Dynamics and properties of waves in a modified Noguchi electrical transmission line

We consider a modified Noguchi electrical transmission line and examine the effects of a linear capacitance $C_s$ on the wave characteristics while considering the semidiscrete approximation. It appears that wave modulations in the network are governed by a dispersive nonlinear Schrödinger equation whose coefficients are shown to be a function of $C_s$. We show that the use of this linear capacitance makes the filter more selective. We also show that the width of the unstable regions increases while that of the stable regions decreases with $C_s$ adding consequently the width of the frequency domain where bright solitons exist. Furthermore, we establish the existence of one more region (compared to the work of Marquié et al. [Marquié et al., Phys. Rev. E 49, 828 (1994)]) in the dispersion curve that allows the motion of envelope solitons of higher frequency in the system. Numerical and experimental investigations done on the model confirm our analytical predictions.

Figure 1

Schematic representation of one unit cell of a modified discrete Noguchi electrical transmission line. The network possesses $N$ identical unit cells.

Figure 2

Linear dispersive curve showing evolution of the frequency $f=\omega /2\pi$ as a function of the wave number $k$ for given values of $\lambda$. This curve is constructed for the following line parameters defined by Eqs. (3). The lower cutoff frequency is $f_0=411\mathrm{kHz}$. The upper cutoff frequencies are, respectively, $f_{max}=1268\mathrm{kHz}$ for $\lambda =0$ and $f_{max}=972.5\mathrm{kHz}$ for $\lambda =0.175$. We note that the frequency domain $[f_0,f_{max}]$ of the filter decreases with the growth of $\lambda$ and the filter becomes more selective.

Figure 3

Group velocity $V_g$ versus $k$ for the line parameters of Fig. 2. This velocity decreases when $\lambda$ increases. This result suggests that a wave packet moving in the network will travel for a long distance before it vanishes.

Figure 4

Behavior of the dispersive coefficient $P$ as a function of the frequency of the carrier $f$ for different values of $\lambda$ and the line parameters of Fig. 2. These curves show that $P$ vanishes for a frequency $f_z$ whose value decreases with the growth of $\lambda$.

Figure 5

Plot of the nonlinear coefficient $Q$ in terms of $f$ for the parameters of Fig. 2. We note that the maximum value of $Q$ passes from $7.92\times 10^4\mathrm{rad}/\mathrm{s}\mathrm{V}$ for $\lambda =0$ to the value $3.575\times 10^4\mathrm{rad}/\mathrm{s}\mathrm{V}$ for $\lambda =0.175$.

Figure 6

Variations of the product $\mathit{PQ}$ as a function of $k$ for given values of $\lambda$. Points $A1(k_{A1}=0.74)$, $B1(k_{B1}=1.04)$, and $C1(k_{C1}=1.89)$ are zeros of $\mathit{PQ}$ for $\lambda =0$, while this product vanishes at $A2(k_{A2}=0.68)$, $B2(k_{B2}=1)$, and $C2(k_{C2}=1.80)$ for $\lambda =0.175$. We define the quantities $z_1=k_{B1}-k_{A1}=0.3$, $z_2=k_{B2}-k_{A2}=0.32$, $z_3=k_{C1}-k_{B1}=0.85$, and $z_4=k_{C2}-k_{B2}=0.8$. Since $z_2>z_1$, the modulational instability zone increases when $\lambda$ grows. On the other hand, we have $z_4<z_3$, which implies that the modulational stability zone decreases with $\lambda$. In both cases, we note that the curve can be divided into four regions that deal with the signs of $\mathit{PQ}$.

Figure 7

Behavior of the nonlinear coefficient $M_{23}$ versus the frequency $f$ of the carrier for the line parameters of Fig. 2 in the case $\lambda =0$. This curve shows that $M_{23}$ does not possess neglected values in spite of the bandpass character of our filter and also establishes the existence of a maximum value at $f_m=789.4\mathrm{kHz}$.

Figure 8

Evolution of the nonlinear coefficient $Q$ in terms of the frequency $f$ for the parameters of Fig. 2 in the case $\lambda =0$. We note that nonlinear effects vanish for two values of the frequency $f_q=590\mathrm{kHz}$ and $f_Q=1062\mathrm{kHz}$. The second frequency $f_Q$ was absent in many works carried out on this bandpass filter [6, 25, 26, 27].

Figure 9

Variations of the product $\mathit{PQ}$ as a function of the frequency $f$ for the parameters of Fig. 2 with $\lambda =0$. We obtain three characteristic frequencies where this product vanishes, that is, $f_q=590\mathrm{kHz}$, $f_z=720\mathrm{kHz}$, and $f_Q=1062\mathrm{kHz}$, which divide the curve into four regions that deal with the signs of $\mathit{PQ}$.

Figure 10

Dispersion curve of our bandpass filter in the case $\lambda =0$. The allowed band $[f_0,f_{max}]$ is divided into four regions concerning the stability of the system, which depends on the sign of the product $\mathit{PQ}$. Previous works [6, 25, 26, 27] exhibit only three domains of frequency.

Figure 11

Signal voltage as a function of time showing stability of the plane wave at frequency $f_P=460\mathrm{kHz}$ belonging to domain I of the dispersion curve. The initial amplitude $V_m=0.2\mathrm{V}$, while the modulation rate $b=1\%$ and modulation frequency $f_m=8.75\mathrm{kHz}$.

Figure 12

Signal voltage versus time exhibiting the stability of the plane wave for the frequency $f_p=810\mathrm{kHz}$ (domain III) with the parameters of Fig. 11.

Figure 13

Modulational instability phenomenon of the plane wave for the frequency $f_p=622\mathrm{kHz}$ that belongs to domain II of the dispersion curve with the parameters of Fig. 11.

Figure 14

Modulational instability phenomenon of the plane wave observed at the frequency $f_p=1205\mathrm{kHz}$ that belongs to region IV with the parameters of Fig. 11.

Figure 15

Propagation of the bright soliton solution of the NLS equation in the network for $f_p=622\mathrm{kHz}$ with the initial amplitude $V_m=0.3\mathrm{V}$.

Figure 16

Transmission of the bright soliton solution of the NLS equation through the network for $f_p=1205\mathrm{kHz}$ and $V_m=0.3\mathrm{V}$.

Figure 17

Schematic representation of the experimental arrangement.

Figure 18

Experimental stability of the plane wave with the frequency $f_p=460\mathrm{kHz}$ belonging to region I of the dispersion curve (Fig. 10) observed in cell 16.

Figure 19

Experimental stability of the signal with a carrier wave frequency chosen in domain III ($f_p=810\mathrm{kHz}$) with the parameter of Fig. 18.

Figure 20

Experimental observation of the instability under modulation of the plane wave at frequency $f_p=622\mathrm{kHz}$ belonging to domain II of the dispersion curve where $\mathit{PQ}>0$. The signal is at cell 16.

Figure 21

Experimental observation of the MI phenomenon of the signal of frequency $f_p=1205\mathrm{kHz}$ belonging to domain IV of the dispersion curve with the parameter of Fig. 20.