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Subharmonic oscillation stands as a critical concern in the context of ripple-based constant-on-time (COT) controllers. While this issue can be mitigated through the application of the virtual inductor current (VIC) technique, it comes at the cost of load transient response. To achieve both high stability and rapid transient response, this paper introduces an Output Capacitor Voltage Ripple Compensation (VRC) technique. This technique minimizes the phase delay attributed to output capacitance by introducing a virtual output ripple ( $\text{V}_{\mathrm {VOR}})$ inversely related to the feedback voltage. The VVOR serves to expedite the load transient response by counteracting the additional virtual inductor current injection during load transients. Utilizing a $0.13\mu \text{m}$ BCD technology, a synchronous buck converter is integrated with the proposed VRC-COT controller, showcasing exceptional stability across a load current range of 0 to 8A, even when equipped with a 30% $88\mu \text{F}$ ceramic output capacitor. Additionally, the load transient response exhibits reduced overshoot and undershoot, measuring at 120mV and 130mV, respectively, in response to load steps from 0A to 8A within a $10\mu \text{S}$ timeframe.

The unexpected subharmonic oscillation with an evident asymmetric feature is one of the most salient dynamical behaviors in the grid-connected inverter, which will bring disaster to normal system operation. In view of time-varying equilibrium points and complex dynamical response of the system, this asymmetrical subharmonic oscillation is difficult to predict through traditional analysis methods. In this paper, a ripple-based current model is proposed to describe the special subharmonic oscillation of single-phase grid-connected DC–AC inverter with the One-Cycle Compensation (OCC) method. Based on the proposed model, the influence of dynamic response and the time-varying equilibrium points on the stability of the system are comprehensively analyzed, and the internal mechanism of asymmetric subharmonic oscillation is revealed. Furthermore, the critical points of the subharmonic oscillation at the beginning and the end of the whole circuit cycle are predicted. Particularly, a global stable operation boundary is identified from the ripple-based analysis to optimize circuit design. Finally, circuit experiments are performed to verify these numerical and theoretical results. [ABSTRACT FROM AUTHOR]

Traditional local-averaged state-space modeling for peak current mode (PCM) controls fails to explain the subharmonic oscillation phenomenon when the spectrum is higher than half of the switching frequency. To address this problem, this paper presents a small-signal modeling method in the z-domain, and builds a discrete linear model for the current loop of a full-bridge DC-DC converter. This discrete model is converted into a second-order continuous model that is able to represent the system performance with a wider frequency range. A frequency-domain analysis shows that this model can be used to explain the subharmonic oscillations and unstable characteristics. This provides an engineering guideline for the practical design of slope compensation. The effectiveness of the proposed modeling method has been verified by simulation and experimental results with a prototype working in the Buck mode. [ABSTRACT FROM AUTHOR]

In this brief a constant on–time (COT) buck converter with high efficiency and low offset voltage is presented. It was fabricated with a TSMC 0.18 μm 1P6M CMOS process. An additional current feedback path (ACFP) technique was incorporated to avoid subharmonic oscillation of the converter without a large equivalent series resistance (ESR) capacitor, thus reducing the output voltage ripple. However, the ACFP causes an offset voltage on the converter’s output voltage dc level with loading changes. The proposed offset cancellation can remove the dc level from current sensor in ACFP. Hence, the offset voltage can be reduced through a proposed offset cancellation technique. The experimental results revealed that the offset voltage of the proposed converter is 8mV under a 0.1A to 1A loading changes. The converter’s maximum power efficiency is 93.4 %, and its output voltage ripple is less than 18mV. [ABSTRACT FROM AUTHOR]

