Open access peer-reviewed chapter

# Waveguide Port Approach in EM Simulation of Microwave Antennas

Written By

Faik Bogdanov, Irina Chochia, Lily Svanidze and Roman Jobava

Submitted: 29 January 2022 Reviewed: 01 February 2022 Published: 24 March 2022

DOI: 10.5772/intechopen.102996

From the Edited Volume

## Recent Microwave Technologies

Edited by Ahmed Kishk and Kim Ho Yeap

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## Abstract

This chapter generalizes a recently proposed MoM-based approach to waveguide port excitation (WPE) problems on arbitrary conducting and composite geometries. This approach combines the canonical aperture coupling approach with the EFIE-PMCHWT formulation for composite structures. Each WPE problem in this approach is divided into equivalent sub-problems for internal and external regions, which are solved using the MoM. Internal WPE problems are solved using waveguide modal expansion in the port plane, while external problems are solved using the equivalence principle to reduce these problems to the systems of algebraic equations for unknown electric and magnetic currents. The developed approach is validated on radiation and coupling problems for coaxial ports by comparing simulated results with those obtained by other approaches and measurements. An excellent agreement between the simulated and measured results is demonstrated. Finally, this approach is applied to practical EMC problems for microwave antennas fed by coaxial ports.

### Keywords

• coupling problem
• coaxial port
• equivalence principle
• method of moments (MoM)
• waveguide modal expansion
• waveguide port

## 1. Introduction

The excitation problem is of increasing importance at microwave frequencies [1]. Microwave antennas and other microwave devices are often fed by waveguides (rectangular, circular, and coaxial) or transmission lines (such as microstrips). In general, these devices are composite structures consisting of both conductive and dielectric elements. Therefore, the appropriate modeling of the waveguide excitation of such structures is of great interest. Such modeling in numerical methods is usually done by truncating the feed waveguide to create a waveguide port and formulate suitable boundary conditions (BC) imposed on the port. Such BCs should be able to launch an incident wave into the waveguide and absorb the reflected (in active mode) or received (in passive mode) wave without spurious reflections [2].

To date, most approaches to solving the waveguide port excitation (WPE) problem are based on volume discretization methods, such as the finite-element method (FEM) [1, 2], finite difference time domain (FDTD) [3, 4, 5], discontinues Galerkin time-domain (DGTD) [6], contour integral method (CIM) [7], etc. Most of these works use various modal absorbing boundary conditions (MABC) [4, 5], developed for time-domain methods as termination conditions imposed on the port.

At present, many electromagnetic (EM) problems are solved using surface integral equations (SIE) together with the method of moments (MoM) [8]. Within the framework of SIE, the WPE problem was first formulated as an aperture coupling problem for a conducting geometry, and the MoM solution for magnetic currents was obtained in the presence of a short-circuited conductive sheet [9]. This approach was then modified using the pseudo-image method for magnetic currents in addition to electric currents [10]. Further, MoM was applied to the waveguide port problems [11] and antenna radiation problems with aperture port excitation [12]. However, until recently, a MoM-based solution to the WPE problem for arbitrary geometries has been poorly represented in the literature. In our recent works [13, 14, 15], such a solution was obtained for radiation and coupling problems for various types of geometries.

This chapter generalizes the recently proposed MoM-based approach to WPE problems [13, 14, 15] on arbitrary conducting and composite geometries. The obtained approach combines the canonical aperture coupling approach with the EFIE-PMCHWT formulation for composite structures [16, 17, 18, 19, 20, 21, 22]. Each WPE problem in this approach is divided into equivalent sub-problems for internal and external regions, which are solved using the MoM. The internal WPE problems are solved using waveguide modal expansion in the port plane, while the external problems are solved using the equivalence principle to reduce these problems to the systems of algebraic equations for unknown electric and magnetic currents. The obtained solution also considers the problem of material junctions between adjacent surfaces, considered in [19, 20, 21, 22].

The developed approach is validated on radiation and coupling problems for coaxial ports by comparing the simulated results with those from other approaches and measurements. In addition, this approach is applied to practical EMC problems for microwave antennas fed by coaxial ports. The MoM calculations were performed using the TriD numerical code incorporated in the EMCoS Studio software package [23].

