# Study of Standing-Wave Thermoacoustic Electricity Generators for Low-Power Applications
## 4 Results and Discussion
### Two-Stage Engine with Loudspeaker Coupled in Side Branch Arrangement  
As the two-stage engine with push-pull LA is unable to provide the required power output, the LA coupling typology is switched to a side-branch configuration. In this configuration the alternator is placed at the end of a relatively long duct while the other end (the branch input section) is connected to the main engine trunk in sideway (see Figure 6). The design process of the two-stage TAEG starts by firstly optimizing the branch/LA component. The value of the input acoustic impedance of the optimized branch, \(Z_{b}=R_{b}+jX_{b}\), is then used in the optimization process of the engine. Making the choice of operating the LA at an acoustic-to-electric conversion efficiency of at least 70%, the plots of Figure 1 allow us to determine the corresponding values of the operation frequency (\(f=194.5\) Hz) and load resistance (\(R_{L}=19\)\(\Omega\)) compatible with the target output power (100 W) and maximum allowable stroke (\(\pm 6\) mm). The driving pressure needed to match the target output power is \(\mid\!p_{1}\!\mid\approx 3\) kPa, which corresponds to a volume flow rate amplitude of \(\mid\!U\!\mid=0.0968\) m/s). The branch duct comprise a narrow straight segment, a divergent cone, and a wide straight segment which accommodates the LA/housing assembly (see Figure 7). Geometrical dimensions of the branch should be selected in order the following two conditions be met at the system working frequency:  
* The impedance at the branch input section should be sufficiently higher than the local trunk impedance to preserve the acoustic wave in the trunk from being distorted and the branch from absorbing too great amounts of power that could lead to unwanted high onset temperatures or, worse, to a not-starting engine. At the same time, the impedance should be not too large to allow the target acoustic power to enter the branch with (as far as possible) low driving pressures (which reduce termoviscous dissipation on the resonator walls proportional to \(\mid\!p\!\mid^{2}\)). So, a trade-off between these two requirements has to be found. * The impedance at the other side of the branch, where the LA is installed, should match the low impedance of the transducer. The latter can be adjusted, as in the one-stage case, by regulation of the back volume compliance with the aim of decreasing \(\phi_{pU}\). The back volume compliance, however, also affects the input impedance of the branch and this leads again to the issues discussed at the previous point. Hence, also in this case a trade-off between different requirements has to be found. Figure 6: Schematic of the two-stage engine with loudspeaker coupled in side-branch mode. Selecting the branch configuration in order to create a near \(\lambda/4\) acoustic field distribution (\(\lambda\) being the sound wavelength) can fulfill both these conditions. This is clearly illustrated in Figure 7 which reports the pressure and velocity distributions along the branch when the pressure amplitude at the input section is about 10 kPa, the module and phase angle of the input impedance of the branch being \(2.58\times 10^{5}\) Pa s/m\({}^{3}\) and \(44.5^{\circ}\), respectively (\(R_{b}=1.84\times 10^{5}\) Pa s/m\({}^{3}\); \(X_{b}=1.81\times 10^{5}\) Pa s/m\({}^{3}\)). The back volume, modeled by a compliance, is adjusted to bring the resonance frequency of the composed LA/housing near the target operation frequency (194.5 Hz). For the selected back volume (equal to 0.0015 m\({}^{3}\) after subtraction of the LA volume) the resonance frequency of the LA/housing system is about 187.6 Hz and the phase angle between \(p_{1}\) and \(U\) is near \(10^{\circ}\). For this configuration, the alternator extracts 100 W of electric power when 153.6 W of acoustic power flow through the branch input section, the peak-to-peak stroke of the loudspeaker being 12 mm.