\section{Introduction}
\vspace*{-4mm}

In many experimental acoustofluidic platforms, the device is actuated by an attached piezoelectric transducer, driven by a sine-wave generator through a power amplifier. To describe the performance of the acoustofluidic actuation, the operating conditions are typically expressed in terms of the voltage amplitude or the electric power dissipation together with quantities such as the acoustic energy density, the acoustic focusing time, or achievable flow rates \cite{Barnkob2010, Barnkob2012, Lickert2021}. Often, it is however left unclear under which conditions and at which point in the electric circuit, the relevant quantities such as voltage amplitude or power dissipation have been measured. Recent studies compare device performance at constant average power for different placements of the transducer \cite{Barani2021, Qiu2022}. Dubay \textit{et~al.}~\cite{Dubay2019} performed thorough power and voltage measurements for the evaluation of their acoustofluidic device, however, they noted that the actual power delivered to the transducer might reduce to only a fraction (as low as 10\%) of the reported value. The likely cause of this reduction is that the transducer is acting as a large capacitive load, where electrical impedance matching between source and load impedance is difficult to accomplish \cite{Lissandrello2018, Dubay2019}.

Whereas optimization of the driving circuit is customary in other fields, such as ultrasonic transducers for cellular applications \cite{Kim2016c}, non-destructive testing \cite{Garcia2010}, and pulse-echo systems \cite{Emeterio2002}, this has not been given much consideration in the field of acoustofluidics, where the focus often lies on optimizing the acoustic impedance matching \cite{Leibacher2014, Ohlsson2018}, while neglecting the impact of the driving circuit. A recent work, though, considers topics such as electrical impedance matching in the context of developing low-cost and possibly hand-held driving circuits for acoustofluidics \cite{Huang2021}. To our knowledge, studies have not yet been performed, in which the impact of different electrical excitation methods on a transducer in a given acoustofluidic device is compared with respect to the resulting acoustophoretic particle focusing.

In the case of bulk piezoelectric transducers, where the electrical impedance ranges over several orders of magnitude as a function of frequency, the voltage amplitude across the transducer can differ severely from the amplitude expected by simply considering the voltage input at the amplifier. Suitable voltage compensation circuits or voltage correction methods should be used to achieve the desired voltage amplitude directly at the transducer. Furthermore, a standard has not yet been established whether it is more beneficial to run frequency sweeps at a constant voltage or at a constant power. We therefore in this work investigate the impact of three different actuation approaches during a frequency sweep: (1) Constant input voltage into the amplifier, (2) constant voltage at the transducer, and (3) constant power dissipation in the transducer. We compare experimental findings with our numerical model. The aim of this paper is to establish guidelines on which actuation approach is preferable for acoustofluidic applications using bulk piezoelectric transducers to generate acoustophoresis in bulk acoustic waves.

The paper is structured in the following way: In \secref{theory} a brief summary is given of the governing equations for the pressure field, the displacement field, and the electric potential in our acoustofluidic device. \secref{methods} gives an overview of our experimental setup, and the procedure used for the measurement of the particle velocities is described step by step. In \secref{num_model} we describe the numerical approach used in our study, and in \secref{results} we compare several aspects of the obtained results for the device under study: a comparison between the electrical characteristics of the device, as well as the numerically and experimentally observed acoustophoretic particle velocities are given. Furthermore, some details of the simulated fields are shown. Finally, the paper concludes in \secref{conclusion} with a short summary and some guidelines on the actuation of piezoelectric transducers for acoustofluidic applications.

\section{Theory}
\seclab{theory}
\vspace*{-4mm}

The theoretical approach follows our previous work \cite{Skov2019, Skov2019b, Steckel2021b, Lickert2021}, in which the computational effort in the simulations are reduced by employing the effective-boundary-layer theory derived by Bach and Bruus  \citep{Bach2018}. We assume time-harmonic first-order fields with angular frequency $\omega = 2\pi f$ for the acoustic pressure $\tilde{p}_1(\myvec{r},t) = p_1(\myvec{r})\:\mathrm{e}^{-{\ii\omega t}}$, the electric potential $\tilde{\varphi}(\myvec{r},t) = \varphi(\myvec{r})\:\mathrm{e}^{-{\ii\omega t}}$, and the displacement field $\tilde{\myvec{u}}(\myvec{r},t) = \myvec{u}(\myvec{r})\:\mathrm{e}^{-{\ii\omega t}}$. Derived through a perturbation approach, these fields represent tiny perturbations of the unperturbed zero-order fields.



\subsection{Governing equations}
\vspace*{-4mm}

For a fluid with speed of sound $c_0$, density $\rho_0$, dynamic and bulk viscosity of the fluid $\eta_0$ and $\eta_0^\mr{b}$, damping coefficient $\Gamma_0$, and the isentropic compressibility $\kappa_0 = (\rho_0 c_0^2)^{-1}$, the acoustic pressure $p_{1}$ is governed by the Helmholtz equation, and the acoustic velocity $\myvec{v}_1$ is a gradient field,
 \bsubal{EquMotionFluid}
 \eqlab{p1_Helmholtz}
 \nabla^2\ p_1 &= -\frac{\omega^2}{c_0^2} \big(1+\mathrm{i}\Gamma_0\big)\, p_1,
 \text{ with }
 \Gamma_0=\Big(\frac{4}{3}\eta_0+\eta_0^\mr{b}\Big)\,\omega\kappa_0,
 \\
 \eqlab{v1_grad_p1}
 \myvec{v}_1 &= -\mathrm{i}\:\frac{1-\mathrm{i}\Gamma_0}{\omega\rho_0}\:\boldsymbol{\nabla} p_1
 \esubal
For an elastic solid with density $\rho_\mr{sl}$, the displacement field $\myvec{u}$ is governed by the Cauchy equation
 \beq{CauchyEq}
 -\omega^2\rho_\mr{sl}\:\myvec{u} = \boldsymbol{\nabla} \cdot \bm{\sigma},
 \end{equation}
where $\bm{\sigma}$ is the stress tensor. In the Voigt notation, the $1\times 6$  stress $\bm{\sigma}$ and strain $\myvec{s}$ column vectors are given by the $6\times 1$ transposed row vectors $\bm{\sigma}^\textsf{T} = (\sigma_{xx}, \sigma_{yy}, \sigma_{zz}, \sigma_{yz}, \sigma_{xz}, \sigma_{xy})$ and $\myvec{s}^\textsf{T} = (\partial_x u_x,\partial_y u_y,\partial_z u_z, \partial_y u_z +\partial_z u_y, \partial_x u_z +\partial_z u_x, \partial_x u_y +\partial_y u_x)$, respectively, and $\bm{\sigma}$ is related to $\myvec{s}$ by the $6\times 6$ stiffness tensor $\myvec{C}$ having the elastic moduli $C_{ik}$ as components. For a linear, isotropic, elastic solid of the $\infty mm$-symmetry class the relation is,
 \beq{StressStrainSolid}
 \bm{\sigma} = \myvec{C}\cdot\myvec{s},
 \qquad
 \myvec{C} = \pare{
 \begin{array}{cccccc}
 C_{11} & C_{12} & C_{13} &0 &0 &0  \\
 C_{12} & C_{11} & C_{13} &0 &0 &0  \\
 C_{13} & C_{13} & C_{33}  &0 &0 &0  \\
 0 & 0 & 0 & C_{44} & 0 & 0 \\
 0 & 0 & 0  & 0 & C_{44} & 0 \\
 0 & 0 & 0 & 0 & 0 & C_{66} \\
 \end{array}
 }.
 \end{equation}
Here, the components $C_{ik} = C'_{ik} + \mathrm{i} C''_{ik}$ are complex-valued with real and imaginary parts relating to the speed and the attenuation of sound waves in the solid, respectively. In this work we assume the glass and the glue layer to be isotropic, yielding the following relations $C_{33} = C_{11}$, $C_{66} = C_{44}$ and $C_{13} = C_{12} = C_{11} - 2C_{44}$. This leaves the two independent complex-valued parameters $C_{11}$ and $C_{44}$, relating to the longitudinal and transverse speed of sound and attenuation in the glass and glue layer. For a lead zirconate titanate (PZT) transducer, $C_{66} = \frac{1}{2} (C_{11} - C_{12})$, which leaves five independent complex-valued elastic moduli, $C_{11}$,  $C_{12}$,  $C_{13}$,  $C_{33}$,  and $C_{44}$.

