# Using a Lossy Electrical Transmission Line Model for Optimizing Straw Phonation Configurations
## 2 Materials and Methods  
To compare the validity of the electrical circuit method for impedance computation to the existing chain matrix method, data of 18 vocal tract shapes were collected from previous research performed by [PERSON] et al. [16]. Each of these shapes discretized the vocal tract into tubelets of equal length (0.396825 cm), so each shape had varying amounts of tubelets ranging from 40 to 46. To prevent undefined results from the calculations, tubelets listed with \"0.00\" cross-sectional area were increased to 0.01 cm\({}^{2}\). Since the largest tubelet was about 9 cm\({}^{2}\), an increase of 0.01 cm\({}^{2}\) for the tubelets with no cross-sectional area is small in magnitude and should not produce biased results. No other measurements from the glottal area data were altered. The formulae provided in [PERSON] et al. are copied below:  
\[ \begin{pmatrix}P_{out}\\ U_{out}\end{pmatrix} = \begin{pmatrix}\cosh(\frac{el}{c})&\frac{-\rho c\gamma}{ara }\sinh(\frac{el}{c})\\ \frac{-\alpha\alpha\gamma}{\rho c\gamma}\sinh(\frac{el}{c})&\cosh(\frac{el}{c} )\end{pmatrix}\begin{pmatrix}P_{in}\\ U_{in}\end{pmatrix} \]  
\[\gamma=\sqrt{\frac{r+j\omega}{\beta+j\omega}};\qquad\sigma=\gamma(\beta+j \omega);\qquad\beta=\frac{j\omega(2\pi F_{T})^{2}}{(j\omega+r)j\omega+(2\pi F _{\omega})^{2}}+\sqrt{j\omega q} \tag{1}\]  
where \(F_{T}=200\) Hz, \(F_{\omega}=15\) Hz, \(r=408\) rad/s, \(q=4\) rad/s. For the electrical circuit model, formulae for inductance, capacitance, resistance, and shunt conductance of the tubelet equivalent circuits were taken from Flanagan from 1972 [17] and listed below. \[L=\frac{\rho}{A};\qquad\quad C=\frac{A}{\rho c^{2}};\qquad\quad R=\frac{S}{A^ {2}}\sqrt{\frac{\omega\rho\mu}{2}};\qquad\quad G=S\frac{\eta-1}{\rho c^{2}} \sqrt{\frac{\lambda\omega}{2C_{p}\rho}} \tag{2}\]  
where \(\omega\) is the angular frequency in radians per second, defined as \(\omega=2\pi f\); \(A\) is the cross-sectional area of tubelet; and S is the circumference of the tubelet, which can be calculated from the geometric relation to the area such that \(S=\sqrt{4\pi A}\). It is important to note that there is no term for the length of the tubelet in any of the equations, which will be important for calculating impedance further on. Definitions for the other variables and their standard values used in these calculations are listed in Table 2. Since this model includes values for resistance and shunt conductance, it is lossy, as opposed some other models for artificial speech synthesis [18]. Moreover, the Flanagan model was also chosen for its simplicity by calculating the total amount of each electrical value for each tubelet, instead of multiple electrical components [19; 20; 21]. Similar to [PERSON] et al., impedance can be calculated from a series of 2 \(\times\) 2 matrices where the elements of each matrix are  
\[ \begin{pmatrix}E_{out}\\ I_{out}\end{pmatrix} = \begin{pmatrix}\cosh(\gamma\ell)&-Z_{0}\sinh(\gamma\ell)\\ -Y_{0}\sinh(\gamma\ell)&\cosh(\gamma\ell)\end{pmatrix}\begin{pmatrix}E_{in} \\ I_{in}\end{pmatrix}  \tag{3}\]  
where \(Z_{0}\) is the characteristic impedance, \(Y_{0}\) is the characteristic admittance, and \(\gamma\) is the propagation constant (note that this \(\gamma\) is different from the variable used in the [PERSON] et al. equations) of that tubelet such that [22]  
\[ \begin{split} Z_{0}&=\sqrt{\frac{R+j\omega L}{G+j\omega C}}\\ Y_{0}&=\frac{1}{Z_{0}}=\sqrt{\frac{G+j\omega C}{R+j\omega L}}\\ \gamma&=\sqrt{(R+j\omega L)(G+j\omega C)}\end{split}  \tag{4}\]  
The equations above highlight the novelty of the transmission line model. While CFD can only find pressure and volume velocity as real numbers which actually occur in the vocal tract, the transmission line yields complex values which indicate both the realized aerodynamic values but also the stored energy to be later expended.