# Novel Real-Time Acoustic Power Estimation for Dynamic Thermoacoustic Control
## 2 Materials and Methods
### RC Equivalent Model in Electrical Analogy  
The acoustic wave interactions in thermoacoustic devices are difficult to analyse, either because of their physical complexity or because of their dependence on the medium in which the wave propagates, in many cases requiring the use of very advanced tools. It is possible to establish an electric-acoustic correspondence thanks to the similarity of the fundamental equations that govern both disciplines [1; 32]. The analogies relevant to this article are shown in Table 1. This similarity enables the representation of complex acoustic systems as simplified electrical circuits, facilitating the use of electrical analysis tools and simulators for efficient system evaluation and optimization. The acoustic RC load in the power extraction branch of a TA-SLiCE, depicted in Figure 2, consists of a small tank of volume \(V_{C}\) (compliance) connected through a needle valve to the mechanical branch resonator. The expression describing the process is given by Equation (1):  
\[P_{E}(t)=P_{V}(t)+P_{C}(t), \tag{1}\]  
According to the electro-acoustic analogies shown in the table above and the physical problem under consideration, the pressure in the resonator, \(P_{E}(t)\), is analogous to the total voltage or source voltage, \(V_{g}(t)\). Similarly, the pressure in the capacitor \(P_{C}(t)\) corresponds to the capacitor voltage \(V_{C}(t)\) and the pressure in the valve, \(P_{V}(t)\), is analogous to the resistor voltage \(V_{g}(t)\). On the other hand, the analogue to the volume flow rate, \(U(t)\), is the circuit current, \(I(t)\). This results in Equation (2) and the corresponding electrical model shown in Figure 3.  
\[V_{g}(t)=V_{R}(t)+V_{C}(t), \tag{2}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Physical Variable** & **Electrical Analogy** \\ \hline Acoustic pressure (P) & Electrical voltage (V) \\ Volume flow rate (U) & Current (I) \\ Resistance (R) & Resistance (R) \\ Compliance (C) & Capacitance (C) \\ Acoustic power \(\left(\dot{E}_{load}\right)\) & Electric power (P\({}_{\text{elec}}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Electro-Acoustic analogous variables. Figure 2: Diagram of a thermoacoustic resistance–capacitance (RC) load. \(P_{C}\), \(P_{E}\), and \(P_{V}\) are the pressures at the compliance, the resonator, and the valve, respectively. \(U\) is the volume flow rate through the valve. On the other hand, the characteristic equation of a capacitor is  
\[i_{C}(t)=C\frac{dV_{C}(t)}{dt}\, \tag{3}\]  
where \(C\) is the capacitance of the capacitor and \(i_{C}(t)\) is the current flowing through it. As the elements are in series, the current flowing through the whole circuit would be the same:  
\[I(t)=i_{C}(t)=C\frac{dV_{C}(t)}{dt}, \tag{4}\]  
If the circuit analysis is performed in steady state, phasor notation can be used, leading to  
\[\overset{\rightarrow}{V_{g}}=\overset{\rightarrow}{V_{R}}+\overset{ \rightarrow}{V_{C}}, \tag{5}\]  
where \(\overset{\rightarrow}{V_{g}}\), \(\overset{\rightarrow}{V_{R}}\), and \(\overset{\rightarrow}{V_{C}}\) are the voltage phasors of the source, the resistor, and the capacitor, respectively. \[\overset{\rightarrow}{I}=jwC\overset{\rightarrow}{V_{C}}, \tag{6}\]  
The \(\overset{\rightarrow}{I}\) denotes the current phasor. The capacitor voltage, as a function of the current flowing through it, is expressed in Equation (7):  
\[\overset{\rightarrow}{V_{C}}=\frac{1}{jwC}\overset{\rightarrow}{I}=-\frac{ j}{wC}\overset{\rightarrow}{I}, \tag{7}\]  
The phasor of a number in polar form is expressed by its magnitude and phase, such that \(\overset{\rightarrow}{I}=I\angle\varphi_{I}\), \(\overset{\rightarrow}{V_{R}}=V_{R}\angle\varphi_{VR}\), and \(\overset{\rightarrow}{V_{C}}=V_{C}\angle\varphi_{VC}\) are the phasors, expressed in polar form, of the current, the voltage across the resistor, and the voltage across the capacitor, respectively.