New methodology for determining the dielectric ...

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Received: 11 August 2017

Accepted: 7 November 2017

DOI: 10.1111/jace.15338

ORIGINAL ARTICLE

New methodology for determining the dielectric constant of a piezoelectric material at the resonance frequency range Hossein Daneshpajooh1

| Husain N. Shekhani2 | Minkyu Choi1 | Kenji Uchino1

1 International Center for Actuators and Transducers, The Pennsylvania State University, University Park, Pennsylvania

Abstract A new methodology is proposed to measure the dielectric constant and loss of a

2 Washington University, St Louis, Missouri

piezoelectric at the resonance frequency range based on the burst excitation method. Using a k31 type soft PZT rectangular specimen, we investigated the “force” and “voltage” factors carefully under the short- and open-circuit condi-

Correspondence Hossein Daneshpajooh, International Center for Actuators and Transducers, The Pennsylvania State University, University Park, PA. Email: [email protected]

tions of the burst method, in terms of the ratio of the ring-down current and voltage with the plate end vibration velocity and the displacement, and their phase lags. We provide the obtained material properties, including loss parameters, in particular, dielectric properties at the resonance frequency range.

Funding information Office of Naval Research, Grant/Award Number: N00014-17-1-2088

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KEYWORDS burst mode, dielectric loss, ferroelectric materials, high power, piezoelectric material/properties

| INTRODUCTION

Currently, high power driving of a piezoelectric under its electromechanical resonance condition is primarily focused in the development of various devices, such as piezoelectric transformers, ultrasonic motors, and ultrasonic medical applications.1-3 Also, the electromechanical behavior of a piezoelectric element at resonance is important in new switching sensor methods for high-precision measurements, where sensing element is external capacitor or inductor.4,5 However, the bottleneck of piezoelectric high-power applications is the heat generation originated from the losses of the piezoelectric materials. There are, in general, three losses in a piezoelectric material; dielectric, elastic, and piezoelectric losses. e

X


¼ e ð1  j tan d Þ

E


¼ s ð1  j tan / Þ

s

X

E

0

0

d
¼ dð1  j tan h0 Þ

1 QA;31 1 1 ¼ þ QB;31 QA;31

(1)

(3)

¼ tan ;011

1þ




(4) 2

 k31

2

X2b;31   tan d033 þ tan ;011  2 tan h031 ;

(2)

Where, ɛ, s, and d, are the permittivity, elastic compliance, and piezoelectric coefficient, respectively. Also, J Am Ceram Soc. 2017;1–9.

superscripts X and E correspond to constant stress and constant electric field conditions, respectively. The total loss for generating heat in the material is provided by a combination of these three losses. Previously, our group proposed the antiresonance frequency usage in the PZT actuators, rather than the resonance frequency drive, in order to reduce the required input electric power or to suppress the generated heat.6,7 We derived phenomenologically the mechanical quality factors QA at resonance and QB at antiresonance to be expressed in the case of k31 rectangular plate by1,2,7:

1 k31

(5)

where QA, QB and k31 are the resonance and antiresonance quality factors and piezoelectric coupling factor, respectively. Also, Ωb = xbl/2v is the normalized frequency, and /0 , d0 , and h0 correspond to elastic, dielectric, and

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piezoelectric losses, respectively, and “prime” denotes the “intensive” loss, because of the E constant condition on the k31 type specimen. As long as the piezoelectric loss tan h031 is larger than the average of dielectric and elastic losses 0 1 0 2 tan d33 þ tan ;11 in the PZT compositions, we can expect QB > QA, that is, the required input electric energy should be lower at the antiresonance than that at the resonance in order to generate the same output mechanical energy. Most recently, we further proposed that much less input electric energy seems to be required, if we choose an optimized drive frequency in between the resonance and antiresonance frequencies.8-11 Using the recent equivalent circuits with three losses embedded correctly, we can estimate this optimized drive frequency, as long as three loss values are correctly assigned.6 In the experiments, there are two methods used in high power characterization of piezoelectric materials at resonance/anti-resonance condition; (i) resonance/antiresonance admittance/impedance spectrum analysis,2,12 and (ii) burst mode method. The main disadvantage of continuous electric spectroscopy method is the considerable heat generation during continuous driving condition. However, in the burst mode, the effect of temperature rise on high power resonance/antiresonance behavior of piezoelectric ceramics can be eliminated, by introducing a short-circuit or an opencircuit system immediately after a few cycles of driving force (less than 0.1 milliseconds). Therefore, burst mode is useful for decoupling of temperature effects in the loss characterization, thus higher driving levels can be achieved. By measuring the vibration signal’s decay, material properties, and loss at the resonance and antiresonance can be calculated.13 In the burst method, the short-circuit current and vibration velocity are related using force factor (A), and the open-circuit voltage and vibration displacement are related through voltage factor (B).13 In our previous papers,2,7,14 we suggested how to determine the three losses separately: 1. Obtain tand0 from an impedance analyzer or a capacitance meter at a frequency away from the resonance or anti-resonance range; 2. Obtain the following parameters experimentally from an admittance/impedance spectrum around the resonance (A-type) and antiresonance (B-type) range: xa, xb, QA, QB (from the 3 dB bandwidth method), and the normalized frequency Ωb = xbl/2v; 3. Obtain tan/0 from the inverse value of QA (quality factor at the resonance) in the k31 mode; 4. Calculate electromechanical coupling factor k from the xa and xb with the IEEE Standard equation in the k31 mode15:   k31 2 p xb pðxb  xa Þ ¼ tan ; (6) 2xa 1  k31 2 2 xa

