original coupler circuit. It is implemented in a circuit simulator (here, ADS) and consists of duplicated compact cell EM simulation data as well as circuit theory‐based feeding line models. Our primary optimization routine is a trust‐region‐embedded gradient search algorithm. To further reduce the design cost, the system response Jacobian is estimated at the level of the low‐fidelity model, which is sufficient due to good correlation between the low‐ and high‐fidelity models. The coupler is explicitly optimized for size reduction, whereas electrical performance parameters are controlled using a penalty function approach. The presented methodology is demonstrated through the design of a 1‐GHz wideband microstrip branch‐line coupler. Numerical results are supported by experimental validation of the fabricated coupler prototype.     

Cost‐efficient design methodology for compact rat‐race couplers

In this article, a reliable and low‐cost design methodology for simulation‐driven optimization of miniaturized rat‐race couplers (RRCs) is presented. We exploit a two‐stage design approach, where a composite structure (a basic building block of the RRC structure) is first optimized using a pattern search algorithm, and, subsequently, the entire coupler is tuned by means of surrogate‐based optimization (SBO) procedure. SBO is executed with the underlying low‐fidelity model implemented as cascaded response surface approximations (RSAs) of the composite structure. Full‐wave analysis of the entire coupler is required at the tuning stage only. By combining SBO with coupler decomposition and RSA surrogates, the overall cost of the design process corresponds (in terms of CPU time) to less than three electromagnetic simulations of the compact RRC, and results in highly miniaturized structure (82% footprint reduction compared to conventional coupler) that exhibits perfect return loss and isolation (almost ?60 dB at the operating frequency), as well as a strong harmonic and spurious suppression (below ?20 dB) in, approximately, 3–9.5 GHz frequency band. © 2014 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:236–242, 2015.     

Performance‐driven modeling of compact couplers in restricted domains

Fast surrogate models can play an important role in reducing the cost of Electromagnetic (EM)‐driven design closure of miniaturized microwave components. Unfortunately, construction of such models is challenging due to curse of dimensionality and wide range of geometry parameters that need to be included in order to make it practically useful. In this letter, a novel approach to design‐oriented modeling of compact couplers is presented. Our method allows for building surrogates that cover wide range of operating conditions and/or material parameters, which makes them useful for design purposes. At the same time, careful definition of the model domain permits dramatic (volume‐wise) reduction of the of the design space region that needs to be sampled, thus, keeping the number of training data samples at acceptable levels. The proposed technique is demonstrated using a compact rat‐race coupler modeled for operating frequencies from 1 to 2 GHz and power split of ?6 to 0 dB. Benchmarking and application examples for coupler design optimization as well as experimental validation are also provided.     

Phase‐spacing optimization of linear microstrip antenna arrays using simulation‐based surrogate superposition models

A technique for simulation‐driven optimization of the phase excitation tapers and spacings for linear arrays of microstrip patch antennas is presented. Our technique exploits two models of the array under optimization: an analytical model which is based on the array factor, as well as an electromagnetic (EM) simulation‐based surrogate model of the entire array. The former is used to provide initial designs which meet the design requirements imposed on the radiation response. The latter is used for tuning of the array radiation response while controlling the array reflection response as well as for validation of the final design. Furthermore, the simulation‐based surrogate model allows for subsequent evaluation of the array responses in the beam scanning operation at negligible computational costs. The simulation‐based surrogate model is constructed with a superposition of simulated radiation and reflection responses of the array under design with only one radiator active at a time. Low computational cost of the surrogate model is ensured by the EM‐simulation data computed with coarse meshes. Reliability of the model is achieved by means of suitable correction carried out with respect to the high‐fidelity array model. The correction is performed iteratively in the optimization process. Performance, numerical efficiency, and accuracy of the technique is demonstrated with radiation pattern synthesis of linear arrays comprising 32 microstrip patch antennas by phase‐spacing optimization. Properties of the optimal designs in the beam scanning operation are then studied using the superposition models and compared to suitably selected reference designs. The proposed technique is versatile as it also can be applied for simulation‐based optimization of antenna arrays comprising other types of individually fed elements. © 2015 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:536–547, 2015.     