## 2. Waveguide port approach for conducting geometry

### 2.1 Dividing the original problem into equivalent problems

Figure 1a illustrates a canonical waveguide port problem for conducting geometry. This geometry consists of a semi-infinite waveguide 1 with perfect electric conducting (PEC) walls and a microwave structure 2, which is yet supposed to be conductive. We intend to create port P in waveguide 1 to divide the geometry into two regions (A and B) to truncate the mesh in the region A and impose appropriate termination conditions in the port plane.

For this purpose, we follow the classical approach for the aperture problem [9] to divide the original problem into two equivalent problems, as shown in Figure 1b and c. We introduce a perfectly conducting surface Sa into the port plane P to separate regions A and B and consider two equivalent sub-problems: internal (for region A) and external (for region B). In addition, we introduce equivalent magnetic currents and M on both sides of Sa to restore the tangential electric fields on the boundary surface Sa. Let us consider these equivalent problems separately.

### 2.2 Formulation of the internal equivalent problem

Consider an internal equivalent problem for the region A. The total EM field in the region A is composed of the incident field Einc,Hinc and the reflected field Eref,Href generated by magnetic currents in the presence of a conductor. According to the equivalence principle [24], these currents are related to the total electric field EASa on the port surface Sa by the relation:

M=n×EASa=n0×EASaE1

where n is the internal normal in the region A, and n0=n is the propagation direction of the incident wave.

Equation (1) relates the total electric field at the port surface Sa to magnetic currents depending on the geometric and material properties of the external region B. The internal equivalent problem is to find the modal expansion of the total EM field at the port surface Sa through these currents.

### 2.3 MoM solution of the internal equivalent problem

The total EM field in the region A on the port surface Sa can be generally written as the sum of the incident (+) and reflected (−) TEM (if exists), TE, and TM modes [2, 13]:

EASa=a0++a0e0TEM+s=1NTEas++asesTE+s=1NTMbs++bsesTME2
HASa=1Za0+a0n0×e0TEM+s=1NTE1ZsTEas+asn0×esTE+s=1NTM1ZsTMbs+bsn0×esTME3

where as± and bs± are the mode amplitudes, e0TEM, esTE and esTM are the transverse modal functions of TEM, TE and TM waves with wave impedances Z=μ/ε, ZsTE=ωμ/γs and ZsTM=γs/ωε, respectively, ε and μ are the permittivity and permeability of the medium, Γs is the propagation constant, and NTE and NTM are the numbers of accounted TE and TM modes, respectively.

Next, we use the MoM to relate the mode amplitudes of the reflected fields in (2), (3) to the magnetic currents M. The port surface Sa is discretized into planar patches, and the unknown magnetic currents are approximated as

M=n=1NaMnfnE4

where fn are linear independent basis functions (BFs), Mn are unknown expansion current coefficients, and Na is the number of these BFs on the surface Sa. Substituting now (4) and (2) into (1), multiplying both sides by n0×e0TEM, n0×esTE and n0×esTM, respectively, and integrating over the port surface Sa, we relate the amplitudes of the reflected waves with those of the incident waves and magnetic current coefficients:

where

Tn0=Safnn0×e0TEMdS,Tns=Safnn0×esTEdS,Tns=Safnn0×esTMdSE6
R0=San0×e0TEMn0×e0TEMdS,Rs=San0×esTEn0×esTEdS,Rs=San0×esTMn0×esTMdS.E7

Substitution of (5) into (2), (3) determines the total electric and magnetic fields on the port surface in region A through the still unknown magnetic currents.

### 2.4 Formulation of the external equivalent problem for conducting geometry

Consider now an external equivalent problem for conducting geometry. The scattered EM field in the external region B in Figure 1c of the conducting geometry is produced by electric currents J flowing over surfaces Sa and Sc and equivalent magnetic currents M at the surface Sa, which can be written as:

EBscJM=LEJJ+LEMME8
HBscJM=LHJJ+LHMME9

where LEJ, LEM, LHJ and LHM are the linear integro-differential operators of electric and magnetic fields applied to the electric and magnetic currents, respectively. Applying the boundary conditions for the tangential electric and magnetic fields on the surfaces Sa and Sc, we obtain the following system of integral equations for the unknown electric and magnetic currents J and M

EBscJMtanSa+Sc=0just outsideSaE10
HBscJMtanSa+Sc=HASatanjust insideSaE11

### 2.5 MoM solution of the external equivalent problem for conducting geometry

To obtain the MoM solution to the BC (10) and (11), we consider, along with Eq. (4), the following expansion for an unknown electric current J:

J=n=1Na+NcInfn,E12

where fn are the BFs taken the same as for the expansion of magnetic currents in (4), In are the unknown expansion current coefficients on the surfaces Sa and Sc, and Na and Nc are the numbers of these BFs on these surfaces. Substitution of expansions (4) and (12) in (8) and (9) gives the following expressions for the EM field in region B:

EBscJM=n=1Na+NcInLEJfn+n=1NaMnLEMfn,E13
HBscJM=n=1Na+NcInLHJfn+n=1NaMnLHMfn.E14

Substituting now (3), (5), (13) and (14) into (10) and (11), introducing the boundary operators L̂JJ=LEJSa+Sc, L̂JM=LEMSa+Scoutside, L̂MJ=LHJSainside and L̂MM=LHMSa and testing the resulting equations with appropriate weighting functions w1r, w2r,…, wmr leads to the following system of linear algebraic equations

ZmnJJZmnJMZmnMJZmnMMInMn=0VmWE15

with elements defined as:

ZmnJJ=wmL̂JJfn,ZmnJM=wmL˜JM+12n×fn,E16
ZmnMJ=wmL˜MJ+12n×fn,ZmnMM=wmL̂MMfn+QmnW,E17
QmnW=T̂m0WTn0R0+s=1NTET̂msWsTMTnsRs+s=1NTMT̂msWsTMTnsRs,E18
VmW=2T̂m0Wa0++s=1NTET̂msWsTMas++s=1NTMT̂msWsTMbs+,E19

where L˜JM and L˜MJ are the regular parts of the boundary operators L̂JM and L̂MJ, the notation wf=swfdS is used for the scalar product, and

T̂m0=Sawmn0×e0TEMdS,T̂ms=Sawmn0×esTEdS,T̂ms=Sawmn0×esTMdSE20

In the case of Galerkin’s procedure wm=fm, and coefficients (20) and (6) become the same. The MoM system (15) determines the solution to the waveguide port problem in the conducting geometry.

### 2.6 Validation of the developed approach for conducting geometry

The developed approach has been validated to simulate the scattering characteristics of a flanged coaxial line as proposed in [25, 26, 27]. Such structures are frequently used in biomedical engineering for non-destructive testing of various materials [25, 26, 27].

When modeling a coaxial line, it is convenient to choose the port plane at the output of the line to provide fast damping of evanescent waves. In this case, it can be assumed NTE=NTM=0 in (2) and (3) to take taken into account only fundamental, TEM mode with the modal function e0TEM=eρ/ρlnD/d, where ρ is the radial distance, eρ is the unit radial vector, and D and d are the outer and inner diameters of the coaxial waveguide.

Figure 2 shows a flanged coaxial line consisting of a coaxial waveguide section with an outer radius D/2 = 4.725 mm, an inner radius of d/2 = 1.4364 mm, and a length L = 10 mm, ended with a circular disc with a diameter 2R = 200 mm. The bottom plane of the waveguide is accepted as a waveguide port, and the structure is excited in this port by TEM mode. To validate the developed approach for conducting geometry, we analyze the case when both the waveguide and outer space have the same permittivity εr=2.05.

Figure 3a and b show the magnitude and phase of the reflection coefficient at the end of the coaxial line used as the reference plane. We compare the simulation results obtained using the developed approach, the mode-matching technique [25], and the matrix pencil method [26] with measurement data [27]. Note that infinite flanges are assumed in [25, 26]. Phase data conforms to the time convention expt.

Comparison of various results shows excellent agreement between them. However, the phase characteristics obtained by our approach agree somewhat more accurately with the measurement data. Thus, the obtained results validate the developed approach to modeling a coaxial waveguide port for conducting geometries.

## 3. Waveguide port approach for composite geometry

### 3.1 Equivalent problems for composite geometry

Figure 4 shows the geometry of the problem, consisting of a composite structure composed of k-1 homogeneous regions Di, i = 1,2,…, k−1, located in the free space region D0 and exposed to waveguide excitation from the waveguide region B, which will be considered as k-th region of the problem. The region Dk is a finite section of the waveguide, confined by the PEC walls, the port surface Sa, and the dielectric surface Skd, through which the structure is fed. The port surface Sa separates the region Dk (B) from the semi-infinite waveguide region A with incident waveguide excitation. In addition, each region Di is excited, in general, by the incident field Eiinc, Hiinc.