The electrical potential $\varphi$ inside the PZT transducer is governed by Gauss's law for a linear, homogeneous dielectric with a zero density of free charges,
 \beq{GaussLaw}
 \boldsymbol{\nabla} \cdot \myvec{D} = \boldsymbol{\nabla} \cdot (-\myvec{\varepsilon}\cdot \boldsymbol{\nabla} \varphi) = 0,
 \end{equation}
where $\myvec{D}$ is the electric displacement field and $\myvec{\varepsilon}$ the dielectric tensor. Furthermore in PZT, the complete linear electromechanical coupling relating the stress and the electric displacement to the strain and the electric field is given as,
 \bsubal{ConstitPZT}
 \eqlab{sigmaDPZT}
 \pare{\begin{array}{c} \bm{\sigma} \\ \myvec{D} \end{array}}
 &=
 \pare{\begin{array}{cc}\myvec{C} & -\myvec{e}^\textsf{T}  \\ \myvec{e} & \bm{\varepsilon} \end{array}}
 \,
 \pare{\begin{array}{c} \myvec{s}\\ \myvec{E} \end{array}},
 \\
 \eqlab{ePZT}
 \text{with }\; \myvec{e}  &= \pare{
 \begin{array}{cccccc}
 0 &0 &0 &0 &e_{15} &0  \\
 0 &0 &0 &e_{15} &0 &0  \\
 e_{31} &e_{31} &e_{33} &0 &0 &0
 \end{array}
 }\;
 \text{ and }\;
 \bm{\varepsilon}  = \pare{
 \begin{array}{ccc}
 \varepsilon_{11} &0 &0\\
 0 &\varepsilon_{11}  &0\\
 0 &0 &\varepsilon_{33}
 \end{array}
 }\!.
 \esubal

\subsection{The acoustic radiation force and the acoustophoretic particle velocity}
\seclab{FradVacoust}
\vspace*{-4mm}

We consider polystyrene particles with density $\rho_\mathrm{ps}$, compressibility $\kappa_\mathrm{ps}$, and a radius $a$, which is much larger than the viscous boundary layer and much smaller than the acoustic wavelength. In this case, the acoustic radiation force $\myvec{F}^\mathrm{rad}$ on the particles placed in water is given by the negative gradient of the Gorkov potential $U^{\mathrm{rad}_{}}$, \cite{Settnes2012}
 \bsubal{FradEq}
 \myvec{F}^\mathrm{rad} &= -\boldsymbol{\nabla} U^{\mathrm{rad}_{}}, \; \text{ with}
 \\
 U^{\mathrm{rad}_{}} &=
 \pi a^3\Big(\frac13 f_0\: \kappa_0  |p_\mr{1}|^2
 - \frac12 f_1\:\rho_0 |\myvec{v}_\mr{1}|^2 \Big),
 \;\;
 f_0 = 1 - \frac{\kappa_\mathrm{ps}}{\kappa_0},\; \text{ and }\;
 f_1 = \frac{2(\rho_\mathrm{ps}-\rho_0)}{2\rho_\mathrm{ps} + \rho_0}.
 \esubal
If a (polystyrene) microparticle of radius $a$ is placed in a fluid of viscosity $\eta_0$ flowing with the local velocity $\myvec{v}_0$, the presence of $\myvec{F}^\mathrm{rad}$ imparts a so-called acoustophoretic velocity $\myvec{v}_\mr{ps}$ to the particle. As inertia is negligible, $\myvec{v}_\mr{ps}$ is found from a balance between $\myvec{F}^\mathrm{rad}$ and the viscous Stokes drag force $\FFF^{\mathrm{drag}_{}}$, \cite{Skov2019}
  \beq{vAcoust}
  \myvec{v}_\mr{ps} = \frac{1}{6\pi\eta_0 a}\:\myvec{F}^\mathrm{rad} + \vvv_0.
  \end{equation}


\vspace*{10mm}


\subsection{Electrical impedance and power dissipation}
\vspace*{-4mm}

For a PZT transducer with an excited top electrode and a grounded bottom electrode set by the respective potentials $\varphi = \varphi_\mr{pzt}$ and $\varphi = 0$~V, the electrical impedance $Z$ is given by the ratio of
$\varphi_\mr{pzt}-0~$V and the surface integral of the polarization current density $\myvec{D} +\epsilon_0\boldsymbol{\nabla}\varphi$ as, \citep{Skov2019b}
 \beq{impedanceSim}
 Z =  \frac{\varphi_\mr{pzt}}{I},
 \;\text{ wtih }\;
 I = -\mathrm{i}\omega\int_{\partial\Omega} \myvec{n} \cdot (\myvec{D} +\epsilon_0\boldsymbol{\nabla}\varphi)\:\mathrm{d} a.
 \end{equation}
The electrical power dissipation $P_\mathrm{pzt}$ in the PZT transducer is given by
 \beq{avgPowerDiss}
 P_\mathrm{pzt} = \frac{1}{2}\mathrm{Re}\big[(\varphi_\mr{pzt}) I^{\star}\big]
 = \frac{1}{2}\big|\varphi_\mr{pzt}\big|\; \big|I\big|\: \cos\theta,
 \;\text{ with }\;
 \theta = \arg(Z).\\[-2mm]
 \end{equation}

