Constant-power versus constant-voltage actuation in frequency sweeps for acoustouidic applications Fabian LickertHenrik Bruusyand Massimiliano Rossiz

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Constant-power versus constant-voltage actuation in

frequency sweeps for acoustoﬂuidic applications

Fabian Lickert,∗Henrik Bruus,†and Massimiliano Rossi‡

Department of Physics, Technical University of Denmark,

DTU Physics Building 309, DK-2800 Kongens Lyngby, Denmark

(Dated: 4 October 2022)

Supplying a piezoelectric transducer with constant voltage or constant power during a frequency

sweep can lead to diﬀerent results in the determination of the acoustoﬂuidic resonance frequencies,

which are observed when studying the acoustophoretic displacements and velocities of particles sus-

pended in a liquid-ﬁlled microchannel. In this work, three cases are considered: (1) Constant input

voltage into the power ampliﬁer, (2) constant voltage across the piezoelectric transducer, and (3)

constant average power dissipation in the transducer. For each case, the measured and the simu-

lated responses are compared, and good agreement is obtained. It is shown that Case 1, the simplest

and most frequently used approach, is largely aﬀected by the impedance of the used ampliﬁer and

wiring, so it is therefore not suitable for a reproducible characterization of the intrinsic properties

of the acoustoﬂuidic device. Case 2 strongly favors resonances at frequencies yielding the lowest

impedance of the piezoelectric transducer, so small details in the acoustic response at frequencies

far from the transducer resonance can easily be missed. Case 3 provides the most reliable approach,

revealing both the resonant frequency, where the power-eﬃciency is the highest, as well as other

secondary resonances across the spectrum.

Keywords: acoustoﬂuidics, microparticle acoustophoresis, general defocusing particle tracking,

particle-velocity spectroscopy.

1. INTRODUCTION

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 [1–3]. Often, it is however left unclear under which conditions

and at which point in the electric circuit, the relevant quantities such as voltage am-

plitude or power dissipation have been measured. Recent studies compare device per-

formance at constant average power for diﬀerent placements of the transducer [4,5].

Dubay et al. [6] 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 [6,7].

Whereas optimization of the driving circuit is customary in other ﬁelds, such as ultra-

sonic transducers for cellular applications [8], non-destructive testing [9], and pulse-echo

systems [10], this has not been given much consideration in the ﬁeld of acoustoﬂuidics,

where the focus often lies on optimizing the acoustic impedance matching [11,12], 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 [13]. To our knowledge, studies have not

yet been performed, in which the impact of diﬀerent electrical excitation methods on

∗fabianl@dtu.dk

†bruus@fysik.dtu.dk

‡rossi@fysik.dtu.dk

arXiv:2210.02311v1 [physics.flu-dyn] 5 Oct 2022

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 dur-

ing 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 Section 2a brief summary is given

of the governing equations for the pressure ﬁeld, the displacement ﬁeld, and the electric

potential in our acoustoﬂuidic device. Section 3gives an overview of our experimental

setup, and the procedure used for the measurement of the particle velocities is described

step by step. In Section 4we describe the numerical approach used in our study, and in

Section 5we 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 Section 6

with a short summary and some guidelines on the actuation of piezoelectric transducers

for acoustoﬂuidic applications.

2. THEORY

The theoretical approach follows our previous work [3,14–16], in which the compu-

tational eﬀort in the simulations are reduced by employing the eﬀective-boundary-layer

theory derived by Bach and Bruus [17]. We assume time-harmonic ﬁrst-order ﬁelds with

angular frequency ω= 2πf for the acoustic pressure ˜p1(r, t) = p1(r) e−iωt, the electric

potential ˜ϕ(r, t) = ϕ(r) e−iωt, and the displacement ﬁeld ˜

u(r, t) = u(r) e−iωt. De-

rived through a perturbation approach, these ﬁelds represent tiny perturbations of the

unperturbed zero-order ﬁelds.

2.1. Governing equations

For a ﬂuid with speed of sound c0, density ρ0, dynamic and bulk viscosity of the ﬂuid

η0and ηb

0, damping coeﬃcient Γ0, and the isentropic compressibility κ0= (ρ0c2

0)−1, the

acoustic pressure p1is governed by the Helmholtz equation, and the acoustic velocity v1

is a gradient ﬁeld,

∇2p1=−ω2

01 + iΓ0p1,with Γ0=4

3η0+ηb

0ωκ0,(1a)

v1=−i1−iΓ0

ωρ0

∇p1(1b)

For an elastic solid with density ρsl, the displacement ﬁeld uis governed by the Cauchy

equation

−ω2ρsl u=∇·σ,(2)

where σis the stress tensor. In the Voigt notation, the 1×6 stress σand strain scolumn

vectors are given by the 6 ×1 transposed row vectors σT= (σxx, σyy , σzz, σyz, σxz, σxy)

and sT= (∂xux, ∂yuy, ∂zuz, ∂yuz+∂zuy, ∂xuz+∂zux, ∂xuy+∂yux), respectively, and σis

related to sby the 6 ×6 stiﬀness tensor Chaving the elastic moduli Cik as components.

