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

^{*}

## Abstract

**:**

_{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.

## 1. Introduction

_{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.

_{01}mode in the measurement of a liquid material [7] and in a high-power microwave application [8]. In these applications, a broadband coaxial-to-circular waveguide transition, with a reflection coefficient of less than −20 dB would be very helpful.

_{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).

_{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.

_{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 9.25–16.26 GHz. Yun and co-workers used a printed dipole to realize an in-line CCWA operating at 13.6–17.9 GHz [20].

_{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.

## 2. Adapter Design Methodology

_{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.

_{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.

_{11}, TE

_{21}and TE

_{31}modes, and their cutoff frequencies are given by the following equations [21]:

_{r}is the dielectric constant of the material filling the coaxial cable. 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.

_{mn}modes with a broad wall width of a and narrow wall height of b is given by the following equation:

_{10}mode, with the cutoff frequency given by

_{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}.

_{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}.

_{10}mode, a stepped-ridge transformer is employed. Existing structures for TEM-TE

_{10}mode conversion include an L probe [22], a stepped L probe [23], a continuously tapered ridge transformer [24] and a stepped-ridge transformer [25,26,27,28,29,30,31,32,33,34]. A survey of the literature reveals that the stepped-ridge transformer is a preferred geometry for broadband transitions, which has also been adopted in this work.

_{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]:

_{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 higher-order modes to less than −25 dB relative to the dominant mode as will be shown later. Figure 3, Figure 4 and Figure 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 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.

_{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 E- and 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.

## 3. Simulation-Based Design of the Adapter

^{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-to-circular 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.

_{11}and TE

_{12}modes in the circular waveguide; TE

_{11}, TM

_{11}and TE

_{30}modes in the rectangular waveguide).

_{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.

^{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 7.8–23.0 GHz. 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 8.00–22.95 GHz (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.

## 4. Analysis of the Dominant and Higher-Order Modes

_{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.

_{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 8.00–22.95 GHz. Above 22.95 GHz, the level of higher-order modes is greater than −25 dB. In Figure 8, 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.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Structure of the proposed coaxial-to-circular waveguide adapter. (

**a**) Transparent view and (

**b**) cutaway view.

**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.**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.**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.**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.**Reflection and transmission coefficients from the time- and frequency-domain analyses of the proposed adapter.

**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.

**Figure 9.**Electric (

**a**) and magnetic (

**b**) fields of the dominant mode inside the proposed adapter at 15 GHz.

Waveguide | Modes/Cutoff Frequency (GHz) |
---|---|

W | TE_{11}, TM_{11}, TE_{12}/7.88, 16.39, 22.80 |

R_{1} | TE_{10}, TE_{11}, TE_{30}/7.87, 17.68, 23.61 |

R_{2} | TE_{10}, TE_{11}, TE_{30}/7.70, 20.17, 23.10 |

R_{3} | TE_{10}, TE_{11}, TE_{30}/7.54, 24.03, 22.61 |

C | TEM, TE_{11}, TE_{21}/0.00, 26.27, 48.09 |

Parameter | Value | Parameter | Value |
---|---|---|---|

a_{1}, b_{1}, L_{1} | 19.05, 8.04–9.47, 6.73 | d, L_{b}, F, E | 22.31, 55.04, 1.91, 5.62 |

a_{2}, b_{2}, L_{2} | 19.47, 6.57–8.04, 18.22 | L_{a}, L_{S}, t, G | 48.19, 21.96, 1.57, 1.06 |

a_{3}, b_{3}, L_{3} | 19.89, 6.57, 2.89 | Ridge width/height | 3.62/1.12, 2.14, 3.8, 5.33 |

Coax. cable | 1.27/4.11, ε_{r} = 2.08 | Probe hole | Dia. = 2.95, Len. = 1.57 |

Work | Type | Frequency (GHz) | Reflection (dB) | Ratio Bandwidth | Complexity |
---|---|---|---|---|---|

[11] | Right-angle | 7.34–17.77 | −20 | 2.42 | Low (conical probe and tapered transformer) |

[16] | Right-angle | 0.7–3.0 | −10 | 4.29 | Medium (double ridge) |

[19] | In-line | 9.2–16.2 | −20 | 1.76 | High (coaxial-to-rectangular-to-circular) |

[20] | In-line | 13.6–17.9 | −15 | 1.32 | Medium (dielectrically filled; contains air gap) |

This work | In-line | 8.00–22.95 | −20 | 2.87 | Medium (stepped ridge and spline taper converter) |

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**MDPI and ACS Style**

Altanzaya, E.; Heo, J.; Xu, S.; Lee, C.-S.; Ahn, B.-C.; Kim, S.-S.; Choi, S.-G.
Computational Design of an In-Line Coaxial-to-Circular Waveguide Adapter with More Than an Octave Bandwidth. *Symmetry* **2024**, *16*, 304.
https://doi.org/10.3390/sym16030304

**AMA Style**

Altanzaya E, Heo J, Xu S, Lee C-S, Ahn B-C, Kim S-S, Choi S-G.
Computational Design of an In-Line Coaxial-to-Circular Waveguide Adapter with More Than an Octave Bandwidth. *Symmetry*. 2024; 16(3):304.
https://doi.org/10.3390/sym16030304

**Chicago/Turabian Style**

Altanzaya, Erdenesukh, Jiwon Heo, Songyuan Xu, Chan-Soo Lee, Bierng-Chearl Ahn, Sung-Soo Kim, and Seong-Gon Choi.
2024. "Computational Design of an In-Line Coaxial-to-Circular Waveguide Adapter with More Than an Octave Bandwidth" *Symmetry* 16, no. 3: 304.
https://doi.org/10.3390/sym16030304