We propose to use memristors as memory nonlinear circuits to build behavioral models useful in the simulation of passive inter-modulation in RF and microwave devices such as filters, antennas and in general connections

Leon Chua introduced memristors in 1971 [1] as an ideal two-terminal circuit element in complement to the already known three basic circuit elements: resistor, inductor and capacitor (RLC). Memristors are defined by a non-linear memristance that relates the flux (or integral of voltage across the device) to the charge (or integral of current in the device). Because of this definition the memristor will generate passive intermodulation products and their power will depend on the memory of the past current that is contained in the device.

Non-analytic behavioral models of passive non linearity presented in 2013 are used to define and propose new specifications for multicarrier passive intermodulation in antennas. A margin of up to 10 dB is obtained in some cases and can be used to relax 2-carrier PIM specifications.

Passive inter-modulation products between transmitted signals may prevent the correct operation of satellite receivers using the same antenna. In many cases, these passive products do not obey the classical rule of 3 dB/dB slope as a function of carrier input power and they cannot be modeled using the classical theory based on polynomials. This has prevented the exact computation of carrier to inter-modulation ratio in multicarrier conditions from 2-carrier measurements. This has led to the use of higher than necessary margins. We present non-analytic models that generate non-integer inter-modulation slopes identical to that obtained in measurements and permit to predict multicarrier results.

Discontinuities at origin have been used to better approximate measured curves in recent papers but generally not explicitly and their physical validity has not always been demonstrated. In this communication, we show that these discontinuities can be explained by physically acceptable discontinuities in the real physical device. We propose simple criteria to accept or reject these discontinuities, in either passive or active devices, depending on the order of the discontinuity. In addition, we show that models having such discontinuities behave differently from classical models. In particular, these discontinuities explain non-integer dB/dB slopes of harmonic power and intermodulation power as a function of input power. Recent and older measurements of intermodulation products in passive devices, telephony base-station and RF transistors show such a behavior so that supposed lack of measurement cannot be used as a reason to reject discontinuities as non-physical.

Les dispositifs passifs ne sont pas parfaitement linéaires aux très fortes puissances. Ils créent des harmoniques et des produits d’intermodulation pairs et impairs. Ces produits peuvent perturber fortement les performances des récepteurs utilisant les mêmes équipements passifs ou simplement installés sur le même site que les émetteurs. Ces produits n’obéissent pas à la loi classique de croissance en fonction de la puissance d’entrée avec une pente en dB/dB égale à l’ordre du produit. Ceci a interdit, jusqu’à présent, de modéliser correctement ces dispositifs. On propose une modélisation par des fonctions non analytiques qui permet de mieux reproduire cette caractéristique avec un nombre minimal de coefficients et à partir d’un nombre minimal de mesures.

Many measurement results on passive intermodulation products exhibit slopes of third order intermodulation product level as a function of input level different from the classical 3 dB/dB slope. Even-integer and real values between 1 and 3 are commonly reported for telephony base station towers antennas and filters. No classical model has been able to approximate these measurements up to now. We propose a non-analytic model that explains this behaviour and may serve as theoretical basis to find a physical model.

Nonlinear power amplifiers (transistors and tubes) are used in telecom transmission because of their power efficiency. These amplifiers produce intermodulation noise (in same bandwidth and in adjacent bandwidth) that impairs the spectrum efficiency. When using a linearizer, a telecom engineer will get a better optimum between spectrum and energy efficiency. Nobody has been able to write a test specification for a good non linearity except that it should be easy to linearize and give good intermodulation results when linearized. In the last 2 decades, cellular telephony base station manufacturers have specified that RF non linear components should be measured under best linearization conditions. Test benches now include an adaptive linearizer for these measurements. After AM/AM and AM/PM measurements of the device, the test bench computes the best possible pre-distortion for the device and then measures efficiency and distortion (such as ACPR)using RF signals that have been pre-distorted by a digital linearizer. The base station manufacturer will then build its power amplifier together with a practical linearizer (real time analogue or digital) targeting this best linearized performance. In the next years, this will be applied also to RF and microwave amplifiers and systems such as telecom satellite links and microwave LANs. Future benches will be more demanding: - Higher centre frequency (up to 60 GHz) - Higher bandwidth (up to some GHz) and much higher sample frequency (keeping the same memory length and depth or more) - Higher spectrum efficiency signals (time and frequency packing, low roll-off, single carrier, multicarrier or OFDM) - Best compromise between energy and spectrum efficiency taking into account nominal RF power and operating point of the device, thermal noise, intermodulation noise and interference (SNIR or SNIIR). The first 2 points will need new hardware, particularly high frequency 12 bits ADC and DAC and fast memory storage. The last ones will need improvements in the simulation of new test signals, the computation of best linearization curves (particularly in the case of memory non-linearity) and the measurement of ACPR, EVM or NPR.

NPR (noise power ratio) and EVM (error vector magnitude) measurements are used to characterize linear or nonlinear distortions and degradations in digital modulators, microwave or RF amplifiers and transmission links. These methods are also used in system simulations. Each of these methods results in an equivalent noise power that can be added to the thermal noise in the link budget in order to compute the BER (bit error rate). In this paper we examine the conditions necessary for these two measurements (or simulations) to give the same value for this equivalent noise. From this identity, it is possible to use either one or the other method with the goal to simplify simulation software or test benches particularly in the case of wide bandwidth measurements.