# Computational Electromagnetics (CEM)

CEM group concentrates on theoretical research in classical electromagnetic theory. Among the currently developed topics belong radiation properties of electrically small radiators and scatterers and their optimizations. Members of the group have for many years been also involved in classical antenna theory and field propagation in artificial composite materials.

Current research activities are strongly entangled with a “source concept” paradigm, in which all involved field quantities as well as engineering metrics are described solely in terms of electromagnetic sources. The idea of the source concept is sketched in the following figure.

CEM team collaborates with other department’s teams. Theses of various topics are supervised by members of the group.

### Past Members

Vit Losenicky
Michal Masek
Jakub Liska
Jonas Tucek
Vojtech Neuman

Martin Strambach
Lamyae Akrou

## Projects

• Virtual Prototyping and Validation of Electromagnetic Systems (2018-2021, TH04010373)
• Fundamental Bounds on Electromagnetic Radiation and Scattering Phenomena and Associated Realizable Subforms (2019-2021, GA19-06049S)
• Tools for Synthesis of Antennas and Sensors (2014-2017, TA04010457)
• Source Concept of Electrically Small Antenna Synthesis  (2015-2017, GA15-10280Y)

## Active External Collaborations

• Brno University of Technology, Czech Republic (Petr Kadlec)
• University of Lund, Sweden (Mats Gustafson, Casimir Ehrenborg)
• University of Santa Clara, USA (Kurt Schab)
• KTH Royal Institute of Technology, Sweden (Lars Johnsson)
• Catholic University of Leuven, Belgium (Xuezhi Zhang, Guy Vandenbosch)
• Université Nice Sophia Antipolis (Fabien Ferrero)
• MECAS ESI s.r.o., Czech Republic (Jaroslav Rymus)

