# Dynamical Networks Modelling Applied to Low Voltage Lines with Nonlinear Filters

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation and Contribution

#### 1.2. Literature Review

#### 1.3. Paper Structure

## 2. Electromagnetic Interference on Low Voltage Lines—Network Model

#### 2.1. A short on Electromagnetic Interference

- 1)
- Eliminate EMI next to the source. If the source can be accessed, suppressing EMI could be done by, for instance, moving it away, replacement, shielding, using output filters and other suppressor devices, or by altering its internal design of circuits and components. So far, this is the best to do and the source will not be a cause of interference for the whole environment. However, in general, the source is not under control and cannot be accessed.
- 2)
- Eliminate EMI in the path. The second possibility is doing adjustments on the transmitting path. Electromagnetic waves can be transmitted through conductive means, such as cables or “earth”, (conducted EMI) or can be radiated through the environment. The alternatives to prevent the interference are the removal of the recipient equipment, or shielding it against radiated interference, and applying various types of suppressors and filters for undesired conducted disturbances.
- 3)
- Eliminate EMI in the recipient. If it is neither possible to deal with the source nor the path, the last alternative is the elimination of EMI within the recipient. This could be a difficult task and will involve the study of the environment, the existing noise, possible transients, and surges, which will lead to a set of solutions such as physical system layout, distributed filters, suppressors and shielding.

#### 2.2. Robustness of a Simple Communication Network System

**structural**and

**functional**. The structure depends on the features related to the system physical structure, and the functional factor has to be with the processes running on that structure. Intrinsically interrelated, the two factors could define degrees of the system robustness. Defining a degree of system robustness is a subject that escapes from the purpose of this paper and will not be addressed in this work. Instead, a model to dynamically deal with the system is developed later, showing ways to enhance the low voltage line robustness and sustainability.

#### 2.3. Low Voltage Lines—Network Model

_{n}and receivers Rn (Figure 6, bottom). This approach is called the transmission line method (TLM) and is well established [8,9,10,11,12], standing as a finite difference method in the time domain. The finite difference methods are based on space discretization (grids) that allows computational solutions based on matrix algebra techniques [13].

_{n}is seen as a transmission line itself connected to the previous (N

_{n}

_{−1}) and the next transmission line (N

_{n}

_{+1}), from the source to the receiver. The signal moves from node to node in time-steps—that is, from one transmission line segment to the following transmission line segment. The basics of the whole model, in one, two and three-dimensional networks, were described in [9,11], whereas a different three-dimensional cell was presented by [12].

_{n}comprises these parameters around the point x, which is modelled as an impedance Z

_{0}with losses represented by R and G (respectively resistance and conductance for the segment with length Δx (Figure 7 bottom). This representation is called the transmission line method model, where the node N

_{n}is a transmission line itself, with the incident and reflected voltages for each time-step. A complete description of the one-dimensional method and its equations were given in [8,12]. Below, we summarize its most important concepts and equations, which will be useful to model the system’s dynamical behaviour and the nonlinear filter in the sections following.

_{0}is given by

_{d}and C

_{d}are the distributed line inductance and capacitance, respectively. The wave speed propagation v, for this line, which is also the speed propagation in each line segment, is

_{n}and a receiver R

_{n}(we call it node N

_{n}). The Thévenin equivalent for this node N

_{n}shows voltage sources VE

_{k,n}, and VD

_{k,n}and line parameters G, R and Z

_{0}, where k is the iteration and n is the node (Figure 8, right). Losses caused by line resistance and conductance are considered and calculated within the node. This is an important feature of this method when we consider the node as an element of a dynamical network. It means that the network behaviour, as signals travel between nodes through existent connections, will be modified by the specific characteristics of each node, and not only by the general structure of the network (in other words, it is not a view of a graph in its static mode).

_{n}, detailed in [8], are:

^{i}) and reflected (V

^{r}) voltages at iteration k for the node n can be obtained by adding their contributions, which determines its total voltage on the left (VE

_{k,n}) and right (VD

_{k,n}), as seen in the following equations:

_{n}will be the incident voltages to the adjacent nodes N

_{n}

_{−1}and N

_{n}

_{+1}on the next time step k + 1, as follows:

_{k,n}(Equation (7)). The source S is considered in the first node, whilst the receiver R in the last one. The detailed explanation and development of equations can be found in [9] and [12].