\subsection{Butterworth--Van Dyke circuit model}
\seclab{BVDmodel}
\vspace*{-4mm}


\begin{figure}[b!]
    \centering

    \vspace*{-3mm}
    \includegraphics[width=13.5cm]{fig_01.pdf}
    \caption{\figlab{device_overview}
    (a) A schematic overview of the electrical circuit driving the transducer. The transducer, represented by the BVD-model with a resistor $R_1$, an inductor $L_1$,  and two capacitors $C_0$ and $C_1$, is coupled in series with the parasitic wire resistance and inductance. (b) A disk-shaped piezoelectric transducer is glued to a long, straight glass capillary tube. The tube is connected to a 3D-printed sample holder (green), and inlet/outlet tubing is glued to the ends of the tube. (c) The acoustofluidic device is mounted above the microscope lens and iselectrically connected via two spring-loaded pins on each side of the transducer. (d) Using the symmetry planes $x$-$z$ and $y$-$z$, only a quarter of the actual geometry needs to be simulated numerically. The different domains of the model: PZT (gray), glass (light blue), water (dark-blue), and the thin glue layer (orange). The dimensions The dimensions are $r_\mr{pzt} = 5.02$~mm, $h_\mr{pzt} = 506~\textrm{\textmu{}m}$, $w_\mr{fl} = 2060~\textrm{\textmu{}m}$, $h_\mr{fl} = 200~\textrm{\textmu{}m}$, $h_\mr{cpl} = 39~\textrm{\textmu{}m}$, $w_\mr{cap} = 2324~\textrm{\textmu{}m}$, and $h_\mr{cap} = 483~\textrm{\textmu{}m}$. In the simulation the reduced lengths are $l_\mr{cap} = 6.44$~mm and $l_\mr{pml} = 839~\textrm{\textmu{}m}$. (e) The cross-section in the $y$-$z$-plane showing the glass tube, the water, and the glue layer.}
\end{figure}

To describe the electrical response of the transducer around its thickness resonance frequency, we use a single-frequency Butterworth-Van Dyke (BVD) model. We furthermore include the impact of the wiring and the parasitic effects of the circuit leading to the PZT transducer in our model. An equivalent circuit of our model is shown in \figref{device_overview}(a). It consists of the parasitic wire resistance $R_\mr{wire}$ and inductance $L_\mr{wire}$ in series with a PZT circuit having the transducer capacitance $C_0$ in parallel with an transducer LCR-circuit $R_1$-$L_1$-$C_1$. The four parameters $R_1$, $L_1$, $C_1$, and $C_0$ can be obtained from the PZT admittance spectrum $Y(f) = 1/Z(f)$ at the resonance frequency $f_\mr{r}$ and anti-resonance frequency $f_\mr{a}$ \cite{Rathod2019, Huang2011},
 \beq{BVD_param}
 C_0 = \frac{\mathrm{Im}{Y(f_\mr{r})}}{2\pi f_\mr{r}},
 \quad
 R_1 = \frac{1}{\mathrm{Im}{Y(f_\mr{r})}},
 \quad
 C_1 = C_0 \Bigg[\frac{f_\mr{a}^2}{f_\mr{r}^2} - 1 \Bigg],
 \quad
 L_1 = \frac{1}{(2\pi f_\mr{r})^2 C_1}.
 \end{equation}
We perform simulations of the BVD-circuit using the \textit{SPICE}-based circuit simulator software \textit{LTspice} with parameters for the circuit components obtained via \eqref{BVD_param} and the measured values of the wire resistance $R_\mr{wire}$ and inductance $L_\mr{wire}$.


\section{Materials and methods}
\seclab{methods}
\vspace*{-4mm}

\subsection{The experimental setup}
\seclab{device_overview}
\vspace*{-4mm}

In this work, an acoustofluidic device is used, that consists of a 483-$\textrm{\textmu{}m}$-thick, 2324-$\textrm{\textmu{}m}$-wide and 50.9-mm-long glass capillary (VitroTubes, VitroCOM, Mountain Lakes, USA) containing a 200~$\textrm{\textmu{}m}$ high and 2.06~mm wide microchannel. The device is glued to a cylindrical piezoelectric transducer disk (Pz27, Meggitt A/S, Kvistgaard, Denmark), made from PZT, of thickness $506~\textrm{\textmu{}m}$ and diameter 10.045~mm with a nominal resonance frequency at around 4~MHz. The capillary tube is glued to the transducer by a thin ($39~\textrm{\textmu{}m}$) layer of UV-curable glue (NOA 86H, Norland Products, Jamesburg, USA). An overview of the device is shown in \figref{device_overview}(b). Using silicone glue, the device is mounted on a 3D-printed sample holder, and rubber tubing is glued to the glass capillary tube on both ends. The electrical connection to the piezoelectric transducer is made via four spring-loaded pins, as can be seen in \figref{device_overview}(c). These pins both minimize the clamping force on the transducer and enables four-probe measurement of the electric voltage across the transducer.

A schematic overview of the electrical circuit, including signal generator and amplifier is shown in \figref{device_overview}(a). As signal generator an Analog Discovery 2 (Digilent, Pullman, USA) in connection with the power amplifier TOE 7607 (TOELLNER Electronic Instrumente GmbH, Herdecke, Germany) was used to drive the piezoelectric transducer. The output of the amplifier is connected to the spring-loaded pins via a coaxial cable followed by $30~\textrm{cm}$ hookup wire. The wire is considered as a short transmission line with negligible capacitance, but with non-negligible parasitic resistance $R_\mathrm{wire}$ and inductance $L_\mathrm{wire}$. In our simplified circuit model, we only consider the thickness resonance at around 4~MHz of the transducer, and model the transducer via the BVD-model of \eqref{BVD_param}.



\subsection{Fabrication and characterization of the devices}
\vspace*{-4mm}

The device is assembled in a step-by-step procedure, and after each fabrication step the electrical impedance spectrum $Z(f)$ of the piezoelectric transducer is recorded with the Vector Network Analyzer Bode 100 (OMICRON electronics GmbH, Klaus, Austria) in the range from 500~Hz to 5~MHz. Device dimensions were measured using an electronic micrometer (RS Pro, RS Components, Corby, UK) with an accuracy of $\pm 4~\textrm{\textmu{}m}$. The assembly process consisted of the followings seven steps:\\[-5mm]

\begin{enumerate}
  \item Measure the dimensions of the capillary tube and the transducer.\\[-7mm]
  \item Measure the initial impedance spectrum $Z_\mr{init}(f)$ of the Pz27 disk.\\[-7mm]
  \item Fit the Pz27 material parameters using ultrasound electrical impedance spectroscopy (UEIS), following the method described in Ref. \cite{Bode2022}.\\[-7mm]
  \item Glue the capillary tube onto the transducer and UV-curing using an exposure time of 168~s at a UV-intensity of $15~\mr{mW}/\mr{cm}^2$ and a wavelength of 365~nm.\\[-7mm]
  \item Measure the total device thickness to obtain the glue layer thickness.\\[-7mm]
  \item Mount the device on a 3D-printed sample holder and connection to rubber tubing using silicone glue.\\[-7mm]
  \item Measure the impedance spectrum $Z_\mr{sys}(f)$ of the combined capillary-glue-transducer system, both air- and fluid-filled.\\[-5mm]
\end{enumerate}

