Computation of all the Floquet eigenfunctions in autonomous circuits

Abstract

SUMMARYAnalysis methods based on the Floquet theory have gained a large interest to compute small signal and noise effects in non-autonomous and autonomous circuits (oscillators). A reliable and straightforward method to compute all the eigenfunctions related to the Floquet analysis is discussed here. The method has been implemented in a circuit simulator and tested on several circuits. Numerical results are presented.

Periodic noise analysis of electric circuits: Artifacts, singularities and a numerical method

“…Therefore, as phase noise, we define the projection of the total noise along the eigenfunction u 1 (t) corresponding to the Floquet multiplier theoretically equal to 1 (see Section 2.1). As amplitude noise we define the difference between the total noise and the phase noise [6,7,[10][11][12].…”

“…The method proposed in [3] considers non-autonomous circuits, but its extension to autonomous ones is as straightforward as that described in [2]. The TAL method was later reinterpreted and completely justified through Floquet theory [5] in [6][7][8][9][10].…”

Selective Determination of Floquet Quantities for the Efficient Assessment of Limit Cycle Stability and Oscillator Noise

“…In fact, the loss of stability of the limit cycle corresponds to Re {μ k } = 0 for some k and, as shown in [7], for stable oscillators, the FEs with larger magnitude are expected to provide a negligible contribution to the oscillator noise spectrum. FEs are mostly determined in time domain; several techniques [9]- [12] are available, often exploiting advanced multistep integration algorithms to guarantee a good accuracy. However, frequency domain approaches such as harmonic balance (HB), although, in general, leading to larger size numerical problems, have significant practical importance chiefly because there are circuit elements, such as transmission lines, whose frequency domain description is more efficient [13].…”

Extension of the variational equation to analog/digital circuits: numerical and experimental validation

“…This condition cannot be satisfied, for example, in the simulation of analog/digital circuits and in circuit models using ideal switching elements. A correct fundamental matrix is also a key aspect of the periodic small signal analysis (PAC) and periodic noise analysis (PNOISE) of circuits working on a stable limit cycle [4][5][6]. If the Newton method is employed in conventional SH to solve the non linear algebraic equations implementing the periodicity constraint, an incorrect fundamental matrix is found.…”

“…This leads to an incorrect Jacobian matrix [2,3] and, possibly, non-convergence of the Newton method. A correct fundamental matrix is also a key aspect of the periodic small signal analysis (PAC) and periodic noise analysis (PNOISE) of circuits working on a stable limit cycle [4][5][6]. These analyses are implemented, possibly with different names, for example, in the commercial RF circuit simulators SPECTRE ™ by Cadence [4,7] and ELDO ™ by Mentor Graphics.…”