# Accounting for dielectric losses in piezoelectric shunt tuning

Jens D. Richardt, Boris Lossouarn, Jan Høgsberg, Jean-François Deü

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Jens D. Richardt, Boris Lossouarn, Jan Høgsberg, Jean-François Deü. Accounting for dielectric losses in piezoelectric shunt tuning. Journal of Sound and Vibration, 2026, 625, pp.119598. 10.1016/j.jsv.2025.119598. hal-05459939

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# Accounting for dielectric losses in piezoelectric shunt tuning

Jens D. Richardt ID a,*, Boris Lossouarn ID b, Jan Høgsberg ID a, Jean-François Deü ID b

## ARTICLE INFO

### Keywords:

- Vibration control

- Piezoelectric shunt damping

- Dielectric losses

- Shunt calibration

## ABSTRACT

Effective vibration reduction of mechanical structures by piezoelectric shunt damping requires a precise design of the electrical shunt circuit. In this context, the influence of dielectric losses on the accuracy of shunt calibration is analyzed. Numerical analysis of an SDOF structure shows that dielectric losses can result in a significantly increased vibration amplitude when not accounted for in the calibration of a resonant shunt. To address this, a new correction method based on impedance matching is proposed, enabling the use of existing tuning expressions not accounting for dielectric losses. It is demonstrated numerically and experimentally that the influence of dielectric losses is more pronounced for low electromechanical coupling factors. The proposed correction method is shown to provide an accurate correction for the influence of dielectric losses on the shunt tuning.

## 1. Introduction

Piezoelectric shunt damping has been extensively studied for the reduction of mechanical vibrations. Hagood and von Flotow [1] demonstrated that a simple electrical circuit consisting of a resistor shunted to a piezoelectric transducer (R-shunt) can be used for vibration mitigation. They also showed that the energy dissipation for a single vibration mode can be significantly enhanced by adding an inductor in series with a resistor (series RL-shunt) when the inductor is tuned to the targeted vibration mode. Since then, various electrical circuits have been proposed. For instance, Wu [2] introduced a resonant shunt with a resistor and inductor in parallel (parallel RL-shunt). At the same time, circuits targeting multiple vibration modes have been developed, using a single [3] or multiple piezoelectric transducers [4]. Different tuning criteria have also been proposed for shunt calibration. Explicit calibration expressions for resonant shunts have, for instance, been provided by Høgsberg and Krenk [5], Thomas et al. [6], Soltani et al. [7], and Ikegame et al. [8], based on optimality criteria, such as maximum modal damping or minimum response amplitude.

Some shunt circuits are sensitive to mistuning. For example, Andreaus and Porfiri [9] show that the performance of a series RL-shunt is highly sensitive to mistuning of the inductor. Additionally, Lossouarn et al. [10] showed that resonant shunts may require custom made inductors to obtain a large inductance with a small series resistance. An accurate initial calibration of the electrical components is therefore beneficial for experimental implementation and practical engineering applications with piezoelectric shunts. Most of the existing shunt calibration methods are based on a single-mode truncation of a conservative mechanical structure, neglecting the intrinsic losses of the electromechanical system. The influence of non-targeted (or residual) modes on the calibration of piezoelectric shunts attached to a flexible structure has been demonstrated by Gardonio and Casagrande [11] for an increasing

<footnote>a Department of Civil and Mechanical Engineering, Technical University of Denmark, Koppels Alle Building 404, Kongens Lyngby, 2800, Denmark</footnote>

<footnote>b Laboratoire de Mécanique des Structures et des Systèmes Couplés, Conservatoire national des arts et métiers (Cnam), 292 Rue Saint-Martin, Paris, 75003, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: jdari@dtu.dk (J.D. Richardt), boris(lossouarn@lecnam.net (B. Lossouarn), jahog@dtu.dk (J. Høgsberg), jean-francois.deu@lecnam.net (J.-F. Deü).</footnote>
number of vibration modes. Høgsberg and Krenk [12] introduced a quasi-static correction to account for the influence of residual modes in calibration of a single shunt targeting a single vibration mode, while Berardengo et al. [13] proposed a similar correction by a modal capacitance. Furthermore, Raze et al. [14] included the influence of lower and higher mechanical modes in the calibration of piezoelectric shunts targeting multiple vibration modes.

The losses of piezoelectric materials can be modeled in different ways. The data sheets of piezoelectric transducers typically provide the mechanical quality factor $Q'$ describing the sharpness of the fundamental resonance peak with the electrodes of the piezoelectric transducer in the open circuit (OC) condition together with a dielectric loss factor $tan(\delta')$ describing the ratio of resistive to reactive power of the piezoelectric material at constant stress, usually measured at 1 kHz [15] on a number of test specimens. The losses of piezoelectric materials are often represented by introduction of complex coefficients in the constitutive equations of the material [16–19]. The complex coefficients introduce a phase lag between strain, stress, and the electrical fields, thereby introducing mechanical, dielectric and piezoelectric loss terms, which provide a valid approximation for small losses within a narrow frequency band [20]. The piezoelectric loss term is often neglected, although, as described by Wild et al. [20], it can have a significant influence on the total power dissipation. Furthermore, Wild et al. [20] describe that the contribution from each loss term varies with the form of the constitutive equations, whereby the three loss terms represent a combination of individual dissipation mechanisms.

Few studies have examined the effects of losses when a piezoelectric material is bonded to a mechanical structure and used for piezoelectric shunt damping. Structural damping has been incorporated in calibration expressions based on a pole placement criterion for series RL-shunts, parallel RL-shunts, and RLC shunts by Caruso [21], while Soltani et al. [22] include structural damping in an approximate closed-form solution for $H_\infty$ optimization of a series RL-shunt. Similarly, Ikegame et al. [8] provide calibration expressions that account for structural damping for parallel and series RL-shunts considering different calibration objectives. Finally, Toftekær and Høgsberg [23] propose a calibration for series- and parallel RL-shunts that accounts for structural damping and structural loading in the tuning of the inductor, utilizing that the shunt performance is insensitive to small changes of the resistor. To the authors' knowledge, dielectric losses in the piezoelectric transducer have only been considered for piezoelectric shunt damping by Yamada et al. [24], where a correction for dielectric losses is included in the calibration of the resistance for series and parallel RL-shunts. Furthermore, the influence of dielectric losses has been found by Liang et al. [25] to have an effect in energy harvesting applications using piezoelectric transducers. This influence has been further studied numerically and experimentally by Morel et al. [26].

Piezoelectric shunt implementation typically employs the constitutive equations on the e-form, where strain and electric field are the independent variables with a mix of intensive and extensive coefficients. In Raze et al. [4], the required parameters for shunt calibration, with intrinsic losses included as structural damping, are determined by injecting a current in the piezoelectric transducer and measuring the voltage. The parameters are then determined using the PolyMax identification method [27]. A similar method is applied by Dietrich [28], while a simpler identification method is introduced by Richardt et al. [29] where two points on the measured dynamic capacitance curve are used for the parameter identification. The above mentioned references do not include piezoelectric and dielectric losses. However, models with complex coefficients in the constitutive equations on the e-form including piezoelectric and dielectric losses have been applied to piezoelectric materials based on impedance measurements and finite element (FE) simulations [30,31].

This paper provides a critical analysis of the influence of dielectric loss on the calibration of piezoelectric shunts and introduces a generalized correction for dielectric losses for a single piezoelectric shunt targeting a single vibration mode. To enable experimental validation, a characterization procedure for mechanical, piezoelectric, and dielectric losses is proposed using experimental modal analysis with impedance measurements performed on a piezoelectric transducer bonded to the mechanical structure, including losses by complex coefficients in the constitutive equations. The proposed method is then demonstrated experimentally through the calibration of a series RL-shunt attached to a composite blade.

Section 2 presents the governing equations for a piezoelectric shunt attached to a single-degree-of-freedom (SDOF) structure, used for numerical analysis of the influence of dielectric losses on shunt calibration. A correction for the influence of dielectric losses is presented in Section 3, also demonstrating its accuracy by numerical analysis. Section 4 provides an experimental example involving vibration mitigation of a composite blade for varying values of the effective electromechanical coupling factor. Final conclusions and perspectives are presented in Section 5.

## 2. Governing equations

The electromechanical system shown in Fig. 1 is considered in the following sections to describe the influence of dielectric loss on the calibration of a piezoelectric shunt targeting a single vibration mode.