To formulate the waveguide port excitation problem through the port surface Sa, we first consider the aperture coupling problem between the waveguide regions A and B [9]. Thus, we cover the port surface Sa with a PEC sheet and introduce equivalent magnetic currents M and M on both sides of Sa to divide the excitation problem into two different equivalence problems: the internal problem for region A, and the external problem for region B (Dk), as done in Section 2.1. Then, the internal equivalent problem is identical to that formulated in Section 2.2 and solved in Section 2.3. The external equivalent problem requires consideration of equivalent problems for each boundary surface in regions Di, i = 1,2,…, k, including the port surface Sa.

### 3.2 Formulation of the external equivalent problem for composite geometry

An external equivalent problem for composite geometry is reduced to a set of equivalent problems for each conducting and dielectric boundary Sic and Sid of free space region D0 (i = 0), composite structure regions Di (i = 1,…,k-1), and finite waveguide region Dk (i = k). In turn, each surface Sid comprises a set of boundary surfaces sij=DiDj (ij), being the interfaces between the regions Di and Dj.

Per the equivalence principle [24], the total EM field inside the i-th region Di can be expressed as the sum of the incident field Eiinc, Hiinc and that induced by the total surface currents distributed over its boundary surface Si and radiating into a homogeneous medium with constitutive parameters εi and μi of the region Di. The total electric currents Ji on the boundary surface Si consist of conducting currents Jic, flowing on the inner sides of conducting boundaries Sic, and equivalent electric currents Jid, flowing on the inner sides of dielectric boundaries Sid. Magnetic currents in the region Di are equivalent currents Mid, flowing on dielectric boundaries Sid. In addition, in the waveguide region Dk there are equivalent magnetic currents M on the port surface Sa.

Unknown electric and magnetic currents can be found using the boundary conditions at the conducting boundaries of the composite structure:

Eiinc+EiscJicJidMidtanSic=0,i=0,1,,k1E21

dielectric boundaries of regions Di (i,j=0,1,,k,ij):

Eiinc+EiscJicJidMidMδiktansij=[Ejinc+Ejsc(JjcJjdMjdMδjk)]tansij,E22
Hiinc+HiscJicJidMidMδiktansij=[Hjinc+Hjsc(JjcJjdMjdMδjk)]tansij,E23

and on the port surface Sa and the conducting boundary Skc of the k-th region:

Ekinc+EkscJkcJkdMkdMtanSa+Skc=0just outsideSaE24
Hkinc+HkscJkcJkdMkdMtanSa=HASatanjust insideSa,E25

where δik is the Kronecker delta, which shows that magnetic currents M radiate only in a waveguide region Dk. The magnetic field on the right-hand side of (25) is expressed by Eq. (3). The scattered EM fields in (21)–(25) can be expressed in terms of electric and magnetic currents Ji and Mi in the dielectric region Di as

EiscJiMi=LiEJJiLiEMMiE26
HiscJiMi=LiHJJiLiHMMiE27

where LiEJ, LiEM, LiHJand LiHM are linear integro-differential operators of EM fields applied to currents radiated in the i-th region. It can also be shown [19, 20, 21, 22] that the equivalent currents on opposite sides of the dielectric boundaries are related as:

Jid=Jjd,Mid=MjdonsijE28

Equation (21)(25) together with relations (26)(28) and expansions (3) represent the general (EFIE-PMCHWT) form of integral equations for a composite structure with an arbitrary excitation, including the waveguide port.

### 3.3 MoM solution of the external equivalent problem for composite geometry

To solve the coupled system of integral Eqs. (21)(28), we use the MoM to discretize the geometry of all boundary surfaces of the regions Di (i=1,…,k) into the planar patches and to consider the following expansions for the unknown currents:

Jkc=n=1Na+NkcIncfn,M=n=1NaMn,Jici=0k1=n=1NCInCfn,E29
Jidi=0k=n=1NdIndfn,Midi=0k=n=1NdMndfn,E30

where fn are the suitable BFs, Inc, Mn, InC, Ind and Mnd are the unknown expansion current coefficients, and Na, Nkc, NC and Nd are the numbers of BFs on the surfaces Sa, Skc, Sici=0k1, if any, and Sidi=0k1, respectively. Expansions (29) and (30) take into account relations (28) for unknown equivalent currents on opposite sides of the dielectric boundaries. They also consider the ratios for adjacent currents at material junctions, which are the boundaries between several media [22].

