Chung-Chieh Fang

Chung-Chieh Fang, Eyad H. Abed

Period doubling bifurcation in buck converters is studied by using the harmonic balance method. A simple dynamic model of a buck converter in continuous conduction mode under voltage mode or current mode control is derived. This model consists of the feedback connection of a linear system and a nonlinear one. An exact harmonic balance analysis is used to obtain a necessary and sufficient condition for a period doubling bifurcation to occur. If such a bifurcation occurs, the analysis also provides information on its exact location. Using the condition for bifurcation, a feedforward control is designed to eliminate the period doubling bifurcation. This results in a wider range of allowed source voltage, and also in improved line regulation.

Chung-Chieh Fang

A general critical condition of subharmonic oscillation in terms of the loop gain is derived. Many closed-form critical conditions for various control schemes in terms of converter parameters are also derived. Some previously known critical conditions become special cases in the generalized framework. Given an arbitrary control scheme, a systematic procedure is proposed to derive the critical condition for that control scheme. Different control schemes share similar forms of critical conditions. For example, both V2 control and voltage mode control have the same form of critical condition. A peculiar phenomenon in average current mode control where subharmonic oscillation occurs in a window value of pole can be explained by the derived critical condition. A ripple amplitude index to predict subharmonic oscillation proposed in the past research has limited application and is shown invalid for a converter with a large pole.

Chung-Chieh Fang, Eyad H. Abed

Three local bifurcations in DC-DC converters are reviewed. They are period-doubling bifurcation, saddle-node bifurcation, and Neimark bifurcation. A general sampled-data model is employed to study the types of loss of stability of the nominal (periodic) solution and their connection with local bifurcations. More accurate prediction of instability and bifurcation than using the averaging approach is obtained. Examples of bifurcations associated with instabilities in DC-DC converters are given.

Chung-Chieh Fang

By transforming an exact stability condition, a new Nyquist-like plot is proposed to predict occurrences of three typical instabilities in DC-DC converters. The three instabilities are saddle-node bifurcation (coexistence of multiple solutions), period-doubling bifurcation (subharmonic oscillation), and Neimark bifurcation (quasi-periodic oscillation). In a single plot, it accurately predicts whether an instability occurs and what type the instability is. The plot is equivalent to the Nyquist plot, and it is a useful design tool to avoid these instabilities. Nine examples are used to illustrate the accuracy of this new plot to predict instabilities in the buck or boost converter with fixed or variable switching frequency.

Chung-Chieh Fang

Dynamics and stability of average current control of DC-DC converters are analyzed by sampled-data modeling. Orbital stability is studied and it is found unrelated to the ripple size of the orbit. Compared with the averaged modeling, the sampled-data modeling is more accurate and systematic. An unstable range of compensator pole is found by simulations, and is predicted by sampled-data modeling and harmonic balance modeling.

Chung-Chieh Fang

Sampled-data analysis and harmonic balance analysis are applied to analyze switching DC-DC converters under constant on-time control. Design-oriented boundary conditions for the period-doubling bifurcation and the saddle-node bifurcation are derived. The required ramp slope to avoid the bifurcations and the assigned pole locations associated with the ramp are also derived. The derived boundary conditions are more general and accurate than those recently obtained. Those recently obtained boundary conditions become special cases under the general modeling approach presented in this paper. Different analyses give different perspectives on the system dynamics and complement each other. Under the sampled-data analysis, the boundary conditions are expressed in terms of signal slopes and the ramp slope. Under the harmonic balance analysis, the boundary conditions are expressed in terms of signal harmonics. The derived boundary conditions are useful for a designer to design a converter to avoid the occurrence of the period-doubling bifurcation and the saddle-node bifurcation.

Chung-Chieh Fang

A unified model of voltage mode control (VMC) and current mode control (CMC) is proposed to predict the saddle-node bifurcation (SNB). Exact SNB boundary conditions are derived, and can be further simplified in various forms for design purpose. Many approaches, including steady-state, sampled-data, average, harmonic balance, and loop gain analyses are applied to predict SNB. Each approach has its own merits and complement the other approaches.