### 2.1. Piezoelectric transducer

To establish a connection with the experimental analysis in Section 4, the various losses are initially introduced for a linear piezoelectric transducer based on the 31-effect, where axial strain $S_1$ is related to a transverse electric field $E_3$. The constitutive equations for the piezoelectric material are obtained from the IEEE Standard on Piezoelectricity [32] with the independent variables $S_1$ and $E_3$ as

$$T_1 = c_{11}^E S_1 - e_{31} E_3 \qquad (1)$$

$$D_3 = e_{31} S_1 + \epsilon_{33}^S E_3 \qquad (2)$$
<figure><img src="image_4.png" /><figcaption>Fig. 1. SDOF structure with a single piezoelectric transducer connected to an electrical circuit with impedance Z(ω).</figcaption></figure>

where $D_3$ is the transverse electric displacement, and $T_1$ is the axial stress. In the two equations, $c_{11}^E$ is the elastic stiffness at constant electric field, $e_{31}$ is the piezoelectric constant, and $\epsilon_{33}^S$ denotes the dielectric permittivity at constant strain.

For low losses and a narrow frequency band, the losses in the piezoelectric material can be modeled by complex coefficients in the constitutive equations, which introduces a phase lag between the mechanical and electrical components [17,20]. For the constitutive equations in Eqs. (1)–(2), this introduces the complex parameters

$$ \tilde{c}_{11}^E = c_{11}^E(1 + i\eta_c) \qquad (3) $$

$$ \tilde{e}_{31} = e_{31}(1 + i\eta_e) \qquad (4) $$

$$ \tilde{\epsilon}_{33}^S = \epsilon_{33}^S(1 - i\eta_e) \qquad (5) $$

where $i = \sqrt{-1}$ and $\tilde{\gamma}$ denotes the complex version of the constant $\gamma$. The loss factors ($\eta$) are referred to as the mechanical loss $\eta_c$, the piezoelectric loss $\eta_e$, and the dielectric loss $\eta_\epsilon$, with reference to their associated constants. The sign of the complex coefficients $\eta_c$ and $\eta_\epsilon$ are chosen such that positive coefficients ensure energy dissipation for zero electric field ($E_3 = 0$) or zero strain ($S_1 = 0$), respectively. Similarly, the sign of the piezoelectric loss coefficient $\eta_e$ is chosen to ensure energy dissipation when the transducer is in the OC condition ($D_3 = 0$). However, as shown by Wild et al. [20], while the total power dissipation must be positive, the contribution from the piezoelectric loss term depends on the phase lag between $E_3$ and $S_1$, whereby a positive total power dissipation can potentially be realized by a negative $\eta_\epsilon$.

Although the piezoelectric transducers in the subsequent experiments have a more complex structural attachment, the loss factors' influence on shunt tuning is initially investigated for the idealized configuration in Fig. 1, where a single transducer acts directly on an SDOF structure with displacement variable $x$. Upon integration across the piezoelectric material, replacing the constants $c_{11}^E$, $e_{31}$, and $\epsilon_{33}^S$ in Eqs. (1)–(2) with their corresponding complex coefficients in Eqs. (3)–(5) leads to the global constitutive equations for the piezoelectric transducer (see, for instance, [33] or [34]). For a piezoelectric transducer with the length $l_p$, width $w_p$ and thickness $t_p$, the force $f_p$ acting on the concentrated mass $m$ in Fig. 1 and the charge $Q$ at one of the electrodes are given by the expressions

$$ f_p = k_p(1 + i\eta_c)x - \theta(1 + i\eta_e)V \qquad (6) $$

$$ Q = \theta(1 + i\eta_e)x + C^\infty(1 - i\eta_\epsilon)V \qquad (7) $$

where $V$ is the voltage across the electrodes, $k_p = c_{11}^E t_p w_p / l_p$ is the mechanical stiffness with short circuit (SC) electrodes, $\theta = w_p e_{31}$ is a piezoelectric coefficient, and $C^\infty = w_p l_p / t_p \epsilon_{33}^S$ is the blocked capacitance.

## 2.2. Dissipative electromechanical system

For assumed harmonic motion with angular frequency $\omega$, the equation of motion for the SDOF structure in Fig. 1 can be written as

$$ (-\omega^2 m + i\omega c + k)x = -f_p + f \qquad (8) $$

introducing the structural mass $m$, viscous damping coefficient $c$, mechanical stiffness $k$, and the external load $f$. When inserting the expression for the piezoelectric transducer force $f_p$ from Eq. (6), the equation of motion in Eq. (8) is given as

$$ (-\omega^2 m_s + i\omega c_s + k_s)x - \theta(1 + i\eta_e)V = f \qquad (9) $$

in which $k_s = k + k_p$ accounts for the mechanical stiffness $k_p$ from the piezoelectric transducer with SC electrodes, while $m_s = m + m_p$ includes any added mass $m_p$. As the dissipated energy from mechanical losses in the main structure is expected to be significantly larger than the dissipated energy from mechanical losses in the piezoelectric transducer, the total mechanical loss is conveniently represented by a common equivalent viscous damping coefficient $c_s = c + \eta_c k_p / \omega_s$, where $\eta_c k_p / \omega_s$ is the equivalent viscous damping for the mechanical hysteresis loss in Eqs. (3) and (6). The natural frequency $\omega_s$ with the piezoelectric transducer in the SC condition,
the damping ratio $\zeta_s$ for mechanical losses, and the squared modal coefficient $\chi_s^2$ are then introduced as

$$ \omega_s = \sqrt{\frac{k_s}{m_s}}, \quad \zeta_s = \frac{c_s}{2\sqrt{k_s m_s}}, \quad \chi_s^2 = \frac{\theta_s^2}{k_s} \qquad (10) $$

whereby, the governing equations of the electromechanical system in Fig. 1 can be obtained from Eqs. (7), (9), and (10) as

$$ (-\omega^2 + 2i\omega\zeta_s\omega_s + \omega_s^2)x - \frac{\omega_s^2\chi_s}{\sqrt{k_s}}(1 + i\eta_e)V = \frac{\omega_s^2}{k_s}f \qquad (11) $$

$$ \sqrt{k_s \chi_s (1 + i \eta_e) x + C^\infty (1 - i \eta_e) V} = Q \qquad (12) $$

Shunting an electrical circuit with the impedance $Z = Z(\omega)$ to the piezoelectric transducer, as illustrated in Fig. 1, introduces the relation

$$ V = -ZI = -i\omega ZQ \qquad (13) $$

between the current $I = i\omega Q$ at one of the terminals of the piezoelectric transducer to the voltage $V$ across the two terminals. By inserting Eq. (13) and combining the two governing Eqs. (11) and (12), the complex frequency response function (FRF) of the SDOF structure is obtained as

$$ \frac{xk_s}{f} = \frac{\omega_s^2[i\omega Z C^\infty(1 - i\eta_e) + 1]}{(-\omega^2 + 2i\omega\zeta_s\omega_s + \omega_s^2)[i\omega Z C^\infty(1 - i\eta_e) + 1] + i\omega Z\omega_s^2\chi_s^2(1 + i\eta_e)^2} \qquad (14) $$

The associated dynamic amplification factor (DAF) can then be obtained as the magnitude of the FRF, i.e. $|xk_s/f|$. The losses of the electromechanical system are characterized by the losses introduced by the electrical shunt impedance $Z$, as well as the intrinsic losses in the host structure and piezoelectric transducer represented by the damping ratio $\zeta_s$, the piezoelectric loss factor $\eta_e$, and the dielectric loss factor $\eta_e$.

## 2.3. Effective electromechanical coupling factor

The squared modal (or generalized) electromechanical coupling factor (MEMCF) can be defined as

$$ K_s^2 = \frac{\chi_s^2}{C^\infty} \qquad (15) $$

and is commonly used in explicit calibration expressions for piezoelectric shunts attached to SDOF structures [1,5,6]. The MEMCF is often obtained experimentally in terms of an effective electromechanical coupling factor (EEMCF) relating the natural frequency $\omega_s$ with open circuit (OC) electrodes of the piezoelectric transducer to the natural frequency $\omega_s$ with SC electrodes. The squared EEMCF $\kappa_s^2$ is defined as

$$ \kappa_s^2 = \frac{\frac{\omega_s^2 - \omega_s^2}{\omega_s^2}}{\omega_s^2} \qquad (16) $$

which corresponds to the squared MEMCF for a piezoelectric transducer attached to an SDOF structure [35]. Toftekær et al. [36] demonstrated that the MEMCF and EEMCF are not identical for flexible structures with multiple vibration modes. In fact, in [29], the influence of the non-targeted (residual) modes is accounted for in the shunt calibration by replacing the MEMCF $K_s$ with the EEMCF $\kappa_s$, and substituting the blocked capacitance $C^\infty$ with a blocked modal capacitance $C_s^\infty$. Similar to the relation in Eq. (15), which defines the squared MEMCF $K_s^2$ from the blocked capacitance $C^\infty$, the squared EEMCF $\kappa_s^2$ can be obtained from the blocked modal capacitance $C_s^\infty$ as

$$ \kappa_s^2 = \frac{\chi_s^2}{C_s^\infty} \qquad (17) $$

The use of the two parameters, $\kappa_s$ and $C_s^\infty$, in shunt calibration corrects for the influence of residual vibration modes. Hereby, the derivations and findings for an SDOF structure in the present paper, can be directly transferred to more realistic flexible structures with multiple vibration modes.

