**7. Observation of the solitons in the line**

#### **7.1. Soliton trains**

It is established for a long time that when a wave is subjected to a modulation instability, it eventually splits up in a wave train depending solitons type [12–15]. To observe this behavior in our bi-inductance line, we introduced at the entrance of the line a wave whose frequency is located in the region of modulation instability of BF and the mode, which presents an amplitude sufficiently high to cause a strong disturbance that will trigger the phenomenon of MI. We take a signal of frequency as *f* = 491.5 kHz and amplitude VM = 4 V. We put two probes at the cell 74 and the cell 86 to observe the evolution of the wave train. Knowing the frequency of the wave and the wave number through the dispersion curve, we determine the phase of the wave speed thanks to the equation (Eq. (1)), we obtain: *f* = 491.5 KHz, *k* = 1.21 rad./cel., by the dispersion curve, and *v<sup>ϕ</sup>* = 2.55 × 106 cel/s.

This indicates that the wave crosses a cell in 0.39 × 10 −6 s and (86–74) cells or 12 cells closer than 5 ms. This value allows us to identify a particular point of the wave collected at the 74th cell and determine its new position on the waveform of the 86th cell. We present, in **Figure 21**, the waveform of signals collected at the cells 74 and 86. We see in the figure, the train of solitons, which run through the line. We identify a shoulder (cursor C1) which is the waveform corresponding to the signal collected at the 74th cell, which corresponds to two solitons which follow. The cursor positioned on the soliton of greater amplitude. The cursor positioned at 5 ms, (cursor C2) shows the new position of the two solitons. We find that the highest amplitude of soliton exceeded the lowest amplitude of soliton. This observation is in agreement with the results of simulations and observations made on the solitons confirms that the speed of propagation of a soliton is more important than its amplitude [2].

#### **7.2. Propagation of a solitary wave in the line**

The image of the optical solitons, which are the natural optoelectronics bits, can be designed as electric solitons that present a sech2 profile in the time domain. We have therefore built a signal whose profile is given by the relationship (Eq. (4)):

$$V(t) \triangleq A \sec h^2(at) \tag{4}$$

where *A* and *α* are adjustable parameters and *t* time.

We inject into the bi-inductance line a signal in sech2 profile, and we follow the evolution of the signal in the line. We observe, in **Figure 22**, the evolution of signal at input cell 72 and input cell 144. We note that the shape of the signal remains intact during its spread in the line. However, we are seeing a weakening of the amplitude that is bound to the dissipative nature of the line. On the other hand, just like Remoissenet [2], we observe an oscillatory tail that accompanies the solitary wave in its spread.

#### **7.3. Shape modification of the signal**

As we previously announced in [18], the LTNL can be used to modify the forms of signals. One example is in the case of the generation of radar abrupt front signals to obtain systems that are more accurate. Even our experimental line is not designed and optimized to produce that effect. We have been able to observe some modification in waveforms by nonlinearity.

#### *7.3.1. Signal compression*

**Figure 20.** Variation of the return period expressed in number of cell based on the inverse of the square root of the

It is established for a long time that when a wave is subjected to a modulation instability, it eventually splits up in a wave train depending solitons type [12–15]. To observe this behavior in our bi-inductance line, we introduced at the entrance of the line a wave whose frequency is located in the region of modulation instability of BF and the mode, which presents an amplitude sufficiently high to cause a strong disturbance that will trigger the phenomenon of MI. We take a signal of frequency as *f* = 491.5 kHz and amplitude VM = 4 V. We put two probes at the cell 74 and the cell 86 to observe the evolution of the wave train. Knowing the frequency of the wave and the wave number through the dispersion curve, we determine the phase of the wave speed thanks to the equation (Eq. (1)), we obtain: *f* = 491.5 KHz, *k* = 1.21 rad./cel., by

amplitude of the signal applied to a frequency *f* = 475 KHz wave and voltage polarization V0 = 1.5 V.

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

= 2.55 × 106

gation of a soliton is more important than its amplitude [2].

signal whose profile is given by the relationship (Eq. (4)):

**7.2. Propagation of a solitary wave in the line**

cel/s.

