## 2 Materials and Methods
### RC Load Method  
In this section, a brief overview of the physical background behind the employed method for acoustic power measurement is provided. It is crucial to introduce the concept of lumped-element approximation, initially proposed by [PERSON] [1], which stablishes an analogy between the acoustic impedance (\(Z_{a}\)) and the impedance of an electrical circuit (\(Z\)) [4]:  
\[Z_{a}=\frac{p_{1}}{U_{1}}=R_{a}+j\bigg{(}\omega L_{a}-\frac{1}{\omega C_{a}} \bigg{)}\quad\leftrightarrow\quad Z=\frac{V_{1}}{I_{1}} \tag{2}\]  
where \(R_{a}\) is a resistive acoustic load (resistance), \(L_{a}\) is an inductive acoustic load (inertance), \(C_{a}\) is a capacitive acoustic load (compliance), and \(\omega\) is the acoustic frequency. \(p_{1}\) and \(U_{1}\) are the pressure wave amplitude and the volumetric flow rate, respectively. \(V_{1}\) and \(I_{1}\) are the electric AC voltage and the electric AC current, correspondingly.  
It is interesting to note that the compliance introduces a positive phase shift of \(90^{\circ}\) between \(\left|p_{1}\right|\) and \(\left|U_{1}\right|\), meaning it acts as a capacitive load so that the flow advances \(90^{\circ}\) in relation to the pressure. Resistance, however, does not cause a phase change between \(\left|p_{1}\right|\) and \(\left|U_{1}\right|\). Accordingly, in Equation (2), the compliance is purely imaginary, whereas the resistance is purely real.  
Assuming the short duct length approximation, any compliance can be written [21]:  
\[C=\frac{V}{\gamma P_{m}} \tag{3}\]  
where \(V\), \(P_{m}\), and \(\gamma\) are the volume of the compliance, mean pressure, and specific heat ratio.  
A variable RC acoustic load is connected to the power extraction branch of the TA-SLiCE under study, as shown in Figure 6, to indirectly measure the volumetric flow rate through the resistance.  
Assuming simple harmonic sound at angular velocity \(\omega\):  
\[U_{1}=j\omega CP_{C}=j\omega\frac{V}{\gamma P_{m}}\;P_{C} \tag{4}\]  
Hence, the dissipated acoustic power at the resistance is determined using the RC load method described by [PERSON] et al. [11]. A variable RC load can be constructed by connecting a small volume tank to the mechanical resonator of the power extraction branch through a needle valve, shown in Figures 4 and 5. This method requires little equipment, two pressure sensors; one of them is placed close to the inlet of the needle valve, while another is installed inside the tank. By measuring the acoustic signals from these two sensors, it is possible to determine the instantaneous amount of acoustic power that is lost due to dissipation in the load:  
\[\dot{E}_{RC}=\frac{1}{2}\frac{\omega V}{\gamma P_{m}}Im\Big{(}P_{E}\widetilde {P}_{C}\Big{)}=\frac{1}{2}\frac{\omega V}{\gamma P_{m}}|P_{E}||P_{C}||sin\theta _{EC}, \tag{5}\]  
Figure 6: RC load representation through the analogy between (**a**) electrical and (**b**) thermoacoustic.  
The acoustic power dissipated in the RC load is determined as the time-averaged product of the volumetric flow rate through the valve and the two pressures (\(P_{E}\) and \(P_{C}\)) across the valve. \(V\), \(P_{m}\), and \(\gamma\) are volume of the tank, mean pressure, and specific heat ratio. \(\theta_{EC}\) is the phase shift in radians between the pressure measured at the mechanical resonator, \(P_{E}\), and the pressure measured at the volume tank, \(P_{C}\). This algebraic expression represents a revision and simplification of previous models and expressions from the two-pressure sensor method by analyzing pressure measurements at two points a specific distance apart at the power extraction location. This broadly used method, which yields a refined expression of the power available for extraction in thermoacoustic devices, only needs to measure the two pressure signals \(P_{C}\) and \(P_{E}\) (amplitude, phase, and acoustic frequency \(\omega\)).