# Calculation and Analysis of Characteristic Parameters for Lossy Resonator

^{*}

## Abstract

**:**

_{010}are numerically calculated. With the proposed method, the impact of various dielectric structures and characteristics parameters on the resonant properties of the TM

_{010}mode may be thoroughly examined by taking into account the influence of the thickness as well as the materials of the lossy layer in the z direction. The relative error between the theoretical and the simulated results is below 0.7% at different structures and lossy dielectrics, which indicates that the general calculation approach, as well as crucial data and structure references, is suitable for a related device design in TM

_{010}mode.

## 1. Introduction

_{010}mode field distribution in the lossy resonator using the Borgnis function [21] and the mode-matching method [21,22,23].

_{1}and z = −h

_{1}.

_{010}mode is thoroughly examined. The findings demonstrate that the approach is accurate and stable and that it can be applied generically to the calculation and analyses of such structures.

## 2. Calculation and Analysis Model

_{010}mode [21].

- Developing electromagnetic field expressions and dividing regions. The electromagnetic field components of each zone in the entire model were determined by the Borgnis function, which was separated into numerous regular portions;
- Acquiring characteristic equations. By using the mode-matching method, some equations were created at the interface between various dielectrics;
- Numerical simulation. This process used the secant method to numerically solve for the resonant frequency and Q value, which can result in rapid convergence.

#### 2.1. Establishment of Electromagnetic Field Components in Each Region

_{1}and the length was 2 h

_{2}. The thickness of the cap-shaped dielectric coated longitudinally was h

_{2}− h

_{1}, and the whole structure was symmetric about the z-axis. The calculation model was divided into 3 regions. Region 1 was regarded as a vacuum with ε

_{0}permittivity, region 2 and region 3 were cap-shaped lossy dielectric coated on both ends of the cavity, respectively, and the corresponding permittivities were ε

_{2}and ε

_{3}. Moreover, the relative permeability of all regions equaled 1.

_{0i}mode was even in the ϕ direction [27], it could be obtained that the field component in the ϕ direction equaled 1. In addition, the PCW was set at the outermost layer (surface at r = r

_{1}, r = −r

_{1}, z = h

_{2}, and z = −h

_{2}) for the numerical computation. Thus, the z-direction component of the U function in region 2 and region 3 could be written as a hyperbolic cosine function to meet the boundary conditions, while the component in region 1 was the superposition of forward and backward waves. Moreover, the ratio of the resonator length to the radius was maintained at 0.7, which meant the first mode of the cavity was the TM mode. In summary, according to the Borgnis function, the V function was 0 and the U function of each region could be described as follows:

#### 2.2. Boundary Conditions and Characteristic Equations

_{1}[29]. Thus, Equation (16) could be obtained using the above conditions as well as Equations (4)–(7). Meanwhile, the relationship between ${k}_{ci}^{\left(1\right)}$ and ${\gamma}_{i}^{\left(1\right)}$ was defined as Equation (17).

_{2}and z = −h

_{2}, and they could be described as follows for n = 1.

#### 2.3. The Solution Method of Characteristic Equations

_{H}and mat

_{L}. When the mat

_{H}fails to fulfill the accuracy standard, we compute the difference coefficient, abbreviated diff. At the same time, a new iteration value, abbreviated f

_{new}, is generated, and the loop is restarted. The convergence requirement is met and the final numerical results are achieved if mat

_{H}is less than the tolerance. In this paper, the tolerance equaled 10

^{−6}.

## 3. Numerical Calculation Results and Analysis of the Influence of Dielectric Structure on Resonance Characteristics

#### 3.1. Analysis of the Influence of Dielectric Thickness on Resonance Characteristics

_{1}= 29.85 mm, 2 h

_{2}/r

_{1}= 0.7, ε

_{r}

_{2}= ε

_{r}

_{3}= 11.4, tan δ

_{2}= tan δ

_{3}= 0.0027. In the tables, delta h represents the thickness of the lossy dielectric in the z direction. f

_{rc}and Q

_{0c}are numerical calculation results while f

_{rs}and Q

_{0s}are simulation results. The assumption was that the first five terms of the electromagnetic field equations represented the Bessel function series in all calculations. The results demonstrated that the proposed method for determining the resonant frequency and unloaded Q value was accurate under these circumstances. The error of the resonant frequency calculation was within 0.1% and it was within 0.7% for the Q

_{0}value calculation. On the other hand, it can be concluded that the resonant frequency and unloaded Q factor of the TM

_{010}mode decreased significantly with the increase of the thickness. The former decreased by about 1.12 GHz and the latter was reduced by about 30,000 when the thickness reached 5.5 mm.