Using the four-probe setup, shown in \figref{device_overview}(a),(c), the voltage amplitudes $\varphi_\mr{gen}$ at the signal generator, $\varphi_\mr{amp}$ at the amplifier, and $\varphi_\mr{pzt}$ directly across at the transducer were recorded during the measurements. The time-averaged dissipated power $P_\mathrm{pzt}$ for a given frequency $f$ was calculated from \eqref{avgPowerDiss} as
\beq{target_diss_power}
     P_\mathrm{pzt}(f) =
     \frac{1}{2} \frac{\varphi^2_\mr{pzt} \cos\big[\theta_\mr{sys}(f)\big]} {|Z_\mr{sys}(f)|},
     \; \text{ with }\; \theta_\mr{sys} = \arg(Z_\mr{sys}).
\end{equation}
A feedback control system was implemented to actuate the transducer at the desired constant power or constant voltage during the frequency sweeps. In the following analysis, we consider the following three case:

\begin{description}
    \item[(Case 1)] Constant voltage at the generator, $\varphi_\mr{gen} = 1$ V.\\[-6mm]
    \item[(Case 2)] Constant voltage at the transducer, $\varphi_\mr{pzt} = 0.5$ V.\\[-6mm]
    \item[(Case 3)] Constant power dissipation in the transducer, $P_\mathrm{pzt} = 50$ mW.\\[-9mm]
\end{description}


\subsection{Determination of acoustofluidic resonance frequencies\\ by particle tracking velocimetry}
\vspace*{-4mm}

The acoustofluidic resonance frequencies were determined by measuring the average velocity of particles focusing under acoustofluidic actuation. For the particle focusing experiment, we used a neutrally buoyant solution of 10-$\upmu$m-diameter fluorescent polystyrene spheres (microparticles GmbH, Berlin, Germany) in a fluid consisting of 83\%~(v/v) of ultrapure water  (Direct-Q3 System, Merck) and 17\%~(v/v) OptiPrep (Density Gradient Medium, Sigma-Aldrich). The particle concentration was about 500 particles/\textmu L.

The acoustic focusing was studied using single-camera 3D particle tracking performed with the general defocusing particle tracking (GDPT) method~\cite{Barnkob2015}. GDPT determines the depth position of defocused particle images from the analysis of the corresponding defocusing patterns, previously mapped with a proper calibration procedure \cite{Rossi2020}. The particle images were recorded using a high-sensitive sCMOS camera (pco.edge 5.5, Excelitas PCO GmbH, Kelheim, Germany) with an optical system consisting of a 5$\times$ microscope objective (EC EPIPlan, Zeiss AG, Oberkochen, Germany) and a cylindrical lens in front of the camera sensor to enhance the defocusing patterns. The images were processed using the open-source software \textit{defocustracker} version 2.0.0 \cite{Rossi2021}.

In each experiment, 200 images were recorded at 25~frames/s, and the signal generator was switched on precisely 1~s after the camera had started to record the first frame using an electrical trigger. The frequency sweeps were performed at frequencies in the range from 3.3 to 4.3~MHz in steps of 10~kHz. After the GDPT evaluation, we obtained a set of $N$ measured three-dimensional particle trajectories $\myvec{s}^{(j)}(t) = \{x(t),y(t),z(t) \}^{(j)}$ for each frequency. We then proceed to compute the three components $s_i^\mr{exp}(t)$ of the average cumulative particle displacement vector $\myvec{s}^\mr{exp}(t)$ as
\bal
\eqlab{avgDisp}
    s_i^\mr{exp}(t) &= \frac{1}{N} \sum_{j=1}^{N}|s^{(j)}_i(t) - s^{(j)}_i(t_0)|, \quad \text{for }i=x,y,z,
\eal
where $t_0$ is the time when the acoustics are turned on. The average acoustophoretic speed  $v_\mr{exp}$ of the particles is then calculated at time $t_\mr{exp} = 40~\mr{ms}$ after turning on the acoustics, as
\beq{particle_vel}
    v_\mr{exp} = \sqrt{\sum_{i=x,y,z}\left (\partial_t s_i^\mr{exp} |_{t=t_\mr{exp}}\right )^2}.
\end{equation}


\section{Numerical model}
\seclab{num_model}
\vspace*{-4mm}

\subsection{Description of the modeled system}
\vspace*{-4mm}

We perform numerical simulations of the device described in \secref{device_overview} using the software COMSOL Multiphysics 6.0, following the implementation in Ref.~\cite{Skov2019, Bode2021, Lickert2021}. By using the $x$-$z$ and $y$-$z$ symmetry planes, only a quarter of the actual geometry is modeled. In the model we consider the piezoelectric transducer, a thin coupling layer, and the water-filled glass capillary tube. To further minimize the computational complexity, we apply a perfectly matched layer (PML) at the end of the glass capillary \cite{Bode2021}. The PML mimicks perfect absorption of all outgoing waves, and it allows to reduce the length of the capillary tube. In our experimental setup, the damping at the edge of the tube is ensured by the silicone glue connecting the tube to the sample holder. A sketch of the system is shown in \figref{device_overview}(d,e). Simulations were performed on a workstation with a 12-core, 3.5-GHz central processing unit and 128 GB random access memory. Details on the mesh convergence analysis and the material parameters used for the simulation can be found in Appendix \secnoref{conv_test} and \secnoref{mat_param}.


\subsection{Numerical simulation of the particle velocity}
\seclab{particle_vel_sim}
\vspace*{-4mm}

In our simulation model, we use the "\textit{Particle Tracing for Fluid Flow}" module and compute the particle trajectories of 1000 randomly distributed particles. The wall condition is set to "\textit{Stick}" to mimic stuck particles, which were also commonly observed in the experimental setup. The force acting on the particles is the simulated radiation force $\myvec{F}^\mathrm{rad}$, see \eqref{FradEq}. Acoustic streaming is neglected due to the size of the particles, and the influence of gravity is neglected due to the use of a neutrally-buoyant solution. Similarly to what is done for the experimental data, we obtain the velocity  $\myvec{v}_i$ of each particle $i$ at time $t_\mr{sim}$ and compute the average speed $v_\mr{sim}$ of the particle as
\beq{vel_sim}
    v_\mr{sim} = \frac{1}{N} \sum_{i=1}^{N} \big|\myvec{v}_i(t_\mr{sim})\big|,
\end{equation}
at time $t_\mr{sim} = (40\: t_\mr{foc}^\mr{sim}/t_\mr{foc}^\mr{exp}) ~\mr{ms}$, where $t_\mr{foc}^\mr{sim}/t_\mr{foc}^\mr{exp}$ is the ratio between the numerical and experimental focusing time at resonance.

