Using Nyquist or Nyquist-like plot to predict three typical instabilities in DC–DC converters

Abstract

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.

A Time-Domain Asymptotic Approach to Predict Saddle-Node and Period Doubling Bifurcations in Pulse Width Modulated Piecewise Linear Systems

“…Although apparently the condition (22) derived in [4] and [10] and that in (21) do not coincide, they just happen to be the same conditions but expressed differently. Note however that the expression (21) is simpler than the one derived in [4] and [10].…”

“…It has to be mentioned here that in [4], [10] a slightly differently expressed condition has been obtained for the same boundary condition which is reported and adapted here for comparison…”

“…Note however that the expression (21) is simpler than the one derived in [4] and [10]. It must be pointed out that in [6], the steady-state asymptotic approach was only applied to a dc-dc switching buck converter for which A 1 = A 2 obtaining the stability condition only for similar particular cases.…”

“…Most of the PWL systems studied in the literature are characterized by switching among linear subsystems when certain time-varying and T − periodic boundaries in the state space are reached. This is the case of Pulse Width Modulation (PWM) systems like switching dc-dc power converters [3], [4], [5], [6], [9], [10], dc-ac inverters [11], temperature control systems [12], switched capacitor networks and chaos generators [13] and hydraulic and fluid valve drivers [14], [15]. In steady-state, during a switching period of length T , a trajectory of these systems starts at time instant nT and is described by the vector field f 1 (x) = A 1 x + B 1 u, intersects a switching boundary described by the equation σ(x(t), t) := Fx(t) − r(t) = 0 at switching instant t s = DT , and then goes to another linear system described by the vector field f 2 (x) = A 2 x + B 2 u, where r is a time-varying T −periodic external signal, x ∈ R n is the vector of the state variables, n is the order of the system A i ∈ R n×n and B i ∈ R n×m , i = 1, 2 are the system state matrices for phase i (i = 1, 2) and u ∈ R m is the vector of the system inputs in both the plant and controller, m being the number of the external inputs to the system which are supposed to be constant within a switching cycle.…”

“…The dynamical behavior and stability analysis of this kind of systems can either be tackled by long-time integration of the continuous-time switched model, discrete-time model and its Jacobian matrix or Floquet theory with Fillipov technique to compute the mondromy matrix [17], [18], [19]. After obtaining the Jacobian or monodromy matrix, critical boundary conditions for some singularities like saddlenode (SN) bifurcation or period-doubling (PD) can be obtained by imposing in the characteristic equation in the zdomain that one of the eigenvalues is equal to +1 or −1 respectively [4]. Another approach recently used in [6] for locating these boundaries which was wrongly called harmonic balance is by expanding the feedback signal into a [6].…”