## 2.4. Influence of loss terms

The qualitative effect of neglecting the three losses $\zeta_s$, $\eta_e$, and $\eta_e$ in the shunt tuning is initially illustrated for the series RL-shunt with the impedance

$$ Z = R + i\omega L \qquad (18) $$

using the calibration from an $H_\infty$ optimization for minimum displacement given in [8] and derived in [7]. This calibration is chosen as it results in two peaks of equal amplitude in the structure's frequency response, which allows for a direct evaluation of potential detuning from the intended shunt calibration. The calibration expressions for inductance $L$ and resistance $R$ are summarized in Appendix A.
<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Frequency response of an SDOF structure with a series RL-shunt for increasing losses (gray curves, with lighter shades corresponding to increased losses) and the ideal undamped baseline (red curve). The introduced losses are structural damping $ζ_s = (0 : 0.002 : 0.01)$ in (a), piezoelectric loss $η_e = (0 : 0.01 : 0.05)$ in (b), and dielectric loss $η_d = (0 : 0.01 : 0.05)$ in (c). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Three shunt circuits: R (a), parallel RL (b), and series RL (c).</figcaption></figure>

Fig. 2 shows the ideal DAF (red solid curve) for the optimally tuned shunt attached to an undamped structure with the modal parameters $ζ_s = η_e = η_ℓ = 0$, $ω_s = 100$ rad/s, $κ_s = 0.05$, and $C^∞ = 100$ nF. The figure also shows the individual response for five levels of increasing structural damping $ζ_s$ (a), piezoelectric losses $η_e$ (b), and dielectric losses $η_d$ (c) (gray curves, with lighter shades corresponding to increased losses). The range of realistic loss levels is provided in the figure caption. In accordance to [22,23], increased structural damping $ζ_s$ in (a) is seen to reduce the amplitude of the SDOF mass, while the magnitude of the two peaks becomes slightly skewed, indicating a slight deviation from the optimal inductance. Increased piezoelectric loss $η_e$ in (b) is also seen to introduce a deviation from the optimal inductance as the magnitude of the left peak increases while the magnitude of the right peak decreases. Finally, increased dielectric loss $η_d$ in (c) is seen to transform the two distinct peaks in the ideal response (red solid curve) into a single peak, indicating a significant deviation from the optimal tuning of the resistor. Besides the significant reduction in vibration mitigation performance, the single peak in the frequency response curve also complicates experimental fine-tuning of the shunt, as it is not directly visible whether and how the inductance should be changed to recover the desired equal (double) peak property. In the present example, dielectric losses are thus identified as the most influential damping effect for the calibration of a resonant shunt.

### 3. Dielectric losses in shunt calibration

The structural damping $ζ_s$, piezoelectric loss $η_e$, and dielectric loss $η_d$ terms in Eqs. (11) and (12) all influence the calibration accuracy for piezoelectric shunts, as illustrated for a series RL-shunt in Section 2.4. In the example, the dielectric loss $η_d$ is identified as the main source to shunt mistuning for realistic losses. While the other loss types may be dominating in other examples, the present paper focuses entirely on the influence from dielectric losses $η_e$ on the shunt calibration, thus deliberately neglecting the two other sources by assuming $ζ_s = η_e = 0$. In the following, a correction for the influence of dielectric losses is introduced based on impedance matching between an electromechanical system with and without dielectric losses to enable the use of calibration expressions derived for undamped systems. The method is then analyzed and validated for the three fundamental shunt circuits in Fig. 3.

#### 3.1. Impedance matching

The governing equations for a piezoelectric shunt attached to an SDOF structure are given in Eqs. (11) and (12), with the impedance $Z = Z(ω)$ of the shunt circuit included by Eq. (13). The equivalent electrical circuit to the electromechanical system is shown in Fig. 4(a) in the absence of external loading ($f = 0$), structural damping ($ζ_s = 0$), piezoelectric losses ($η_e = 0$), and dielectric losses ($η_d = 0$). The optimal impedance of the shunt circuit attached to the undamped electromechanical system is defined as $Z(ω) = \tilde{Z}(ω)$.
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 4. Equivalent electrical circuit for a piezoelectric shunt attached to an SDOF structure with no losses (a) and including dielectric losses (b).</figcaption></figure>

Including dielectric losses in the electromechanical system ($\eta_e > 0$) by the complex capacitance $C^\infty(1 - i\eta_e)$ corresponds to adding a frequency-dependent resistance $1/(\omega\eta_e C^\infty)$ in parallel to the blocked capacitance $C^\infty$ in the equivalent circuit of the electromechanical system, shown in Fig. 4(b). Comparison of the two circuits in Fig. 4 shows that the electromechanical systems, with and without dielectric losses, are identical if the admittance of the two highlighted parts of the circuits are identical, corresponding to

$$ \frac{1}{\bar{Z}(\omega)} = \omega\eta_e C^\infty + \frac{1}{Z(\omega)} \qquad (19) $$

The present paper is limited to shunts targeting a single mode, whereby the relation in Eq. (19) must only be accurate in a sufficiently narrow frequency-band around the structure's natural frequency ($\omega \approx \omega_s$). This suggests that Eq. (19) should be valid at $\omega$ equal to the SC natural frequency $\omega_s$,

$$ \frac{1}{\bar{Z}(\omega_s)} = \omega_s \eta_e C^\infty + \frac{1}{Z(\omega_s)} \qquad (20) $$

which in the following is used to account for dielectric losses in piezoelectric shunt tuning. First, an ideal shunt tuning is performed without dielectric losses (for $\eta_e = 0$), determining the components in $\bar{Z}(\omega)$ at $\omega = \omega_s$, whereafter, the corrected tuning parameters in the actual impedance $Z(\omega)$ are determined by the impedance matching relation in Eq. (20).

## 3.2. R-shunt

The correction for dielectric losses by impedance matching, introduced in the previous Section 3.1, is implemented for an R-shunt when the piezoelectric transducer is attached to the SDOF structure in Fig. 1. The R-shunt in Fig. 3(a) consists of a single resistor $R$, whereby the impedance is

$$ Z(\omega) = R \qquad (21) $$

Application of the correction for dielectric losses in Eq. (20) leads to the updated resistance

$$ R = \frac{\bar{R}}{1 - \omega_s \eta_e C^\infty \bar{R}} \qquad (22) $$

as a function of the optimal resistance $\bar{R}$ calibrated for an ideal SDOF structure without including dielectric losses.

By introduction of the non-dimensional resistance $\rho = \omega_s C^\infty R$ and $\bar{\rho} = \omega_s C^\infty \bar{R}$, the ratio between the ideal and corrected non-dimensional resistance is found as

$$ \frac{\rho}{\bar{\rho}} = \frac{1}{1 - \eta_e \bar{\rho}} \qquad (23) $$

The dielectric loss and the non-dimensional ideal resistance are both positive, i.e. $\eta_e \ge 0$ and $\bar{\rho} \ge 0$. From Eq. (23) it is then seen that the existence of dielectric losses ($\eta_e > 0$) increases the optimal resistance of an R-shunt compared to the resistance in an ideal lossless system. The non-dimensional resistance $\bar{\rho}$ typically only depends on the EEMCF $\kappa_s$, whereby the correction for dielectric losses in Eq. (23) only depends on $\kappa_s$ and $\eta_e$. It is seen from Eq. (23), that a negative non-dimensional resistance $\rho$ is required when $\eta_e \bar{\rho} > 1$. Therefore, an R-shunt with pure passive components requires $\eta_e \bar{\rho} < 1$. In the calibration expressions for R-shunts given by Thomas et al. in [6], a first-order Taylor series expansion around $\kappa_s = 0$ gives the non-dimensional resistance as $\bar{\rho} = 1 + \mathcal{O}(\kappa_s^2)$. With the approximation $\bar{\rho} \approx 1$, the requirement for an R-shunt with passive components is $\eta_e < 1$, which is satisfied for realistic dielectric losses.

