On the Behavior of a Non-Linear Bandpass Filter with Self Voltage-Controlled Resistors

Abstract

In this work, we explore the behavior of a classical RLC resonance-based bandpass filter, which includes two resistors (one of which is associated with a non-ideal inductor), when either of these resistors is self voltage-controlled. In particular, self-feedback control is achieved by using the voltage developed across the inductor or the capacitor to dynamically change the value of the controlled resistor. This results in a multiplication-type non-linearity, which transforms the linear filter into a non-linear filter described by a set of non-linear differential equations. When gradually increasing the strength of the non-linearity, a notch-like behavior is observed at twice the resonance frequency. However, the non-linear filter can lose its stability with excessive feedback. Simulations and experimental results are provided to support the theory.

1. Introduction

The study of non-linear resonance is a fundamental topic in circuit theory due to its numerous applications [1,2,3]. Non-linear resonators are used to widen the bandwidth of wireless power transfer systems [4] and reduce their sensitivity to position mismatch [5] because they can have more than one peak power frequency and can also show hysteresis behavior [6,7]. Non-linear resonance networks have also been widely used to model oscillatory systems such as the Van der Pol and Duffing oscillators [8,9], among others. However, little attention has been given to the filtering behavior and applications of these networks, as briefly demonstrated in our recent work [10]. In general, non-linear analog filters are not widely used, although some work has demonstrated their value in noise suppression [11,12,13,14]. Non-linear analog filters should not be confused with analog filters realized using non-linear analog circuits such as Log-domain circuits [15].

It is important to recall that non-linear resonators have been playing an increasingly important role in advancing wireless power transfer technologies and maintaining high power transfer efficiency despite changes in environmental parameters, such as the distance between transmitter and receiver or their relative alignment. Traditional linear resonators often show strong sensitivity to these variations, with large associated losses in power transfer efficiency. By contrast, non-linear resonant circuits, like parity-time symmetric or Duffing resonators, are quite robust and can intrinsically accommodate parameter variations. For example, in [16], a non-linear parity-time-symmetric circuit utilized in a wireless power transfer system shows that an operation window is quite wide, wherein efficiency remains close to unity without the need for frequency tuning or internal coupling parameters. The result is that such systems become highly valuable in dynamic environments, where the relative positioning of transmitter and receiver changes, like in wireless charging of moving vehicles or medical devices. Additionally, the non-linear Duffing-based resonators can be integrated into wireless power transfer systems to improve their operational bandwidth [17]. This type of non-linear resonator has the ability to maintain a wide bandwidth and attaining a similar amplitude level compared to linear resonators, without the associated frequency sensitivity.

It is thus concluded that non-linear circuits play an important role in compensating for problems created by frequency detuning in wireless power transfer systems. Detuning can result from system parameter variations, such as inductance or capacitance, which result in a loss in transmission efficiency. In particular, non-linear resonant circuits developed to counter the influence of such effects are mainly based on the Van der Pol equation. These circuits exhibit a broadened frequency response that allows them to maintain high output power even when the working frequency is far from the resonant frequency. These make them suitable for applications in which a fine control of the frequency is difficult to realize or in systems that go through different loads and environmental conditions [18,19].

It is important to recall that a large number of non-linear resonators are modeled in normalized form by the forced Liénard second-order differential equation [20]

$$\ddot{x}+f\left(x\right)\dot{x}+g\left(x\right)=f\left(t\right).$$

(1)

The classical linear resonance network is described by (1) with $f\left(x\right)=1/Q_r$ and $g\left(x\right)={\omega }_r^{2}x$, where $Q_r$ is the quality factor of the resonance network and ${\omega }_r$ is its resonance frequency.

