Computational Design of an In-Line Coaxial-to-Circular Waveguide Adapter with More Than an Octave Bandwidth

: This paper presents a computer-simulation-based design of an in-line, coaxial-to-circular waveguide adapter for converting the coaxial transverse electromagnetic (TEM) mode to the circular waveguide TE 11 mode over more than a one-octave bandwidth. The proposed adapter consists of a coaxial-to-rectangular waveguide transformer employing a stepped-ridge converter and a rectangular-to-circular waveguide transformer employing a curved transition. The proposed adapter has been optimized using a commercial simulation tool. The dimensions of the designed adapter are given so that it can be verified by anyone who is interested. The designed adapter operates from 8.00 GHz to 22.95 GHz (2.87:1 bandwidth) with a reflection coefficient of less than − 20 dB and a higher-order mode level of less than − 25.0 dB.


Introduction
Circular waveguides are widely used in high-frequency signal transmission as well as in various microwave components, including rotary joints [1], high-power microwave devices [2], radiating elements [3][4][5][6], measurement fixtures [7] and power transmission using the low-loss TE 21 mode [8].In all these applications, it is often necessary to launch a signal from a coaxial cable to a circular waveguide for which a high-performance adapter is required.In many applications, it is advantageous for an adapter to work with as large a bandwidth as possible.
Coaxial-to-circular waveguide adapters (CCWAs) have been employed in a rightangle form for a prime-focus reflector feed [3], a conical horn [4], a double-ridge horn [5] and a quad-ridge horn [6].An in-line CCWA has been used to convert the coaxial TEM mode to the circular TM 01 mode in the measurement of a liquid material [7] and in a high-power microwave application [8].In these applications, a broadband coaxial-tocircular waveguide transition, with a reflection coefficient of less than −20 dB would be very helpful.
In a coaxial-to-metallic waveguide adapter, the conversion between the coaxial TEM mode to the fundamental waveguide mode has been realized using a simple cylindrical probe [9], a disc probe [10], a conical probe [11] and a stepped-ridge converter [12,13].In most of the existing works on CCWAs, right-angle structures have been employed, where the coaxial cable axis is perpendicular to the circular waveguide axis, as indicated by the studies of [14][15][16].Bang and Ahn presented a wideband, right-angle CCWA that operates from 1.04 to 2.53 times the TE 11 -mode cutoff frequency of the circular waveguide, with a reflection coefficient of less than −20 dB [11].Schönfield, Tsai and Chu presented a right-angle CCWA with double ridges in a circular waveguide [16].Their adapter operates from 0.83 GHz to 2.8 GHz, with a reflection coefficient of less than 0.3 (−10 dB).For some applications, it is advantageous to use an in-line adapter instead of a rightangle one.In-line CCWAs have mostly been studied for TEM-TM 01 mode conversion [17,18].A thorough literature survey reveals that only two previous works [19,20] have been published on in-line TEM-TE 11 mode converters with broadband performance.It is also important to point out that no commercial in-line CCWA product is available at the time of writing.
Tribak and co-workers designed an in-line CCWA as a part of their circular waveguide TM 01 -TE 11 mode converter [19].Their design employs a coaxial-to-rectangular waveguide transition and a rectangular-to-circular waveguide transition.Their adapter converts the circular waveguide TM 01 mode to the circular waveguide TE 11 mode at 9. 25-16.26GHz.Yun and co-workers used a printed dipole to realize an in-line CCWA operating at 13.6-17.9GHz [20].
The purpose and goal of this paper is to design a broadband, in-line adapter connecting a coaxial cable to a circular waveguide.Our work is built upon the work in [11], where a single rectangular waveguide section was employed between a coaxial cable and a circular waveguide for improved broadband performance.This paper proposes an advancement in the related method by employing three rectangular waveguide sections and a smooth splineshaped transition between the rectangular and circular waveguides to ensure the lower operating frequency limit approaches the cutoff frequency of the circular waveguide's TE 11 mode.The upper end of the operating frequency is extended to its maximum by making the structure as symmetrical as possible so that the higher-order modes are suppressed as much as possible.The end result is an in-line, coaxial-to-circular waveguide adapter operating from 8.00 GHz to 22.95 GHz (a ratio bandwidth of 2.87 or a 96.6% bandwidth).To the best of our knowledge, there is no published work on in-line, coaxial-to-circular waveguide adapters having such a wide bandwidth.CST Studio Suite TM V2023 was employed in the design of the proposed adapter.
The remainder of this paper is organized as follows.Section 2 describes the design methodology of the adapter that has been adopted in this paper.Section 3 presents a simulation-based design of the proposed adapter.Section 4 contains an interpretation of the design results.Section 5 presents a discussion of the design results.Finally, the conclusion of this paper is presented in Section 6.

