4.1. Equivalent Circuit for Modeling the Developed Device
The developed device was modeled using an equivalent circuit to better understand its characteristics. The electric equivalent model can be divided into three parts for investigation: electric, mechanical, and acoustic fields (
Figure 9). Electric and mechanical field modeling can be found in the literature [
26,
27,
28]. Acoustic field modeling was first proposed. Between the electric and mechanical domains, the input voltage Vin was a sinusoidal signal set by the function generator and the imported current on the device was I. V1 represents the effective voltage that converted the force through the piezoelectric effect of the PZT film. The conversion ratio was modeled as an ideal transformer with a turn ratio ϕ, that is,
Fp = ϕ × V
1. The electric domain was considered as the device operation under fundamental resonance conditions. The force
Fp caused a vibration of the PZT membrane with a velocity of u. Meanwhile, the resulting air pressure against the vibrated PZT membrane was modeled as the force
Fa.
Fa is the force to actuate the surrounding air at the top and bottom surfaces of the PZT membrane and is affected by environmental impedance.
Fp due to the piezoelectric effect converted from V
1.
Fp-
Fa means that the force piezoelectric effect generated minus the force actuating the surrounding air, which caused the motion of the piezoelectric membrane itself and modeled as a C
m, R
m, and L
m equivalent circuit.
P1 represents the air pressure on the top surface of the PZT membrane caused by the PZT membrane-produced force Fa. In the equivalent circuit for the acoustic field, P1 was the input pressure to the Helmholtz resonator. The neck portion of the Helmholtz resonator was modeled as the effect of the resistance (Re) consuming energy and the inductor (Le) conserving kinetic energy. The cavity is modeled as a capacitor (Ce). We used the vibrational area of the PZT membrane as the conversion ratio, that is, Fa = Av × P1. P2 is pressure inside the Helmholtz cavity. P3 is the output pressure close to the opening region of the PMUT membrane.
If we converted the mechanical ports to acoustic ports with radiation impedance, we should consider the different surrounding conditions on the top and bottom of the four-beam PMUT membrane once the Helmholtz cavity was added. The four openings of the PMUT vibrational membrane induced the interaction of the air between the top and bottom surroundings of the membrane. This made the traditional three-port piezoelectric model with one electrical port and two mechanical ports more complicated. Using two-port representation to describe a system similar to our investigation can be found in [
20,
28,
29,
30].
We will discuss the evaluation of the elements labeled in the electric, mechanical, and acoustic-domain equivalent circuits in detail below.
Electric part:
In the electric domain, C0 represents the intrinsic capacitance of the PZT film, and R0 represents the dielectric loss in the PZT film. Because C0 and R0 are frequency dependent, we estimated these two values at the condition in which the frequency was close to the fundamental resonance and the vibration of the PZT membrane approached zero, that is u ≅ 0. Under this condition, the driving current from the input can be modeled by simply passing it through R0 and C0. In this study, we determined R0 and C0 by impedance measurement at a frequency that was approximately three times the bandwidth of the fundamental resonance below the peak frequency of the fundamental resonance.
Figure 10 shows the measured electrical impedance for the open-cavity case. The resonant frequency was 23.7 kHz, and the bandwidth was approximately 300 Hz. Thus, we selected the impedance as 22.8 kHz, which has a magnitude of 4098 Ω and an angle of −78.62°, to derive R
0 and C
0. The resulting R
0 was 808.6 Ω, and C
0 was 1.752 nF.
Mechanical Part:
Considering the PZT membrane vibration as a lumped mass–damper–spring system, the displacement of the mass was set as the displacement at the center of the membrane. When the membrane vibrates, the equation of motion is formulated as follows:
where
M is the mass,
B is the viscous damper, and
Ks is the spring constant. We can obtain the transfer function of the displacement
X(
s) vs. the force difference Δ
F(
s), which is the force difference provided by the driving force
fp due to piezoelectric conversion from the applied driving voltage V
1 and
fa. This overcomes the force
fa due to the surrounding air exerted on the vibrating membrane, as follows:
Because the membrane performs a harmonic motion, its velocity and associated force can be expressed as
u(t) = ue
jωt =
jωxe
jωt and
f(t) = Fe
jωt. Thus, we have
This expression allows us to be analogical to the inductor L
m, resistor R
m, and capacitor C
m elements in the equivalent circuit with L
m = M, R
m = B, and C
m = 1/Ks (
Figure 9).