Improved trust‐region gradient‐search algorithm for accelerated optimization of wideband antenna input characteristics

Full‐wave electromagnetic (EM) simulation models are ubiquitous in carrying out design closure of antenna structures. Yet, EM‐based design is expensive due to a large number of analyses necessary to yield an optimized design. Computational savings can be achieved using, for example, adjoint sensitivities, surrogate‐assisted procedures, design space dimensionality reduction, or similar sophisticated means. In this article, a simple modification of a rudimentary trust‐region‐embedded gradient search with numerical derivatives is proposed for reduced‐cost optimization of input characteristics of wideband antennas. The approach exploits information and history of relative changes of the design (as compared with the trust region size) during algorithm iterations to control the updates of components of the antenna response Jacobian, specifically, to execute them only if necessary. It is demonstrated that the proposed framework may lead to over 50% savings over the reference algorithm with only minor degradation of the design quality, specifically, up to 0.3 dB (or <3%). Numerical results are supported by experimental validation of the optimized antenna designs. The presented algorithm can be utilized as a stand‐alone optimization routine or as a building block of surrogate‐assisted procedures.     

Low‐cost multiband compact branch‐line coupler design using response features and automated EM model fidelity adjustment

Design closure of compact microwave components is a challenging problem because of significant electromagnetic (EM) cross‐couplings in densely arranged layouts. A separate issue is a large number of designable parameters resulting from replacement of conventional transmission line sections by compact microstrip resonant cells. This increases complexity of the design optimization problem and requires employment of expensive high‐fidelity EM analysis for reliable performance evaluation of the structure at hand. Consequently, neither conventional numerical optimization algorithms nor interactive approaches (e.g., experience‐driven parameters sweeps) are capable of identifying optimum designs in reasonable timeframes. Here, we discuss application of feature‐based optimization for fast design optimization of dual‐ and multiband compact couplers. On one hand, design of such components is difficult because of multiple objectives (achieving equal power split and good matching and port isolation for all frequency bands of interest). On the other hand, because of well‐defined shapes of the S‐parameter responses for this class of components, feature‐based optimization seems to be well suited to control multiple figures of interest as demonstrated in this work. Two‐level EM modeling is used for further design cost reduction. More importantly, we develop a procedure for automated determination of the low‐fidelity EM model coarseness that allows us to find the fastest possible model that still ensures sufficient correlation with its high‐fidelity counterpart, which is critical for robustness of the optimization process. Our approach is illustrated using two dual‐band compact couplers. Experimental validation is also provided.     

An X‐band compact and low‐profile waveguide magic‐T

An X‐band low‐profile waveguide magic‐T is presented as a combination of two compact and co‐planar E‐plane and H‐plane T‐junctions. Two architectures are proposed according to the sum and difference ports lying on the same side or on opposite sides. This can be an attractive feature in the design of compact architectures. A proof of concept is manufactured with low‐cost 3D printing technology and electroplated. Good agreement between simulated and measured results is observed.     

Reduced‐cost microwave filter modeling using a two‐stage Gaussian process regression approach

A technique for the reduced‐cost modeling of microwave filters is presented. Our approach exploits variable‐fidelity electromagnetic (EM) simulations, and Gaussian process regression (GPR) carried out in two stages. In the first stage of the modeling process, a mapping between EM simulation filter models of low and high fidelity is established. The mapping is subsequently used in the second stage, making it possible for the final surrogate model to be constructed from training data obtained using only a fraction of the number of high‐fidelity simulations normally required. As demonstrated using three examples of microstrip filters, the proposed technique allows us to reduce substantially (by up to 80%) the central processing unit (CPU) cost of the filter model setup, as compared to conventional (single‐stage) GPR—the benchmark modeling method in this study. This is achieved without degrading the model generalization capability. The reliability of the two‐stage modeling method is demonstrated through the successful application of the surrogates to surrogate‐based filter design optimization. © 2014 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:453–462, 2015.     

