## 1 Introduction  
Self-excited thermoacoustic instability (SETAI) is a very undesirable phenomenon for gas- and liquid-fired equipment [1; 2; 3]. The accompanying strong vibration causes large noise, equipment damage, failure, and even personal safety accidents [4]. SETAI is more likely to occur in high-intensity combustion systems, where the released heat cannot be transferred away immediately, such as gas turbines, aero-engines, low-nitrogen boilers, and even compact gas water heaters. While SETAI has been investigated for more than 200 years, this tough problem is still the main challenge limiting the development of combustion technology in practice. The famous Rayleigh criterion [5] proposed that the coupling between pressure oscillation and reaction heat release oscillation is the main cause of SETAI, and this coupling will encourage SETAI if the phase difference is less than 90\({}^{\circ}\). This criterion has been widely confirmed and acknowledged by various scholars [1; 2; 6; 7; 8]. However, the oscillation phase difference is very microscopic, and it is virtually impossible to use operation parameters for SETAI prediction via the Rayleigh criterion. SETAI behavior has been proven to be affected by the equipment's structure and operation parameters, which provided the chance for SETAI control, but conflicting results have been observed with regard to the effect of the heat output [9; 10; 11], equivalent ratio (\(\varphi\)) [12; 13; 14], and system structure [15; 16; 17; 18]. Due to insufficient knowledge, SETAI control is generally achieved via empirical or semi-empirical methods, mainly classified into two kinds of approaches. One is to increase the system's acoustic damping [3; 15; 19; 20], such as by adding quarter-wave tubes, half-wave tubes, Helmholtz resonators, perforated plates, and perforated liners. The other is to affect the coupling between the sound pressure oscillation and heat release oscillation to obtain an out-of-step oscillation by changing the combustion parameters or structure dimensions [4; 9; 11; 13; 20; 21], or by adopting a flow jet [15; 22; 23]. These detailed control strategies always require a number of costly tentative retrofits of the combustion system, which also influence the combustion or thermal performance of the equipment. More unfortunately, when the operation condition is changed, the verified control approach could become invalid. To accurately and conveniently control SETAI, it is necessary to clarify the mechanism by which the phase relation between the sound pressure oscillation and reaction heat release oscillation is affected during combustion. Fully relating this phase difference to the operation parameters and structure dimensions is the key to solving this tough problem. Our previous work demonstrated that the premixed supply duct length (\(L_{\text{P}}\)), fuel flow rate (\(V_{\text{F}}\)), and \(\varphi\) have a coupled effect that allows for the identification of the phase difference between pressure and heat release oscillation [24; 25] by introducing the flow rate oscillation at the combustion chamber inlet. This flow rate oscillation is caused by the pressure oscillation, and their phase difference [8] is determined by the sound pressure mode shape within the premixed duct [2], which is further collaboratively influenced by \(L_{\text{P}}\) and \(V_{\text{F}}\). However, this flow rate oscillation could cause heat release oscillation, and there is also a phase difference between them due to the time needed for the combustion reaction. Moreover, the heat release delay time relative to the flow rate oscillation (\(\tau\)) [26] is mainly influenced by the combustion \(\varphi\). In such a mode, these three parameters ultimately affect the coupling between sound pressure and heat release oscillation, thus determining SETAI behavior. This mechanism has been proven by many experiments conducted on a hedge premixed combustion system with a symmetrical air/fuel supply.