Stability Analysis of Non-Foster Impedance Inverters

Abstract

Recently, active impedance inverters based on non-Foster negative capacitors have been proposed for applications in widely tunable filters. These designs use a traditional Linvill’s topology of the negative capacitor. Unfortunately, the range of external loads needed for the stable operation of such active inverters is rather limited. However, there is also the negative capacitor based on a recently proposed loss-compensated passive structure. This novel design promises stability-robust behavior for an extremely wide range of external loads. In this study, we compare the stability properties of both approaches and show that the design based on the loss-compensated passive structure is more robust.

1. Introduction

An ideal impedance inverter (K-inverter) is a fictitious two-port device that behaves similarly to a λ/4 transmission line segment (λ being the signal wavelength) with characteristic impedance K. However, the behavior of an ideal K-inverter does not depend on the frequency at all (i.e., it is a dispersionless device) [1,2]. When loaded at the output port with Z_L, its input impedance is Z_in = K²/Z_L, with Z_in and Z_L being the input and load impedances, respectively, and K being the “transformation” factor. Similarly, an ideal admittance inverter has the “transformation” factor J determined as Y_in = J²/Y_L, with Y_in and Y_L being the input and load admittance. Such inverters are routinely used in multistage filters to isolate neighboring resonators by reducing the loading effect. Due to their ability to convert high impedance/admittance to low impedance/admittance and vice versa, inverters are essential for applications in filter design, phase shifters, antenna matching, artificial transmission lines, and metamaterials [1,2,3,4,5,6].

Here, we focus on the impedance inverter (K inverter), which consists of a T-circuit with three capacitive reactances (Z₁ = 1/jωC₁, Z₂ = 1/jωC₂, C₁ > 0, C₂ < 0, Z₂ = −Z₁) (Figure 1a,b) [1]. The dual implementation with three inductances (sometimes called an inductive-T “dualizer”) is used for the broadband transformation of small radiation resistance of short dipole antennas to real system impedance [3]. It is important to notice that generic implementations of an impedance inverter (Figure 1a,b) always contain at least an ideal (dispersionless) negative reactive element (negative capacitor or negative inductor). However, such ideal negative elements are physically unrealizable.

A common way to go around this problem is the use of impedance inverter designs in which the negative impedance of physically unrealizable negative elements is “absorbed” by the positive impedance of adjacent stages that contain ordinary elements. Another possibility is to use a transmission line segment or a lumped element whose positive impedance is used to “emulate” a negative shunt capacitance (C₂ = −C₁ = −1/ω²L) (Figure 1c). Unfortunately, all these implementations suffer from a narrow bandwidth (approximately up to 20% [6]).

Narrow bandwidth is caused by dispersion, as a positive inductance only behaves like a negative capacitance at a single frequency. This is a consequence of the very fundamental Foster theorem, which states that the input reactance/susceptance of every passive and lossless network increases monotonically with frequency [7,8].

On the other hand, there are active electronic circuits that mimic the behavior of hypothetical negative capacitors and negative inductors, the so-called non-Foster (NF) elements [9,10,11,12]. The dispersion function of NF elements can be approximated as the inverse of the dispersion function of ordinary reactive elements [8,13]. For example, the dispersion of the ordinary capacitance can be compensated by the dispersion of the negative capacitance (NC), which significantly improves the bandwidth. Practically speaking, classical non-Foster (NF) elements are based on a special amplifier (the so-called negative impedance converter, NIC) that contains a load to be “negated”, which is located in the positive feedback loop. The superposition of the input signal and the amplified signal being fed back changes the ratio between input voltage and current, resulting in negative input impedance [13,14]. NF elements have been investigated for many years since they can assure the broadband operation of numerous devices such as small antennas [3,15,16,17,18,19,20,21], active metamaterials [8,22,23], active metasurfaces [24,25,26,27,28], amplifiers [29,30,31], voltage controlled oscillators [32,33,34], and phase shifters [35,36,37], to mention a few. The most serious drawback of the NF elements is their proneness to instability, caused by the inevitable use of a positive feedback loop of the associated NIC [38,39,40,41,42,43,44,45,46,47,48,49]. Although significant progress has been achieved in recent years, maintaining stable operation is still a difficult task that prevents the widespread use of NF elements.

