# Fast Calculation Method for Transient Response of Transmission Line Based on Chebyshev Pseudospectral–Two-Step Three-Order Boundary Value Coupled Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Chebyshev Pseudospectral Method

## 3. Two-Step Three-Order Boundary Value Method

## 4. Calculation Method for Transient Response of Transmission Line Based on Chebyshev Pseudospectral–Two-Step Three-Order Boundary Value Coupled Method

## 5. Simulation Test

#### 5.1. Test for Example 1

#### 5.2. Test for Example 2

#### 5.3. Calculation of Transient Response of Nondestructive Transmission Line

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Li, Z.; Tao, Y.; Abu-Siada, A.; Masoum, M.A.; Li, Z.; Xu, Y.; Zhao, X. A New Vibration Testing Platform for Electronic Current Transformers. IEEE Trans. Instrum. Meas.
**2019**, 68, 704–712. [Google Scholar] [CrossRef] - Lee, S. Dynamics of Trapped Solitary Waves for the Forced KdV Equation. Symmetry
**2018**, 10, 129. [Google Scholar] [CrossRef] - Kokkinos, T.; Sarris, C.D.; Eleftheriades, G.V. Periodic finite-difference time-domain analysis of loaded transmission-linenegative-refractive-index metamaterials. IEEE Trans. Microw. Theory Tech.
**2005**, 53, 1488–1495. [Google Scholar] [CrossRef] - Zhang, Y. Finite-difference time-domain analysis of integrated ceramic ball grid array package antenna for highly integrated wireless transceivers. IEEE Trans. Antennas Propagat.
**2004**, 52, 435–442. [Google Scholar] [CrossRef] - Jiang, L.; Chen, Z.; Mao, J. On the numerical stability of the Precise Integration Time-Domain (PITD) method. IEEE Microw. Wirel. Compon. Lett.
**2007**, 17, 471–473. [Google Scholar] [CrossRef] - Tang, M.; Mao, J.; Gao, C.; Chen, B.; Chen, H. A precise Time-Step Integration method for transient analysis of lossy nonuniform transmission lines. IEEE Trans. Electromagn. Compat.
**2008**, 50, 166–174. [Google Scholar] [CrossRef] - Feng, X.; Chen, L.; Wei, H.; Wu, K. Fast and accurate transient analysis of buried wires and its applications. IEEE Trans. Electromagn. Compat.
**2014**, 56, 188–199. [Google Scholar] [CrossRef] - Xu, Q.; Li, Z.; Wang, J. Modeling of transmission lines by the differential quadrature method. IEEE Microw. Guided Wave Lett.
**1999**, 9, 145–147. [Google Scholar] [CrossRef] - Bagci, H.; Yilmaz, A.E.; Michielssen, E. An FFT-accelerated time domain multiconductor transmission line simulator. IEEE Trans. Electromagn. Compat.
**2010**, 52, 199–214. [Google Scholar] [CrossRef] - Thielen, B.L.A.; Vandenbosch, G.A.E. Fast transmission line coupling calculation using a convolution technique. IEEE Trans. Electromagn. Compat.
**2001**, 43, 11–17. [Google Scholar] [CrossRef] - Wang, T.; Guo, B. Composite generalized Laguerre–Legendre pseudospectral method for Fokker-Planck equation in an infinite channel. Appl. Numer. Math.
**2008**, 58, 1448–1466. [Google Scholar] [CrossRef] - Ghoshi, S. Relative Effects of Asymmetry and Wall Slip on the Stability of Plane Channel Flow. Fluids
**2017**, 2, 66. [Google Scholar] [CrossRef] - Izadkhah, M.M.; Saberi, J.; Toutounian, F. An extension of the Gegenbauer pseudospectral method for the time fractional Fokker-Planck equation. Math. Method. Appl. Sci.
**2018**, 41, 1301–1315. [Google Scholar] [CrossRef] - Arribas, D.G.; Rivo, M.S.; Arnedo, M.S. Optimization of path-constrained systems using pseudospectral methods applied to aircraft trajectory planning. IFIC Pap.
