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Abstract
Solitary waves (solitons) are fundamental modes supported by nonlinear guiding structures and propagate without any distortion due to dispersion. They can be used in the synthesis of waveforms with desired spectral or time-domain characteristics. However, the nonlinearity inherent to solitons complicates their analysis on general guiding structures. This paper proposes an iterative algorithm that can compute soliton solutions supported by general periodic electrical networks loaded by nonlinear capacitors. The proposed technique accounts for the exact dispersion of the structure and allows the capacitance to take on an arbitrary function of voltage. The algorithm computes the solution throughout the entire structure while only modeling a single unit cell, thus reducing the computational cost significantly. It is implemented within an in-house solver and verified against a commercially available circuit simulator. Different circuits which include lumped inductors, ideal transmission lines, and co-planar waveguides are simulated and shown to agree with commercial harmonic balance solvers. The presented technique will allow microwave and optical engineers to design and optimize a wide variety of nonlinear guiding structures efficiently.
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1. Introduction
Nonlinear devices and materials allow control over the temporal characteristics of electromagnetic waves. In the context of guided-wave propagation, this enables unique electromagnetic functions such as parametric amplification, pulse-shaping, frequency-mixing, and non-reciprocity. Nonlinear structures with applied modulation signals have recently received significant attention in the realization of electromagnetic metamaterials and metasurfaces whose properties are a function of both space and time. The analysis of space-time modulation (STM) is accompanied by a unique set of challenges which arise from the coupling between space-time harmonics. When an applied electromagnetic signal is sufficiently small with respect to the modulation, an STM structure can be treated as a linear device, analogous to a multi-port network interconnecting the various frequency harmonics [1–5]. This feature is often exploited in the analysis of STM systems for frequency mixing [6–9], bandwidth enhancement [10–12], amplification [13–15], and nonreciprocity [16–18]. However, when the electromagnetic signal is sufficiently large, superposition no longer holds and the applied signal itself modulates the system. This complicates the analysis procedure since the system and the solution are interdependent. Despite their complexity, nonlinear systems are regularly employed in the large-signal regime for amplitude filtering/detection [19–21], oscillator design [22,23], and pulse/frequency-comb generation [24–28].
Pulsed signals are used in a wide range of applications, including biomedical imaging and radar range determination. With respect to continuous wave signals, the reflection of a short pulse from stratified biological materials contains additional information regarding the underlying layers [29,30]. Similarly, pulsed radio waves are regularly used in radar systems to determine the range of a scanned object. The accuracy and probability of detecting a scanned object increases with the number of pulses per second [31,32]. Therefore, reliable pulsed sources with a high repetition rate are desirable for radar systems. Recently, ultra-wideband (UWB) systems have gained traction in common consumer products like handheld cellular phones and tracking devices. UWB systems utilize RF impulse-like waveforms with wide bandwidth to accurately calculate distance between objects. As shown in [33], the accuracy of the system is directly related to the bandwidth of the pulse. However, the bandwidth of these pulses are limited in practice by spectral requirements imposed by regulatory agencies, such as the Federal Communications Commission (FCC) in the United States. Thus, the RF pulses for high-efficiency UWB systems should maximize the spectrum within the allocated bandwidth subject to these out-of-band constraints as shown in [34]. Finally, wide-bandwidth pulse waveforms can be exploited to generate high-order frequency harmonics. Thus, devices which generate these pulses can be designed as sub-harmonic mixers and frequency comb synthesizers [35].
Nonlinear guided-wave systems are frequently used in generating electromagnetic pulses from the microwave to the optical domain. One such structure available at microwave frequencies is a nonlinear transmission line (NLTL). NLTL structures capable of efficiently producing impulse-like signals with wide bandwidths have been demonstrated [24,36,37]. At optical frequencies, ultrashort pulses are often generated using nonlinear crystals [38]. These materials are analogous to the NLTLs available at microwave frequencies. Ultrashort optical pulses have been shown to enable precise frequency characterization and time synchronization among other applications [39–41]. The recently commissioned Zettawatt Equivalent Ultrashort pulse laser System (ZEUS) at the University of Michigan, for example, will soon enable the generation of ultrashort high power laser pulses designed to support advanced research in high-energy physics [42].
