Optimized design of harmonic-injection dividers

A new formulation is presented for the efficient harmonic-balance analysis of the division bandwidth of frequency dividers by a high order. The procedure is based on some mathematical properties of the solution curve under low-amplitude of the input generator. Through a simple fitting technique, it is possible to determine the variation of the synchronization bandwidth versus any design parameter, while keeping constant the central frequency of the division band. The procedure also enables a prediction of the frequency-division interval for any value of the input-generator amplitude within the region of linear behavior with respect to the input source. It has been applied to the optimization of the input matching network in a frequency divider by 10, which uses a nonlinear transmission line (NLTL) as a feedback network.Spanish project TEC2011-29264-C03-01 for financial support

Advances in the analysis of injection-locked oscillators

This work reviews recent advances on the realistic analysis of injection–locked oscillators, for an efficient prediction of their complex multi–valued solution curves. The oscillator or its active core is modeled with a nonlinear admittance function extracted from harmonic balance, whereas other system elements are introduced at a second analysis stage. The analytical modeling of the external elements provides insight into their effect on the locking bands and other aspects of the behavior. In purely numerical simulations, the solution curves are traced through contour plots that make use of the nonlinear admittance function. The methods are illustrated with state–of–the–art applications, including compact transmitters and receivers, wireless–power transfer, and active sensing.Work funded by the Spanish Ministry of Science and Innovation (MCIN/AEI/10.13039/501100011033) – PID2020 116569RB-C3

Generalized analysis of systems based on two nonlinear resonators

Circuits containing two nonlinear resonators have been recently proposed for a variety of applications, such as nonlinear isolators, robust wireless power transfer, and sensors. However, their simulation is difficult due to the presence of hysteresis phenomena, associated with turning points in the solution curve, and even disconnected curves, as will be shown in this work. Here, we will present a general analysis methodology, compatible with commercial harmonic balance (HB) and able to provide all the coexisting periodic solutions. It is based on the use of two auxiliary generators (AGs), one per nonlinear resonator. The first AG acts like an independent source and controls the second one, which also depends on the input source, unlike a previously presented formulation. This extra dependence enables a systematic and broad-scope application but demands a conceptually different analysis strategy, presented in this work. Besides its numerical capabilities (in combination with HB), the new formulation will provide insight into the complex behavior of systems composed by two nonlinear resonators. It will be illustrated through its application to a Lorentzian-Fano nonlinear isolator and a system for robust near-field wireless power transfer, in which the new formulation particularizes to the previous one.This work was supported by the Spanish Ministry of Science and Innovation (MCIN/AEI/10.13039/501100011033) under Grant PID2020-116569RB-C31 and Gobierno de Cantabria (Contrato Programa Gobierno de Cantabria—UC)

Harmonic-injection divider based on feedback through a nonlinear transmission line

An application of nonlinear transmission lines (NLTLs) to the design of harmonic-injection frequency dividers by high division order is presented. The NLTL, composed by inductance-varactor cells, is used as a feedback block of an active-device circuit, which gives rise to a free-running oscillation. Due to its highly nonlinear behavior, the NLTL can be optimized to enhance the harmonic components required for high order division through mixing with the input signal. The frequency bandwidth is further increased through a combination of phase-locking and frequency-locking effects. This is done extracting an error baseband signal from the active-device output, which, after suitable amplification, is applied to the varactor diodes of the NLTL. A design procedure is also presented to enable the switching of the division order between N and N-1 through control of the bias voltage. The novel configuration can have the advantage of a low complexity, low dc power consumption, and scalability. The new concept has been applied for the design a dual-order frequency divider by 10 and 9 with about 2% bandwidth

Two-level stability analysis of complex circuits

A new methodology is proposed for the small- and large-signal stability analysis of complex microwave systems, containing multiple active blocks. It is based on a calculation of the system characteristic determinant that ensures that this determinant does not exhibit any poles on the right-hand side (RHS) of the complex plane. This is achieved by partitioning the structure into simpler blocks that must be stable under either open-circuit (OC) or short-circuit (SC) terminations. Thus, the system stability is evaluated using a two-level procedure. The first level is the use of pole-zero identification to define the OC- or SC-stable blocks, which, due to the limited block size, can be applied reliably. In large-signal operation, the OC- or SC-stable blocks are described in terms of their outer-tier conversion matrices. The second level is the calculation and analysis of the characteristic determinant of the complete system at the ports defined in the partition. The roots of the characteristic determinant define the stability properties. The Nyquist criterion can be applied since, by construction, the determinant cannot exhibit any poles in the RHS. In addition, one can use pole-zero identification to obtain the values of the determinant zeroes. Because the determinant is calculated at a limited number of ports, the analysis complexity is greatly reduced.This work was supported in part by the Spanish Ministry of Science and Innovation and the European Regional Development Fund / Fondo Europeo de Desarrollo Regional (ERDF/FEDER) under research project TEC2017-88242-C3-1-R