## 3. Linear and Nonlinear Filters

#### 3.1. Overview

#### 3.2. Metal-Oxide Varistor

#### 3.3. The Dynamic Model of A ZnO Varistor Filter

## 4. Results, Discussion and Perspectives

#### 4.1. Results of Practical Experiments and Simulations

#### 4.2. Discussions, Perspectives and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Results for the ZnO varistor model S05K11 [24], which is composed with a metallic disk of 5 mm, nominal voltage 11 V and capacitance 1.6 nF. The left side pictures show squared pulses achieving the load (blue line, 50–150 V) and the resultant voltages after the nonlinear filter clamping. The right side shows atmospheric surges and correspondent clampings. Note the consistency of the clamping values for both surge types and how it smoothly decreases as the surge goes down for the atmospheric example.

## Appendix B

**Figure A2.**Results for the ZnO varistor model S10K11 [24], which is composed by a metallic disk of 10 mm, nominal voltage 11 V and capacitance 6.8 nF. The left side pictures show squared surge pulses and the right side shows atmospheric surges and correspondent clampings. Note the capacitance influence on the surge steep rise and fall, which is quite interesting to prevent damages on subsequent loads. This additional filtering effect is obtained by the higher capacitance, which is a very simple constructive improvement compared to the previous S05K11. As the diameter is bigger, also its energy absorption is better.

## Appendix C

**Figure A3.**Results for the ZnO varistor model S10K14 [24], which is composed by a metallic disk of 10 mm, nominal voltage 14 V and capacitance 5.2 nF. The left side pictures show squared surge pulses and the right side shows atmospheric surges and correspondent clampings. Note here again the capacitance influence on the surge steep rise and fall. However, its nominal voltage is slightly higher than S10K11 that results on a higher voltage clamping.