In contrast to linear guided-wave structures, the fundamental modes supported by nonlinear systems like NLTLs and nonlinear crystals are not necessarily sinusoidal in time. The modal solutions often correspond to a precise time-domain waveform, referred to as solitary waves (or solitons). Solitons were first discovered in 1834 by John Scott Russell in the context of hydrodynamic waves [43]. It has since been shown that solitons can propagate through nonlinear materials without any distortion caused by the dispersion in the guiding structure [44]. Figure 1 depicts a graphical illustration of a soliton mode as it propagates undistorted through a ladder network of series inductors and shunt nonlinear varactor diodes. The frequency spectrum of these modes depends both on the nonlinearity and dispersion of the transmission line. In theory, the nonlinearity and dispersion of the transmission line can be tuned to provide a desired frequency spectrum [24]. These circuits can produce very sharp pulses with high spectral content.
Fig. 1. Illustration of a soliton mode propagating through a spatially-discretized nonlinear transmission line consisting of series inductors and shunt varactor diodes loading a co-planar waveguide transmission line.
In this work, an iterative technique is developed for computing the soliton modes supported by general periodic electrical networks loaded by nonlinear capacitors. The technique fully accounts for the exact frequency dispersion of the constituent cells and allows for an arbitrary nonlinear dependence of the capacitance on applied voltage. Furthermore, the proposed algorithm utilizes a single unit cell to compute the supported soliton modes thus significantly decreasing the computational cost. The paper is structured as follows: Section 2 reviews the fundamental concepts regarding solitons and existing techniques for computing soliton modes. Section 3 outlines the general structure of the proposed technique for computing soliton solutions supported by general periodic electrical networks loaded with nonlinear capacitors. Numerical results using the proposed technique are reported and validated using commercially available solvers in Section 4. Finally, the conclusion is provided in Section 5.
2. Review of solitons and solvers
From the telegrapher’s equations, it can be concluded that a monochromatic signal ($\cos \omega t$) is a supported mode on a linear transmission line. When nonlinearity is introduced in the structure, it is evident that $\cos \omega t$ is no longer a supported mode as any given frequency is coupled with its harmonics. Since the guiding medium itself depends on the local amplitude of voltage or current, the phase velocity of the wave on the line becomes modulated by the wave. The pulse profile for a given soliton mode is such that the frequency dispersion of the velocity, which typically spreads out an RF pulse, is balanced by the spatial modulation of the velocity. As a result, soliton modes propagate through nonlinear materials without any distortion. This phenomenon was first seen in a narrow water channel by John Scott Russell, where a soliton wave of water maintained its form and continued to propagate with minimal dissipation for several kilometers despite the highly dispersive nature of hydrodynamic waves [43]. Since then, significant effort has been dedicated to the analytical characterization of these soliton modes in the context of electronic circuits. For an artificial transmission line constructed using series linear inductors, $L$, and shunt nonlinear capacitors, $C(v)$, the telegrapher’s equations can be written as $\frac {\partial {v}}{\partial {x}} = -L' \frac {\partial {i}}{\partial {t}}$ and $\frac {\partial {i}}{\partial {x}} = - \frac {\partial }{\partial {t}}\{C'(v)v\}$, where $L'\equiv L/\delta$, $C'\equiv C/\delta$, and $\delta$ is the physical length of the unit cell. Note that these relations assume that the electrical delay through a given unit cell is small (the continuum limit). If the structure is weakly dispersive [44], then the nonlinear wave equation governing the transmission line is written as
3. Iterative technique
The proposed iterative formulation closely follows the generalized eigenvalue problem derived in [8] for spatially discrete traveling-wave modulated electrical networks. Consider the general nonlinear periodic electrical network depicted in Fig. 2. The derivation provided in this work is carried out for the case where the varactor diode is connected in shunt between two arbitrary linear, time-invariant (LTI) unit cells. However, this can be modified (as in Section 4) to account for alternative connections to the diode. The solution is assumed to be periodic in time with period $T=2\pi /\omega$. Thus, the current and voltage at the input of cell $n$ can be expanded into frequency harmonics, $l\omega$, as ${{v}_{n}}\left ( t \right )=\sum\limits _{l=-\infty }^{\infty }{{{V}_{n,l}}{{e}^{jl\omega t}}}$, and ${{i}_{n}}\left ( t \right )=\sum \limits_{l=-\infty }^{\infty }{{{I}_{n,l}}{{e}^{jl\omega t}}}$. The varactor diodes are modeled as nonlinear capacitors, whose charge is given by ${{q}_{n}}\left ( t \right )={{C}_{d}}\left ( {{v}_{d;n}}( t) \right ){{v}_{d;n}}\left ( t \right )$. Given the voltage $v_{d;n} (t)$ over the diode, an equivalent time-dependent capacitance can be defined as ${{C}_{d;n}}\left ( t \right )={{C}_{d}}\left ( {{v}_{d;n}}( t) \right )$. Since the solution is time-periodic, the capacitance and charge are time-periodic as well, and are given by
