Exploration of the effect of fractional elements in nonlinear transmission lines

The paper is concerned with the effect of the inclusion of fractional elements on the behaviour of nonlinear transmission lines (NLTLs). NLTLs are an important element of high-frequency circuit design as they have many applications including high-power RF pulse generators (Nikoo et al., 2018), electrical oscillators (Ricketts et al., 2006, 2019), pulse compressors in radar systems amongst others (Nouri et al., 2017) and medical applications (Abe and Minamitani, 2012; Gardner et al., 2024). Nonlinear lumped element transmission lines with nonlinear capacitors, nonlinear inductors and hybrid lines with both nonlinear capacitors and inductors have been investigated by several authors (e.g. Kuek et al., 2012; Fairbanks et al., 2020). At high frequencies, fractional elements may be considered more realistic models for capacitors and inductors (R-Smith et al., 2017; Liu et al., 2016 and Antonini et al., 2021). Fractal derivatives and local fractional derivatives as distinct from non-local fractional derivatives are also encountered in relation to circuits and transmission lines (Wang and Liu, 2023a; Wang, 2023; Wang et al., 2023). The differences and connections between the fractal and fractional form of derivatives are discussed in Deppman et al. (2023). This paper shall explore the effect on the dynamic behaviour of the NLTL if fractional effects are included. Work in this area has largely focussed on determining mathematical solutions to partial differential equations used to model the transmission line for certain forms of nonlinearity such as Iqbal (2021), Aziz et al. (2020) and Paul et al. (2023). Wang et al. (2023) also examine the partial differentiation equation used to model the NLTL under certain assumptions. They explore the effect of local fractional derivatives and present interesting results. In the present contribution, realistic nonlinear capacitor models shall be used to explore how nonlocal fractional derivatives affect transient and steady-state behaviour. There are several forms of fractional derivative, for example, the Riemann-Liouville, the Marchaud and the Caputo derivatives (de Oliveira and Machado, 2014). However, the present work shall use the Caputo fractional derivative. This is because the Caputo derivative does not require fractional-order initial conditions (Garrappa et al., 2019). The Laplace transform of the Caputo derivative also is consistent with the usual definition for integer-order derivatives (Gómez-Aguilar et al., 2014). The paper shall consider analysis of low-pass NLTLs with fractional nonlinear capacitors, fractional-order inductors and when both types of element are governed by fractional differential equations. Changes in the transient results and on soliton behaviour shall be presented first. Several observations shall be commented on. The effect on steady-state behaviour when the amplitude of sinusoidal inputs is varied shall also be explored. Small changes in the fractional order are considered to show the corresponding significant changes in behaviour. The values are also in line with the practical values determined from experimental work given in Shang et al. (2013).

Many approaches have been proposed for simulating fractional-order differential equations, for example, spectral methods (Papadopoulos and Olver, 2024), spectral collocation approaches (Zayernouri and Karniadakis, 2014) and methods using Mittag–Leffler functions (Garrappa et al., 2017). Each has their merits. In this paper, the product-integration trapezoidal approach of Garrappa (2018) is used owing to its stability properties and ease of implementation for large nonlinear systems. However, all methods should yield similar results if the accuracy requirements are the same. While the speed of the simulation is not the focus of the paper, it should be mentioned that numerical simulation of fractional differential equations requires considerable care and various suggestions for improving simulation times have been suggested in Diethelm et al. (2020).

The findings of the paper shall be useful for design and analysis of systems involving NLTLs in communications and medical applications. It also permits exploration of methods to control the behaviour of such systems. It is educational as it exhibits the significant changes in dynamical behaviour when fractional derivatives are included.

Consider the lowpass NLTL shown in Figure 1 which is similar to that in Kuek et al. (2012). n is the number of sections in the transmission line. R_L is the series resistance of a section, L is the inductance of a section, R_gen is the resistance of the source and R_load is the resistance of the load.

The nonlinear capacitors are modelled using the following equation:

where v is the voltage across the capacitor, and a and b are constants. C₀ is the capacitance with zero voltage. This is a realistic model for nonlinear capacitors (Nikoo and Hashemi, 2017; Kuek et al., 2012). The model parameters are determined experimentally in Kuek et al. (2012).

As in Kuek et al. (2012), the equations governing the circuit are as follows:

where i = 1,2,…, N − 2.