In this work, we focus our attention on a classical RLC bandpass filter and use self feedback to induce non-linearity, resulting in a non-linear bandpass filter. In particular, the voltage $v_L$ developed across the inductor or the voltage $v_C$ across the capacitor are used to dynamically change the value of the two resistors in the filter structure. This yields four possible non-linear filters, as depicted in Figure 1. However, none of these non-linear filters can be described by (1) as opposed to the circuits in [10]. In the filter of Figure 1a, the value of the resistance $r_L$ is controlled via the capacitor voltage $v_C$, whereas in Figure 1b the feedback control uses the inductor voltage $v_L$. In Figure 1c,d, it is resistance R that is feedback-controlled. We show that as the strength of the feedback control voltage increases (i.e., magnitude of the non-linear term increases), a notch-like behavior is observed in the magnitude response of the bandpass filter at twice the resonance frequency, and a second notch develops at three times the resonance frequency with further increase in non-linearity. This notch behavior appears due to the inter-modulation terms resulting from the multiplication non-linearity. The non-linear filter’s quality factor, and hence its bandwidth, is also shown to be modulated by the non-linearity. However, these non-linear filters have a narrow stability range and are not always stable. Numerical simulations are used to study the filters and experimental results support the predicted theory.

The work is organized as follows: the circuit analysis is performed in Section 2, while design examples are given in Section 3 and the obtained experimental results confirm the finding of this work.

2. Circuit Analysis

Consider the circuit shown in Figure 1a, which is a classical bandpass filter based on a parallel LC resonance network. The transfer function of this filter is given by:

$$H_L\left(s\right)={\displaystyle \frac{Qs+ϵ}{s^2+(Q+\frac{ϵ}{Q})s+(1+ϵ)}}={\displaystyle \frac{1}{1+\frac{s}{Q}+\frac{1}{Qs}}}{|}_{ϵ=0},$$

(2)

where we define $1/Q=R\sqrt{C/L}\mathrm{and}ϵ=r_L/R$. This transfer function reduces to an ideal bandpass filter when the inductor is ideal and hence $r_L$ tends to zero (i.e., $ϵ\to 0$). Note that $s=j{\omega }_n$ in (2) is normalized with respect to the resonance frequency ${\omega }_r=1/\sqrt{LC}$ (${\omega }_n=\omega /{\omega }_r$), hence setting the filter’s ideal center frequency at ${\omega }_n=1$.

2.1. Voltage-Controlled $r_L$

Now, let us consider that $r_L$ is replaced with a physical resistance composed of a fixed part $r_f$ and a voltage-controlled part $r_v$; i.e.,

$$r_L=r_f\pm r_v\left({\displaystyle \frac{v_C}{v_{ref}}}\right),$$

(3)

where $v_C$ is the voltage on the capacitor and $v_{ref}$ is an arbitrary reference voltage. This change in the circuit structure transforms the filter into a non-linear filter, which can be seen by writing the state-space model of the circuit (see Figure 1a):

$$C{\displaystyle \frac{d{v}_C}{dt}}={\displaystyle \frac{v_i-v_C}{R}}-i_L,L{\displaystyle \frac{d{i}_L}{dt}}=v_C-i_L{r}_L,$$

(4)

where $i_L$ is the inductor current. We introduce the dimensionless variables $x=v_C/v_{ref},$ $y=i_{L}R/v_{ref},{ϵ}_1=r_f/R,{ϵ}_2=r_v/R$, transforming (4) into:

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{Q}}\dot{x}=-x-y+f\left(t\right)\end{array}$$

(5)

$$\begin{array}{c}\hfill Q\dot{y}=x-{ϵ}_{1}y\mp {ϵ}_{2}xy\end{array}$$

(6)

where $f\left(t\right)=v_i/v_{ref}$ and time is normalized as $t=t·{\omega }_n$. The non-linearity in (6) appears in the multiplication term ${ϵ}_{2}x·y$, which will disappear only when ${ϵ}_2=0$, i.e., $r_L$ becomes a fixed resistance. Numerical simulation results of the above set of differential equations with $f\left(t\right)=Asin\Omega t$ are plotted in Figure 2 for $Q=A=\Omega =1,{ϵ}_1=0.1$ and for four different values of ${ϵ}_2$, namely ${ϵ}_2=(0,0.1,0.5,1)$ (see first two columns of the figure). It is clear from these results that a stable limit cycle is observed in all cases, indicating that the filter remains stable despite the increasing strength of the non-linear term ${ϵ}_2$. The filter remains stable until ${ϵ}_2\approx 2$, and beyond this value, the trajectories diverge.