### Journal Papers (Last Three Years)

#### Current and Antenna Optimization

 M. Gustafsson and M. Capek: "Maximum Gain, Effective Area, and Directivity", 2019, eprint arXiv: 1812.07058. [Online]. Available: https://arxiv.org/abs/1812.07058 Abstract Fundamental bounds on antenna gain are found via convex optimization of the current density in a prescribed region. Various constraints are considered, including self-resonance and only partial control of the current distribution. Derived formulas are valid for arbitrarily shaped radiators of a given conductivity. All the optimization tasks are reduced to eigenvalue problems, which are solved efficiently. The second part of the paper deals with superdirectivity and its associated minimal costs in efficiency and Q-factor. The paper is accompanied with a series of examples practically demonstrating the relevance of the theoretical framework and entirely spanning wide range of material parameters and electrical sizes used in antenna technology. Presented results are analyzed from a perspective of effectively radiating modes. In contrast to a common approach utilizing spherical modes, the radiating modes of a given body are directly evaluated and analyzed here. All crucial mathematical steps are reviewed in the appendices, including a series of important subroutines to be considered making it possible to reduce the computational burden associated with the evaluation of electrically large structures and structures of high conductivity. M. Capek, L. Jelinek, M. Gustafsson: "Inversion-Free Evaluation of Nearest Neighbors in Method of Moments", IEEE Antennas and Wireless Propagation Letters, vol. xx, pp. xx - xx, 2019. Abstract A recently introduced technique of topology sensitivity in method of moments is extended by the possibility of adding degrees-of-freedom (reconstruct) into underlying structure. The algebraic formulation is inversion-free, suitable for parallelization and scales favorably with the number of unknowns. The reconstruction completes the nearest neighbors procedure for an evaluation of the smallest shape perturbation. The performance of the method is studied with a greedy search over a Hamming graph representing the structure in which initial positions are chosen from a random set. The method is shown to be an effective data mining tool for machine learning-related applications. M. Capek, L. Jelinek, K. Schab, M. Gustafsson, B.L.G. Jonsson, F. Ferrero, and C. Ehrenborg: "Optimal Planar Electric Dipole Antenna", IEEE Antenna and Propagation Magazine, vol. xx, pp. xx - xx, 2019. Abstract Considerable time is often spent optimizing antennas to meet specific design metrics. Rarely, however, are the resulting antenna designs compared to rigorous physical bounds on those metrics. Here we study the performance of optimized planar meander line antennas with respect to such bounds. Results show that these simple structures meet the lower bound on radiation Q-factor (maximizing single resonance fractional bandwidth), but are far from reaching the associated physical bounds on efficiency. The relative performance of other canonical antenna designs is compared in similar ways, and the quantitative results are connected to intuitions from small antenna design, physical bounds, and matching network design. M. Capek, L. Jelinek, and M. Gustafsson: "Shape Synthesis Based on Topology Sensitivity", IEEE Trans. Antennas Propag., vol. xx, pp. xx - xx, 2019. Abstract A method evaluating the sensitivity of a given parameter to topological changes is proposed within the method of moments paradigm. The basis functions are used as degrees of freedom which, when compared to the classical pixeling technique, provide important advantages, one of them being impedance matrix inversion free evaluation of the sensitivity. The devised procedure utilizes port modes and their superposition which, together with only a single evaluation of all matrix operators, leads to a computationally effective procedure. The proposed method is approximately one hundred times faster than contemporary approaches, which allows the investigation of the sensitivity and the modification of shapes in real-time. The method is compared with known approaches and its validity and effectiveness is verified using a series of examples. The procedure can be implemented in up-to-date EM simulators in a straightforward manner. It is shown that the iterative repetition of the topology sensitivity evaluation can be used for gradient-based topology synthesis. This technique can also be employed as a local step in global optimizers. M. Gustafsson, M. Capek, and K. Schab, "Tradeoff Between Antenna Efficiency and Q-Factor", IEEE Trans. Antennas Propag., vol. 67, pp. 2482 - 2493, 2019. Abstract The trade-off between radiation efficiency and antenna bandwidth, expressed in terms of Q-factor, for small antennas is formulated as a multi-objective optimization problem in current distributions of predefined support. Variants on the problem are constructed to demonstrate the consequences of requiring a self-resonant current as opposed to one tuned by an external reactance. The resulting Pareto-optimal sets reveal the relative cost of valuing low Q-factor over high efficiency, the cost in efficiency to require a self-resonant current, and other insights. L Jelinek, K. Schab, and M. Capek, "The Radiation Efficiency Cost of Resonance Tuning", IEEE Trans. Antennas Propag, vol. 66, pp. 6716 - 6723, 2018. Abstract Existing optimization methods are used to calculate the upper-bounds on radiation efficiency with and without the constraint of self-resonance. These bounds are used for the design and assessment of small electric-dipole-type antennas. We demonstrate that the assumption of lossless, lumped, external tuning skews the true nature of radiation efficiency bounds when practical material characteristics are used in the tuning network. A major result is that, when realistic (e.g., finite conductivity) materials are used, small antenna systems exhibit dissipation factors which scale as (ka)^(−4), rather than (ka)^(−2) as previously predicted under the assumption of lossless external tuning. M. Capek, M. Gustafsson, and K. Schab, "Minimization of antenna quality factor", IEEE Trans. Antennas Propag., vol. 65, pp. 4115 - 4123, 2017. Abstract The optimal currents on arbitrarily shaped radiators with respect to the minimum quality factor Q are found using a simple and efficient procedure. The solution starts with a reformulation of the problem of minimizing quality factor Q as an alternative, so-called dual, problem. Taking advantage of modal decomposition and group theory, it is shown that the dual problem can easily be solved and always results in minimal quality factor Q. Moreover, the optimization procedure is generalized to minimize quality factor Q for embedded antennas, with respect to the arbitrarily weighted radiation patterns, or with prescribed magnitude of the electric and magnetic near-fields. The obtained numerical results are compatible with previous results based on composition of modal currents, convex optimization, and quasi-static approximations; however, using the methodology in this paper, the class of solvable problems is significantly extended. L. Jelinek and M. Capek, "Optimal currents on arbitrarily shaped surfaces," IEEE Trans. Antennas Propag., vol. 65, pp. 329 - 341, 2017. Abstract An optimization problem has been formulated to find a resonant current extremizing various antenna parameters. The method is presented on, but not limited to, particular cases of gain G, quality factor Q, gain to quality factor ratio G/Q, and radiation efficiency $\eta$ of canonical shapes with conduction losses explicitly included. The Rao-Wilton-Glisson basis representation is used to simplify the underlying algebra while still allowing surface current regions of arbitrary shape to be treated. By switching to another basis generated by a specific eigenvalue problem, it is finally shown that the optimal current can, in principle, be found as a combination of a few eigenmodes. The presented method constitutes a general framework in which the antenna parameters, expressed as bilinear forms, can automatically be extremized. M. Capek and L. Jelinek, "Optimal composition of modal currents for minimal quality factor Q", IEEE Trans. Antennas Propag., vol. 64, pp. 5230 - 5242, 2016. Abstract This work describes a powerful, yet simple, procedure how to acquire a current approaching the lower bound of quality factor Q. This optimal current can be determined for an arbitrarily shaped electrically small radiator made of a perfect conductor. Quality factor Q is evaluated by Vandenboschs relations yielding stored electromagnetic energy as a function of the source current density. All calculations are based on a matrix representation of the integro-differential operators. This approach simplifies the entire development and results in a straightforward numerical evaluation. The optimal current is represented in a basis of modal currents suitable for solving the optimization problem so that the minimum is approached by either one mode tuned to the resonance, or, by two properly combined modes. An overview of which modes should be selected and how they should be combined is provided and results concerning rectangular plate, spherical shell, capped dipole antenna and fractal shapes of varying geometrical complexity are presented. The reduction of quality factor Q and the G/Q ratio are studied and, thanks to the modal decomposition, the physical interpretation of the results is discussed in conjunction with the limitations of the proposed procedure.

