**3. Curve dispersion**

The first step in the study of an electrical transmission line is to determine its ability to convey electric waves. Indeed, the electric nonlinear transmission line like all wave guides presents a different response depending on the type of the wave introduced.

#### **3.1. Wavelength determination**

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. 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.

162 Advanced Electronic Circuits - Principles, Architectures and Applications on Emerging Technologies

**2. Overview of the experimental device**

**Photo 1:** Experimental device.

**3. Curve dispersion**

**Computer**: for control, acquisition, and data processing;

**GPS-3030d**: generator of analog tension to polarize the line;

**Voltmeter**: for the control of the tension of polarization of the line; **Amperemeter**: for the control of the current, which crosses the line.

different response depending on the type of the wave introduced.

**HS3–100**: which accommodates the arbitrary signal generator and digital oscilloscope;

The first step in the study of an electrical transmission line is to determine its ability to convey electric waves. Indeed, the electric nonlinear transmission line like all wave guides presents a

**NLTL**: electrical nonlinear transmission line;

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 wavelength expressed in terms of cell number.

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

$$k = \frac{2\pi}{\lambda} \tag{1}$$

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

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 more precise by calculating the phase velocity of the wave.

#### **3.2. Determination of the phase velocity**

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

**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.

$$k = \frac{\omega}{\upsilon\_{\rho}} = \frac{2\pi f}{\upsilon\_{\rho}} \tag{2}$$

*3.3.2. High frequency mode*

*3.3.3. Forbidden band*

line in various proportions.

ing into the line of very low amplitude waves.

**4.1. Effect of dissipation**

ally halved.

**4.2. Effect of the nonlinearity**

and the third cut-off frequency *f*

1 and *f* 2

area, other physical laws [1] govern the wave propagation.

**4. Nonlinearity and dissipation effects on the signal**

High frequency (HF) mode is the branch of the curve of dispersion which lies between the second

that has managed to determine the curve dispersion in HF mode with much precision than that presented in the present work. This is due to the fact that, on the one hand, in HF mode signals are weak compared to those of the bi-frequency (BF) mode, and, on the other hand, the quality of the experimental device impacts the measures as we have used a more accurate methodology.

Dissipation and dispersion phenomena affect the wave propagation in nonlinear electrical

Because of the presence of dissipative element in the line, signals introduced at the entrance of the line undergo a weakening that increases with the distance traveled in the line. Impairment affects the wave in a uniform manner, it is a linear phenomenon that leads to a global change in the amplitude of the wave; however, the overall shape of the wave remains intact. In order to observe the effects of wave dissipation in the nonlinear electrical line, we have to diminish the effects of the nonlinearity. To do so, we are in an almost linear approximation by introduc-

We see, in **Figure 3**, the sine wave introduced at the entrance of the line keeps its intact shape to the 144th cell; however, we note a weakening of the signal, which sees its amplitude virtu-

Note, finally, that the weakening of the signal affects all signals introduced in the line. It is important to note that the HF mode signals are more sensitive to the effects of dissipation; also explained by the fact that high-frequency inductors have impedances higher than in BF mode. This often makes the phenomena more difficult to observe in the HF mode than in BF mode.

To observe the effects of nonlinearity in the nonlinear transmission electric line, the amplitude of the signal introduced into the line is increased. Indeed, by increasing the amplitude of the signal,

the characteristic C(V) (**Figure 4**) point. The various points of the signal, not meeting the same value of the capacity, then move at different velocity, thus leading to a dispersion of the signal.

leading their operating point to move on a significant range of value around the Q0

the voltage varies significantly around the tension of polarization V0

The forbidden band is a range of frequency between the cut-off frequencies of *f*

, respectively. It is noted that practically there is no science

Experimental Studies of the Electrical Nonlinear Bimodal Transmission Line

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

165

2 and *f* 3 . In this

of the diodes varactor thus

resting on

where *f* is the frequency of the incident wave and *v<sup>ϕ</sup>* the phase velocity.

We determine then the wave number *k*, for all frequencies available by the system, which allow us to determine the curve dispersion of the line.