**2015**, 48, 192–197. [Google Scholar] [CrossRef] - Argyros, I.K.; Shakhno, S.; Yarmola, H. Two-Step Solver for Nonlinear Equations. Symmetry
**2019**, 11, 128. [Google Scholar] [CrossRef] - Zhang, Y.; Tan, Z. On the existence of solutions to the Navier-Stokes-Poisson equations of a two-dimensional compressible flow. Math. Method Appl. Sci.
**2011**, 30, 305–329. [Google Scholar] [CrossRef] - Truhlar, D.G. Finite difference boundary value method for solving one-dimensional eigenvalue equations. J. Comput. Phys.
**2015**, 10, 123–132. [Google Scholar] [CrossRef] - Zhu, B.; Zeng, Q.; Chen, Y.; Zhao, Y.; Liu, S. A dual-input high step-up DC/DC converter with ZVT auxiliary circuit. IEEE Trans. Energy Convers.
**2019**, 34, 161–169. [Google Scholar] [CrossRef] - Chen, Z.; Liu, C.; Simos, T.E. New three–stages symmetric two step method with improved properties for second order initial/boundary value problems. J. Math. Chem.
**2018**, 56, 2591–2616. [Google Scholar] [CrossRef] - Jagannadha Rao, G.V.V.; Padhan, S.K.; Postolache, M. Application of Fixed Point Results on Rational F-Contraction Mappings to Solve Boundary Value Problems. Symmetry
**2019**, 11, 70. [Google Scholar] [CrossRef] - Wattanasakulpong, N.; Pornpeerakeat, S.; Chaikittiratana, A. Chebyshev Collocation Solutions for Vibration Analysis of Circular Cylindrical Shells with Arbitrary Boundary Conditions. Int. J. Struct. Stab. Dyn.
**2017**, 17, 1750020. [Google Scholar] [CrossRef] - Noor, M.A.; Al-Said, E.A. Finite-Difference Method for a System of Third-Order Boundary-Value Problems. J. Optim. Theory Appl.
**2002**, 112, 627–637. [Google Scholar] [CrossRef] - Babolian, E.; Fattahzadeh, F. Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration. Appl. Math. Comput.
**2007**, 188, 417–426. [Google Scholar] [CrossRef] - Qiu, L.; Yu, Y.; Xiong, Q.; Deng, C.; Cao, Q.; Han, X.; Li, L. Analysis of electromagnetic force and deformation behavior in electromagnetic tube expansion with concave coil based on finite element method. IEEE Trans. Appl. Supercond.
**2018**, 28. [Google Scholar] [CrossRef] - Doha, E.H.; Abd-Elhameed, W.M.; Youssri, Y.H. Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane–Emden type. New Astron.
**2013**, 24, 113–117. [Google Scholar] [CrossRef] - Xiong, Q.; Tang, H.; Wang, M.; Huang, H.; Qiu, L.; Yu, K.; Chen, Q. Design and implementation of tube bulging by an attractive electromagnetic force. J. Mater. Process. Tech.
**2019**. [Google Scholar] [CrossRef] - Koksa, M.E. An operator-difference method for telegraph equations arising in transmission lines. Discret. Dyn. Nat. Soc.
**2011**, 2011, 561015. [Google Scholar] [CrossRef] - Yang, N.; Huang, Y.; Hou, D.; Liu, S.; Ye, D.; Dong, B.; Fan, Y. Adaptive Nonparametric Kernel Density Estimation Approach for Joint Probability Density Function Modeling of Multiple Wind Farms. Energies
**2019**, 12, 1356. [Google Scholar] [CrossRef] - Danesh, M.; Farajpour, A.; Mohammadi, M. Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method. Mech. Res. Commun.
**2012**, 39, 23–27. [Google Scholar] [CrossRef]

**Figure 7.**The calculation results of end terminal load voltage of nondestructive transmission line. (

**a**) PSM-BVM3; (

**b**) PSM-BVM2; (

**c**) PSM-TR.

**Table 1.**Synthesis of transient response calculation approach. Finite difference time domain method (FDTD); precise integration method (PIM); improved FDTD (IFDTD); differential quadrature method (DQM); fast Fourier transform (FFT); convolution technique (CT); pseudo spectral method (PSM); boundary value method (BVM).