The linear filter transfer function (2) can be retrieved only when ${ϵ}_2=0$. When ${ϵ}_2\ne 0$, we obtain the following transfer function:

$$H_N^{-1}\left(s\right)=1+{\displaystyle \frac{s}{Q}}+{\displaystyle \frac{1}{Qs+{ϵ}_1}}\mp \left({\displaystyle \frac{{ϵ}_2}{Qs+{ϵ}_1}}\right)\left({\displaystyle \frac{\mathcal{L}x\left(t\right)y\left(t\right)}{X\left(s\right)}}\right),$$

(7)

where $\mathcal{L}x\left(t\right)y\left(t\right)$ is the Laplace transform of the non-linear term. Note that (7) for ${ϵ}_1=0$ (i.e., ideal inductor) can also be re-written as:

$$H_N^{-1}\left(s\right)=1+{\displaystyle \frac{s}{Q}}+{\displaystyle \frac{1}{Q_N\left(s\right)s}},$$

(8)

where $Q_N\left(s\right)=Q/(1\mp {ϵ}_2{\displaystyle \frac{\mathcal{L}x\left(t\right)y\left(t\right)}{X\left(s\right)}})$. When compared to the linear bandpass filter transfer function (2), it is clear that the non-linear term modulates the quality factor of the filter. To evaluate the magnitude and phase responses of (7), we perform the following steps:

• solve numerically the differential equations of the system for different values of ${ϵ}_2$;
• obtain the best fit expressions for $x\left(t\right),y\left(t\right)$, and hence compute $x\left(t\right)·y\left(t\right)$;
• find $X\left(s\right)$ and $\mathcal{L}x\left(t\right)y\left(t\right)$ from the best fit expressions;
• use (7) to obtain the magnitude and phase responses of $H_N\left(s\right)$.

Following the above procedure, we show in Figure 2 (columns 3 and 4) plots of the filter’s magnitude and phase responses when ${ϵ}_2=(0,0.1,0.5,1)$. We note that the ideal bandpass response with center frequency located at the normalized value ${\omega }_n=1$ is observed. However, for higher values of ${ϵ}_2$, a notch-like behavior appears at twice the normalized center frequency, i.e., at ${\omega }_n=2$ when ${ϵ}_2=(0.1,0.5)$ and then a second notch appears at ${\omega }_n=3$ when ${ϵ}_2=1$ (very strong non-linearity). Sharp changes can also be seen in the phase response at these critical frequencies. To verify this behavior further, an input signal $f\left(t\right)=sin\left({\omega }_{n}t\right)+sin\left(2{\omega }_{n}t\right)$ was used with ${\omega }_n$ varied in the range $(0.1-10)$. At each frequency, we isolated the first and second harmonics of $x\left(t\right)$, measuring their amplitudes. The results are plotted in Figure 3, where, consistently, we observed an attenuation in the amplitude of the second harmonic that intensifies with increasing ${ϵ}_2$.

In

Table 1. Transfer functions obtained using (7), corresponding to the magnitude and phase responses plotted in Figure 2 when ${ϵ}_1=0.1$.

${\mathit{ϵ}}_2=$	Transfer Function
0	${\displaystyle \frac{s+0.1}{s(s+1.1)+1.1}}$
0.1	${\displaystyle \frac{(s+0.1)\left(\frac{0.91}{s^2+1}+\frac{0.004}{s}\right)}{(s(s+1.1)+1.1)\left(\frac{0.91}{s^2+1}+\frac{0.004}{s}\right)-0.1\left(\frac{0.041}{s}-\frac{0.82}{s^2+4}\right)}}$
0.5	${\displaystyle \frac{(s+0.1)\left(\frac{0.90}{s^2+1}+\frac{0.019}{s}\right)}{(s(s+1.1)+1.1)\left(\frac{0.90}{s^2+1}+\frac{0.019}{s}\right)-0.5\left(\frac{0.041}{s}-\frac{0.79}{s^2+4}\right)}}$
1	${\displaystyle \frac{(s+0.1)\left(\frac{0.88}{s^2+1}+\frac{0.04}{s}\right)}{(s(s+1.1)+1.1)\left(\frac{0.88}{s^2+1}+\frac{0.04}{s}\right)-\left(\frac{0.043}{s}-\frac{0.726}{s^2+4}\right)}}$

