Circuit analysis of radiation reaction in metamaterials by retarded electromagnetic coupling

Because radiation is essential in high ‐ frequency circuits, such as those used in meta-materials and plasmonics, the investigation of radiation loss is important. This study describes the characteristics of radiation loss, which is a radiation reaction in circuits with retarded electromagnetic couplings. The structure of wired metallic spheres is used to demonstrate metamaterial equivalent circuits, where charges and current exist on the spheres and wires, respectively. An inductance matrix and a potential coefficient matrix with retarded electromagnetic couplings are defined to address the radiation reaction. Subsequently, based on the topology of the wires and spheres, an equivalent circuit equation with retardation is formulated to discuss the losses in the resonant circuit caused by the inductive and capacitive elements. Thereafter, the relationship between the resonant frequency and radiation loss caused by the retarded couplings is demonstrated and the difference between the retarded couplings and couplings with transmission lines is clarified. Furthermore, we indicate that retarded coupling generates singularity on a dispersion curve for a one ‐ dimensional array of resonant circuits. Thus, the circuit with retarded couplings generates novel characteristics of radiation reactions that are not represented by the circuit without retardation. This circuit analysis is expected to afford new aspects in studies on topics, such as metamaterials and plasmonics.


| INTRODUCTION
Equivalent circuit models are powerful tools for analysis and design of electromagnetic phenomena.Recently, the design and synthesis of electromagnetic properties in complex metallic structures were conducted for metamaterials and plasmonics.The effectiveness of the equivalent circuit models was reported [1][2][3].Such equivalent circuit models can provide physical insight into resonance [4,5], coupling [6], and propagation [7] in electromagnetic phenomena.However, the radiation loss caused by the radiation reaction is not easily described because the mechanism of the radiation loss is different from the ohmic loss [8][9][10], despite radiation being an important factor in highfrequency circuits.
To represent the radiation loss in the equivalent circuit, it is important to include the delay caused by the retarded potentials [11][12][13].Hence, the radiation reaction is represented as feedback using retarded electromagnetic couplings.The partial element equivalent circuit (PEEC) is a full-wave circuit model with meshes of metallic structures [14][15][16], representing the radiation loss as the retardations between the meshes [17,18].Recently, the topic of the passivity of PEEC [19] and the discussion on its validity as a model of radiation has been reported in [20,21].Another method has been proposed to derive a simple circuit by specified structures, consisting of capacitive spheres and inductive wires [22].In this structure, the topology of the wires corresponds to the topology of the circuit, which simplifies the design of the topology and the radiation loss is modelled by retarded couplings [23].Thus, lumped element circuits with retardation are essential for modelling the electromagnetic radiation losses.
The radiation reaction caused by the retarded electromagnetic couplings provides not only the radiation loss in the lumped circuit model but it also identifies other properties in the circuits.For example, the resonant frequencies are affected by the radiation reactions and the retarded couplings contribute to the relationship between the radiation loss and the resonant frequency [23].Furthermore, the radiation reaction can alter the dispersion curves in the metamaterials [7,24,25].Thus, it is important to clarify these properties of the radiation reaction using the equivalent circuit models with retarded couplings.
This study proposes the circuit analysis of the radiation reaction using a lumped element circuit with retarded couplings.To clarify the effects of the retardation, we use the wired metallic sphere structure [22] and consider the retarded inductive and capacitive couplings [23].The validity of the models with retardation was confirmed in [23] through the comparison with electromagnetic simulations.However, because the derivation of the circuit equation was based on Green's function and the boundary condition on conductors, its relationship with Kirchhoff's law in ordinary circuit analysis is unclear.Therefore, in this study, we present the derivation of the circuit equation using Kirchhoff's law after the circuit elements are given.We used the potential coefficient matrix instead of the capacitance matrix to describe capacitive couplings because the charges generate the potential as given by Maxwell's equation.Based on the wire and sphere topologies, we formulate the circuit equation and the relationship with energy using Tellegen's theorem.
Next, we demonstrate the advantages of the circuit analysis with retardations over the conventional circuit without retardation.We discuss the losses caused by the retardation of the inductance and potential matrices using a simple resonator.The losses are explained by the rotation of impedance in a complex plane caused by the retardation.We show the relationship between the radiation loss and resonant frequency in coupled resonators and clarify the difference between the retarded coupling and transmission line couplings, which also provide coupling delays.Furthermore, we show the generation of the singularity on the dispersion curve with the onedimensional (1-D) periodic array of resonators and explain the radiation loss discontinuity on the light line analytically.
In the circuit analysis of the radiation reaction, the main contributions of this work are fourfold: (1) proposing a power balance according to Tellegen's theorem in Section 2; (2) depicting the difference in the contribution of capacitive and inductive elements to radiation in Section 3; (3) describing the essential difference between the retarded coupling and delays in transmission line in Section 4; and (4) proving the singularity on a dispersion curve created by the retarded coupling in Section 5. Since this circuit analysis clearly describes the power balance including radiation reaction, the potential applications include the design of antennas with complex structures, wireless power transfer, and transmission lines that do not have explicit return paths.