DANESHPAJOOH

ET AL.

(5). Finally obtain tanh0 by the following Equation in the k31 mode:  tan d0 þ tan /0 1 1 1 þ tan h0 ¼ þ 4 QA QB 2 " # (7)  2 1 2 1þ  k31 Xb : k31 However, we found a sort of dilemma in Item (1) on the above suggested loss parameter determination process; that is, the dielectric loss has merely studied at off-resonance frequency.16-18 We had a frustration in the case that the dielectric constant and loss factor exhibit the frequency dependence or dielectric relaxation at resonance/ antiresonance condition. This is our motivation on this research. In addition to the three losses, dielectric (tand0 ), elastic (tan/0 ), and piezoelectric (tanh0 ), we need to distinguish each loss parameter to be “intensive” or “extensive” types, which are presented with prime and non-prime symbols, respectively. The intensive parameters, such as stress and electric field, do not depend on material’s volume and can be controlled externally, while extensive parameters, such as strain and electrical displacement (or roughly polariza19-21 tion), are dependent Extensive  x
tox material’s volume.  dielectric loss e ¼ e ð1  j tan dÞ is obtained by the phase lag between the electric field and electric displacement under a completely clamped (strain x constant) condi
tion; while extensive elastic loss sD ¼ sD ð1  j tan /Þ is obtained by the phase lag between the stress and strain under a completely open (electric displacement D constant) condition, theoretically. Precise measurement of extensive and intensive properties, including extensive and intensive losses, is necessary in order to study comprehensively both the physical microscopic phenomenology of loss mechanism and device design/simulation applications. However, the measurement under constraint conditions is not easy at a low operating frequency, and extensive loss parameters are usually obtained using intensive loss parameters using the following K matrix with the coupling factor, which results in large errors7,19: 2

3 2 3 tan d tan d0 4 tan u0 5 ¼ ½K4 tan u 5; tan h tan h0 2

1 4 12 ½K ¼ k 1  k2 1 k2 ¼

k2 1 1

3 2k 2 2k 2 5; 1  k 2

d2 : sE ðe X e0 Þ

(8)

(9)

(10)

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Several researchers have used the burst mode to measure the properties of piezoelectric ceramics. Noting that due to the short excitation period in the burst mode the temperate rise can be eliminated. Therefore, this method was used widely to study the vibration level dependency, electrical and mechanical dc bias and temperature at high power conditions.20-23 Although, some researchers used nonlinear approaches to study the resonance behavior of piezoceramics, the linear approaches showed good reliability and simplicity. Previously, we provided a comprehensive approach to study the real properties of piezoceramics, using burst mode.13 While some researchers have used burst mode to investigate the mechanical loss, and dissipated energy of piezoelectric transducers, using the open-circuit equivalent circuit,21,24 according to author’s knowledge no analytical or experimental research was conducted to characterize the loss parameters at resonance conditions so far. In next section, we propose a new analytical approach to characterize the dielectric, elastic and piezoelectric losses at high power conditions.