This paper focuses on the steady-behavior of a differential boost inverter used for generating a sinewave AC voltage in rural areas. The analysis of its dynamics will be performed using an accurate approach based on discrete time models and Floquet theory and adopting a quasi-static approximation. In particular, the undesired subharmonic oscillation exhibited by the inverter will be analyzed and its boundary in the parameter space will be predicted and delimited. Combining analytical expressions and computational procedures to determine the quasi-static duty cycle, subharmonic oscillation is accurately predicted. It is found that subharmonic oscillation takes place at critical values of the sinewave voltage reference cycle, which can cause distortion to the input current and degrade the harmonic content of the output voltage. The results provide useful information for the design of the boost inverter to avoid distortion caused by subharmonic oscillation. Namely, the minimum value of the compensation slope and the maximum proportional gain of the AC output voltage controller guaranteeing a pure sinewave voltage and clean inductor current during the entire AC cycle will be determined. Numerical simulations performed on the switched model implemented using PSIM© software confirm the theoretical predictions. [ABSTRACT FROM AUTHOR]

Summary: A unified small‐signal modeling method is proposed for current controlled converters using discrete‐time analysis method. The relationship between the controlled current and the duty ratio is performed by discrete‐time analysis which automatically incorporates the sample‐and‐hold effect. The proposed small‐signal mode is accurate when the modulation frequency varies from zero to half the switching frequency. It is very easy to apply the proposed model to all current controlled converters. When an amplitude peak in the current‐loop gain is observed at half the switching frequency, subharmonic oscillation can be predicted. Based on the proposed model, a time‐length compensation algorithm for elimination of subharmonic oscillation can be analyzed while other small‐signal modeling methods cannot be applied in this condition. Simulations and experiments are committed to verify the analysis. [ABSTRACT FROM AUTHOR]

We review recent advances in the research on quantum parametric phenomena in superconducting circuits with Josephson junctions. We discuss physical processes in parametrically driven tunable cavity and outline theoretical foundations for their description. Amplification and frequency conversion are discussed in detail for degenerate and nondegenerate parametric resonance, including quantum noise squeezing and photon entanglement. Experimental advances in this area played decisive role in successful development of quantum limited parametric amplifiers for superconducting quantum information technology. We also discuss nonlinear down-conversion processes and experiments on self-sustained parametric and subharmonic oscillations. [ABSTRACT FROM AUTHOR]

Linear ramp slope compensation (LRC) and quadratic slope compensation (QSC) are commonly implemented in peak-current-mode-controlled dc–dc converters in order to minimize subharmonic and chaotic oscillations. Both compensating schemes rely on the linearized state-space averaged model (LSSA) of the converter. The LSSA ignores the impact that switching actions have on the stability of converters. In order to include switching events, the nonlinear analysis method based on the Monodromy matrix was introduced to describe a complete-cycle stability. Analyses on analog-controlled dc–dc converters applying this method show that system stability is strongly dependent on the change of the derivative of the slope at the time of switching instant. However, in a mixed-signal-controlled system, the digitalization effect contributes differently to system stability. This paper shows a full complete-cycle stability analysis using this nonlinear analysis method, which is applied to a mixed-signal-controlled converter. Through this analysis, a generalized equation is derived that reveals for the first time the real boundary stability limits for LRC and QSC. Furthermore, this generalized equation allows the design of a new compensating scheme, which is able to increase system stability. The proposed scheme is called polynomial curve slope compensation (PCSC) and it is demonstrated that PCSC increases the stable margin by 30% compared to LRC and 20% to QSC. This outcome is proved experimentally by using an interleaved dc–dc converter that is built for this work. [ABSTRACT FROM AUTHOR]

