# Novel Real-Time Acoustic Power Estimation for Dynamic Thermoacoustic Control
## 1 Introduction  
The thermoacoustic Stirling-like cycle engine (TA-SLiCE) represents an innovative class of external combustion engines designed to convert thermal energy into mechanical work and ultimately into direct current (DC) electrical energy [1]. These systems feature a thermoacoustic core with a regenerator between two heat exchangers, integrated into a looped acoustic network modelled as a lumped-element circuit due to its short-length component dimensions [2]. Over the years, the scientific community has extensively investigated the underlying thermoacoustic phenomena governing the operation of thermoacoustic engines, primarily aiming to improve the efficiency and performance of these systems [3]. [PERSON] [4] established the fundamental theoretical framework for studying low-amplitude thermoacoustic phenomena under laminar flow conditions, providing the governing equations that describe the propagation of acoustic waves coupled with thermal effects in confined fluids. Later, [PERSON] introduced the analogy between a traveling-wave thermoacoustic engine and a lumped-element electrical circuit [5], recognizing that the working fluid undergoes a thermodynamic cycle analogous to that of a Stirling engine [6]. This perspective facilitated the development of more efficient configurations and simplified the mathematical representation of these systems. More recently, [PERSON] [7] refined andextended this approach through lumped-parameter models, providing more accessible and practical tools for the design and optimization of real thermoacoustic engines [8]. Thermoacoustic measurements often provide high signal-to-noise ratios, making the need for statistical analysis unnecessary [1]. While significant progress has been made in understanding the theoretical foundations and in optimizing the design of thermoacoustic engines [9; 10], a variety of passive and active control strategies have been explored to improve the acoustic wave quality of these systems [11]. In this context, acoustic power is relatively easy to measure when volumetric velocity can be reliably inferred, with simpler techniques based on pressure measurements [12] or the motion of a solid object [13] being widely adopted in practice. More high-cost and sophisticated methods for direct velocity measurement, such as hot-wire anemometry particle, image velocimetry, or laser Doppler anemometry techniques, fall beyond the scope of this study and are covered in detail elsewhere [14]. The two-microphone measurement technique [15; 16] is widely used to determine acoustic power from pressure amplitude and phase measurements at two locations in a resonator. This well-established method has been widely validated, particularly under laminar flow conditions, where it aligns with the analytical model developed by [PERSON] et al. [17], the so-called resistance-capacitance (RC) load method. These methodologies, supported by extensive and rigorous work [2], ensure reliable and robust power estimation across varying operational parameters (e.g., loads [18], temperatures, and pressures [19]), reinforcing the applicability of the proposed measurement framework to real-world thermoacoustic systems [20]. In all those studies, the measurement of acoustic power in thermoacoustic engines has been constrained by methods that rely on time-averaged estimations [21; 22]. Although effective in many contexts, these methods do not account for potential fluctuations in acoustic amplitude within the mechanical resonator, which can reduce their effectiveness in applications such as control systems [23]. The ability to dynamically estimate acoustic power [24] and make immediate adjustments has been a long-standing challenge in thermoacoustic systems, where traditional optimization methods have typically relied on structural modifications [25] or temperature-based estimates [26]. Integrating real-time acoustic power measurements enables the system to respond to changes in performance as they occur, thereby ensuring more precise and adaptive control.