The effect of the correction in Eq. (23) is illustrated in Fig. 5 for the calibration of an R-shunt attached to an SDOF structure with the modal parameters $\zeta_s = \eta_e = 0$, $\omega_s = 100$ rad/s, and $C^\infty = 100$ nF. The literature on piezoelectric shunt damping typically considers EEMCFs $\kappa_s$ in the range from 1 to 25%. At the same time, Morel et al. [26] summarize different losses of piezoelectric materials characterized by values of tan($\delta'$) ranging from approximately 0.2% for hard PZT up to approximately 2.5% for soft PZT. The example in Fig. 5 therefore considers EEMCFs and dielectric losses in the range $\kappa_s \in [0.01, 0.25]$ and $\eta_e \in [0, 0.05]$. The R-shunt is calibrated using the calibration expressions given by Thomas et al. [6] for minimum displacement, summarized in Appendix A. The calibration is performed without correction for dielectric losses (assuming $\eta_e = 0$) in Fig. 5(a,c) and with correction for dielectric losses in Fig. 5(b,d).
<figure><img src="image_9.png" /><figcaption>Fig. 5. SDOF structure with an R-shunt tuned without (a,c) and with (b,d) correction for dielectric loss ηε. The increase of maximum amplitude is shown in percent for κs ∈ [0.01, 0.25] and ηε ∈ [0, 0.05] in (a,b) relative to an ideal undamped baseline system with ηε = 0. Examples are shown for the frequency response for κs = 0.05 with increasing ηε (gray curves, with lighter shades corresponding to increased losses) and the undamped baseline (red curve) in (c,d). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

losses ηε by Eq. (23) in Fig. 5(b,d). In (a,b) the percentage-wise amplitude increase for the maximum displacement is shown with respect to the ideal baseline case with ηε = 0. A maximum amplitude increase of 0.12% for the uncorrected case in (a), directly shows that the influence of dielectric losses is insignificant for the maximum response amplitude with an R-shunt. However, the correction in Eq. (23) is still seen to reduce the amplitude increase to less than 10⁻⁴% in (b).

To evaluate the qualitative influence of ηε on the frequency response, Fig. 5(c,d) show the dynamic amplification factor (DAF) of the SDOF structure with κs = 0.05 in the ideal baseline case where ηε = 0 (red solid curve) and for increasing dielectric loss (gray curves, with lighter shades corresponding to increased losses). The magnitude of ηε is illustrated by circles in Fig. 5(a,b). Similar to the influence on the maximum amplitude in (a,b), increasing dielectric loss only has a small effect on the frequency response curves in the uncorrected case (c) compared to the corrected case (d). While the correction for dielectric losses in Eq. (23) is found to be accurate, the influence of dielectric losses is therefore considered as negligible for practical implementations of the R-shunt.

### 3.3. Parallel RL-shunt

The correction for dielectric losses in Section 3.1 is now considered for the parallel RL-shunt with the inductance L and resistance R shown in Fig. 3(b). The admittance of the parallel RL-shunt is

$$ \frac{1}{Z} = \frac{1}{i\omega L} + \frac{1}{R} \qquad (24) $$

Applying the correction for dielectric losses by Eq. (20) leads to the relation

$$ \frac{1}{i\omega_s \bar{L}} + \frac{1}{\bar{R}} = \omega_s \eta_e C^\infty + \frac{1}{i\omega_s L} + \frac{1}{R} \qquad (25) $$

between the corrected parameters L and R, and the parameters L̄ and R̄ calibrated for an ideal SDOF structure without including dielectric losses.

Separation of the real and imaginary parts in Eq. (25), and introduction of the non-dimensional coefficients λ = ωs²C∞L, ρ = ωsC∞R, λ̄ = ωs²C∞L̄, and ρ̄ = ωsC∞R̄, give the following ratios between the corrected and ideal parameters,

$$ \frac{\lambda}{\lambda} = 1 \qquad (26) $$

$$ \frac{\rho}{\rho} = \frac{1}{1 - \eta_e \rho} \qquad (27) $$

It is seen from Eq. (26) that the parallel inductance L is unaffected by dielectric losses ηε, while the correction in Eq. (27) for the resistance R is identical to the correction in Eq. (23) for the R-shunt. Similar to the R-shunt, increased dielectric loss ηε, increases the
<figure><img src="image_10.png" /><figcaption>Fig. 6. SDOF structure with a parallel RL-shunt tuned without (a,c) and with (b,d) correction for dielectric loss $\eta_e$. The increase of maximum amplitude is shown in percent for $\kappa_s \in [0.01, 0.25]$ and $\eta_e \in [0, 0.05]$ in (a,b) relative to an ideal undamped baseline system with $\eta_e = 0$. Examples are shown for the frequency response for $\kappa_s = 0.05$ with increasing $\eta_e$ (gray curves, with lighter shades corresponding to increased losses) and the undamped baseline (red curve) in (c,d). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

optimal resistance R and passive implementation requires $\eta_e \bar{\rho} < 1$. Common calibration expressions for the non-dimensional parallel resistance $\bar{\rho}$ (e.g. in [2,5,21]) are inversely proportional to the EEMCF. From Eq. (27) it then follows that the detuning introduced by dielectric losses increases for parallel RL-shunts with low EEMCFs. For example, the calibration in [2] gives $\bar{\rho} = 1/(\sqrt{2}\kappa_s)$, leading to the condition $\kappa_s > \eta_e/\sqrt{2}$ for implementation with passive components. As the constant $\sqrt{2}$ varies with the used calibration method, an approximate, conservative, and easy to remember condition for the EEMCF may be $\kappa_s > \eta_e$.

The effect of the proposed correction for dielectric losses in the tuning of a parallel RL-shunt is evaluated in Fig. 6 by considering the same electromechanical system as for the R-shunt in Section 3.2 and Fig. 5 with varying values of $\kappa_s$ and $\eta_e$. The parallel RL-shunt is calibrated based on the tuning expressions from an $H_\infty$ optimization for minimum displacement given in [8], based on derivations in [37] and summarized in Appendix A. For the ideal system with no dielectric losses ($\eta_e = 0$), the calibration results in two peaks of equal amplitude in the frequency response curve (red solid curve in (c,d) with $\kappa_s = 0.05$). When not accounted for in the calibration, an increased dielectric loss $\eta_e$ is seen in (c) to lead to a single peak with a significantly increased amplitude (gray curves, with lighter shades corresponding to increased losses). The influence on the amplitude increase relative to the ideal system with $\eta_e = 0$ is seen in (a) to increase for low EEMCFs $\kappa_s$ and large dielectric losses $\eta_e$, leading to a maximum amplitude increase of more than 100% for the considered values of $\kappa_s$ and $\eta_e$. With the correction for dielectric loss in Eqs. (26)–(27), the response is seen in (d) to be nearly unaffected by an increasing dielectric loss $\eta_e$. Furthermore, it is found in (b) that the correction reduces the amplitude increase to less than 0.5%. Thereby, the proposed correction is shown to have a significant effect for the calibration of the resistance in a parallel RL-shunt.

## 3.4. Series RL-shunt

The correction of dielectric losses introduced in Section 3.1 is finally applied to the series RL-shunt in Fig. 3(c) with the inductance L and resistance R, defining the impedance

$$Z = i\omega L + R \qquad (28)$$

Inserting Eq. (28) into Eq. (20) gives the correction for dielectric losses by the impedance matching relation

$$\frac{1}{i\omega_s \bar{L} + \bar{R}} = \omega_s \eta_e C^\infty + \frac{1}{i\omega_s L + R} \qquad (29)$$

from which the inductance L and resistance R are determined by the corresponding parameters $\bar{L}$ and $\bar{R}$ for an ideal SDOF structure with no dielectric losses. Separation of the real and imaginary parts in Eq. (29), followed by the introduction of the non-dimensional
<figure><img src="image_11.png" /><figcaption>Fig. 7. SDOF structure with a series RL-shunt tuned without (a,d), with a simplified (b,e) and with a full (c,f) correction for dielectric loss $\eta_e$. The increase of maximum amplitude is shown in percent for $\kappa_s \in [0.01, 0.25]$ and $\eta_e \in [0, 0.05]$ in (a,b,c) relative to an ideal undamped baseline system with $\eta_e = 0$. Examples are shown for the frequency response for $\kappa_s = 0.05$ with increasing $\eta_e$ (gray curves, with lighter shades corresponding to increased losses) and the undamped baseline (red curve) in (d,e,f). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

coefficients $\lambda = \omega_s^2 C^\infty L$, $\rho = \omega_s C^\infty R$, $\bar{\lambda} = \omega_s^2 C^\infty \bar{L}$, and $\bar{\rho} = \omega_s C^\infty \bar{R}$ gives the ratio between the corrected and ideal calibration as

$$ \frac{\lambda}{\bar{\lambda}} = \frac{1}{\eta_e^2 \bar{\lambda}^2 + \eta_e^2 \bar{\rho}^2 - 2\eta_e \bar{\rho} + 1} \qquad (30) $$

$$ \frac{\rho}{\bar{\rho}} = \frac{1 - \frac{\eta_e}{\bar{\rho}}(\bar{\lambda}^2 + \bar{\rho}^2)}{\eta_e^2 \bar{\lambda}^2 + \eta_e^2 \bar{\rho}^2 - 2\eta_e \bar{\rho} + 1} \qquad (31) $$

The non-dimensional parameters $\bar{\lambda}$ and $\bar{\rho}$ typically only depend on the EEMCF $\kappa_s$ (see, e.g. [1,5]), whereby the correction for dielectric losses for the series RL-shunt only depends on the magnitude of the dielectric loss $\eta_e$ and the EEMCF $\kappa_s$. A Taylor series expansion around $\kappa_s = 0$ for the optimal non-dimensional calibration parameters in [5] and for minimum amplitude in [1], gives $\bar{\lambda} = 1 + O(\kappa_s^2)$ and $\bar{\rho} = \sqrt{2\kappa_s} + O(\kappa_s^3)$. Even though other calibration expressions can give different Taylor series expansions, the non-dimensional inductance is expected close to unity $\bar{\lambda} \approx 1$ and the non-dimensional resistance is assumed in the order of a few percent $\bar{\rho} \ll 1$ as it is proportional to the EEMCF $\kappa_s \ll 1$. The denominator in Eqs. (30)–(31) is then seen to decrease with increasing dielectric loss $\eta_e$. Therefore, the inductance $L$ increases due to the influence of dielectric losses. As both the numerator and denominator in Eq. (31) decreases for increased dielectric loss, the influence from dielectric loss on the resistance $R$ is not directly obvious.