Abstract The control and dynamic stability analysis of roll oscillation of the strongly nonlinear semisubmersible system are not widely investigated in the frequency domain. In particular, the suppression of undesirable nonlinear responses like period doubling route to chaos, subharmonic oscillation etc. of such a system under harmonic excitation has not been attempted so far. The nonlinear semisubmersible system is characterized by a fairly strong quintic nonlinearity arising out of the restoring action. In the present study, the incremental harmonic balance method along with arc-length continuation technique (IHBC) is employed to identify the primary and higher order subharmonic responses of the system. Stability of the responses is checked by the Floquet's theory. A reformulation of the IHBC is then presented to control the responses of the semisubmersible system with quintic nonlinearity under state feedback control with time delay. The stability of the controlled responses is examined by applying the semi-discretization method for delay differential equation. It is shown that by considering appropriate feedback gains and time delay, (i) appreciable reduction of the amplitudes of primary responses, (ii) exclusion of all higher order subharmonics 2T, 3T, 4T, 8T, and (iii) the reduction of the extent of domain of various types of instability phenomena like, bifurcation of solutions, jump phenomena, chaotic responses etc. are possible. Further, it is found that the negative velocity feedback gain is much more effective than the full state feedback gains or varying the time delay parameter for better control of the roll oscillation of the system. Highlights • Studied roll oscillation of nonlinear semisubmersible under state feedback control with time-delay. • Employed IHB method with continuation technique. • Applied Floquet theory and semi discretization technique for the stability of periodic motions. • Investigated the effects of control parameters on the suppression of responses. • Validated results with numerical integration. [ABSTRACT FROM AUTHOR]

This paper focuses on the steady-behavior of a differential boost inverter used for generating a sinewave AC voltage in rural areas. The analysis of its dynamics will be performed using an accurate approach based on discrete time models and Floquet theory and adopting a quasi-static approximation. In particular, the undesired subharmonic oscillation exhibited by the inverter will be analyzed and its boundary in the parameter space will be predicted and delimited. Combining analytical expressions and computational procedures to determine the quasi-static duty cycle, subharmonic oscillation is accurately predicted. It is found that subharmonic oscillation takes place at critical values of the sinewave voltage reference cycle, which can cause distortion to the input current and degrade the harmonic content of the output voltage. The results provide useful information for the design of the boost inverter to avoid distortion caused by subharmonic oscillation. Namely, the minimum value of the compensation slope and the maximum proportional gain of the AC output voltage controller guaranteeing a pure sinewave voltage and clean inductor current during the entire AC cycle will be determined. Numerical simulations performed on the switched model implemented using PSIM© software confirm the theoretical predictions.

Two main methods for controlling switching converters exist in the literature. The direct one is the voltage mode control, which suffers from some disadvantages such as slow response to load variations and an input voltage-dependent total loop gain. The current mode control can overcome these problems but at the expense of extra cost and more complex control design. V1 concept is a new promising control technique for designing voltage mode control of buck-type converters with an optimal response similar to current mode control. In this paper, the dynamics and the stability of buck converters under V1 control are studied. In particular, subharmonic oscillation limits in the parameter space are addressed. First, a closed-loop state-space model is derived and then used to formulate an analytical matrix-form expression for predicting the stability limit of the system. Using this expression, multi-parametric stability boundaries are obtained. It is shown that the equivalent series inductance of the output capacitor can narrow the stability region. It is also demonstrated that the integral action in the feedback loop of a V1-controlled buck converter has a negligible effect on the subharmonic oscillation boundary. The theoretical analysis is validated through numerical simulation of the circuit-level switched model of the system. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Closed-form subharmonic oscillation conditions are derived and experimentally verified for the current-controlled boost converter in continuous conduction mode with closed voltage feedback loop. Typical voltage-loop phase-lag and type-II compensators are analyzed. The well-known open-voltage-loop instability condition becomes a special case. It is found that, unlike in the buck converter, the ripple at the output of the compensator can improve the stability. The analysis is extended to other boost-like converters with pulsating output currents, such as buck–boost, flyback, and SEPIC converters. [ABSTRACT FROM AUTHOR]

A ternary parametron is a logical element consisting of a resonator whose three oscillation phases are controlled by the seed signal applied before the resonator is excited. It is realized by a subharmonic oscillator of order three by forcing. It is known that for sinusoidal forcing, a forcing period exactly one-third of the free period is impossible. To overcome this difficulty, an oscillation amplitude-dependent forcing, or impulsive forcing, is proposed, and by numerical simulation, the stable oscillation and the phase control by seed signals are shown. It is found that the rising edge of the keyed forcing signal triggers the definite phase of oscillation. This is a biasing to the seed signal and can be regarded as a type of noise for the seed signal. Its level can be found by numerical simulation. The seed signal level needs to be sufficiently larger than the noise level for proper operation. [ABSTRACT FROM AUTHOR]