\subsection{Boundary conditions between liquid, solid, and PZT}
\vspace*{-4mm}

In the simulations, we assume a time-harmonic voltage amplitude of $\varphi_\mathrm{pzt}$ at the top surface of the piezoelectric transducer, while the bottom surface is grounded to $\varphi_\mr{gnd} = 0$. We furthermore assume continuous stress between the different domains, a normal component of the dielectric displacement field $\myvec{D} \cdot \myvec{n} = 0$ at interface PZT--air and zero normal stress at the outer surfaces of the solid domains \cite{Skov2019}. For the interface between solid and fluid, we implement the effective  boundary conditions derived by Bach and Bruus \citep{Bach2018}. Here, the fields inside the very thin boundary layers of thickness $\delta_\textrm{fl} = \sqrt{2\eta_0/(\rho_0\omega)} \approx 0.5~\textrm{\textmu{}m}$ are taken into account analytically. The pressure $p_1$ at the fluid-air interface is set to zero. The boundary conditions between the different domains and their corresponding boundary are summarized in \tabref{tab_BCs}.

\begin{table}[b]
\caption{\tablab{tab_BCs} The boundary conditions used in the numerical simulations with the surface normal vector $\myvec{n}$ pointing away from the respective domain.  We use the solid velocity $\vvv_\mr{sl} = -\mathrm{i} \omega \myvec{u}$, and the complex-valued shear-wave number $k_\mathrm{s} = (1+\mathrm{i})\:\delta_\textrm{fl}^{-1}
= (1+\mathrm{i})\:\sqrt{\rho_0\omega/(2\eta_0)}$.}
\begin{tabular}{l@{\hspace{10mm}}ll@{\hspace{10mm}}l@{\hspace{10mm}}}
\toprule
\textbf{Domain} & $\leftarrow$ & \textbf{Boundary}   &  \textbf{Boundary condition}
\\ \hline
PZT & $\leftarrow$  & top electrode  & $\varphi = \varphi_\mathrm{pzt}$ \\

PZT & $\leftarrow$  & bottom electrode  & $\varphi = 0$ \\

PZT & $\leftarrow$  & air  & $\myvec{D} \cdot \myvec{n} = 0$ \\

Solid & $\leftarrow$  & air    & $\bm{\sigma}\cdot \myvec{n} = \boldsymbol{0}$ \\

Solid & $\leftarrow$  & fluid   & $\bm{\sigma} \cdot \myvec{n} = - p_1\: \myvec{n} + \mathrm{i}k_\mathrm{s}\eta_0(\vvv_\mr{sl} - \myvec{v}_1\big)$   \\

Fluid & $\leftarrow$  & solid  &    $\myvec{v}_1 \cdot \myvec{n}  =  \vvv_\mr{sl}\cdot\myvec{n} + \frac{\mathrm{i}}{k_\mathrm{s}} \boldsymbol{\nabla}_\parallel\cdot\big(\vvv_\mr{sl} - \myvec{v}_1\big)_\parallel$ \\

Fluid & $\leftarrow$  & air  & $p_1=0$ \\
\toprule
\end{tabular}
\end{table}

We assume symmetry of all simulated fields at the $yz$-plane at $x = 0$ and at the $xz$-plane at $y = 0$. The symmetry boundary conditions therefore are implemented as follows:
\bsubalat{BC_symmetry}{4}
 \text{Symmetry at } & x=0: &&
 \nonumber \\
 u_x &= 0, \qquad  &  \sigma_{yx} &= \sigma_{zx} = 0,
 \qquad
 & \partial_x p_\mr{1}  &= 0,\qquad & \partial_x \varphi &= 0.
 \\
 \text{Symmetry at } & y=0: &&
 \nonumber \\
 u_y &= 0, \qquad &  \sigma_{xy} &= \sigma_{zy} = 0,
 \qquad
 & \partial_y p_\mr{1} &= 0,  \qquad & \partial_y \varphi &= 0.
 \esubalat

\newpage

\section{Results and discussion}
\seclab{results}
\vspace*{-4mm}

\subsection{Electrical impedance measurements}
\vspace*{-4mm}

\begin{figure}[b!]
    \centering

    \vspace*{-11mm}
    \includegraphics[width=13.5cm]{fig_02.pdf}

    \vspace*{-6mm}
    \caption{\figlab{impedance_Pz27}
    (a) Measured (black) and simulated (green) electrical impedance spectrum $|Z(f)|$ in the frequency range 0.5 - 5000~kHz of the unloaded Pz27 disk. (b) Measured (black) and simulated $|Z(f)|$ for 3.3 - 4.3~MHz, using either UEIS fitted parameters (red) or the parameters from Ref.~\cite{Bode2022} (blue) of the full system consisting of Pz27 disk, glue layer and liquid-filled glass capillary tube.  (c) The measured $|Z(f)|$ (black) for 0.5 - 5000~kHz of the full system (Pz27 disk, glue layer, and liquid-filled glass capillary tube), and the computed $|Z(f)|$ (red) based on the single-frequency BVD-model of \secref{BVDmodel}. (d) The three measured voltage amplitudes versus frequency $f$ at different points in the circuit: $\varphi_\mr{pzt}$ (dark-green) obtained by four-probe measurements directly across the piezoelectric transducer, $\varphi_\mr{amp}$ (dark-orange) measured at the output of the amplifier, $\varphi_\mr{gen}$ (dark-blue) measured at the signal generator. Also shown are the two \textit{LTspice}-simulated voltage amplitudes: $\varphi_\mr{pzt}^\mr{BVD}$ (light green) and $\varphi_\mr{amp}^\mr{BVD}$ (light orange) computed from the BVD-model.}
\end{figure}

The electrical impedance spectrum $Z(f)$ of the unloaded and loaded Pz27 transducer was measured after each step in the fabrication procedure. The material parameters of the specific Pz27 transducer used in this study, were obtained by electrical impedance spectroscopy (UEIS), following the procedure described in Ref.~\cite{Bode2022}, based on the measured $Z(f)$ of the unloaded transducer. The result is shown in \figref{impedance_Pz27}(a), were it is seen that the fitted spectrum $|Z(f)|$ agrees well with the measured one. The piezoelectric parameters obtained from this UEIS fitting, were then subsequently used together with the remaining material parameters listed in \secref{mat_param} of the Appendix to simulate numerically the pressure field $p_1$, the displacement field $\myvec{u}$, and the electric potential $\varphi$ of the transducer-glue-capillary-tube system. The measured and the simulated impedance spectrum of the full system in the frequency range 3.3 to 4.3~MHz are shown in \figref{impedance_Pz27}(b), and good agreement is found. Numerical simulations were performed using both the fitted values, and the values for Pz27 given in Ref.~\cite{Bode2022}. The discrepancy between the two resulting spectra emphasizes the need of obtaining fitted material parameters for the specific transducer used in the study. The remaining deviations from the measured impedance spectrum stem from uncertainties in the glass material parameters, which where taken from literature and not fitted by UEIS.