| Inductance matrix with retarded coupling
Inductances with retardation are represented by complex numbers in the frequency domain.We review an example of self-inductance with retardation using the metallic structure in Figure 1 [23].Here, D is the length between the centres of the two spheres and the radii of the wire and spheres are denoted by a and b, respectively.Assuming that the current in the wire is uniform and a ≪ b ≪ D, the inductance of the wire is approximated as ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where c is the light speed.The details of the derivation using Green's function in 3D with retardation is shown in [23].The complex inductance is caused by the phase shift of e −jω ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi between the position z 0 of the current on the central axis of the wire and the position z of the estimated electric field on the surface of the wire.We consider this property of retardation in this study.
Mutual inductance with retardation, of the structure in Figure 2, can be approximated as ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where η = D/h and the derivation is presented in [23].The phase shift is caused by the distance h between the wires.If η ≪ 1 is satisfied, the mutual inductance can be approximated as and the mutual inductance has the factor 1 h e −jω h c , which corresponds to the Green's function in 3D with retardation.We present a circuit analysis of the properties of elements with this type of retarded coupling.If there are N inductive elements, such as wires, we can define N � N inductance matrix L by the retarded inductances.

| Potential coefficient matrix with retarded coupling
Because the charge contributes to the potential in Maxwell equations, we introduce potential coefficients instead of capacitances.First, the self-potential coefficient of a metallic sphere, shown in Figure 1, is represented as Thus, the self-potential coefficient has a factor 1 b e −jω b c , which corresponds to the Green's function in 3D with retardation.
If b ≪ D is satisfied in Figure 1, the mutual potential coefficient P 12 is represented as and has the factor 1 D e −jω D c .If there are M capacitive elements such as spheres, we can define the M � M potential coefficient matrix P by the retarded potential coefficients.
Although we consider examples of the wired metallic spheres, an inductance matrix L and a potential coefficient matrix P can be defined for inductive elements of thin wire structures with uniform currents and capacitive elements of massive structures with uniform voltages based on the method in [23].In this study, we present the circuit analysis by Kirchhoff's laws after such circuit elements with retardations are given.

| Circuit equation by Kirchhoff's laws
The circuit equation with the inductance matrix L and potential coefficient matrix P of N inductive elements and M capacitive elements, respectively is formulated as follows: We assume that each inductive element is terminated by the capacitive elements, as shown in Figure 3.Then, we set nodes on the capacitive elements and define M � (N + M) incidence matrix U as where U l is the incidence matrix of the inductive elements and 1 t is the unit matrix which corresponds to the capacitive elements.For example, the incidence matrix of the structure in Figure 2 is We define the N + M branch voltages V by the N branch voltages V l of the wires and M node potentials b V of the spheres as