New exact critical conditions for predicting subharmonic instability in switching regulators are approximated by simple design-oriented expressions valid under practical conditions. These simplified expressions contain the ripple and slope information of the feedback control signal. Depending on the converter topology, the controller used and values of parasitic parameters, either the slope or the ripple can be dominant in predicting instability. A discussion on the validity of this interpretation is illustrated through six different examples of switching regulators using the concept of the spectral radius and the relative degree of the system loop. Using this approach, the boundary between the desired stable region and the subharmonic instability can be easily obtained. The theoretical results are validated by means of numerical simulations. [ABSTRACT FROM PUBLISHER]

General closed-form subharmonic oscillation conditions are obtained for V^2 -controlled buck converters. Both constant switching period and constant on-time operations are analyzed with the outer voltage loop closed or open. Once an arbitrary linear feedback is given, the associated closed-form stability condition of the converter is readily obtained. Past research results based on the describing function technique and the Floquet theory become special cases. This paper provides an alternative and easier method to analyze subharmonic instability. [ABSTRACT FROM PUBLISHER]

This paper is an extension of the author's recent research in which only buck converters were analyzed. Similar analysis can be equally applied to other types of converters. In this paper, a unified model is proposed for buck, boost, and buck-boost converters under peak or average current mode control to predict the occurrence of subharmonic oscillation. Based on the unified model, the associated stability conditions are derived in closed forms. The same stability condition can be applied to buck, boost, and buck-boost converters. Based on the closed-form conditions, the effects of various converter parameters including the compensator poles and zeros on the stability can be clearly seen, and these parameters can be consolidated into a few ones. High-order compensators such as type-II and PI compensators are considered. Some new plots are also proposed for design purpose to avoid the instability. The instability is found to be associated with large crossover frequency. A conservative stability condition, agreed with the past research, is derived. The effect of the voltage loop ripple on the instability is also analyzed. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We investigate the nonlinear responses of graphene-matrix composite to harmonic and subharmonic resonances. Assuming anisotropic mechanical properties of graphene, we derive size-dependent governing equations of motion for graphene resting on a matrix based on the von Karman hypotheses and nonlocal elasticity theory. Response of graphene oscillation under a uniform pressure is obtained using the averaging method. We study the effects of length scale as well as the presence of the elastic matrix on harmonic and subharmonic oscillations of graphene. Our results reveal that subharmonic oscillation of order 1/3 can occur when the ratio of excitation to natural frequencies exceeds three. Also, the subharmonic oscillation of the system is triggered in an appropriate initial condition. [ABSTRACT FROM AUTHOR]

This paper presents the derivation and experimental verification of the closed-form subharmonic instability condition of the constant-frequency peak-current-controlled buck converter with closed voltage feedback loop. The well-known open-voltage-loop instability condition becomes a special case. The instability limits in the converter parameter space are readily obtained from the result. It is found that closing the voltage feedback loop tends to degrade the stability margin. One can clearly see the effects of all converter parameters (such as the pole, zero, and gain of the voltage-loop compensator, the equivalent series resistance of the output capacitor, the presence of a second output capacitor, and the switch turn-off delay) on the instability. [ABSTRACT FROM AUTHOR]

Subharmonic oscillation is an undesired phenomenon in switching converters. In past studies, its prediction has been mainly tackled by explicitly deriving a discrete-time model and then linearizing it in the vicinity of the operating point. However, the results obtained from such an approach cannot be applied for design purpose except for simple cases such as peak or valley current-mode control (CMC). Alternatively, in this brief, this phenomenon is analyzed by using a unified formal symbolic approach that can be applied for different control strategies. This approach is based on expressing the condition for subharmonic oscillation occurrence using Fourier series and then converting the result into a matrix-form expression that explicitly depends on the system parameters, making the results directly applicable for design purpose. Under certain practical conditions concerning these parameters, the matrix-form expression can be approximated by standard polynomial functions depending on the operating duty cycle. The results presented in this work clearly generalize the well-known stability condition of peak/valley CMC. [ABSTRACT FROM PUBLISHER]

