## 4 Results and Discussion  
Figure 4 compares the patterns of printed drops with various pulsation frequencies for nozzle A and nozzle B at the minimum pulse voltage of jetting. The two nozzles have the same diameter \(d_{n}\) of 30 \(\upmu\)m and the same natural frequency \(f_{c}\) of 224 Hz, while they have different damping ratios, as shown in Tables 1 and 2. For nozzle A, shown in Figure 4a, where the estimated damping ratio is 2.0 (overdamped), ink drops are printed at the designed drop spacing of 40 \(\upmu\)m for 3, 10, 30 Hz, and even for 1 kHz. The pulsed voltage in Figure 4a was 150 V. On the other hand, for nozzle B, shown in Figure 4b, where the damping ratio is 0.78 (underdamped) and the pulsed voltage is 300 V, the drop spacing becomes multiples of 40 \(\upmu\)m as pulsation frequency increases. Figure 5 shows the measured jetting frequency as a function of pulsation frequency. When the system is overdamped (\(\Gamma>1\)), as shown for nozzle A in Figure 5, jetting frequency is in good agreement with pulsating frequency because the ink refill time is sufficiently small [15] and the meniscus stabilizes quickly after jetting. However, when system is underdamped (\(\Gamma<1\)), as shown for nozzle B in Figure 5, jetting frequency does not increase with pulsation frequency but is limited to a few Hz. This is possibly due to the capillary oscillation of the ink inside the nozzle after jetting. This result demonstrates that the overdamped EHD system is advantageous for high frequency DOD jetting. In an overdamped system, the meniscus shape and capillary force gradually change to have periodic equilibrium for a given pulsation wave. However, in an underdamped system with high frequency, there occurs instability due to an interference of capillary and pulsating waves because the capillary force varies sinusoidally in between the pulsations. \begin{table}
\begin{tabular}{l l l l} \hline \hline \multirow{2}{*}{**Parameter**} & \multicolumn{2}{c}{**Nozzle \#**} \\ \cline{2-4}  & **A** & **B** & **C** \\ \hline Natural frequency inside the capillary, \(f_{c}\) (Hz) & 224 & 224 & 448 \\ Damping ratio of the system, \(\Gamma\) & 2.0 & 0.78 & 1.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculated natural frequency and damping ratio of three different nozzles. Figure 4: Printed drops of Ag ink with various pulsation frequencies of 1, 10, 30, and 1000 Hz for (**a**) overdamped system (Nozzle A; \(\Gamma=2.0\)) and (**b**) underdamped system (Nozzle B; \(\Gamma=0.8\)) at the minimum pulse voltages of jetting of 150 V and 300 V, respectively. The two nozzles have the same diameter \(d_{n}\) of 30 \(\upmu\)m and the same natural frequency \(f_{c}\) of 224 Hz, while they have different damping ratios. Interestingly, Figure 5 also shows that, even for the overdamped nozzle A, jetting frequency deviates from the pulsation frequency near the estimated natural oscillation frequency, \(f\). In order to investigate the effect of \(f\). on DOD jetting, Figure 6 shows the jetting frequency as a function of the natural frequency ratio for overdamped nozzles A and C. As shown in Tables 1 and 2, the inner diameter of nozzle C is 15 \(\upmu\)m (one half the diameter of nozzle A) and, therefore, the estimated natural frequency is 448 Hz (double the natural frequency of nozzle A). The natural frequency ratios of pulsation \(\alpha\) and jetting \(\beta\) are the frequencies normalized by the natural frequency, respectively. In Figure 6, pulsed voltages \(V_{p}\) of either 150 V for nozzle A or 220 V for nozzle C are close to the minimum voltage that allows DOD jetting shown in Figure 2 and Equation (1). The bias voltage  
Figure 5: Measured jetting frequency as a function of pulsation frequency. Jetting frequency is in good agreement with pulsating frequency for overdamped system (Nozzle A; \(\Gamma\) = 2.0), while jetting frequency does not increase with pulsation frequency at a few Hz for underdamped system (Nozzle B; \(\Gamma\) = 0.8).