Substituting (29) and (30) into (21)(25) taking into account (3), (5), (26)(28) and testing the resulting equations with weighting functions w1r, w2r,…, wmr, defined in the range of the respective boundary operators, we obtain the following MoM system of linear algebraic equations:

ZmnJcJcZmnJcM0ZmnJcJdZmnJcMdZmnMJcZmnMM+QmnW0ZmnMJdZmnMMd00ZmnJCJCZmnJCJdZmnJCMdZmnJdJcZmnJdMZmnJdJCZmnJdJdZmnJdMdZmnMdJcZmnMdMZmnMdJCZmnMdJdZmnMdMdIncMnInCIndMnd=VmcVmM+VmWVmCVmdVmHdE31

where the matrix elements are defined as Zmnαβ=wmL̂αβfn, L̂αβ is the respective boundary integral operator, superscripts α,β=JcMJCJdMd; Vmc=wmEkinc, VmM=wmHkinc, VmCi=wmEiinc, Vmdij=wmEiincEjinc, VmHdij=wmHiincHjinc are the voltage elements due to the incident wave in i-th and j-th media, and the elements QmnW and VmW are the same as those expressed by (18) and (19) and determine the additional inclusions in the matrix and voltage elements due to the waveguide ports.

The MoM system (31) generalizes the solution (15) of the canonical waveguide port problem to the case of composite geometry. In the structure of the MoM matrix of this solution, blocks of waveguide excitation, complex structure, and couplings between these objects through dielectric interfaces are clearly seen.

### 3.4 Validation of the developed approach for composite geometry

The developed approach has been validated to simulate the scattering characteristics of a single monopole antenna, fed by waveguide excitation from a flanged coaxial line with dielectric filling. Figure 5a shows a schematic view of such antenna with a height ha = 10 mm placed above a square metallic plate of 20 mm × 20 mm, which serves as a reflector. The coaxial line has an outer diameter D = 6.98 mm, an inner diameter d = 2 mm, and a length hb = 15 mm. The line bottom end is accepted as a waveguide port, and the input impedance of the antenna at this port is simulated for various dielectric fillings of the line.

Figure 6 shows a comparison of the input impedances, calculated by the developed approach for the model of Figure 5a with εr=1.0001, by the WPE approach for the conducting model of Figure 5b, and by discontinuous Galerkin time-domain (DGTD) method [28]. An excellent agreement between the obtained results is seen, which confirms the equivalence and correctness of both WPE approaches (for conducting and composite geometries) for very low dielectric fillings of coaxial lines.

Figure 7 shows a comparison of the input impedances, calculated for the model of Figure 5a with εr=2.25 using the developed approach and DGTD method. An excellent agreement between both results is seen, which validates our approach to treat arbitrary dielectric and geometric parameters of composite structures with waveguide port excitation.

Comparison of Figures 6 and 7 shows that the use of dielectric filling of the coaxial line shifts the resonances of the input impedance to lower frequencies. In addition, this leads to a change in the line’s characteristic impedance from 75 Ω in Figure 6 to 50 Ω in Figure 7. Thus, the developed WPE approach for composite geometries covers a wider area of geometries and provides more control over the characteristics of the analyzed structures.

## 4. Waveguide port approach in coupling problems

### 4.1 Problem formulation

Consider the coupling problem between several composite structures fed by waveguide excitations. Although each structure can be formed from an arbitrary number of dielectric regions, for simplicity, we will consider only one-region structures with composite (dielectric and conducting) boundaries. Figure 8 shows the geometry of the problem consisting of N waveguides Wi radiating into dielectric regions Di, i = 1,2,…N, surrounded by closed surfaces SDi with partially conducting boundaries SDic and inward unit normal nDi. Waveguides Wi are filled, in general, by dielectrics with permittivities εi and permeabilities μi, and the regions Di are filled by dielectrics with parameters εDi and μDi. An outer space region D0 is a free space with material parameters ε0, μ0.

The waveguide ports Pi in cross-sections Sia divide the waveguides Wi into semi-infinite regions Ai and finite regions Bi to truncate the mesh in regions Ai with incident waveguide excitation and act as excitation sources of composite regions Di through the dielectric boundaries SDiBid between the regions Di and Bi. Each region Bi, Di and D0 is also excited, for generality, by the impressed EM field Eαinc, Hαinc, α=Bi,Di,D0.