Adapter Design Methodology
Our goal in this paper is to design an in-line, coaxial-to-circular waveguide adapter operating over more than a one-octave bandwidth.Direct probe-feeding of a circular waveguide does not yield broadband performance.Thus, this paper adopts the approach of Bang and Ahn [11].First, a coaxial probe excites a rectangular waveguide, which is then transformed to a circular guide using a linear taper.
Figure 1 shows the structure of the proposed adapter.It consists of a circular waveguide W; a circular-to-rectangular waveguide transformer T; the first (R 1 ), second (R 2 ) and third (R 3 ) rectangular waveguides; a four-step ridge converter S; a coaxial probe P passing through a hole H in the back short B; an impedance-matching cavity M; and a coaxial cable C.
The coaxial probe P excites the dominant TE 10 mode in the third waveguide R 3 via a four-step converter S. The impedance matching between the coaxial cable C and the third rectangular waveguide R 3 is predominantly facilitated by the stepped-ridge converter S. The reflection coefficient is reduced to less than −20 dB by adjusting the dimensions of the matching cavity M and the coaxial probe hole H in the waveguide back short B as well as that of the stepped-ridge transformer.Three rectangular waveguide sections, R 1 , R 2 and R 3 , are employed to extend the operating frequency range of the adapter.The first rectangular waveguide R 1 is smoothly transformed to the output circular waveguide W using the spline surface geometry designed in CST Studio Suite TM V2023.
Figure 2 shows the dimensional parameters of the proposed adapter.Figure 2 is drawn to scale, which helps in understanding the design concept.The design of the adapter starts with determining the rough dimensions of the coaxial cable, the rectangular waveguide and the circular waveguide.Figure 2 shows the dimensional parameters of the proposed adapter.Figure 2 is drawn to scale, which helps in understanding the design concept.The design of the adapter starts with determining the rough dimensions of the coaxial cable, the rectangular waveguide and the circular waveguide.For the coaxial cable, a probe of a standard connector is employed, with the diameters of center and outer conductors being 2a and 2b.In a coaxial cable, the fundamental mode is the TEM mode with a cutoff frequency of zero and the next three higher-order modes are the TE 11 , TE 21 and TE 31 modes, and their cutoff frequencies are given by the following equations [21]: where c is the speed of the light in vacuum and ε r is the dielectric constant of the material filling the coaxial cable.In an air-filled rectangular waveguide, the cutoff frequency of the TE mn modes with a broad wall width of a and narrow wall height of b is given by the following equation: The dominant mode in a rectangular waveguide is the TE 10 mode, with the cutoff frequency given by In a rectangular waveguide energized by a coaxial probe, the probe can be placed at the center of the waveguide's broad wall but not at the center of the waveguide's narrow wall.In this case, the low-order modes that can be easily excited are the TE 11 and TE 30 modes.The antisymmetric TE 20 mode has a zero field at the center of its broad wall and is not excited by a symmetrically placed coaxial probe.For a waveguide with a width-to-height ratio of 2, the cutoff frequencies of the TE 11 and TE 30 modes are 2.24 and 3.00 times the TE 10 dominant mode's cutoff frequency.To maximize the bandwidth of the transition, it is necessary to reduce the narrow wall's height so that the TE 11 mode is cut off at the same frequency as the TE 30 mode in the waveguide R 3 .
In designing the rectangular waveguides R 1 , R 2 and R 3 , the broad wall width a 1 of the waveguide R 1 is determined first.The broad wall width a 3 of the waveguide R 3 is made about 5 percent larger than a 1 so that the adapter's lower operating frequency limit is as close as possible to the TE 10 -mode's cutoff frequency of the waveguide R 1 .The waveguide section R 2 is used for impedance matching between R 1 and R 3 .