The equations for the final section are:

The Caputo fractional derivative is defined as follows:

where α is the fractional order, m = [α] is the smallest integer greater or equal to α and y^(m)(t) is the mth derivative of y(t), y(t) is such that y^(m−1)(t) is absolutely continuous.

The equations for the ith section of the low-pass NLTL when fractional effects are considered are:

These equations indicate how the fractional order α relates to the circuit in Figure 1.

The product-integration implicit trapezoidal rule for fractional differential equations given in Garrappa (2018) is used to simulate the equations. The rule is as follows. For a differential equation:

where m = [α] and y^k(t₀) is the kth derivative of y(t).

where y_n is the numerical solution at the nth time step, h is the time step and T_m−1(t) is the Taylor polynomial of degree m for y(t) centred at t₀. The method is based on reformulating the differential equation as a Volterra integral equation and subsequently discretising it:

f(τ,y(τ)) is then approximated with a first-order polynomial:

This method is chosen owing to the improved stability of implicit methods and its suitability for large nonlinear systems of equations. Newton’s method is used to solve the nonlinear equations. The order of convergence of the implicit trapezoidal method is 1 + α or |y(t_n) − y_n| = O(h^min{1+α,2}) (Garrappa, 2018).

The following are the parameter values used for the current studies unless otherwise stated:

The input is a rectangular pulse with a duration of 400 ns. There are zero initial conditions, V_1..N(t₀), I_1..N (t₀), $V1..N′(t0)$⁠,$I0..N′(t0)=0$⁠.

The first study shall examine the variation in the load voltage when fractional effects are included for each nonlinear capacitor. The differential equation governing the capacitors in Figure 1 is given by equation (10). Figure 2(a) shows the results for the load voltage when the order α is varied from 0.95–1.05. The value α = 1 corresponds to the integer derivative and is included to highlight the changes in the responses when non-integer values are employed. The same value of α is used in each section as each section of the NLTL is expected to be the same in practical applications (Kuek et al., 2012; Nouri et al., 2017).

A time step of h = 1 × 10⁻⁹ s was selected as it was sufficient for convergence. The time step is reduced until the absolute difference between two results is less than the specified tolerance.

Consider the first case when fractional-order capacitors are present and consider equation (11) and in particular, the term. $hα(a˜n(α)f0+∑j=1nan−j(α)f(tj,yj))$ Since h < 1 and if 0 < α < 1, $hαa˜n(α)$ and $hαan−j(α)$ increase as α decreases and the contribution of the state equation corresponding to the nonlinear capacitor is increased. This has a similar effect to a decrease in the capacitance value as the state equation $f=Ii−Ii+1C$ is inversely proportional to C. Hence, since the speed of a wave is proportional to $1C$⁠, the speed of the wave is increased. This is evident from the results with the shorter delay when α is reduced. Because the $a˜n(α)$ and $an−j(α)$ are dependent on n, even in the case of a linear capacitor, this is not equivalent to a fixed change in the value of C₀ in equation (1). Similarly, when fractional-order inductors are considered, the effect is similar to a decrease in the inductance and consequently, this leads to a shorter delay. When both fractional-order capacitors and inductors are present, both contribute to a reduction in the delay.

The fractional effects also increase the damping effect when 0 < α < 1 owing to the change in the values of the weighting coefficients $a˜n(α)$ and $an−j(α)$ when compared to those when α = 1. Consider the linear equation $dydt=−y$⁠. When the integer-order trapezoidal rule is applied with a time step h, the value of y at the first time step is:

When the fractional-order trapezoidal rule is applied:

Consider, for example, h = 0.01 and α = 1, $y1=1−h/21+h/2y0=0.990y0$⁠. Now consider the case for h = 0.01 and α = 0.95,$y1=1−hαa˜1(α)1+hαa0(α)y0=0.987y0$⁠. When fractional integration with α < 1 is used, there is increased damping of the output that continues for the subsequent time steps. This feature is also true for the NLTL transmission line equations as evidenced by the results in the previous figures. This result would also indicate that reducing α has a stabilizing effect. If one considers the relationship between y_n+1 and y_n as a difference equation, then the fractional effect tends to move the “pole” away from the unit circle. Further to this, when α > 1, fractional integration has the opposite effect. Consider h = 0.01 and α = 1.05, $y1=1−hαa˜1(α)1+hαa0(α)y0=0.992y0$⁠. When fractional integration with α > 1 is used, there is a decreased damping of the output that continues for the subsequent time steps. This decreased damping can be seen in Figure 2(a)–(c) where the output exhibits larger oscillations and a longer delay when α is increased beyond 1. Consequently, α > 1 has a destabilizing effect.