EM‐simulation‐driven design optimization of compact microwave structures using multi‐fidelity simulation models and adjoint sensitivities

Reliable design of miniaturized microwave structures requires utilization of full‐wave electromagnetic (EM) simulation models because other types of representations such as analytical or equivalent circuit models are of insufficient accuracy. This is primarily due to considerable cross‐coupling effects in tightly arranged layouts of compact circuits. Unfortunately, high computational cost of accurate EM analysis makes the dimension adjustment process challenging, particularly for traditional methods based on parameter sweeps, but also for conventional numerical optimization techniques. In this article, low‐cost simulation‐driven designs of compact structures were demonstrated using gradient search with adjoint sensitivities as well as multi‐fidelity EM simulation models. The optimization process was arranged sequentially, with the largest steps taken at the level of coarse‐discretization models. Subsequent fine tuning was realized with the models of higher fidelity. Switching between the models was realized by means of adaptively controlled termination conditions. This allowed for considerable reduction of the design cost compared with single‐level optimization. The approach was illustrated using a compact microstrip rat‐race coupler with two cases considered, that is, (i) bandwidth enhancement, and (ii) minimization of the structure size. In both cases, the optimization cost corresponded to a few high‐fidelity EM simulations of the coupler structure. © 2016 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:442–448, 2016.     

Multi‐objective design optimization of antennas for reflection,size, and gain variability using kriging surrogates and generalized domain segmentation

Cost‐efficient multi‐objective design optimization of antennas is presented. The framework exploits auxiliary data‐driven surrogates, a multi‐objective evolutionary algorithm for initial Pareto front identification, response correction techniques for design refinement, as well as generalized domain segmentation. The purpose of this last mechanism is to reduce the volume of the design space region that needs to be sampled in order to construct the surrogate model, and, consequently, limit the number of training data points required. The recently introduced segmentation concept is generalized here to allow for handling an arbitrary number of design objectives. Its operation is illustrated using an ultra‐wideband monopole optimized for best in‐band reflection, minimum gain variability, and minimum size. When compared with conventional surrogate‐based approach, segmentation leads to reduction of the initial Pareto identification cost by over 20%. Numerical results are supported by experimental validation of the selected Pareto‐optimal antenna designs.     

Compact cell topology selection for size‐reduction‐oriented design of microstrip rat‐race couplers

In this work, we address the problem of compact cell topology selection for miniaturization of rat‐race couplers. The principal objective of the design process is to achieve the smallest possible footprint of the coupler, while maintaining the required levels of electrical parameters imposed on its components. Our approach permits identification of the minimum achievable coupler area, provided that the circuit is composed of a given compact cell and folded lines. This allows for the quantitative assessment of a set of considered cells with respect to the miniaturization capabilities they exhibit under certain design specifications. The proposed method is validated using 6 different cells with unified parameterization to identify the smallest rectangular‐like rat‐race coupler described by 2 design specifications. The obtained results attest that circuit topology and electrical parameters of the reference design are critical factors determining the final miniaturization rate. The proof‐of‐concept prototype devices occupy merely 8% of the conventional coupler area, while preserving fractional bandwidths (20% and 13.5%) of their conventional counterparts. The experimental results confirm the claims inferred from the numerical data.     

Rapid surrogate‐assisted design optimization of minimum‐size broadband branch‐line couplers with variable topology

This work discusses simulation‐driven design of miniaturized wideband branch‐line couplers with a variable topology. Size reduction is enabled here by replacing uniform transmission lines of the original coupler with slow‐wave structures in the form of cascaded compact cells and meander lines. The primary goal is to determine a number of cells in the cascade and particular cell dimensions for which the minimum size of the coupler as well as its required operating conditions are ensured. To this end, we employ a surrogate‐assisted technique involving a trust‐region gradient search framework. Computational efficiency of the design process stems from estimating the Jacobian of circuit responses at the level of a low‐fidelity model of the cascade. The latter is composed in a circuit simulator from duplicated EM‐evaluated data blocks of a single cell and is well correlated with the corresponding high‐fidelity model. The key advantage of this work is the utilization of a reconfigurable, cheap, and well‐aligned low‐fidelity model. The proposed approach is demonstrated through design of a minimum‐size two‐section branch‐line coupler with quasi‐periodic dumbbell‐shaped cells and meander lines. Excellent circuit performance as well as its small size showcase the reliability and usefulness of the presented method. Experimental verification is also provided.     