The current flowing through the diode in cell $n$ is the time-derivative of the charge: ${{i}_{d;n}}\left ( t \right )=\frac {d{{q}_{n}}}{dt}$. Substituting our harmonic expansions for the charge yields an expression for the frequency harmonics of the current flowing through the diode in cell $n$:
Fig. 2. A general 1D nonlinear periodic electrical network loaded by a varactor diode. The frequency harmonics of voltage and current at the input of cell $n$ are denoted with the vectors $\bar {V}_n$ and $\bar {I}_n$ respectively. Meanwhile, the frequency harmonics of charge and voltage on the varactor diode in cell are denoted $\bar {Q}_n$ and $\bar {V}_{d;n}$ respectively.
The voltage and current harmonics at the input of the unit cell are related to the equivalent current source, $\tilde I_{s;n}$, as well as the voltage and current harmonics at the output of the unit cell as
The matrix equation above can be used to generate updated solutions $\left [ \begin {smallmatrix} \tilde V_n \\ \tilde I_n \end {smallmatrix}\right ]$ for a given equivalent current source ${\tilde I}_{s;n}$ and time delay $t_0$. This solution, in turn, can be used to calculate the voltage seen by the diode and update ${\tilde I}_{s;n}$ as well as ${\overline {\overline Y} }_n$, both of which are dependent on $C_{d;n}(t)$.
Fig. 3. An equivalent representation of a single unit cell of the nonlinear periodic electrical network shown in Fig. 2. The varactor diode has been replaced by a parallel combination of a current source and an admittance matrix. Note that the current source is now exciting a linear equivalent network.
Computing the soliton solution using (10) requires a known time-dependent diode capacitance $C_{d;n}(t)$. This capacitance is, itself, dependent on the voltage drop across the diode. Therefore, in addition to (10), the soliton solution must also satisfy ${C_{d;n}}\left ( t \right ) = {C_d}\left ( {{v_{d;n}}(t)} \right )$, where ${{v_{d;n}}(t)}$ is computed as
The circular dependence between (10) and (11) forms the basis for the proposed iterative procedure depicted in Fig. 4. At the beginning of the algorithm, an initial guess is assumed for the voltage drop ${{v_{d;n}}(t)}$ over the diode in cell $n$. This voltage is used to compute ${\overline {\overline Y} }_n$ that defines the (now linear) equivalent network depicted in Fig. 3. The current source, $i_{s;n}(t)$, is also computed to be used as an excitation for the equivalent network subject to the boundary condition specified in (9). Next, a solution is obtained using (10) for the voltage and current at the input of cell $n$. Finally, this solution is substituted into (11) to compute ${{v_{d;n}}(t)}$ for the next iteration. The algorithm continues until the change in the soliton solution between two successive iterations is sufficiently small. In the realization of the algorithm reported in this work, the convergence criterion is stipulated in terms of the $l^2$-norm of the difference in the complex power spectrum between iterations ($i$) and ($i$-$1$). The error between any two solutions $A$ and $B$ is defined
Two key parameters required in formulating the proposed algorithm are the dc voltage, $V_{dc}$, and the soliton delay, $t_0$. In practical implementations of the algorithm, the dc voltage can be specified by the initial guess for $v_{d;n}(t)$. The solver developed in this work to validate the proposed algorithm uses a Gaussian pulse train as the initial guess $v_{d;n}^{(0)}(t)$. A Gaussian pulse and its Fourier transform take the form
Setting $l=0$ yields the dc value of the voltage to be included in the iterative algorithm: ${V_{dc}} = \sqrt {2\pi } \left ( {{V_p} - {V_\infty }} \right )\frac {{{\sigma _t}}}{T} + V_\infty$. The delay of the soliton mode, $t_0$, is directly related to its velocity, $\upsilon _w$, and the size of the unit cell, $d$, as $t_0= d/\upsilon _w$. The velocity of soliton modes is impacted by both the dispersion and nonlinearity of the unit cell. This implies that a range of soliton delays will lead to nontrivial solutions. A guess for a particular value of $\upsilon _w$ that supports a soliton mode can be made using the small signal model of the unit cell. In this work, an initial guess of $\upsilon _w=\upsilon _g (\bar {v}_{d;n}, \omega )$ is assumed, where $\upsilon _g (\bar {v}_{d;n}, \omega )$ is the small signal group delay at angular frequency $\omega$ when the unit cell is biased with a dc voltage $\bar {v}_{d;n}$. The value of $\bar {v}_{d;n}$ is computed by averaging the initial guess for $v_{d;n}(t)$ over the pulse width. Assuming a Gaussian pulse for the initial guess, the average value of $v_{d;n}(t)$ from $-3\sigma _t$ to $3\sigma _t$ is ${{\bar {v}}_{d;n}}\approx \sqrt {\frac {\pi }{18}}\left ( {{V}_{\rm {peak}}}-{{V}_{\infty }} \right )+{{V}_{\infty }}$.
