The Effective Coupling Coefficient for a Completed PIN-PMN-PT Array

Abstract

The computation of the electromechanical coupling coefficient (EMCC) of a fully assembled medical ultrasound transducer array is directly computed with closed form expressions. The Levenberg-Marquardt non-linear regression algorithm (LMA) is employed to help confirm the EMCC calculated prediction (k_EFF) and provide statistical insights. The complex electrical impedance spectra of a 1-3 composite array with two matching layers operating at a 3.75 MHz center frequency using PIN-PMN-PT single crystal material is measured in air both before and after oven heating at 160 °C for 15 minutes. The oven heating produces changes in the EMCC of −4.9%, clamped dielectric constant of −11%, and effective transducer longitudinal velocity of −2.5%. Utilizing the pre- and post-heating array impedance data, the calculated EMCC values from the new closed form expressions agree well with the complete KLM model based LMA, and also exhibit approximately one tenth the error as compared to the formulas for a flat, unloaded transducer.

1. Introduction

The electromechanical coupling coefficient (EMCC) is one of the most critically important transducer parameters since its square is proportional to both energy conversion [1], and the effective bandwidth achievable in a design [2]. The equation that predicts the EMCC for a simply electroded piezoelectric material block has been used for over 50 years. Early expressions of EMCC appeared in the 1950’s [3, 4] and led to the commonly used EMCC expression used today for an unloaded flat resonator [5–7]

$$k_t^2=\raisebox{1ex}{$(\pi $}\!\left/ \!\raisebox{-1ex}{$2)$}\right.\raisebox{1ex}{$(f_r$}\!\left/ \!\raisebox{-1ex}{$f_a)$}\right.\tan (\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.(1-\raisebox{1ex}{$f_r$}\!\left/ \!\raisebox{-1ex}{$f_a$}\right.)).$$ (1)

The EMCC of (1) is based on the electrical input impedance of an unloaded device with the incorporation of Mason’s electromechanical transformer while recognizing the impedance minimum (at f_r) and open circuit condition (at f_a). Based on its KLM based derivation in [7], (1) predicts the coupling for any flat, unloaded piezoelectric resonator. This would include the coupling for a 1-3 composite format, however in this case the k_t would be replaced with the ${\overline{k}}_t$ of Smith and Auld [8], which can approach k₃₃ in ideal conditions (Fig. 1b).

Fig. 1.

The electromechanical coupling coefficient for four general shape formats. Formats (a), (c) and (d) exist in ideal (i.e. no support matrix polymer) conditions which permit directly calculated (generalized) EMCC example values using two types of commercially available single crystal material (i.e. PMN-30%PT [29], and PIN-PMN-PT [30, 31]). The piezo blocks for each case are poled in dimension #3 with electrodes to support fields in tins dimension. See the Supplement for variable definitions.

An “effective” EMCC has also been created [9] as

$${k_{eff}}^2=\left({f_a}^2-{f_r}^2\right)/{f_a}^2$$ (2)

which is generally applied to unloaded piezoelectric block material [10–14], and as well, assembled arrays [1, 15]. This simplified expression agrees with (1) at EMCC values up to 60%, however it shows an increasing deviation compared to (1) of +9.5% at 70%, and +20.4% at 80% (Fig. 2). Clearly the utility of (2) with high coupling single crystal material 1-3 composite (Fig. 1b) is limited.

Fig. 2.

The comparison of EMCC predictions using of (1) as a solid line and (2) as dashed showing marked differences above EMCC values of 60%. In particular, (1) predicts 70% and 80%,while (2) predicts 9.5% and 20.4% above these values, respectively.

The application of the EMCC expressions above can be valuable to the extent the core piezoelectric material can be qualified at the first step in the process of creating an efficient transducer. It is difficult however, to determine an accurate estimate of the EMCC for a completed array assembly due to the effects of many assembly components and final array geometry. For example, the antiresonance frequency (f_a) used in (1) for an unassembled transducer depends only upon the effective piezoelectric thickness and its acoustic velocity. The antiresonance frequency for a completed assembly however will typically not show a precise half wave resonance, but will depend on the matching layers as well as the general shape of the resonator as a plate, or a long thin rod, or thin rectangular blade-like configuration [16, 17]. It is also well known that the coupling coefficient is dependent on the element dimensions (Fig. 1) [18], on kerf fill materials [19, 20], and on the etched or sawn vertical element slope [21]. The array design using a 1-3 composite approach also must make adjustments in array effective impedance and velocity based on polymer to piezoelectric element volume ratios [8].

Since new designs for ultrasound arrays are often integrated with components such as application specific integrated circuits (ASIC) as well as flex circuits, there are potential heating steps in assembly that can degrade the piezo coupling. This is especially a concern when relatively low Curie temperature single crystal materials such as PMN-PT are used in the transducer stack. We have developed a transducer design based upon a modular array concept [22–24]. Testing these modules in air for basic performance by using an electrical measurement is potentially much easier than a wet lab pulse echo measurement.

To enable the derivation of EMCC of a completed stack we start with the use of the analytic expression for the electrical impedance produced directly from the KLM 1D model [2, 25–28]. To provide confidence in our result the Levenberg-Marquardt non-linear regression algorithm (LMA), which uses the full KLM model without any simplifying assumptions, is employed to confirm the analytic expression we derive for EMCC.

The purpose of this work is to describe a method to compute an accurate EMCC of a completely assembled array, or k_EFF. Current EMCC estimation for completed arrays (if attempted at all) is generally based on manual methods with limited accuracy. Use of the approach described in this work may help enable future automated methods to determine not only the EMCC but many critical parameters. We demonstrate our method with a 1-3 composite PIN-PMN-PT array design. This design is modeled before and after oven heating to test the relative accuracy of our closed form expressions against a fully parameterized KLM model, and in addition perform a simple heat stress experiment on our single crystal material as a finished array.

2. Methods

The KLM model provides the basis for a closed form analytic expression for the electrical impedance of an array. A reduced complexity expression for EMCC of a complete array with matching layers can emerge with simplifications assumed. These assumptions include zero values for all losses (elastic, coupling, and dielectric), and for the backing material impedance. Following the estimation of the relevant model parameters for our subject array design of a PIN-PMT-PT 1-3 composite with two matching layers, an estimate for the effective EMCC is produced. Note that EMCC is used in this work as a general term, while k_t and k_eff are defined by (1) and (2), respectively. The closed form EMCC solution derived in this work is k_EFF.

