A unified stability analysis method for spontaneous oscillation condition s in thermoacoustic systems via measured frequency response data with application to standing- and traveling-wave engines

Abstract

In this paper, a unified stability analysis method for thermoacoustic systems is proposed using the Nyquist stability criterion for MIMO (multi-input multi-output) causal stable open-loop systems. Toward this end, the conventional frequency response function in the thermoacoustic instability utilization literature, the transfer matrix, is replaced with a function for systems with a causal input–output relation between the incoming and outgoing traveling-wave pressure components, the scattering matrix, known in the thermoacoustic instability suppression literature, in order to represent whole subsystems including thermoacoustic cores and connected tubes as TITO (two-input two-output) stable open-loop systems. This makes it possible for the Nyquist stability criterion to be applied to the open-loop frequency response to analyze the closed-loop stability. The method provides a simpler procedure than the conventional one that is based on the Nyquist stability criterion for SISO (single-input single-output) systems in the thermoacoustic instability suppression literature because TITO open-loop systems can be assumed to be stable for thermoacoustic systems when traveling-wave pressure components are adopted as their input and output signals. It is shown that the spontaneous oscillation condition s for both standing- and traveling-wave thermoacoustic engines with various settings can be analyzed with a unified procedure, and the actual oscillation conditions are successfully estimated with the proposed method, indicating reasonable stability and/or instability margins in the same manner as in the Nyquist stability criterion for stable open-loop systems. That is, the feasibility of spontaneous oscillation is simply indicated by encirclement(s) of the origin by the Nyquist locus calculated from the measured frequency responses; multiple encirclements correspond to a distorted waveform in the measured pressure signal; the oscillation frequencies can be estimated by the outermost portion of the locus; and the distance between this portion and the origin corresponds to the steady-state pressure amplitude.