Recently, it has been shown that it is possible to use NF elements for the design of active broadband impedance inverters as part of tunable multistage bandpass filters [6,50,51,52,53]. For example, the filters in [6,51] showed a bandwidth of 30% centered at 2.45 GHz and a tunability of up to 50%, which is a remarkable result. However, this broadband tunability was achieved by optimizing the NIC circuit and controlling four design parameters simultaneously. In addition, stability properties have only been tested to a limited extent. Indeed, it was silently presumed that the stability properties of a one-port NF element would be transferred to a two-port device using such an NF element (in this case, an impedance inverter) instead of testing the stability properties of the entire NF-based device.

Furthermore, a new NF passive-structure-based loss-compensated design was also presented recently [54,55,56]. It was demonstrated that these new NF elements can also be used in tunable filters, yielding very good results (20% tunability and 30% bandwidth), but with only two design parameters [55]. A very preliminary study of the stability properties of such a design (with quite a limited range of values of external loading impedances) was published in [56].

To summarize, it appears that NF-based impedance inverters are a promising technology, especially in the context of tunable filters. However, their stability properties are currently unclear.

Here, we analyze the stability properties of a system consisting of an active inverter based on two positive capacitors and an NF shown in Figure 1b, loaded with an external passive network (Figure 2). The NC is implemented in different ways (three different versions of the well-known Linvill’s NIC [14] as well as a recently introduced loss-compensated design based on a passive structure [54,55,56]).

The rest of the paper is organized as follows: Section 2 (Materials and Methods) describes the NF negative capacitors used in the design of the active impedance inverter, whose stability is tested. Section 3 (Results and Discussion) shows the results of the stability analysis, while Section 4 draws the conclusions.

2. Materials and Methods

As mentioned in the Introduction, classical NF negative capacitors are based on reversing the sign of the usual positive reactance through the use of NICs. The simplest NIC implementation contains an ideal voltage amplifier with a voltage gain A and a load (in this case, the simple capacitor C_F) that is in a positive feedback loop (Figure 3a) [8,12,14]. A very simple analysis shows that this amplifier reverses the voltage drop across the capacitor, which in turn reverses the direction of the input current, resulting in input capacitance that is a scaled “negative image” of a load capacitance ($C_{in}=\left(1-A\right)C_F$). In most cases in practice, one chooses $A=2$, resulting in $C_{in}={-C}_F$. This simplified explanation presumes that a NIC contains a dispersionless amplifier with an infinite bandwidth. Unfortunately, such a system would clearly not be causal and therefore not physical. The gain in every realistic amplifier decays with frequency and can be described by a general two-pole (2P) dispersion model:

$$A\left(\omega \right)={\displaystyle \frac{A_0}{\left(1+\mathrm{j}\omega /{\omega }_{\mathrm{p}1}\right)\left(1+\mathrm{j}\omega /{\omega }_{\mathrm{p}2}\right)}}$$

(1)

The term A₀ is the gain at very low frequencies, while ω_p1 and ω_p2 are the angular frequencies of the first and second poles, respectively. There is also a simpler version of (1), the so-called one-pole model (1P), in which ${\omega }_{\mathrm{p}2}\to \infty$. These two models are commonly used in the design and stability predictions of classical NIC-based NF elements [3,11,12,13,15,36]. It is important to mention that one of the well-known Linvill designs from [14] is merely a floating version of the basic principle shown in Figure 3a. The other Linvill designs in [14] use current amplifiers, but the basic principle is again very similar to that shown in Figure 3a and is described by (1). All the above-mentioned implementations provide “negative dispersion”, which is indicative of NF elements (right side of Figure 3a).

