Propagating intrinsic localized mode in a cyclic, dissipative, self-dual one-dimensional nonlinear transmission line

A well-known feature of a propagating localized excitation in a discrete lattice is the generation of a backwave in the extended normal mode spectrum. To quantify the parameter-dependent amplitude of such a backwave, the properties of a running intrinsic localized mode (ILM) in electric, cyclic, dissipative, nonlinear 1D transmission lines, containing balanced nonlinear capacitive and inductive terms, are studied via simulations. Both balanced and unbalanced damping and driving conditions are treated. The introduction of a unit cell duplex driver, with a voltage source driving the nonlinear capacitor and a synchronized current source, the nonlinear inductor, provides an opportunity to design a cyclic, dissipative self-dual nonlinear transmission line. When the self-dual conditions are satisfied, the dynamical voltage and current equations of motion within a cell become the same, the strength of the fundamental, resonant coupling between the ILM and the lattice modes collapses, and the associated fundamental backwave is no longer observed.

Figure 1

Ideal cyclic nonlinear transmission line. The unit cell consists of a single hard nonlinear capacitor, a single hard nonlinear inductor, and no dissipation.

Figure 2

2D-FT schematic drawing of stationary and traveling ILMs superimposed on the solid NLTL dispersion curve. Solid and open circles indicate driving points for stationary and traveling ILMs, respectively. An ILM is expressed as a dispersion line (DL) in $k$ space because it is a localized wave packet in physical space. The solid horizontal line is for a stationary ILM, while the dashed sloped line is for an ILM traveling to the right.

Figure 3

One unit cell of the experimental cyclic NLTL with different $C_e\left(V\right)$ and $L_e\left(I\right)$ nonlinearities. $R_p$ is the parallel resistor of the capacitor, $R_L$ is the series resistor of the inductor, and $V_{d,n}$ the voltage driver with driving capacitor $C_d$.

Figure 4

Comparison of experimental and simulation ILM resonance properties. (a) Experimental voltage pattern versus time for a traveling ILM in a capacitor driven nonlinear transmission line. (b) 2D-FT result of the voltage pattern in (a) gives the ILM dispersion line strength (DLS). Solid red curve indicates the NLTL linear dispersion curve. Cross identifies driver waven umber $14\pi /16$ and driver frequency 43 kHz. The ILM DL intersects the NLTL dispersion curve at the solid arrow. (Signal at middle, dashed arrow, is an experimental ghost.) (c) Solid curve: ILM DLS versus wave number. Dashed curve is obtained from a simulation with the same conditions as the experiment. Long dashed curves beneath the resonances give approximate baselines.

Figure 5

Unit cell of the NLTL with duplex (V, I) driving and identical $C_e$ and $L_e$ nonlinearities. See Eq. (3). $R_p$ is the parallel resistor of the capacitor, $R_L$ is the series resistor of the inductor. The two driving terms for the duplex driver are balanced with the additional condition on the mutual inductance $M_d$ given by Eq. (17).

Figure 6

A traveling, balanced self-dual ILM showing its identical envelope shape at capacitively centered and inductively centered lattice locations in a 32-element NLTL ring. (a) Simulated voltage pattern versus time for a traveling ILM, driven at 50 kHz. Voltage pattern is shown in blue-white-red scale. (b) Time dependence of voltage and current for sites from 15 to 17. (c) Voltage and current as a function of lattice site at $\text{time}=0.5125$ ms, identified by left dashed line in (b). Solid circles identify voltage at integer sites, and open circles identify current at half-integer sites. (d) Voltage and current as a function of lattice site at time identified by right vertical dashed line in (b). Peak is now at the current site. Peak distance between (c) and (d) is 1.5 lattice sites. See text for more details.

Figure 7

2D-FT of running, balanced self-dual ILM voltage pattern presented in Fig. 6. ILM DLS versus $k$ and frequency is expressed by a gray scale. Solid cross identifies driving wave number $k_d=13\pi /16$, frequency ($f_d=50\mathrm{kHz}$) point. Dashed curve: NLTL linear dispersion curve. Arrow: Intersection point of the ILM DL and the NLTL dispersion curve. Running ILM to the right, NLTL backwave to the left. In this extended frequency figure, third harmonic and fifth harmonic ILM DLSs are also seen. Dashed crosses are guides to eye, indicating third and fifth harmonic points of the $(k_d,f_d)$-driven ILM.

Figure 8

ILM DLS of $V$ (red) and $Z_{0}I$ (blue) data for the single driven (model 1) and duplex driven (model 2) cases. Driving frequency 50 kHz. Up arrow, driving point: $k\uparrow =13\pi /16$. Down arrows $k\downarrow =-1.39$, same location in the four frames, identify the intersection point of ILM DL and the NLTL dispersion curve. (a) Model 1, damping balanced. (b) Model (1), damping unbalanced. Nearby red and blue features show the actual resonance responses. (c) Model 2, damping balanced. (d) Model 2, damping unbalanced. No resonance features are observed for (c), the completely balanced self-dual model.

Figure 9

Combined ILM DLS for the same eight sets of data presented in Fig. 8. Results calculated from Eq. (19), with Fig. 8 data: $|\tilde{V}\left(k\right)|$ and $|\tilde{I}\left(k\right)|$. A simple resonance feature now is evident in three of the four panels.

Figure 10

Three examples of resonance strength suppression for balanced nonlinearities (data from Fig. 9). Panel (1), log ordinate. The dashed trace, model 2(c), shows no indication of a resonance and is used as the baseline in panel (2) to obtain the relative strengths of curves (b), (d), (a). Model 1(b): Largest resonance strength, single V driver with unbalanced damping. Model 2(d): Somewhat weaker, duplex driver with damping unbalanced. Model 1(a): Weakest resonance strength, single driver with balanced damping.

Figure 11

ILM-DL strength as a function of wave number and driver frequency. Logarithmic strength is displayed by red-white-blue color. For the unbalanced resistors and capacitive driving case (model 1) panel (a), resonance response (horizontal arrow) is observed at all frequencies. For the all balanced duplex (V, I) driver case (model 2) shown in (b), no resonance signal is observed over the same wide driving frequency range.