1. Introduction
For electrical engineers, nonlinearity and dispersion are generally undesirable, but a nonlinear transmission line (NLTL) can balance these two effects to generate soliton-like pulses. As stable and localized nonlinear waves, solitons can propagate along NLTL in the form of bell-shaped short-duration voltage pulses [
1]. In the last few decades, solitons have become the subjects of numerous studies in various fields of physics and engineering since nature offers various examples of this phenomenon. It is stated in the literature that the first observed soliton was a mono-pulse water wave in a narrow canal where the shallow water possessed both dispersion and nonlinearity [
2].
As a nonlinear electronic circuit, the NLTL must contain electrical components with nonlinear permittivity or permeability in order to achieve pulse sharpening [
3]. There are generally two possible NLTL designs. The first one uses a ladder network of repeating lumped inductors and capacitors, where the inductors or capacitors are nonlinear in their response to current or voltage, respectively [
1,
2,
3]. For experimental research, this implementation of the equivalent NLTL circuit is most common in which special diodes (varactors or Schottky diodes) are used as nonlinear capacitors placed in shunt form [
4]. The second one, more difficult to implement, realizes the NLTL by using a periodically loaded traditional transmission line with lumped varactors [
2]. In both NLTL topologies, the dispersion derives from the structural periodicity, while the nonlinearity originates from the varactors—the most commonly used nonlinear components whose capacitance changes with the applied voltage [
5]. On the other hand, losses in NLTL are inevitably present due to the finite conductivity of the conductors (resistive losses) as well as the dielectric insulator imperfection (dielectric losses). These losses do not conserve energy and generally have a different effect than the dispersion, which rearranges energy but conserves it [
6,
7]. Namely, it is known that losses cause a decreases in the amplitude of localized voltage pulses, but the impact of losses has been insufficiently studied in the available literature, especially at microwave frequencies.
In electronics and especially in the microwave domain, NLTLs are of particular interest because of their wide range of possible applications. Since NLTL provides shaping and sharpening of signals, these lines can find important applications for data transmission in high-speed digital circuits [
6]. Additionally, due to the possible shrinking in the pulse rise and fall times, NLTL can be applied in high-speed sampling oscilloscopes and other instrumentation for microwave systems [
8]. All these studies have shown that NLTL, as a two-port network, requires a high frequency signal at the input to generate a sharp soliton at its output. However, a recent experimental study demonstrated the use of NLTL in an electrical soliton oscillator, as a one-port system, that self-generates the soliton pulses from ambient noise [
9]. In addition to NLTL, an amplifier was used in the realization of this soliton oscillator to compensate for losses and to stabilize the oscillations.
The main goal of this paper is to present a new analytical model for the study of the voltage soliton waves in the lossy NLTL with a special focus on the impact of losses on the propagation. The justification of our approach is clearly confirmed by the results of numerical simulations and an experiment that was performed on an NLTL prototype fabricated at the discrete board level. In
Section 2, we start with a description of the lossy NLTL model with varactors and we derive a voltage equation for the presented model. In this section, we also determine the parameters of the commercially available varactor used in our study. In
Section 3, we provide a detailed theoretical analysis and demonstrate that the NLS equation with loss term can describe the soliton dynamics along the lossy NLTL. The results of the numerical simulations and their comparison with the theoretical predictions are given in
Section 4. In
Section 5, we describe the experiment and present the results related to the formation of voltage bright solitons, as well as the changes in their amplitude and half–maximum width due to losses. Finally,
Section 6 summarizes the key results of this comparative study.
3. Analytical Results
In this paper, the well–known reductive perturbation method [
12,
13] is used for studying the behavior of short–wavelength envelope electrical solitons. Under the assumption of weak nonlinearity, this method is applied to the characteristic voltage Equation (8) of equivalent lossy NLTL circuit model. To begin this procedure, the envelope spatial and temporal variables are introduced as follows:
respectively, where
is the small perturbative parameter,
denotes the NLTL cell number, and
represents the group velocity of the wave’s envelope [
12]. In the literature [
14], it is emphasized that parameter
measures the smallness of the modulation frequency and the amplitude of the input waves. Now, voltage
of
nth unit cell at time
can be expanded to a power series of small quantity
in the following way [
15]
where
represents the rapidly varying phase,
refers to a higher-harmonic wave component and
. denotes complex conjugate. Additionally, parameters
and
are the wavenumber and the angular frequency, respectively.