EM‐based yield optimization exploiting trust‐region optimization and space mapping technology

Design centering is a design problem which looks for nominal values of circuit parameters that maximize the probability of satisfying the design specification (yield function). Direct yield optimization of electromagnetic (EM)‐based circuits is obstructed by the high expense of EM simulations required in the yield estimation process. Also, the absence of any gradient information represents an obstacle against the optimization process. In this article, a new approach for design centering and yield optimization of EM‐based circuits is introduced. In the proposed approach, the generalized space mapping (SM) technique is incorporated with a derivative‐free trust region optimization method (NEWUOA). Moreover, a variance reduction sampling technique is implemented in the yield estimation process. Two techniques suitable for the microwave circuit design centering process are introduced. The first technique exploits the surrogate developed using any circuit optimizer, for example, minimax optimizer, in the yield maximization process. While the second technique iteratively constructs and updates an SM surrogate during the yield optimization process. Our novel approach is illustrated by practical examples showing its efficiency. One of the examples is entirely designed within the sonnet em environment. © 2014 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:474–484, 2015.     

Analysis of scalable two‐π equivalent‐circuit model for on‐chip spiral inductors

High‐accuracy inductor model is vital for the success of RF/mm‐wave circuit design. In this article, the development of two‐π scalable model with four ladder skin effect structure has been described in detail. For the scalable compact circuit modeling, a set of formulas by which all of the compact circuit elements can be calculated according to the components geometric dimensions and process parameters will be given. The proposed modeling method is regarded as full scalable as all the component parameters are calculated by physical equations or revise equations. A series of spiral inductors with various geometries have been fabricated with 0.13 μm SiGe BiCMOS aluminum process to verify the model. Excellent agreements are obtained between the measured data and calculation form the proposed model up to frequencies above self‐resonant. This scalable 28‐element two‐π model enables to accurately characterize RF behaviors of on‐chip spiral inductors and optimize the inductor performance. © 2014 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:93–100, 2015.     

Reliable design optimization of microwave structures using multipoint‐response‐correction space mapping and trust regions

Space mapping (SM) is one of the most efficient simulation‐driven design technologies used in microwave engineering to date. It includes so‐called output SM that ensures exact matching between the EM‐evaluated microwave structure under consideration (fine model) and its surrogate at the current design. The standard, single‐point output SM exploits the fine model data at a single design and is not able to align the models' sensitivity. Here, a multipoint response correction is proposed that generalizes the concept of output SM. By using a design‐variable‐dependent correction term and exploiting all available fine model information, the proposed technique provides exact match between the surrogate and the fine model at several designs. This retains the benefits of output SM but also enhances sensitivity matching between the two models, which results in improved performance of the SM optimization process. The efficiency of the propose approach is demonstrated using several microwave design problems. © 2011 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2011.     

Rapid design and size reduction of microwave couplers using variable‐fidelity EM‐driven optimization

In this article, fast electromagnetic (EM) simulation‐driven design optimization of compact microwave couplers is addressed. The main focus is on explicit reduction of the circuit footprint. Our methodology relies on the penalty function approach, which allows us to minimize the circuit area while ensuring equal power split between the output ports and providing a sufficient bandwidth with respect to the return loss and isolation around the operating frequency. Computational efficiency of the design process is achieved by exploiting variable‐fidelity EM simulations, local response surface approximation models, as well as suitable response correction techniques for design tuning. The technique described in this work is demonstrated using two examples of compact rat‐race couplers. The size‐reduction‐oriented designs are compared with performance‐oriented ones to illustrate available design trade‐offs. Final design solutions of the former case illustrate ～92% of miniaturization for both coupler examples (with corresponding fractional bandwidths of 16%). Alternative design solutions pertaining to the latter case show a lesser size reduction (～90% for both examples), but present a much wider bandwidths (～25% for both couplers). The overall computational cost of the design procedure corresponds to about 20 and 10 high‐fidelity coupler simulations for the first and second design example, respectively. Numerical results are also validated experimentally. © 2015 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:27–35, 2016.     