\subsection{Impact of cable and circuit resonances on measured voltage amplitude}
\vspace*{-4mm}

Using the measured impedance spectrum $Z(f)$ of the full system around the 4-MHz resonance, we obtain the required parameters to describe the transducer through the BVD model of \secref{BVDmodel}, and we find $C_0 = 1.28$~nF, $C_1 = 207~$pF, $L_1 = 7.69~${\textmu}H, and $R_1 = 2.59~\Omega$. We estimate for each of the two wires connecting the amplifier and the transducer that $R_\mr{wire} = 1~\Omega$ and $L_\mr{wire} = 411~$nH. In \figref{impedance_Pz27}(c), we compare  $Z(f)$ computed from the BVD model with the measured $Z(f)$. It is seen that the BVD model captures well the characteristics around the 4-MHz resonance of the transducer, and it can therefore aid the understanding of the circuit characteristics.

When comparing the voltage amplitudes at various points in the circuit using a constant generator voltage amplitude $\varphi_\mr{gen} = 0.5~$V, we find as shown in \figref{impedance_Pz27}(d) that for most of the frequencies in the range 0.5 - 5000~kHz, the voltage amplitude $\varphi_\mr{pzt}$ across the transducer is larger than the voltage amplitude $\varphi_\mr{amp}$ right after the amplifier. This may seem counter-intuitive, but given the resonant nature of the circuit, charge may build up on the capacitive circuit elements. We furthermore find two frequencies were the voltage amplitude is minimal: At $f_\mr{amp} = 3.56~$MHz, the impedance of the full circuit has an impedance minimum, and at $f_\mr{pzt} = 3.98~$MHz, the impedance of the transducer has a minimum. The down-shift of $f_\mr{amp}$ by 0.42~MHz from $f_\mr{pzt}$ is due to the parasitic inductance of the wire connecting the amplifier and the transducer. We note that if the voltage amplitude is recorded right after the amplifier, and not directly across the transducer, a wrong estimate of the voltage amplitude and power dissipation of the transducer may result. Furthermore, the parasitic inductance minimizes the power transfer from the amplifier to the transducer, and therefore it is in general beneficial to minimize this inductance by use of shortened and shielded cables. To minimize the ratio $r = f_\mr{amp}/f_\mr{pzt}$, the inductance of the wire $L_\mr{wire}$ should be minimized according to
\beq{wire_ind}
    L_\mr{wire} <  \frac{(1-r^2)}{2r^2}\frac{C_1 L_1 }{C_1 + C_0 (1 - r^2)}.
\end{equation}

In our circuit, it is required that $L_\mr{wire} < 70~$nH to keep the mismatch of $f_\mr{circuit}$ and $f_\mr{pzt}$ below 1~\%. This is typically hard to achieve, as it requires very thin and short wires. Alternatively, a capacitor $C_\mr{comp} = (2\pi f_\mr{comp})^{-2}L_\mr{wire}^{-1}$ in series with the wire could be used to counteract the impact of $L_\mr{wire}$ at frequency $f_\mr{comp}$. Further improvements of the circuit could be obtained by impedance-matching the load impedance $Z_\mr{load}$ to the source impedance $Z_\mr{source}$, by adding circuit components to the load such that $Z_\mr{source} = Z_\mr{load}^*$ \cite{Rathod2019}.\\[-10mm]


\subsection{Voltage and power dissipation}
\vspace*{-4mm}

\begin{figure}[t]
    \centering
    \includegraphics[width=13.5cm]{fig_03.pdf}

    \vspace*{-7mm}
    \caption{\figlab{diff_voltages}
    The measured voltage amplitudes $\varphi_\mr{pzt}$ (green, left axis) and $\varphi_\mr{gen}$ (blue, left axis) as well as the measured average power dissipation $P_\mathrm{pzt}$ (black, right axis) plotted versus frequency for 3.3 - 4.3~MHz for the three cases (a) constant voltage $\varphi_\mr{gen}$ at the signal generator, (b) constant voltage $\varphi_\mr{pzt}$ across the piezoelectric transducer, and (c) $P_\mathrm{pzt}$ constant average power dissipation.\\[-7mm]}
\end{figure}

The voltage amplitudes and power dissipation in a frequency sweep are depending significantly on the chosen electrical excitation method in the actuation process. In \figref{diff_voltages}(a)-(c) are shown frequency sweeps of the the voltage amplitudes $\varphi_\mr{gen}$ and $\varphi_\mr{pzt}$, and the average dissipated power $P_\mr{pzt}$ for the three considered cases of constant $\varphi_\mr{gen}$, constant $\varphi_\mr{pzt}$, and constant $P_\mr{pzt}$.

The first case with constant $\varphi_\mr{gen} = 1~$V is shown in \figref{diff_voltages}(a). It is seen that $\varphi_\mr{pzt}$ and $P_\mr{pzt}$ are minimal at or close to the transducer impedance minimum at $f_\mr{pzt} = 3.98$~MHz. This is not an ideal situation, because many systems are designed with a resonance close to the nominal transducer resonance in mind. Instead, it is beneficial to stabilize either $\varphi_\mr{pzt}$ or $P_\mr{pzt}$.

The second case with constant $\varphi_\mr{pzt} = 0.5$~V is shown in \figref{diff_voltages}(b). Both $\varphi_\mr{gen}$ and $P_\mr{pzt}$ have a narrow peak near $f_\mr{pzt} = 3.98$~MHz. As will be discussed in \secsref{avr_speed}{sim_results}, this might not be ideal for particle focusing experiments when comparing device performances over wider frequency ranges.

The third case with constant $P_\mr{pzt} = 50~$mW is shown in \figref{diff_voltages}(c). To stabilize $P_\mr{pzt}$, the voltage $\varphi_\mr{gen}$ needs to be adjusted according to the electrical impedance spectrum of the transducer. The voltage $\varphi_\mr{gen}$ needs to be higher when running the transducer on-resonance, compared to off-resonance. As in the first case, the voltage amplitude $\varphi_\mr{pzt}$ is minimal at the resonance $f_\mr{pzt} = 3.98$~MHz. Other effects, such as the non-linear gain of the amplifier and non-linearity of the piezoelectric transducer, may lead to increased discrepancies between $\varphi_\mr{gen}$ and $\varphi_\mr{pzt}$, which furthermore emphasizes the need to monitor $\varphi_\mr{pzt}$ and $P_\mr{pzt}$, and to specify which of them, if any, is kept constant.