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| ANALYTICAL METHODOLOGY

Let us review the analytical equations for the burst mode method.13,25 We consider here a k31 type piezoceramic rectangular plate with thickness a width blength L, poled along the thickness (see Figure 1). As k31 mode sample has symmetric geometry and boundary condition (Figure 1), the mode shape of resonator will be symmetric. Generally, the k31 mode shapes with stress-free condition, for small vibrations is: uðx; t Þ ¼ u0

sinðX31 xÞ sinðxt Þ sinðX31 L=2Þ

properties” nonlinearity. A more detailed derivation process for the real parameters for the “force” and “voltage” factors can be found in Ref. [13]. The difference shown below is the phase lag inclusion for these two factors. The constitutive equation describing the electric displacement D3 of a piezoelectric k31 resonator is: D3 ðt Þ ¼

d31 ou þ e0 ex331 E3 ðt Þ; sE11 ox

(12)

where E3 is the electric field. It should be noticed that ex331 is dielectric permittivity at constant strain condition along the longitudinal axis (i.e., longitudinally clamped (LC) permittivity), and sE11 corresponds to the elastic compliance under constant electric field condition (because of the electrode along the longitudinal axis). In order to consider the hysteresis losses in dielectric, elastic and piezoelectric coupling energy, we introduced complex intensive and extensive loss parameters. After exciting mechanical vibration, electrical shut-down is conducted by a short-circuit or an open-circuit condition in the burst mode method, and the following ring-down behavior of the vibration, current or voltage are monitored.

2.1 | Short circuit condition/resonance condition By imposing the short circuit boundary condition on sample, the external electrical field should be zero. The current L=2 R R D_ 3 dx and: can be calculated using iðt Þ ¼ D_ 3 dAe ¼ b L=2

Ae

i0 ¼

(11)

Where, X31 ¼ x=mE11 , v is the speed of sound, and u0 is the amplitude of the edge displacement of the rectangular plate at the fundamental resonance frequency. External electrical and mechanical loads can create nonlinearities in piezoelectric behavior, however, in k31 mode the external biases are negligible. Therefore, we have assumed no material

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2d31
u0 bxA ; E S11

(13)

Thus, the force factor as a ratio of short circuit current and vibration velocity can be defined as: A31 ¼

  i0 d
d31  (14) ¼ 2b E31
¼ 2b E 1 þ j tan /011  tan h031 v0 s11 s11

2.2 | Open-circuit condition/antiresonance condition The total electric displacement is zero at open circuit con  L=2 R D3 ðt Þdx ¼ 0 . Therefore, assuming the unidition L=2

form electric field distribution across the thickness (E3(x,t) = V(t) / a), we have: Z F I G U R E 1 Geometry of k31 mode piezoelectric resonator

L=2

L=2

Z E3 ðx; tÞdx ¼ 

L=2 L=2

VðtÞ=a dx ¼ 


d31 s11 ex331
e0 E


Z

ou dx L=2 ox L=2

(15)

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Then, V0 ¼


2ad31 u0
x 1
E Ls11 e33 e0

(16)

Here, u0 is the amplitude of the edge displacement of the rectangular plate at the antiresonance frequency. The voltage factor (B31) is defined as the ratio of open-circuit voltage and vibration displacement at the antiresonance frequency, given by: B31 ¼


V0 2a d31 2a d31 ¼ ¼
x 1
E L s11 e33 e0 L sE11 e0 ex331 u0    1 þ j tan /011 þ tan d33  tan h031

(17)

It should be noted that, as the antiresonance mode is activated at an open-circuit boundary condition, the extensive dielectric loss appears in Equation (17). As the force factor and voltage factor are measured under the resonance and antiresonance conditions, respectively, using the burst mode we can measure permittivity and dielectric loss directly:

   A31 a e0 eX33 1  k2 ¼ e0 ex331 ¼ Re (18) B31 bL

tan d33

A31 ¼ Im B31

 

bL x1 e0 e33 a

(19)

Note that comparing Equations (14) and (17), Equation (19) is obtained when ðtan /011  tan h031 Þ does not change between the resonance and antiresonance frequencies. Equations (18) and (19) are essential in this paper, which indicates that the dielectric constant and loss can be determined by measuring the force and voltage factors precisely; that is, measuring the current and vibration velocity (at the plate edge) under a short-circuit condition with their phase lag, and the voltage and vibration displacement (at the plate edge) under an open-circuit condition with their phase lag.

