# Study of Standing-Wave Thermoacoustic Electricity Generators for Low-Power Applications
## 4 Results and Discussion
### One Stage Engine  
The simplest configuration of a TAEG is a straight half-wavelength (\(\lambda\)/2)-type resonator enclosing a stack/HXs assembly on one end and an LA on the other end (see Figure 2). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbol** & **Value** \\ \hline Mechanical resistance & \(R_{m}\) (kg/s) & 0.562 \\ Moving mass & \(m\) (kg) & 0.014 \\ Mechanical stiffness & \(K_{m}\) (N/m) & 3099.3 \\ Coil electrical resistance & \(R_{e}\) (\(\Omega\)) & 5.4 \\ Force factor & \(Bl\) (Tm) & 10.8 \\ Coil inductance & \(L_{e}\) (H) & 0.0006 \\ Resonance frequency & \(f_{0}\) (Hz) & 75 \\ Maximum stroke & \(x_{max}\) (mm) & \(\pm\)6 \\ Diaphrag area & \(A_{d}\) (m\({}^{2}\)) & 0.0132 \\ Maximum transduction efficiency & \(\eta_{max}\) (\%) & 72.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of the loudspeaker. The parameter \(\eta_{max}\) is calculated through Equation (13). As is well known, in such kinds of resonators the acoustic impedance grows continuously from the central section to the end sections so the impedance matching requires in this case that the LA behave as an high-impedance transducer. Furthermore, since on the right side of the resonator the imaginary part of the impedance is negative (the pressure lags the velocity by a \(90^{\circ}\) phase angle) we have to select proper values of the back volume (and eventually of \(R_{L}\)) to make the imaginary part of the input impedance of the coupled LA/housing negative. \[Z_{a}=R_{a}+jX_{a}=\frac{1}{A_{d}^{2}}\bigg{[}R_{m}+\frac{(B)^{2}(R_{c}+R_{L}) }{(R_{c}+R_{L})^{2}+X_{c}^{2}}\bigg{]}+j\bigg{\{}\frac{1}{A_{d}^{2}}\bigg{[} \Big{(}\omega m-\frac{K_{m}+K_{m}^{\prime}}{\omega}\Big{)}-\frac{(B)^{2}X_{c}} {(R_{c}+R_{L})^{2}+X_{c}^{2}}\bigg{]}\bigg{\}} \tag{25}\]  
Note that in this configuration the acoustic power irradiated in the back volume is negligible since it simply corresponds to the power dissipated on the housing surfaces by thermal hysteresis. Taking into account these circumstances, the TAEG has been numerically modeled by DeltaEC and for each working frequency \(f\), the engine parameters (stack location, stack length, LA location) and the LA parameters (load resistance \(R_{L}\) and back volume \(V_{b}\)) have been varied in order to find the optimal matching between the LA and the engine. In each simulation, the output electrical power and the peak-to-peak stroke of the diaphragm are fixed to 50 W and 12 mm, respectively, while the HHX and AHX lengths are modeled to match exactly the local peak-to-peak acoustic particle amplitude [1]. 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 3 where the power values \(W_{a}\), \(W_{e}\), the efficiencies \(\eta_{lu}\), \(\eta_{ac}\), \(\eta_{lu}\), and the impedance parameters \(1Z_{a}\), \(\phi_{pU}\) are plotted as a function of \(f\). Figure 2: Schematic of the one-stage standing-wave thermoacoustic electricity generator (TAEG) model. AHX: ambient heat exchanger; HHX: hot heat exchanger. From an inspection of the graphs it can be deduced that the heat-to-acoustic conversion efficiency \(\eta_{ha}\) of the engine exhibits a maximum at \(f\approx 180\) Hz (with \(R_{L}\) = 42.2 \(\Omega\)). The behavior of \(\eta_{ha}\) vs. \(f\) can be interpreted observing that decreasing the frequency starting from high values has the effect of reducing the absolute value of the phase angle \(\phi_{pU}\). This should have the positive effect of increasing the amount of acoustic power absorbed by the LA. This circumstance explains the observed increase of \(\eta_{ha}\). Further reduction of the frequency (below 180 Hz) has however the detrimental effect of decreasing \(\eta_{ha}\). This trend reflects the worsening of the stack performance in converting heat to sound. For a stack with square pores of hydraulic radius \(R_{h}\), in fact, the best performances should be reached when \(R_{h}\approx\delta_{x}\)[12].