To formulate the waveguide port excitation problems through the port surfaces Sia, we consider the aperture coupling problems between the regions Ai and Bi to divide an original problem into two sets of equivalence problems: internal problems for regions Ai and external problems for regions Bi, Di and D0. For this purpose, we cover the port surfaces Sia with PEC sheets and introduce equivalent magnetic currents Mi and Mi on both sides of Sia to restore tangential electric fields on the port surfaces Sia.

### 4.2 Solution of the internal equivalent problem

The internal equivalent problems for the considered geometry are similar to those formulated in Section 2.2 and implemented in Section 2.3. According to the equivalence principle [24], the magnetic currents in the regions Ai are related to the total electric field EAiSia on the port surface Sia by the relation:

Mi=ni×EAiSia=n0i×EAiSiaE32

where ni is an inward normal in the regionAi, and n0i=ni is the propagation direction of the incident wave. Thus, the solution of the internal problem is expressed by formulas analogous to those obtained in Section 2.3 with adding the index i, when necessary.

### 4.3 Formulation of the external equivalent problem

When considering the external equivalent problem, let SBic be the conducting boundary of the region Bi, including the inner sides of the waveguide walls and the conductive part of the boundary surface between the regions Bi and Di; SDic is the conductive part of the boundary surface SDi, and SD0cis the conducting boundary of the region D0, including the outer sides of the waveguide walls and all conducting boundaries between the regions D0 and Di. Further, SDiBid is the dielectric boundary between the regions Di and Bi, and SDiD0d is the dielectric boundary between the regions Di and D0. Per the equivalence principle [24], the dielectric boundaries between different regions can be replaced by oppositely directed equivalent electric and magnetic currents flowing on both sides of the dielectric interfaces.

The EM field in the waveguide region Bi is created by electric currents JBic flowing along the port surface Sia and conducting surface SBic, equivalent electric and magnetic currents JDiBid and MDiBid flowing along the dielectric interfaces SDiBid, and equivalent magnetic currents Mi flowing along the port surface Sia. The EM field in the region Di is created by electric currents JDic flowing along the conducting surfaces SDic, equivalent currents JDiBid and MDiBid flowing on dielectric boundaries SDiBid between the regions Di and Bi, and equivalent currents JDiDd0 and MDiDd0 flowing on dielectric boundaries SDiD0d between the regions Di and D0. The field in the free space region D0 is created by electric currents SD0c flowing along the total conducting boundary of the region D0, and equivalent currents JDiDd0 and MDiDd0 at dielectric boundaries between the regions Di and D0.

The unknown currents JBic,Mi,JDic,JDiBid,MDiBid,JD0c,JDiD0d,MDiD0dcan be found from the boundary conditions on the port surface and the conducting boundaries of the waveguide region Bi:

EBiinc+EBiscJBicMiJDiBidMDiBidtanSia+Sic=0just outsideSiaE33
HBiinc+HBiscJBicMiJDiBidMDiBidtanSia=HAiSiatanjust insideSia,E34

and the boundary conditions on the conducting and dielectric boundaries of the regions Di and D0:

EDiinc+EDiscJDicJDiBidMDiBidJDiD0dMDiD0dtan=0onSDicE35
EBiinc+EBiscJBicMiJDiBidMDiBidtan=EDiinc+EDiscJDicJDiBidMDiBidJDiD0dMDiD0dtanonSDiBidE36
HBiinc+HBiscJDicMiJDiBidMDiBidtan=HDiinc+HDiscJDicJDiBidMDiBidJDiD0dMDiD0dtanonSDiBidE37
ED0inc+i=1NED0scJD0cJDiD0dMDiD0dtan=0onSD0cE38
EDiinc+EDiscJDicJDiBidMDiBidJDiD0dMDiD0dtan=[ED0inc+ED0scJD0c+i=1NED0sc(JDiD0dMDiD0d)]tanonSDiD0dE39
HDiinc+HDiscJDicJDiBidMDiBidJDiD0dMDiD0dtan=[HD0inc+HD0scJD0c+i=1NHD0sc(JDiD0dMDiD0d)]tanonSDiD0dE40

The scattered EM fields in (33)(40) are related to the equivalent electric and magnetic currents by Eqs. (26) and (27). After substituting (26) and (27) into (35)(40), Eqs. (35)(40) represent a coupled system of integral equations in terms of unknown currents for solving the coupling problem between several composite structures.