After substituting the expression of
given with Equation (10) into the discrete nonlinear Equation (8) and using Equation (9), we can transform voltage Equation (8) into a system of equations in different powers of the small perturbative parameter
. Namely, the terms with the same power of
are collected and set to zero. In that manner, we obtain first-, second- and third-order sets of equations. For the first harmonic (
), collecting terms at the first-order of the parameter
leads to the following linear dispersion relation
where
represents the cutoff angular frequency. In
Figure 3, the dispersion curve is given, defined by the Equation (11), for the lossy NLTL under study. This curve shows the evolution of frequency
as a function of the wavenumber
, which is taken in the Brillouin zone,
[
16]. For the characteristic capacitance of the varactor
and linear inductance
, the cutoff frequency of
was obtained. Furthermore, based on the linear dispersion relation given by Equation (11), the corresponding group velocity
is determined by the following expression
On the other hand, for the first harmonic collecting terms at the third-order of the parameter
, we obtain the following equation going over to continuum approximation
Thus, propagation the time–space voltage waves along the lossy NLTL can be modeled by a nonlinear Schrödinger (NLS) equation incorporating a linear loss term [
17,
18]
where are, respectively the dispersion, nonlinearity and dissipation coefficients given by:
It is clear from Equation (15) that the sign of the dispersion coefficient
is always negative. However, the sign of the nonlinearity coefficient
depends on the nonlinear characteristics of the voltage-dependent capacitance, as well as the carrier wave angular frequency,
. Therefore, the focusing or the defocusing effect and the corresponding generation of bright or dark solitons [
19] in lossy NLTL can be described and understood by means of the NLS Equation (14) with losses. Namely, the sign of the product
can be positive or negative, which leads to formation of a bright soliton (
) or a dark soliton (
), respectively [
12,
13].
Figure 4 shows the wavenumber (frequency) dependence of the product
for the lossy NLTL under study. Namely, two different regimes are observed: a dark soliton regime (
) at low frequencies (light green region), and bright soliton regime (
) at high frequencies (dark green region). According to determined parameters of commercially available varactor (SMV1234–079LF) [
11], see
Figure 2, and commercial linear inductor (MLG1005S2N0BTD25) [
20] used in this study, our analytical results predict that the threshold value of the carrier wave frequency is
, above which a formation of bright solitons will occur. Below, we focus on the numerical and experimental study of bright soliton generation and its propagation at higher frequencies (
in accordance with the results of our theoretical analysis.
In the absence of losses (
), Equation (13) reduces to the standard NLS equation, which possesses numerous symmetries and respective conservation laws such as the total pulse intensity and total energy [
18]. A well-known analytical bright soliton solution of the standard NLS equation for the envelope function
is given with the following expression [
17]
where
denotes a real amplitude that depends on the initial conditions. Furthermore, from Equation (16), it is obtained that the spatial width of the soliton defined at half maximum can be determined as [
16]
It is obvious that the increase in nonlinearity causes greater localization of soliton waves and the area covered by the envelope function
, given by Equation (16), remains preserved. For example, if the carrier wave frequency
, which is in the previous determined frequency range
for bright soliton formation in lossy NLTL under study, one finds
and
with the use of known varactor and inductor parameters. Theoretically, it provides that the width at half maximum calculated by Equation (17) is
. This result indicates that the degree of localization ensures that the exploited continuum approximation can be fairly justified.
On the other hand, when losses are present in the NLTL (
), it is necessary to include the impact of a dissipation on the soliton wave dynamics. According to soliton perturbation theory, the losses do not affect the velocity of the soliton but its amplitude, which becomes a decaying function of time [
17]. Specifically, the amplitude of the bright soliton, defined by the lossy NLS Equation (14), will decrease exponentially as
[
18]
while its spatial width increases exponentially as
.
Regarding a modulational instability, study [
16] showed that a plane-wave solution of NLS Equation (14) is modulationally stable for
. On the other hand, for
, the system can be modulation unstable if the frequency of the perturbation
satisfies the following condition
where
is the corresponding wavenumbers determined by
In this case, the modulation increases as time progresses and the continuous wave breaks into a periodic pulse or envelope soliton train. With the use of the lossy NLTL parameters, the theoretical frequency of the envelope modulation for
V lies in the instability domain
, predicted by Equation (19). Furthermore, in a mechanical analogue of NLS equation with viscous dissipation, it has been experimentally demonstrated that dissipation generally prevents modulational instability [
21]. Namely, modulational instability might still occur, but losses can stop the growth of perturbations before nonlinear interactions become significant. In this case, if the perturbations are small enough initially, then they never grow large enough for nonlinear interactions to become important.
