# An efficient and compact supply for electroactive polymers

Morgan Almanza<fnref n="1" />, Christophe Baron<fnref n="1" />, Guojun Lan<fnref n="1" />, Yiming Wei<fnref n="1" />, Yoan Civet<fnref n="2" />, Yves Perriard<fnref n="2" />, Martino LoBue<fnref n="1" />

**Abstract**—New trends in electroactive-polymers (EAP) based devices for soft actuators and for electrocaloric cooling revive studies on compact, efficient, reversible and low power supply. High electrode resistance, high voltage and small capacitance of EAPs represent a major hindrance for an efficient charge/discharge. Because Joule losses in EAP are entwined with the voltage profile of the converter, we study an inductor based coupling, focusing on a multistep voltage profile produced by a Solid-State Marx Modulator. The energy stored in the inductor, its volume, and the efficiency of the charge/discharge process are studied as a function of the number of steps and of the RLC damping factor. Furthermore, to avoid detrimental current surge related to resonance, a pre and post charge protocol are implemented.

The tradeoff between inductor size, efficiency and the number of steps is discussed using an analytical model, and through comparison with measurements performed using a specifically developed Solid-State Marx Modulator. This study shows that the multistep charge of EAPs represents an efficient tool for tackling the Joule losses vs. volume reduction issue offering a key tool to guide the design of the new generation of EAP based devices.

**Index Terms**—Electroactive polymer, Dielectric devices, Losses minimization, Capacitors charge, Multilevel converters

## I. INTRODUCTION

From soft robotics with for example augmented aorta cardiac assist device [1] to more efficient and flexible solid-state cooling technologies [2], [3], electroactive-polymers (EAPs) based devices are nowadays between the most promising functional materials. In the cardiac assist application, the EAP film is rolled in a tube, and squeezes and expands to pump the blood. On the other hand, cooling applications rely on the reversible temperature/entropy change induced by application or removal of an electric field. EAPs are made of a polymer film sandwiched between two electrodes. To properly operate they need thin and compliant electrodes to avoid hindering the deformation and the thermal transfer. Carbon based electrodes are commonly used because of their mechanical compliance, thermal stability, and oxidation resistance [1]–[3]. Unfortunately, their poor electrical conductivity brings-in a high access resistance R in series with the capacitance C related to the EAP. For the aforementioned applications the values of C are of the order of dozens of nanofarads, with access resistance R up to few kilo ohms, and maximum voltages V<sub>C</sub> up to 10 kV. The energy delivered to the EAP spans from few dozens of mili-Joules to half a Joule.

The energy E<sub>EAP</sub> (e.g. mechanical work for a dielectric elastomer, or heat for an electrocaloric heat pump) converted by the active material represents 20% (or less) of the electrical energy E<sub>C</sub> that has been stored in the EAP capacitance, the remanent being recovered during the discharge [4], [5]. This can be described through a coupling factor α where E<sub>EAP</sub> = αE<sub>C</sub>.

EAPs are cycled between zero and their maximum voltage up to several kV, at a frequency of few Hertz. The high currents associated with the short charging/discharging time (viz. of the order of 10 ms) produce significant Joule losses in R during the charge E<sub>J</sub><sup>ch</sup> and the discharge E<sub>J</sub><sup>dis</sup> processes. The high resistance associated with low coupling factor in a cyclic charge/discharge pose a serious challenge for the total EAP efficiency.

The overall EAP efficiency η<sub>EAP</sub>, is defined as the ratio between the energy converted by the active material E<sub>EAP</sub> = αE<sub>C</sub>, and the net electrical work W performed by the generator over a full cycle, η<sub>EAP</sub> = E<sub>EAP</sub>/W. Considering the Joule losses in the access resistance as the main source of dissipation, the net electrical work is W = E<sub>EAP</sub> + E<sub>J</sub><sup>ch</sup> + E<sub>J</sub><sup>dis</sup>. Thence,

$$ \eta_{EAP} = \frac{\alpha E_C}{\alpha E_C + E_J^{ch} + E_J^{dis}} \quad (1) $$

To achieve an EAP efficiency up to 84%, η<sub>EAP</sub> ≥ 0.84, the term (E<sub>J</sub><sup>ch</sup> + E<sub>J</sub><sup>dis</sup>)/E<sub>C</sub> must shrink below 4%, whence the relevance, in order to improve the efficiency, to drastically minimize the total Joule losses. The objective here is to reduce the ratio between Joule losses and the energy transferred to the capacitance E<sub>C</sub> by optimising the charge/discharge process. In order to focus this study on the charge/discharge mechanism, the non-linear nature of the physics underlying energy conversion in EAPs is neglected. Moreover, when reversing the voltage step keeping the same profile, the current is reversed, leading to the same loss ratio at charge and at discharge. Within this linear approximation [6], [7], the loss ratio is:

$$ \frac{E_J}{E_C} = \frac{2RC}{T} \frac{i_{rms}^2}{i_{avg}^2}, \quad (2) $$

where i<sub>rms</sub> and i<sub>avg</sub> are the root-mean-square (RMS), and the average current over the charging time T respectively. i<sub>rms</sub><sup>2</sup>/i<sub>avg</sub><sup>2</sup> is the current factor, while 2RC/T represents a time factor that in usually small. In the case considered here 2RC/T ≈ 0.001 (i.e 0.1%). Equations (1), and (2) show that the charging profile, resulting from the power supply, is a key feature in order to maximize the total EAP's efficiency. For a