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| EXPERIMENTAL PROCEDURE

We have used five k31 mode samples of commercial soft PZT, PIC 255 (PI Ceramic GmbH, Germany), with dimensions 40 mm 95 mm 9 1 mm. Also, each sample was used twice; 0° and 180° between polarization and electric field. The samples were supported at the nodal point by adjustable spring holders to minimize the vibration interference. Figure 2 shows the setup diagram schematic. Samples were excited by a function generator (Siglent, SDO0342, Shenzen, China) through a power

F I G U R E 2 Experimental setup schematic diagram

amplifier (NF Corporation, NF4010, Yokahoma, Japan). The current was measured using a current probe amplifier (Tektronix, TCPA300, Beaverton, OR) and 15 turns wire loop (homemade). And the voltage was measured by a 1009 voltage probe (Tektronix, P1500, Address is missing). Also, to measure the vibration velocity/displacement, a laser doppler vibrometer (Polytech, OFV 3001), was used. Due to the transient nature of decaying signals, the averaging signal processing methods cannot be used for phase lag measurements. Therefore, we have used small sampling data intervals with constant frequency approximation to measure the phase and frequencies. Due to the resistive behavior of the sample at the resonance/antiresonance condition, capacitive/inductive components in the measuring system can affect the accuracy of the measurement. At open circuit condition, the elimination of inductive loads, introduced by wire loops and equipment, is feasible. However, at short circuit condition extra loads are inevitable, which makes the accurate phase measurements hard. In burst method, the samples are excited mechanically for several 10 seconds of cycles (less than 0.1 milliseconds) near the resonance or antiresonance, then short-circuit or open-circuit condition by removal of the driving force. Therefore, the sample vibrates in specimen’s natural frequency. The constraint condition, short- or open-circuit, generates a different resonance frequency. However, in order to minimize the higher harmonic excitation, it is important to adjust the initial excitation frequency near the natural resonance/antiresonance frequency. Figure 3A shows the short circuit condition, in which after removal of driving force the current and vibration velocity decay. Similarly, after imposing the open circuit condition using a mechanical relay, the voltage and vibration displacement decays, however, at antiresonance depending on the driving period and frequency jump a DC bias voltage can be seen.25

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F I G U R E 3 Mechanical decaying waveform after imposing (A) short circuit condition; (B) open circuit condition

F I G U R E 4 (A) Resonance (fa) and antiresonance (fb) frequency; (B) elastic compliance (sE11 ), piezoelectric coefficient (d31) and coupling factor (k31); dependency on vibration level

4 | CHARACTERIZATION OF DIELECTRIC CONSTANT, ELASTIC COMPLIANCE, AND PIEZOELECTRIC CONSTANT First, we can obtain the resonance and antiresonance frequencies from the time-cyclic pitch from the short- and open-circuit conditions as a function of the vibration velocity (see Figure 4A). Knowing the material’s density, we can determine the elastic compliance sE11 , as shown in Figure 4B.13 Also, using the measured force factor and elastic compliance, we can determine the piezoelectric coefficient (d31) (Figure 4B).13 Second, from the resonance and antiresonance frequencies, we can derive the electromechanical coupling factor k31 as a function of the vibration velocity (Figure 4B).7,13 As shown in Figure 4, the elastic compliance and piezoelectric coefficient magnitude increase

at higher vibration velocity conditions. Also, the k31 increased with increasing the vibration velocity. Third, we now obtain the dielectric constant/permittivity from the force and voltage factors from Eq. (18), which is shown in Figure 5. Both free eX33 and clamped permittivity ex331 increase with vibration velocity. The off-resonance permittivity measured at 10 kHz is also plotted with a star, which is close to the free permittivity extrapolating to a low power condition.

5 | CHARACTERIZATION OF DIELECTRIC LOSS, ELASTIC LOSS AND PIEZOELECTRIC LOSS Fourth, using the relative rate of the vibration amplitude decay, the mechanical quality factor for a damped linear system can be describes as:

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stabilized so that the systematic phase lag can be eliminated. Therefore, the calibration was performed by mounting precisely a high-Q (1000) PZT sample (PIC144) with negligible phase lag around the desired frequency range. As the force factor and voltage factor have different circuits and frequency ranges, they require separate calibrations, and the systematic phase delay cannot be eliminated by arithmetic approaches. As shown in Figure 6A,B, it is obvious that the intensive piezoelectric loss is less than the sum of the intensive elastic and extensive dielectric loss, and the intensive piezoelectric loss is larger than the average of the intensive elastic and dielectric losses: 

F I G U R E 5 Free and longitudinally clamped dielectric constant dependency on vibration velocity

Q¼

2pf
 v1 2 ln v2 =ðt2  t1 Þ

(20)

As seen in Figure 6, the piezoceramic quality factor is degraded gradually with increasing vibration velocity. In order to measure the force factor phase lag (UA31 ¼ tan /011  tan h031 ) and voltage factor phase lag (UB31 ¼ tan /011 þ tan d33  tan h031 ), the system should be