The diameter d of the circular waveguide W is chosen such that the circular TE 11 mode is cut off at the same frequency as the TE 10 mode of the rectangular waveguide R 1 .When a circular waveguide is excited by the TE 10 mode of a rectangular waveguide through a smooth transition, such a circular waveguide TE mode is easily excited as the one whose electric field is symmetric in the H plane and non-zero at the center of the waveguide cross-section.The circular waveguide modes compatible with the rectangular TE 10 mode include the fundamental TE 11 mode and higher-order TM 11 , TE 12 and TM 12 modes, whose cutoff frequencies are 2.08, 2.90 and 3.81 times the TE 11 -mode cutoff frequency, respectively.The cutoff frequencies of these modes are given by the following equations [21]: Table 1 shows the cutoff frequencies of the first three modes including the dominant mode in the coaxial cable C; three rectangular waveguides R 1 , R 2 , R 3 ; and the circular waveguide W in the proposed adapter, whose dimensions are given in the next section.
From Table 1, one can see that the adapter's operating frequency can be limited by the onset of the TM 11 mode in the circular waveguide and the TE 11 mode in the rectangular waveguide.With a careful design of the structure, one can suppress the level of higherorder modes to less than −25 dB relative to the dominant mode as will be shown later.Figures 3-5 show the electric fields at 50 GHz of the modes listed in Table 1 on the crosssection of the coaxial cable C, the rectangular waveguide R 3 and the circular waveguide W, respectively.The frequency of analysis is set to 50 GHz to ensure all the modes in Table 1 are above the cutoff.
cross-section.The circular waveguide modes compatible with the rectangular TE10 mode include the fundamental TE11 mode and higher-order TM11, TE12 and TM12 modes, whose cutoff frequencies are 2.08, 2.90 and 3.81 times the TE11-mode cutoff frequency, respectively.The cutoff frequencies of these modes are given by the following equations [21]: Table 1 shows the cutoff frequencies of the first three modes including the dominant mode in the coaxial cable C; three rectangular waveguides R1, R2, R3; and the circular waveguide W in the proposed adapter, whose dimensions are given in the next section.From Table 1, one can see that the adapter's operating frequency can be limited by the onset of the TM11 mode in the circular waveguide and the TE11 mode in the rectangular waveguide.With a careful design of the structure, one can suppress the level of higher-order modes to less than −25 dB relative to the dominant mode as will be shown later.Figures 3-5 show the electric fields at 50 GHz of the modes listed in Table 1 on the cross-section of the coaxial cable C, the rectangular waveguide R3 and the circular waveguide W, respectively.The frequency of analysis is set to 50 GHz to ensure all the modes in Table 1    Finally, for the design of the transition from the rectangular waveguide R 1 to the circular waveguide W, a smooth spline surface is employed, which is supported by the simulation tool.Compared with a flat surface or a linear transition, the reflection coefficient is lower with a spline taper.The E-plane taper or the taper in the rectangular waveguide's narrow wall starts from the middle of the waveguide R 2 , while the H-plane taper or the taper in the rectangular waveguide's broad wall starts at the end of the waveguide R 2 .Due to different transformation ratios in the Eand H-plane cross-sections of the circular and rectangular waveguides, the E-plane taper length L b is larger than the H-plane taper length L a , as shown in Figure 2. The following section presents a simulation-based design of the adapter based on the design methodology presented above.Finally, for the design of the transition from the rectangular waveguide R1 to the circular waveguide W, a smooth spline surface is employed, which is supported by the simulation tool.Compared with a flat surface or a linear transition, the reflection coefficient is lower with a spline taper.The E-plane taper or the taper in the rectangular waveguide's narrow wall starts from the middle of the waveguide R2, while the H-plane taper or the taper in the rectangular waveguide's broad wall starts at the end of the waveguide R2.Due to different transformation ratios in the E-and H-plane cross-sections of the circular and rectangular waveguides, the E-plane taper length Lb is larger than the H-plane taper length La, as shown in Figure 2. The following section presents a simulation-based design of the adapter based on the design methodology presented above.