To perform an initial test of this direct computational method for k_EFF, the experimental array was heated in an oven to effectively modify key parameters which would in turn modify the k_EFF. The LMA is used to provide some additional confidence in the estimation for k_EFF. The KLM model used by the LMA in this work includes all known parameters and loss terms, as well as the actual impedance of the backing material in order to provide a “gold standard” comparison against our closed form estimation of k_EFF.

2.1. KLM basis for the closed form solution

Although it is an indirect method, reasonable KLM model estimates for transducer velocity and EMCC can be found by using iterative methods with lab data (typically impedance spectra) as the guide. Other variables (e.g. matching layer acoustic impedances, thicknesses, velocities) can be measured a-priori. This feedback-loop method to determine operational metrics has been demonstrated in piezoelectric transducer topology optimization design [12], and in MEMS modeling of 1-3 composites subjected to electromechanical loads [32].

The steps to determine the new EMCC estimate, k_EFF, are summarized below. The methods chosen are very similar to the classical EMCC derivation approach [6], [33], [7], but with considerably more terms due to the inclusion of front matching layers. The coupling and velocity variables are isolated by setting the imaginary part of the simplified impedance function to zero at f_r, and f_a respectively. Low loss material such as PIN-PMN-PT allows reasonable use of f_m and f_n as surrogates for f_r and f_a, respectively [34].

Considering only the model of the acoustic stack of our prototype design, there are seven two port transmission matrices (see Supplement, section A) to multiply in order to form the transmit transfer function (TTF) which is the principal function derived in the use of the KLM model [28], [27], [7, 35]. A representation of the TTF(f) calculation is given in (3), which is equivalently TTF(f) = [N1] [N2] [N3] [N4] [N5] [N6] [N7]. These seven networks are

$$TTF\left(f\right)=\left[\begin{array}{cc}1& Z_{C_0}\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& Z_{C^{\prime }}\\ 0& 1\end{array}\right]\left[\begin{array}{cc}\varnothing & 0\\ 0& {\varnothing }^{-1}\end{array}\right]\left[\begin{array}{cc}1& 0\\ ZB{a}^{-1}& 1\end{array}\right]\left[\begin{array}{cc}AX2& BX2\\ CX2& DX2\end{array}\right]\left[\begin{array}{cc}AM1& BM1\\ CM1& DM1\end{array}\right]\left[\begin{array}{cc}AM2& BM2\\ CM2& DM2\end{array}\right].$$ (3)

The product of all the 2-port networks [36] results in a single 2 x 2 matrix which represents TTF(f). Assuming the output of the TTF(f) network is loaded with air so the acoustic load on the second matching layer is a virtual short circuit (Fig. 3b), the ratio of two terms of (3) produce the input electrical impedance spectrum,

$$Z_{in}\left(f\right)=\raisebox{1ex}{${\left(TTF\right)}_{1,2}$}\!\left/ \!\raisebox{-1ex}{${\left(TTF\right)}_{2,2}$}\right..$$ (4)

Fig. 3.

The KLM model electromechanical circuit and block diagram used in a closed form solution for input impedance of a completed array in air. The large arrows show the input electrical port; the acoustic impedance ZFa is the impedance looking into the matching layers from the middle of the composite; similarly, the ZBa is the parallel impedance looking towards the backing material included in network N4.

2.2. Closed form expressions for input impedance

The loss terms are neglected in the development of a closed form electrical input impedance expression without substantial loss of accuracy in the prediction of the resonance and antiresonance frequencies. This is justified by calculations indicating a Q > 50 (as defined by eq. 132 on page 50 [13]) for the RLC equivalent circuit model components for the design.

To implement this simplification, all the lossy acoustic transmission line cosh(x) and sinh(x) terms are converted to lossless cos(y) and sin(y) terms, respectively [37]. The hyperbolic terms in the KLM model, which have arguments in the form of γL = α(f)L + jβ(f)L, are converted to real functions of frequency by neglecting the loss part of the argument and using the relations j sin (Im(x))≅sinh (x) and cos (Im(x))≅cosh (x).

The relation ${\omega }_0=\pi \frac{vXD}{tXD}$, and propagation constants $\gamma X\left(f\right)=\pi f\frac{tXD}{vXD}=\frac{\pi 2\pi f}{2{\omega }_0}$, $\gamma M1\left(f\right)=2\pi f\frac{tM1}{vM1}$, $\gamma M2\left(f\right)=2\pi f\frac{tM2}{vM2}$, and variables defined in Table I are utilized in (3) as the frequency dependent KLM two port matrices [7].

Table I.

Design Values for the PIN-PMN-PT 1-3 Composite 6.4 mm square Array

Specification	Value	Specification	Value	Specification	Value
Center Frequency, CF	3.75 MHz	Impedance, 1-3 Composite (Zc)	24.2 MRayl	M1 Impedance (ZM1)	6.45 MRayl
Pitch, Azimuth, 16 ele.	400 μm	Thickness, 1-3 Composite (tXD)	318 μm	M1 Thickness (tM1)	232 μm
Pitch, Elevation, 16 ele.	400 μm	Velocity, 1-3 Composite (vXD)	3680 m/s	M1 Speed (vM1)	2892 m/s
Pitch, 1-3 Composite	200 μm	Electromechanical coupling, k33	~0.80	M1 Attenuation (αM1, at CF)	160 Np/m
Kerf, 1-3 Composite	20 μm	Dielectric constant, $\raisebox{1ex}{${ϵ}_{33}^S$}\!\left/ \!\raisebox{-1ex}{${ϵ}_0$}\right.$	488	M2 Impedance (ZM2)	2.79 MRayl
Active Area (Axd, N = 256)	41 x 10−6 m2	Electrical loss, tan-δ	0.02	M2 Thickness (tM2)	136 μm
Capacitance (C0, N = 256)	556 pF	Mechanical Q at CF, Qm	21	M2 Speed (vM2)	2270 m/s
Impedance, backing (ZB)	5.46 MRayl	Attenuation (αXD, at CF)	147 Np/m	M2 Attenuation (αM2, at CF)	74 Np/m

The assumed physical termination impedance of the acoustic medium (i.e. air) is a zero-impedance shunt which represents a zero-pressure condition at the outer surface of the M2 matching layer. The N4 matrix represents the parallel connected impedance, ZBa(f) (i.e., inverse of [N4]_2,1), which includes the backing impedance, ZB. For purposes of simplification, we can make ZB effectively zero with little error (i.e. in preserving the spectral resonance and antiresonance frequencies) if the condition ZB/Zc < 0.25 is satisfied. This 0.25 limit appears to be valid from in silico modeling studies, and thus assumed reasonable for the purposes of this work; further review of tins is provided in the Discussion section. Under this condition, the parallel connected impedance in N4 (i.e. ZBa(f)) is simply reduced to jZc tan(γX(f)).