, the corresponding filter transfer functions $H_N\left(s\right)$ (obtained using (7)) in each case are provided. It is important to mention here that the magnitude and phase of $H_N\left(s\right)$ cannot be experimentally obtained using classical frequency-sweep analyzers. This is confirmed later in the experimental results section.

Now, consider the circuit shown in Figure 1b, where the inductor voltage $v_L$ is used to control the resistor $r_L$ (recall Equation (3)) instead of the capacitor voltage $v_C$. It can be shown that the circuit is hence described by the equations:

$${\displaystyle \frac{1}{Q}}\dot{x}=-x-y+f\left(t\right),Q\dot{y}={\displaystyle \frac{x-{ϵ}_{1}y}{1\pm {ϵ}_{2}y}}.$$

(9)

Numerical simulations of these differential equations with $f\left(t\right)=Asin\Omega t$ are plotted in Figure 4 when $Q=A=\Omega =1,{ϵ}_1=0.1$ and for three different values of ${ϵ}_2$, namely ${ϵ}_2=(0,0.1,0.5)$ (see first two columns of the figure). A stable limit cycle is observed, indicating that the filter is stable, until ${ϵ}_2\approx 0.65$, where the trajectories diverge and the filter becomes unstable. The stability range of this filter is therefore narrower than the one of Figure 1a, and its non-linear transfer-like function is given by

$$H_N^{-1}\left(s\right)=H_L^{-1}\left(s\right)\pm \left({\displaystyle \frac{{ϵ}_2}{s+{ϵ}_1/Q}}\right)\left({\displaystyle \frac{\mathcal{L}y\left(t\right)\dot{y}\left(t\right)}{X\left(s\right)}}\right).$$

(10)

Following a similar procedure to that described above, in Figure 4, a plot of the non-linear filter’s magnitude and phase responses compared to the linear filter’s response (${ϵ}_2=0$) is shown. Again, it is noted that a notch-like response appears at twice the normalized resonance frequency as ${ϵ}_2$ is increased with a corresponding change in the phase response.

2.2. Voltage-Controlled R

Consider the circuit shown in Figure 1c, where $v_C$ controls the resistor R, which is composed of a fixed part $R_f$ and a variable part $R_v$ such that:

$$R=R_f\pm R_v\left({\displaystyle \frac{v_C}{v_{ref}}}\right).$$

(11)

Defining $x=v_C/v_{ref},y=i_L{r}_L/v_{ref},{ϵ}_1=R_f/r_L,{ϵ}_2=R_v/r_L$, and $1/Q=r_L\sqrt{C/L}$, it can be shown that this filter is described by the set of equations:

$${\displaystyle \frac{1}{Q}}\dot{x}={\displaystyle \frac{f\left(t\right)-x}{{ϵ}_1\pm {ϵ}_{2}x}}-y,Q\dot{y}=x-y.$$

(12)

where the non-linearity is clearly of the form ${ϵ}_{2}x\dot{x}$ and ${ϵ}_{2}xy$. Therefore, it is too complicated to proceed forward with computing the transfer function for this non-linear system.

Similarly, for the circuit in Figure 1d, where $v_L$ controls resistor R, the describing equations are:

$${\displaystyle \frac{1}{Q}}\dot{x}={\displaystyle \frac{f\left(t\right)-x}{{ϵ}_1\pm {ϵ}_2(x-y)}}-y,Q\dot{y}=x-y.$$

(13)

The non-linearity in this case is also complex. We investigated the stability of the above two filters numerically when $Q=A=\Omega =1,{ϵ}_1=0.1$ and found that both have a very limited stability range with respect to increasing ${ϵ}_2$ (${ϵ}_2<0.15$). Figure 5 shows the numerical simulations of (13) for ${ϵ}_2=(0,0.1)$. The limited stability range can be improved at different values of Q, but this requires a more detailed study. In what follows, we focus our attention on the realization and experimental verification of the circuit in Figure 1a.