Figure 3. (a) The simplest NIC implementation of NF NC based on a positive feedback loop that flips the positive capacitor into a negative capacitor with the comparison of dispersion of positive capacitance (red) and negative equivalent capacitance (green) generated by such a circuit. (b) Passive loss-compensated NF NC. Parallel R-L network with negative-capacitance behavior (C_eq) above the cut-off frequency (ω > R/L) (the behavior of only the reactance (X) is shown in the graph). Equivalent negative capacitance is connected in series with R_eq, which models the losses. (c) A full model of loss-compensated NF NC, which shows the compensation of losses from the R-L network by a series circuit with negative resistance generated by NIC (R_NIC).

Figure 3. (a) The simplest NIC implementation of NF NC based on a positive feedback loop that flips the positive capacitor into a negative capacitor with the comparison of dispersion of positive capacitance (red) and negative equivalent capacitance (green) generated by such a circuit. (b) Passive loss-compensated NF NC. Parallel R-L network with negative-capacitance behavior (C_eq) above the cut-off frequency (ω > R/L) (the behavior of only the reactance (X) is shown in the graph). Equivalent negative capacitance is connected in series with R_eq, which models the losses. (c) A full model of loss-compensated NF NC, which shows the compensation of losses from the R-L network by a series circuit with negative resistance generated by NIC (R_NIC).

The main difference between the traditional Linvill’s designs and the newly proposed loss-compensated approach [54] is the relaxation of the stability conditions. In Linvill’s approach, the load that is converted to its negative counterpart (using either a one-pole/low-pass or a two-pole/bandpass design) is purely reactive. Simply, as detailed previously, in classical negative-capacitance applications, one uses a lumped capacitor as the load to be “inverted”, which is located in a positive feedback loop. Unfortunately, stability conditions are rather demanding in the presence of purely reactive loads [51,52]. Conversely, in the newly proposed design [54], NC behavior results from a passive parallel RL network. As detailed in [54], that network behaves like a lossy NC above its cut-off frequency (f_c = f_RL = R/(2πL)) (Figure 3b).

We start with the input impedance of a shunt RL network:

$$Z_{RL}={\displaystyle \frac{R\frac{{\omega }^2}{{\omega }_{RL}^2}+jR\frac{\omega }{{\omega }_{RL}} }{1+\frac{{\omega }^2}{{\omega }_{RL}^2}}},$$

(2)

Here, ω_RL = R/L. For the case of high-frequency behavior (ω ≫ ω_RL), one obtains

$$Z_{RL}\left(\omega \gg {\omega }_{RL}\right)\approx R+j{\displaystyle \frac{R}{\frac{\omega }{{\omega }_{RL}}}}=R+j{\displaystyle \frac{R^2}{\omega L}}=R_{eq}+{\displaystyle \frac{1}{-j\omega \left|C_{eq}\right|}},$$

(3)

with C_eq = −L/R. Thus, at the frequencies above f_RL, the impedance of the shunt RL network (Figure 3b) shows a negative slope of the reactance X, but with a considerable real part (high losses). This behavior can be modeled as a series connection of the resistance (R_eq) and a negative capacitance (C_eq < 0). The (unwanted) losses are then compensated by an additionally added series circuit with negative resistance (Figure 3c). This negative resistance is generated by an additional NIC circuit that flips a positive resistance into a negative one, thus achieving a loss-compensated passive NF NC. This design, therefore, shows a band-limited behavior with a very broad operating bandwidth (up to 1:10 [54]). Furthermore, it lowers the Q factor due to the presence of losses, which reduces the sensitivity of the whole circuit, which in turn also improves stability properties.