_{010}mode tended to decrease linearly. For every 0.5 mm increment of the cap-shaped dielectric thickness, the value of f

_{r}reduced by about 0.11 GHz. This was because the TM

_{010}mode had a strong longitudinal electric field in the axial region of the resonator. As a result, when the lossy dielectric coating was too thick in the longitudinal direction, it absorbed the energy of the electric field, increasing the loss of the resonant cavity and lowering the resonant frequency of the mode. Meanwhile, the increase of the dielectric thickness in the z direction meant that the dielectrics at both ends were closer, which may have led to the increase of equivalent capacitance of the cavity. This phenomenon can also cause the reduction of f

_{r}. Thus, the position of the lossy dielectric in the z-direction has a significant impact on f

_{r}. It is also important to keep in mind that the accuracy significantly decreased when the cap-shaped dielectric was quite thick, even though the overall calculation results were still very precise. However, when the dielectric was thin, the equivalent capacitance of the cavity was small and the simulation process might be less influenced by it, so that the error of the former was smaller.

_{010}mode in this cavity was primarily concentrated in the central axis area [31], which is directly shown in Figure 5. Meanwhile, a thin dielectric meant a small loss, and the simulation process might ignore the loss from the area of weak electric activity. Therefore, the error of the latter was larger. Similarly, the conclusion would be opposite in a thick dielectric situation.

#### 3.2. Analysis of the Influence of Dielectric Loss on Resonance Characteristics

_{010}mode. Consequently, we chose 10 different kinds of lossy dielectrics and studied how they would affect the characteristic parameters of the resonator under the situation that all of them were of the same thickness. Additionally, the 10 materials are listed in detail in Table 3 and Table 4; they had a variety of relative permittivities and tangent deltas. Based on these parameters, accurate results for both calculations and simulations could be produced. Notably, the variation of the characteristic equation’s root due to a change in permittivity prevented the procedure from converging when the initial values were incorrect.

_{1}= 0.02985 m, 2 h

_{2}/r

_{1}= 0.7, delta h = 1 mm. It can be concluded from the two tables that the equivalent capacitance of the cavity became larger when the real part of ε

_{r}

_{2}increased. The situation was similar to the increase of the dielectric thickness in the z direction and resulted in the decrease of f

_{r}. Moreover, the raise of tan δ signified an increase of dielectric loss so that the Q factor decreased. In conclusion, the two tables demonstrate that our technique had a good accuracy when calculating the characteristic parameters for various dielectric losses. When the loss was significant, the computed result and the simulation result were extremely similar.

_{r}changed slightly at the loss range we set. The real part of the complex permittivity had the greatest influence on the resonance frequency value, and their relationship was inversely proportional. As a result, the calculation’s findings followed electromagnetic rules. Additionally, the results of the calculation resembled the simulation more closely when the loss of the cap-shaped dielectric was quite substantial. This finding suggested that variable dielectric losses only altered the resonance frequency of the TM

_{010}mode a little, when the thickness of the lossy dielectric in the z direction was known and thin. This phenomenon may be utilized for device designs when f

_{r}is limited in a specific range while others are apparently different.

_{0}decreased as the dielectric loss increased. According to Figure 5, it can be assumed that the dissipation of the longitudinal electric field was the main cause of loss. In addition, excessive dielectric loss in the z-direction restrained the start oscillation of the TM

_{010}mode. For instance, when type 9 and type 10 were present, the Q

_{0}was below 1000, which resulted in a significant loss of energy. On the other hand, in large loss conditions, the error tended to be less. The findings above demonstrated that the materials had a significant impact on the Q value of this cavity. The value of Q

_{0}will also noticeably decrease when a material with a high tangent delta is coated in the z-direction, preventing the commencement of this mode even though the lossy dielectric is quite thin. Overall, our approach was very accurate and useful for resolving eigenmode issues for lossy resonators with the structure we suggested.

