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

Dissipation and dispersion phenomena affect the wave propagation in nonlinear electrical line in various proportions.

#### **4.1. Effect of dissipation**

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 introducing into the line of very low amplitude waves.

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 virtually halved.

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.

#### **4.2. Effect of the nonlinearity**

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 voltage varies significantly around the tension of polarization V0 of the diodes varactor thus leading their operating point to move on a significant range of value around the Q0 resting on 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.

**Figure 3.** Evolution of an input signal of small amplitude sinusoidal shape (*f* = 20 KHz, V signal = 50 mV) after crossing a stretch of 144 cells of the nonlinear electrical line bi-inductance. Note the linear weakening of the signal keeps sinusoidal shape intact.

**Figure 5.** Nonlinearity caused by crossing a stretch of line of 144 cells induces a very strong dissymmetry of the signal,

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**Figure 6.** Illustration of the spreading of the wave in the spectral domain due to the effects of nonlinearity after crossing a section of power line nonlinear of 144 cells by a sinusoidal signal of frequency *f* = 20 kHz, amplitude Vsignal = 1.5 V for

which then presents a flat forehead.

a polarization of the V0

line voltage = 1.5 V.

**Figure 4.** Vsignal of amplitude input voltage is superimposed on the bias of the varactor diode voltage to impose on them an operating point located on the one hand and the resting point Q0 on the other hand.

**Figure 5.** Nonlinearity caused by crossing a stretch of line of 144 cells induces a very strong dissymmetry of the signal, which then presents a flat forehead.

**Figure 3.** Evolution of an input signal of small amplitude sinusoidal shape (*f* = 20 KHz, V signal = 50 mV) after crossing a stretch of 144 cells of the nonlinear electrical line bi-inductance. Note the linear weakening of the signal keeps sinusoidal

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

**Figure 4.** Vsignal of amplitude input voltage is superimposed on the bias of the varactor diode voltage to impose on

on the other hand.

them an operating point located on the one hand and the resting point Q0

shape intact.

**Figure 6.** Illustration of the spreading of the wave in the spectral domain due to the effects of nonlinearity after crossing a section of power line nonlinear of 144 cells by a sinusoidal signal of frequency *f* = 20 kHz, amplitude Vsignal = 1.5 V for a polarization of the V0 line voltage = 1.5 V.

experimentally that a uniform continuous wave train could be unstable to the disruptions that modulate its envelope [7, 8]. The impact of the work of Benjamin and Feir, often called

In a previous work, Pelap [1] conducted a theoretical study of a power nonlinear transmission line bi-inductance. He first showed that the wave propagation in the line governed by an equation of type Ginzburg Landau complex (GLC) then sought solutions of the equation in discrete semi-approximation. The approach was different from that adopted by Lange and Newell for hydrodynamic fields [16] to establish a criterion of instability for a plane wave propagating through a nonlinear bi-inductance linear and weakly dissipative inductance. The

*Pr Qr* + *Pi Qi* > 0 (3)

If the pseudo-product (Eq. (3)) is positive, it means then that the wave is unstable under the

We study, for our electrical nonlinear bi-inductance transmission line, the evolution of the

Knowledge of the values of the critical wave in the HF mode allows us to clarify if areas of the curve dispersion of the wave are stable under the modulation or not (**Figure 10**). Thus, we are building a decision tool that allows us in our different investigations to determine the frequency of the signals that we send in the line for the observation of specific phenomena properly.

**Figure 8.** Shape of the pseudo-product in BF mode for the inductance bi line. The value of the wave number is critical in this mode which marks the separation between the area of stability, and the zone of instability is *kcl* = 1.23305 rad./cel.

are, respectively, the real and imaginary coefficients of dispersion terms

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wave is unstable under modulation if the pseudo-product (Eq. (3)) is positive that is:

instability of Benjamin-Feir, is mainly in hydrodynamics.

where *Pr*

, *Pi* , *Qr*

*P* and of nonlinearity *Q* .

, and *Qi*

modulation and the system will be the seat of an MI.

pseudo-product in HF mode (**Figure 8**) and the BF mode (**Figure 9**).

**5.1. Criterion of modulation instability in the electrical bi-inductance line**

**Figure 7.** Destruction of the sine wave kept after crossing a section of electrical nonlinear line aft 288th cells by a sinusoidal signal of frequency *f* = 20 kHz, amplitude Vsignal = 1.5 V, and a polarization of the line voltage V0 = 1.5 V.

The signal can then undergo a significant distortion as it penetrates into the line. We present a characteristic effect related to the effects of nonlinearity of the signals (in **Figure 5**). It is obvious that the 144th cell wave become very asymmetrical, as phases of the various points of the wave velocity are different.

In fact, everything happens as if parts of the wave with large amplitudes are moving faster than low amplitude. The wave starts to break down; it shows more and more overtones, thus reflecting the complexity of the shape and explains why nonlinearity leads to a spreading of the wave (**Figure 6**).

Ultimately, the wave front flattens completely; called shock wave similar to the phenomenon observed in the aerodynamic field when a mobile starts to move at a speed greater than the speed of sound. At this moment, the wave breaks (**Figure 7**).
