TMTT-2020-07-0823.R1 1 Abstract—A concurrent dual–band self–oscillating mixer (SOM), based on a ring-shaped stepped-impedance resonator, is proposed and analyzed in detail. Taking advantage of the ring even and odd resonances, the circuit can operate in concurrent dual quasi–periodic mode and injection–locked mode. In the second case, it behaves as a dual-band zero–IF mixer. Initially an analytical study of the SOM behaviour in the two modes is presented. Then a variety of accurate numerical methods are used for an in-depth investigation of the main aspects of its performance, including stability, conversion gain, linearity, and phase noise. The recently proposed contour-intersection technique and the outer-tier perturbation analysis are suitably adapted to the SOM case. A method is also presented to distinguish the parameter intervals leading to heterodyne and to zero-IF operation at both the lower and upper frequency bands. In the zero-IF SOM, the possible instantaneous unlocking in the presence of modulated input signals is investigated and avoided. The methods have been applied to a dual mixer at the frequencies 2.4 GHz and 4.1 GHz. Index Terms— harmonic balance, oscillators, phase-noise analysis, stability analysis. I. INTRODUCTION he recent works [1]-[3] present a novel concept to obtain zero-IF frequency conversion using an injection-locked oscillator, suitably designed and biased to enhance its mixing capabilities. The so-called zero-IF self-oscillating mixer (SOM) enables the implementation of compact transmitters and receivers. The concept is of general interest in systems requiring a small size and weight, such as the µRFID tag proposed in [1]. Considering also the increasing demand of multiband wireless systems, the work [4] proposed a concurrent dual-band zero-IF SOM, which extends the compact frequency conversion to simultaneous operation in two frequency bands. This requires a concurrent dual- frequency oscillator at two incommensurable fundamental frequencies ω1 and ω2 [5]-[7], which corresponds to a doubly autonomous quasi–periodic solution. In the concurrent dual- frequency zero-IF SOM, each oscillation gets locked to its corresponding RF signal [4], so the circuit behaves as a concurrent dual-frequency injection-locked oscillator. A major challenge in the design of concurrent dual-frequency oscillators is the robustness of this concurrent operation mode [8], which mathematically coexists with two periodic solutions, at ω1 and ω2, respectively, and with the DC solution. For a reliable concurrent operation, the quasi– periodic solution at ω1 and ω2 must be the only stable one. Manuscript received July 26, 2020; revised, September 30, 2020, accepted, November 7, 2020. This work was supported by the Spanish Ministry of Economy and Competitiveness through the European Regional Development Fund (ERDF)/ Fondo Europeo de Desarrollo Regional (FEDER) and under Project TEC2017-88242-C3-(1/2)-R. This article is an expanded version from the IEEE MTT-S International Microwave Recently, a concurrent dual-frequency oscillator based on a stepped-impedance loop resonator [9]-[11] has been proposed [12]-[14]. This element enables a compact implementation of two orthogonal resonances at ω1 and ω2 1 with high quality factor and an excellent isolation. It has the advantage of a compact size, usually much smaller than the one resulting from two independent resonators with similar quality factors. With this element, low phase noise can be achieved, as shown in [15], where a ring-shaped step- impedance resonator was used to reduce the phase-noise spectral density of a single-frequency voltage-controlled oscillator (VCO). Departing from a dual-frequency oscillator based on a ring- shaped resonator, the work [4] presented a double functionality SOM, able to operate both as a zero-IF SOM and as a heterodyne SOM [16]-[18]. In heterodyne mode each of the two concurrent oscillations ( ω1 and ω2) mixes with its corresponding input signal, that is, ωin1 mixes ω1 and ωin2 mixes with ω2, to provide two distinct intermediate-frequency (IF) outputs. This work expands [4] with a nonlinear analysis of the SOM in both heterodyne and zero-IF modes. Initially, an analytical investigation of the SOM behavior in the two modes is presented. It provides insight into the conversion gain, linearity, and oscillation extinction, in the heterodyne case, and the injection-locked operation and mechanisms for the amplitude and frequency demodulation, in the zero IF case. Then, the transistor-based dual-band SOM is addressed with a variety of numerical methods based on harmonic balance (HB). The first goal will be the detailed a stability analysis of the ring-resonator dual-frequency oscillator, considering the DC solution, the distinct periodic oscillations at ω1 and ω2, and the concurrent quasi-periodic oscillation at ω1 and ω2. In heterodyne mode, the SOM will be analyzed through an extension of the contour-intersection method proposed in [19]-[20] to quasi-periodic regime. The variation of the conversion gain is analyzed exporting a gain surface that must be interpolated at the solution points obtained through the contour intersection. The phase noise is analyzed with a perturbation formulation based on an outer-tier admittance- type description of the SOM circuit. The input-generator values (in terms of frequency and power) for operation as a heterodyne SOM and as zero-IF SOM are distinguished through a bifurcation analysis [21]-[25]. In particular, one must obtain the loci at which the self-oscillation is extinguished (Hopf locus) [21]-[25] and the locus at which this oscillation is locked to the input source (turning-point Symposium (IMS 2020), Los Angeles, CA, USA, August 4-6, 2020. (Corresponding author: Mabel Pontón.) The authors are with the Departamento de Ingeniería de Comunicaciones, Universidad de Cantabria, Santander, 39005, Santander, Spain (e-mail: mabel.ponton@unican.es; herreraa@unican.es; suareza@unican.es). Double functionality concurrent dual–band self– oscillating mixer Mabel Pontón, Member, IEEE, Amparo Herrera, and Almudena Suárez, Fellow, IEEE T