Transient Response Calculation Approach | Advantages | Drawbacks |
---|---|---|

FDTD | Simple and easy to implement and employ in practical application | Calculation timestep limitation, low computation efficiency |

PIM | Good numerical stability, high precision | Calculation of numerical matrix exponent increase due to increase of system dimension |

IFDTD | Higher calculation accuracy than conventional FDTD | Calculation burden due to its small calculation timestep |

DQM | Accuracy can be guaranteed due to its global approximation | Complex calculation |

FFT | Can preferably reduce electromagnetic interference and has good compatibility | Susceptible to numerical dispersion and performed with deficiency |

CT | Good computation efficiency | Stability limitation, numerical oscillation |

PSM | Requires fewer discrete nodes to obtain higher precision | Singularity at the boundary, non-periodic |

BVM | High accuracy and great stability | Heavy and deficient calculation |

**Table 2.**The calculation error of the Chebyshev pseudospectral–two-step two-order boundary value coupled method (PSM-BVM2, the Chebyshev pseudospectral–two-step three-order boundary value coupled method (PSM-BVM3), and the Chebyshev pseudospectral–two-step differential quadrature method (PSM-DQM2). $\left({N}^{\prime}=8,h=0.1\mathrm{ms}\right)$.

$\mathit{T}$ (s) | PSM-BVM2 $(\times {10}^{-8})$ | PSM-BVM3 $(\times {10}^{-11})$ | PSM-DQM2 $(\times {10}^{-8})$ | |||
---|---|---|---|---|---|---|

AE | RE | AE | RE | AE | RE | |

0.5 | 4.3670 | 8.4341 | 3.1342 | 6.2379 | 5.2916 | 9.0227 |

1 | 1.5147 | 8.7979 | 1.1578 | 6.4569 | 1.9340 | 9.6047 |

2 | 0.1238 | 8.8508 | 0.1058 | 6.5736 | 0.1605 | 9.6879 |

**Table 3.**The calculation error of PSM-BVM3, PSM-BVM2 and PSM-DQM2. $\left({N}^{\prime}=12,h=0.1\mathrm{ms}\right)$.

$\mathit{T}$ (s) | PSM-BVM2 $(\times {10}^{-8})$ | PSM-BVM3 $(\times {10}^{-11})$ | PSM-DQM2 $(\times {10}^{-8})$ | |||
---|---|---|---|---|---|---|

AE | RE | AE | RE | AE | RE | |

0.5 | 2.8129 | 4.1365 | 1.2515 | 2.9344 | 7.4130 | 6.2612 |

1 | 1.9138 | 5.3662 | 0.9821 | 3.5907 | 3.0820 | 7.6619 |

2 | 0.1017 | 5.7901 | 0.1048 | 3.6808 | 0.2664 | 7.8543 |

${\mathit{N}}^{\prime}$ | PSM-BVM2 | PSM-BVM3 | PSM-DQM2 |
---|---|---|---|

8 | 0.936 s | 1.172 s | 5.386 s |

12 | 1.013 s | 1.948 s | 5.742 s |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, Z.; Hu, T.; Tao, Y.; Zhang, T.; Li, Z.
Fast Calculation Method for Transient Response of Transmission Line Based on Chebyshev Pseudospectral–Two-Step Three-Order Boundary Value Coupled Method. *Symmetry* **2019**, *11*, 721.
https://doi.org/10.3390/sym11050721

**AMA Style**

Li Z, Hu T, Tao Y, Zhang T, Li Z.
Fast Calculation Method for Transient Response of Transmission Line Based on Chebyshev Pseudospectral–Two-Step Three-Order Boundary Value Coupled Method. *Symmetry*. 2019; 11(5):721.
https://doi.org/10.3390/sym11050721

**Chicago/Turabian Style**

Li, Zhenhua, Tinghe Hu, Yuan Tao, Tao Zhang, and Zhenxing Li.
2019. "Fast Calculation Method for Transient Response of Transmission Line Based on Chebyshev Pseudospectral–Two-Step Three-Order Boundary Value Coupled Method" *Symmetry* 11, no. 5: 721.
https://doi.org/10.3390/sym11050721