Simulation-Based Design of the Adapter
This section presents a simulation-based design of the proposed adapter.The coaxialto-rectangular waveguide transformer and the rectangular-to-circular waveguide transformer are designed separately and then combined.The combined structure is optimized to achieve low reflection over a wide frequency range.
The adapter's dimensional parameters, shown in Figure 2, are optimized using the simulation tool.In designing the coaxial-to-rectangular waveguide transformer, the length, height and thickness of the stepped-ridge converter S, the dimensions of the matching cavity M and the diameter and length of the probe hole H have been optimized using CST Studio Suite TM V2023.The lengths L1, L2, L3; widths a1, a2, a3; and heights b1, b2,  Finally, for the design of the transition from the rectangular waveguide R1 to the cir cular waveguide W, a smooth spline surface is employed, which is supported by the simulation tool.Compared with a flat surface or a linear transition, the reflection coefficien is lower with a spline taper.The E-plane taper or the taper in the rectangular waveguide's narrow wall starts from the middle of the waveguide R2, while the H-plane taper or the taper in the rectangular waveguide's broad wall starts at the end of the waveguide R2.Due to different transformation ratios in the E-and H-plane cross-sections of the circular and rectangular waveguides, the E-plane taper length Lb is larger than the H-plane taper length La, as shown in Figure 2. The following section presents a simulation-based design of the adapter based on the design methodology presented above.

