PPV-based modeling and event-driven simulation of injection-locked oscillators in SystemVerilog

Coupled oscillator networks (CONs) occur in various domains such as biology and electronics. The nonlinear phenomenon of “injection locking” leads to frequency synchronization among the oscillators in a CON. In this chapter, we first obtain simplified tools to analyze the phenomenon of injection locking and present the Generalized Adler’s equation. The Generalized Adler’s equation is applicable for analyzing the injection locking in any type of oscillator. We demonstrate the use of Generalized Adler’s equation for injection locking analysis on a ring oscillator and a Hodgkin-Huxley neuronal oscillator. Next, we describe efficient system-level simulation techniques (synchronized steady state and transient) for coupled oscillator networks using the Perturbation Projection Vector (PPV) phase macromodel of the oscillator. The coupled oscillator simulation techniques are generic in nature and can be applied for the simulation of any coupled oscillator network, like the ones found in biological or electronic systems. We apply our system-level techniques on a −G m LC (harmonic) and a Hodgkin-Huxley (non-harmonic) CON.

This paper presents an event-driven simulation methodology for injection-locked oscillators. The proposed method adopts a phase-domain macromodel based on a perturbation projection vector (PPV), which expresses the oscillator's phase response using a nonlinear, time-varying differential equation. By expressing the PPV using a piecewise polynomial function, the PPV model can be simulated in an event-driven fashion on a digital logic simulator like SystemVerilog. While this piecewise polynomial approximation introduces a trade-off between the simulation accuracy and speed, our results show that the simulation accuracy quickly reaches its optimal value as the piecewise interval number increases even with first-order polynomials. The experimental results with an LC oscillator, ring oscillator, and burst-mode CDR models operating with both periodic and aperiodic inputs demonstrate that the proposed models achieve speeds up to 2 ~ 30× that of transistor-level Spectre simulations with the same accuracy.

A V-band CMOS injection-locked oscillator (ILO) based on a cross-coupled oscillator configuration containing nMOS and pMOS devices is implemented by using 0.18 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mu</i> m CMOS process. This ILO has a free-running output frequency around 60 GHz while the double push-push technique was used as a frequency multiplier. It also has a wide output locking bandwidth of up to 3.64 GHz (from 59.36 to 63 GHz) as injecting a signal near 15 GHz with the fundamental injection-locked behavior. Controlling the output frequencies by the injected signals, the output signal has a phase noise of 108 dBc/Hz at a 1 MHz offset with consuming only 9.8 mW dc power under a 2 V supply.

This work presented a fundamental injection-locked oscillator (ILO), based on a modified common collector Colpitts topology. An injection buffer improves the sensitivity and reverse isolation and a common base output buffer reduces loading effects. The circuit achieves a wide tuning range of 50 GHz, with a high output power of 2.5 dBm and has a free-running oscillation frequency of 185.8 GHz. The total dc power consumption is 30 mW from a single 2.5 V supply, resulting in a dc-to-RF efficiency of 5.9 %. The circuit is intended for mm-wave communication systems operating around 180 GHz and can be combined with a frequency multiplier to achieve a spectral-clean reference signal, with excellent noise performance. The final chip occupies an area of 0.32 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and is implemented in a 130 nm SiGe BiCMOS process, which offers a maximum oscillation frequency f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> of 450 GHz. This circuit demonstrates one of the highest efficiencies and locking ranges for an ILO, compared to the state-of-the-art.

We present a new gate-level approach to current simulation. We use a symbolic model of current pulses that takes accurately into account the dependence on the switching conditions. We then construct current waveforms during event-driven logic simulation by means of pulse composition. We obtain satisfying accuracy on time-domain current waveforms and on peak current estimates, while maintaining performance comparable with traditional gate-level simulation.