For a linear, isotropic, elastic solid of the ∞mm-symmetry class the relation is,

σ=C·s,C=





C11 C12 C13 000

C12 C11 C13 000

C13 C13 C33 000

000C44 0 0

0000C44 0

00000C66







.(3)

Here, the components Cik =C0

ik +iC00

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 C33 =C11,C66 =C44 and C13 =C12 =C11 −2C44. This leaves the two

independent complex-valued parameters C11 and C44, relating to the longitudinal and

transverse speed of sound and attenuation in the glass and glue layer. For a lead zirconate

titanate (PZT) transducer, C66 =1

2(C11 −C12), which leaves ﬁve independent complex-

valued elastic moduli, C11,C12,C13,C33, and C44.

The electrical potential ϕinside the PZT transducer is governed by Gauss’s law for a

linear, homogeneous dielectric with a zero density of free charges,

∇·D=∇·(−ε·∇ϕ)=0,(4)

where Dis the electric displacement ﬁeld and εthe 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,

σ

D=C−eT

e ε  s

E,(5a)

with e=



0000e15 0

000e15 0 0

e31 e31 e33 0 0 0 

and ε=



ε11 0 0

0ε11 0

0 0 ε33 

.(5b)

2.2. The acoustic radiation force and the acoustophoretic particle velocity

We consider polystyrene particles with density ρps, compressibility κps, 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 Frad on the particles placed in water

is given by the negative gradient of the Gorkov potential Urad , [18]

Frad =−∇Urad ,with (6a)

Urad =πa31

3f0κ0|p1|2−1

2f1ρ0|v1|2, f0= 1 −κps

κ0

,and f1=2(ρps −ρ0)

2ρps +ρ0

.(6b)

If a (polystyrene) microparticle of radius ais placed in a ﬂuid of viscosity η0ﬂowing with

the local velocity v0, the presence of Frad imparts a so-called acoustophoretic velocity

vps to the particle. As inertia is negligible, vps is found from a balance between Frad

and the viscous Stokes drag force Fdrag , [14]

vps =1

6πη0aFrad +v0.(7)

2.3. Electrical impedance and power dissipation

For a PZT transducer with an excited top electrode and a grounded bottom electrode

set by the respective potentials ϕ=ϕpzt and ϕ= 0 V, the electrical impedance Zis given

by the ratio of ϕpzt −0 V and the surface integral of the polarization current density

D+0∇ϕas, [15]

Z=ϕpzt

I,wtih I=−iωZ∂Ω

n·(D+0∇ϕ) da. (8)

The electrical power dissipation Ppzt in the PZT transducer is given by

Ppzt =1

2Re (ϕpzt)I?=1

2ϕpztIcos θ, with θ= arg(Z).(9)

2.4. Butterworth–Van Dyke circuit model

To describe the electrical response of the transducer around its thickness resonance fre-

quency, 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 Fig. 1(a). It

consists of the parasitic wire resistance Rwire and inductance Lwire in series with a PZT

circuit having the transducer capacitance C0in parallel with an transducer LCR-circuit

R1-L1-C1. The four parameters R1,L1,C1, and C0can be obtained from the PZT

admittance spectrum Y(f)=1/Z(f) at the resonance frequency frand anti-resonance

frequency fa[19,20],

C0=Im Y(fr)

2πfr

, R1=1

Im Y(fr), C1=C0"f2

−1#, L1=1

(2πfr)2C1

.(10)

We perform simulations of the BVD-circuit using the SPICE -based circuit simulator

software LTspice with parameters for the circuit components obtained via Eq. (10) and

the measured values of the wire resistance Rwire and inductance Lwire.

FIG. 1. (a) A schematic overview of the electrical circuit driving the transducer. The trans-

ducer, represented by the BVD-model with a resistor R1, an inductor L1, and two capacitors C0

and C1, 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.

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 rpzt = 5.02 mm, hpzt = 506 µm, wﬂ= 2060 µm,

hﬂ= 200 µm, hcpl = 39 µm, wcap = 2324 µm, and hcap = 483 µm. In the simulation the reduced

lengths are lcap = 6.44 mm and lpml = 839 µm. (e) The cross-section in the y-z-plane showing

the glass tube, the water, and the glue layer.

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Constant-powerversusconstant-voltageactuationinfrequencysweepsforacoustouidicapplicationsFabianLickert,HenrikBruus,yandMassimilianoRossizDepartmentofPhysics,TechnicalUniversityofDenmark,DTUPhysicsBuilding309,DK-2800KongensLyngby,Denmark(Dated:4October2022)Supplyingapiezoelectrictransducerwithconsta...

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Constant-power versus constant-voltage actuation in frequency sweeps for acoustouidic applications Fabian LickertHenrik Bruusyand Massimiliano Rossiz

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