## References

- Paul, C.R. Introduction to Electromagnetic Compatibility; John Wiley & Sons: New York, NY, USA, 2006. [Google Scholar]
- Goedbloed, J. Electromagnetic Compatibility; Prentice Hall: New York, NY, USA, 1992. [Google Scholar]
- Christopoulos, C. Principles and Techniques of Electromagnetic Compatibility; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Violette, N. Electromagnetic Compatibility Handbook; Springer: New York, NY, USA, 2013. [Google Scholar]
- Perez, R. Handbook of Electromagnetic Compatibility; Academic Press: San Diego, CA, USA, 2013. [Google Scholar]
- Kaiser, K.L. Electromagnetic Compatibility Handbook; CRC Press: Boca Raton, FL, USA, 2004. [Google Scholar]
- Ott, H.W. Electromagnetic Compatibility Engineering; John Wiley & Sons: New York, NY, USA, 2011. [Google Scholar]
- Fazion Filho, M. O Método da Modelagem por Linhas de Transmissão (TLM) e Aplicações em Compatibilidade Eletromagnética (EMC). Master’s Thesis, Federal University of Santa Catarina, Florianopolis, Brazil, 1997. [Google Scholar]
- Christopoulos, C. The Transmission-Line Modeling Method TLM; IEEE Press: New York, NY, USA, 1995. [Google Scholar]
- Johns, P.B.; O’Brien, M. Use of the transmission-line modelling (tlm) method do solve nonlinear lumped networks. Radio Electron. Eng.
**1980**, 50, 59–70. [Google Scholar] [CrossRef] - Hoefer, W.J.R. The Transmission Line Matrix (TLM) Method. In Numerical Techniques for Microwave and Millimeter-Wave Passive Structures; Itoh, T., Ed.; Wiley: New York, NY, USA, 1989. [Google Scholar]
- Fazion Filho, M. Estudos de Modelagem Eletromagnética com o Método das Linhas de Transmissão (TLM). Ph.D. Thesis, Federal University of Santa Catarina, Florianopolis, Brazil, 2001. [Google Scholar]
- Grossmann, C.; Roos, H.; Stynes, M. Numerical Treatment of Partial Differential Equations; Springer: Cham, Switzerland, 2005; ISBN 978-3-540-71584-9. [Google Scholar]
- Johns, P.B.; Beurle, R.L. Numerical solution of 2-dimensional scattering problems using a transmission-line matrix. Proc. IEEE
**1971**, 118, 1203–1208. [Google Scholar] [CrossRef] - Akhtarzad, S.; Johns, P.B. Solution of Maxwell’s equations in three space dimensions and time by the T.L.M method of numerical analysis. Proc. IEEE
**1975**, 122, 1344–1348. [Google Scholar] [CrossRef] - Akhtarzad, S.; Johns, P.B. Generalized elements for T.L.M method of numerical analysis. Proc. IEEE
**1975**, 122, 1349–1352. [Google Scholar] - Johns, P.B. A Symmetrical Condensed Node for the TLM Method. IEEE Trans. Microw. Theory Tech.
**1987**, 35, 370–377. [Google Scholar] [CrossRef] - Herring, J.L. Developments in the Transmission-Line Modelling Method for Electromagnetic Compatibility Studies. Ph.D. Thesis, University of Nottingham, Nottingham, UK, May 1993. [Google Scholar]
- Trenkic, V. The Development and Characterization of Advanced Nodes for the TLM Method. Ph.D. Thesis, University of Nottingham, Nottingham, England, November 1995. [Google Scholar]
- Righi, M. Diakoptic Solution Techniques for Electromagnetic Fields in Time and Frequency Domain. Ph.D. Thesis, University of Victoria, Victoria, BC, Canada, 1995. [Google Scholar]
- Menezes, L.R.A.X. Modeling of General Medium Constitutive Relationships in the Transmission Line Matrix Method (TLM). Ph.D. Thesis, University of Victoria, Victoria, BC, Canada, 1996. [Google Scholar]
- Johns, D.P. Transmission line matrix method (TLM). In Proceedings of the 2017 IEEE International Symposium on Electromagnetic Compatibility & Signal/Power Integrity (EMCSI), Washington, DC, USA, 7–11 August 2017; pp. 1–34. [Google Scholar]
- Mir, H.; Arya, V.; Höffler, H.; Brand, A. A Novel TLM Analysis for Solar Cells. IEEE J. Photovolt.
**2019**, 9, 1336–1342. [Google Scholar] [CrossRef] - SIOV Metal Oxide Varistors; EPCOS AG, TDK Group Company: Munich, Germany. 2018. Available online: https://www.tdk-electronics.tdk.com/download/531268/8954d4a78154a9da5c70a7119fa03e86/siov-general.pdf (accessed on 11 December 2019).
- SIOV Metalloxid-Varistoren, Siemens Matsushita Components; Siemens Matsushita Components GmbH: Dusseldorf, Germany, 1997; pp. 15–50.
- Fazion Filho, M. varistorGeneralModel.py. 2019. Available online: https://github.com/mfaccioni/varistorGeneralModel/blob/master/varistorGeneralModel.py (accessed on 11 December 2019).
- Bley, M., Jr.; Fazion Filho, M.; Raizer, A. Modeling transient discharge suppressors. IEEE Potentials
**2004**, 23, 43–45. [Google Scholar] [CrossRef] - Fazion Filho, M.; Ferreira, G.S.; Raizer, A. Nonlinear Device Modeling for Transient Suppression. In Proceedings of the Eighth Biennial IEEE Conference on Electromagnetic Field Computation, Tucson, AZ, USA, 1–3 June 1998. [Google Scholar]
- Fazion Filho, M.; Ferreira, G.S.; Raizer, A. Simulation of Nonlinear Protection Device Using TLM Method. In Proceedings of the EMC’98 International Symposium on Electromagnetic Compatibility Proceedings, Rome, Italy, 14–18 September 1998; pp. 403–407. [Google Scholar]
- Fazion Filho, M.; Ferreira, G.S.; Raizer, A. Modelagem Numérica de Filtros Não-Lineares para Transientes Eletromagnéticos. In Proceedings of the III Congresso Brasileiro de Eletromagnetismo–CBMag’98, São Paulo, Brazil, 28–25 October 1998; pp. 18–21. [Google Scholar]
- Fazion Filho, M.; Bley, M., Jr.; Raizer, A. Modelling nonlinear suppressors using tlm. In Proceedings of the 5th International Workshop on Computational Electromagnetics in Time Domain-TLM, FDTD and Other Techniques (CEM-2003), Halifax, NS, Canada, 17–19 June 2003. [Google Scholar]
- Tektronix Oscilloscope. Available online: https://www.tek.com/oscilloscope (accessed on 11 December 2019).
- Ladyman, J.; Lambert, J.; Wiesner, K. What is a complex system? Eur. J. Philos. Sci.
**2013**, 3, 33–67. [Google Scholar] [CrossRef] - Holland, J.H. Emergence: From Chaos to Order; Perseus Books: Cambridge, MA, USA, 1998; ISBN 0-7382-0142-1. [Google Scholar]
- Costa, L.F.; Rodrigues, F.A.; Travieso, G.; Villas Boas, P.R. Characterization of complex networks: A survey of measurements. Adv. Phys.
**2007**, 56, 167–242. [Google Scholar] [CrossRef] [Green Version] - Fortunato, S. Community detection in graphs. Phys. Rep.
**2010**, 486, 75–174. [Google Scholar] [CrossRef] [Green Version] - Wasserman, S.; Faust, K. Social Network Analysis: Methods and Applications; Cambridge University Press: Cambridge, UK, 1994. [Google Scholar]

**Figure 2.**A simple one-dimensional model where a squared signal is sent through a line from a source S to a receiver R (

**top**). The same line (

**bottom**) considered as a set of “n” nodes N interconnected by line-segments L, where the squared signal travels from node to node in time-steps.