We demonstrate an interferer suppression method based on feed-forward cancellation that uses an injection-locked oscillator (ILO) to extract interferers from the incident spectrum in an auxiliary receiver. The technique is expected to be useful in environments where a strong narrowband interferer appears along with a wideband desired signal, such as ultra-wideband (UWB) and emerging cognitive-radio applications. The ILO is further embedded within a phase-locked loop which allows self-tuning of the ILO center frequency and automatic phase alignment between the main signal path and auxiliary path. An IC that uses this approach is implemented in a 0.18 mum CMOS process. In measurement, the chip demonstrates 20 dB suppression for phase- and frequency-modulated interferers while maintaining around 18 dB power gain and a noise figure below 5 dB, including an off-chip balun for the desired signal. Techniques for canceling amplitude modulated interferers, though not included in the fabricated chip, can also be implemented by adding an amplitude tracking gain-control loop.

A 423.5-450 MHz low-power OOK/FSK/QPSK wireless transmitter is proposed with multi-phase injection locking and frequency multiplication techniques. A low-frequency phase-rotation-based frequency synthesizer is employed to provide multichannel support and FSK modulation. The far-out phase noise caused by the quantization noise of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta \Sigma $ </tex-math></inline-formula> modulator is filtered through a proposed N-path filter and two-stage injection-locked ring oscillators (ILROs). Phase modulation is achieved by changing the offset frequency between the oscillator free-running frequency and the injected frequency to control the steady-state phase offset of the injection-locked oscillator. Finally, a current mode class-D edge-combining power amplifier with high energy efficiency is used to achieve 5X frequency multiplication to generate the required carrier frequency. A prototype transmitter is fabricated in TSMC 90 nm CMOS process. It is shown that the device delivers an output power of −7 dBm with a power consumption of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$870 \mu \text{W}$ </tex-math></inline-formula> under a 0.75 V supply voltage and supports data rates of 40/5/80 Mbps under OOK/BFSK/QPSK modulation, respectively, achieving the energy efficiency of 11 pJ/bit and 23% global efficiency.

