# Design and Impedance Control of a Traveling Wave Ultrasonic Motor with a Spherical Rotor
## 4 Output Behaviors of the TWUM  
The output behaviors of the TWUM were determined using the vibration state of the stator. The alternating voltages applied on the PZT board can be expressed as:  
\[\begin{array}{l}U_{A}=A\sin(2\pi ft)\\ U_{B}=A\sin(2\pi ft+\varphi)\end{array} \tag{4}\]  
where \(A\) is the amplitude, \(f\) is the frequency, and \(\varphi\) is the phase difference. Therefore, there were three optional parameters to control. As mentioned previously, the output torque control was uncertain, but we could control the speed of the TWUM by adjusting the three parameters. Table 1 summarizes the speed regulation mechanisms and characteristics of the three control parameters [19].  
Due to the narrow linear range and the difficulty of achieving a drive circuit, the amplitude modulation mode was not considered in this paper. The frequency modulation and phase shift could be easily achieved using PSoC technology. Experience shows that phase difference modulation has a better stability and compliance than frequency modulation for the control of a TWUM and is smoother when frequent reversing is required [20]. Hence, the phase difference modulation based on a certain frequency was adopted in this study. Figure 5 shows the relationship between the steady-state speed and frequency in different directions. Because the stator drove the rotor with teeth on its inner ring, the frequency range was only about 1.5 kHz. As shown in Figure 5, there was a speed gap when the motor was operated in different directions, which was likely due to the different contact conditions in the forward and reverse directions. Since the stator of the TWUM had a low resistance characteristic near the mechanical resonance frequency, and thus had a better efficiency, the working frequency should be near the right side of the resonance frequency. However, if it is very close to the resonance frequency, there will be a risk of damage to the PZT board. Consequently, the working frequency was selected to be 44.5 kHz.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Control Mode** & **Mechanism** & **Advantage** & **Disadvantage** \\ \hline Amplitude & \begin{tabular}{c} Change the amplitude of \\ the traveling wave \\ Change the resonance \\ state of the stator \\ Change the elliptic trace \\ of the particle motion on \\ the stator surface \\ \end{tabular} & Good linearity & \begin{tabular}{c} Narrow range, large dead zone, \\ low torque at low speed \\ \end{tabular} \\ Frequency & \
\begin{tabular}{c} Change the resonance \\ state of the stator \\ Change the elliptic trace \\ of the particle motion on \\ the stator surface \\ \end{tabular} & High response & Nonlinear, poor stability \\ Phase difference &
\begin{tabular}{c} Change the elliptic trace \\ of the particle motion on \\ the stator surface \\ \end{tabular} & Stable, reversing smoothly & Certain dead zone at low speed \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of speed regulation mechanism and characteristics.  
Figure 5: Steady-state speed versus frequency.  
Figure 6 shows the effects of the phase difference modulation on the steady-state rotary speed of the rotor under various external torques. From the figure, some features can be observed. The output rotary speed decreased with a decrease of the phase difference, and there was a control dead zone of about \(\pm 20^{\circ}\). Furthermore, as the load increased, the dead zone expanded slightly. When the load increases to 0.1 Nm, the dead zone expands to about \(\pm 35^{\circ}\). On the other hand, when the speed was the same, the output torque tended to increase with the increase of the phase difference, although the relationship was not very linear. This meant that it was possible to control the output torque directly using phase difference modulation.  
The main properties of the designed TWUM with a spherical rotor are shown in Table 2.