Abstract
Applying numerical routines based on trapezoidal rule of integration (Heun’s method for numerical integration), simple models of transmission lines are used to analyze and simulate the propagation of communication signals in PLC-type systems (power line communication systems). Such systems are shared by the same systems for the transfer of electrical power and signal transmission. For the mentioned routines, the main objectives are: simulate the propagation of electromagnetic transients in these systems and analyze the interference of such phenomena in the transmitted signal. Such simulations are performed with classical structures that represent infinitesimal units of transmission lines. Modifications in the structure of such units are analyzed to improve the results obtained by the mentioned simulations.
Keywords
- waveguide
- electromagnetic transient
- eigenvalues and eigenfunctions
- linear systems
- numerical analysis
- simulation
- state space methods
- numerical integration method
- transmission line modeling
1. Introduction
Systems of conductors for signal transmission or power transmission, that are, in general, classified as waveguide systems, systems of protection, coordination and control of the main system, the waveguide system, are as important as the conductors that are used. For projecting these accessory systems, analysis of short-circuit levels, overvoltages, as well as the duration of transient phenomena are very important [1, 2, 3, 4, 5, 6, 7, 8, 9]. In several situations, it is not possible to perform tests related to the occurrence of transient electromagnetic phenomena in actual transmission systems [10]. One situation for this is when the systems are in the design phase and have not yet been built or fabricated. Another situation is where the system cannot be shut down for maintenance or testing, for example, in the case of transmission lines responsible for interconnecting great power plants to great consumer centers. Because the theory and equations related to the propagation of electromagnetic fields in systems of conductors can be related to power and signal transmissions, different transmission systems are modeled as transmission lines or waveguides [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. For example, systems with low voltage, low power and very high frequency for signal transmission and systems with very high voltage, very high power and low frequency can be modeled as transmission lines or waveguides. In analyses of these types of electrical systems affected by electromagnetic transient phenomena, time-domain and frequency-dependent models are considered efficient and accurate for applications in this field [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Ways to improve these models have been researched yet, searching for increasing the accuracy of the results and the efficiency of the applied methods. For the analysis of the propagation of transient electromagnetic phenomena in electrical networks using transmission line theory, the waveguides can be decomposed into infinitesimal parts modeled by π circuits or T circuits [3, 4, 5, 6, 7, 8, 9, 10, 11, 17]. Simple numerical routines for this type of analysis can be good tools for undergraduate students to investigate and simulate these types of phenomena [11, 18, 19, 20] and to test improvements in the numerical model applied to the mentioned analyses. On the other hand, for more complex numerical or simpler numerical models, to a greater or lesser degree, respectively, numerical routines are influenced by numerical errors [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].
Considering the simplified representation of a transmission line by π circuit cascades, the solution of this system is obtained with the application of trapezoidal integration, and the results are affected by numerical oscillations or Gibbs’ oscillations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. It is possible to minimize the influence of numerical oscillations or Gibbs’ oscillations, in the obtained results, by means of structural modifications of these circuits [11, 17]. The proposed modification initially involves adding damping resistors (
2. Trapezoidal rule
The trapezoidal rule or Heun’s method is a numerical integration method based on the transformation of differential equations into their algebraic equivalents. The integral of a function is approximated by the first-degree function related to the original function (the area of a trapezoid) where the endpoints are approximated by points of intersection between the original and the first-degree functions. By improving approximation accuracy, a large range of independent variable values can be subdivided into equally small portions, called integration steps (Figure 1).
Applying the trapezoidal rule, the equation below is obtained:
The time step is
Using Eq. (1), Eq. (3) is obtained:
Considering a model of a physical system (or physical phenomenon), x is a vector composed by state variables of the mentioned system. Also, considering that the physical system is described by the first-order differential linear system, Eq. (4) is obtained:
In this case, the
From Eqs. (3)–(5), for very small time step, Eq. (6) is obtained:
Simplifying Eq. (6) and considering that
In this case,
Using Eq. (7), it is possible to determine the next state (
3. Transmission line equivalent circuit model
Analyzing the propagation of waves in transmission lines or waveguides, these systems can be decomposed into infinitesimal portions that can be represented by π circuits. For representing the whole system, there is a need to use a cascade with a large number of π circuits. The generic unit of π circuits is shown in Figure 2.