3. Circuit Design and Experimental Results

The non-linear bandpass filter in Figure 1a was realized using the circuit in Figure 6a. The resistor $r_L$ in that circuit is composed of $r_f$ in series with a MOSFET transistor, functioning as the variable resistor $r_v$. The voltage across the capacitor $v_C$ is sensed by an operational amplifier, which subsequently controls the resistance of the MOSFET. The measurement results of this circuit were carried out with $L=78$ mH (with a quality factor of 20 at 10 kHz), $C=3.2$ nF, and $R=5$ k$\Omega$. The LMC662 operational amplifier (powered by ±5 V supplies) was used and was configured with $R_1=R_2=$100 k$\Omega$. An NMOS from a CD4007 transistor array was used and, for proper operation, a biasing circuit consisting of $C_b=1$ nF, $R_b=10$ k$\Omega$ and $V_b=2.5$ V was used.

The measurement setup used for circuit validation is illustrated in Figure 6a. In the circuit of Figure 6b, $r_f$ accounts for all fixed resistances, including a physical resistor of $100\Omega$ (needed to measure the inductor current), a parasitic resistor of $245\Omega$ associated with the non-ideal inductor and a fixed MOSFET resistance of $805\Omega$. These circuit components and biasing voltage values correspond to a non-linear filter with ${ϵ}_1=0.23$ and ${ϵ}_2=0.15$. Figure 6c shows the experimentally measured waveforms of $x\left(t\right)=v_C\left(t\right)$ and $y\left(t\right)=i_L\left(t\right)R$ with an input signal of $v_{in}\left(t\right)=0.5sin\left(2\pi {10}^{4}t\right)$.

Figure 7a shows the measured magnitude and phase responses of the circuit in Figure 6a. The transfer function $H\left(s\right)$ was directly obtained from data measured using a spectrum analyzer, while $H_N\left(s\right)$ was obtained by evaluating (7) with the measured waveforms of $x\left(t\right)$ and $y\left(t\right)$. This result shows a notch-like behavior around twice the center frequency, as theoretically expected. The notch-like behavior was not captured by the spectrum analyzer, which employs a frequency sweeping method, because the transfer function (7) of the non-linear filter is dependent on both $X\left(s\right)$ and $\mathcal{L}x\left(t\right)y\left(t\right)$. Thus, the notch-like behavior can only be identified through post-processing the measured time-domain data of $x\left(t\right)$ and $y\left(t\right)$ or alternatively using a spectrum analyzer with a wide-band excitation signal rather than with frequency sweep. Figure 7b illustrates the bandwidth dependency of the non-linear filter on the input signal amplitude. Specifically, the 3-dB bandwidths were found to be 7.17 kHz, 7.30 kHz, and 7.43 kHz for input signal amplitudes of 1 V, 0.5 V, and 0.1 V, respectively. The filter bandwidth decreases with increased amplitude.

The transfer functions in Table 1, which correspond to the cases of ${ϵ}_1=0.1$ and ${ϵ}_2=0,0.5,1$ were also implemented using the Field Programmable Analog Array (FPAA) AN231E04 device provided by Anadigm [21]. The clock frequency is equal to $f_{clk}$ = 250 kHz. Using the Anadigm Designer^® ver.2 software provided by Anadigm [21], the resulting design for implementing the transfer function for ${ϵ}_2=0$ is depicted in Figure 8a, where the FilterBiquad Configurable Analog Modules (CAMs) have been employed. This originates from the fact that this biquadratic transfer function is decomposed as as sum of second-order band-pass and low-pass filter functions

$$H\left(s\right)={\displaystyle \frac{G_{BP1}\left(\frac{2\pi f_{01}}{Q}\right)s+G_{LP1}{\left(2\pi f_0\right)}^2}{s^2+\frac{2\pi f_{01}}{Q_1}s+{\left(2\pi f_{01}\right)}^2}}.$$

(14)