Essentially, a very good stability robustness is achieved due to the fact that the active part (NIC) performs the negation of only the real load, which is in line with the analysis of the stability of negative resistances [12]. Finally, it should also be mentioned that the novel loss-compensated approach is, in a sense, similar to the NF elements based on loss-compensated negative-group-delay (NGD) networks [57,58]. However, there are two important differences: (i) NF elements based on an NGD approach have a narrower operating bandwidth (1:1.45), due to the inevitable use of RLC resonators; (ii) NF elements based on loss-compensated NGD approach are inherently unidirectional, which seriously limits their applications. We performed stability analysis using four different NC designs. These designs were the “traditional” Linvill’s NIC in one-pole (low-pass) and two-pole (bandpass) configurations from [51], the improved Linvill’s NIC from [6], and the new loss-compensated NF NC design [54,55,56]. The entire stability analysis was performed using linear equivalent circuits that were prepared in two ways. The first three designs were simulated as linear causal circuit-theory models with ideal positive and negative circuit elements from [6,59] for the one- and two-pole NF NC, and from [6] for the improved two-pole NF NC design. The simulation of loss-compensated NF NC was based on a small-signal equivalent circuit extracted from the SPICE model in [55]. This SPICE model had been originally used in [52,55] for Cadence Virtuoso^TM simulations of the NIC part in 40 nm TSMC CMOS technology. The newly proposed structure [54,55,56] is inherently a bandpass design (at higher frequencies, the dispersion of the NIC causes the system shown in Figure 3b to behave like a positive inductance). Therefore, we compared it to similar realizations of traditional NF NCs: the one-pole (low-pass) and two-pole (bandpass) designs from [6,51]. To make a fair comparison, we adjusted the NF NC bandwidth from the design in [51] to match the bandwidth from a newly proposed design in [55] (0 GHz to 6.72 GHz for one-pole and 1.34 GHz to 6.72 GHz for both two-pole designs and loss-compensated design, respectively). The impedance inverter transformation factor for the analyzed inverters (K = 1/(ω$\left|{\mathrm{C}}_1\right|$) = 1/(ω$\left|{\mathrm{C}}_2\right|$)) was set to 56.48 Ω at 2.45 GHz, with C₁ = +1.15 pF and C₂ = −1.15 pF [19,23,24]. Furthermore, the stability-improved two-pole Linvill’s NF NC from [6] had K = 44 Ω at 2.45 GHz, with C₁ = +1.48 pF and C₂ = −1.48 pF.

At first, the impedance inversion of all the mentioned active impedance inverters was cross-checked by circuit-theory simulations in the Keysight ADS^TM (version 2024) CAD environment. Then, the stability properties of all four realizations of active impedance inverter, loaded with five different passive networks (Figure 2), were investigated using a two-step process.

In the first step, we used the Amcad Stability Analysis Tool STAN^TM (version 3.1), which is based on pole-zero identification. We used STAN^TM as it allows us to work with experimental/measured data, as well as SPICE-based models, when the full model of the system is unknown. Of course, there are other software packages that could have been used for stability analysis. For example, pole-zero identification is available in freeware packages such as Python (using the control systems library) [60] and MATLAB (via the System Identification app) [61].

By investigating the (unstable) RHS poles in the frequency domain of the current transfer function (I_out/I_in) and the impedance function (V_in/I_in), stability maps were obtained, which are presented in Figure 4. A polynomial of a sufficiently high degree was used in order to span the whole range of the two-dimensional parameter space presented in each subfigure of Figure 4.

In the second step, the above results were cross-checked using a Keysight ADS^TM transient solver, where the unstable behavior shows an exponential growth of the output voltage (V_out) and current (I_out) in the time domain, while stable behavior was verified by decaying time signals, under impulse excitation.

All simulations were performed with a current source excitation (Figure 2), as the NF NCs used belong to the open-current stable (OCS) types. This is also the reason why we only analyzed the scenario in which the input and output loads were RLC tanks connected in shunt. There are also known implementations of multistage bandpass filters with inverters connected between series tank circuits [6,51,55]. One may conclude that the active inverters in these filters should be of the SCS type. Whether the stability properties are different in these scenarios is currently unclear and would need to be investigated in a separate study.