### 4.4 MoM solution of the external equivalent problem

To solve the boundary problem (35)(40), we use the following MoM expansions for the unknown currents:

JBici=1N=n=1Na+NBcIncBfn,Mii=1N=n=1NaMnfn,JDici=1N=n=1NDcIncDfn,JD0c=n=1ND0cIncD0fn,E41
JDiBidJDiD0di=1N=n=1NdIndfn,MDiBidMDiD0di=1N=n=1NdMndfn,Mii=1N=n=1NaMnfnE42

where fn are the suitable BFs, IncB,IncD,IncD0,Ind,Mnd and Mn are the unknown expansion current coefficients, and Na,NBc,NDc,ND0candNd are the numbers of these BFs on the surfaces Siai=1N, SBici=1N, SDici=1N, SD0c and SDiBidSDiD0di=1N, respectively. Substituting now (41) and (42) into (35)(40) with an accounting of (3), (5), and (26) and (27) for each i-th region and testing the obtained equations with weighting functions w1r, w2r,…, wmr, defined in the range of the respective boundary operators, we obtain the following MoM system of linear algebraic equations:

ZJBcJBcZJBcM00ZJBcJdZJBcMdZMJBcZMM+QW00ZMJdZMMd00ZJDcJDc0ZJDcJdZJDcMd000ZJD0cJD0cZJD0cJdZJD0cMdZJdJBcZJdJDcZJdJDcZJdJD0cZZJdJdZJdMdZMdJBcZMdMZMdJDcZMdJD0cZMdJdZMdMdIcBMIcDIcD0IdMd=VcBVM+VWVcDVcD0VdVHdE43

where the elements of the block matrices are defined as: Zmnαβ=wmL̂iαβfn, L̂αβ is the respective boundary integral operator, superscripts α,β=JBcMJDcJD0cJdMd, voltage elements are defined in the same way as in Eq. (31), and the elements of the block matrices QW and VW are expressed by (18) and (19) for each i-th feeding waveguide and determine the additional inclusions in the matrix and voltage elements due to the waveguide ports. The MoM system (43) defines a solution to the coupling problem between several composite geometries. In the structure of the MoM matrix of this solution, blocks of waveguide excitations, complex geometries, and couplings between them are clearly seen.

### 4.5 Validation of the developed approach for coupling problems

The developed approach has been validated on a two-element antenna array fed by coaxial waveguide ports by comparing the simulation results obtained using the developed MoM approach and the DGTD method [28]. Figure 9 shows a schematic view of two identical monopole antennas flanged over the PEC plate and fed by coaxial waveguides with generally different diameters and dielectric fillings. The monopoles located at a distance La = 40 mm from each other have the same height ha = 10 mm above the PEC plate with a width W = 40 mm and a length L = 80 mm, which serves as a reflector. Coaxial waveguides have the same inner diameter d1 = d2 = 2 mm, but generally different outer diameters D1 and D2 and relative permittivities ε1 and ε2. The depth of each coaxial waveguide under the flange is hb = 15 mm, and its end is taken as the reference plane of the waveguide port.

Figure 10 shows the real and imaginary parts of the transmission coefficient S21=a02/a01+ between waveguide ports 1 and 2 with the same radii and dielectric fillings: D1/2 = D2/2 = 6.65 mm, and εr1 = εr2 = 5.17, which leads to the same characteristic impedances: Zc1=Zc2 = 50 Ω. The developed MoM approach and the DGTD method are compared. The first antenna in these simulations is considered active, and the second is passive. Comparison of these results shows very good agreement between them over a wide frequency range from 1 GHz up to 10 GHz. This validates the developed approach in modeling coupling problems for coaxial waveguide ports with the same characteristic impedance.

Figures 11 and 12 show a comparison of the transmission coefficient S21=a02/a01+Zc1/Zc2 between waveguide ports 1 and 2, calculated by the MoM and DGTD method for different parameters of coaxial waveguides. Figure 11 is made for the same fillings of waveguides: εr1 = εr2 = 2.25, but with different outer radii: D1/2 = 3.49 mm and D2/2 = 6.52 mm, while Figure 12 is performed for different fillings: εr1 = 4 and εr2 = 1.78, but with the same outer radii D1/2 = D2/2 = 5.3 mm. Both cases result in characteristic impedances of waveguides Zc1 = 50 Ω and Zc2 = 75 Ω. Comparison of the MoM and DGTD results again shows very good agreement between both simulated results, which validates the developed approach to modeling coupling problems for coaxial waveguide ports with different characteristic impedances.