Employing the partial fraction expansion tool for decomposing the associated transfer functions that correspond to the cases of $=0.5,1$, the resulting simplified expression is

$$H\left(s\right)={\displaystyle \frac{G_{BP1}\left(\frac{2\pi f_{01}}{Q}\right)s+G_{LP1}{\left(2\pi f_{01}\right)}^2}{s^2+\frac{2\pi f_{01}}{Q_1}s+{\left(2\pi f_{01}\right)}^2}}+{\displaystyle \frac{G_{BP2}\left(\frac{2\pi f_{02}}{Q_2}\right)s+G_{LP2}{\left(2\pi f_{02}\right)}^2}{s^2+\frac{2\pi f_{02}}{Q_2}s+{\left(2\pi f_{02}\right)}^2}}+{\displaystyle \frac{G_{LP3}\left(2\pi f_{03}\right)}{s+2\pi f_{03}}},$$

(15)

and the resulting design is demonstrated in Figure 8b, where the FilterBiquad and FilterLowFreqBilinear CAMs are utilized.

Denormalizing the utilized transfer functions to the frequency 10 krad/s, the values of the characteristics of the filters, described by (14) and (15), are summarized in

Table 2. Values of the characteristics of the filters, described by (14) and (15).

Characteristic	${\mathit{ϵ}}_1=0.1$ and ${\mathit{ϵ}}_2=0$	${\mathit{ϵ}}_1=0.1$ and ${\mathit{ϵ}}_2=0.5$	${\mathit{ϵ}}_1=0.1$ and ${\mathit{ϵ}}_2=1$
$f_{01}$ (kHz)	10.49	20.78	21.58
$Q_1$	0.953	20.6	12.99
$G_{BP1}$	0.91	0.483	0.711
$G_{LP1}$	0.091	−0.042	−0.066
$f_{02}$ (kHz)	—	10.68	10.78
$Q_2$	—	1.08	1.21
$G_{BP2}$	—	0.961	0.984
$G_{LP2}$	—	0.139	0.170
$f_{03}$ (Hz)	—	3.66	8.39
$G_{LP3}$	—	4.65	3.89

.

The obtained input and output waveforms are demonstrated in Figure 9, Figure 10 and Figure 11, while the measured values of the gain and phase at specific frequencies are summarized in

Table 3. Values of the characteristics of the filters described by (14) and (15).

Characteristic	${\mathit{ϵ}}_1=0.1$ and ${\mathit{ϵ}}_2=0$	${\mathit{ϵ}}_1=0.1$ and ${\mathit{ϵ}}_2=0.5$	${\mathit{ϵ}}_1=0.1$ and ${\mathit{ϵ}}_2=1$
gain at 1 kHz (dB)	−17.9 (−17.7)	−18.8 (−18.5)	(−18.7)
phase at 1 kHz (°)	41 (39.2)	29 (28.2)	(16)
gain at 10 kHz (dB)	−0.9 (−0.8)	−0.9 (−0.8)	(−0.8)
phase at 10 kHz (°)	0 (0)	0 (0)	(0)

. The corresponding theoretically predicted values are given between parentheses and confirm the accuracy of the presented implementations.

4. Conclusions

In this study, we explored the dynamics of a non-linear RLC resonance-based bandpass filter with self-voltage-controlled resistors. Our investigations show that the non-linearity modulates the quality factor of the filter, and hence its bandwidth, but does not change the resonance frequency. The bandwidth is thus a function of the strength of the feedback signal and is dependent on the input signal amplitude. Also, a pronounced notch-like behavior appears at twice the resonance frequency. The technique presented here for transforming a linear filter into a non-linear one is not limited to the studied prototype example in Figure 1 and can be easily extended to other types of filters. Our future work will target transforming other passive and active filters from linear to non-linear filters using a similar self feed-back control technique while examining the effects of this transformation on the filter parameters. It is needless to say that such a study cannot be automated and may or may not lead to identical findings, since the transformed filters are described by non-linear differential equations, which need to be solved numerically. It is also possible to use current-controlled resistors instead of voltage-controlled ones to obtain the non-linear filters.