The explicit equation manipulations of the product of N1 through N7 and the subsequent production of reduced complexity spectral impedances are too unwieldly to present here, but the major steps are presented below. In general, each simplification step removes terms that do not significantly alter the resonance (f_r) and antiresonance (f_a) frequencies of the KLM model. The use of lossless propagation constants, the assumption that ZB ~ 0, and the subordinate impedance functions ZFa(f) and ZBa(f) help in the initial reduction in computational complexity.

The transmission matrices at the front port are used to produce a series acoustic impedance looking from the middle of the transducer towards the front port loaded by air; see Fig. 3. This impedance is evaluated as

$$ZFa\left(f\right)=\frac{\frac{jZM1\sin \left(\gamma S\left(f\right)\right)}{A\mathit{cos}\left(\gamma D\left(f\right)\right)+\mathit{cos}\left(\gamma S\left(f\right)\right)}+j\mathit{Zc}\mathit{tan}\left(\gamma X\left(f\right)\right)}{1-\frac{ZM1\tan \left(\gamma X\left(f\right)\right)\mathit{sin}\left(\gamma S\left(f\right)\right)}{Zc\left(A\mathit{cos}\left(\gamma D\left(f\right)\right)+\mathit{cos}\left(\gamma S\left(f\right)\right)\right)}}$$ (5)

which is computed without the loss terms. To make the expression more compact we have introduced $\gamma S\left(f\right)=2\pi f(\raisebox{1ex}{$tM1$}\!\left/ \!\raisebox{-1ex}{$vM1$}\right.+\raisebox{1ex}{$tM2$}\!\left/ \!\raisebox{-1ex}{$vM2$}\right.)$, $\gamma D\left(f\right)=2\pi f(\raisebox{1ex}{$tM1$}\!\left/ \!\raisebox{-1ex}{$vM1$}\right.-\raisebox{1ex}{$tM2$}\!\left/ \!\raisebox{-1ex}{$vM2$}\right.)$, $S\left(f\right)=\sin \left(\gamma X\left(f\right)\right)$, $C\left(f\right)=\cos \left(\gamma X\left(f\right)\right)$, and $A=\raisebox{1ex}{$\left(ZM1-ZM2\right)$}\!\left/ \!\raisebox{-1ex}{$\left(ZM1+ZM2\right)$}\right.$.

Continuing the process to obtain the closed form spectral impedance expression, we assert that the electrical input spectral impedance using the KLM model (4) without loss terms is an adequate estimate of the spectral impedance. Greatly reducing the number of terms of (4), a simplified closed form version of the electrical input spectral impedance is found to be

$$Z_{ie1}\left(f\right)=\frac{Az\left(f\right)+Bz\left(f\right)+{\omega }_0{\left(k_{EFF}\right)}^2\cos \left(\raisebox{1ex}{${\pi }^{2}f$}\!\left/ \!\raisebox{-1ex}{${\omega }_0$}\right.\right)\sin \left(\raisebox{1ex}{${\pi }^{2}f$}\!\left/ \!\raisebox{-1ex}{${\omega }_0$}\right.\right)\left[jZFa\left(f\right)C\left(f\right)-ZcS\left(f\right)\right]}{2{\pi }^3{f}^2{C}_0\left[ZFa\left(f\right)C\left(f\right)+ZBC\left(f\right)+j\text{S}\left(\text{f}\right)\left(\left(\raisebox{1ex}{$ZB$}\!\left/ \!\raisebox{-1ex}{$Zc$}\right.\right)ZFa\left(f\right)+Zc\right)\right]}$$ (6)

where

$$\begin{array}{l}Az\left(f\right)={\pi }^{2}fZcS\left(f\right)-j2{\left(k_{EFF}\right)}^2{\omega }_{0}ZFa\left(f\right)S\left(f\right)\cos {\left(\raisebox{1ex}{${\pi }^{2}f$}\!\left/ \!\raisebox{-1ex}{${\omega }_0$}\right.\right)}^2\\ Bz\left(f\right)=-j{\pi }^{2}fZFa\left(f\right)C\left(f\right)+j2{\left(k_{EFF}\right)}^2{\omega }_{0}ZFa\left(f\right)S\left(f\right).\end{array}$$

This formulation is made with attention to preserving both the spectral locations of resonance (f_r ~ f_m), and at antiresonance (f_a ~ f_n). Further simplification can be made, but at the cost of preserving only the resonance spectral location with reasonable accuracy. With the assumption that ZB is small (i.e. ZB/Zc < 0.25), (6) becomes

$$Z_{iek}\left(f\right)=\frac{{\pi }^2\text{f}\mathrm{Zc}S\left(f\right){\left(ZFa\left(f\right)\right)}^{-1}+\text{j}\left(2{\left(k_{EFF}\right)}^2{\omega }_0\text{S}\left(\text{f}\right)-{\pi }^2\text{f}C\left(f\right)\right)}{2{\pi }^3{f}^2{C}_0}.$$ (7)

The closed form (7), being of reduced complexity for the purpose of deriving k_EFF, is only appropriate for use at the series resonance frequency. Since the acoustic stack resonance frequency, f_r, occurs by definition as the imaginary part of the spectral impedance crosses zero, we can obtain k_EFF, which is

$$k_{EFF}=\pi \sqrt{\frac{f_r}{2{\omega }_0}\left(\mathit{Im}\left(\frac{-Zc}{ZFa\left(f_r\right)}\right)+\frac{1}{\mathit{tan}\left(\gamma X\left(f_r\right)\right)}\right)}.$$ (8)

We note that Im(Z_iek) = 0 occurs at a frequency which is not exactly the resonance f_r of the KLM model. However, since this zero-crossing frequency is very close to the lab observable f_m frequency, we will use the functional equivalency f_r ≈ f_m. It can be easily shown that (8) becomes (1) in the absence of matching layers (see Supplement, section B).