  1 tan /011 þ tan d33 [ tan h031 [ tan /011 þ tan d0 33 (21) 2

Due to the external inductive/capacitive loads in short-circuit condition, force factor phase lag ΦA31 measurement is hard. Therefore, direct measurement of dielectric loss using Equation 18, is not feasible. However, using ΦB31 phase lag (Equation 17), resonance/antiresonance quality factors (Equations 4 and 5), and K matrix (Equations 8-10), the intensive and extensive dielectric and piezoelectric loss can be calculated as follow: tan h031

  1 1 1 1 1  k2 2 ¼  þ X  UB31  2 QA QB 1  k2 k2 (22)

F I G U R E 6 (A) QA and QB variation by vibration velocity. (B) Voltage factor phase lag at open-circuit condition

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tan d033

  1 1 1 1 þ k2 1  k4 2 ¼  þ X  2 QA QB 1  k2 k2

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(23)

 tan /011  2UB31 tan d33 ¼ tan h031  tan /011  UB31

F I G U R E 7 Frequency dependency of dielectric loss; measured by LCR meter at off-resonance (black) and burst method at resonance frequency (Blue)

(24)

In conventional methods, the dielectric loss is measured at off-resonance (100 Hz) and considered to be constant at resonance condition. The off-resonance dielectric loss can be measured using LCR meters at constant voltage condition. While using the Equations 23 and 24, the free and clamped dielectric loss can be measured at the resonance condition. Figure 7 shows the dielectric loss dependency on driving frequency. It should be noted that the dielectric loss measured at low power resonance condition exhibits an almost equal value to the extrapolated one from the offresonance measurement. The dielectric loss measurement at low-power condition can be accurately conducted by LCR meter, however, the LCR meter has limited frequency range and power. As can be seen, the dielectric loss logarithmically increased with an increase in driving frequency.

F I G U R E 8 (A) Vibration level dependency of elastic compliance (sE11 ) and elastic loss (tan /011 ); (B) Vibration level dependency of

piezoelectric constant (d31) and piezoelectric loss (tan h031 Þ; (C) Vibration level dependency of free/clamped dielectric constants (eE33 , eLC 33 ); (D) Vibration level dependency of intensive/extensive dielectric loss (tan d033 , tan d33)

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DANESHPAJOOH

Finally, the intensive and extensive, elastic loss, and piezoelectric losses are plotted in Figure 8. As seen in Figure 8A, the elastic compliance and elastic loss increased with increasing the vibration velocity. Also, the magnitude of piezoelectric constant and piezoelectric loss increased with increasing the vibration velocity. Moreover, according to authors’ knowledge, using the proposed methodology the dielectric constant and loss were measured at the resonance range for the first time. As shown in Figure 8C, the measured low-power dielectric constant was in good agreement with that of measured at off-resonance (10 kHz using LCR meter), while the measured low-power dielectric loss is smaller than that of the value at the off-resonance. In contrast with constant dielectric loss supposition considered in the conventional loss characterization methodologies, the dielectric loss increased around the resonance frequency range, and also increased with increasing the vibration velocity.

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| CONCLUSION

In this paper, a new methodology for investigation of the “voltage” and “force” factor phase lags, using burst mode was validated. The proposed methodology can be used for measuring the dielectric constant and loss at the resonance/antiresonance condition. The vibration level dependency of elastic, piezoelectric, and dielectric constants and their corresponding loss parameters, was measured at high vibration amplitude conditions. Also, for the first time, the dielectric loss was measured experimentally at the resonance condition, using voltage factor phase lag. The dielectric loss at low-power conditions under the resonance was slightly larger than that at the offresonance (low frequency) dielectric loss measured with the conventional method, which suggests the frequency dependence of dielectric loss, and the dielectric loss increased with increasing the vibration level. The reported dielectric loss behavior was in contrast with the “constant dielectric loss assumption” in the conventional methods. ACKNOWLEDGMENT This research was supported by Office of Naval Research, under Grant No. N00014-17-1-2088. ORCID Hossein Daneshpajooh 9739

http://orcid.org/0000-0002-6152-

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How to cite this article: Daneshpajooh H, Shekhani HN, Choi M, Uchino K. New methodology for determining the dielectric constant of a piezoelectric material at the resonance frequency range. J Am Ceram Soc. 2017;00:1–9. https://doi.org/10.1111/jace.15338

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