## 5. Application of waveguide port approach

The obtained approach has been applied to practical EMC problems for microwave antennas fed by coaxial waveguides. Such waveguides are the most commonly used to excite microwave antennas and electronic devices. This excitation usually uses microwave coaxial connectors, such as BNC and SMA.

### 5.1 Modeling of two branches feeding large printed UWB antenna

First, based on the measurement data [28], a printed ultra-wideband (UWB) antenna is considered. Figure 13 shows a schematic view of a large printed UWB antenna with a two-branch-feed, the bottom of which is connected to the core of a 50 Ω SMA connector with waveguide excitation, the covering of which is connected to a metal plate serving as a reflector. The bottom end of the connector is accepted as a waveguide port, and the input impedance of the UWB antenna at the waveguide port is measured and simulated.

The model of a printed UWB antenna is a square metal patch with a length La = 40 mm and a width Wa = 40 mm, printed on a dielectric substrate with a length Lb = 43 mm, a width Wb = 47.5 mm, a thickness t = 1.5 mm and material parameters εrd = 4.4 and tanδd = 0.02. The antenna is connected to a two-branch-feeding strip with a total width t1 = 15 mm, a distance between the branches t2 = 11 mm, and a height of the branches h1 = 3.5 mm. The UWB antenna is placed at a height h2 = 3 mm above a metallic plate of a length L = 275 mm and a width W = 207 mm and is connected to the SMA connector. The model of the SMA connector is represented by a coaxial waveguide with an outer radius D/2 = 2.125 mm, inner radius d/2 = 0.635 mm, and a length Lcon = 6.8 mm, filled with a polyethylene dielectric with relative permittivity εr = 2.24 and loss tangent tan δ = 0.005.

Figure 14 shows a comparison of simulated input impedances of a printed UWB antenna at the waveguide port with measurement results [28]. Comparison of the simulated results with measurement data shows a good agreement between them in a wide frequency range from 1 to 10 GHz. This validates the developed approach to modeling the composite antenna geometries fed by coaxial waveguides with dielectric filling.

### 5.2 Coupling problem between GPS and SDARS antennas

In conclusion, based on the measurement data [28], the coupling between the GPS and SDARS patch antennas was analyzed in the frequency range from 1 GHz to 3 GHz. Figure 15 shows the measurement setup (a) and its schematic view (b) for studying the coupling between active GPS and passive SDARS antennas, separated by a distance of d = 4 cm. Both antennas are fed by 50 Ohm coaxial lines with standard SMA connectors with parameters described in Section 5.1.

The parameters of the setup are the following. The SDARS antenna is a square metallic patch of 32 mm × 32 mm size with two opposite cut corners, printed on a dielectric substrate with dimensions 34 mm × 34 mm × 3.25 mm and εr = 4.1. The GPS antenna is constructed by a square metallic patch of 21 mm × 21 mm size with truncated corners, printed on a 25 mm × 25 mm × 4 mm dielectric substrate with εr1 = 20.34. Both patch antennas are mounted on a 190 mm × 145 mm metal plate.

Figure 16 shows a comparison of the transmission coefficient between active GPS and passive SDARS patch antennas, obtained by the developed MoM approach and measurements. A pretty good agreement between the simulated results and measured data in the frequency range of 1–3 GHz is observed. This comparison validates the developed waveguide port approach with measurements to model coupling problems between different composite geometry antennas with coaxial waveguide ports.

## 6. Conclusion

The MoM-based waveguide port approach was developed to model waveguide port excitation problems on arbitrary conducting and composite geometries. The developed approach was validated for modeling radiation and coupling problems for coaxial ports by comparing the simulated results with those obtained by other approaches and measurements. The approach has been applied to practical EMC problems for microwave antennas fed by coaxial connectors. A good agreement between the simulated and measured results has been demonstrated. The efficiency of the developed approach for solving various complex problems with waveguide excitation has been verified.

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Written By

Faik Bogdanov, Irina Chochia, Lily Svanidze and Roman Jobava

Submitted: 29 January 2022 Reviewed: 01 February 2022 Published: 24 March 2022