The unitless effective EMCC, k_EFF, for a fully assembled transducer can be calculated directly, but (8) critically depends on a) the matching layer characteristics represented by ZFa, and b) the tangent argument. Both of these depend on an accurate estimate for the transducer velocity, vXD. In the case of a finished array where either the velocity is unknown or has been altered by heating, etc., a reasonable estimate for transducer velocity (with ρ as the transducer composite density) can be produced,

$$vXD=\frac{\pi f_{a}tXD}{\pi -\mathrm{atan}\left(\left|\frac{2\rho vXD\left(A\mathit{cos}\left(\gamma Da\right)+\mathit{cos}\left(\gamma Sa\right)\right)}{ZM1\mathit{sin}\left(\gamma Sa\right)}\right|\right)}$$ (9)

where we have solved for velocity in a reduced version of (6) (see Supplement, section B) by setting imaginary part to zero while the frequency is at antiresonance, f_a, and using the relations $\gamma Sa\left(f\right)=2\pi f_a\left(\raisebox{1ex}{$tM1$}\!\left/ \!\raisebox{-1ex}{$vM1$}\right.+\raisebox{1ex}{$tM2$}\!\left/ \!\raisebox{-1ex}{$vM2$}\right.\right)$, $\gamma Da\left(f\right)=2\pi f_a\left(\raisebox{1ex}{$tM1$}\!\left/ \!\raisebox{-1ex}{$vM1$}\right.-\raisebox{1ex}{$tM2$}\!\left/ \!\raisebox{-1ex}{$vM2$}\right.\right)$. In a similar manner as with (8), we make the equivalency of f_a ≈ f_n for the purposes of evaluation of (9), where f_n is the laboratory acquired antiresonance frequency which occurs at a maximum of the |Z|. The solution of (9) requires a few iterative cycles (< 5) to solve since vXD appears to be dependent (weakly) on itself.

2.3. Estimation of relevant model parameters

The known (pre-calculated or measured) parameters values which include matching layer dimensions and velocities, piezoelectric impedance, dielectric constant, etc. are all known a-priori from lab measurements.

Our early array design [22] is based upon a 1-3 composite construction with PIN-PMN-PT single crystal piezoelectric; this early work lead to an ad hoc test array which was built by Sonic Concepts (Sonic Concepts Inc., Bothell, WA), (Fig. 4). The two-layer matching was designed to help utilize the broad band potential of the single crystal. The 1-3 composite design method of Smith and Auld [8] was used to adjust the array acoustic impedance from 37 to 24.2 MRayl, and the velocity from 4653 to 3680 m/s. The test device consists of an array of 256 elements which have been electrically connected together to function as a single large element with a 6.4 x 6.4 mm active aperture. With a-priori knowledge of layer thicknesses and properties the critical unknowns of vXD, k₃₃, and ${ϵ}_{33}^S$ were estimated with the final (pre-heating) model values are shown in Table I. For the purposes of this work we will use this prototype array to a) refine our KLM model, and b) compare model and lab results before and after a 160 °C heating of the array device in an oven.

Fig. 4.

The prototype array with a double matching layer 1-3 composite design using PIN-PMN-PT single crystal. For modeling purposes, all 256 composite elements were electrically connected together.

2.3.1. Pre-heating model parameters

All relevant array dimensions and materials (in pre-heated state) including the matching layers and backing are well known a-priori by either measurement or design (Table I). The primary KLM modeling parameter estimates (Table II) however can be computed by methods described below to permit reasonable input parameter values. Note that the KLM parameters in Table II have been scaled in area (10⁶) and velocity (10⁻⁶) to achieve better matrix condition numbers (for use in LMA); as a consequence, the spectral frequency has been scaled by 10⁻⁶. With regard to our particular selection of modeling parameters, we have chosen (as surrogates), Zc and C0, rather than the elastic stiffness constant, $C_{33}^D$, and the clamped dielectric constant, ${ϵ}_{33}^S$, respectively [38], [34].

Table II.

Scaled KLM Model Values for the PIN-PMN-PT 1-3 Composite Array

KLM Model Parameter	Scaled Value used in LMA		Scaling
	Pre-Heating	Post-Heating	area	vel.	Net Factor
Composite propagation constant, γXD	0.272 - j0.0065	0.279 - j0.0095		✓	106
Electromechanical coupling, kEFF	0.795	0.756			1
Composite bulk capacitance, C0	0.000560 - j4 x 10−5	0.000495 - j7 x l0−5	✓		106
Composite acoustic impedance, Zc	990 - j23.4	965 - j30	✓	✓	1
First matching layer acoustic impedance, ZM1	264 - j5.65	(same)	✓	✓	1
Second matching layer acoustic impedance, ZM2	114.278 - j0.8185	(same)	✓	✓	1
First matching layer propagation constant, γM1	0.50404 - j0.01008	(same)		✓	106
Second matching layer propagation constant, γM2	0.3764 - j0.002635	(same)		✓	106
Backing material acoustic impedance, ZB	223.6	(same)	✓	✓	1

2.3.1a. Composite acoustic impedance, Zc, and composite acoustic propagation constant, μXD

The effective bare 1-3 composite density and velocity can be accurately calculated a-priori [8]. The assembled array composite velocity is estimated with (9). Regarding composite elastic loss, our lab derived estimate of αdB = 3.5 dB/cm/MHz, or αXD = 147 Np/m at the resonance frequency, produces a Q_m of 21 using $Q_m=\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$\left(\alpha XD\lambda \right)$}\right.$, where λ is vXD/CF. This is similar to a Q_m of 15 reported for a PMNT 1-3 composite array [39]. The acoustic impedance of our composite is then estimated by $Z_c=\rho \left(vXD\right)Axd\left(1-\raisebox{1ex}{$j$}\!\left/ \!\raisebox{-1ex}{$2Q_m$}\right.\right)$. A similar calculation is made for ZM1 and ZM2 with loss terms from lab derived loss estimates. As well, the effective complex propagation constant for the composite array is calculated using the same velocity and Q_m by the relation $\gamma XD=\pi \left(\raisebox{1ex}{$tXD$}\!\left/ \!\raisebox{-1ex}{$vXD$}\right.\right)\left(1-\raisebox{1ex}{$j$}\!\left/ \!\raisebox{-1ex}{$2Q_m$}\right.\right)$. The matching layer propagation constants are estimated in a similar fashion.

