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Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas Miloslav ˇ Capek 1 , Luk´ ınek 1 , Mats Gustafsson 2 , and V´ y 1 aˇ s Jel´ ıt Losenick´ 1 Department of Electromagnetic Field, Czech Technical University in Prague, Czech Republic miloslav.capek@fel.cvut.cz 2 Department of Electrical and Information Technology, Lund University, Sweden October 2, 2019 ICECOM, Dubrovnik, Croatia Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 1 / 33

Outline 1. Bounds on Radiation Efficiency 2. Utilizing Integral Equations a 3. Solution to QCQP Problems for Radiation Efficiency 4. Solution for a Spherical Shell and Scaling of the Problem 5. Algebraic Representation with Volumetric MoM 6. A New Numerical Method Hybridizing MoM & T-Matrix 7. Concluding Remarks Electrically small antenna inside a circumscribing sphere of a radius a . ◮ Document available at capek.elmag.org . ◮ To see the graphics in motion, open this document in Adobe Reader! Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 2 / 33

Bounds on Radiation Efficiency Radiation Efficiency and Dissipation Factor Radiation efficiency 1 : P rad 1 η rad = = (1) P rad + P lost 1 + δ lost Dissipation factor 2 δ : δ lost = P lost (2) P rad ◮ fraction of quadratic forms (can be scaled with resistivity model). 1 145-2013 – IEEE Standard for Definitions of Terms for Antennas , IEEE, 2014 Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 3 / 33

Bounds on Radiation Efficiency Radiation Efficiency and Dissipation Factor Radiation efficiency 1 : P rad 1 η rad = = (1) P rad + P lost 1 + δ lost Dissipation factor 2 δ : δ lost = P lost (2) P rad ◮ fraction of quadratic forms (can be scaled with resistivity model). 1 145-2013 – IEEE Standard for Definitions of Terms for Antennas , IEEE, 2014 2 R. F. Harrington, “Effect of antenna size on gain, bandwidth, and efficiency,” J. Res. Nat. Bur. Stand. , vol. 64-D, pp. 1–12, 1960 Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 3 / 33

Bounds on Radiation Efficiency Radiation Efficiency and Dissipation Factor: Example A wire dipole of length ℓ = 5 m made of copper wire of 2 . 055 mm: 100% η rad 80% analytic simulation 60% ℓ 1 2 5 10 20 f (MHz) ( Z 0 /R s ) ( ka ) 2 δ lost 2600 2400 analytic simulation 2200 10 − 1 10 0 ka Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 4 / 33

Bounds on Radiation Efficiency What Is This Talk About? Questions to be investigated. . . 1. What are the fundamental bounds on radiation effiency? 2. What are other costs (self-resonance, trade-offs)? 3. Are these bounds feasible? Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 5 / 33

Bounds on Radiation Efficiency What Is This Talk About? Questions to be investigated. . . 1. What are the fundamental bounds on radiation effiency? 2. What are other costs (self-resonance, trade-offs)? 3. Are these bounds feasible? Tools we have: ◮ Circuit quantities (equivalent circuits). ◮ Field quantities (spherical harmonics). ◮ Source currents (eigenvalue problems). L mn = � ψ m ( r ) , L [ ψ n ( r )] � ε 0 a L =[ L mn ] µ 0 a Z 0 E AI = λ IB Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 5 / 33

Bounds on Radiation Efficiency A Little History of the Problem. . . Circuit Quantities ◮ Circuit quantities (equivalent circuits): 1. C. Pfeiffer, “Fundamental efficiency limits for small metallic antennas,” IEEE Trans. Antennas Propag. , vol. 65, pp. 1642–1650, 2017. 2. H. L. Thal, “Radiation efficiency limits for elementary antenna shapes,” IEEE Trans. Antennas Propag. , vol. 66, no. 5, pp. 2179–2187, 2018. ε 0 a µ 0 a Z 0 Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 6 / 33

Bounds on Radiation Efficiency A Little History of the Problem. . . Field Quantities ◮ Field quantities (spherical harmonics): 1. R. F. Harrington, “Effect of antenna size on gain, bandwidth, and efficiency,” J. Res. Nat. Bur. Stand. , vol. 64-D, pp. 1–12, 1960. 2. A. Arbabi and S. Safavi-Naeini, “Maximum gain of a lossy antenna,” IEEE Trans. Antennas Propag. , vol. 60, pp. 2–7, 2012. 3. K. Fujita and H. Shirai, “Theoretical limitation of the radiation efficiency for homogenous electrically small antennas,” IEICE T. Electron. , vol. E98C, pp. 2–7, 2015. 4. A. K. Skrivervik, M. Bosiljevac, and Z. Sipus, “Fundamental limits for implanted antennas: Maximum power density reaching free space,” IEEE Trans. Antennas Propag. , vol. 67, no. 8, pp. 4978 –4988, 2019. E Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 7 / 33

