# Study of Standing-Wave Thermoacoustic Electricity Generators for Low-Power Applications
## 3 Acoustic Impedance Matching  
The impedance matching between the alternator and the engine is a crucial task for designing high-performing devices. Optimal matching means that the presence of the alternator does not significantly alter the structure of the acoustic field (resonance frequency, pressure and velocity distributions, and their phase) which maximizes the engine performance in converting heat to acoustic power. Likewise, the structure of the acoustic field at the transducer location should make it absorb useful amounts of acoustic power, converting it to electrical power with high efficiency. In general, the optimal location of an LA in an acoustic field depends on the impedance of the transducer itself. This statement can be easily interpreted observing that the pressure drop (\(p_{1}-p_{2}\)) across the LA and the local volume velocity (\(U\)) must always satisfy Equation (6). This requires high-impedance (or non-compliant) transducers, with high masses and suspension stiffness, to be preferably installed near high-impedance regions of the acoustic field and, conversely, low-impedance (or ultra-compliant) transducers, with low masses and suspension stiffness, to be preferably installed near low impedance regions of the acoustic field [1]. Standard LAs are generally low impedance-type transducers. For this kind of devices the above requirements on impedance matching can be effectively met by using enclosed housing or ducts of proper length (branches) which act as impedance-matching transformers. Since the acoustic impedance of an LA is a function of frequency, optimal matching requires preliminary a selection of its optimal working frequency ranges. This task can be accomplished by a simple design strategy using as inputs the target output power, \(W_{e}\), and the maximum diaphragm excursion \(\mid\!\!x_{max}\!\!\mid\). Observing that at the operation angular frequency \(\omega\) the following two equations have to be verified  
\[\mid\!\!U\!\mid=A_{d}\omega|x_{max}\!\!\!\mid,\text{ and} \tag{21}\]  
\[W_{e}=\frac{1}{2}\frac{\mid\!\!U\!\mid^{2}}{A_{d}^{2}}\left[\frac{(Bl)^{2}R_{L }}{(R_{e}+R_{L})^{2}+X_{e}^{2}}\right], \tag{22}\]we can substitute the first into the second for \(\mid U\mid\) and then resolve the resulting equation with respect to \(\omega\). The result is  
\[f=\frac{1}{2\pi}\sqrt{\frac{2W_{e}(R_{e}+R_{L})^{2}}{(Bl)^{2}R_{L}|x_{max}|^{2}-2 W_{e}L_{e}^{2}}}, \tag{23}\]  
which allows us to determine, for a given load resistance \(R_{L}\), the operation frequency \(f\) compatible with the target output power and the maximum allowable stroke of the diaphragm. The \(f\), and \(\eta_{ae}\) values calculated through Equations (23) and (11) respectively, corresponding to the target output powers \(W_{e}\) (=50 and 100 W), are reported in Figure 1 as a function of \(R_{L}\). The volume flow rates generating the target power outputs can then be calculated by substituting these frequencies in Equation (21). For maximizing the amount of electric power generated by the LA it is desirable, according to Section 2, that its natural resonance frequency falls within the useful working frequency ranges above calculated. This brings the phase angle between \(p_{1}\) and \(p_{2}\) and \(U\) near zero, that enhances the acoustic power absorbed by the LA. For configurations involving LAs with enclosed housings, this condition could be met by adjusting the combined LA/housing impedance. The gas-spring effect caused by the back volume \(V_{b}\) of the housing is equivalent, in fact, to an increment of the suspension stiffness of the LA given by  
\[K_{m}^{\prime}=\frac{\gamma P_{0}A_{d}^{2}}{V_{b}}, \tag{24}\]  
which allows regulations of \(K_{m}^{\prime}\) through variations of \(V_{b}\). The adjustment operated on \(Z_{a}\) (=\(R_{a}+jX_{a}\)) to match the above condition has however a strong impact on the engine efficiency in converting heat to acoustic power (\(\eta_{ha}\)) since \(Z_{a}\) acts as an acoustic load to the TA engine.