Harmonic Balance Simulation and Analysis

Abstract

The contribution reviews the use of the Harmonic Balance technique for the numerical determination of the steady‐state working point of nonlinear circuits operated in periodic or quasi‐periodic conditions. Details are also provided on the main steps for the implementation of harmonic balance in EDA tools.

Memcomputing NP -complete problems in polynomial time using polynomial resources and collective states

“…Notably, if we wanted to simulate our machine by including the readout unit, the computational complexity would be O(2 p ) [close to the standard dynamic programming that is O(n2 p ) (17)]. From the Nyquist-Shannon sampling theorem (18), the minimum number of samples of the shifted collective state (that is, the outputs of the frequency shift module) must be equal to the number of frequencies of the signal [in our case of O(2 p )] to accurately evaluate even one of the harmonic amplitudes (19). The last claim can be intuitively seen from this consideration: the DC voltage V DC up must be calculated in the simulation by evaluating the integral V DC up ¼ T −1 ∫ T 0 v up ðtÞdt, and this requires at least O(2 p ) samples for an accurate evaluation (18).…”

“…From the Nyquist-Shannon sampling theorem (18), the minimum number of samples of the shifted collective state (that is, the outputs of the frequency shift module) must be equal to the number of frequencies of the signal [in our case of O(2 p )] to accurately evaluate even one of the harmonic amplitudes (19). The last claim can be intuitively seen from this consideration: the DC voltage V DC up must be calculated in the simulation by evaluating the integral V DC up ¼ T −1 ∫ T 0 v up ðtÞdt, and this requires at least O(2 p ) samples for an accurate evaluation (18). On the other hand, the multimeter of the hardware implementation, being essentially a narrow low-pass filter, performs an analog implementation of the integral over a continuous time interval T (independent of n and p), directly providing the result and thus avoiding the need to sample the waveform and compute the integral.…”

“…The measurement process consists of acquiring the output waveforms and applying the fast Fourier transform in software. The collective state being a purely periodic signal, that is, a signal containing only frequency multiples of f 0 and with known maximum frequency f max = Af 0 , from the discrete Fourier transform theory and the Nyquist-Shannon sampling theorem (8,18), we need to sample the interval [0, 1/f 0 ] into N = 2f max /f 0 + 1 subinterval of width Dt = (Nf 0 ) −1 to compute the exact spectrum of g(t). That is, we need samples g(t j ) with t j = kDt and k = 0, …, N − 1 to compute the exact spectrum of g(t).…”

A Circuit Theory Perspective on the Modeling and Analysis of Vibration Energy Harvesting Systems: A Review

“…The total number of unknowns for the HB technique remains the same for both autonomous and non-autonomous systems.) [86,87].…”

Interferência de RF em circuitos integrados de potência inteligentes com ênfase em aplicações industriais, automotivas e de energia renovável

“…Descrição mais aprofundada do método pode ser encontrada em Gilmore e Steer (1991),Maas (2003, Cap. 3),Bonani et al (2014),Agilent (2004), Keysight (2020) e AWR (2020).…”