2.3.1b. EMCC for the composite array, k_EFF

With f_a (i.e. ~f_n), vXD (9), Zc, and γXD estimated, a value of f_r (i.e. ~f_m), will allow a direct estimation of k_EFF with (8). The coupling loss can be included, however it is small enough to be neglected; k_EFF is expressed in this work as a real number.

2.3.1c. Composite bulk capacitance, C0

Using the Im(Z) of the lab data at a frequency of one tenth f_r, we can estimate the low frequency, free permittivity [40] (with knowledge of transducer area and thickness) as $\raisebox{1ex}{${ϵ}_{33}^T$}\!\left/ \!\raisebox{-1ex}{${ϵ}_0$}\right.=1298$. The relation $\raisebox{1ex}{${ϵ}_{33}^S$}\!\left/ \!\raisebox{-1ex}{${ϵ}_{33}^T$}\right.=1-{k_{33}}^2$ (letting k₃₃ = k_EFF) produces a clamped permittivity estimate of $\raisebox{1ex}{${ϵ}_{33}^S$}\!\left/ \!\raisebox{-1ex}{${ϵ}_0$}\right.=488$. Various dielectric loss tangent fractions has been reported: a slab of PIN-PMN-PT is 0.4% [41], a similar single crystal material found ~ 9% [42], and a PMNT 1-3 composite was reported to be 1.2% [39]. We have used a loss tangent estimate of 2%.

2.3.2. Post-heating model parameters

Using methods developed earlier by the authors to examine the thermal properties of PIN-PMN-PT arrays [43], the subject array in this work was exposed to 160 °C for 15 minutes in order to produce a noticeable change in the EMCC without risking depolarization. Matching layer #1 was carefully characterized for its acoustic properties in the lab first without heating stress, and again after a 160 °C oven exposure. Both ZM1 and γM1 were not measurably affected by the heat stress. Although unmeasured, we assume the second matching layer has similar stress resistance.

The composite manufacturer (TRS Technologies, Inc., State College, PA) does not provide specific glass transition temperatures for its composites, but it does provide data for the pre-diced crystals. The composite post material, TRS X4B (PIN-PMN-PT) [31], has characteristics of dielectric loss: 0.2-0.8 %, coercive field Ec: 4.5-6 kV/cm, transition temperature Trt: 120-130 °C, and Curie temperature Tc: 160-200 °C. This indicates that 160 °C is above the transition temperature, but near the lower edge of the Curie temperature range. Since the composite does have a plastic component (likely an epoxy similar to EpoTek 301), it may affect the behavior of the material, however the specifications for the crystal material itself are still generally valid for the 1-3 composite. The kerf fill epoxy likely has a glass transition temperature (Tg) which is approximately 65 °C. Above this glass transition temperature, the polymer becomes rubbery which may affect the general performance parameters of the composite; however, it is very likely the composite Trt, and Tc temperatures are likely dictated by the crystal material itself. The primary goal in heating the (completely assembled array) composite in this work was to force changes in the device (especially the EMCC) which could be determined by our characterization methods.

2.3.2a. Composite acoustic impedance, Zc, and composite acoustic propagation constant, γXD

The post-heating lab impedance spectrum location of f_a reveals a slower transducer velocity, vXD (9), which suggests a reduced Zc. Although one group found a velocity decline of ~ 5% with a 160 °C thermal stress [44] in slab form PIN-PMN-PT, we found the velocity (9) declines about 2.47% for our 1-3 composite. Based on general trends reported for slab material at 100 °C [42], we expect the Q_m to decrease by perhaps 25% thus we chose a post-heating Q_m of 15 (pre-heating Q_m = 21). With the updated vXD, the effective post-heating γXD and Zc are estimated similarly to the computation shown section 2.3.1a above.

2.3.2b. EMCC for the composite array, k_EFF

As with the pre-heating estimate, the post-heating estimate is made with a direct estimation of kEFF with (8), which is influenced by the new velocity, as well as f_r and f_a frequency shifts.

2.3.2c. Composite bulk capacitance, C0

Following the same pre-heating case methodology, the post-heating estimate of C0 is obtained. Based on a report that slab material PIN-PMN-PT exhibits a sharp rise in loss tangent at 100 °C (9% to 27%) [42], we increased our estimated post-heating loss tangent factor by 3 times to 6%.

2.4. Levenberg-Marquardt non-linear least squares method (LMA)

Regarding medical ultrasound transducer characterization, there have been iterative modeling methods [45], [46], simulated annealing methods [47], and variations of the Gauss-Newton (GN) method [34]), [48], [49] to derive piezoelectric parameters. Some groups have used the LMA [38], [50], [51]. All these efforts have focused on bare slab or disk material, or arrays without matching layers. Our use of the LMA is made with the full KLM model of the complex electrical input impedance, including backing and matching layers for use in comparison as a “gold standard” against the EMCC closed form expression prediction.

The LMA (Fig. 5) is based on the Gauss-Newton (GN) method which uses a truncated Taylor series expansion to iteratively match a model function $\widehat{y}$ comprising an array of parameters p to a lab measured data set y. The recursive algorithm incrementally improves the model function match by using $\widehat{y}\left(p+h\right)\approx \widehat{y}\left(p\right)+\raisebox{1ex}{$(\partial \widehat{y}$}\!\left/ \!\raisebox{-1ex}{$\partial p)$}\right.h=\widehat{y}+Jh$ , where h is a parameter update factor, and J is the Jacobian of partial derivatives of the modeled function with respect to each parameter. The spectral impedance sample points for both the model function and lab data comprise m points with n parameters under consideration.

Fig. 5.

The LMA algorithm is shown (for real valued data, using (10), for brevity). The model parameters of array (p) are the desired outputs with accompanying statistical metrics which help establish confidence in their estimation. The complete LMA routine (MATLAB R2019a, MathWorks, Natick, MA) for complex data is described in the Section J of the Supplement.