Electromagnetic waves at millimeter-wave/sub-Terahertz (mm-Wave/sub-THz) frequencies (60 GHz ~ 300 GHz) can interact uniquely with gas molecules, and by detecting the frequency responses of the absorption gas cells, the unique spectral patterns of the gas molecules can be identified and detected with high accuracy. As a result, mm-Wave/sub-THz frequency range has attracted growing interest for rotational spectroscopy for its great potential in environment safety, homeland security and health care applications such as air quality monitoring, toxic gas detection and human breath analyses. With a continuous scaling of CMOS technologies, it is feasible to consider CMOS to significantly reduce the cost and size of rotational spectroscopy systems. An ultra-wideband and low-phase-noise mm-Wave and sub-THz signal generation is essential in a rotational spectroscopy system to improve its detection sensitivity. In this dissertation, sub-THz signal generation systems with novel circuit techniques are proposed in CMOS process to generate high purity local oscillator (LO) signals with ultra-wideband from 60 GHz to 280 GHz. Firstly, magnetic-tuning technique for varactor-less mm-Wave/sub-THz oscillators is analyzed and discussed. Based on the results, a dual-band voltage-controlled oscillator (VCO) prototype is proposed to achieve a frequency tuning range from 95.7 GHz to 110.5 GHz with FoM&lt;sub&gt;T&lt;/sub&gt; of -181.7 dBc/Hz while consuming 6.2 mW. In addition, for quadrature signals generation, a transformer coupling quadrature VCO (TC-QVCO) prototype is proposed with a frequency tuning range from 89.4 GHz to 100.9 GHz and phase error less than 2.6°. Moreover, a fundamental sub-THz oscillator prototype is designed and demonstrated near the f&lt;sub&gt;max&lt;/sub&gt; with a peak output power of -12.1 dBm and frequency tuning range from 157.8 GHz to 166.3 GHz, and a 340-GHz push-push oscillator above the f&lt;sub&gt;max&lt;/sub&gt; is also presented. Secondly, an ultra-wideband sub-THz signal generation system is proposed for rotational spectroscopy employing a magnetic-tuning varactor-less quad-band oscillator (QB-VCO), a locking-range-enhanced dual-mode injection-locked frequency divider (DM-ILFD), a power-efficient injection-locked oscillator (ILO) as a driver, and self-mixing-based frequency multipliers (FMs) for frequency extension. The QB-VCO demonstrated a continuous frequency tuning range of 32% at around 70 GHz. The proposed novel dual-mode DM-ILFD achieves a locking-range of 31.8% in the divide-by-2 mode and 41.2% in the divide-by-4 mode, respectively. The signal generation system achieves a record frequency tuning range from 58.8 GHz to 275.6 GHz with FoM&lt;sub&gt;T&lt;/sub&gt; of -190.4 dBc/Hz while consuming only 54 mW. Thirdly, a harmonic extraction technique is proposed to enhance the third-harmonic amplitude and further improve the phase noise performance of the sub-THz LO generation. Based on this technique, a scalable cascaded injection-locked frequency multiplier chain architecture is proposed, which can generate sub-THz signal from a low RF tone. A prototype of a cascaded two-stage ILFM-based sub-THz LO generation is demonstrated with ultra-low power consumption of 2.5 mW. The prototype achieves a frequency tuning range of 23.7% from 153.9 GHz to 195.3 GHz with a phase noise of -106.8 dBc/Hz at 10-MHz offset, corresponding to FoM of -186.5 dBc/Hz and FoM&lt;sub&gt;T&lt;/sub&gt; of -194.1 dBc/Hz. Finally, by employing a radio frequency (RF) sub-sampling PLL (SS-PLL) cascaded with an injection-locked-based mm-Wave LO generation chain, and a sub-THz mixer and push-push extraction for frequency extension, a fully integrated wideband and low phase noise sub-THz frequency synthesizer is proposed for the rotational spectroscopy system. Third-harmonic and fourth-harmonic enhancement methods are proposed for the ILFMs with different multiplication ratios. In addition, a novel frequency tracking loop (FTL) with automatic frequency and amplitude calibration is proposed for the ILFMs. Moreover, a distributed biased technique is also proposed to improve the linearity of the wideband sub-THz oscillator. Designed with 65-nm CMOS process, the sub-THz frequency synthesizer measures a close-loop frequency tuning range from 61.2-to-100.8 GHz, 122.4-to-136.8 GHz and 198.5-to-273.6 GHz, measures a close-loop phase noise from -79.3 dBc/Hz to -95.4 dBc/Hz at 1-MHz offset across the whole tuning range, achieves an integrated jitter of 124 fs, demonstrates an output power of -11 dBm and DC-to-RF efficiency of 0.16% at the frequency above 200 GHz.

The aim of this paper is to highlight the aspects of the baroreflex control of the cardiovascular system that could be relevant to the analysis and modelling of cardiovascular oscillations and regulation. In particular, complex and/or controversial issues of the baroreflex control are addressed on the basis of results obtained in previous studies by others as well as by our group. Attention has been focused on time-variant and nonlinear characteristics of the baroreflex function and on the influence of this physiological mechanism on different frequency regions of blood pressure and heart rate spectra.

Synchronisability of limit cycle oscillators has been measured by the width of the synchronous frequency band, known as the Arnold tongue, concerning external forcing. We clarify a fundamental limit on maximizing this synchronisability within a specified extra low power budget, which underlies an important and ubiquitous problem in nonlinear science related to an efficient synchronisation of weakly forced nonlinear oscillators. In this letter, injection-locked Class-E oscillators are considered as a practical case study, and we systematically analyse their power consumption; our observations demonstrate the independence of power consumption in the oscillator from power consumption in the injection circuit and verify the dependency of power consumption in the oscillator solely on its oscillation frequency. These systematic observations, followed by the mathematical optimisation establish the existence of a fundamental limit on synchronisability, validated through systematic circuit simulations. The results offer insights into the energetics of synchronisation for a specific class of injection-locked oscillators.

