Analysis of noise and dynamical effects in zero-IF self-oscillating mixers M. Pontón 1 , S. Sancho 2 , A. Herrera 3 , A. Suárez 4 University of Cantabria, Spain 1 mabel.ponton@unican.es, 2 sanchosm@unican.es, 3 herreraa@unican.es, 4 suareza@unican.es Abstract—Zero-IF self-oscillating mixers (SOMs), based on injection locking, enable a compact direct frequency conversion and substantially mitigate the phase noise problem of heterodyne SOMs. Despite these advantages, their operation is complex and susceptible to exhibit a variety of dynamical effects. Here the need for an optimum selection of the operation point in the static injection-locked curve is demonstrated, as well as the inability of the oscillator to follow the variations of the input signal from a certain modulation frequency. Then a noise analysis of the down- converted signal is presented, based on a conversion-matrix formalism applied to a semi-analytical formulation. Keywords—injection-locking, noise analysis, envelope transient I. INTRODUCTION The zero-IF self-oscillating mixers (SOMs) presented in [1]-[2] enable a direct frequency conversion that avoids the need for both an oscillator and a mixer. As a result, they reduce power consumption and size, of general interest for the implementation of compact transmitter and receivers, such as those used in RFID and sensor systems. Zero-IF SOMs are based on the injection locking of an oscillator by an RF signal, which substantially mitigates the phase noise problem of heterodyne SOMs. Zero-IF SOMs can be used under both amplitude and frequency modulations, and the demodulation can be easily carried out by amplifying the voltage drop in a resistor of the bias circuitry. Their interest and potential are demonstrated in the recent work [2], which presents a sourceless bank of zero-IF SOMs to receive and demodulate simultaneously different data streams. Despite their demonstrated usefulness, the operation of modulated injection-locked oscillators is complex and susceptible at exhibiting a variety of dynamical effects. The work [3] presented an analysis of the boundaries between zero- IF and heterodyne SOM operation in terms of the input frequency and power, as well as an analysis of possible instantaneous unlocking under amplitude modulations. However, other distortion mechanisms associated with their oscillatory nature have not yet been investigated. Their knowledge and accurate prediction should allow an optimized design of these compact circuits to take advantage of their full potential. Here we will demonstrate the need for a proper choice of the operation point in the static injection-locked curve to avoid a relevant distortion of the demodulated signal. The impact of the modulation frequency will also be analyzed, showing the inability of the oscillator to follow the variations of the input signal above a certain value of the modulation frequency. Then a noise analysis of the down-converted signal will be presented, based on a conversion-matrix formalism applied to a semi- analytical formulation of the injection-locked oscillator. Initially, an analytical study of a simple cubic-nonlinearity oscillator, enabling insight into the problem, will be carried out. Then, a FET-based oscillator at 800 MHz will be considered. II. DYNAMICAL EFFECTS AT THE MODULATION SCALE The cubic-nonlinearity oscillator of Fig. 1 is injection locked with an independent source  ௜௡ ሺሻ of amplitude  ௜௡ and frequency  ௜௡ . Limiting the analysis to DC and the fundamental frequency, it is described with the equations: ሺ0ሻ ଴ ൅ ሺ0ሻ ଴ ሺ ଴ ,  ଵ ሻൌ 0, ሺ ௜௡ ሻ ଵ ൅ ሺ ௜௡ ሻ ଵ ሺ ଴ ,  ଵ ሻൌ ௜௡  ௝ఝ (1) where  ଴ ,  ଵ and  ଴ ,  ଵ are the components at DC and  ௜௡ of the voltage ሺሻ and the current ሺሻ,  ଵ is an amplitude and  is the phase shift between  ௜௡ ሺሻ and ሺሻ . The frequency dependent functions are ሺሻ ൌ ሺ ൅ ሻ ൅ 1 and ሺሻ ൌ  ൅ . Because the two equations are coupled,  ଴ will vary with  ௜௡ . One obtains a closed curve  ଴ ሺ ௜௡ ሻ, where  ௜௡ ൌ  ௜௡ /2 , though only the stable lower section of this curve has been represented in Fig. 2(a) and (b). Fig. 1. Cubic-nonlinearity oscillator. The circuit parameter values are  ଵ ൌ െ0.2 Ω ଵ ,  ଶ ൌ 0.01 A/V 2 ,  ଷ ൌ 0.02 A/V 3 , ൌ 0.1 nH, ൌ 0.1 Ω, ൌ 0.1 nF. The injection source amplitude is  ௜௡ ൌ 0.02 V. When considering an input modulation, the state variables become time variant, and proceeding like in [4] one obtains: ሺ0ሻ ଴ ሺሻ ൅ ሺ0ሻ ଴ ሺሻ െ  ఠ ሺ0ሻ ሶ ଴ ሺሻ െ  ఠ ሺ0ሻ ሶ ଴ ሺሻ ൌ 0, ሺ ௜௡ ሻ ଵ ሺሻ ൅ ሺ ௜௡ ሻ ଵ ሺሻ െ  ఠ ሺ ௜௡ ሻ ሶ ଵ ሺሻ െ  ఠ ሺ ௜௡ ሻ ሶ ଵ ሺሻ ൌ  ௜௡ ሺሻ ௝ఝሺ௧ሻ (2) where  ଵ ൌ  ଵ  ௝థሺ௧ሻ and the subscript  indicates frequency differentiation. As a meaningful example, we will consider an injection locking source at the free-running frequency  ௜௡ ൌ 2 ௢ , where  ௢ ൌ 1.583 GHz is the free-running frequency,