4. Numerical simulations of nonlinear unit cells
In this section, the proposed iterative procedure is demonstrated for three representative examples of nonlinear unit cells. A solver based on the proposed technique is developed and validated using a commercial circuit solver: Keysight Advanced Design System (ADS). In the first example, an LC ladder network with an effective nonlinear capacitance (see Fig. 5(a)) is studied. Rather than a direct shunt connection of the varactor diode as shown in Fig. 2, the shunt networks contain series capacitors, $C_s$, dc biasing inductors, $L_{\rm {dc}}$, and parallel capacitors, $C_p$. In contrast to existing works on nonlinear LC ladder networks [46–50], the proposed algorithm accounts for the exact frequency dispersion of the unit cell. Further, the technique is able to accurately model the nonlinear dependence between the diode capacitance and the applied voltage. In the second example, the series inductor is exchanged with an ideal transmission line as shown in Fig. 5(b). This example highlights the ability of the proposed formulation to model unit cells with complex frequency dispersion. In the final example, the transmission lines in Fig. 5(b) are assumed to be conductor-backed co-planar waveguide, or grounded co-planar waveguide (CPWG). The CPWG lines are characterized through a full-wave simulation in Ansys High-Frequency Structure Simulator (HFSS), and integrated into the iterative technique to obtain the soliton mode.
Fig. 5. 1D nonlinear periodic electrical networks with spatially-discrete unit cells. (a) An LC ladder network consisting a series inductors ($L_0 /2$) and shunt nonlinear networks. (b) A transmission line loaded periodically by shunt nonlinear networks.
The circuit layout for the nonlinear shunt networks in all examples depicted in Fig. 5 are identical. The series capacitance $C_s$ is dominant with respect to $C_d$ when the voltage drop over both elements becomes sufficiently negative (pushing the diode towards forward bias). A large inductor, $L_{\rm {dc}}$, is introduced to allow a dc bias to be applied to the varactor diode through the series elements. Finally, the parallel capacitance $C_p$ is added to the network to allow the small signal phase delay and impedance of the network to be tuned. The varactor diodes are assumed to be MACOM MAVR-000120-1411 [51]. In the developed solver, the varactor diodes are modeled as a nonlinear capacitance, whose junction capacitance is given by ${C_d}\left ( {{v_d}} \right ) = A / {{{\left ( {{v_d} - {V_0}} \right )}^m}}$ in reverse bias, where $V_0$ is the contact potential of the diode, $m$ is the grading coefficient, and $A$ is a constant [52]. In the solver, this expression is modified to avoid the singularity in $C_d$ at $v_d = V_0$, and becomes
Each of the structures depicted in Fig. 5 are simulated using an ADS harmonic balance circuit solver in addition to the in-house solver. Prior to incorporating the SPICE diode model in ADS, a cascade of unit cells which include an ideal equation-based nonlinear diode model is simulated in order to validate the in-house solver. This ADS simulation consists of 40 cascaded nonlinear unit cells. As shown in Fig. 6, this finite structure is terminated with ${Z_{B}(l\omega )} \equiv \frac {{{V_{n,l}}}}{{{I_{n,l}}}}$ to maintain the boundary conditions at both ends of the cascade which mimic an infinite periodic network. The cells are excited by a voltage generator $v_g(t)=2 v_n (t)$. The complex power flowing into the terminating impedance is then extracted from the ADS simulation and compared to the results obtained from the in-house solver for $\frac {1}{2} V_{n,l} I_{n,l}^*$. Finally, the error between the solution obtained from the ADS simulation, (ADS), and the in-house solver based on the iterative technique, (I.T.), is computed using (12) as ${\rm {Err}}\left ( {(\rm {ADS}),(\rm {I.T.})} \right )$.
