# Using a Lossy Electrical Transmission Line Model for Optimizing Straw Phonation Configurations
## 3 Results
Figure 3: Impedance curves for four vocal tract shapes with three length of straws each with 1 cm\({}^{2}\) cross-sectional area: 0 cm (black), 15 cm (blue), and 30 cm (red). Impedance is in units of dyne-s/cm\({}^{5}\). (**a**) /t/ (**b**) /i/ (**c**) /u/ (**d**) /n/. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Chain Matrix** & \multicolumn{3}{c}{**Transmission Line**} \\ \hline
**Phoneme** & \(\mathbf{Z_{max}}\) & **Frequency** & \(\mathbf{Z_{max}}\) & **Frequency** \\ \hline /t/ & 8.836 & 670 & 8.993 & 100 \\ /i/ & 8.634 & 980 & 7.601 & 220 \\ /u/ & 8.193 & 860 & 7.435 & 250 \\ /n/ & 7.441 & 350 & 7.422 & 80 \\ /m/ & 7.260 & 240 & 7.103 & 70 \\ /p/ & 6.278 & 240 & 9.188 & 140 \\ /s/ & 5.916 & 430 & 7.309 & 480 \\ /u/ & 5.724 & 420 & 8.663 & 350 \\ /l/ & 5.591 & 490 & 6.645 & 430 \\ /m/ & 5.573 & 200 & 8.238 & 120 \\ /k/ & 5.536 & 270 & 8.254 & 170 \\ /o/ & 5.493 & 380 & 7.176 & 340 \\ /u/ & 5.456 & 390 & 8.549 & 450 \\ /c/ & 5.318 & 360 & 8.270 & 600 \\ /s/ & 4.839 & 340 & 7.497 & 640 \\ /ae/ & 4.504 & 330 & 6.989 & 680 \\ /s/ & 4.474 & 280 & 10.317 & 1000 \\ /a/ & 4.436 & 310 & 8.749 & 660 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Frequencies at which maximum impedance occurs versus maximum impedance value on log scale for both impedance calculation methods for various phonemes. Maximum impedance (\(Z_{max}\)) is in units of dyne-s/cm\({}^{5}\) and frequency is units of Hz. The plots in Figure 3 each demonstrate that the addition of a straw to the phonation of a given phoneme alter the impedance levels and location of the fundamental frequency F0. As the length of the straw is increased from 0 cm to 15 cm to 30 cm, F0 is decreased monotonically, although the amount of such change is not apparently related to the length of the straw. However, the change in the impedance value at F0 is not the same for all of the phonemes considered. Beyond F0, the highly nonlinear nature of phonation and aerodynamic vocal tract wall interaction appear to overwhelm any effect that the introduction of the straw may have, but since as discussed below, the fundamental frequency is of greater importance than other relative frequencies, this may not be a worthwhile argument against the effectiveness of straw phonation. Furthermore, another goal of this paper is to gain more knowledge of acoustic power and its relation to impedance, which was calculated from the electrical equation for power dissipated. An example graph of power is shown in Figure 4, with the same tubelet configuration of 17.5 cm long, 1 cm\({}^{2}\) area straw as Figure 2.  
It is clear that spikes in the power graph correspond to the same frequencies at which local extrema of the impedance graph exist--both minima and maxima. The case of the power peak near the impedance peak makes sense; because the numerator of the power equation is of cubic order, while the denominator is only quadratic, maximizing resistance and minimizing reactance will overall maximize power. This is precisely the case where impedance is at a maximum as described already. However, the peak near the minimum of impedance results from the properties of radiation impedance. While radiation resistance is not directly correlated to the frequency (as there is the same power of \(\omega\) in both the numerator and denominator), radiation reactance changes inversely with \(\omega\), so resistance is relatively unchanged while reactance goes to zero.