\subsection{Average acoustophoretic particle speed}
\seclab{avr_speed}
\vspace*{-4mm}

\begin{figure}[t]
    \centering
    \includegraphics[width=13.5 cm]{fig_04.pdf}

    \vspace*{-5mm}
    \caption{\figlab{vel_magn}
    (a) Plot of the measured average acoustophoretic particle speed $v_\mr{exp}$ versus frequency $f$ in the range 3.3 - 4.3~MHz. (b) The measured power dissipation $P_\mathrm{pzt}$ in the transducer versus $f$ for 3.3 - 4.3~MHz for the three cases of constant voltage $\varphi_\mr{gen}$ at the signal generator, constant voltage $\varphi_\mr{pzt}$ across the piezoelectric transducer, and constant average power dissipation $P_\mathrm{pzt}$. (c) The experimental (black) and simulated (gray) normalized average particle speed $v^\mr{P}/v^\mr{P}_\mr{max}$ versus frequency $f$ at constant-power dissipation $P_\mr{pzt} = 50~$mW.  (d) The experimental (dark-green) and simulated (light-green) normalized average particle speed $v^\varphi/v^\mr{P}_\mr{max}$ versus frequency $f$ at constant transducer voltage $\varphi_\mr{pzt} = 0.5~$V.\\[-8mm]}
\end{figure}

When the acoustic pressure field $p_1$ is switched on via the Pz27 transducer in our setup shown in \figref{device_overview}, polystyrene microparticles inside the water-filled capillary tube acquire an acoustophoretic velocity $\myvec{v}_\mr{ps}$, see \eqssref{vAcoust}{particle_vel}{vel_sim}, proportional to the acoustic radiation force $\myvec{F}^\mathrm{rad}$, see \eqref{FradEq}. For the above three cases, the experimental results for the average particle speed $v_\mr{exp}$ are shown in \figref{vel_magn}(a), and the corresponding results for $P_\mr{pzt}$ are shown in \figref{vel_magn}(b). In the case of constant $\varphi_\mr{gen} = 1~$V, we observe the highest particle speed at $f = f_\mr{res}^\mr{gen} = 4.24~$MHz with $v_\mr{exp} \approx 46~\SImum\,\SIs^{-1}$. For constant $\varphi_\mr{pzt} = 0.5~$V, we find the highest particle speed at $f = f^\varphi_\mr{exp} = 3.98~$MHz with $v_\mr{exp} \approx 71~\SImum\,\SIs^{-1}$. Finally, for constant $P_\mr{pzt} = 50~$mW, a resonance appears at $f = f^\mr{P}_\mr{exp} = 4.03$~MHz with $v_\mr{exp} \approx 77~\SImum\,\SIs^{-1}$.  Furthermore, for $\varphi_\mr{pzt} = 0.5~$V and $P_\mr{pzt} = 50~$mW, we observe a local resonance at $f = f^{(10)}_\mr{width} = 3.51~$MHz.  This resonance relates to an acoustic mode with 5 wavelengths along the $y$-direction, leading to particle focusing in 10 nodal lines parallel to length and evenly distributed across the width of the microchannel above the center region of the Pz27 transducer, as discussed further in \secref{sim_results}.

When analyzing the measurements in \figref{vel_magn}(b) of the power dissipation in the three cases, we find that, when driving the transducer at constant $\varphi_\mr{pzt}$, a clear maximum in $P_\mr{pzt}$ appears at $f_\mr{exp}^\varphi = 3.98~$MHz, but conversely, $P_\mr{pzt}$ has a minimum at the same frequency for constant $\varphi_\mr{gen}$. The reason is that at this frequency, the transducer has an intrinsic resonance and thus a minimum in its impedance. Lastly, we note that experimentally it is difficult to perfectly stabilize $P_\mr{pzt}$ near the transducer resonance $f_\mr{exp}^\varphi = 3.98~$MHz. This difficulty is likely due to on-resonance heating effects of the transducer.


\subsection{Comparing numerical simulations with experiments}
\seclab{sim_results}
\vspace*{-4mm}

In \figref{vel_magn}(c) using constant $P_\mathrm{pzt} = 50~\mr{mW}$, the measured $v^\mr{P}_\mr{exp}/v^\mr{P}_\mr{max}$ and the simulated $v^\mr{P}_\mr{sim}/v^\mr{P}_\mr{max}$ average acoustophoretic speed, normalized by the measured maximum speed $v^\mr{P}_\mr{max} = \mr{max}|\myvec{v}^\mr{P}_\mr{exp}|$, are plotted versus frequency for 3.3 - 4.3~MHz. The agreement between the two curves is good, and they both show a resonance at nearly the same frequency $f = f_\mr{exp}^P = 4.03~$MHz and $f = f_\mr{sim}^P = 4.04~$MHz, respectively. A similar plot is shown \figref{vel_magn}(d), but now for the case of constant $\varphi_\mr{pzt} = 0.5~\mr{V}$, namely the measured $v^\varphi_\mr{exp}/v^\mr{P}_\mr{max}$ and the simulated $v^\varphi_\mr{sim}/v^\mr{P}_\mr{max}$ versus frequency with the same normalization $v^\mr{P}_\mr{max}$ as before. Again, the agreement between simulation and experiment is good, and both curves have a maximum at $f = f_\mr{exp}^\varphi = f_\mr{sim}^\varphi = 3.98~$MHz, about 50~kHz lower than the constant-power resonance frequency $f_\mr{exp}^P = 4.03~$MHz.

Some interesting features are seen in the measured and simulated spectrum of the constant-voltage acoustophoretic velocity spectrum $v^\varphi_\mr{exp}(f)$ in \figref{vel_magn}(d). Its maximum, obtained at $f^\varphi_\mr{exp}= 3.98~$MHz, is 8\% less then the one obtained in the constant-power velocity spectrum $v^\mr{P}(f)$ in \figref{vel_magn}(c),  $v^\varphi_\mr{max} = 0.92\: v^\mr{P}_\mr{max}$. Moreover, far from being at the maximum, $v^\varphi(f_\mr{exp}^\mr{P}) = 0.27\:v^\mr{P}_\mr{max}$ is a local minimum. Clearly, the optimal operating condition for acoustophoresis is to run the system at $f_\mr{exp}^\mr{P}$ with constant-power actuation. Operating directly at the transducer resonance at $f_\mr{exp}^\varphi = 3.98~$MHz, is not equally efficient due to the low impedance of the transducer and the resulting high power dissipation at this frequency. Constant-voltage frequency sweeps can be misleading in that regard.

\begin{figure}[t!]
    \centering
    \includegraphics[width=13.5 cm]{fig_05.pdf}\\[-7mm]
    \caption{\figlab{simulated_fields}
    Color plot from $-150$~kPa (blue) to 150~kPa (red) of the simulated acoustic pressure field $p_1$ inside the fluid channel at (a) $f = f^{(10)}_\mr{width} = 3.51~$MHz with a standing-5-wavelength resonance mode in the $y$-direction (10 nodal lines above the center region of the transducer), and (b) at $f = f_\mr{exp}^P = 4.03~$MHz with a standing-$\frac12$-wavelength resonance mode in $z$-direction. (c) Micrograph of the particles focused in 10 nodal lines (marked by red arrows) inside the microfluidic channel after 4~s at the resonance frequency $f^{(10)}_\mr{width} = 3.51~$MHz. (d) Micrograph of the particles focused in 1 nodal plane (the $xy$-plane) inside the microfluidic channel after 4~s at the resonance frequency $f_\mr{exp}^P = 4.03~$MHz.\\[-8mm]}
\end{figure}