Chung-Chieh Fang

Dynamics and stability of average current-mode control of buck converters are analyzed by sampled-data and harmonic balance analyses. An exact sampled-data model is derived. A new continuous-time model "lifted" from the sampled-data model is also derived, and has frequency response matched with experimental data reported previously. Orbital stability is studied and it is found unrelated to the ripple size of the current-loop compensator output. An unstable window of the current-loop compensator pole is found by simulations, and it can be accurately predicted by sampled-data and harmonic balance analyses. A new S plot accurately predicting the subharmonic oscillation is proposed. The S plot assists pole assignment and shows the required ramp slope to avoid instability.

Chung-Chieh Fang

A recent paper [1] (El Aroudi, 2012) misapplied a critical condition (Fang and Abed, 2001) to a well-known example. Even if the mistake is corrected, the results in [1] are applicable only to buck converters and period-doubling bifurcation. Actually, these results are known in Fang's works a decade ago which have broader critical conditions applicable to other converters and bifurcations. The flaws in [1] are identified.

Chung-Chieh Fang

Design-oriented boundary conditions for subharmonic oscillations are of great interest recently. Based on a subharmonic oscillation boundary condition reported in a PhD thesis more than a decade ago, extended new boundary conditions are derived in closed forms for general switching DC-DC converters. Sampled-data and harmonic balance analyses are applied and generate equivalent results. It is shown that equivalent series resistance causes the boundary conditions for voltage/current mode control to have similar forms. Some recently reported boundary conditions become special cases in view of the general boundary conditions derived. New Nyquist-like design-oriented plots are proposed to predict or prevent the occurrence of the subharmonic oscillation. The relation between the crossover frequency and the subharmonic oscillation is also analyzed.

Chung-Chieh Fang

Nonlinear systems, such as switching DC-DC boost or buck converters, have rich dynamics. A simple one-dimensional discrete-time model is used to analyze the boost or buck converter in discontinuous conduction mode. Seven different control schemes (open-loop power stage, voltage mode control, current mode control, constant power load, constant current load, constant-on-time control, and boundary conduction mode) are analyzed systematically. The linearized dynamics is obtained simply by taking partial derivatives with respect to dynamic variables. In the discrete-time model, there is only a single pole and no zero. The single closed-loop pole is a linear combination of three terms: the open-loop pole, a term due to the control scheme, and a term due to the non-resistive load. Even with a single pole, the phase response of the discrete-time model can go beyond -90 degrees as in the two-pole average models. In the boost converter with a resistive load under current mode control, adding the compensating ramp has no effect on the pole location. Increasing the ramp slope decreases the DC gain of control-to-output transfer function and increases the audio-susceptibility. Similar analysis is applied to the buck converter with a non-resistive load or variable switching frequency. The derived dynamics agrees closely with the exact switching model and the past research results.

Chung-Chieh Fang

A general and exact critical condition of saddle-node bifurcation is derived in closed form for the buck converter. The critical condition is helpful for the converter designers to predict or prevent some jump instabilities or coexistence of multiple solutions associated with the saddle-node bifurcation. Some previously known critical conditions become special cases in this generalized framework. Given an arbitrary control scheme, a systematic procedure is proposed to derive the critical condition for that control scheme.

Chung-Chieh Fang

Saddle-node bifurcation occurs in a boost converter when parasitic inductor resistance is modeled. Closed-form critical conditions of the bifurcation are derived. If the parasitic inductor resistance is modeled, the saddle-node bifurcation occurs in the voltage mode control or in the current mode control with the voltage loop closed, but not in the current mode control with the voltage loop open. If the parasitic inductor resistance is not modeled, the saddle-node bifurcation does not occur, and one may be misled by the wrong dynamics and the wrong steady-state solutions. The saddle-node bifurcation still exists even in a boost converter with a popular type-III compensator. When the saddle-node bifurcation occurs, multiple steady-state solutions may coexist. The converter may operate with a voltage jump from one solution to another. Care should be taken in the compensator design to ensure that only the desired solution is stabilized. In industry practice, the solution with a higher duty cycle (and thus the saddle-node bifurcation) may be prevented by placing a limitation on the maximum duty cycle.

Chung-Chieh Fang

In a review paper [1] (El Aroudi, et al., 2005), two stability conditions for DC-DC converters are presented. However, these two conditions were published years earlier at least in a journal paper [2] (Fang and Abed, 2001). In this note, the similar texts of [1] and [2] are compared.