Stability criteria of two-port networks, application to thermo-acoustic systems

System theory methods are developed and applied to introduce a new analysis methodology based on the stability criteria of active two-ports, to the problem of thermo-acoustic instability in a combustion appliance. The analogy between thermo-acoustics of combustion and small-signal operation of microwave amplifiers is utilized. Notions of unconditional and conditional stabilities of an (active) two-port, representing a burner with flame, are introduced and analyzed. Unconditional stability of two-port means the absence of autonomous oscillation at any embedding of the given two-port by any passive network at the system's upstream (source) and downstream (load) sides. It has been shown that for velocity-sensitive compact burners in the limit of zero Mach number, the criteria of unconditional stability cannot be fulfilled. The analysis is performed in the spirit of a known criterion in microwave network theory, the so-called Edwards-Sinsky's criterion. Therefore, two methods have been applied to elucidate the necessary and sufficient conditions of a linear active two-port system to be conditionally stable. The first method is a new algebraic technique to prove and derive the conditional and unconditional stability criteria, and the second method is based on certain properties of Mobius (bilinear) transformations for combinations of reflection coefficients and scattering matrix of (active) two-ports. The developed framework allows formulating design requirements for the stabilization of operation of a combustion appliance via purposeful modifications of either the burner properties or/and of its acoustic embeddings. The analytical derivations have been examined in a case study to show the power of the methodology in the thermo-acoustics system application.