Meanwhile, the measured displacement of the vibrating membrane around the first resonant frequency had a relatively good fit to a second-order system. Because the actual magnitude of the measured displacement was proportional to the magnitude of the fast Fourier transform (FFT) result, we utilized parameter
K2 to represent the ratio:
where
ζ is the damping ratio,
ωn is the natural frequency, and
K1 is a constant gain. By adjusting the
ζ, ω
n, and
K1 values, we created a frequency response of a second-order system
G(
s) to match the FFT resulting curve converted from the measured displacement.
We normalized the amplitude of the FFT curves shown in
Figure 7 with the near-zero slope region below the resonance (i.e., 21–22 kHz) to approximately 1 by dividing by 0.0025. This facilitates determining the damping ratio, natural frequency, and suitable gain of a corresponding second-order system to match this normalized FFT curve.
Figure 8a shows that the transfer function
s) matches the normalized curve of the open-cavity case
Xfft(s). Moreover, using the relation ∆
F(
s)/
M =
K1K2ωn2, we can further derive
At resonance, the displacement can be obtained by measurement,
Hence, the force applied to the membrane at resonance can be expressed as
Because the mass of the vibrating membrane can be determined by the dimension of the membrane and the densities of the composed materials, we used M = 316.6 × 10−9 kg as the mass of the vibrating membrane for the following analysis. Thus, the ∆F could be obtained from the given mass M and ωn (=2π × 23,701 rad/s), measured displacement at resonance 1.65 μm, Xfft@resonance = 9.32, and for the case of open cavity: ∆F (s = jωp) = Fp(s)−Fa(s) = 1.714 × 10−4 N.
For the Helmholtz resonating cavities of varying volumes, we can find the individual
G(s) to fit the corresponding FFT results of the measured displacement responses (
Figure 11b–f). The parameters
K1,
ωn, ω
p, and
ζ of
G(
s) are listed in
Table 1. Based on these parameters and the measured displacement at resonance, we can compute the force difference Δ
F(
s) for cases with different cavity lengths using the above equation.
Moreover,
Fa(s) could be estimated as follows: according to [
31], if we consider the investigated vibrating membrane of diameter a and surface vibration velocity of
v0 as a small piston performing repeated back-and-forth motion and the generated sound wave as possessing the relation of the wave number,
k, multiplying the diameter of membrane satisfies
ka <<1, and the pressure along the axis normal to the surface of the vibrating membrane at the antinode can be expressed as
where
h represents the distance from the membrane surface to the observation point,
v0 is the vibration velocity of the membrane, c is the speed of sound, and
ρ0 is the density of surrounding air. If the observation point is in the near field,
h <<
a, and satisfies the condition
ka < 1, the pressure can be written in the following form:
By dividing Equation (8) by (9), we can obtain the ratio of the pressure
Because the measured pressure data were 15 mm above the antinode, the pressure on the antinode can then be evaluated as
Thus, we can obtain Fa according to the derived pressure at the antinode of the membrane
Moreover, the ratio of 0.0565 is applied to the following studied cases of the Helmholtz cavity with varied volumes to describe the pressure attenuation from the point of measured pressure to a point very close to the vibrational membrane.
The radiation impedance of the fluid surrounding the PMUT top surface was considered using Equation (10). The microphone was placed 15 mm away from the surface of the PMUT top surface. The pressure at the position very close to the PMUT surface is about 24.96 dB of the measured sound pressure (at microphone position) (paxis/paxis,VNF = 0.0565 = −24.96 dB). In this study, the condition of h = 15 mm and a = 1.5 mm was used, some error could be induced by the assumption h << a. However, using the condition as a first-order approximation would allow us to analyze the experimental data.
