# Using a Lossy Electrical Transmission Line Model for Optimizing Straw Phonation Configurations
## 3 Results  
The first goal of this paper was to determine whether the electrical circuit analogy of the vocal tract as a transmission line is a viable alternative to the existing standard model from [PERSON]. While the two models essentially are equivalent for vocal tracts solely described as hard-walled straws, they diverge for realistic soft-walled vocal tracts. Example graphsof impedance, resistance, and reactance are given for both models based on a 17.5 cm long, 1 cm\({}^{2}\) area straw in Figure 2.  
Several notable features of impedance are apparent from these graphs. First and foremost, both impedance calculation methods generate remarkably similar plots for each of the values considered, and so much so that the overlap of the graphs appears to show only one curve. The plots only appear to separate at local extrema--both minima and maxima. The relative values at each of the extrema are not necessarily of importance because of the different units used between the two calculation methods, but rather, the shape of the general plot and the location of the local extrema are better indicators of the agreement between the two methods. The peak value for impedance occurs at resonance, and more specifically the fundamental frequency F0. This frequency also coincides with the maximum resistance and somewhat counterintuitively, where the reactance curve crosses the horizontal axis. As explained by [PERSON] et al., this is plausible because resonance occurs when there is a sharp increase in the amount of sound transmitted, indicating an ease in phonation. While the energy being used for phonation can either be in the form of resistance or reactance, reactance is the storage of energy meaning that there is less energy to be transmitted to the atmosphere. Therefore, it is quite reasonable to expect that resonance occurs at this frequency. However, the models produce disparate results when computing the input impedance of actual vocal tract shapes. Table 3 lists the maximum impedance values along with the frequency at which each of these impedance values occurred for all 18 vocal tract shapes, calculated by both models, listed in order of decreasing value of maximum impedance calculated by the chain matrix method. The discrepancies between the two models are expected, however. From the outset, [PERSON] indicates that the values for electric parameters were derived for a hard walled tube with losses proportional to I\({}^{2}\)R and E\({}^{2}\)G. Thus, the losses due to the elasticity of the vocal tract wall itself are unaccounted for by [PERSON]. As such, henceforth, only the chain matrix method will be used to determine the best configuration of phoneme and straw. Since increased impedance is generally considered beneficial, the four vocal tract shapes with the highest maximum impedance as listed in Table 3 (/t/, /i/, /u/, and /n/, respectively) are plotted in Figure 3. Each phoneme impedance curve is calculated with no straw, a 15 cm straw, and a 30 cm straw as described above in the methods. Only the range of frequencies between 0 and 300 Hz are considered here because the most important information to gather from the impedance curve relates to the fundamental frequency which is found in this range for each of the phonemes. Note  
Figure 2: Demonstration of similarity between the chain matrix and transmission line impedance calculation methods for a hard walled 17.5 cm long tube with constant 1 cm\({}^{2}\) cross-sectional area. Impedance, resistance, and reactance are in units of dyne-s/cm\({}^{5}\) and frequency is in Hz. The resistance and reactance plots have a frequency range of 300–700 Hz to highlight resonance and each asymptotically approach 0 resistance and reactance from both directions. Calculations done with the chain matrix method are red lines and those done with the transmission line method are blue for the following values: (**a**) Impedance (**b**) Resistance (**c**) Reactance. that the maximum impedance value listed in Table 3 may not be achieved in the range of these frequencies.