**Figure 3.**An interference originated by an external source strikes the line at a random node, propagating through the line. It may happen at the time when the squared signal is passing, distorting the signal that reaches the receiver R (

**top**) or maybe dislocated in time, changing the original signal sequence at R (

**middle**). If a noise hits the node (

**bottom**), the squared signal could be completely distorted.

**Figure 4.**Other examples of interference, such as a wave caused by lightning, adding a signal that changes a large sequence of bits (

**top**), or multiplying the voltage of the original square by an undefined factor (

**bottom**). External interferences, their magnitude, and shapes can be unexpected and out of control.

**Figure 5.**Line structural variations could modify the signal sent by S in such a way that R could misunderstand it (

**top**) or even do not receive it (

**bottom**). These inherent characteristics affect the communication of internal factors. A metallic line has a different and increasing capacitance from node N5 and beyond, significantly distorting the signal (top example), or a failure that intermittently interrupts the line-segment between N5 and N6 (bottom example).

**Figure 6.**Metallic transmission line connecting source S to receiver R, considering n nodes N (

**top**). The transmission line segments have distributed inductances and capacitances, their specific structural characteristics (

**middle**). Each line segment can be seen as a transmission line itself, considering also distributed sources S

_{n}and receivers R

_{n}(

**bottom**). This line segment is called a “node” of the network.

**Figure 7.**Segments of a transmission line (x, x + Δx) with distributed parameters L, C, G and R (top). The same segments represented as transmission line method models, where the node “n” is a transmission line itself, with incident V

^{i}and reflected V

^{r}voltages for each time-step (bottom).

**Figure 8.**Each line segment (the node N

_{n}) is a transmission line with a source S

_{n}and a receiver R

_{n}(

**left**). Its electrical model - Thévenin equivalent (

**right**) consists of voltage sources VE

_{k,n}and VD

_{k,n}and line parameters G, R and Z

_{0}, where k is the iteration.

**Figure 9.**A nonlinear filter applied to one or more points of the metallic line (

**top**) and its representation as applied to the respective network node (

**bottom**). The filter will be temporarily part of the structure according to unexpected functional feature variances (for example, alien attacks).

**Figure 10.**Typical voltage/current (V/I) symmetrical curve of metal-oxide varistors (not dependent on the polarity of the voltage applied, linear scale).

**Figure 11.**Typical V/I curve of a commercial metal-oxide varistor [9].

**Figure 12.**Section of a varistor showing the ZnO grains and the composition of “microvaristors” encapsulated between metallic discs.

**Figure 14.**One, two and three-dimensional grids, where an excited node (input) sends signals to its neighbours, starting an iterative time-step incident/reflection process. These examples show static network structures, with dynamical processes running on top of them.

**Figure 15.**The system as a static network structure with dynamical processes running on top of it. The dynamical approach considers reflected signals (time-step t) and incident signals (time-step t + 1), dependent on the connections and node functions f

_{n}.

Line Parameters (Distributed) | ||
---|---|---|

R (Ω/m) | C (F/m) | L (H/m) |

1.0 × 10^{−5} | 1.0 × 10^{−10} | 2.5 × 10^{−7} |

Varistor Model [tdk] | Metal Oxide | Disk Diammeter (mm) | C (nF) | Nominal Voltage (V) | Assumed Values According to Technical Specifications | |||||
---|---|---|---|---|---|---|---|---|---|---|

V_{a} (V)
| I_{a} (A) | V_{b} (V)
| I_{b} (A)
| α | β | |||||

S05K11 | ZnO | 5 | 1.6 | 11 | 23 | 0.01 | 34 | 1 | 11.78 | 9.04 |

S10K11 | ZnO | 10 | 6.8 | 11 | 23 | 0.01 | 30 | 0.5 | 14.72 | 8.93 × 10^{−23} |

S10K14 | ZnO | 10 | 5.2 | 14 | 28 | 0.01 | 36 | 0.5 | 15.56 | 2.97 × 10^{−25} |

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Fazion Filho, M.
Dynamical Networks Modelling Applied to Low Voltage Lines with Nonlinear Filters. *Appl. Syst. Innov.* **2020**, *3*, 18.
https://doi.org/10.3390/asi3020018

**AMA Style**

Fazion Filho M.
Dynamical Networks Modelling Applied to Low Voltage Lines with Nonlinear Filters. *Applied System Innovation*. 2020; 3(2):18.
https://doi.org/10.3390/asi3020018

**Chicago/Turabian Style**

Fazion Filho, Mauro.
2020. "Dynamical Networks Modelling Applied to Low Voltage Lines with Nonlinear Filters" *Applied System Innovation* 3, no. 2: 18.
https://doi.org/10.3390/asi3020018