Recently, constant on-time V2 control, and its variety named constant on-time control, or constant on-time ripple-based control is more and more popular in industry products due to features of high light-load efficiency, simple implementation, and fast transient response. However, subharmonic oscillation occurs when using multilayer ceramic caps due to the lagging phase of the capacitor voltage relative to the inductor current. External ramp compensation is one simple solution to solve the instability issue. However, the characteristics of constant on-time V2 control with external ramp are not fully understood and no explicit design guidelines for the external ramp are provided. This paper investigates the small-signal characteristics of constant on-time V2 control with external ramp compensation by providing a factorized, easy-to-use small-signal model. The external ramp is a critical parameter because it directly affects the position and damping of two pairs of double poles. Based on this fact, design guidelines of the external ramp for optimal dynamic performance are provided. Moreover, the effect of duty cycle is investigated. Finally, the small-signal experimental results and load transient performance are presented to verify the small-signal analysis and proposed design guideline. [ABSTRACT FROM AUTHOR]

A general closed-form subharmonic stability condition is derived for the buck converter with ripple-based constant on-time control and a feedback filter. The turn-on delay is included in the analysis. Three types of filters are considered: low-pass filter (LPF), phase-boost filter (PBF), and inductor current feedback (ICF) which changes the feedback loop frequency response like a filter. With the LPF, the stability region is reduced. With the PBF or ICF, the stability region is enlarged. Stability conditions are determined both for the case of a single output capacitor and for the case of two parallel-connected output capacitors having widely different time constants. The past research results related to the feedback filters become special cases. All theoretical predictions are verified by experiments. [ABSTRACT FROM AUTHOR]

Highlights: [•] The dynamics and stability of DC–DC converters fed from voltage source is wellknown. [•] We have investigated the nonlinear phenomena of a boost converter fed from PV panel. [•] Four different types of controllers (e.g., peak current mode) have been used. [•] For all the cases the range of stability in the parameter space have been obtained. [•] The results have been validated through experimental observation. [ABSTRACT FROM AUTHOR]

The complex dynamics and coexisting fast-slow scale instability in current-mode controlled buck converter with constant current load (CCL), operating in both continuous conduction mode (CCM) and discontinuous conduction mode (DCM), are investigated in this paper. Via cycle-by-cycle computer simulation and experimental measurement of current-mode controlled buck converter with CCL, it is found that a unique fast-slow scale instability exists in the second-order switching converter. It is also found that a unique period-doubling accompanied by Neimark-Sacker bifurcation exists in this simple second-order converter, which is different from period-doubling or Neimark-Sacker bifurcations reported previously. Based on a nonlinear discrete-time model and the corresponding Jacobian, the effects of CCL and input voltage on the dynamics of current-mode controlled buck converter are investigated and verified theoretically. Fixed point analysis for slow-scale low-frequency oscillation is also given to verify the dynamics and the coexisting fast-slow scale instability. [ABSTRACT FROM AUTHOR]