This indicates that the wave crosses a cell in 0.39 × 10 −6 s and (86–74) cells or 12 cells closer than 5 ms. This value allows us to identify a particular point of the wave collected at the 74th cell and determine its new position on the waveform of the 86th cell. We present, in **Figure 21**, the waveform of signals collected at the cells 74 and 86. We see in the figure, the train of solitons, which run through the line. We identify a shoulder (cursor C1) which is the waveform corresponding to the signal collected at the 74th cell, which corresponds to two solitons which follow. The cursor positioned on the soliton of greater amplitude. The cursor positioned at 5 ms, (cursor C2) shows the new position of the two solitons. We find that the highest amplitude of soliton exceeded the lowest amplitude of soliton. This observation is in agreement with the results of simulations and observations made on the solitons confirms that the speed of propa-

The image of the optical solitons, which are the natural optoelectronics bits, can be designed as electric solitons that present a sech2 profile in the time domain. We have therefore built a

**7. Observation of the solitons in the line**

**7.1. Soliton trains**

the dispersion curve, and *v<sup>ϕ</sup>*

To illustrate the phenomenon of signal compression, we inject a sinusoidal wave amplitude V = 1.75 V for a voltage polarization of the line V0 = 1.5 V and frequency *f* = 169.4 KHz, at the entrance of the line. We observe at the 101th cell, the signal compared to the starting signal as a factor of compression order 2 (**Figure 23**).

**Figure 21.** Observation of a train of soliton produced by the fragmentation of a plane wave subject to modulation instability for the BF of 491 kHz frequency and amplitude 4 V. The positions of the C1 and C2 sliders allow seeing the evolution of a pair of soliton in the line at cell 74 and cell 86.

**Figure 22.** Evolution of a solitary wave sech2 profile in the bi-inductance line. Note the conservation of cohesion of the wave; the decrease amplitude is due to losses by Joule effect and the presence of the oscillatory tail.

For what concerns us, we show the feasibility of the project by proposing another approach. We had shown in Section 2 that the nonlinearity could produce a shock wave on a sinusoidal signal injected in the NLTL (**Figure 7** in Section 4). If we bring this phenomenon, we show that it is possible to decompose a given plane wave of frequency *f* = 20 KHz at wave train alone. We show (**Figure 24**) that the half-life of the initial sine wave consists of the solitary wave train, which was observed at the 175th cell. Everything happens if the system increases the frequency of the original wave by a new wave train, which has the frequency greater than the starting frequency.

**Figure 24.** Decomposition of a half period of a sine wave of amplitude V = 1.5 V and *f* = 20 KHz frequency, in a solitary

Experimental Studies of the Electrical Nonlinear Bimodal Transmission Line

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

181

wave train observed the cell 175 for a voltage polarization V0 = 1.5 V.

We present in this work a selection of experimental results which we reached and which concerns essentially the system of study of our choice, worth knowing, the electrical nonlinear

We notice that our experimental device was proved to be a powerful tool of work characterized by its flexibility and its robustness. Qualities essentially owed to our strategic choice who allowed us to make our investigations on sections of lines in the modest size by comparison to other systems constituted by hundreds or even thousands of cells. This arrangement also allowed us to reduce considerably the sources of drift or of artifacts, which increase with the

From the point of view of the experimental results, we determined, at first, experimentally the dispersion curve of the line by an original method, which consists of determining the phase

**8. Conclusion**

transmission line.

number of components of the line.

**Figure 23.** Compression of a signal effect observed in the cell 101 (full lines) for an input signal of sinusoidal form *f* = 169.4 KHz frequency and amplitude V = 1.75 V (dashed), polarization of the line voltage is V0 = 1.5 V.

#### *7.3.2. Frequency multiplier*

Generally to raise the frequency in electronic systems, it is necessary to use oscillators which present increasingly raised (brought up) frequencies of vibration. Since a few years, the NLTL combined with an amplifier, which amplifies the signals stemming from the noise of the electronics.

**Figure 24.** Decomposition of a half period of a sine wave of amplitude V = 1.5 V and *f* = 20 KHz frequency, in a solitary wave train observed the cell 175 for a voltage polarization V0 = 1.5 V.

For what concerns us, we show the feasibility of the project by proposing another approach. We had shown in Section 2 that the nonlinearity could produce a shock wave on a sinusoidal signal injected in the NLTL (**Figure 7** in Section 4). If we bring this phenomenon, we show that it is possible to decompose a given plane wave of frequency *f* = 20 KHz at wave train alone. We show (**Figure 24**) that the half-life of the initial sine wave consists of the solitary wave train, which was observed at the 175th cell. Everything happens if the system increases the frequency of the original wave by a new wave train, which has the frequency greater than the starting frequency.