4. Results of Numerical Simulations
In order to verify our theoretical predictions, numerical simulations were performed on a lossy NLTL with 30 identical cells, see
Figure 1. For this purpose, we considered that each NLTL cell consists of varactors (SMV1234–079LF) [
11] with an average serial loss of
, and fixed linear SMD inductors
(MLG1005S2N0BTD25) [
20] with an average loss of
. Additionally, the varactor parameters used in our numerical analysis were determined by fitting the data taken from the varactor datasheet, see
Figure 2. A circuit simulator is implemented in the MATLAB software package using a well–known finite-difference method in the time domain.
For an input pulse of amplitude 1 V and a carrier wave frequency
selected in the passband of the NLTL, see
Figure 4, the numerically obtained envelope waveform is presented in
Figure 5. It is obvious that a bright soliton is formed, preserving its shape, which is in good agreement with our theoretical predictions since the product of the dispersion and nonlinearity coefficients is positive for the parameters of the NLTL and accordingly the bright soliton should occur. The direct comparison of analytical and numerical results for envelope soliton at the 20th cell of the lossy NLTL is given in
Figure 6. Evidently, a good match between the obtained waveforms confirms that the lossy NLS approximation, defined by Equation (14), is the suitable model for describing the generation and propagation of soliton waves along NLTL with losses.
Based on the obtained numerical results shown in
Figure 5, the impact of losses present in NLTL on the soliton amplitude is obvious and in agreement with our theoretical predictions. According to soliton perturbation theory [
16,
17], the amplitude of the envelope soliton is an exponentially decreasing function of time, see Equation (18). To confirm these predictions, we have studied in detail how the amplitude of the envelope bright soliton with carrier wave frequency
decays along the NLTL. The obtained results are illustrated in
Figure 7. These data were fitted with an exponential function and close values for the damping constant were calculated. Namely, according to our theoretical predictions and numerical results, the damping constants are
and
, respectively. These values are in good agreement, as well as with the experimental results (
) that will be discussed in the section below.
5. Experimental Results
In order to validate our analytical and numerical results, experiments were carried out on an NLTL prototype with 30 identical cells, which was fabricated at the discrete board level, as shown in the inset of
Figure 8. In this prototype, commercially available SMD linear inductors (MLG1005S2N0BTD25) [
20] and varactors (SMV1234–079LF) [
11], which serve as nonlinear capacitances, were used. The parameter values of these components are the same as the parameters used in the analytical and numerical study. The fabricated NLTL prototype was fed by a signal generator (Agilent E4438C ESG), and soliton waveforms were analyzed using a high-performance
oscilloscope (Infiniium DSO90604A) with probes which have high impedance and small capacitance in order to avoid parasitic reflections. Generally, the main experimentally adjustable parameters are: the pulse frequency, amplitude and width. In this experimental study, a voltage pulse of amplitude
and frequency
was applied at the input of the NLTL.
Figure 8 displays the measured voltage response at three different NLTL nodes. Specifically, the upper, middle and bottom traces correspond to the response at nodes 10, 20 and 30, respectively. It is obvious that the bright soliton profile possesses a well-defined envelope and remains localized as it travels along NLTL even though it loses some of its energy due to dissipation. The measured values of the amplitude and half–maximum width at node 10 were
and
, respectively, whereas the corresponding values at node 30 (the end of the NLTL) were
and
, respectively. This obtained decrease in the amplitude of the envelope soliton is well-matched with theoretical analysis and results of numerical simulations, as can be seen in
Figure 7. The small discrepancy is most likely caused by additional losses in conductors which are not included in the NLTL model, as well as the fact that the parameter values of components vary according to their tolerances. Additionally, the bright soliton with higher amplitude has smaller half-maximum width, which is in good agreement with our analytical result given by the Equations (17) and (18). This is a clear affirmation of our theoretical prediction that the proposed model defined by NLS Equation (14) with losses is able to excellently predict the soliton waveforms.