All tests were conducted using a standard PC (Windows 10, Intel i5 processor with 64 GB of RAM and a 1 TB SSD drive (Intel, Santa Clara, CA, USA)). The settings for the tools used in this study were kept at default values. For all cases, behavior was examined in a range of 1 MHz to 300 GHz (in order to cover frequencies much higher than the operating frequency of several GHz). In ADS^TM transient analysis, a sufficient timestep and total length were used in order to assess the increase or decrease in signal behavior. For STAN^TM, polynomials of a sufficiently high degree were used in order to span the whole range of the 2D parameter sweep up to 300 GHz, in order to achieve a good fit.

3. Results and Discussion

The results obtained by the above-described two-step procedure (pole-zero stability analysis in STAN^TM and transient simulations in ADS^TM) are presented in Figure 4, with all simulation parameters given in the caption. Briefly, two different practical scenarios were analyzed. The case with varied pure real loads R_S and R_L (Figure 4a–d, second row of Figure 4) represents a realistic scenario of the inverters used as impedance tapers or transformers. In the third row (Figure 4e–h), the resistors R_S and R_L were varied again, but connected in series with the LC resonant circuit (Figure 2), which simulates the filter scenarios from [6,51,55]. Additional results for the scenario of filters with varying Q factor are shown in the fourth, fifth, and sixth rows in Figure 4. In these cases, the loading capacitance and inductance were varied across a very broad range of values with several different values of fixed resistance. The range of element values used in this parametric analysis was chosen to represent all possible practical scenarios. First, we took the initial values from published designs of practical filters with active impedance inverters [6,51,55]. Then, the range of these values was extended by two orders of magnitude. In this way, all other practical applications with high and low impedance, such as non-Foster-based small dipole antenna and non-Foster-based small loop antenna, are covered by this analysis. The results for the impedance inverter with simple one-pole NF NC (Figure 4a–g, first column) show limited performance. This realization is unstable for pure real loads (Figure 4a), stable in a filter environment (Figure 4e), and unaffected by varying Q factors of external resonant networks (Figure 4i–q). However, this design is rather sensitive. Indeed, any change in loading capacitance larger than 0.699 pF pushes the system into unstable behavior. In the case of the two-pole Linvill’s NF NC-based impedance inverter (Figure 4b–r, second column), the system shows stable behavior only for rather small values of loading external impedances. Unfortunately, this system is unstable over almost the entire range of the K-curve and only shows stable behavior for very large values of the load resistance (Figure 4b,f). Similarly, this system is also stable for low values of external capacitance and inductance, with better stability characteristics for smaller Q factors (Figure 4j–r). In classical broadband filters with low-Q design, the better stability properties observed under a low Q factor are an advantage. The same applies to a low-Q tunable bandpass filter with constant fractional bandwidth. The stability properties improve in the enhanced realization of the two-pole system in [6], which shows that it is possible to achieve stability for a large group of external loads through careful modifications (Figure 4c–s, third column).

Furthermore, we calculated the logarithmic sensitivity [62] of the envelope decay of the impulse response current (${~e}^{-\sigma t}$, σ being the real part of complex frequency s = σ + jw) for the modest-Q case (Figure 4m–p), defined with respect to the value of C at the operating point, which is defined as R = 50 Ω and L = 10 nH. Those values were chosen since most RF systems have a Z₀ = 50 Ω, and L = 10 nH was chosen in order to have the same inductance for all four cases of some medium value. The envelope decay of the impulse response current is presented in Figure 5, with calculations of sensitivity (|S| = (σ5% − σ1%)/(C5% − C1%)) presented in

Table 1. Fitted parameters of impulse response envelope (σ, modelling e^−σt) for different distances from stability boundary (dsb) (10%, 5% and 1% of value of C_s) and sensitivity S = |(σ5% − σ1%)/(C5% − C1%)|.