#### Characteristic Modes

 M. Masek, M. Capek, L. Jelinek, K. Schab, "Modal Tracking Based on Group Theory", 2018, eprint arXiv: 1812.03006. [Online]. Available: https://arxiv.org/abs/1812.03006 Abstract Modal tracking in the presence of crossings and crossing avoidances between eigenvalue traces is solved via the theory of point groups. The von Neumann-Wigner theorem is used as a key factor in predictively determining mode behavior over a frequency range. The capabilities of the proposed procedure are demonstrated on a particular case of characteristic mode decomposition. The procedure is, nevertheless, general and can be applied to an arbitrarily parametrized eigenvalue problem. A treatment of modal degeneracies is included. Several examples are presented to illustrate modal tracking improvements and the immediate consequences of improper modal tracking. An approach leveraging a symmetry-adapted basis to accelerate computation is also discussed. A relationship between geometrical and physical symmetries is demonstrated on a practical example D. Tayli, M. Capek, L. Akrou, V. Losenicky, L. Jelinek, M. Gustafsson, "Accurate and Efficient Evaluation of Characteristic Modes", IEEE Trans. Antennas Propag., vol. 66, pp. 7066 - 7075, 2018. Abstract A new method to improve the accuracy of characteristic modes decomposition for perfectly conducting bodies is presented. The method uses the expansion of the Green dyadic in spherical vector waves. This expansion is utilized in the method of moments solution of the electric field integral equation to improve the numerical range of the real part of the impedance matrix, R, that determines the number of obtainable modes from characteristic modes decomposition. Computation speed of the R matrix and characteristic modes are improved. The method can easily be integrated in existing method of moments solvers. Several structures are investigated illustrating the improved accuracy and performance of the new method. M. Capek, V. Losenicky, L. Jelinek, and M. Gustafsson, "Validating the characteristic modes solvers," IEEE Trans. Antennas Propag., vol. 65, pp. 4134 - 4145, 2017. Abstract Characteristic modes of a spherical shell are found analytically as spherical harmonics normalized to radiate unitary power and to fulfill specific boundary conditions. The presented closed-form formulas lead to a proposal of precise synthetic benchmarks which can be utilized to validate the method of moments matrix or performance of characteristic mode decomposition. Dependence on the mesh size, electrical size and other parameters can systematically be studied, including the performance of various mode tracking algorithms. A noticeable advantage is the independence on feeding models. Both theoretical and numerical aspects of characteristic mode decomposition are discussed and illustrated by examples. The performance of state-of-the-art commercial simulators and academic packages having been investigated, we can conclude that all contemporary implementations are capable of identifying the first dominant modes while having severe difficulties with higher-order modes. Surprisingly poor performance of the tracking routines is observed notwithstanding the recent ambitious development. M. Capek, P. Hazdra, M. Masek, and V. Losenicky, "Analytical representation of characteristic modes decomposition," IEEE Trans. Antennas Propag., vol. 65, pp. 713 - 720, 2017. Abstract Aspects of the theory of characteristic modes (CMs), based on their variational formulation, are presented and an explicit form of a related functional, involving only currents in a spatial domain, is derived. The new formulation leads to deeper insight into the modal behavior of radiating structures as demonstrated by a detailed analysis of three canonical structures: a dipole, an array of two dipoles and a loop, cylinder, and a sphere. It is demonstrated that knowledge of the analytical functional can be utilized to solve important problems related to the theory of CM decomposition such as the resonance of inductive modes or the benchmarking of method of moments code.