In the simulation and experiment with constant $P_\mathrm{pzt}$, see \figref{vel_magn}(a, c), a local maximum in the average particle speed $v$ is observed at $f = f^{(10)}_\mr{width} = 3.51~$MHz. in \figref{simulated_fields}(a,b), we compare the simulated pressure field at this frequency with the simulation results at $f = f_\mr{exp}^P = 4.03~$MHz. Images of the particles after 4~s at the two corresponding frequencies are shown in \figref{simulated_fields}(c) and (d). Both in the numerically simulated pressure field, as well as in the measured particle positions, we observe in the $x$-$y$ plane at $f = f^{(10)}_\mr{width} = 3.51~$MHz, the formation  of 10 nodal lines parallel to the tube axis along the $x$ direction, and with an equidistant distribution across the width, see \figref{simulated_fields}(c). In contrast, at the main resonance at $f_\mr{exp}^P = 4.03~$MHz shown in \figref{simulated_fields}(d), we observe particle focusing  in the $x$-$y$ center plane of the glass capillary tube, above the center region of the Pz27 transducer, caused by the standing half-wave in the vertical $z$-direction. No transverse nodal lines are observed here.\\[-7mm]


\section{Conclusion}
\seclab{conclusion}
\vspace*{-4mm}

Monitoring power dissipation in and voltage across the piezoelectric transducer is important and helpful for understanding and optimizing the performance of acoustofluidic systems. As shown by our measurements on the setup shown in \figref{device_overview}, the voltage can differ significantly between amplifier output and transducer, due to the varying impedance of the transducer at different frequencies. In this work, we compared the performance of an acoustofluidic device using three types of actuation: (Case 1) Supplying a constant voltage amplitude $\varphi_\mr{gen}$ to the amplifier input from the signal generator, (Case 2) driving the piezoelectric transducer using constant-voltage actuation $\varphi_\mr{pzt}$, and  (Case 3) keeping the power dissipation $P_\mr{pzt}$ in the transducer constant. The acoustofluidic performance was evaluated in terms of the average acoustophoretic particle speed $v$ in the microfluidic channel measured with 3D particle tracking velocimetry and computed numerically, see \figref{vel_magn}.

Case 1, performing frequency sweeps with constant  $\varphi_\mr{gen}$, which is typically used for acoustofluidic devices, may result in a misleading identification of the ideal actuation frequency. The reason is that the power dissipation in the transducer is dependent of the impedance of the transducer as well as the resonant behavior of the cables connecting amplifier and transducer. Instead, keeping a constant power $P_\mr{pzt}$ is a better choice for obtaining a reproducible characterization of the intrinsic properties of acoustofluidic devices.

Case 2, frequency sweeps with constant $\varphi_\mr{pzt}$ often result in high power dissipation at the transducer resonance frequency $f^\varphi_\mr{exp}$, where the impedance of the transducer is at a minimum. Therefore, the strongest acoustofluidic response will be observed closest to this frequency, but it is likely not the most power-efficient frequency, as it results in increased heating and comes at the cost of high input powers. Acoustofluidic applications, however, are often constrained by power-limitations of the frequency generator or the amplifier, as well as the requirement of maintaining a defined temperature to enable the processing of biological samples.

Case 3, frequency sweeps with constant $P_\mr{pzt}$ appear to be a better measure to compare device performance across frequencies, as this compensates for the decrease in impedance at the transducer resonance. As a consequence, also finer details in the acoustic fields that occur at frequencies further away from the transducer resonance frequency can be observed. This is exemplified by the transverse resonance in the width direction at $f = f^{(10)}_\mr{width} = 3.51~$MHz, see \figref{simulated_fields}(a) and (c); a resonance clearly visible as a strong peak in the constant-power spectrum in \figref{vel_magn}(c), but not visible in the constant-voltage spectrum  in \figref{vel_magn}(d). Keeping $P_\mr{pzt}$ constant, enhances the intrinsic properties of the device performance, as it does not depend on the wiring.
In conclusion, frequency sweeps with constant power may help to identify the most power-efficient actuation frequencies leading to the desired acoustofluidic response, and it allows a more clear-cut comparison of different acoustofluidic resonances across the frequency spectrum.

Lastly, the external circuitry, see \figref{device_overview}, may have an impact on the resonance behavior of the setup. In this work, the parasitic impact of the wire inductance, connecting the amplifier and the transducer, was observed. We note that by fine-tuning the impedance of the external circuitry to match the impedance of the transducer at resonance, the power transfer to the transducer can be increased. Such an impedance matching  is common in many other fields. Considering the whole circuit, rather than just the piezoelectric transducer in an acoustofluidic setup, therefore can be beneficial to further improve system performance in various acoustofluidic applications.\\[-7mm]



\section*{Acknowledgements}
\vspace*{-4mm}

\noindent
\textbf{Author contributions}: Conceptualization, F.L., H.B., and M.R.; methodology, F.L., H.B., and M.R.; software, F.L. and M.R.; validation, F.L., H.B., and M.R.; formal analysis, F.L., H.B., and M.R.; investigation, F.L., H.B., and M.R.; resources, F.L. and M.R.; data curation, F.L.; concluding discussions,  F.L., H.B., and M.R.;  writing---original draft preparation, F.L.; writing---review and editing, F.L., H.B. and M.R.; visualization, F.L., H.B., and M.R.; supervision: H.B. on theory and simulation, M.R. on experiment; project administration, H.B. and M.R.; funding acquisition, H.B. and M.R. All authors have read and agreed to the published version of the manuscript.

\textbf{Funding}: This work is part of the Eureka Eurostars-2 joint programme E!113461 AcouPlast project funded by Innovation Fund Denmark, grant no.~9046-00127B, and Vinnova, Sweden's Innovation Agency, grant no.~2019-04500, with co-funding from the European Union Horizon 2020 Research and Innovation Programme. MR acknowledges the financial support by the VILLUM foundation, grant no.~00036098.

\textbf{Data availability statement}: The data presented in this study are available on request from the corresponding author.

\textbf{Conflicts of interest}: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the~results.

\noindent
\textbf{Abbreviations}\\
The following abbreviations are used in this manuscript:\\

\noindent
\begin{tabular}{@{}ll}
BVD & Butterworth--Van Dyke\\
GDPT \hspace*{5mm}& General Defocusing Particle Tracking\\
PML & Perfectly Matched Layer\\
UEIS & Ultrasound Electrical Impedance Spectroscopy
\end{tabular}