The correction in Eqs. (30)–(31) can be consistently simplified using the assumptions $\bar{\lambda} \approx 1$, $\bar{\rho} \ll 1$, and $\eta_e \ll 1$, whereby $(\eta_e^2 \bar{\lambda}^2 + \eta_e^2 \bar{\rho}^2 - 2\eta_e \bar{\rho} + 1) \approx 1$ and $(\bar{\lambda}^2 + \bar{\rho}^2) \approx 1$. Thereby, a simplified correction for dielectric losses in the calibration of the series RL-shunt is obtained as

$$ \frac{\lambda}{\bar{\lambda}} \approx 1 \qquad (32) $$

$$ \frac{\rho}{\bar{\rho}} \approx 1 - \frac{\eta_e}{\bar{\rho}} \qquad (33) $$

Opposite to the full correction in Eqs. (30)–(31), the inductance from Eq. (32) is unaffected by the dielectric loss $\eta_e$, while the resistance from Eq. (33) decreases for increasing $\eta_e$, and remains positive only when $\eta_e/\bar{\rho} < 1$. Inserting the full Taylor series expansion gives the condition $\kappa_s > \eta_e/\sqrt{2}$ for implementation of series RL-shunts with passive components. The constant $\sqrt{2}$ is replaced by unity to secure $\kappa_s > \eta_e$ as a conservative and simple condition for the series RL-shunt, which is furthermore identical to that obtained for the parallel RL-shunt in Section 3.3.
<figure><img src="image_12.png" /><figcaption>**Fig. 8.** Picture of the composite blade with 8 rows of piezoelectric patches in (a), a picture of the digital shunt hardware in (b), and a schematic diagram of the digital shunt unit in (c).</figcaption></figure>

The influence of the corrections for dielectric losses on the calibration of a series RL-shunt is investigated in Fig. 7 for the electromechanical system considered in Sections 3.2 and 3.3. An ideal calibration is applied without accounting for dielectric losses, using the calibration based on $H_\infty$ optimization for minimum displacement presented by Soltani et al. [7], summarized in both [8] and Appendix A. The response for the ideal baseline with $\eta_e = 0$ (red solid curve in (d,e,f)) with $\kappa_s = 0.05$) leads to two peaks of equal magnitude around the targeted mode $\omega_s$. Similar to the example with the parallel RL-shunt in Section 3.3, increased dielectric loss $\eta_e$ (gray curves, with lighter shades corresponding to increased losses) is seen in (d) to give a single peak with a significant increase in amplitude when dielectric loss is not accounted for in the calibration. On the other hand, the response with the full correction using Eqs. (30)–(31) in (f) is nearly unaffected. The response based on the simplified correction in Eqs. (32)–(33) in (e) is also very effective, although introducing a small frequency detuning compared to the ideal (red curve) baseline without dielectric loss.

The amplitude increase relative to the ideal baseline with $\eta_e = 0$ is shown in Fig. 7(a,b,c) for varying $\kappa_s$ and $\eta_e$. Like the example with the parallel RL-shunt in Section 3.3, it is seen in (a) that low EEMCFs $\kappa_s$ and large dielectric losses $\eta_e$ can lead to an amplitude increase of more than 100% when not correcting for dielectric losses in the calibration of the series RL-shunt. Additionally, it is seen, that the amplitude increase is less than 0.5% with the full correction for dielectric loss in (c), while it is less than 10% when using the simplified correction in (b). Although the simplified correction is not as accurate as the full correction, it still leads to a significant reduction compared to not including any correction for dielectric loss in the calibration. In many practical applications, the simplified correction in Eqs. (32)–(33) is expected to be sufficient, unless a very accurate correction is needed, in which case the full correction in Eqs. (30)–(31) can be applied.

## 4. Experimental validation

The influence of dielectric losses on shunt calibration and the accuracy of the correction method presented in Section 3 are demonstrated experimentally through the calibration of a series RL-shunt tuned to target a single mode of the woven carbon-epoxy composite blade in Fig. 8(a), previously considered in [29]. The composite blade is clamped at one end, with 40 piezoelectric transducers (DuraAct P-876.A15, PI Ceramic) distributed in a 5 by 8 array on one side. One of the terminals of each piezoelectric transducer is grounded, while the other terminal is connected to a switch allowing the piezoelectric transducer to be in OC or to be connected in parallel with any of the other transducers. The piezoelectric transducers are of a soft PZT material (PIC255) with a dielectric loss factor given in the product catalog from PI Ceramics as $\tan(\delta') = 0.02$ [38]. The experiments target the second bending mode, with a natural frequency of 167 Hz for all piezoelectric transducers in OC condition.

The electrical components of the series RL-shunt are implemented through a digital shunt unit, synthesizing the admittance of the shunt circuit by measuring the voltage across the electrodes of the piezoelectric transducer and prescribing the corresponding current. Several different designs of the digital shunt unit have been applied in the literature. For instance, the original design by Fleming et
al. [39] is based on a voltage compensating voltage-controlled-current-source, while the design by Nečásek et al. [40] relies on the Howland current pump. The digital shunt circuit applied in this paper is described in [29] and [41]. It contains a dSpace MicroLabBox as the digital control unit (DCU), and additional hardware based on the setup in [42]. A picture of the shunt hardware is shown in Fig. 8(b), with the circuit schematic provided in Fig. 8(c). The digital shunt hardware contains a series resistance $R_{ser} = 10$ kΩ and a parallel resistance $R_{par} = 1$ MΩ. As described in [41], the resistor $R_{par}$ is considered as part of the structural model, whereby the current $I$ generated by the digital shunt unit is

$$I = -\frac{1}{R_{ser}}V_{out} \qquad (34)$$

with $V_{out}$ being the output voltage of the DCU. The input voltage $V_{in}$ of the DCU corresponds directly to the voltage $V$ across the electrodes of the piezoelectric transducers.

To demonstrate the influence of dielectric losses on the shunt calibration for different values of the EEMCF $\kappa_s$, the experimental validation is conducted for a range of combinations of the piezoelectric transducers connected in parallel. The two lowest rows in Fig. 8(a) are used to excite the blade, while combinations of the piezoelectric transducers in rows p3 to p8 are used for shunt damping. A parenthesis is used to specify when all five piezoelectric transducers in each row are not used simultaneously. For instance, p8(3/5) specifies that only three out of the five piezoelectric transducers in row eight are used.

## 4.1. Modal parameter estimation

Implementation of a series RL-shunt attached to the composite blade, and subsequent tuning with the correction for dielectric losses proposed in Section 3, requires an estimation of the modal parameters $\omega_s$, $C_s^\infty$, $\kappa_s$, and $\eta_e$. In this section, the modal parameters required for shunt calibration are determined from measurements of the charge-to-voltage ratio $Q/V$ obtained using the digital shunt unit. In the absence of an external load $f = 0$, eliminating the displacement $x$ between Eqs. (11) and (12) gives the charge-to-voltage ratio as

$$\frac{Q}{V} = C^{\infty}(1 - i\eta_e) + \frac{\omega_s^2 \chi_s^2 (1 + i\eta_e)^2}{-\omega^2 + 2i\omega\zeta_s\omega_s + \omega_s^2} \qquad (35)$$

The charge-to-voltage ratio $Q/V$ is obtained experimentally by applying a Gaussian noise current at one of the terminals of the piezoelectric transducer and measuring the voltage across the two terminals for approximately one minute. The measurements are obtained using the hardware of the digital shunt unit as in [29] with a sampling frequency of $f_{sample} = 10$ kHz and with a delay correction as described in [29] with $\tau = 0.053$ ms. The measured charge and voltage are then converted into a charge-to-voltage ratio in the frequency domain using the MATLAB toolbox ABRAVIBE [43]. In [29], the charge-to-voltage ratio is transferred into an equivalent dynamic capacitance $C(\omega)$ and conductance $1/R(\omega)$ in parallel, corresponding to

$$C(\omega) = \mathrm{Re}\left(\frac{Q}{V}\right), \quad \frac{1}{R(\omega)} = -\omega\mathrm{Im}\left(\frac{Q}{V}\right) \qquad (36)$$

The modal parameters are then determined directly using the two extremum points on the curve for the dynamic capacitance. Although the dynamic capacitance $C(\omega)$ is independent of dielectric losses $\eta_e$, the method could be directly extended to include the estimation of dielectric losses from the dynamic conductance $1/R(\omega)$. While this approach is simple and computationally efficient, the present study considers calibration for very low EEMCFs, which requires the improved parameter accuracy provided by curve fitting. A range of frequency-based modal parameter estimation techniques are available [44], all representing potential alternatives to the two-point approach. However, in the following, a Linear Square Complex Frequency Estimator (LSCF) and a Least-Squares Frequency-Domain estimator (LSFD) are used to estimate the modal parameters, using the MATLAB toolbox [45].