The LMA improves the GN method by introducing both a scalar damping factor, λ, to govern the speed of convergence, and a weighting matrix, W, to normalize the calculations and facilitate the use of the Chi-square statistic to monitor the convergence. The general LMA iterative computation [52] is represented by

$$\left[J^{T}WJ+\lambda diag(J^{T}WJ)\right]h_{LM}=J^{T}W\left(y-\widehat{y}\right)$$ (10)

where the Jacobian J is a (m x n) matrix (and J^T is its transpose) with the assumption that all rows and columns are linearly independent; J is comprised of spectral point-by-point partial derivatives of the impedance model function $\widehat{y}$ with respect to each parameter of p considered. The (n x n) J^TWJ matrix must be invertible. The measured data y and model points $\widehat{y}$ are (l x m) matrices. The weighting matrix W is a diagonal (m x m) matrix whose m diagonal members are the inverted error variance assumed for each data point. The W matrix is generally set to be $I/{\widehat{\sigma }}_y^2$. Since we do not typically know the expected statistical variance at each point, we can find an initial single value estimated variance as

$${\widehat{\sigma }}_y^2=S{\sigma }^{-1}{\left(m-n+1\right)}^{-1}{\left(y-\widehat{y}\left(p\right)\right)}^T\left(y-\widehat{y}\left(p\right)\right)$$

where Sσ is an assumed variance scalar with typical values used in this work of 20 to 50. Large Sσ values can be chosen if excellent data-model matching can be realized as a result of well determined impedance models without over-parameterization. In contrast however, the Sσ value may need to be modest if the LMA input parameters are close to their optimum values, or if the LMA model minimum error output variance is not very small. The latter case is a common LMA modeling limitation if the impedance model used is not a well determined (transducer array) system description.

The reduced chi-square statistic is defined as ${\chi }_{\upsilon }^2={\left(m-n+1\right)}^{-1}{\left(y-\widehat{y}\left(p\right)\right)}^{T}W\left(y-\widehat{y}\left(p\right)\right)$ , which takes on initially large values (approximately Sσ) and approaches unity near the minimum error output variance achievable.

Fig. 5 describes the LMA applicable to real data (for brevity), however the LMA in tins work is performed with complex impedance data. Because of this, both the measured and modeled data are handled as real and imaginary parts; the Jacobian, J, and weighting matrix, W, matrices are generated separately. This results in the introduction of new variables, $H_r=J{r}^{T}WrJr$ and $H_i=J{i}^{T}WiJi$ with (10) above now expressed as

$$\left[H_r+H_i+\lambda diag(H_r+H_i)\right]h_{LM}=J{r}^{T}Wr(yr-\widehat{yr})+J{i}^{T}Wi(yi-\widehat{yi})$$ (11)

where yr and yi, and $\widehat{yr}$ and $\widehat{yi}$, are real and imaginary lab data, and real and imaginary model points respectively. The LMA begins with the damping factor λ as a relatively large number and is progressively diminished as the convergence proceeds. Each LMA loop calculates an inversion of the (n x n) matrix, or HLM, in the brackets on the left side of (11) to find the (n x l) element array, h_LM, which is used to update the parameters.

The real and imaginary parameter variance-covariance matrices are H_r⁻¹, H_i⁻¹ respectively. The square root of the sum of their diagonals produce the standard error array for the n number of output parameter values

$$SE=real\left[\sqrt{diag\left(inv\left(H_r\right)\right)+diag\left(inv\left(H_i\right)\right)}\right].$$ (12)

If (e.g.) the number of degrees of freedom (i.e. m − n + 1) is greater than 100 for a two-tailed t-distribution, the expected range for each of the n parameters considered for a statistical significance level of 0.05 is 1.645 x (SE_n). [53]

3. Results

3.1. Application of the closed form expression for kEFF

The pre-heating and post-heating impedance spectra were measured with an Agilent 4396B in the range of 0.1 to 8 MHz. The resonance (f_r ≈ f_m) and antiresonance f_a ≈ f_n frequencies were found in each case. These spectral locations together with (5), (9), and (8) have produced the estimates for vXD and k_EFF, shown in Table III.

Table III.

Closed Form Calculations of vXD and k_EFF

Case	Fr	Fa	vXD	Δ vXD	kEFF	Δ kEFF
	(MHz)	(MHz)	(m/s)	%		%
Pre-Heating	3.640	6.228	3680	0	0.795	0
Post-Heating	3.697	6.085	3589	−2.47	0.756	−4.91

3.2. Narrow band, single parameter LMA

The primary purpose for the use of the LMA in this work is to aid the validation of the closed form calculation of the effective EMCC, as k_EFF. for a completed array with two matching layers. Since the k_EFF result using (8) depends greatly on the spectral location of f_r [33] we will focus the impedance spectrum regression around f_r. The predicate assumption for the proper use of (8) assumes as well that we have a) known matching layer characteristics, b) a low enough backing material impedance, c) an air front loading, and d) the true transducer material velocity (9) which is a function of f_a. In contrast, the impedance model for the LMA is constructed directly from the full KLM model itself with all terms and all losses. The LMA model includes, for example, the actual ZB value rather than a zero valued backing assumed for the k_EFF derivation.

With a relatively narrow spectral band application of the LMA around the resonance frequency (2.944 to 4.0105 MHz comprising 28 spectral points at 0.0395 MHz spacing), and with a single parameter (k_EFF) we can assure a well-conditioned recursion problem without over-parameterization. Good conditioning is promoted if the inverted matrix HLM has full rank and a low condition number. With only a single parameter in this application of the LMA, the inversion matrix is a single value and thus well-conditioned by definition.

The narrow band LMA routine was applied 17 times (loops in the LMA routine) for each of the pre-heating and post-heating cases to determine the “input capture range” of the LMA given a particular seeded EMCC input between 0.6 and 1.0 (Fig. 6). Each run required between 16 and 22 seconds with MATLAB running on a 1.8 GHz i7 CPU with 16 GB of memory.

Fig. 6.

The LMA derived EMCC for both pre- and post-heating given wide ranging EMCC seed value inputs. The two diamond points are the directly calculated prediction of EMCC (i.e. k_EFF) for each case; the fine dash curved lines are the standard error ranges for p = 0.05.