Figure 3 shows the evolution of a soliton for three values of α ∈ {0.95, 1.0, 1.05}. For α = 0.95 as shown in Figure 3(a), the output after 1(µs) exhibits smaller oscillations compared to the equivalent for α = 1.0 as illustrated in Figure 3(b). However, when α is increased beyond 1.0 to α = 1.05, the output after 1(µs) exhibits higher oscillations as shown in Figure 3(c). Hence, this demonstrates the increased damping and faster velocity of the wave with α < 1 and the reduction in damping and slower velocity of the wave with α > 1.

In this section, the effect of the inclusion of fractional derivatives on the nonlinear behaviour of the lowpass NLTL is investigated when the input voltage is a sinusoid of various amplitudes. This type of study was completed for a lossless transmission line in Balyakin and Ryskin (2001).

Figure 4(a) and (b), shows the output of the NLTL when the input is a 0.5 (MHz) sinusoid of unity amplitude. This frequency is well below the Bragg frequency $2LC$ of the NLTL. In Figure 4(a), α = 1, and hence, no fractional derivative is considered. The output voltage is periodic and repeats with the same period as the input sinusoid, i.e. every 2(µs). However, the output voltage is distorted with signals of a higher frequency superimposed on top of the expected output sinusoid, i.e. harmonic content is present on the output owing to the nonlinear capacitors. However, when fractional derivatives are considered in Figure 4(b) with α = 0.95, the output repeats every 2(µs) and no harmonic content is present. This demonstrates that when fractional derivatives with α < 1 are considered, the inclusion of the fractional derivative terms dampens out the harmonics and resonance that are present when α = 1.

Figure 4(c) and (d), examines the behaviour when the amplitude of the input sinusoid is varied from 0.1 to 1.8. When steady state is reached, the peak values of the output wave are noted and plotted against the corresponding value of the amplitude of the sinusoid. For example, consider Figure 4(b), there are five peaks presented all at V ≈ 1(V), these are denoted by the red stars. Hence, when they are plotted against the amplitude value of the sinusoid, a single point appears on the plot as they are all at the same voltage. This is the expected output when no harmonic content is present. Consider Figure 4(a), there are 20 peaks present, these are denoted by the red stars. These peaks appear at four different values. Hence, when they are plotted against the amplitude value of the sinusoid, four points appear on the plot. This is the expected output when harmonic content is present.

Figure 4(c) shows the behaviour of the low pass NLTL with α = 1.0 as the amplitude of the sinusoid varies. For low values, a single point can be seen on the plot. This indicates that no harmonic content is present in the output of the system. At an amplitude value of 0.75 (V), two points appear on the plot. At 0.8(V), a third point appears, and at 0.95 (V), a fourth point appears. Further to this, consider at an amplitude of 1 (V), the peak load voltage is at 1.76 (V) and the load voltage ripple is ∼3.1 (V). This demonstrates that as the amplitude of the sinusoid increases, harmonic content is introduced onto the output of the system. This distorts the desired output voltage and increases the peak-to-peak load voltage ripple.

Figure 4(d) shows the behaviour of the low pass NLTL with α = 0.95 as the amplitude of the sinusoid varies. For low values, a single point can be seen on the plot. Harmonic content is not introduced until the amplitude reaches 1.45 (V). When the input sinusoid has an amplitude of 0.75 (V), 0.8 (V) and 0.95 (V), a single point can be seen. Unlike when α = 1.0, no harmonic content is present. At an amplitude of 1 (V), the load voltage oscillates between approximately −1(V) and +1(V). Hence, the fractional-order terms (α ≠ 1) introduce damping into the NLTL and there is no significant increase in the peak-to-peak load voltage ripple.

Figure 4(a)–(d) demonstrates that the inclusion of fractional derivatives dampens the output of the system when sinusoidal inputs are considered.

The paper has explored the effect on the transient and steady-state behaviour of a practical NLTL when fractional derivatives are included. The fractional derivatives result in an increase in the speed of the wave and in increased damping when α < 1 and a reduction in speed and an amplification of the harmonics when α > 1. Relatively small deviations from the integer derivative (α = 1) lead to a significant change in behaviour. These features are important in the design, analysis and stability assessment of NLTLs for use in high-speed measurement and high-frequency generation which have applications in communications and medical devices. Accurate models can result in more effective designs with resultant cost benefits.