# Computation of the Characteristic Parameters of Coaxial Waveguides Used in Precision Sensors

## Abstract

**:**

## 1. Introduction

## 2. Multilayer Waveguide

## 3. Internal Impedance of the Outer Conductor

^{th}conductive layers, whose i

^{th}inner and outer boundaries are limited by ${r}_{i}$ and ${r}_{i+1}$, respectively. The innermost boundary is indicated by ${r}_{0},$ while the outermost boundary is ${r}_{N}$. The physical properties of each i

^{th}layer are given by electrical conductivity ${\sigma}_{i}$, absolute magnetic permeability ${\mu}_{i}$ and absolute electrical permittivity ${\epsilon}_{i}$. They allow the computation of the complex wave propagation constant given by [33]:

^{th}layer, at radial distance $r$ from the center of the wire, are expressed by the solutions of the second order Maxwell differential equations in the cylindrical coordinate system [33]:

^{th}, ${\overline{K}}_{n}$ is the complex-valued modified Bessel function of the second kind of order n

^{th}, ${\overline{C}}_{i}$ and ${\overline{D}}_{i}$ are complex-valued constants and $I$ is the total current flowing along the longitudinal direction of the waveguide.

^{th}layer are given by:

^{th}layers:

## 4. Results

#### 4.1. The Validation of the Algorithm

^{6}S/m. The first layer is assumed to be equal to 10 μm thick copper, whose relative permeability is considered to be 0.99994 and electrical conductivity is 5.96 ∙10

^{7}S/m. The second material is 5 μm thick gold whose relative permeability is equal to 0.999966 and electrical conductivity is 4.4 ∙10

^{7}S/m. The relative permittivity was assumed to be equal to unity for all the materials used. The outer cylindrical conductor is made of the same materials as the inner but in reverse order: the first (counting from the inside) is 5 μm thick gold, then 10 μm thick copper and finally stainless steel pipe, equal to a 100 μm thick wall. The innermost radius of the outer structure is 1.6 mm. The dielectric placed between the conductors is air, whose relative permeability and permittivity is equal to 1.00054 and 1, respectively. The electrical conductivity of air was ignored.

_{c}|, R p.u.l. and L p.u.l. obtained with the newly presented algorithm and the algorithm from [30]. Both analytical algorithms provide similar results to those obtained from the FEM model for lower frequencies. However, the discrepancy between analytical and FEM results starts to be visible for frequencies higher than 1 MHz, while the skin effect becomes more significant. A detailed discussion of the FEM model at higher frequencies can be found in [37]. The results presented in this subsection confirm the correctness of the algorithm developed.

#### 4.2. Comparison of the Computation Time for the Analytical Algorithms

## 5. Discussion

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Comparison of the computation time of the characteristic impedance for the algorithm [30] and newly developed equations.

f, Hz | |Z_{c}|, Ω | ||
---|---|---|---|

New Algorithm | Algorithm from [30] | COMSOL FEM | |

10^{0} | 25,424 | 25,424 | 25,428 |

10^{1} | 8039.9 | 8039.9 | 8041.1 |

10^{2} | 2542.4 | 2542.4 | 2543.2 |

10^{3} | 804.00 | 804.00 | 804.32 |

10^{4} | 254.34 | 254.34 | 254.31 |

10^{5} | 83.227 | 83.227 | 83.259 |

10^{6} | 50.382 | 50.382 | 50.382 |

10^{7} | 48.737 | 48.737 | 48.742 |

10^{8} | 48.534 | 48.534 | 48.542 |

10^{9} | 48.368 | 48.368 | 48.379 |

4·10^{9} | 48.318 | 48.318 | 48.328 |

f, Hz | R, Ω/m | ||
---|---|---|---|

New Algorithm | Algorithm from [30] | COMSOL FEM | |

10^{0} | 0.2806685 | 0.2806685 | 0.2806685 |

10^{1} | 0.2806685 | 0.2806685 | 0.2806685 |

10^{2} | 0.2806685 | 0.2806685 | 0.2806685 |

10^{3} | 0.2806685 | 0.2806685 | 0.2806685 |

10^{4} | 0.2806714 | 0.2806714 | 0.2806714 |

10^{5} | 0.2809538 | 0.2809538 | 0.2809538 |

10^{6} | 0.3003694 | 0.3003694 | 0.3003694 |

10^{7} | 0.3733909 | 0.3733909 | 0.3733909 |

10^{8} | 0.8968540 | 0.8968540 | 0.8968594 |

10^{9} | 3.0633201 | 3.0633201 | 3.0635652 |

4·10^{9} | 9.6498060 | 9.6498060 | 9.6366165 |

f, Hz | L, nH/m | ||
---|---|---|---|

New Algorithm | Algorithm from [30] | COMSOL FEM | |

10^{0} | 170.8578 | 170.8578 | 170.8578 |

10^{1} | 170.8578 | 170.8578 | 170.8578 |

10^{2} | 170.8578 | 170.8578 | 170.8578 |

10^{3} | 170.8578 | 170.8578 | 170.8578 |

10^{4} | 170.8575 | 170.8575 | 170.8575 |

10^{5} | 170.8273 | 170.8273 | 170.8273 |

10^{6} | 168.7718 | 168.7718 | 168.7718 |

10^{7} | 164.0388 | 164.0388 | 164.0388 |

10^{8} | 162.7767 | 162.7767 | 162.7769 |

10^{9} | 161.6674 | 161.6674 | 161.6794 |

4·10^{9} | 161.3358 | 161.3358 | 161.3464 |

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Kubiczek, K.
Computation of the Characteristic Parameters of Coaxial Waveguides Used in Precision Sensors. *Sensors* **2023**, *23*, 2324.
https://doi.org/10.3390/s23042324

**AMA Style**

Kubiczek K.
Computation of the Characteristic Parameters of Coaxial Waveguides Used in Precision Sensors. *Sensors*. 2023; 23(4):2324.
https://doi.org/10.3390/s23042324

**Chicago/Turabian Style**

Kubiczek, Krzysztof.
2023. "Computation of the Characteristic Parameters of Coaxial Waveguides Used in Precision Sensors" *Sensors* 23, no. 4: 2324.
https://doi.org/10.3390/s23042324