SUMMARY Exact and approximate sampled-data models in closed forms are derived for switching DC-DC converters under peak/valley current-mode control. The corresponding sampled-data poles and zeros in closed forms are also derived. The location and stability conditions of the poles and zeros, boundary conditions of subharmonic instability, and nulling of the audio-susceptibility are also derived. It is proved that the stable operating range of the source voltage is linearly proportional to the ramp slope. The sampled-data models agree with previous experiment results and accurately predict the subharmonic instability. The different view from the sampled-data model about the number and stability (minimum phase) of pole and zero does not necessarily invalidate the traditional continuous-time averaged model. However, this different view gives better prediction about converter dynamics and is useful for the analog or digital controller design for DC-DC converters. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents a new digital current-mode control technique for single-inductor multiple-output converters in continuous-conduction mode. This predictive control method can calculate all the required duty cycles corresponding to the outputs for the next switching cycle. Since the calculation is based on every current reference representing the load current, all current control loops corresponding to different outputs are theoretically independent, which can greatly reduce the cross regulation. To avoid subharmonic oscillation, proper predictive digital current control is adopted. Inductor current estimation is used to remove the current sensing circuit, and corresponding calibration techniques are presented. Simulation and experimental results are given to verify the effectiveness of this control technique. [ABSTRACT FROM AUTHOR]

This paper presents a multi-MHz buck regulator for portable applications using an auto-selectable peak- and valley-current control (ASPVCC) scheme. The proposed ASPVCC scheme can enable the current-mode buck regulator to reduce the settling-time requirement of the current sensing by two times. In addition, the dynamically biased shunt feedback technique is employed to improve the sensing speed and the sensing accuracy of both the peak and valley current sensors. With both ASPVCC scheme and advanced current sensors, the buck regulator can thus operate at high switching frequencies with a wide range of duty ratios for reducing the required off-chip inductance. [ABSTRACT FROM AUTHOR]

Abstract: The present paper describes an improved version of the elliptic averaging method that provides a highly accurate periodic solution of a non-linear system based on the single-degree-of-freedom Duffing oscillator with a snap-through spring. In the proposed method, the sum of the Jacobian elliptic delta and zeta functions is used as the generating solution of the averaging method. The proposed method can be used to obtain the non-odd-order solution, which includes both even- and odd-order harmonic components. The stability analysis for the approximate solution obtained by the present method is also discussed. The stability of the solution is determined from the characteristic multiplier based on Floquet’s theorem. The proposed method is applied to a fundamental oscillator in a non-linear system. The numerical results demonstrate that the proposed method is very effective for analyzing the periodic solution of half-swing mode for systems based on Duffing oscillators with a snap-through spring. [ABSTRACT FROM AUTHOR]

This paper aims at the representation and analysis of the mathematical model for the tracking of a receiver in the spread spectrum communication using the deterministic differential equation. The tracking system is based on the 2Δ-type baseband delay-lock loop using the pseudorandom waveform by M-sequence. Since the M-sequence has the periodicity, the correlator output is approximated by the Fourier series expansion with the power series coefficients and is treated as a function of the time and the phase difference between the pseudorandom waveforms. By those approaches, the basic equation for the tracking system is derived as a second-order nonautonomous nonlinear differential equation for the phase difference, including the nonlinear restoring force, the forced input, and the parameter excitation. A computer simulation is applied to the derived nonlinear differential equation, and it is verified that the synchronization process represented clearly as a trajectory on the phase plane. The behavior in the neighborhood of the phase difference zero is analyzed, and it is shown that the delay-lock loop may become unstable, resulting in the one-half subharmonic oscillation. The Poincaré map is constructed by a computer simulation to verify the forementioned property. [ABSTRACT FROM AUTHOR]

Parametric oscillations which occur in a nonlinear two-phase uircuit with nonlinear reactors and capacitors are applied to many kinds of power devices. When the devices are operating under light load, the output voltage of these devices exhibits unstable oscillations, i.e., subharmonic oscillations, quasi-periodic oscillations or chaotic oscillations. This paper analyzes the abnormal oscillations and confirms the validity of the analysis experimentally. In this paper, new circuit equations are presented that can be applied to the analysis of a two-phase nonlinear circuit in which reactor saturation curves are sharp. It is clarified by this analysis that the parametric oscillations of the circuit which contains reactors of moderate saturation characteristics become subharmonic or quasi-periodic with the existence of resistances of the power supply side. Also, when the nonlinearity of the reactors in the circuit is sharp, the parametric oscillations may change into chaotic oscillations in addition to the aforementioned oscillations. [ABSTRACT FROM AUTHOR]