#### Stored Energy and Quality Factor Q

 K. Schab, L. Jelinek, M. Capek, C. Ehrenborg, D. Tayli, G. A. E. Vandenbosch, M. Gustafsson, "Energy Stored by Radiating Systems," IEEE Access , vol. 6, pp. 10553 - 10568, 2018. Abstract Though commonly used to calculate Q-factor and fractional bandwidth, the energy stored by radiating systems (antennas) is a subtle and challenging concept that has perplexed researchers for over half a century. Here, the obstacles in defining and calculating stored energy in general electromagnetic systems are presented from first principles as well as using demonstrative examples from electrostatics, circuits, and radiating systems. Along the way, the concept of unobservable energy is introduced to formalize such challenges. Existing methods of defining stored energy in radiating systems are then reviewed in a framework based on technical commonalities rather than chronological order. Equivalences between some methods under common assumptions are highlighted, along with the strengths, weaknesses, and unique applications of certain techniques. Numerical examples are provided to compare the relative margin between methods on several radiating structures. K. Schab, L. Jelinek, and M. Capek, "Recoverable energy of dissipative electromagnetic systems," 2017, eprint arXiv: 1701.06313. [Online]. Available: https://arxiv.org/abs/1701.06313 Abstract Ambiguities in the definition of stored energy within distributed or radiating electromagnetic systems motivate the discussion of the well-defined concept of recoverable energy. This concept is commonly overlooked by the community and the purpose of this communication is to recall its existence and to discuss its relationship to fractional bandwidth. Using a rational function approximation of a system's input impedance, the recoverable energy of lumped and radiating systems is calculated in closed form and is related to stored energy and fractional bandwidth. Lumped circuits are also used to demonstrate the relationship between recoverable energy and the energy stored within equivalent circuits produced by the minimum phase-shift Darlington's synthesis procedure. M. Capek and L. Jelinek, "Comments on On Stored Energies and Radiation Q," IEEE Trans. Antennas Propag., vol. 64, pp. 4575 - 4576, 2016. Abstract The commented paper [1] claims to provide a new expression for an energy stored around a general radiator. The major purpose of this comment is to show that the claim is unjustified. Alongside with this issue, it is pointed out that some of the core formulas of [1] are not completely correct, and that their correct form has in fact been derived elsewhere, though for the purpose of evaluating the quality factor QZ and not the stored energies. M. Capek, L. Jelinek, and G. A. E. Vandenbosch, "Stored electromagnetic energy and quality factor of radiating structures," Proc. R. Soc. A, vol. 472, pp. 1 - 16, 2016. Abstract This paper deals with the old yet unsolved problem of defining and evaluating the stored electromagnetic energy - a quantity essential for calculating the quality factor, which reflects the intrinsic bandwidth of the considered electromagnetic system. A novel paradigm is proposed to determine the stored energy in the time domain leading to the method, which exhibits positive semi-definiteness and coordinate independence, i.e. two key properties actually not met by the contemporary approaches. The proposed technique is compared with an up-to-date frequency domain method that is extensively used in practice. Both concepts are discussed and compared on the basis of examples of varying complexity.

#### Artificial Materials

 L. Jelinek, O. Kratky, and M. Capek, "An evaluation of polarizability tensors of arbitrarily shaped highly conducting bodies," IET Microw. Antenna P., vol. 11, pp. 852 - 858, 2017. Abstract A full-wave numerical scheme of polarizability (polarisability) tensors evaluation is presented. The method accepts highly conducting bodies of arbitrary shape and explicitly accounts for the radiation as well as ohmic losses. The method is verified on canonical bodies with known polarizability tensors, such as a sphere and a cube, as well as on realistic scatterers. The theoretical developments are followed by a freely available code whose sole user input is the triangular mesh covering the surface of the body under consideration.

/Edited 06. 06. 2019, MC/