F I G U R E 3
Resonator structures with all wires terminated by spheres [22] NAKATA ET AL.
-313 We assume that the external excitation V E l is applied to the wires and define the N + M voltage source vector V E as Then, we use the Kirchhoff's voltage law to show the existence of the node voltages represented as where 'T' indicates the transpose matrix.We define the N + M branch currents I by the N branch currents I l of wires and M branch currents I t = jωQ of spheres as Then, the Kirchhoff's current law can be represented as The relationship between the branch voltages V and currents I is given as From Equations ( 12), (14), and (15), the equation of the wire currents I l is given as Since the coefficient matrix of I l is an impedance matrix and describes the properties of the circuits, we can derive the characteristics of the circuits using the impedance matrix.

| Power balance by Tellegen's theorem
Because the circuit equation are formulated by Kirchhoff's law, the power balances of complex powers in circuits are given by Tellegen's theorem [26], which are derived using Kirchhoff's laws Equations ( 12) and ( 14) as follows.
where bar represents the complex conjugate.If we use the inductive currents I l and capacitive currents I t = jωQ, the relation is This equation defines the contribution of each element in the power balance.The imaginary parts of L and P generate the real parts of the complex powers which correspond to the radiation loss.The coefficients jω and 1/jω of the first and second terms, respectively, give the essential difference contributing to the loss.

| RADIATION LOSS
To clarify the property of the radiation loss in circuits, we review the behaviour of the resonator in Figure 1 by the circuit approach.The equivalent circuit Equation ( 16) of the resonator is given as: P s ðωÞ ≡ P s1 ðωÞ − P s2 ðωÞ; ð21Þ If we set a = 0.01 mm, b = 1 mm and D = 7 mm in Figure 1, the real and imaginary parts of the frequency characteristics of the impedance Z s are shown in Figures 4 and 5, respectively.Because the loss is caused by retardation, the Re [Z s ] without retardation (dotted line) equals 0 in Figure 4, where Re[⋅] denotes the real part of '⋅'.The ω* satisfying Im [Z s (ω*)] = 0 in Figure 5 is the resonant angular frequency since the real impedance Z s means resonance, where Im[⋅] is the imaginary part of '⋅'.
To clarify the meaning of the loss caused by the impedance Z s with the retardation, we show the phase relation of the impedances in a complex plane at the resonant frequency ω* in Figure 6.Although the impedance has only the imaginary components in the case without retardation, jω*L s and P s1 / jω*, P s2 /jω* rotate on a complex plane because of the retardation factors and they become complex numbers.The real parts of the capacitive elements are negative; however, the total real part Re[jω*L s + (P s1 − P s2 )/jω*] becomes positive because of the inductive element.
The second order Taylor polynomials of the impedances [23] are Re � jωL s ðωÞ Re " The real part of the capacitive impedance Equation ( 24) is negative.When b ≪ D is satisfied, one-third of the real part by the inductive elements is cancelled by the capacitive elements.The total real parts of the impedance agrees with the radiation resistance by a small dipole [7]. Figure 7 shows the frequency characteristics of the current |I| when V E = 1.Although the frequency characteristics without retardation approach infinity at the resonant frequency, the case with retardation is finite and the value of the current is confirmed based on the electromagnetic analysis in [23].Thus, the total contributions of the positive [Equation ( 23)] and negative [Equation (24)] real parts give the radiation loss in the circuits.The power balance [Equation (18)] in this case is written by At the resonant angular frequency ω*, the V E and I are in phase, and the supplied complex power V E � I is real.The real part of the inductive and capacitive parts is 41.5 |I| 2 and (− 60.0 + 46.6)|I| 2 in Figure 6, respectively.If we substitute the current value I = 0.0356 in Figure 7, we can confirm that the balance is satisfied.