This paper presents a low-power small-area DLL-based frequency multiplier. Instead of using edge combiner-based clock synthesis scheme, the proposed frequency multiplier utilizes a ring oscillator, which is controlled by a DLL. An injection-locked slave ring oscillator is used for jitter suppression. The implementation of the proposed structure in 130-nm CMOS occupies an area of 0.02 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . It operates in the frequency range of 100 MHz to 1.5 GHz while consuming 24-mW power from a 1.2-V supply at 1.5 GHz. The measured output phase noise at 1.5 GHz is -100.1 dBc/Hz at a 4-MHz frequency offset.

This paper presents an LO phase shifter with frequency tripling for 28-GHz 5G transceivers. The phase shifting and frequency tripling are achieved using an injection-locked oscillator and injection-locked frequency tripler, respectively. A phase detector based on third harmonic mixing is also implemented and is used to detect the applied phase shift, supporting automatic calibration of the phase shifter. Additionally, an algorithm to automatically tune the oscillators to their respective locking frequency is presented. To test the phase shifter, a 24–30-GHz sliding-IF receiver is implemented. Simulations show that a > 360^circ tuning range over the full 24–30 GHz span is achieved, with a gain variation of 0.11 dB or less, and that the phase detector has an rms phase error of < 2.5^circ . The circuit is implemented in a 28nm FD-SOI CMOS process and the entire chip measures 1080 upmu hbox {m}times 1080 upmu hbox {m}, including pads, and consumes 27–29 mW from a 1 V supply.

This work investigates the nonlinear dynamics of an injection-locked power oscillator inductively coupled to an external resonator. This allows a high-efficiency power transfer while ensuring a constant oscillation frequency versus the coupling factor, unlike free-running implementations. The investigation focuses on the impact of the external-resonator elements on the locking range, output power, efficiency, and phase noise. The aim is to derive a strategy for an optimum selection of these elements. Initially, the effect of the coupled resonator is theoretically studied using a simple oscillator model, based on a cubic nonlinearity. For practical oscillators, two kinds of analysis methods, compatible with the use of commercial harmonic-balance (HB) simulators, are presented. The first one is semianalytical and is based on the extraction of a phase-dependent nonlinear admittance function from HB simulations. The system response is predicted in a flexible and computationally efficient manner, but coupling effects are considered at the fundamental frequency only. The second set of methods is fully based on HB and relies on the combination of a nonlinear immittance function and a Thevenin/Norton equivalent. The impact of the external resonator on the stability properties is analyzed through bifurcation detection. The phase-noise spectrum is predicted with a semianalytical formulation that demonstrates the benefit of the injection-locked operation. For validation, the methods have been applied to a Class-E oscillator at 13.56 MHz.

In this paper, injection-locked MESFET oscillators are analyzed using several numerical models. The injection-locking behavior of the van der Pol equation and of a more complex representation using the Curtice-Cubic MESFET model are investigated. Analysis and experimental results are compared for an NE71083 transistor oscillator operating at 0.5 GHz. The deficiencies of using a van der Pol oscillator model are pointed out. Time-domain results from the complex model exhibiting multicycle and apparently chaotic behaviors are also examined, and point to problems with common nonlinear simulation techniques for these circuits.

SAW oscillators can provide fundamental frequency operation to above 1·5 GHz, with stability and FM noise performance approaching that offered by bulk crystal oscillator technology. Their high fundamental frequency, small size and rugged construction gives SAW technology a unique capability at UHF and microwave frequencies.The low FM thermal noise floor associated with fundamental frequency operation can be combined with the stability and low close‐to‐carrier noise of multiplied bulk crystal oscillators by locking a high frequency SAW oscillator to a bulk crystal reference. SAW oscillator stability is compatible with conventional phase‐locked‐loop techniques and also with injection lock stabilization, and their own low close‐to‐carrier FM noise ensures that such locked sources exhibit minimum phase noise. Furthermore, locked oscillator phase noise is not significantly degraded when extreme operating conditions, such as those experienced in space applications, demand a reduced SAW device Q for reliable locking using either technique.Use of a PLL avoids any need for reference frequency multiplication, and provides additional design flexibility with respect to reference frequency selection and phase noise optimization. Injection locking offers design simplicity and uses fewer frequency control components, which can contribute additional noise in PLL sources.