	One-Pole Non-Foster Capacitor	Two-Pole Non-Foster Capacitor	Improved Two-Pole Non-Foster Capacitor	Loss-Compensated Non-Foster Capacitor
dsb = 10%	s = 1.06 × 109	s = 4.35 × 108	s = 1.57 × 108	s = 2.48 × 109
dsb = 5%	s = 4.93 × 108	s = 2.05 × 108	s = 2.05 × 108	s = 2.47 × 109
dsb = 1%	s = 9.97 × 107	s = 3.92 × 107	s = 3.92 × 107	s = 2.46 × 109
	S = 1.41 × 10−2 Hz/F	S = 1.28 × 10−2 Hz/F	S = 1.8 × 10−3 Hz/F	S = 9.587 × 10−5 Hz/F

. As expected, this calculation resulted in the lowest sensitivity for the loss-compensated design of NC.

Thus, it can be concluded that the newly proposed passive, loss-compensated design shows an extremely robust stability behavior for all the investigated scenarios, both for real and complex loads, which is evident by consistently stable behavior with changes in all the parameters presented in Figure 4. The summary of the stability analysis of the four different NF NC implementations of impedance inverters from Figure 4 is given in

Table 2. Areas of external impedances for which the NC-based impedance inverter is stable under various terminations.

Loads\Implementations	One-Pole	Two-Pole	Improved Two-Pole	Loss-Compensated
varying resistance	unstable	<20% of the area is stable	<50% of the area is stable	100% of the area is stable
variable resistance with a fixed series LC circuit	100% of the area is stable	<20% of the area is stable	100% of the area is stable	100% of the area is stable
high-Q LC circuit with varying reactance	<50% of the area is stable	<20% of the area is stable	<25% of the area is stable	100% of the area is stable
modest-Q LC circuit with varying reactance	<50% of the area is stable	<25% of the area is stable	<50% of the area is stable	100% of the area is stable
low-Q LC circuit with varying reactance	<50% of the area is stable	100% of the area is stable	>90% of the area is stable	100% of the area is stable

.

The presented results show several important insights into the design of NF NCs. First, a two-pole bandpass design that eliminates the unstable DC pole [50,52] improves the stability properties of the associated system. Second, the stability properties can be further improved by careful improvement of a “traditional” two-pole Linvill’s NF NC [6]. Third, the largest improvement in stability properties can be achieved by incorporating a loss-compensated passive network into the NC design [55,56]. The physical background of stability robustness is based on the simultaneous application of two different phenomena. The first is the band-limited (high-pass) operation, which prevents the occurrence of an unstable DC pole, and the second is the presence of losses, which, although externally compensated by a negative resistance, reduces the Q factor of the loss.compensated NF NC and thus the overall sensitivity.

Finally, it should be emphasized that the presented stability analysis was performed on the basis of linear equivalent circuits in order to allow for a fair comparison of all the known designs of NF-based impedance inverters. The analysis examines the basic physical phenomena that can cause instability. The focus of this study is on understanding circuit theory models and their influence on stability, not on various technologies that could be used to achieve such systems. It is also assumed that possible influences of the unavoidable parasitics, component tolerances, aging, and temperature variations can be compensated by standard methods such as the use of additional negative feedback and microelectronic technology [6,8,52,63]. The analysis of these technological parameters on the stability characteristics, as well as the practical realization of microelectronic loss-compensated NF NC in CMOS technology, could be a topic of our future study.

4. Conclusions

A numerical investigation of the stability of different realizations of NF-based active impedance inverters was carried out using both a pole-zero analysis and a transient analysis. Both methods showed that the stability properties of the active inverter based on a passive loss-compensated NF structure were superior to those of all “traditional” designs based on Linvill’s NICs. The stability characteristics of the passive loss-compensated NF structure were also significantly better than those of loss-compensated NGD-based designs. Apart from tunable filters, impedance inverters based on loss-compensated NF structures can also be used in other RF systems that require stable behavior across widely varying impedance values. These include, for example, actively matched small broadband antennas and active metamaterials/metasurfaces.