The dashed lines in Fig. 6 show the p = 0.05 statistical confidence boundaries for the LMA predictive ability, given a wide range of EMCC input seed values. Using this narrowband, single parameter LMA as the “gold standard,” Table IV shows differential predictions for k_EFF (8) as compared to k_t (1) and k_eff (2). The calculated k_EFF compares well with LMA derived values, and the standard equations show their (expected) variance with the LMA values.

Table IV.

Differential EMCC Comparisons using LMA as Gold Standard

Case	kEFF Calculated Eq (8)	EMCC from LMA	Δ %	kt Eq (1)	Δ %	keff Eq (2)	Δ %
	A	B	C = A/B -1	D	E = D/B-1	F	G = F/B-1
Pre-heating	0.795	0.793	0.25%	0.838	5.67%	0.811	2.27%
Post-heating	0.756	0.755	0.13%	0.822	8.87%	0.794	5.17%

Table V shows the LMA precision in EMCC prediction and the statistically significant LMA capture ranges for both pre- and post-heating cases. The calculated EMCC (i.e. k_EFF) change from heating of −4.91% in Table III compares favorably with the LMA prediction of −4.79% in Table V.

Table V.

EMCC Estimation by LMA Only Using the Narrow Band, Single Parameter

Case	EMCC	EMCC Δ %	EMCC input capture limit (min.)	EMCC input capture limit (max.)	Std Dev within the capture range	Std Dev as fraction of EMCC
Pre-heating	0.793	− 4.79 %	0.650	0.850	0.001927	0.24 %
Post-heating	0.755		0.600	0.825	0.001981	0.26 %

3.3. Wide band, multiple parameter LMA

A secondary purpose of the LMA in this work is to introduce its potential use for finding model parameters which are otherwise imprecisely assumed. The principal benefit to this endeavor is the generation of error statistics which can assist in the relative confidence in each of the estimated parameter outputs. Great care was taken to limit the LMA noise floor scaling factor, Sσ, which determines the targeted minimum mean square error within the LMA routine.

A very challenging 8 parameters were used in the wide band application of the LMA; the intent here is to demonstrate the LMA potential for future use. HLM matrix conditioning was improved by scaling the spectrum from MHz to Hz, which means that the effective transducer area was scaled by 1E+6 and the effective transducer velocity was scaled by 1E−6. The matrix condition number however was quite large; both the pre- and post-heating HLM condition numbers were above 1E+13.

The results of the wide band application of the LMA with 8 parameters are shown in Table VI for the two cases created from heating the array. The 4.7% drop in the EMCC post-heating in Table VI agrees well with the −4.91% closed form expression (9) prediction in Table III, and the narrow band prediction of −4.79% in Table V. Our calculated prediction of the change in the clamped dielectric constant (i.e. the change in the C0) agrees well with the LMA output at − 11%. The pre/post heating LMA impedance spectra results are shown in Fig. 7.

Table VI.

Parameter Estimation using the Wide Band, Multiple Parameter LMA

	Pre-Heating LMA Parameters (scaled)				Range p = 0.05		Post-Heating LMA Parameters (scaled)				Range p = 0.05		Post/Pre Output Δ
	Input		Output				Input		Output
param.	Re	Im	Re	Im	(%)		Re	Im	Re	Im	(%)		(%)
γXD	0.272	−6.5E-03	0.273	−6.2E-03	1.0%		0.279	−9.5E-03	0.279	−9.0E-03	1.7%		2.2
EMCC	0.795		0.790		0.5%		0.756		0.753		0.8%		−4.7
C0	5.56E-04	−4.0E-05	5.56E-04	−3.9E-05	2.6%		4.95E-04	−7.0E-05	4.95E-04	−6.9E-05	3.7%		−11.0
Zc	990	−23.4	990	−24.7	4.5%		965	−30.0	935	−63.7	6.5%		−5.6
ZM1	264	−5.65	249	−5.73	11.9%		264	−5.65	231	−13.10	21.9%
ZM2	114	−0.82	110	−2.24	9.2%		114	−0.82	102	−6.68	13.8%
γM1	0.504	−1.0E-02	0.489	−8.6E-03	7.1%		0.504	−1.0E-02	0.485	−3.7E-03	12.7%
γM2	0.376	−2.6E-03	0.387	−4.3E-03	6.3%		0.376	−2.6E-03	0.393	−9.1E-03	11.2%

i) parameters scaled: Axd · 1E + 6, vXD · 1E − 6

ii) λ start = 100, λ inc./dec. = 2, Sσ = 35

i) parameters scaled: Axd · 1E + 6, vXD · 1E − 6

ii) λ start = 100, λ inc./dec. = 3, Sσ = 25

Fig. 7.

The air-loaded magnitude and phase electrical impedance spectra of the lab measurements compared to the corresponding LMA least squares results, with (a) at pre-heating, and (b) at post-heating.

The comparison of statistical confidence (at p = 0.05) among all 8 parameters considered in the wide band LMA indicates a reasonably high level of confidence (i.e. ranges of 0.5% (pre), and 0.8%(post)) for EMCC. The transducer propagation constant, γXD, shows a 2.2% increase which indicates a vXD decrease by that much, but this only modestly agrees with our calculated (9) prediction of a −2.47% change in vXD. As well, the impedance, Zc, shows a net 5.6% decrease while a 2.47% decrease was expected. These discrepancies are likely a reflection of the statistical confidence for each parameter. E.g., the confidence ranges in the Zc predictions are approximately 8 times worse (i.e. 4.5%, and 6.5%) than that for the EMCC prediction.

4. Discussion

4.1. Closed form expression for k_EFF

The closed form expression for the fully assembled electrical transducer spectral impedance (6) is a close approximation to the KLM model result which uses all the terms in (3) and (4). This expression (6) was obtained firstly by ignoring all the loss terms (which were small for this design) and subsequently discarding any remaining terms in the impedance sum that did not affect the spectral impedance compared to the KLM model impedance across the band of interest, especially the band from resonance to antiresonance. To further simplify the spectral impedance expression (6) to obtain (7) two assumptions were used: a) ZB could be assumed to be zero if ZB < 0.25Zc, and b) the resonance frequency of (7) in particular is accurate enough if it agrees with the KLM model prediction within a spectral accuracy of (~ 1%). In contrast, as shown in Fig. 8, if the full KLM model impedance (4) is used with ZB exceeding 0.25Zc, the predicted k_EFF from (8) will produce significant errors as a result of resonance frequency shifts to lower frequencies with elevated backing impedance values. Thus, the use of (7) which produced the expression for k_EFF (8) is reasonable if ZB is small.