| Resonant angular frequencies of two modes
To clarify the property of the retarded coupling, we review the behaviour of the coupled resonators in Figure 2 by circuit analysis.The circuit Equation ( 16) of the structure is given as where L m ≡ L 12 in Equation ( 4) and P m is defined as Using the symmetry in Equation ( 25), we decompose the equation into even and odd modes as where If we define mode-impedances of even and odd modes as

| Resonant frequency and loss by retarded coupling
To understand the relation between the resonant angular frequency and loss, we estimate the mutual impedance Z m in Equation (30) under the condition of D ≪ h.The mutual inductance L m = L 12 in Equation ( 4) is approximated by L 12 in Equation ( 5) and has the factor 1 h e −jω h c .Similarly, P m in Equation ( 27) is approximated by and has the factor 1 h 2 e −jω h c .Because the inductive elements are dominant, Z m defined by Equation ( 30) is approximated as Thus, the circuit Equation ( 28) is expressed as the following approximation, where Ĩe ≡ I 1 þ I 2 and Ĩo ≡ I 1 − I 2 .
Figure 10 shows the original Equation (30) and approximated Equation (34) and Z m on a complex plane with the parameter h.Thus, the phase of the retarded factor rotates the mutual impedance depending on the distance between metaatoms h and contributes to the relationship between the resonant frequency and loss in Figures 8 and 9.That is, when Im

| Comparison with delay in transmission line
To clarify the property of the retarded coupling, we compare the retarded coupling of resonators in Figure 2 with coupling through conducting wires of a transmission line shown in Figure 11.The difference between the retarded coupling and the coupling through the conducting wire is well known in the discussion of electromagnetic fields.However, it is important to clarify the essential difference of circuit elements that contain delays.In this discussion, we assume that the transmission line has no losses and its characteristic impedance, propagation constant, and length are Z 0 , jβ, and h, respectively.We define the lumped circuit parameters as follows: where ω* is the resonant angular frequency of Figure 1.
The lumped circuit equation of each resonator is expressed as The coupling by the transmission line is cos βh jZ 0 sin βh j Z 0 sin βh cos βh If we decompose Equations ( 38) and (39) into two current modes in the same way as the retarded couplings, the circuit equation of each eigenmode is expressed by where we set β ¼ ω c and Because Equations ( 42) and (43) correspond to Equations (35) and (36), respectively, the impedance Z m in Equation (34) is replaced by Although the mutual impedance Z m rotates in the complex plane, depending on h (shown in Figure 10), Z m 0 for the transmission line is always a pure imaginary number depending on h (shown in Figure 12) and does not exhibit coupling loss.Thus, the couplings in Figures 2 and 11 are essentially different with regard to the energy loss and the retarded coupling provides new elements in the circuit analysis.

| Dispersion characteristics with retardation
As an example of where the retarded coupling plays an essential role, this section addresses the radiation reactions in the 1D array structure of meta-atoms shown in Figure 13.Relationships between the propagation modes and resonant frequencies are represented by the dispersion characteristics, and these properties are often used in the design of metamaterials.We show the singularity on a dispersion curve [7] created by the retarded coupling theoretically using circuit analysis.
To derive the dispersion curve, we introduce the periodic boundary condition, such that L and U T l PU l become circulant matrices.Then, the impedance matrix Z is also a circulant matrix, where Z 0 = Z s and Z n (n = 1, 2, … , N − 1) is defined by ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi The circulant matrix (46) is diagonalised by a discrete Fourier transform (DFT) matrix, whose column vectors are written by h 1; e j2π m N ; e j2π 2m N ; …; e j2π ðN−1Þm The mth eigenvector corresponds to the wave number of mth eigenmode and is expressed as k m ¼ 2πm Nh .Using the eigenvectors in Equation (49), we derive the eigenvalue, that is, the mth mode impedance as Solving the equation of ω yields the resonant condition of the mth mode, we obtain the resonant angular frequency ω � m and real part of the impedance, which represent the radiation loss in the resonant angular frequency of each wave number k m .The dispersion  The dispersion curve has a singularity on the light line and the real part of mode impedances changes discontinuously on the light line.This property is well known as the fact that the Bloch mode is no longer fully guided after crossing the light line [27]; however, the property is not described by circuit analysis without retardation.
To discuss the relation between Figures 14 and 15, we show the real and imaginary parts of the mode impedance Zðω; k m Þ in Figures 16 and 17, respectively.The cross point of the plane Im½ Z� ¼ 0 in Figure 17 corresponds to the dispersion curve in Figure 14, while the real parts Re½ Zðω � m ; k m Þ� on the curve in Figure 16 represent the curves in Figure 15.The discontinuity in Figure 15 corresponds to the discontinuity on the light line in Figure 16.Thus, the radiation loss and resonant frequency are represented by the real and imaginary parts of the mode impedance Zðω; k m Þ.

