# Multi-Physics Coupling Simulation Technique for Phase Stable Cables

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## Abstract

**:**

## 1. Introduction

## 2. Theories of Multiple Physical Fields and Their Couplings

#### 2.1. Electromagnetic Field

_{eav}and w

_{mav}are, respectively, the average energy density of electric and magnetic fields; q

_{eav}is the average dissipation power density caused by electric field in a cycle; $\overrightarrow{{D}^{\ast}}$, $\overrightarrow{{H}^{\ast}}$, and $\overrightarrow{{E}^{\ast}}$ are, respectively, the conjugates of the electric displacement vector, magnetic field intensity vector, and electric field intensity vector. It is noted that the dissipation power density caused by magnetic field, known as the magnetic hysteresis loss, is neglected due to the nonferromagnetic material of interest.

_{l}is the leaking conductivity of the material; δ is the loss angle of the material; and tan δ is the loss angle tangent, which is also named the dissipation factor.

_{0}is the reference temperature, which is often selected as 20 °C or 25 °C; σ

_{0}is the conductivity at T

_{0}; α

_{c}is called the linear temperature coefficient of resistibility, which is usually regarded as being independent with the temperature. Anyhow, the permittivity of a conductor material is always taken as a constant, which is equal to the permittivity of vacuum.

#### 2.2. Thermal Flow Field

_{p}is the specific heat capacity at constant pressure; T, as aforementioned, is temperature; t is time; λ is the heat conductivity; and q

_{e}is the dissipation power of electromagnetic heat. Given that the time constant of the thermal field is much greater than that of the electromagnetic field, q

_{e}is replaced by q

_{eav}, the mean of q

_{e}in one cycle.

_{f}is the kinematic viscosity of the fluid. Equation (13). reveals the conservation of mass, and Equation (14), which is known as the Navier–Stokes equation, demonstrates the conservation of momentum. In fact, ρ is a function of T, and for an approximate ideal gas, it can be expressed as follows:

_{A}is the absolute pressure; M is the average molar mass of the gas; and R is the universal gas constant, which is equal to 8.314 J∙K

^{−1}∙mol

^{−1}.

_{e}is the surface emissivity; σ

_{r}is Stefan-Boltzmann constant, which is equal to 5.67 × 10

^{−8}J∙m

^{−2}∙s

^{−1}∙K

^{−4}; T

_{amb}is the ambient temperature; and $\overrightarrow{n}$ is the unit normal vector of the surface.

#### 2.3. Thermal Mechanics Field

## 3. Modelling of Corrugated Phase Stable Cable

#### 3.1. Geometric Model

_{1}and s

_{2}are parameters; and (x

_{1}, y

_{1}, z

_{1}) and (x

_{2}, y

_{2}, z

_{2}) are the coordinates of the points, respectively, in two parts of the surface, which have normal vectors in different directions. The external surface of the outer conductor can be described similarly.

#### 3.2. Physical Model

- As the skin depth is far less than radii and thicknesses of the conductors at frequencies in interest, the Joule heat dissipated from the conductors can be regarded to yield only on the laminas proximate to the insulation;
- The contact thermal resistance between the insulation and the conductors is ignored and the radiation heat transfer of the outer surface of the cable is ignored;
- The surrounding air has a temperature of 20 °C and the temperature of the upstream air of natural convection keeps 20 °C;
- The velocity of the surrounding air is limited and the flow mode is laminar flow;
- The surrounding air is regarded as viscous incompressible fluid, with a Mach number lower than 0.3;
- The deformation of the cable satisfies the infinitesimal deformation hypothesis such that linear solid mechanics always holds;
- The gravity is ignored when studying the thermal deformation of the cable in solid mechanics.

_{rP}(T), is measured according to IEC 61196-125 and presented in Figure 3, which is to be utilized in the simulation model with interpolation.

