# Study of Standing-Wave Thermoacoustic Electricity Generators for Low-Power Applications
## 2 Theoretical Modeling of Linear Alternators
\begin{array}{c} W_{e}=\frac{1}{2}|I|^{2}R_{L}=\frac{1}{2}\frac{|U|^{2}}{A_{d}^ {2}}\Bigg{[}\frac{\left(Bl\right)^{2}R_{L}}{\left(R_{e}+R_{L}\right)^{2}+X_{ e}^{2}}\Bigg{]}=\\ \\ =\frac{1}{2}|p_{1}-p_{2}|^{2}A_{d}^{2}\frac{(Bl)^{2}R_{L}}{\left[\left(Bl \right)^{2}-X_{m}X_{e}+R_{m}\left(R_{e}+R_{L}\right)\right]^{2}+\left[X_{m} \left(R_{e}+R_{L}\right)+R_{m}X_{e}\right]^{2}},\end{array}  \tag{9}\]  
which shows that for delivering high electric power levels to the load, high oscillation velocities of the diaphragm are required. The acoustic-to-electric conversion efficiency is  
\[\eta_{ae}=\frac{W_{e}}{W_{a}}=\frac{|I|^{2}R_{L}}{|U|^{2}R_{a}}=\frac{\left( Bl\right)^{2}R_{L}}{R_{m}[\left(R_{e}+R_{L}\right)^{2}+X_{e}^{2}]+\left(Bl \right)^{2}\left(R_{e}+R_{L}\right)}, \tag{10}\]  
From the above equations it results that the performance of an LA (expressed by the parameters \(\eta_{ae}\), \(W_{a}\), and \(W_{e}\)) grows with increasing force factor (\(Bl\)) and with decreasing mechanical (\(R_{m}\)) and electrical (\(R_{e}\)) resistances (in the ideal case \(R_{m}=R_{e}=0\) the device would have an acoustic-to-electric efficiency of 100%). The performance also increases when the electrical reactance (\(X_{e}\)) is decreased. An additional condenser could be added to the coil circuit to use the alternator in favorable electrical resonant conditions (\(X_{e}=0\)) at the selected operation frequency [9] (even if the inductance of the voice coil, \(\omega L_{e}\), is generally negligible at the typical working frequencies of TA devices for standard woofers and sub-woofers). Finally, the amount of acoustic power absorbed by the alternator also grows with decreasing mechanical reactance \(X_{m}\) so the performance of the device is highest when it works in both mechanical and electrical resonant conditions (\(X_{m}=0\), \(X_{e}=0\)) for which the alternator acoustical impedance becomes real (\(p_{1}-p_{2}\) in phase with \(U\)). The optimal value of the load resistance maximizing the efficiency can be found from Equation (10) when written (neglecting the electrical reactance term) in the form  
\[\frac{1}{\eta_{ae}}=1+\frac{R_{e}}{R_{L}}+\frac{1}{\left(Bl\right)^{2}}\frac{R_{ m}}{R_{L}}{\left(R_{e}+R_{L}\right)^{2}}. \tag{11}\]  
Equating to zero the derivative of this equation we find that the optimal load resistance is  
\[R_{L}=R_{e}\sqrt{1+\frac{\left(Bl\right)^{2}}{R_{e}R_{m}}}. \tag{12}\]  
Substituting this value in Equation (11) we find that the maximum achievable efficiency is  
\[\eta_{max}=\frac{\sqrt{\beta+1}-1}{\sqrt{\beta+1}+1}, \tag{13}\]  
where \(\beta=(Bl)^{2}/R_{e}R_{m}\). The value provided by Equation (12) differs from the value of the optimal load resistance that maximizes the electric output power. Using Equation (2), in fact, and indicating with \(\mid x_{max}\mid\) the maximum admissible diaphragm excursion (the corresponding maximum volume velocity being \(\mid U_{max}\mid=\omega A_{d}\mid x_{max}\mid\)), the maximum electrical output power is  
\[W_{e}=\frac{1}{2}|I|^{2}R_{L}=\frac{1}{2}(Bl)^{2}\frac{|U_{max}|^{2}}{A_{d}^{2 }}\frac{R_{L}}{\left(R_{e}+R_{L}\right)^{2}}=\frac{1}{2}(Bl)^{2}\omega^{2}|x_ {max}|^{2}\frac{R_{L}}{\left(R_{e}+R_{L}\right)^{2}}, \tag{14}\]  
which shows that for achieving high power levels, without working at increasingly values of the frequency, LAs characterized by high diaphragm excursions are required. Equation (14) maximizes for  
\[R_{L}=R_{e}, \tag{15}\]  
that, when substituted in Equation (14), provides the maximum electric power obtainable at the operation frequency \(\omega\)  
\[W_{max}=\frac{1}{8}\frac{(Bl)^{2}\omega^{2}|x_{max}|^{2}}{R_{e}}, \tag{16}\]  
To model the TA engine and its coupling to the LA, the Design Environment for Low-Amplitude ThermoAcoustic Engines (DeltaEC) computer code [10], developed at the Los Alamos National Laboratory, is used as design tool in the present work.