Based on the structure of π circuit and Kirchhoff’s laws, the relations of voltages and currents for this generic unit are determined by Eq. (9):
Each π circuit has two state variables: the transversal voltage (
In this case, the structure of the
Considering only one voltage source in the initial of the transmission line or the waveguide, the
If damping resistances are included in π circuits, this is shown in Figure 3. The relations of voltages and currents for the generic unit of π circuits are in Eq. (13):
Based on Figure 3 and Eq. (12), the structure of the
Also, based on Figure 3 and Eq. (12), the structure of the
The damping resistance is determined by
The
4. Numerical computation
In Figure 4, the flowchart applied to obtain the results of electromagnetic transient phenomena simulations without the introduction of damping resistances is shown.
In case of Figure 4, the flowchart is based on Eq. (6) to Eq. (12). The simulations are carried out considering that the analyzed waveguide or the transmission line is connected to an independent step voltage source of 1 pu. The end line is opened, and, because of this, the value of the propagated voltage wave is doubled compared to the voltage value at the initial line. Using the flowchart of Figure 4, the parameter values applied to the obtained results are
In Figure 5, the flowchart related to the inclusion of damping resistances in the π circuits for representing the analyzed waveguide or the transmission line is shown. The values of
5. Effects of the simulation accuracy without damping resistance
Applying the flowchart of Figure 4, the main obtained results are shown in Figure 6. For these results, the time step (
The numerical routine described by the flowchart is simple. Despite this characteristic, the numerical simulations lead to results with errors of 25% independently that the number of π circuits is applied. The increase of the number of π circuits is not related to a correspondent decrease of the numerical errors and numerical oscillations in the obtained results. A proposed alternative is the introduction of damping resistances for decreasing numerical errors and Gibbs’ oscillations in the obtained results. Next, both items show the results obtained with this alternative.
6. Improvement of the simulation accuracy with damping resistance
Applying damping resistances and using the kD factor as 2.5, the obtained results are shown in Figure 7. Comparing Figure 7 to Figure 6, if the number of π circuits is increased, numerical oscillations or Gibbs’ oscillations are decreased. So, considering a constant value for the
7. Effects of kD factor variation
Applying damping resistances, from Figures 9–15, the number of π circuits is changed from 50 to 500 for different values of the kD factor. In Figure 9, the results are related to
Increasing the value of the
Analyzing the results from Figures 9–15, the highest voltage peak for each value of the
8. Number of π circuit variation
Setting the number of π circuits, the results are analyzed considering the
Considering 150 units of π circuits, the range of the
Still analyzing Figures 21–23, it is observed that the relation between the number of the π circuits and the influence of the
9. Other analyses
Based on the results shown in the previous sections, it is concluded that the numerical oscillations and, consequently, the voltage peaks obtained by the proposed model are influenced jointly by two factors: the damping resistance value and the number of π circuits applied to the numerical simulations of electromagnetic phenomena in waveguides or transmission lines. Because of this, the analyses of this joint influence must be based on three-dimensional graphics. In Figure 24, the highest voltage peaks during the first wave reflection on the transmission end line or the receiving end terminal of the waveguide are shown. These peaks depend on the kD factor and the number of π circuits considering the time step as 50 ns. In Figures 25 and 26, the time steps are 10 ns and 200 ns, respectively. These graphics are used for completing the analyses carried out in the previous sections.
Based on the last three sets of obtained results of this chapter (Figures 24–26), for a specific time step, there are sets of the number of π circuits and the
10. Conclusions
Modifications on the classical structure of π circuits for modeling transmission lines are presented. These modified π circuits are applied to obtain a cascade that represents the analyzed transmission lines. Based on the electromagnetic basic concepts, very long circuits for power transmission and circuits for data transmission can be analyzed using the theoretical bases of the transmission lines. So, a numerical routine for simulating electromagnetic transient phenomena in waveguides (transmission lines for power systems or data transmission) is obtained.
In the proposed numerical routine, damping resistances for minimizing Gibbs’ oscillations or numerical oscillations are included. These oscillations are caused by the numerical integration method applied to the solution of the linear system that describes the waveguide. Applying this proposed numerical routine, several results of simulations varying the number of π circuits, the
Based on the obtained results, it is observed that there are ranges of the model parameters adequate for the minimization of numerical oscillations that influence these results. The main model parameters that influence the minimization of numerical oscillations are the number of π circuits and the
Acknowledgments
The authors would like to thank the financial support by FAPESP (The São Paulo Research Foundation). The following processes are related to the results shown in this chapter: 2015/21390-7, 2015/20590-2, 2015/20684-7, 2016/02559-3, 2017/05988-5, 2017/05995-1, and 2017/23430-1.
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