Simulation-Based Design of the Adapter
This section presents a simulation-based design of the proposed adapter.The coaxialto-rectangular waveguide transformer and the rectangular-to-circular waveguide transformer are designed separately and then combined.The combined structure is optimized to achieve low reflection over a wide frequency range.
The adapter's dimensional parameters, shown in Figure 2, are optimized using the simulation tool.In designing the coaxial-to-rectangular waveguide transformer, the length, height and thickness of the stepped-ridge converter S, the dimensions of the matching cavity M and the diameter and length of the probe hole H have been optimized

Simulation-Based Design of the Adapter
This section presents a simulation-based design of the proposed adapter.The coaxialto-rectangular waveguide transformer and the rectangular-to-circular waveguide transformer are designed separately and then combined.The combined structure is optimized to achieve low reflection over a wide frequency range.
The adapter's dimensional parameters, shown in Figure 2, are optimized using the simulation tool.In designing the coaxial-to-rectangular waveguide transformer, the length, height and thickness of the stepped-ridge converter S, the dimensions of the matching cavity M and the diameter and length of the probe hole H have been optimized using CST Studio Suite TM V2023.The lengths L 1 , L 2 , L 3 ; widths a 1 , a 2 , a 3 ; and heights b 1 , b 2 , b 3 of the waveguides R 1 , R 2 and R 3 , respectively, have also been optimized.For the rectangular-tocircular waveguide transformer T, this paper has applied a smooth spline taper with its length L a and L b set as control parameters.L a and L b are the length of the taper from the rectangular waveguide's broad (starting from A and B) and narrow walls to the circular waveguide, respectively.The taper shape is of the spline type provided by CST Studio Suite TM V2023 with smoothness control applied.
Table 2 shows the final dimensions of an optimized adapter.For the coaxial port, an SMA connector is used.With the dimensions given in Table 2, the cutoff frequencies of the fundamental mode and the excitable higher-order modes are shown in Table 1.These values are of fundamental importance in the design of a broadband adapter.It is important to sufficiently suppress the excitation of the higher-order modes (TM 11 and TE 12 modes in the circular waveguide; TE 11 , TM 11 and TE 30 modes in the rectangular waveguide).Figure 6 shows the reflection coefficient from the time-domain analysis of the coaxial-torectangular waveguide transformer and the rectangular-to-circular waveguide transformer.The lengths L a and L b of the rectangular-to-circular waveguide transformer are made sufficiently large so that the reflection coefficient is less than −20 dB at 8-24 GHz.Local peaks due to higher-order mode excitation occur at 19.18, 20.87 and 22.46 GHz.The coaxial-to-rectangular transformer is optimized for reflections < −20 dB at a 8-23 GHz frequency range.A local peak is observed at 21.25 GHz.The reflection coefficient of the final combined structure will closely follow that of the coaxial-to-rectangular waveguide transformer.
from the rectangular waveguide's broad (starting from A and B) and narrow walls to the circular waveguide, respectively.The taper shape is of the spline type provided by CST Studio Suite TM V2023 with smoothness control applied.
Table 2 shows the final of an optimized adapter.For the coaxial port, an SMA connector is used.With the dimensions given in Table 2, the cutoff frequencies of the fundamental mode and the excitable higher-order modes are shown in Table 1.These values are of fundamental importance in the design of a broadband adapter.It is important to sufficiently suppress the excitation of the higher-order modes (TM11 and TE12 modes in the circular waveguide; TE11, TM11 TE30 modes in the rectangular waveguide).Figure 6 shows the reflection coefficient from the time-domain analysis of the coaxialto-rectangular waveguide transformer and the rectangular-to-circular waveguide transformer.The lengths La and Lb of the rectangular-to-circular waveguide transformer are made sufficiently large so that the reflection coefficient is less than −20 dB at 8-24 GHz.Local peaks due to higher-order mode excitation occur at 19.18, 20.87 and 22.46 GHz.The coaxial-to-rectangular transformer is optimized for reflections < −20 dB at a 8-23 GHz frequency range.A local peak is observed at 21.25 GHz.The reflection coefficient of the final combined structure will closely follow that of the coaxial-to-rectangular waveguide transformer.contact resistance in the mating parts.The reflection coefficient is less than −20 dB at 8.00-22.95GHz (2.87:1 bandwidth).In this frequency range, nulls can be observed (at 8.14, 8.86, 10.48, 14.76 and 19.67 GHz) in the reflection coefficient caused by the destructive interference of the reflections at the rectangular-to-circular waveguide transformer and at the coaxial-to-rectangular waveguide transformer.

Analysis of the Dominant and Higher-Order Modes
The performance of the proposed adapter is governed the behavior of the dominant mode as well as that of the higher-order modes.This section presents an analysis of the dominant and higher-order modes in the designed adapter.
The bandwidth of the designed adapter is primarily determined by the transition between the coaxial cable R1 and the rectangular waveguide R3.When the bandwidth performance of the stepped-ridge transformer (S in Figure 2) is large enough, the lower limit of the operating frequency is governed by the broad wall dimensions of the waveguide guide R3, while the upper limit is determined by the excitation of higher-order modes.The lower operating frequency limits of the stepped-ridge transformer as well as that of the rectangular-to-circular waveguide taper (T in Figure 2) are very close to the fundamental mode's cutoff frequency, as shown in Figure 6.The reflection coefficient of the rectangular-to-circular waveguide transition rises rapidly as the frequency approaches the cutoff frequency from above.The transition T shows a reflection coefficient of less than −20 dB from the lower frequency limit and upward.
In any adapter, higher-order modes need to be sufficiently suppressed.Figure 8 shows the coefficient of transmission from the coaxial TEM mode to the fundamental TE11 and higher-order modes in the output circular waveguide along with the reflection coefficient of the TE11 mode.Compared to the level of the dominant TE11 mode, the higherorder mode level is less than −25.0 dB at the operating frequency range of 8.00-22.95GHz.Above 22.95 GHz, the level of higher-order modes is greater than −25 dB.In Figure 8