Fig. 6. A cascade of 40 unit cells terminated by Bloch impedance to maintain boundary conditions of infinite periodic network.
4.1 LC ladder network simulations
The LC ladder network shown in Fig. 5(a) was simulated using the proposed iterative technique and validated by an ADS harmonic balance simulation. One of the key advantages of the proposed technique is that only one unit cell is required in order to compute the soliton mode, dramatically reducing the computational cost. As a result, the formulation can be easily incorporated into an optimization routine in order to obtain a desired mode profile. In this work, the genetic algorithm optimization routine available in the MATLAB Global Optimization Toolbox is used such that spectral power in each of the first 5 harmonics is no less than −6 dB with respect to the fundamental frequency, $f=500$ MHz. The parameters obtained by the optimizer are shown in the first row of Table 1 (labeled "LC Ladder"). The time-domain waveform for the voltage at the input of cell $n$ corresponding to the solution to (10) obtained using the in-house solver is depicted in Fig. 7(a). Assuming the initial guess shown in Fig. 7(a), the algorithm terminates after 59 iterations with ${\rm {Err}}\left ( (i-1),(i)\right ) < 10^{-14}$. The power spectrum of the soliton mode is depicted in Fig. 7(b) and is compared with the power spectrum computed using ADS at the output of the final (i.e. the $40^{\rm {th}}$) unit cell. In both solvers, $L=70$ frequency harmonics are maintained in the simulation. The in-house solver is first compared to an ideal ADS simulation where an equation-based nonlinear component is utilized to model the varactor diodes in Fig. 5(a). As seen in Fig. 7(b), the two solvers are in good agreement and the error as defined in (12) between the in-house solver and ADS result is ${\rm {Err}}\left ( {{\rm {ADS}},{\rm {I}}{\rm {.T}}{\rm {.}}} \right ) = 3.27\times 10^{-5}$. Replacing the ideal, equation-based component for the varactor in the ADS simulation with the SPICE model specified in [53] degrades the agreement between the in-house solver and ADS simulation for higher-order frequency harmonics as shown in Fig. 7(b). This is likely due to the losses introduced by the varactor diode and the deviation from (15) when the varactor approaches forward bias. The error defined in (12) between the in-house solver and ADS result with the SPICE diode model is ${\rm {Err}}\left ( {{\rm {ADS}},{\rm {I}}{\rm {.T}}{\rm {.}}} \right ) = 4.87\times 10^{-2}$.
Fig. 7. Simulated soliton solution obtained for the structure depicted in Fig. 5(a) and described by the first row of Table 1. (a) Initial guess (black dashed line) and final solution (blue dashed line) of the time-domain voltage drop across the varactor diode $v_{d;n}$, and the final soliton mode solution at the output of the $n^{\rm {th}}$ cell (red solid line). (b) The power at each frequency harmonic $l\omega$ as computed by the in-house solver (blue circles), an ideal ADS simulation using an equation-based diode model (red $\times$s), and an ADS simulation which includes the SPICE diode model (black dots).