#### **3.3. Frequency modes**

We present, in **Figure 2**, the curve of dispersion obtained by the theoretical calculation [1] (solid lines) and the experimental points determined by the above-described method (filled triangles). We note a good agreement between our experimental results and the theoretical results. The frequencies of experimentally determined cuts *f <sup>C</sup>*<sup>1</sup> <sup>=</sup> (590 <sup>±</sup> <sup>15</sup>) KHz, *<sup>f</sup> <sup>C</sup>*<sup>2</sup> <sup>=</sup> (830 <sup>±</sup> 20) KHz, and *f <sup>C</sup>*<sup>3</sup> <sup>=</sup> (975 <sup>±</sup> <sup>25</sup>) KHz are in good agreement with the theoretical values *<sup>f</sup> <sup>C</sup>*<sup>1</sup> <sup>=</sup> <sup>550</sup> KHz, *f <sup>C</sup>*<sup>2</sup> <sup>=</sup> <sup>807</sup> KHz, and *<sup>f</sup> <sup>C</sup>*<sup>3</sup> <sup>=</sup> 970 KHz, respectively. These cut-off frequencies delimit several areas of the dispersion curve, which correspond to modes of propagation of individual waves in the transmission line.

#### *3.3.1. Low frequency mode*

Low frequency (LF) mode is the branch of the curve dispersion that lies below the first cutoff frequency.

**Figure 2.** Comparison between the curve dispersion experimental triangles and that obtained by calculation (continuous line) for the bimodal transmission line (*L*<sup>1</sup> = 220 *F*, *L*<sup>2</sup> = 470 *F* with polarization voltage of line V0 = 1.5 V).

#### *3.3.2. High frequency mode*

*k* = \_\_*<sup>ω</sup>*

**3.3. Frequency modes**

KHz, and *f*

frequency.

*<sup>C</sup>*<sup>2</sup> <sup>=</sup> <sup>807</sup> KHz, and *<sup>f</sup>*

transmission line.

*3.3.1. Low frequency mode*

*f*

where *f* is the frequency of the incident wave and *v<sup>ϕ</sup>*

allow us to determine the curve dispersion of the line.

results. The frequencies of experimentally determined cuts *f*

*vϕ*

164 Advanced Electronic Circuits - Principles, Architectures and Applications on Emerging Technologies

We determine then the wave number *k*, for all frequencies available by the system, which

We present, in **Figure 2**, the curve of dispersion obtained by the theoretical calculation [1] (solid lines) and the experimental points determined by the above-described method (filled triangles). We note a good agreement between our experimental results and the theoretical

*<sup>C</sup>*<sup>3</sup> <sup>=</sup> (975 <sup>±</sup> <sup>25</sup>) KHz are in good agreement with the theoretical values *<sup>f</sup>*

the dispersion curve, which correspond to modes of propagation of individual waves in the

Low frequency (LF) mode is the branch of the curve dispersion that lies below the first cutoff

**Figure 2.** Comparison between the curve dispersion experimental triangles and that obtained by calculation (continuous

line) for the bimodal transmission line (*L*<sup>1</sup> = 220 *F*, *L*<sup>2</sup> = 470 *F* with polarization voltage of line V0 = 1.5 V).

*<sup>C</sup>*<sup>3</sup> <sup>=</sup> 970 KHz, respectively. These cut-off frequencies delimit several areas of

<sup>=</sup> <sup>2</sup>*<sup>f</sup>* \_\_\_ *vϕ*

the phase velocity.

*<sup>C</sup>*<sup>1</sup> <sup>=</sup> (590 <sup>±</sup> <sup>15</sup>) KHz, *<sup>f</sup>*

(2)

*<sup>C</sup>*<sup>2</sup> <sup>=</sup> (830 <sup>±</sup> 20)

*<sup>C</sup>*<sup>1</sup> <sup>=</sup> <sup>550</sup> KHz,

High frequency (HF) mode is the branch of the curve of dispersion which lies between the second and the third cut-off frequency *f* 1 and *f* 2 , respectively. It is noted that practically there is no science that has managed to determine the curve dispersion in HF mode with much precision than that presented in the present work. This is due to the fact that, on the one hand, in HF mode signals are weak compared to those of the bi-frequency (BF) mode, and, on the other hand, the quality of the experimental device impacts the measures as we have used a more accurate methodology.

#### *3.3.3. Forbidden band*

The forbidden band is a range of frequency between the cut-off frequencies of *f* 2 and *f* 3 . In this area, other physical laws [1] govern the wave propagation.