| Analytical estimation of radiation loss
To derive the analytical estimation the radiation loss by circuit analysis, we divide the contributions of the inductances and potential coefficients in the real part of the mode impedance as follows: Furthermore, approximating Equation ( 5), we define the imaginary part of the inductance L i n ðn ¼ 0; 1; …; N − 1Þ as which is a Sinc function and is caused by retarded couplings.
Then, the contribution of the inductance Re½ ZL ðω; k m Þ� can be written by which is a window function when N is significantly large.Thus, the discontinuity of Re½ ZL ðω; k m Þ� on the light line is shown analytically by the circuit model.The potential coefficient Equation (48) also consists of the Sinc function; however, the second term has irregular samp ling.The contributions of Re½ Zðω; k m Þ�; Re½ ZL ðω; k m Þ�; Re½ ZP ðω; k m Þ� at ω = 5.14 � 10 10 are shown in Figure 18.We can confirm that Re½ ZL ðω; k m Þ� is approximated by the Sinc function and discontinuity in Re½ Zðω; k m Þ� is caused by inductive couplings.The retarded inductive couplings generate the radiation loss discontinuity on the light line which is an important characteristic of radiation by leaky-waves [28].

| CONCLUSION
We introduced a circuit with retarded couplings to describe the radiation loss resulting from a radiation reaction.
The potential coefficient matrix was used instead of the capacitance matrix because of the retardation.The circuit equations were formulated such that the real parts of the complex power with the retarded couplings describe the radiation loss.Using the simple resonant circuit, we initially demonstrated the radiation loss mechanism by the retardation of the inductance and potential coefficients to clarify the contribution of the inductive and capacitive elements.We then clarified the radiation reaction in the coupled resonant circuit.The relation between radiation loss and resonant frequency was shown by the phase rotation of the retarded coupling.We discussed the essential difference between the retarded coupling and the coupling with the transmission line.We then analysed the 1D array of resonators with all retarded couplings.The radiation loss was discontinuously changed on the light line and the dispersion curve has singularity on the light line.Thus, the circuit with the retarded coupling describes the radiation reaction and generates novel characteristics that are not described by the circuit without retardation.The circuit analysis of the radiation reaction will afford new aspects in studies on topics such as metamaterials and plasmonics.
Structure of coupled two meta-atoms in parallel position, which has two resonators with retarded coupling.We use parameters of wire radii a = 0.01 mm, b = 1 mm and D = 7 mm in this study

F I G U R E 5 F G U R E 7 = 1 NAKATA 4 ,
Frequency characteristics of impedance Z s (imaginary part) with and without retardation F I G U R E Phase relation among impedances.W/O retardation: L s , P s are real numbers; hence, Z s is an imaginary number.retardation: L s , P s are complex numbers; hence, Z s is a complex number Frequency characteristics of current I in Figure 1 with V E Downloaded from https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/cds2.12104 by Cochrane Japan, Wiley Online Library on [24/11/2022].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License A Self-archived copy in Kyoto University Research Information Repository https://repository.kulib.kyoto-u.ac.jpP m ðωÞ ≡ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi D 2 þh 2 p c ! ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi then the resonant angular frequencies ω � e and ω � o are given by the imaginary part of the impedances Im½ Ze ðωÞ� ¼ 0; Im½ Zo ðωÞ� ¼ 0; ð32Þ respectively.The distance h dependencies of ω � e and ω � o are shown in Figure 8.The period of the cross points of the ω � e and ω � o is approximately half of the wavelength λ = 2πc/ ω* = 36.7 mm.In contrast, the radiation losses Re½ Ze ðω � e Þ� and Re½ Zo ðω �o Þ� shown in Figure9also have a similar period, and the cross points in Figure8correspond to the maximal or minimum points in Figure9.That is, the cross points in Figure8satisfy Im½Z m ðω � e Þ� = Im½Z m ðω � o Þ� = 0 as in Equation (31) and at these points the real parts Re½Z m ðω � e Þ� and Re½Z m ðω � o Þ� become maximum or minimum, as shown in Figure9.Such a relation was also confirmed by the electromagnetic simulations and natural frequencies in[23].