When applied to a single vibration mode $s$ with no closely spaced modes, the measured frequency response function $H(\omega)$ is assumed to be on the form

$$H(\omega) = A + \frac{1}{-\omega^2} B + \frac{r_s}{-\omega^2 + 2i\gamma_s\Omega_s\omega + \Omega_s^2} \qquad (37)$$

where the modal parameters obtained using the algorithm are the natural angular frequency $\Omega_s$, the damping ratio $\gamma_s$, the complex residue $r_s$, as well as two complex coefficients $A$ and $B$. By respectively comparing the denominator and numerator in the last terms of Eqs. (37) and (35), the SC natural frequency $\omega_s$, the structural damping ratio $\zeta_s$, the piezoelectric coefficient $\chi_s$, and the piezoelectric loss factor $\eta_e$ are obtained as

$$\omega_s = \Omega_s, \quad \zeta_s = \gamma_s, \quad \chi_s^2 = \frac{|r_s| + \mathrm{Re}(r_s)}{2\omega_s^2}, \quad \eta_e = \frac{\mathrm{Im}(r_s)}{2\omega_s^2 \chi_s^2} \qquad (38)$$

where derivations make use of the relations: $|r_s| = \omega_s^2 \chi_s^2 (1 + \eta_e^2)$ and $\mathrm{Re}(r_s) = \omega_s^2 \chi_s^2 (1 - \eta_e^2)$.

The present paper is limited to shunts targeting a single vibration mode, whereby the expression in Eq. (35) only needs to be accurate in the narrow frequency band around the natural frequency of the structure ($\omega \approx \omega_s$). This suggests that the expressions in Eqs. (35) and (37) should be equal at the SC natural frequency $\omega_s$, whereby comparison of the former terms give the blocked modal capacitance $C_s^\infty$ and the dielectric loss coefficient $\eta_e$ as

$$C_s^\infty = \mathrm{Re}\left(A + \frac{1}{-\omega_s^2} B\right), \quad \eta_e = -\frac{1}{C_s^\infty} \mathrm{Im}\left(A + \frac{1}{-\omega_s^2} B\right) \qquad (39)$$
<caption>**Table 1**</caption>

Estimated modal parameters for different patch combinations on the composite blade.*: A different model order and stability requirement is used for the modal parameter estimation, see Section 4.1.

<table><thead><tr><th>Patch Rows</th><th>ω<sub>s</sub>/2π [Hz]</th><th>C<sub>s</sub><sup>∞</sup> [nF]</th><th>κ<sub>s</sub> [-]</th><th>ζ<sub>s</sub> [-]</th><th>η<sub>e</sub> [-]</th><th>η<sub>ε</sub> [-]</th></tr></thead><tbody><tr><td>3–8</td><td>166.6</td><td>956</td><td>0.079</td><td>0.0026</td><td>-0.014</td><td>0.015</td></tr><tr><td>4–6</td><td>166.8</td><td>481</td><td>0.074</td><td>0.0027</td><td>-0.023</td><td>0.015</td></tr><tr><td>3–6</td><td>166.7</td><td>640</td><td>0.073</td><td>0.0026</td><td>-0.014</td><td>0.015</td></tr><tr><td>5</td><td>167.0</td><td>159</td><td>0.046</td><td>0.0024</td><td>-0.007</td><td>0.018</td></tr><tr><td>6</td><td>167.1</td><td>163</td><td>0.043</td><td>0.0025</td><td>-0.018</td><td>0.017</td></tr><tr><td>3–4</td><td>167.0</td><td>322</td><td>0.040</td><td>0.0025</td><td>-0.017</td><td>0.016</td></tr><tr><td>4</td><td>167.1</td><td>161</td><td>0.038</td><td>0.0025</td><td>-0.018</td><td>0.018</td></tr><tr><td>7</td><td>167.1</td><td>159</td><td>0.032</td><td>0.0025</td><td>-0.007</td><td>0.018</td></tr><tr><td>3</td><td>167.2</td><td>160</td><td>0.019</td><td>0.0027</td><td>-0.007</td><td>0.018</td></tr><tr><td>8(5/5)</td><td>167.2</td><td>159</td><td>0.016</td><td>0.0025</td><td>-0.015</td><td>0.018</td></tr><tr><td>8(3/5)</td><td>167.2</td><td>95</td><td>0.015</td><td>0.0029</td><td>-0.010</td><td>0.021</td></tr><tr><td>8(2/5)<sup>*</sup></td><td>167.1</td><td>64</td><td>0.009</td><td>0.0030</td><td>0.025</td><td>0.026</td></tr></tbody></table>

The modal parameters $ω_s$, $C_s^∞$, $κ_s$, and $η_e$ required for tuning with the correction for dielectric losses in Section 3 can then be obtained from the expressions in Eqs. (17), (38), and (39).

The LSCF and LSFD algorithms are applied with model orders ranging from 10 to 50, over a frequency range of 100 Hz to 400 Hz, which excludes the lower mode at ~30 Hz and the higher modes appearing from ~410 Hz. The tolerance for the stability of the estimated poles is set to $10^{-4}$. However, for the case with only two patches in the top-row (denoted p8(2/5)), the maximum model order is increased to 70, while the tolerance is relaxed to $10^{-3}$. The modal parameters are estimated for 12 combinations of piezoelectric transducers in Table 1, sorted by the estimated EEMCF $κ_s$ in descending order from 0.079 for patches p3-8 to 0.009 for patches p8(2/5). All other patches are kept in the OC condition for each measurement, whereby the natural frequency in the SC condition varies between 166.6 Hz to 167.2 Hz for the different patch combinations. The modal damping ratio $ζ_s$ ranges from 0.0024 to 0.0030, while the piezoelectric loss factor ranges from -0.018 to 0.025 with a negative sign for the first 11 measurements. As described in Section 2, the contribution from the piezoelectric loss term may be negative as long as the total power dissipation is positive. However, as the parameter is neglected in the following shunt tuning, its accuracy has not been studied in further detail. The dielectric loss coefficient $η_e$ shows larger variation than the modal damping ratio $ζ_s$, with values ranging from 0.015 to 0.026. It is seen from Eq. (35) that the dielectric loss factor $η_e$ represents the phase lag between charge $Q$ and voltage $V$ when $ω → ∞$. This high-frequency limit corresponds to a mechanically blocked piezoelectric transducer in a zero strain state. This limit is in accordance with the introduction of the dielectric loss factor $η_e$ in Eq. (5) for the dielectric permittivity at constant strain. As described in the introduction, data sheets of piezoelectric materials often contain the dielectric dissipation factor $tan(δ')$ at constant stress, corresponding to the quasi-static limit. While the two loss terms are not strictly identical, they are close in practice. In particular, considering the uncertainty of the parameter in the datasheet, the measured dielectric loss $η_e$ in Table 1 is close to the value of $tan(δ')$ given as 0.02 in the datasheet [38].

The accuracy of the electromechanical model introduced in Section 2 and Eq. (35) is demonstrated in Fig. 9 for patch combination p3-8 in (a,b) and p8(3/5) in (c,d), where the $Q/V$-ratio is separated into a parallel capacitance $C$ (a,c) and conductance $1/R$ (b,d). The measured data (black dots) is seen to contain significantly more noise for patch combination p8(3/5) in (c,d) compared to p3-8 in (a,b). The estimated frequency response function with the algorithm in Eq. (37) (red dashed curve) is seen to provide a good approximation of the measured capacitance and conductance in the considered frequency range. The $Q/V$-ratio is also determined using the model introduced in Eq. (35) with the modal parameters in Table 1 (blue dot-dashed curve). While the model gives a good fit of the dynamic capacitance for the patches with a large EEMCF in (a,b), it deviates significantly far away from resonance for the patches with a low EEMCF in (c,d). However, in all cases, the model provides a good approximation of the data for a narrow frequency band around $ω_s$, as shown in the close-ups in Fig. 9. As described in Section 2, the model is used for tuning of a shunt targeting a single vibration mode and therefore only needs to be accurate within a narrow frequency band around the targeted natural frequency, i.e. for $ω ≈ ω_s$. Thus, the model and its estimated parameters in Table 1 are considered sufficiently accurate in the present case. For calibration of shunts targeting multiple vibration modes, or structures with closely spaced modes, a model with multiple degrees of freedom is required using more than one mode for the LSCF and LSFD algorithm.