Analysis of the Dominant and Higher-Order Modes
The performance of the proposed adapter is governed by the behavior of the dominant mode as well as that of the higher-order modes.This section presents an analysis of the dominant and higher-order modes in the designed adapter.
The bandwidth of the designed adapter is primarily determined by the transition between the coaxial cable R 1 and the rectangular waveguide R 3 .When the bandwidth performance of the stepped-ridge transformer (S in Figure 2) is large enough, the lower limit of the operating frequency is governed by the broad wall dimensions of the waveguide guide R 3 , while the upper limit is determined by the excitation of higher-order modes.The lower operating frequency limits of the stepped-ridge transformer as well as that of the rectangular-to-circular waveguide taper (T in Figure 2) are very close to the fundamental mode's cutoff frequency, as shown in Figure 6.The reflection coefficient of the rectangular-to-circular waveguide transition rises rapidly as the frequency approaches the cutoff frequency from above.The transition T shows a reflection coefficient of less than −20 dB from the lower frequency limit and upward.
In any adapter, higher-order modes need to be sufficiently suppressed.Figure 8 shows the coefficient of transmission from the coaxial TEM mode to the fundamental TE 11 and higher-order modes in the output circular waveguide along with the reflection coefficient of the TE 11 mode.Compared to the level of the dominant TE 11 mode, the higher-order mode level is less than −25.0 dB at the operating frequency range of 8.00-22.95GHz.Above 22.95 GHz, the level of higher-order modes is greater than −25 dB.In Figure 8 Figure 9 shows the electric and magnetic fields of the dominant mode inside the adapter at 15 GHz.The peak values of the electric and magnetic fields are uniform along the wave propagation direction, indicating a small voltage standing wave ratio (VSWR) and thus a good impedance matching performance.
is less than −40 dB up to 23 GHz.
Figure 9 shows the electric and magnetic fields of the dominant mode inside the adapter at 15 GHz.The peak values of the electric and magnetic fields are uniform along the wave propagation direction, indicating a small voltage standing wave ratio (VSWR) and thus a good impedance matching performance.Figure 9 shows the electric and magnetic fields of the dominant mode inside the adapter at 15 GHz.The peak values of the electric and magnetic fields are uniform along the wave propagation direction, indicating a small voltage standing wave ratio (VSWR) and thus a good impedance matching performance.

Discussion
The adapter designed in this work operates from 8.00 GHz to 22.95 GHz with a reflection coefficient of less than −20 dB and higher-order mode level of less than −25 dB.This design has been made possible by the advanced capabilities of modern simulation tools and fast computing hardware (i.e., CPUs), as well as the adoption of advanced transition geometries in the design of the coaxial-to-rectangular waveguide and the rectangu-

Figure 1 .
Figure 1.Structure of the proposed coaxial-to-circular waveguide adapter.(a) Transparent view and (b) cutaway view.

Figure 2
Figure2shows the dimensional parameters of the proposed adapter.Figure2is drawn to scale, which helps in understanding the design concept.The design of the adapter starts with determining the rough dimensions of the coaxial cable, the rectangular waveguide and the circular waveguide.
Figure2shows the dimensional parameters of the proposed adapter.Figure2is drawn to scale, which helps in understanding the design concept.The design of the adapter starts with determining the rough dimensions of the coaxial cable, the rectangular waveguide and the circular waveguide.

1 Figure 2 .
Figure 2. Dimensional parameters of the proposed adapter.Figure 2. Dimensional parameters of the proposed adapter.

Figure 2 .
Figure 2. Dimensional parameters of the proposed adapter.Figure 2. Dimensional parameters of the proposed adapter.

Figure 3 .
Figure 3. Electric fields at 50 GHz in the coaxial cable C in (a) TEM mode, (b) TE11 mode and (c) TE21 mode.Figure 3. Electric fields at 50 GHz in the coaxial cable C in (a) TEM mode, (b) TE 11 mode and (c) TE 21 mode.

Figure 3 .
Figure 3. Electric fields at 50 GHz in the coaxial cable C in (a) TEM mode, (b) TE11 mode and (c) TE21 mode.Figure 3. Electric fields at 50 GHz in the coaxial cable C in (a) TEM mode, (b) TE 11 mode and (c) TE 21 mode.

Figure 4 .
Figure 4. Electric fields at 50 GHz in the rectangular waveguide R 3 in (a) TE 10 mode, (b) TE 11 mode and (c) TE 30 mode.

Figure 5 .
Figure 5. Electric fields at 50 GHz in the circular waveguide W in (a) TE 11 mode, (b) TM 11 mode and (c) TE 12 mode.