#### 3.3. Simulation Model

- Calculate the electromagnetic field, the thermal field and, the flow field when transmitting electromagnetic wave of certain power by a frequency stationary study;
- Calculate the thermal deformation (solid mechanics field) of the corrugated cable by a stationary study;
- Calculate the electromagnetic field once again to attain the phase stability of the cable by a frequency domain study.

## 4. Analysis of Simulation Results

#### 4.1. The Physical Fields

#### 4.2. Phase Stability

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Cross-section view of the corrugated cable in Figure 1.

**Figure 4.**Distribution of the electric field in the lengthwise cross-section of the cable insulation.

**Figure 5.**Distributions of (

**a**) the thermal field and (

**b**) the flow field when transmitting a 100 W electromagnetic wave at 1 GHz.

**Figure 7.**Diagram of the displacement of every particle in the lengthwise cross-section of the cable.

**Figure 9.**Thermal phase stability of the corrugated phase stable cable obtained by multi-physics coupling simulations and experiments.

**Table 1.**Explanation of the notations in Figure 2.

Notation | Meaning | Value |
---|---|---|

R_{in} | Radius of the inner conductor | 1.28 mm |

R_{out} | Minimum inside radius of the outer conductor | 3.32 mm |

h | Maximum difference of the outer conductor inside radius | 0.3 mm |

l | Pitch of the corrugation | 3 mm |

t_{out} | Thickness of the outer conductor along the radial direction | 0.3 mm |

Material | Copper | ePTFE | Air |
---|---|---|---|

Relative permittivity | 1 | ε_{rP}(T) | — |

Relative permeability | 1 | 1 | — |

Conductivity at 25 °C | 6 × 10^{7} | — | — |

Temperature coefficient of resistibility (K^{−1}) | 3.8 × 10^{−3} | — | — |

Dissipation factor | — | 1 × 10^{−4} | — |

Density at 20 °C (kg∙m^{−3}) | 8960 | 2180 | 1.24 |

Average molar mass (g∙mol^{−1}) | — | — | 29 |

Thermal conductivity (W∙m^{−1}∙K^{−1}) | 400 | 0.24 | 0.027 |

Specific heat capacity (J∙kg^{−1}∙K^{−1}) | 385 | 1000 | 1010 |

Thermal expansion coefficient (K^{−1}) | 1.7 × 10^{−5} | 1 × 10^{−4} | — |

Young’s modulus (Pa) | 1.1 × 10^{11} | 4 × 10^{8} | — |

Poisson’s ratio | 0.35 | 0.46 | — |

Kinematic viscosity (Pa∙s) | — | — | 4 × 10^{−5} |

EMW Power (W) | Temperature (°C) | EMW Power (W) | Temperature (°C) |
---|---|---|---|

61 | 30 | 519 | 90 |

100 | 40 | 602 | 100 |

185 | 50 | 684 | 110 |

270 | 60 | 752 | 120 |

354 | 70 | 827 | 130 |

435 | 80 | 897 | 140 |

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**MDPI and ACS Style**

Zhang, G.; Chen, X.; Yang, D.; Wang, L.; He, X.; Zhang, Z.
Multi-Physics Coupling Simulation Technique for Phase Stable Cables. *Electronics* **2023**, *12*, 1602.
https://doi.org/10.3390/electronics12071602

**AMA Style**

Zhang G, Chen X, Yang D, Wang L, He X, Zhang Z.
Multi-Physics Coupling Simulation Technique for Phase Stable Cables. *Electronics*. 2023; 12(7):1602.
https://doi.org/10.3390/electronics12071602

**Chicago/Turabian Style**

Zhang, Gang, Xiao Chen, Dazhi Yang, Lixin Wang, Xin He, and Zhehao Zhang.
2023. "Multi-Physics Coupling Simulation Technique for Phase Stable Cables" *Electronics* 12, no. 7: 1602.
https://doi.org/10.3390/electronics12071602