Table 1. Simulated Microwave Circuit Parameters
4.2 Nonlinear loaded ideal transmission line
The proposed method is applicable to a wide selection of electrical networks with arbitrary frequency dispersion. To highlight this generality, the loaded transmission line structure shown in Fig. 5(b) is simulated using the proposed iterative technique and validated by an ADS harmonic balance simulation. The parameters that describe the unit cell are obtained through the genetic algorithm routine available in MATLAB. The structure is optimized to obtain a solution such that the spectral power in each of the first 5 harmonics is no less than −6 dB with respect to the fundamental frequency, $f=500$ MHz. The parameters corresponding to this solution are provided in the second row of Table 1 (labeled "Loaded TL"). The time-domain waveform for the voltage at the input of cell $n$ corresponding to the solution to (10) obtained using the in-house solver is depicted in Fig. 8(a). Assuming the initial guess shown in Fig. 8(a), the algorithm terminates after 69 iterations with ${\rm {Err}}\left ( (i-1),(i)\right ) < 10^{-14}$. The power spectrum of the soliton mode is depicted in Fig. 8(b) and is compared with the power spectrum computed using ADS at the output of the final (i.e. the $40^{\rm {th}}$) unit cell. In both solvers, $L=70$ frequency harmonics are maintained in the simulation. The in-house solver is compared to ADS simulations with both an equation-based diode model as well as the SPICE diode model. As seen in Fig. 8(b), the solvers are in good agreement and the error as defined in (12) between the in-house solver and the ADS results using the equation-based nonlinear diode model is ${\rm {Err}}\left ( {{\rm {ADS}},{\rm {I}}{\rm {.T}}{\rm {.}}} \right ) = 4.43\times 10^{-5}$. This error increases when the in-house solver is compared to the ADS results which include the SPICE diode model to ${\rm {Err}}\left ( {{\rm {ADS}},{\rm {I}}{\rm {.T}}{\rm {.}}} \right ) = 2.02\times 10^{-1}$.
Fig. 8. Simulated soliton solution obtained for the structure depicted in Fig. 5(b) and described by the second row of Table 1. (a) Initial guess of time-domain voltage drop across the varactor diode $v_{d;n}$ (black dashed line), final solution (blue dashed line) and the final soliton mode solution at the output of the $n^{\rm {th}}$ cell (red solid line). (b) The power at each frequency harmonic $l\omega$ as computed by the in-house solver (blue circles), an ideal ADS simulation using an equation-based diode model (red $\times$s), and an ADS simulation which includes the SPICE diode model (black dots).
The solution obtained by the proposed method fully accounts for the frequency dispersion of the unit cell. Given the soliton mode, an effective shunt time-varying capacitance can be defined for the two combined nonlinear networks in Fig. 5(b) as $C_{\rm {tot}}(t)\equiv 2 C_p + 2/(C_s^{-1}+C_{d;n}(t)^{-1})$. The time-average value of this capacitance for the solution in Fig. 8(a) is $C_{\rm {av}}=1.93$ pF. Together with the transmission line parameters reported in the second row of Table 1, $C_{\rm {av}}=1.93$ pF corresponds to a cutoff frequency of $2.63$ GHz [54]. This lies in-between the $5^{\rm {th}}$ and $6^{\rm {th}}$ harmonic of the fundamental frequency, meaning that several simulated harmonics fall within the stopband of the approximate linear model of the unit cell. The proposed method allows the frequency dispersion to be leveraged and even designed such that power is concentrated within desired frequency harmonics and reduced outside of the targeted bandwidth.
4.3 Nonlinear loading of grounded co-planar waveguide
As a final example, the nonlinear loaded transmission line in Fig. 5(b) is simulated using a full-wave model for the transmission line interconnects. The transmission lines are assumed to be grounded co-planar waveguides (CPWGs), and a single unit cell of the structure is shown in Fig. 9. The scattering parameters corresponding to the CPWG interconnects is extracted in a full-wave simulation in HFSS and integrated with the in-house solver and ADS simulation. Due to the symmetry of the structure, the computational domain can be reduced by placing perfect magnetic conductor on the $y$-$z$ plane, and simulating only the half space $x>0$. Wave ports are placed on either end of the unit cell in HFSS. These ports are ultimately subjected to the boundary condition in (9) within the in-house solver. Meanwhile, the lumped ports provide a connection to the nonlinear shunt networks. These ports are loaded by the admittance matrix defined in (6) and excited by the equivalent source $i_{s;n}(t)$. Parameters $w$, $g$, $h$, $d$, and $\epsilon _r$ reported in the third row of Table 1 (labeled "Loaded CPWG") are designed to provide a line impedance $Z_h=68.7$ and delay $\tau =49.3$ ps. With the geometry fixed, parameters $C_s$, $C_p$, $V_{\rm {dc}}$, and $t_0$ are