A computer algorithm to determine the steady-state response of nonlinear oscillators

Harmonic Issues Assessment on PWM VSC-Based Controlled Microgrids Using Newton Methods

“…Average models fall within this type of systems, but they have evident limitations to evaluate adverse effect of the harmonic distortion on the performance of the MG. Alternatively, the harmonic domain modeling approach is able to consider the distorted nature of microgrids and also leads to time-invariant representations; however, the proper modeling of closed-loop controls and nonlinearities are still a bottleneck in this domain [20]. In order to deal with these problems, the fast time-domain methods were proposed [21], [22] and based on these seminal papers, other methods have been proposed and some of them will be presented in the following sections. Depending on the kind of system, these techniques are referred as shooting methods, Poincaré map methods or simply Newton methods [15]; in particular for non-autonomous systems, the shooting and the Poincaré map methods are essentially the same.…”

HomSSPICE: a homotopy-based circuit simulator for periodic steady-state analysis of oscillators

“…Aprille and Trick [2] fist proposed an approach to eliminate the additional unknown when applying the Newton-Raphson shooting method to analyze oscillators. This method was subsequently implemented in SSPICE [3].…”

“…Hence, the periodic steady-state circuit response is simulated directly using either the time-domain NewtonRaphson shooting method, or the frequency-domain harmonic-balance method [7]. The time-domain NewtonRaphson shooting method is preferred for simulating highly nonlinear circuits [2,3,6].…”

Fast transient analysis of oscillators using multiple time scales with accurate initial conditions

“…That is easy to implement since the mapping between the two time axes has already been obtained in Eq. (2). As shown in Eq.…”

“…For example [1] provides a method to find the steadystate of high-Q oscillators using the transient analysis in Spice. In [2], a Newton algorithm is used to find the periodic response and the period of oscillators quickly in lightly damped systems. However, the complete transient solution is still not accessible in these methods.…”