Fig. 8.

Apparent EMCC values using (8) with backing impedance variations for the subject array design with an expected EMCC of approximately 0.8. The solid line is a reference value due to the assumption of ZB = 0 and use of (7); the dashed curve uses the complete KLM model which shows an apparent rise (in error) of the EMCC with increasing backing impedance.

Regarding our two matching layer array design, the simplifying assumption of zero backing impedance used to obtain (7) appears to produce an EMCC estimation error on the order of 3.3% at a backing/composite impedance ratio of 25%. Our design however used an impedance ratio of 5.46/24.2, or 22.6% which suggests an EMCC estimation error < 3%. Generally, empirical array designs will typically use lower rather than higher backing impedance due to practical concerns. Problems with high backing impedance include: a) a greater difficulty in the creation of a material with both high absorption and high impedance backing, b) a higher level of acoustic energy injected into the higher Z backing, c) handling and molding in place the higher impedance material, and d) the potential for the backing to act as a heat reservoir because of (b).

The direct calculation of k_EFF pre- and post-heating compared well with the narrow band, single parameter LMA result. The LMA routine used the full KLM model produced by (4) including the actual value of the backing material impedance. As mentioned above, the major two assumptions used in the formulation of (7) and thus (8) are a) low loss, and b) ZB < 0.25 Zc. With advanced materials such as single crystal transducer material the low loss assumption is readily satisfied. Certainly, for array designs with higher losses, or with substantially higher backing impedance, the use of k_EFF can be inappropriate. However, since modern single crystal high performance materials offer low loss, and as well many designs use relatively low backing impedance, the methods outlined here are potentially very useful. Additional work regarding the effects of loss and high backing impedance on the accuracy of k_EFF is left as a future refinement.

Good modeling of array designs has always depended upon a-priori lab measurements of materials and guidance from the literature. However, the modeling quality can now be further refined through the direct observation of three spectral points from lab measurement: a) low frequency Im(Z), b) resonance frequency of |Z| (i.e. f_m), c) anti-resonance frequency of |Z| (i.e. f_n), together with known material characteristics for the computation of the improved estimates for both velocity (9) and k_EFF (8).

The aspirational goal in this work is to produce an improved means for the estimation of critical parameters of a finished array, especially the EMCC. Current reporting practice regarding a finished array might include, e.g., measurement results of insertion loss and bandwidth. However, these metrics are generally only useful in a relative sense because they are very dependent upon the entire test setup which includes both transmit signal generation, cabling, echo target used, amplification, loop losses, etc. Furthermore, a traditional full KLM modeling exercise might permit estimations of EMCC by manually comparing simulation against lab data, but this is a potentially laborious method and can yield errors due to overlapping parameter effects on the spectra in comparison. This overlapping parameter effect problem can be greatly reduced, in substantial part, by using the estimation of k_EFF (8) and a single laboratory spectral data point to produce a very good estimate of the clamped dielectric constant, and thereby the capacitance of the array (this approach is described in section 2.3). Additionally, the new closed form prediction of array velocity (9) helps to reduce error in the estimation of the transducer propagation constant, γXD.

With respect to the universal applicability of the closed form solution for EMCC (8), in silico simulations have been examined for both the a) no-matching layer design case (which Section B in the Supplement shows how the new EMCC closed form solution (8) becomes k_t (1) with M1 and M2 absent), and b) single matching layer design cases. Good predictive results were obtained with the use of the simplified closed form expressions. Based on these abbreviated initial studies there are indications that (8) is a useful predictor of EMCC for low loss designs using modest backing impedances, however further design testing and bench comparisons are needed to fully explore the boundaries of this new approach for EMCC estimation.

4.2. Use of the LMA

The LMA is a potent tool but must be used carefully. It’s ability to accurately predict model parameters is dependent on at least: a) good lab data devoid of artifacts, with low parasitic measurement effects, b) a model with all appropriate parameters and terms to enable a good least squares match to the lab data, and c) parameter variables which provide specific and unique effects on the modeled result across the entire spectrum of points considered. With regards to this last requirement, if there are many parameters considered in the LMA with overlapping impedance modeling effects the accuracy of the LMA predicted parameters will be less trustworthy. Therefore, the range of the measured parameters at p = 0.05 significance in Table VI provides a guide to the quality of parameter estimation using the LMA. We note, e.g., ZM1 is the least trustworthy parameter prediction in the group of 8. However, the a-priori measured, true-value of ZM1 was used as a guide in the selection of the variance scalar, Sσ. Since the example case of the 8 parameter LMA (Section 3.3) is over-parameterized, large values for Sσ (i.e. small expected statistical variance) would lead to better MSE results (and thus better matches to lab spectra in Fig. 7) but also significantly altered final values of high variance parameters such as ZM1. This underscores the need for the prudent use of the LMA.

To arrive at the best LMA estimation of EMCC we employed the LMA with only the single parameter, and narrowed the spectral region of interest to a relatively narrow band around the resonance frequency. Although not pursued in this work for brevity, future work to find other parameter estimates might involve a multiple looping code structure using the LMA to sequentially find specifically identified parameters which prominently effect the impedance in a given spectral band under test.

5. Conclusions

A 1-3 composite array design at 3.75 MHz using PIN-PMN-PT single crystal material has been measured on the bench to demonstrate a method to improve the effective accuracy and economy of the 1D KLM model. A new closed form solution to determine the EMCC (k_EFF) of an assembled array with two matching layers was derived and tested against the standard closed form calculation for EMCC estimation. Utilizing the pre- and post-heating array impedance data, we report in Table IV the aggregate predicted EMCC values produced from the new closed form expressions exhibited approximately one tenth the error as compared to the traditional formula. A new method to determine the transducer velocity of an array with matching layers was also developed and tested, and is a critically important component in the new EMCC calculation method. As a means to evaluate the effectiveness of the model, the effects of heating on the properties of the composite transducer were determined. Only the core composite transducer material appeared to change, which showed reductions in EMCC of 4.9%, clamped dielectric constant of 11%, and effective transducer longitudinal velocity of 2.5%.