Acoustic part:
A conventional Helmholtz resonator consists of neck and cavity portions. The neck of the resonator is an airflow passage that causes friction loss in the airflow. Meanwhile, the air around the neck can be considered as a finite mass with inertia that possesses kinetic energy. Therefore, the neck portion can be modeled as the effect of the resistance consuming energy and the inductor conserving kinetic energy in the equivalent circuit. The air inside the cavity of the resonator could be treated with compressible and possesses potential energy. Hence, the cavity is modeled as a capacitor in the electrical circuit. The equivalent circuit to describe the dynamic response of a Helmholtz resonator is then a resistor, inductor, and capacitor connected in series with input pressure as the input voltage and the output pressure as the voltage across the capacitor (
Figure 12).
The values of the modeled RLC elements can be found in many studies. In this study, we utilized the traditional formulas below [
32]:
The resulting values for
Ct,
Lt, and
Rt are m
3/Pa, kg/m
4, and kg/(m
4·s), respectively. The corresponding equivalent circuit could be expressed as shown in
Figure 12.
This model described that the output sound pressure of the Helmholtz resonator can be amplified from the input sound pressure through an equivalent RLC circuit modeling and the gain can be expressed as a second-order system.
If we substituted the physical parameters listed in
Table 2 using different
Ct, which is a function of volume change by adjusting the length change of the Helmholtz cavity, a constant
Mt and a constant
Rt, the results of the transfer function
Gt(
s) =
P2(
s)/
P1(
s) are displayed in
Figure 13. The results indicate that the large volume change ratio caused the peak frequency to change, ranging from 20.1 to 34.9 kHz and the gain value ranged from 779 to 450.
Examining the frequency response based on the measured output sound pressure, which was a gain of 24.96 dB multiplied by
P3(
s), from the studied Helmholtz resonator integrated PMUT, an approximate second-order system could be observed (
Figure 8a). To further study this response, let us consider the frequency response
P3(
s) as a function of
P2(
s), or more specifically, the relation between
P3(
s) and
P2(
s) follows
, where
is a gain factor as a function of frequency. Thus,
,
, and
P1(
s) was the input sound pressure.
Since the FFT result of acquired output pressure
P3
mea,
P3
fft, was proportional to
P3, we tried to find a suitable second-order system to fit
P3
fft first. The ultrasonic wave signals for different cavity lengths in this investigation were analyzed. We tried to fit the frequency responses
P3
fft of measured ultrasonic wave signals with proper damping ratio
ζ and natural frequency
ωn. For all the fitting results,
P3
fft displayed a standard second-order system multiplied with a factor. Hence, we could express the fitting result of
P3
fft(
s) as a function of
Kp(
s)·
Gs(
s)·
P1
fft(
s). Regarding the parameter
Kp(
s), we could model this factor when the pressure
P2 in the cavity transmits to the outside environment with the pressure
P3. Because the relation between
P3 and
P3
fft(
s) can be employed a factor kf, i.e.,
P3(s) =
P3
fft(
s)·kf and
P3
fft(
s) =
Kp(
s)·
Gs(
s)·
P1
fft(
s), under the same signal processing data points, we could use the same factor kf to relate
P1(
s) and
P1
fft(
s) to better describe the relationship between
P3 and
P1, same as
P3 and
P2. In this study, kf can be evaluated as 3.912 Pa. This allowed us to estimate
P1 over the investigated frequency range with
P3 divided by
Gs(
s) and
Kp(
s).
Figure 14 shows the results of
P3
fft(
s) and
P1
fft(
s) (=
P3
fft(
s)/(
Kp(
s)·
Gs(
s))) in the frequency range of interest. Therefore, the pressure
P2(
s) can be described as
Gs(
s)·
P1(
s). The ratio of
P2 to
P1 then satisfied a standard second-order system with
P2 as the capacitor voltage output, which was described in the traditional Helmholtz resonator model.
Table 3 lists important parameters of
G(
s) and
Kp.