optimized to obtain a solution such that spectral power in each of the first 5 harmonics is no less than than −6 dB with respect to the fundamental frequency, $f=500$ MHz. The time-domain waveform for the voltage at the input of cell $n$ corresponding to the solution to (10) obtained using the in-house solver is depicted in Fig. 10(a). Assuming the initial guess shown in Fig. 10(a), the algorithm terminates after 51 iterations with ${\rm {Err}}\left ( (i-1),(i)\right ) < 10^{-14}$. The power spectrum of the soliton mode is depicted in Fig. 10(b), and is compared with the power spectrum computed using a co-simulation of ADS and HFSS at the output of the final (i.e. the $40^{\rm {th}}$) unit cell. In both solvers, $L=100$ frequency harmonics are maintained in the simulation. The in-house solver is compared to ADS/HFSS co-simulations with both an equation-based model as well as the SPICE model for the varactor diodes. As seen in Fig. 10(b), the solvers are in good agreement and the error as defined in (12) between the in-house solver and the ADS/HFSS results using the equation-based nonlinear diode model is ${\rm {Err}}\left ( {{\rm {ADS}},{\rm {I}}{\rm {.T}}{\rm {.}}} \right ) = 8.42\times 10^{-5}$. This error increases when the in-house solver is compared to the ADS/HFSS results which include the SPICE diode model to ${\rm {Err}}\left ( {{\rm {ADS}},{\rm {I}}{\rm {.T}}{\rm {.}}} \right ) = 1.17\times 10^{-1}$.
Fig. 9. A unit cell of a periodic electrical network consisting of CPWG transmission lines loaded by two shunt nonlinear circuits. The scattering parameters corresponding to the CPWG line are extracted from a full-wave simulation using HFSS. These scattering parameters are then imported into the in-house solver or ADS to compute the soliton modes.
Fig. 10. Simulated soliton solution obtained for the structure depicted in Fig. 9 and described by the third row of Table 1. (a) Initial guess of time-domain voltage drop across the varactor diode $v_{d;n}$ (black dashed line), final solution (blue dashed line) and the final soliton mode solution at the output of the $n^{\rm {th}}$ cell (red solid line). (b) The power at each frequency harmonic $l\omega$ as computed by the in-house solver (blue circles), an ideal ADS/HFSS co-simulation using an equation-based diode model (red $\times$s), and an ADS/HFSS co-simulation which includes the SPICE diode model (black dots).
The example in Fig. 9 illustrates the broad scope of the proposed iterative technique. By incorporating full-wave simulations of the surrounding geometry, soliton modes that are supported by a wide variety of structures with general three-dimensional features can be obtained. In contrast to previous works, only a single cell needs to be characterized in order to determine the solution throughout space. For example, the current supported on the conductor within the top layer of the structure in Fig. 9 is plotted in Fig. 11. This is constructed by computing the excitation within a single unit cell at all frequency harmonics, $l\omega$. The current in neighboring unit cells is then obtained by weighting the solution by a phase of $-l\omega t_0$. While the soliton mode repeats in space after $T/t_0=1/ft_0=24.75\approx 25$ unit cells, the required computational domain (the shaded yellow box in Fig. 11) contains only half of a single unit cell.
Fig. 11. Snapshot in time of the surface current, $\vec {J}_s$, on the top metal layer of the structure depicted in Fig. 9 corresponding to the soliton mode computed by the in-house solver. Note that the computational domain is reduced with respect to the spatial period of the soliton by a factor of $T/t_0\approx 25$.
5. Conclusion
In this work, an iterative technique for computing soliton modes supported by general nonlinear electrical networks is reported. This procedure is implemented in an in-house solver and validated for three representative cases: 1) a lumped element LC ladder network, 2) an ideal transmission line loaded by nonlinear capacitors, and 3) a co-planar waveguide loaded by nonlinear capacitors. All three cases include a non-traditional nonlinear network as a shunt element. The procedure is validated using a commercial harmonic balance circuit solver and shown to be in good agreement for all cases. It is shown that the proposed technique can be applied to a broad class of electrical networks with arbitrary frequency dispersion. Further, the procedure iteratively solves for soliton modes in a single unit cell, making it computationally efficient. This feature allows designers to rapidly tune the parameters of a nonlinear guided-wave structure to obtain an optimal pulse profile without the need to model the entire structure.
Acknowledgments
The authors would like to thank Prof. Zoya Popovic for her thoughtful discussions and ideas.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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