Systematic design of 7‐to‐40‐GHz on‐chip mixer based on optimal impedance deviation coefficient

In this article, a 7‐GHz to 40‐GHz ultra‐wideband passive double‐balanced mixer MMIC using compact wideband Marchand balun (CWMB) is presented. The CWMB is analyzed and designed by introducing a novel optimal impedance deviation coefficient. A trade‐off between the needed bandwidth and the acceptable insertion loss in an ultra‐wideband passive‐doubly‐balanced mixer design can be made through introducing the optimal impedance deviation coefficient. Finally, to verify the proposed methodology, a compact wideband passive double‐balanced mixer monolithic microwave integrated circuit (MMIC) was designed and fabricated using a standard gallium arsenide (GaAs) pHEMT technology according to the process characteristics. Experimental results show that an ultra‐wideband mixer MMIC is realized from 7 GHz to 40 GHz (140% fractional bandwidth) with a measured conversion loss between 9.5 dB~12.5 dB (in‐band fluctuation less than 3 dB) and a LO‐to‐RF isolation larger than 34 dB. The measurement results are in good agreement with the simulation results.     

Reduced‐cost surrogate modeling of input characteristics and design optimization of dual‐band antennas using response features

In this article, a procedure for low‐cost surrogate modeling of input characteristics of dual‐band antennas has been discussed. The number of training data required for construction of an accurate model has been reduced by representing the antenna reflection response to the level of suitably defined feature points. The points are allocated to capture the critical features of the reflection characteristic, such as the frequencies and the levels of the resonances, and supplemented by the additions (infill) points, which is necessary to provide sufficient data that allows restoring the entire response through interpolation. Because the coordinates of the feature points exhibit less nonlinear behavior (as a function of antenna geometry parameters) compared to S‐parameters as a function of frequency, surrogate model construction can be realized with a smaller number of data points. The presented modeling approach is demonstrated using an example of a planar dipole antenna. Also, the feature‐based method is favorably compared to direct modeling of reflection characteristics using kriging. The relevance of the technique is further verified by its application for design optimization.     

Characterization of Si‐BCB transmission line at millimeter‐wave frequency by compact fractional‐order equivalent circuit model

In this article, fractional‐order calculus is introduced to characterize Si‐BCB transmission line up to millimeter‐wave frequency region. Direct calculation method is employed to build this compact fractional‐order equivalent circuit model. Measured results have confirmed that the proposed fractional‐order model is capable of accurately describing the behavior of BCB‐Si T‐Line over a large range of frequencies up to 110 GHz.     

Dynamic output‐feedback fuzzy MPC for Takagi‐Sugeno fuzzy systems under event‐triggering–based try‐once‐discard protocol

The fuzzy model predictive control (FMPC) problem is studied for a class of discrete‐time Takagi‐Sugeno (T‐S) fuzzy systems with hard constraints. In order to improve the network utilization as well as reduce the transmission burden and avoid data collisions, a novel event‐triggering–based try‐once‐discard (TOD) protocol is developed for networks between sensors and the controller. Moreover, due to practical difficulties in obtaining measurements, the dynamic output‐feedback method is introduced to replace the traditional state feedback method for addressing the FMPC problem. Our aim is to design a series of controllers in the framework of dynamic output‐feedback FMPC for T‐S fuzzy systems so as to find a good balance between the system performance and the time efficiency. Considering nonlinearities in the context of the T‐S fuzzy model, a “min‐max” strategy is put forward to formulate an online optimization problem over the infinite‐time horizon. Then, in light of the Lyapunov‐like function approach that fully involves the properties of the T‐S fuzzy model and the proposed protocol, sufficient conditions are derived to guarantee the input‐to‐state stability of the underlying system. In order to handle the side effects of the proposed event‐triggering–based TOD protocol, its impacts are fully taken into consideration by virtue of the S‐procedure technique and the quadratic boundedness methodology. Furthermore, a certain upper bound of the objective is provided to construct an auxiliary online problem for the solvability, and the corresponding algorithm is given to find the desired controllers. Finally, two numerical examples are used to demonstrate the validity of proposed methods.