**Circuit Test and Analysis**

[18] Cheng S, Jin Y, Rao Y, Arnold DP. An active voltage doubling AC/DC converter for lowvoltage energy harvesting applications. IEEE Transactions on Power Electronics. Aug.

[19] Cheng S, Sathe R, Natarajan RD, Arnold DP. A voltage-multiplying self-powered AC/DC converter with 0.35-V minimum input voltage for energy harvesting applications. IEEE

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2011;26(8):2258-2265

**Chapter 8**

**Provisional chapter**

**Experimental Studies of the Electrical Nonlinear**

**Experimental Studies of the Electrical Nonlinear** 

DOI: 10.5772/intechopen.76204

After a few years of calm, the investigations on the dynamic, especially nonlinear, systems returned to the front of the research in non-linear physics. We propose, in this chapter, a study of an electrical nonlinear transmission line, realized in a previous work, to use the latter to highlight certain properties (modulation instability—MI, Fermi-Pasta-Ulam (FPU) recurrence, fragmentation of solitons in wave trains, multiplication(increase) and division of frequencies, etc.), which are observed in several domains in applied physics: hydraulic, artificial neuronal, network physical appearance (physics) of the plasma, and

**Keywords:** nonlinear transmission line, trains of solitons, modulation instability, FPU

Nowadays, the study of electrical nonlinear transmission lines (NLTLs) progresses in both the theoretical field [1, 2] and technology [3–5]. The tools for the simulation of mechanical systems and the study of electrical transmission lines became a major challenge because many electronic systems have nonlinear transmission line (NLTL) modules. The experimental results presented in this chapter are part of an effort to understand the phenomena that occurs in the NLTL. We designed and built the experimental device to perform many investigations on the fundamentals that allow the understanding of the nonlinear effects. Other researchers

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

**Bimodal Transmission Line**

**Bimodal Transmission Line**

http://dx.doi.org/10.5772/intechopen.76204

**Abstract**

the circulation.

**1. Introduction**

recurrence, dispersion curve

Abdou Karim Farota, Mouhamadou Mansour Faye, Bouya Diop, Diène Ndiaye and Mary Teuw Niane

Abdou Karim Farota, Mouhamadou Mansour Faye, Bouya Diop, Diène Ndiaye and Mary Teuw Niane

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

#### **Experimental Studies of the Electrical Nonlinear Bimodal Transmission Line Experimental Studies of the Electrical Nonlinear Bimodal Transmission Line**

DOI: 10.5772/intechopen.76204

Abdou Karim Farota, Mouhamadou Mansour Faye, Bouya Diop, Diène Ndiaye and Mary Teuw Niane Abdou Karim Farota, Mouhamadou Mansour Faye, Bouya Diop, Diène Ndiaye and Mary Teuw Niane

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.76204

#### **Abstract**

After a few years of calm, the investigations on the dynamic, especially nonlinear, systems returned to the front of the research in non-linear physics. We propose, in this chapter, a study of an electrical nonlinear transmission line, realized in a previous work, to use the latter to highlight certain properties (modulation instability—MI, Fermi-Pasta-Ulam (FPU) recurrence, fragmentation of solitons in wave trains, multiplication(increase) and division of frequencies, etc.), which are observed in several domains in applied physics: hydraulic, artificial neuronal, network physical appearance (physics) of the plasma, and the circulation.

**Keywords:** nonlinear transmission line, trains of solitons, modulation instability, FPU recurrence, dispersion curve

#### **1. Introduction**

Nowadays, the study of electrical nonlinear transmission lines (NLTLs) progresses in both the theoretical field [1, 2] and technology [3–5]. The tools for the simulation of mechanical systems and the study of electrical transmission lines became a major challenge because many electronic systems have nonlinear transmission line (NLTL) modules. The experimental results presented in this chapter are part of an effort to understand the phenomena that occurs in the NLTL. We designed and built the experimental device to perform many investigations on the fundamentals that allow the understanding of the nonlinear effects. Other researchers

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

focused more on specialized aspects that allowed us to revisit nonlinear effects in a completely new light of research that have marked the history of nonlinear physics in particular.

**3.1. Wavelength determination**

length expressed in terms of cell number.

**3.2. Determination of the phase velocity**

*k* = \_\_\_ <sup>2</sup>*<sup>π</sup>*

where *k* is expressed in rad/cel and *λ* is a wavelength.

more precise by calculating the phase velocity of the wave.

determine then the number of waves by the relationship (Eq. (2)):

To determine the wavelength of a signal, we introduce one low amplitude sine wave in the line input in order to stay in the linear approximation (50 mV), then we put a first probe at the entrance of a cell of order *n*, then a second probe to a cell located at the position *n* + 1, *n* + 2 until the signals observed from the two probes are in phase. Thus, we determine the wave-

However, this method is quite unclear, and it is rare to see a wavelength that is always equal to an integer multiple of the number of cells. However, this step has the advantage to confirm that the wavelength is greater than the cell, which allows considering the use of a method

Staying in the linear approximation, we introduce the input of line at low amplitude (50 mV) sine wave and visualize the signals collected by two sensors located on two consecutive cells. We then determine the phase of the wave velocity by choosing a point of the wave, which has the same phase (e.g., maximum). This phase velocity is expressed in cell/s (**Figure 1**). We

**Figure 1.** The phase velocity of the wave measured by taking the signals at the entrance of two inductors of the same values that are consecutive. Then we determine the difference of time between these two points, making sure that the

two points of the wave are the same phase; cells 10–12, *f* = 330 kHz, Δ*t* = 810.667 ns, *v<sup>ϕ</sup>* = 2.46 106 cells/s.

*<sup>λ</sup>* (1)

Experimental Studies of the Electrical Nonlinear Bimodal Transmission Line

http://dx.doi.org/10.5772/intechopen.76204

163

We have at the determined level the wave number *k* given in the relationship (Eq. (1)):

In Section 2 of this chapter, we present the experimental device realized in a preceding work. In Section 3, we propose an experimental method of determination of wavelength and the velocity phase and velocity group that allowed us to trace point by point the curve dispersion of the line. The effects of fading and nonlinearity are highlighted in Section 4. The phenomenon of modulation instability (MI) is the object of Section 5. In Section 6, we discuss the follow-ups in the periodic recurrence of Fermi-Pasta-Ulam (FPU) in low and high frequencies. Section 7 is dedicated to some applications to use our experimental platform.