## 4.2. Calibration of series RL-shunt

A series RL-shunt is calibrated to target the second vibration mode of the blade for a range of EEMCFs by designing a shunt for each patch combination in Table 1. The sampling frequency of the DCU is $f_{sample} = 50$ kHz when it is used to synthesize the admittance in the RL-shunt. As demonstrated in [29] and [46], a high sampling frequency is required to ensure the accuracy and stability of digital shunts. In the present paper, the discretization of the transfer function is implemented as in [29] with a Forward Euler integration used for the discretization of the transfer function and a correction for the influence of delay introduced by the updated shunt parameters

$$ \tilde{L} = L\cos(2\omega_s\tau) - \frac{R}{\omega_s}\sin(2\omega_s\tau) \quad (40) $$

$$ \tilde{R} = R\cos(2\omega_s\tau) + \omega_s L\sin(2\omega_s\tau) \quad (41) $$
<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Measurements of the charge-to-voltage ratio Q/V separated into an equivalent capacitance C in (a,c), and a parallel conductance 1/R in (b,d) for all the patches in row 3–8 (p3-8) in (a,b) and 3 patches in row 8 (p8(3/5)) in (c,d). Included are the experimental data (black dots), H(ω) in Eq. (37) (red dashed), and Q/V in Eq. (35) (blue dashed-dotted). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<caption>**Table 2**</caption>

Series RL-shunt tuning for different combinations of patches on the composite blade.
*: A different model order and stability requirement is used for the modal parameter estimation, see Section 4.1.

<table><thead><tr><th rowspan="2">Patch</th><th rowspan="2">κ<sub>s</sub> [-]</th><th rowspan="2">η<sub>e</sub> [-]</th><th colspan="2">Ideal Calibration</th><th colspan="2">Full Correction</th><th colspan="2">Ratio</th></tr><tr><th>L&#773; [H]</th><th>R&#773; [Ω]</th><th>L [H]</th><th>R [Ω]</th><th>L/L&#773;[-]</th><th>R/R&#773;[-]</th></tr></thead><tbody><tr><td>3–8</td><td>0.079</td><td>0.015</td><td>0.943</td><td>96.0</td><td>0.945</td><td>81.6</td><td>1.003</td><td>0.850</td></tr><tr><td>4–6</td><td>0.074</td><td>0.015</td><td>1.872</td><td>178.2</td><td>1.876</td><td>150.2</td><td>1.002</td><td>0.843</td></tr><tr><td>3–6</td><td>0.073</td><td>0.015</td><td>1.409</td><td>131.9</td><td>1.412</td><td>110.6</td><td>1.002</td><td>0.838</td></tr><tr><td>5</td><td>0.046</td><td>0.018</td><td>5.687</td><td>339.2</td><td>5.696</td><td>233.5</td><td>1.002</td><td>0.688</td></tr><tr><td>6</td><td>0.043</td><td>0.017</td><td>5.564</td><td>308.9</td><td>5.572</td><td>207.1</td><td>1.002</td><td>0.670</td></tr><tr><td>3–4</td><td>0.040</td><td>0.016</td><td>2.812</td><td>146.2</td><td>2.815</td><td>100.6</td><td>1.001</td><td>0.688</td></tr><tr><td>4</td><td>0.038</td><td>0.018</td><td>5.611</td><td>275.8</td><td>5.618</td><td>172.0</td><td>1.001</td><td>0.624</td></tr><tr><td>7</td><td>0.032</td><td>0.018</td><td>5.690</td><td>231.8</td><td>5.696</td><td>125.0</td><td>1.001</td><td>0.539</td></tr><tr><td>3</td><td>0.019</td><td>0.018</td><td>5.646</td><td>140.7</td><td>5.649</td><td>36.0</td><td>1.001</td><td>0.256</td></tr><tr><td>8(5/5)</td><td>0.016</td><td>0.018</td><td>5.694</td><td>119.8</td><td>5.696</td><td>13.3</td><td>1.000</td><td>0.111</td></tr><tr><td>8(3/5)</td><td>0.015</td><td>0.021</td><td>9.567</td><td>181.1</td><td>9.570</td><td>-34.4</td><td>1.000</td><td>-0.190</td></tr><tr><td>8(2/5)*</td><td>0.009</td><td>0.026</td><td>14.114</td><td>165.6</td><td>14.113</td><td>-217.9</td><td>1.000</td><td>-1.315</td></tr></tbody></table>

where L̃ and R̃ are the inductance and resistance implemented in the DCU, while τ = 1/(2f<sub>sample</sub>) represents the temporal correction.

The series RL-shunt is calibrated based on the calibration expressions given by Yamada et al. in [24] for minimum acceleration, excluding the influence of dielectric losses (assuming η<sub>e</sub> = 0). The calibration expressions are summarized in Appendix A. The resulting uncorrected inductance L̃ and resistance R̃ are given for all patch combinations in Table 2 along with the corrected inductance L and resistance R obtained by the full correction for dielectric losses from Eqs. (30)–(31). Additionally, Table 2 includes the EEMCFs κ<sub>s</sub> and dielectric loss factors η<sub>e</sub> from Table 1, and the ratios L/L̃ and R/R̃ between the corrected and uncorrected parameters. It is seen that with decreasing EEMCFs, the ratio L/L̃ reduces from 1.003 to 1.000, while R/R̃ reduces from 0.0850 to -1.315. Thus, the relative difference between the corrected and uncorrected inductance is below 0.5%, whereas the resistance is significantly affected by the correction. For the patch combinations p8(3/5) and p8(2/5) with EEMCFs of 0.015 and 0.009, respectively, the corrected resistance becomes negative. Such shunts cannot be made using passive components. However, all the shunts are synthesized in the following using the digital shunt introduced in Section 4, allowing the realization of a negative shunt resistance. Section 3.4 defines the
<figure><img src="image_14.png" /><figcaption>**Fig. 10.** Amplitude reduction by the full correction (solid blue curve in insets with accelerance) relative to an uncorrected calibration without correction for dielectric losses (solid black curve in insets with accelerance). The measured accelerance with the shunts in the OC condition is shown in the insets by the black dashed curves. The accelerance curves for all 12 cases are shown in Fig. B.1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

conservative lower limit $\kappa_s > \eta_e$ for the EEMCF required for implementation of a series RL-shunt with passive components. The three patch combinations in row 8 are found to violate the limit. While the patch combination P8(5/5) has a small positive resistance, the two combinations P8(3/5) and P8(2/5) require a negative resistance. For the present cases, the approximate lower limit is consequently found to be conservative. The negative resistance required for P8(3/5) and P8(2/5) compensates for the equivalent parallel resistance introduced by dielectric losses. No stability issues were observed in the implementation of the negative shunt resistance.

The frequency response of the blade is measured for each shunt combination in Table 2 by measuring the acceleration at the top of the blade under voltage excitation with a Gaussian noise signal applied to the two lowest rows of patches in Fig. 8(a). Fig. 10 shows the relative reduction of the maximum accelerance for the different EEMCFs $\kappa_s$ when correcting for dielectric losses in the shunt calibration (relative to calibration without correction). The figure also shows the accelerance curves for three EEMCF, showing the OC case (dashed black), tuning without correction (solid black), and tuning with correction for dielectric losses (solid blue). The measured frequency response functions for all considered patch combinations are provided in Appendix B.

Fig. 10 shows that both the uncorrected (black solid) and corrected (blue solid) RL-shunts lead to a significant reduction in accelerance compared to the OC case (black dashed). For an ideal model, the applied calibration method is expected to give two peaks with equal magnitude around the targeted resonance frequency. As in the numerical example in Section 3.4, the uncorrected calibration gives two distinct peaks for the highest EEMCF (p3-8) in Fig. 10 but shifts to a single distinct peak as the EEMCF decreases (for both p6 and p8(2/5)). The single peak indicates that the total resistance is too large in the RL-shunt circuit, which may complicate retuning of the shunt, as it is not directly apparent from the single peak whether the inductance tuning is sufficiently accurate. Opposite to the uncorrected case, the corrected calibration (blue solid) maintains the two distinct peaks. Furthermore, the amplitude of the two peaks is not identical, as in the ideal example in Section 3.4, which indicates a slight detuning of the inductance. However, effects such as structural damping, and the location of the external load and accelerometer are not accounted for in the calibration. Therefore, the small asymmetry in the two (blue) resonance peaks is well within the expected accuracy for most practical applications.

