# Study of Standing-Wave Thermoacoustic Electricity Generators for Low-Power Applications
## 2 Theoretical Modeling of Linear Alternators  
The physical behavior of an LA is quantitatively modeled in the frequency domain by the well-known canonical equations for an antireciprocal transducer. These equations are conveniently written in complex notation, which allows us to express any variable \(a(t)\) oscillating harmonically at angular frequency \(\omega\) about its mean value \(a_{0}\) as  
\[a(t)=a_{0}+\text{Re}\Big{\{}Ae^{j\omega t}\Big{\}}, \tag{1}\]  
where \(t\) is the time, of the imaginary unit \(j\), Re( ) signifies the real part, and the complex amplitude \(A\) accounts for both magnitude and phase of the oscillation. Neglecting hysteresis losses and considering that the electric terminals are closed on a load electric resistance \(R_{L}\), the canonical equations assume the form of the following two linear equations [8]  
\[0=(Z_{e}+R_{L})I-\frac{(Bl)}{A_{d}}U, \tag{2}\]  
\[(p_{1}-p_{2})=\frac{(Bl)}{A_{d}}I+\frac{Z_{m}}{A_{d}^{2}}U, \tag{3}\]  
where \(p_{1}\) and \(p_{2}\) are the complex amplitudes of the acoustic pressures acting on the front and back sides, respectively, of the diaphragm of area \(A_{d}\), \(I\) is the complex amplitude of the current flowing in the voice coil, \(U\) is the complex amplitude of the volumetric velocity due to the diaphragm motion, \(Bl\) is the \"force factor\" (the product of the magnetic field, \(B\), times the length of the voice coil wire, \(I\)), and \(Z_{e}\) (the electrical impedance with blocked diaphragm) and \(Z_{m}\) (the mechanical impedance with open electric circuit) are  
\[Z_{e}=R_{e}+jX_{e}=R_{e}+j\omega L_{e}, \tag{4}\]  
\[Z_{m}=R_{m}+jX_{m}=R_{m}+j\bigg{(}\omega m-\frac{K_{m}}{\omega}\bigg{)}, \tag{5}\]with \(R_{e}\) and \(L_{e}\) being the electrical resistance and inductance of the voice coil, \(R_{m}\) the mechanical resistance of the system, \(m\) the mass of the diaphragm and voice coil, and \(K_{m}\) the mechanical stiffness of the elastic suspensions. By solving Equations (2) and (3) the following expression of the acoustic impedance of the LA is found  
\[ \begin{array}{c} Z_{a}=\frac{\left(p_{1}-p_{2}\right)}{U}=\frac{1}{A_{d}^{2 }}\Bigg{[}Z_{m}+\frac{\left(Bl\right)^{2}}{Z_{e}+R_{L}}\Bigg{]}=R_{a}+jX_{a}= \\ \\ =\frac{1}{A_{d}^{2}}\Bigg{[}R_{m}+\frac{\left(Bl\right)^{2}\left(R_{e}+R_{L} \right)}{\left(R_{e}+R_{L}\right)^{2}+X_{e}^{2}}\Bigg{]}+j\frac{1}{A_{d}^{2}} \Bigg{[}X_{m}-\frac{\left(Bl\right)^{2}X_{e}}{\left(R_{e}+R_{L}\right)^{2}+X_ {e}^{2}}\Bigg{]},\end{array}  \tag{6}\]  
The acoustic power absorbed by the alternator can be written in the following equivalent forms  
\[W_{a}=\frac{1}{2}\text{Re}\Big{\{}(p_{1}-p_{2})\widetilde{U}\Big{\}}=\frac{1 }{2}|U|^{2}R_{a}=\frac{1}{2}\frac{|p_{1}-p_{2}|^{2}}{|Z_{a}|^{2}}R_{a}=\frac{ 1}{2}|p_{1}-p_{2}||U|\cos(\ \varphi_{\text{PU}})\, \tag{7}\]  
where \(\varphi_{\text{PU}}\) is the phase angle between the pressure difference (\(p_{1}-p_{2}\)), and \(U\) and the tilde symbol indicate complex conjugation. Substitution of Equation (6) into Equation (7) provides  
\[ \begin{array}{c} W_{a}=\frac{1}{2}\frac{|U|^{2}}{A_{d}^{2}}\Bigg{[}R_{m}+ \frac{\left(Bl\right)^{2}R_{e}}{\left(R_{e}+R_{L}\right)^{2}+X_{e}^{2}}+\frac {\left(Bl\right)^{2}R_{L}}{\left(R_{e}+R_{L}\right)^{2}+X_{e}^{2}}\Bigg{]}=\\ \\ =\frac{1}{2}|p_{1}-p_{2}|^{2}A_{d}^{2}\frac{R_{m}[\left(R_{e}+R_{L} \right)^{2}+X_{e}^{2}]+\left(Bl\right)^{2}\left(R_{e}+R_{L}\right)}{\left[ \left(Bl\right)^{2}-X_{m}X_{e}+R_{m}\left(R_{e}+R_{L}\right)\right]^{2}+\left[ X_{m}\left(R_{e}+R_{L}\right)+R_{m}X_{e}\right]^{2}},\end{array}  \tag{8}\]  
This equation shows that a fraction of the absorbed acoustic power is dissipated by the mechanical resistance \(R_{m}\), another fraction is consumed in the coil electrical resistance \(R_{e}\), and the remaining fraction is extracted by the load \(R_{L}\) as electric power. The last term can be also written as  
\[