According to the above analysis, the experimental frequency responses display a gain factor
Kp(
s) multiplying
Gs(
s), which means a gain factor should be added besides the conventional Helmholtz resonator circuit model including R, L, and C elements. To fully describe the experimental response with a proper electrical circuit model, we utilized an ideal amplifier to represent the gain factor for the modified equivalent circuit (
Figure 15).
When we further examined the peak frequencies calculated based on the conventional model derived from
Gt(
s), the ratio of
P2 to
P1, though some of the peak frequency values were close to the results of
Gs(
s), a noticeable deviation exists for most of the studied cases. Moreover, the computed damping ratios were quite different from the
P2 obtained by experimental results. To better describe these differences, we compared the damping ratio and resonant frequency of the RLC circuit modeled second-order system with input pressure
P1 and output pressure
P2 modeled as the voltage across the capacitor based on conventional theory and experimental results (
Table 4).
Because the obvious deviation existed to use the conventional model to describe our developed PMUT with the integration of the Helmholtz resonator, in this study we proposed the modified values of the equivalent circuit elements of the Helmholtz resonator to replace the element values obtained in the conventional equivalent circuit to better fit the experimental results (
Figure 15). In this modified circuit element, we kept the capacitance as the conventional modeling and the viscous resistance and effective mass were modified as Re and Le, respectively, which is explained in more detail below.
We proposed the concept of natural resonant frequency and damping ratio adjusting factors. This facilitated us to convert the
Rt and
Lt expressed in the conventional circuit model of the Helmholtz resonator to the modified
Re and
Le parameters for the experimental data fitted modified circuit model. Define the natural resonant frequency and damping ratio adjusting factors
α and
β as follows:
where
and
represent the natural resonant frequency derived from the experimental result and calculated from the theoretical model, respectively.
and
stand for the damping ratio from the experimental result and theoretical model, respectively. Moreover, if we set the capacitance
C as a controlled variable, this means the parameters defined in the capacitance of the theoretical model were identical to the modified circuit model. Hence, the modified mass (inductor) term can be expressed as
Similarly, the modified viscous damper (resistance) term can be expressed as
where
γ and
δ are defined as the adjusting factors of the effective mass and viscous damper, respectively. Thus, we can determine the relationship between the effective mass of the Helmholtz resonator based on the experimental results and theoretical model as a function of α and the relationship between the effective viscous damper based on the experimental results and theoretical model as a function of
α and
β.
Table 5 lists the results of adjusting factors of natural resonant frequency, damping ratio, effective mass, and viscous damper for different volumes of the Helmholtz resonant cavity. The corresponding effective mass and viscous damper after considering the adjusting factors to fit the experimental results were also calculated. The modified effective mass and viscous damper were compared with the conventional effective mass and viscous damper. The latter parameters were much lower than the former parameters. This implied that the investigated microliter-sized Helmholtz resonator had stronger effects on viscous damping and larger effective mass compared to the regular milliliter or even greater-sized Helmholtz cavity.
To further explore the trend of the four investigated adjusting factors as a function of the volume change of the Helmholtz resonating cavity, we tried to perform the mathematical analysis by using different categories of curve fittings. A very good curve fitting result can be obtained by using a two-term power series model,
for these four adjusting factors.
Figure 13 shows all fitting curves, exhibiting an R-squared value greater than 0.99. Using these fitting curve equations, we can map the conventional lumped model parameters to a new set of model parameters, allowing us to properly explain the empirical outcomes of the pressure
P2 and
P1 for our designed PMUT with a microliter-sized resonating resonator.
To correlate the output pressure P3 and cavity pressure P2, a circuit model of the ideal amplifier was connected to the capacitor of the acoustic model. Since the input impedance of the ideal amplifier approaches infinity, the loading effect will not result in the voltage drop across the capacitor for this modeling. This described the pressure P2, which was amplified through the Helmholtz cavity, converted to the output pressure P3 simply via a gain factor Kp, to accurately match the experimental results.
Experimental data also revealed a gain factor
Kp variation accompanied by a volume change of the Helmholtz cavity. Interestingly, the relation also well fitted the two-term power series curve,
(
Figure 16).