Efficient analysis methodologies for emerging oscillator configurations

Oscillators enable a low-cost and compact implementation of radio-frequency identification (RFID) readers and radar systems, among other. The function integration comes at the cost of a more complex performance and a demanding analysis since the circuit must respond to several specifications, while maintaining the oscillation condition. Here a review of recently introduced semi-analytical formulations, intended for these oscillator-based circuits, is presented. They are based on the use of a numerical model of the standalone oscillator, extracted from harmonic-balance or envelope-transient simulations, which is introduced in an analytical description of the oscillator interaction with the external elements. Various types of formulations, applied to self-injection locked radar, RFID readers and super-regenerative oscillators, will be described and validated.Author is grateful to S. Sancho, F. Ramírez and M. Pontón, from U. of Cantabria, R. Melville from Emecom, and S. Hernandez from AMCAD Engineering. Work supported by project TEC2017-88242-C3-1-R (AEI,FEDER/UE)

Stability and phase-noise analysis

This contribution emphasizes the relevance of Prof.Vittorio Rizzoli’s works on stability and phase-noise analysis and describes how they have impacted more recent investigations. Regarding the stability analysis, in 1985 he developed a frequencydomain formulation that provided unvaluable insight into the way how the perturbed system should be described and analyzed. This formulation enabled the application, for the first time, of the Nyquist criterion to circuits simulated with harmonic balance (HB). In 1994, he derived a HB formulation for phase-noise analysis, which considered both the frequency modulation, associated with the timing noise, and the frequency conversion effects; it provided a complete prediction of the noisy oscillator spectrum at small and large offset frequencies from the carrier. Departing from these relevant contributions, more recent advances in the two topics will be described.Work supported by the Spanish Ministry of Science and Innovation (MCIN/ AEI / 10.13039/501100011033) under grant PID2020-116569RB-C31

Nonlinear microwave simulation techniques

The design of high performance circuits with short manufacturing cycles and low cost demands reliable analysis tools, capable to accurately predict the circuit behaviour prior to manufacturing. In the case of nonlinear circuits, the user must be aware of the possible coexistence of different steady-state solutions for the same element values and the fact that steady-state methods, such as harmonic balance, may converge to unstable solutions that will not be observed experimentally. In this contribution, the main numerical iterative methods for nonlinear analysis, including time-domain integrations, shooting, harmonic balance and envelope transient, are briefly presented and compared. The steady-state methods must be complemented with a stability steady-state analysis to verify the physical existence of the solution. This stability analysis can also be combined with the use of auxiliary generators to simulate the circuit self-oscillation and predict qualitative changes in the solution under the continuous variation of a parameter. The methods will be applied to timely circuit examples that are demanding from the nonlinear analysis point of view.This work has been supported by the Spanish Government under contract TEC2014-60283-C3-1-R and the Parliament of Cantabria (12.JP02.64069)

Systematic methodology for the global stability analysis of nonlinear circuits

A new methodology for the detection of Hopf, flip, and turning-point bifurcations in nonlinear circuits analyzed with harmonic balance (HB) is presented. It enables a systematic determination of bifurcation loci in terms of relevant parameters, such as input power, input frequency, and bias voltages, for instance. It does not rely on the use of continuation techniques and is able to globally provide the entire loci, often containing multivalued sections and/or disconnected curves, in a single simulation. The calculation of Hopf and flip bifurcations is based on the extraction of a small-signal admittance/impedance function from HB and the calculation of its zeros through a geometrical procedure. The method is ideally suited for the investigation of the global stability properties of power amplifiers and other nonlinear circuits. The turning-point locus, associated with either jump phenomena or synchronization, is obtained by taking into account the annihilation/generation of steady-state solutions that is inherent to this type of bifurcation. A technique is also presented for the exhaustive calculation of oscillation modes in multidevice oscillators and oscillators loaded with multiresonance networks. The new methodologies are illustrated through their application to a power amplifier and a multimode oscillator.This work was supported by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF/FEDER) under research projects TEC2014-60283-C3-1-R and TEC2017-88242-C3-1-R