Bounds on Radiation Efficiency A Little History of the Problem. . . Source Currents ◮ Source currents (eigenvalue problems): 1. M. Uzsoky and L. Solym´ ar, “Theory of super-directive linear arrays,” Acta Physica Academiae Scientiarum Hungaricae , vol. 6, no. 2, pp. 185–205, 1956. 2. R. F. Harrington, “Antenna excitation for maximum gain,” IEEE Trans. Antennas Propag. , vol. 13, no. 6, pp. 896–903, 1965. 3. M. Gustafsson, D. Tayli, C. Ehrenborg, et al. , “Antenna current optimization using MATLAB and CVX,” FERMAT , vol. 15, no. 5, pp. 1–29, 2016. 4. L. Jelinek and M. Capek, “Optimal currents on arbitrarily shaped surfaces,” IEEE Trans. Antennas Propag. , vol. 65, no. 1, pp. 329–341, 2017. L mn = � ψ m ( r ) , L [ ψ n ( r )] � L =[ L mn ] AI = λ IB Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 8 / 33

Utilizing Integral Equations Integral Operators and Their Algebraic Representation Radiated and reactive power: P rad + 2j ω ( W m − W e ) = 1 2 � J ( r ) , Z [ J ( r )] � Lost power (surface resistivity model): P lost = 1 T + l n ρ − P − 2 � J ( r ) , Re { Z s } J ( r ) � n n n ρ + n P + A − n A + n T − n n r ◮ The same approach as with the method of moments 3 (MoM) z y x O � J ( r ) ≈ I n ψ n ( r ) RWG basis function ψ n . n 3 R. F. Harrington, Field Computation by Moment Methods . Piscataway, New Jersey, United States: Wiley – IEEE Press, 1993 Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 9 / 33

Utilizing Integral Equations Algebraic Representation of Integral Operators Radiated and reactive power P rad + 2j ω ( W m − W e ) = 1 2 � J ( r ) , Z [ J ( r )] � (3) Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 10 / 33

Utilizing Integral Equations Algebraic Representation of Integral Operators Radiated and reactive power P rad + 2j ω ( W m − W e ) = 1 2 � J ( r ) , Z [ J ( r )] � ≈ 1 2 I H ZI (3) Electric Field Integral Equation 4 (EFIE), Z = [ Z mn ]: � � � Z mn = ψ m · Z ( ψ n ) d S = j kZ 0 ψ m ( r 1 ) · G ( r 1 , r 2 ) · ψ n ( r 2 ) d S 1 d S 2 . (4) Ω Ω Ω 4 W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves . Morgan & Claypool, 2009 Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 10 / 33

Utilizing Integral Equations Algebraic Representation of Integral Operators Radiated and reactive power P rad + 2j ω ( W m − W e ) = 1 2 � J ( r ) , Z [ J ( r )] � ≈ 1 2 I H ZI (3) Electric Field Integral Equation 4 (EFIE), Z = [ Z mn ]: � � � Z mn = ψ m · Z ( ψ n ) d S = j kZ 0 ψ m ( r 1 ) · G ( r 1 , r 2 ) · ψ n ( r 2 ) d S 1 d S 2 . (4) Ω Ω Ω ◮ Dense, symmetric matrix. ◮ An output from PEC 2D MoM code. 4 W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves . Morgan & Claypool, 2009 Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 10 / 33

Utilizing Integral Equations Algebraic Representation of Integral Operators Lost power P lost = 1 2 � J ( r ) , Re { Z s } [ J ( r )] � (5) Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 11 / 33

Utilizing Integral Equations Algebraic Representation of Integral Operators Lost power P lost = 1 2 � J ( r ) , Re { Z s } [ J ( r )] � ≈ 1 2 I H LI (5) � L mn = ψ m · ψ n d S (6) Ω Surface resistivity model: Z s = 1 + j (7) σδ � with skin depth δ = 2 /ωµ 0 σ . Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 11 / 33

Utilizing Integral Equations Algebraic Representation of Integral Operators Lost power P lost = 1 2 � J ( r ) , Re { Z s } [ J ( r )] � ≈ 1 2 I H LI (5) � L mn = ψ m · ψ n d S (6) Ω Surface resistivity model: Z s = 1 + j (7) σδ � with skin depth δ = 2 /ωµ 0 σ . ◮ Sparse matrix (diagonal for non-overlapping functions { ψ m ( r ) } ). ◮ The entries L mn are known analytically. Miloslav ˇ Capek, et al. Fundamental Bounds on Dissipation Factor for Wearable and Implantable Antennas 11 / 33