Stability and bifurcation analysis of multi-element non-foster networks

A stability and bifurcation analysis of multi-element non-Foster networks is presented, illustrated through its application to non-Foster transmission lines. These are obtained by periodically loading a passive transmission line with negative capacitors, implemented with negative-impedance converters (NICs). The methodology takes advantage of the possibility to perform a stability analysis per subintervals of the perturbation frequency. This will allow an independent analytical study of the low-frequency instability, from which simple mathematical criteria will be derived to prevent bias-network instabilities at the design stage. Then, a general numerical method, based on a combination of the Nyquist criterion with a pole-zero identification of the individual NIC, will be presented, which will enable the detection of both low- and high-frequency instabilities. A bifurcation analysis of the multi-element non-Foster structure will also be carried out, deriving the bifurcation condition from a matrix-form formulation of the multi-element structure. The judicious choice of the observation ports will enable a direct calculation of all the coexisting bifurcation loci, with no need for continuation procedures. These bifurcation loci will provide useful insight into the global-stability properties of the whole NIC-loaded structure.This work was supported in part by the Spanish Ministry of Economy and Competitiveness and in part by the European Regional Development Fund (ERDF/FEDER) under Project TEC2014-60283-C3-1-R and Project TEC2017-88242-C3-1-R

Circuit-level stability and bifurcation analysis of non-foster circuits

An in-depth stability analysis of a typical non-Foster matching network is presented. The investigation is carried out at two levels: considering an ideal implementation of the negative impedance inverter (NIC) and using detailed circuit-level descriptions of all its active and passive components. The ideal NIC model will enable an analytical derivation of the characteristic system and the system poles, which will provide insight into the main instability mechanisms in these configurations. A good qualitative agreement is obtained with the circuit-level analyses, based on pole-zero identification and bifurcation detection methods. The impact of significant parameters, such as the biasing resistors or the value of the reactive component to be negated, is investigated in detail. A circuit-level methodology is proposed to obtain the stability boundaries and margins in an efficient and rigorous manner. For illustration, a non-Foster circuit based on a NIC has been manufactured and measured, obtaining very good agreement with the analysis results.This work has been funded by the Spanish Government under contract TEC2014-60283-C3-1-R, the European Regional Development Fund (ERDF/FEDER) and the Parliament of Cantabria (12.JP02.64069). The authors would like to thank S. Pana, University of Cantabria, for her assistance with the manufacturing process

Analysis and synthesis of hysteresis loops in an oscillator frequency characteristic

A methodology for the analysis and synthesis of multiple hysteresis loops in the frequency characteristic of a voltage-controlled oscillator (VCO) is presented. This is achieved through the coupling of an oscillator inductance to multiple external (passive) resonators with resonant frequencies in the tuning range of the VCO. A possible application to the implementation of a compact chipless radio frequency identification (RFID) system is explored, using the oscillator as a reader and placing the external resonators in the tag. The system takes advantage of the high sensitivity to the tag resonances in the presence of hysteresis, which leads to vertical jumps in frequency versus the tuning voltage. A desired bit pattern would be encoded in the tag by enabling or disabling passive resonances at a sequence of frequencies. In the practical realization, the inductors in the oscillator and the external board are implemented through spiral inductors so that the resonators in the VCO and the tag have strong broadside coupling. The coupling effect is modeled through electromagnetic simulations, from which a linear admittance, representing the coupled subnetwork, is extracted. The multihysteresis oscillator characteristic can also be obtained experimentally through a new methodology able to stabilize the physically unstable sections without altering their steady-state values. Different demodulation methods for reading the tag are discussed.This work was supported in part by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF/FEDER) under research project TEC2017-88242-C3-1-R

Mutual injection locking of oscillator circuits through inductor coupling

This work presents an investigation of mutual injection locking of oscillator circuits through inductor coupling. A realistic analytical formulation provides insight into system behavior, with coexistent oscillation modes. The stability of these modes is determined through a perturbation analysis, extended to the calculation of the phase-noise spectral density. An analytical expression enables an understanding of the phase-noise reduction mechanism. The cases of two coupled oscillators at the fundamental frequency and two distinct oscillators at a 1/3 frequency ratio are considered. Possible applications include the oscillator phase-noise reduction and the implementation of sensors using the phase shift between the two oscillator elements.Work supported by the Spanish Ministry of Science and Innovation (ERDF/FEDER) TEC2017-88242-C3-1-R

Nonlinear analysis of oscillator mutual injection locking through inductor coupling

This work presents an in-depth investigation of the nonlinear behavior of two mutually injection-locked oscillators through inductor coupling. An analytical formulation, solved through an innovative procedure, facilitates the understanding of the qualitative transformations in the system solutions when increasing the coupling factor. The analysis demonstrates that, in a manner similar to the unilaterally injection-locked oscillators, families of disconnected/connected curves are obtained when increasing this factor, although the patterns, associated with distinct operation modes, are more complex. Then, an accurate numerical method to predict the behavior of coupled transistor-based oscillators is presented, based on nonlinear admittance models of the individual oscillators. Mathematical conditions are derived to solve the coupled system through a two-level contour-intersection technique. In this way, all the solutions coexisting for a given set of element and parameter values are calculated simultaneously, in an exhaustive manner. The cases of two coupled oscillators at the fundamental frequency and at 1:3 frequency ratio are considered. Possible applications include the oscillator phase-noise reduction and the implementation of sensors using the phase shift between the two oscillator elements.This work was supported in part by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF/FEDER) under research project TEC2017-88242-C3-1-R