## 4. Conclusions

_{0}value of the TM

_{010}mode in the longitudinal direction. We think that these findings can offer crucial information and a structural benchmark for a TM

_{010}mode device design. Our approach, however, is limited in that it can only be used to solve TM

_{0i0}mode issues in a particular cavity. The next step should be to start investigating the impact of the dielectric’s structural characteristics on high-order modes and a loss from the r direction.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## Appendix B

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**Figure 1.**(

**a**) The longitudinal section of the proposed lossy resonator; (

**b**) the transverse section of the proposed lossy resonator.

**Figure 2.**The flow chart of a computer program for characteristic parameters calculation based on the secant method.

**Table 1.**The calculation results of resonant frequency for TM

_{010}mode changing with dielectric thickness compared with CST simulation.

delta h/mm | f_{rc}/GHz | f_{rs}/GHz | Error |
---|---|---|---|

1.0 | 3.6661 | 3.6651 | 0.027% |

1.5 | 3.5672 | 3.5661 | 0.031% |

2.0 | 3.4600 | 3.4588 | 0.035% |

2.5 | 3.3443 | 3.3429 | 0.042% |

3.0 | 3.2206 | 3.2189 | 0.053% |

3.5 | 3.0905 | 3.0889 | 0.052% |

4.0 | 2.9562 | 2.9544 | 0.061% |

4.5 | 2.8201 | 2.8186 | 0.053% |

5.0 | 2.6846 | 2.6825 | 0.078% |

5.5 | 2.5514 | 2.5496 | 0.071% |

**Table 2.**The calculation results of Q

_{0}for TM

_{010}mode changing with dielectric thickness compared with CST simulation.

delta h/mm | Q_{0c} | Q_{0s} | Error |
---|---|---|---|

1.0 | 31,460 | 31,264 | 0.627% |

1.5 | 16,295 | 16,215 | 0.493% |

2.0 | 9448 | 9418 | 0.319% |

2.5 | 5938 | 5925 | 0.219% |

3.0 | 3997 | 3990 | 0.175% |

3.5 | 2862 | 2861 | 0.035% |

4.0 | 2169 | 2167 | 0.092% |

4.5 | 1724 | 1726 | 0.116% |

5.0 | 1432 | 1434 | 0.139% |

5.5 | 1233 | 1235 | 0.162% |

**Table 3.**The calculation results of resonant frequency for TM

_{010}mode changing with dielectric loss compared with CST simulation.

ε_{r}_{2} | tan δ | f_{rc}/GHz | f_{rs}/GHz | Error |
---|---|---|---|---|

11.4 | 0.00270 | 3.6661 | 3.6651 | 0.027% |

8.2 | 0.00780 | 3.6741 | 3.6731 | 0.027% |

7.5 | 0.01120 | 3.6766 | 3.6756 | 0.027% |

6.1 | 0.01660 | 3.6831 | 3.6822 | 0.024% |

5.3 | 0.01940 | 3.6882 | 3.6873 | 0.024% |

4.1 | 0.02375 | 3.6995 | 3.6986 | 0.024% |

3.5 | 0.02585 | 3.7079 | 3.7070 | 0.024% |

3.2 | 0.02755 | 3.7132 | 3.7123 | 0.024% |

2.8 | 0.02965 | 3.7220 | 3.7211 | 0.024% |

2.5 | 0.03545 | 3.7303 | 3.7295 | 0.021% |

**Table 4.**The calculation results of resonant frequency for TM

_{010}mode changing with dielectric loss compared with CST simulation.

ε_{r}_{2} | tan δ | Q_{0c} | Q_{0s} | Error |
---|---|---|---|---|

11.4 | 0.00270 | 31,460 | 31,264 | 0.627% |

8.2 | 0.00780 | 8698 | 8664 | 0.392% |

7.5 | 0.01120 | 5652 | 5633 | 0.337% |

6.1 | 0.01660 | 3218 | 3210 | 0.249% |

5.3 | 0.01940 | 2439 | 2434 | 0.205% |

4.1 | 0.02375 | 1585 | 1582 | 0.190% |

3.5 | 0.02585 | 1261 | 1259 | 0.159% |

3.2 | 0.02755 | 1089 | 1088 | 0.092% |

2.8 | 0.02965 | 895 | 894 | 0.112% |

2.5 | 0.03545 | 675 | 674 | 0.148% |

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**MDPI and ACS Style**

Cui, J.; Yu, Y.; Lu, Y.
Calculation and Analysis of Characteristic Parameters for Lossy Resonator. *Electronics* **2023**, *12*, 7.
https://doi.org/10.3390/electronics12010007

**AMA Style**

Cui J, Yu Y, Lu Y.
Calculation and Analysis of Characteristic Parameters for Lossy Resonator. *Electronics*. 2023; 12(1):7.
https://doi.org/10.3390/electronics12010007

**Chicago/Turabian Style**

Cui, Jian, Yu Yu, and Yuanyao Lu.
2023. "Calculation and Analysis of Characteristic Parameters for Lossy Resonator" *Electronics* 12, no. 1: 7.
https://doi.org/10.3390/electronics12010007