In the lower plot in Fig. 10, showing the relative amplitude reduction for the corrected calibration, it is seen that accounting for dielectric losses in the calibration leads to an improvement of less than 5% for the shunts with an EEMCF $\kappa_s$ larger than 0.05. However, for the shunts with smaller EEMCFs, the amplitude reduction increases significantly, up to approximately 25% for the shunt with $\kappa_s < 0.01$. This demonstrates that correcting for dielectric losses has a significant effect on the calibration accuracy of the series RL-shunt for piezoelectric transducers with small EEMCFs.

## 5. Conclusions and perspectives

The influence of dielectric losses on the calibration of piezoelectric shunts targeting a single vibration mode is considered in this paper. The proposed correction method, based on impedance matching at the targeted natural frequency includes the influence of dielectric losses in the shunt calibration.
The accuracy of the proposed correction has been validated numerically for an R-shunt, a parallel RL-shunt, and a series RL-shunt. Dielectric losses are seen to have a small effect on the calibration of an R-shunt, whereas the loss can cause significant detuning for parallel and series RL-shunts at low effective electromechanical coupling factors (EEMCFs). The proposed correction is seen to be accurate with a negligible increase in amplitude compared to the ideal system not considering dielectric losses. For large dielectric losses and small EEMCFs, the correction may require a negative shunt resistance which is not possible with passive components. An approximate lower limit for the EEMCF required to ensure a positive shunt resistance is found as $κ_s > η_ε$ for both the parallel and series RL-shunts.

The accuracy of the proposed correction method has been further validated experimentally by calibration of a series RL-shunt attached to a composite blade for EEMCFs ranging from 0.009 to 0.079. It is demonstrated that all modal parameters required for shunt calibration, including the blocked capacitance and the dielectric loss factor, can be determined using experimental modal analysis from impedance measurements on the piezoelectric transducer bonded to the structure. The correction for dielectric losses is seen to give an accurate shunt calibration for the considered EEMCFs. The experimental example confirms that while the influence of dielectric loss is small for the calibration of resonant shunts with a high EEMCF, it is important to account for dielectric losses in the calibration for small EEMCFs approaching the dielectric loss coefficient.

Future work could extend the method to account for dielectric losses in the calibration of shunts or networks targeting multiple vibration modes. Additionally, it would be relevant to study dielectric losses across a wider range of structures and different types of encapsulated piezoelectric transducers, such as Macro Fiber Composites.

### CRediT authorship contribution statement

**Jens D. Richardt:** Writing – review & editing, Writing – original draft, Visualization, Validation, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization; **Boris Lossouarn:** Writing – review & editing, Validation, Supervision, Methodology, Formal analysis, Conceptualization; **Jan Høgsberg:** Writing – review & editing, Supervision, Methodology, Conceptualization; **Jean-François Deü:** Writing – review & editing, Supervision, Methodology, Conceptualization.

### Data availability

Data will be made available on request.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

The composite blade has been manufactured by Safran, using a resin transfer molding (RTM) process. Thanks to Safran for allowing the use of the composite blade for the experimental analysis conducted at Cnam. Also thanks to Augustinus Fonden, Knud Højgaards Fond, and Reinholdt W. Jorck og Hustrus Fond for financially supporting the visit of JD Richardt at Cnam.

## Appendix A. Ideal shunt tuning formulas

The applied explicit shunt tuning formulas are summarized in the following.

### A.1. R-shunt

The used calibration of an R-shunt is given in Thomas et al. [6], based on a transfer function criterion to minimize the peak amplitude of the frequency response for the DAF (displacement over force). The optimal non-dimensional resistance $ρ = ω_s C^∞ R$ is given as

$$ρ = \frac{1}{\sqrt{1 + \kappa_s^2/2}} \qquad (A.1)$$

### A.2. Parallel RL-shunt

The present calibration of a parallel RL-shunt is described by Ikegame et al. [8], based on derivations in Tang et al. [37]. The calibration uses an $H_∞$-norm minimization of the amplitude of the frequency response for the DAF (displacement over force). A range of intermediate parameters are introduced as

$$a_0 = 64 \qquad (A.2)$$

$$a_1 = -16\kappa_s^2 \qquad (A.3)$$
$$a_2 = -64 + 16\kappa_s^2 + 11\kappa_s^4 \qquad (A.4)$$

$$a_3 = 2\kappa_s^2(8 - \kappa_s^2)(2 - \kappa_s^2) \qquad (A.5)$$

$$a_4 = -2\kappa_s^4(2 - \kappa_s^2) \qquad (A.6)$$

$$A_6 = 27(a_0 a_3^2 + a_1^2 a_4) - 9a_2(a_1 a_3 + 8a_0 a_4) + 2a_2^3 \qquad (A.7)$$

$$A_5 = 12a_0 a_4 - 3a_1 a_3 + a_2^2 \qquad (A.8)$$

$$A_4 = \sqrt[3]{\frac{A_6^2 - 4A_5^3 + A_6}{2}} \qquad (A.9)$$

$$A_3 = \frac{1}{2}\sqrt{\frac{a_1^2}{a_0^2} + \frac{4(-2a_2A_4 + A_4^2 + A_5)}{3a_0A_4}} \qquad (A.10)$$

$$A_2 = -\frac{8a_0^2 a_3 - 4a_0 a_1 a_2 + a_1^3}{4a_0^3 A_3} \qquad (A.11)$$

$$A_1 = \frac{3a_1^2 A_4 - 2a_0(4a_2 A_4 + A_4^2 + A_5)}{6a_0^2 A_4} \qquad (A.12)$$

$$r = -\frac{a_1}{4a_0} + \frac{A_3}{2} + \frac{\sqrt{A_1 + A_2}}{2} \qquad (A.13)$$

This results in a required frequency ratio $v$ and damping ratio $\zeta$

$$v = \sqrt{\frac{2r - \kappa_s^2 + \sqrt{16r^2 - 4r\kappa_s^2 + \kappa_s^4}}{6}} \qquad (A.14)$$

$$\zeta = \sqrt{\frac{(r-1)(3r - 2\kappa_s^2)}{r(2r + \kappa_s^2 - \sqrt{16r^2 - 4r\kappa_s^2 + \kappa_s^4})}} \qquad (A.15)$$

The non-dimensional inductance $\lambda = \omega_s^2 C^\infty L$ and resistance $\rho = \omega_s C^\infty R$ are then given as

$$\lambda = \frac{1}{v^2} \qquad (A.16)$$

$$\rho = \frac{1}{\zeta v} \qquad (A.17)$$

### A.3. Series RL-shunt

Two calibrations are considered for the series RL-shunt: one for minimum compliance (displacement over force) in Figs. 2 and 7, and one for minimum accelerance (acceleration over force) in the experimental Section 4.2.

#### A.3.1. Compliance

The calibration of a series RL-shunt for minimum compliance is described by Ikegame et al. [8] based on derivations in Soltani et al. [7]. The calibration is based on an $H_\infty$-norm minimization of the amplitude of the frequency response for the DAF (displacement over force). An intermediate parameter is introduced as

$$r = \frac{\sqrt{64 - 16\kappa_s^2 - 26\kappa_s^4 - \kappa_s^2}}{8} \qquad (A.18)$$

The non-dimensional inductance $\lambda = \omega_s^2 C^\infty L$ and resistance $\rho = \omega_s C^\infty R$ are then obtained as

$$\lambda = \frac{4}{8 + 3\kappa_s^2 - 4r} \qquad (A.19)$$

$$\rho = \frac{2\sqrt{2}\sqrt{64 + 80\kappa_s^2 + 27\kappa_s^4 - 16r(4 + 3\kappa_s^2)}}{(8 + 5\kappa_s^2)\sqrt{8 + 3\kappa_s^2 - 4r}} \qquad (A.20)$$

#### A.3.2. Accelerance

The calibration of a series RL-shunt for minimum accelerance (acceleration over force) is derived by Yamada et al. [24]. The non-dimensional inductance $\lambda = \omega_s^2 C^\infty L$ and resistance $\rho = \omega_s C^\infty R$ are given as

$$\lambda = \frac{1}{(1 + \kappa_s^2)^2} \qquad (A.21)$$
$$ \rho = \sqrt{\frac{3}{2} \frac{\kappa_s^2}{(1 + \kappa_s^2)^3}} \qquad (A.22) $$

## Appendix B. Individual shunt response

The individual measurements of the acceleration at the top of the blade for the 12 different combinations of piezoelectric patches in Tables 1 and 2 are shown in Fig. B.1.

<figure><img src="image_15.png" /><figcaption>Fig. B.1. Acceleration at the top of the blade with the piezoelectric transducer in OC condition (dashed black curve), and with a series RL-shunt calibrated without correction for dielectric losses (solid black curve) and with correction for dielectric losses (solid blue curve). The rows of patches used for shunt damping are given in the title of each figure. It is specified if less than five patches in a row are used, e.g. for p8(3/5) where three out of five patches in row 8 are used. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
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