# Study of Standing-Wave Thermoacoustic Electricity Generators for Low-Power Applications
## 4 Results and Discussion
### Two-Stage Engine with Alternator Coupled in Push-Pull Arrangement  
The achievent of an electric output power of 100 W requires a two-stage engine since the one-stage configuration is unable to match the target power under the constraint imposed by \(x_{max}\) (results not shown). A two-stage engine with a stack/HXs assembly placed on each side of a \(\lambda/2\) standing wave resonator and the LA installed in the middle section (see Figure 4) is firstly considered in this study. Since the middle sections of the resonator are low-impedance regions of the standing wave, the acoustic coupling of the LA to the engine requires that it behave as a low-impedance transducer. This configuration where the acoustic pressures acting on both sides of the LA have the same amplitude \(\mid\!p\!\mid\) but are \(180^{\circ}\) out of phase is called \"push-pull\" coupling mode. For a given acoustic impedance this arrangement should bring the advantage of doubling the acoustic power \(\mathrm{W_{a}}\) absorbed by the alternator compared to the one-stage case when considering that, with \(p_{1}\) and \(p_{2}\) being in phase opposition, \(\mid\!p_{1}-p_{2}\!\mid\!=\!2\!\mid\!p\!\mid\) results, and observing that in the one-stage configuration the LA is driven by the pressure acting only on one side [13]. Compared to the one-stage case, however, this arrangement does not allow regulations of the alternator impedance through regulations of the acoustic compliance introduced by the back volume. In the simulations performed by DeltaEC for each working frequency \(f\), the engine parameters (stack location, stack length, loudspeaker location \(x_{L}\)) and the loudspeaker parameters (load resistance \(R_{L}\)) have been varied in order to find the optimal matching between the LA and the engine. In each run, the peak-to-peak stroke of the diaphragm is fixed to 12 mm while the HHX and AHX lengths are varied, as in the one-stage case, to match exactly the local peak-to-peak acoustic particle amplitude. Furthermore, the distance of the alternator from the AHX (equal on both sides) is modeled to be not lower than 8 cm to allow the accommodation of the alternator in a real technical implementation. The geometrical dimensions of the engine deriving from the optimization procedure are given in Table 2. Results concerning the TAEG performance are shown in Figure 5 where the efficiencies \(\eta_{lu}\), \(\eta_{ae}\), the power values \(W_{a}\), \(W_{e}\), and the impedance parameters \(\mid\!Z_{a}\!\mid\!\), \(\phi_{PUL}\) are plotted as a function of \(f\). The graph shows how the TAEG is characterized by a very low efficiency in converting heat to electricity (\(\eta_{he}\approx 2\%\) at a maximum with \(\eta_{lu}\approx 3\%\) and \(\eta_{ae}\approx 67.6\%\)) and the maximum obtainable electric output power amounts to \(W_{e}\) = 35.7 W (with \(W_{a}=54.1\) W). Note that in this case the efficiencies \(\eta_{lu}\), \(\eta_{ae}\), \(\eta_{he}\) are referred to maximum power output. These performance levels are considerably lower than the ones achieved by the one-stage engine previously considered. This result can be interpreted on the basis of the following observations:  
* The LA is installed in a low-impedance region of the standing wave so the acoustic coupling is realized for relatively low values of \(\mid\!Z_{a}\!\mid\!\) ( \(\mid\!Z_{a}\!\mid\!\approx 10^{4}\) Pa s/m\({}^{3}\) compared to \(\mid\!Z_{a}\!\mid\!\approx 10^{5}\) Pa s/m\({}^{3}\) of the one-stage case). This entails that at a given working frequency the stroke of the alternator reaches the maximum allowable value \(\mid\!x_{max}\!\mid\!\) for low values of the driving pressure \(\mid\!p_{1}-p_{2}\!\mid\!\)  
Figure 4: Schematic of the two-stage engine with loudspeaker coupled in push–pull mode. \((\,|\,p_{1}-p_{2}\,|\,\approx 10^{3}\) Pa compared to \(\,|\,p_{1}\,|\,\approx 10^{4}\) Pa of the one-stage case). This precludes the stacks from generating high acoustic power levels which scale proportionally to \(\,|\,p\,|^{2}\)[1].