F I G U R E 1 0
Comparison of mutual impedance Z m between Equations (30) and (34): changing h at ω* = 5.14 � 10 10 rad/s.The mutual impedance rotates in a complex plane depending on h and generates the relation between the resonant frequency and radiation loss F I G U R E 1 1 Two resonators coupled by a lossless transmission line NAKATA ET AL. -317 17518598, 2022, 4, Downloaded from https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/cds2.12104 by Cochrane Japan, Wiley Online Library on [24/11/2022].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License A Self-archived copy in Kyoto University Research Information Repository https://repository.kulib.kyoto-u.ac.jp

F I G U R E 1 2 1 F I U R E 1 4
Mutual impedance Z m 0 in the circuit by transmission line.The mutual impedance is a pure imaginary number dependent on h.Although the coupling caused by a lossless transmission line has retardation, the coupling does not generate loss F I G U R E 1 3 1D array structure of the meta-atoms.Each meta-atom is a resonator in Figure Dispersion characteristics of 1D array structure (Figure 13).The curve has a singularity on the light line.'Single' is the resonant frequency of the single meta-atom characteristics ðk m ; ω � m Þ and real parts ðk m ; Re½ Zðω � m ; k m Þ�Þ are shown in Figures 14 and 15 .

μ 0 D 2 F I U R E 1 5
ω 4h ðck m < ωÞ μ 0 D 2 ω 8h ðck m ¼ ωÞ 0 ðck m > ωÞ ; ð55Þ Real part of mode impedance on the resonant angular frequency.The curve has a discontinuity where the dispersion curve crosses the light line.'Single' is the real part of the single meta-atom F I G U R E 1 6 Real part of mode impedance (Re [ Zðω; k m Þ]).The impedance has a discontinuity on the light line NAKATA ET AL.

F I G U R E 1 7 14 F I G U R E 1 8
Imaginary part of mode impedance (Im[ Zðω; k m Þ]): the cross point of the plane Im½ Z� ¼ 0 corresponds to the dispersion curve in Figure Components in Equation (52) at ω = 5.14 � 10 10 (rad/s).The inductance loss is approximated as a Sinc function and generates radiation loss discontinuity 320 -NAKATA ET AL. 17518598, 2022, 4, Downloaded from https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/cds2.12104 by Cochrane Japan, Wiley Online Library on [24/11/2022].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License A Self-archived copy in Kyoto University Research Information Repository https://repository.kulib.kyoto-u.ac.jp Distance h dependencies of resonant angular frequencies ω � e and ω � ½Z m ðω � Þ� ¼ 0 is satisfied because Equations (35) and (36), ω � e is equal to ω � o in Figure 8 and the large real parts jRe½Z m ðω � e Þ�j and jRe½Z m ðω � o Þ�j causes the maximum or minimum radiation loss in Figure 9.
o F I U R E 9 Distance h dependencies of the radiation losses Re [ Ze ðω � e Þ] and Re[ Zo ðω � o Þ] on the resonant angular frequency 316 -NAKATA ET AL. 17518598, 2022, 4, Downloaded from https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/cds2.12104 by Cochrane Japan, Wiley Online Library on [24/11/2022].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License A Self-archived copy in Kyoto University Research Information Repository https://repository.kulib.kyoto-u.ac.jp