Analysis of superregenerative oscillators in nonlinear mode

Superregenerative oscillators in a nonlinear mode are investigated in detail using methodologies based on envelope transient, complemented with additional algorithms. A maximum-detection technique is applied to obtain the input-power threshold for nonlinear operation under different implementations of the quench signal. A mapping procedure enables the prediction of hangover and self-oscillation effects. It is based on the detection of the sequence of local maxima in the envelope amplitude after the application of a single input pulse. Using a contour-intersection method, and depending on the analysis time interval, it is possible to quantify the hangover effects and obtain the oscillation boundary, in terms of any two significant parameters. Then, a compact time-variant behavioral model is derived, valid in the absence of hangover and self-oscillation effects. It consists of a single time-variant Volterra kernel and is applicable provided that the amplitude transitions occur outside the sensitivity interval. Various methodologies are tested in a practical FET-based oscillator at 2.7 GHz. The prototype has been manufactured and measured, obtaining good agreement with the analysis results.This work was supported by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF/FEDER) under the research project TEC2017-88242-C3-1-R

Envelope-domain analysis and modeling of super-regenerative oscillators

An envelope-domain methodology for the numerical modeling of super-regenerative oscillators (SROs) is presented. The main advantage is its generality of application to transistor-based oscillators with arbitrary topology. Initially, a stability analysis of the nonoscillatory steady-state solution, forced by the quench signal, is performed. It is based on the calculation of a linear-time-variant (LTV) transfer function, obtained by linearizing the circuit envelope-domain equations about the nonoscillatory regime. Under moderate quench frequencies, it will be possible to estimate the SRO normalized envelope and sensitivity function from the detected dominant pair of complex-conjugate poles. In the general case, the SRO oscillatory response is modeled with a numerical method, valid under linear operation with respect to the input signal. This is based on the calculation of the LTV impulse response from a time-frequency transfer function obtained under a small-signal sinusoidal excitation. The LTV impulse response enables a straightforward determination of the sensitivity time interval and time distance to the envelope maximum. An integral expression, in terms of the LTV transfer function, will provide the SRO response to any small-signal input with any arbitrarily carrier frequency and modulation. The methodology has been successfully validated through its application to an SRO at 2.7 GHz, which has been manufactured and measured.This work was supported by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF/FEDER) under Research Project TEC2014-60283-C3-1-R and Research Project TEC2017-88242-C3-1-

Experimental investigation of bifurcation behavior in nonlinear microwave circuits

We present an experimental technique to study bifurcations in periodically forced nonlinear microwave circuits, including even physically unstable (periodic) steady states. The designer specifies a key node in the circuit being studied (often associated with an active device) and the method synthesizes a voltage waveform to match the waveform at the selected node so that no current flows across the interface. This null condition is maintained while a parameter, such as bias voltage, is varied over a specified range. The addition of the external nulling source is able to stabilize a steady state that would be unstable in the original circuit. Various applications are presented.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)

Analysis of a frequency divider by two based on a differential nonlinear transmission line

A recently proposed frequency divider by two, based on differential nonlinear transmission line, acting like a reflective distributed resonator, is analyzed in-depth. The flip bifurcation locus of a single cell is obtained analytically, which enables an understanding of the divider behavior and an initial estimation of its element values. The possibility to modify the division threshold and bandwidth through the proper selection of an additional linear capacitor is demonstrated. The influence of the number of cells on the division bandwidth and on the generated standing wave at the subharmonic frequency is also investigated. The techniques have been applied to two frequency divider with 1.5 and 2.2 GHz input frequency.Spanish project TEC2011-29264-C03-01 for financial support

Oscillation modes in symmetrical wireless-locked systems

Time synchronization of multiple elements of a wireless network can be achieved through the wireless coupling of their oscillator circuits. Most previous works on wireless locking of oscillators analyze the system in an idealized manner, representing the oscillator elements with phase models and describing the propagation effects with constant scalar coefficients and time delays. Here, a realistic analysis of the wireless system is presented, which relies on the extraction of the oscillator models from harmonic-balance (HB) simulations and takes into account the antenna gains and propagation effects. The most usual network configurations, corresponding to ring, fully connected, and star topologies, are investigated in detail. In symmetric conditions, the oscillation modes are detected through an eigenvalue/eigenvector calculation of an equivalent coupling matrix. For each particular mode, the system is analyzed in the following manners: by means of an analytical formulation, able to provide all the coexistent solutions, and through a circuit-level HB simulation of an equivalent system with a reduced number of oscillator elements. The stability properties are determined by means of a perturbation system of general application to any coupled structure. A specific formulation is also derived to predict the impact of discrepancies between the oscillator elements. All the results have been validated with independent circuit-level simulations and measurements.This work was supported in part by the Spanish Ministry of Economy and Competitiveness under the research project TEC2017-88242-C3-1-R, in part by the European Regional Development Fund (ERDF/FEDER), in part by Juan de la Cierva Research Program under IJCI-2014-19141, and in part by the Parliament of Cantabria under the project Cantabria Explora 12.JP02.64069