# Helmholtz Resonance Frequency of the Synthetic Jet Actuator
## 3 Results  
### Identification of Resonance Frequencies  
The resonance frequencies of the SJ actuator were identified from the measurement of the electric apparent power with the constant voltage \(E\) operation. The apparent power \(P\) supplied by the actuator is a product of the measured effective voltage \(E\) value and the effective electric current \(I\). Figure 4 presents the characteristic of the apparent power consumed by the actuator when supplied with an effective voltage of 3 V in the case of SJ actuator geometries dl-9/V1-dl-9/V9. It can be seen that two local minima of the power occur. These characteristic  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Office** & **Office** & \\
**Case** & **Diameter,** & **Length,** & **I**_/_d_ [-] \\  & \(d\) **[mm]** & \(I\) **[mm]** & \\ \hline dl-1 & 6.20 & 4.78 & 0.77 \\ dl-2 & 9.12 & 4.81 & 0.53 \\ dl-3 & 15.03 & 4.84 & 0.32 \\ dl-4 & 19.95 & 4.80 & 0.24 \\ dl-5 & 23.95 & 4.79 & 0.20  
points correspond to the two resonance frequencies. The first power minimum occurs at the diaphragm resonance frequency equal to \(f_{\rm D}=19\) Hz. This frequency depends on the loudspeaker parameters and the vibrating air mass in the volume of the orifice [13]. The second observed resonance frequency observed is the Helmholtz resonance frequency. In the case of SJ actuator geometry dl-9/V1, the Helmholtz resonance frequency is equal to \(f_{\rm H}=352\) Hz. As the cavity volume increases, the value of \(f_{H}\) decreases, and reaches a value equal to 184 Hz for the case of dl-9/V9. Resonance frequencies were also determined from the frequency-impedance characteristic. The impedance \(Z\) is the ratio of effective voltage \(E\) divided by the effective electric current \(I\) feeding the SJA. Because both impedance and apparent power are directly dependent only on the effective electric current and effective voltage, the resonance frequency will be the same for both characteristics. In the case of the impedance characteristics, the local maxima occur at the diaphragm resonance frequency and at the Helmholtz resonance frequency. Determination of the resonance frequencies is also possible on the basis of velocity measurements. The centerline momentum velocity \(U\) in the orifice exit plane was obtained from the formula:  
\[U=\sqrt{\frac{1}{T}\int_{0}^{T}(u^{2})d\tau}, \tag{1}\]  
where \(T\) is an oscillation period and \(u\) is an instantaneous velocity. The dependence of \(U\) on \(f\) is presented in Figure 5. Maxima of the characteristic velocity occur at the resonance frequencies. The estimation of resonance frequencies from the characteristic velocity was utilized in the reference [11; 12]. This measurement technique has lower precision and results in more scattered data than the impedance measurement technique. In addition, the hot-wire measurement required calibration, room temperature stabilization, and temperature compensation, in addition to precise positioning of the probe. Figure 4: Dependence of SJA apparent power on excitation frequency for effective voltage \(E=3\) V.  
The comparison of the Helmholtz resonance frequencies obtained from impedance measurement\(f_{H}(Z)\) and momentum velocity measurement \(f_{H}(U)\) is presented in Figure 6, which shows a high compliance of the results. As discussed previously, the Helmholtz resonance frequency can be determined from measurements of electrical parameters or measurements of the air velocity. Due to the better precision and repeatability of the electrical measurements in relation to the air velocity measurements with the use of constant temperature anemometry, Helmholtz resonance frequencies necessary for the validation of the mathematical model were determined from impedance measurements. Effects of The Supply Voltage and SJ Actuator Geometrical Parameters on the Helmholtz Resonance Frequency  
To select the effective voltage used to supply the synthetic jet actuator, an experiment was carried out to determine the impedance characteristics.