Figure 6 .
Figure 6.Reflection coefficients from the time-domain analysis of the coaxial-to-rectangular waveguide transformer and the rectangular-to-circular waveguide transformer in the proposed adapter.

Figure 7
Figure 7 shows the reflection and transmission coefficients of the final design of the proposed adapter.They have been obtained using the time-and frequency-domain simulation methods in CST Studio Suite TM V2023.The results from two different simulation schemes agree well with each other.The two transmission coefficients in blue and green

Figure 6 .
Figure 6.Reflection coefficients from the time-domain analysis of the coaxial-to-rectangular waveguide transformer and the rectangular-to-circular waveguide transformer in the proposed adapter.

Figure 7
Figure 7 shows the reflection and transmission coefficients of the final design of the proposed adapter.They have been obtained using the time-and frequency-domain simulation methods in CST Studio Suite TM V2023.The results from two different simulation schemes agree well with each other.The two transmission coefficients in blue and green overlap each other at 7.8-23.0GHz.The transmission coefficient is determined by the reflection coefficient Γ via the relation 1 − |Γ|2 .In a fabricated adapter, the transmission magnitude will be reduced by the finite conductivity of the conducting material and the contact resistance in the mating parts.The reflection coefficient is less than −20 dB at 8.00-22.95GHz (2.87:1 bandwidth).In this frequency range, nulls can be observed (at 8.14, 8.86, 10.48, 14.76 and 19.67 GHz) in the reflection coefficient caused by the destructive interference of the reflections at the rectangular-to-circular waveguide transformer and at the coaxial-to-rectangular waveguide transformer.

Figure 7 .
Figure 7. Reflection and transmission coefficients from the time-and frequency-domain analyses of the proposed adapter.
, the TM11 mode level is the largest, ranging from −40 dB to −25 dB at 17-23 GHz.The next largest mode is the TE31 mode, ranging from −40 dB to −25 dB at 18-23 GHz.The TM01

Figure 7 .
Figure 7. Reflection and transmission coefficients from the time-and frequency-domain analyses of the proposed adapter.
, the TM 11 mode level is the largest, ranging from −40 dB to −25 dB at 17-23 GHz.The next largest mode is the TE 31 mode, ranging from −40 dB to −25 dB at 18-23 GHz.The TM 01 mode steadily increases from −60 dB at 20 GHz to −20 dB at 23 GHz, while the TE 12 mode is less than −40 dB up to 23 GHz.

Figure 8 .
Figure 8. Reflection coefficient of the TE11 mode and the transmission coefficient of higher-order modes in the output circular waveguide of the proposed adapter.

Figure 8 .
Figure 8. Reflection coefficient of the TE 11 mode and the transmission coefficient of higher-order modes in the output circular waveguide of the proposed adapter.

Symmetry 2024 ,
16, x FOR PEER REVIEW 9 of 12 mode steadily increases from −60 dB at 20 GHz to −20 dB at 23 GHz, while the TE12 mode is less than −40 dB up to 23 GHz.

Figure 8 .
Figure 8. Reflection coefficient of the TE11 mode and the transmission coefficient of higher-order modes in the output circular waveguide of the proposed adapter.

Figure 9 .
Figure 9. Electric (a) and magnetic (b) fields of the dominant mode inside the proposed adapter at 15 GHz.

Figure 9 .
Figure 9. Electric (a) and magnetic (b) fields of the dominant mode inside the proposed adapter at 15 GHz.

Table 1 .
Table 1 lists cutoff frequencies in the SMA connector employed in the proposed transition, where 2a = 1.27 mm, 2b = 4.11 mm and ε r = 2.08.The first higher-order TE 11 mode is cut off at 26.27 GHz and the second TE 21 mode is cut off at 48.09 GHz.Waveguide modes and cutoff frequencies in the proposed adapter.

Table 1 .
are above the cutoff.Waveguide modes and cutoff frequencies in the proposed adapter.

Table 2 .
Dimensions of the proposed adapter (unit: mm).

Table 2 .
Dimensions of the proposed adapter (unit: mm).