<footnote n="1">¹Université Paris-Saclay, ENS Paris-Saclay, CNRS, SATIE, 91190, Gif-sur-Yvette, France</footnote>

<footnote n="2">²École polytechnique fédérale de Lausanne (EPFL), Integrated Actuators Laboratory, Neuchâtel, Switzerland</footnote>
constant current (the ideal current profile), the current factor is equal to 1 (i.e. the optimum). Consequently, the loss ratio could go down to 0.1%, at least in principle. On the contrary, when applying a voltage step on the RC circuit, the current profile is a rather inappropriate one making the loss ratio huge (i.e ≈ 1), and reducing η<sub>EAP</sub> down to 10%.

Recently, high voltage (kV), low current (mA), bi-directional and compact converters to power EAPs have risen a considerable interest [3], [8]–[14]. However, two major challenges have emerged: reducing the size of the inductor, and keeping the losses resulting from the parasitic capacitance low. A rather recurrent feature of the existing literature is focusing on the converter efficiency for the charge/discharge alone, neglecting the strong dependence of the EAP efficiency shows over current profile. For instance, connecting a filtering capacitor to decouple the current profile, requires a further volume addition and increases also the energy to be transferred back and forth, inducing an additional reduction of the coupling factor. Besides, connecting EAP in series with an inductance, forming a RLC allows to improve the loss ratio, but at the price of a high value of *R*, and of *L* entailing a bulky, and difficult to design device. Tackling separately either the optimal capacitor charge/discharge (i.e. focusing on the loss ratio only), or the power electronics driving the device, ignoring at the same time EAP's specificities when designing the converter, leads to bulky devices, and to poor total efficiencies. Putting it another way, the converter, and the EAP need to be optimized jointly by taking into account the role of their mutual coupling in the resulting total efficiency. The different voltage profiles resulting from the converter, namely the single step charge [3], [5]

or the successive multistep charging (SMSC) [14], and their potentiality to attain a better loss management, and a better inductor compactness have seldom been studied jointly so far.

The point we shall address here is to manage at once the EAP charge/discharge through an inductor, and a multistep voltage profiles (SMSC) in order to reduce the two key issues that have been pinpointed, namely minimizing over a cycle the following quantities:

* the Joule losses (i.e the loss ratio $E_J/E_C$ );

* the maximum energy stored in the inductance (i.e $E_L/E_C$), and consequently its volume.

Fig. 1 shows the circuit studied where $v_{RLC}(t)$ represents the voltage profile produced by an Solide-State Marx Modulator [14], $L$ is the inductance and the EAP is represented through a resistance $R$ in series with a capacitance $C$. The approach adopted here focuses on the interaction between the EAP and the converter voltage output. Actually, this approach is the more suitable to describe a capacitive load (i.e. an EAP) always kept in a transient state all along its cyclic working operation, with a high access resistance. A rather different state of affairs compared with DC-DC converters and especially switched capacitor converters where the efficiency is studied at a stationary working point, and with a resistor as a load.

The paper is organized as follows. After this introduction, the second section is devoted to an analytical study of the RLC circuit along its charge/discharge in terms of its Joule-loss ratio $E_J/E_C$, number of voltage step (N) and of the maximum

<figure><img src="image_1.png" /><figcaption>Fig. 1. The voltage step generator connected to the RLC circuit where the EAP is represented by the elements within the red dash square.</figcaption></figure>

energy stored in the inductance $E_L/E_C$. In addition, optimized pre- and post-charge protocols are used to avoid detrimental overshoots in $E_L$ and $E_J$ due to resonance.

In the third section, the theoretical results are compared with experiments carried with the Solid-State Marx Modulator presented in [14]. Afterwards, the opportunity offered by a multilevel voltage is discussed as a way to improve the loss while reducing the inductance maximum energy (volume).

In the appendix, the analytic expressions used to obtain the presented results are detailed.

## II. MODELING OF THE MULTI-STEP CHARGING

The RLC circuit shown in Fig. 1, is characterized by its angular frequency $\omega_0 = 1/\sqrt{LC}$ and its damping factor $\xi = \frac{R}{2}\sqrt{\frac{C}{L}}$. The voltage applied through the Solid-State Marx Modulator [14] on the RLC, $v_{RLC}(t)$, is made of successive steps, each lasting a time $t_s$. The steps are: $v_{RLC}(0) = V_s$, $v_{RLC}(t_s) = 2V_s$, $v_{RLC}(kt_s) = (k+1)V_s$ and so on up to the final voltage $v_{RLC}((N-1)t_s) = NV_s = U$ with $U$ the final voltage and $k = [0, 1, 2, ..., N-1]$. Thus, the single step amplitude $V_s$ depends on the number of steps $N$, namely $V_s = U/N$. The delay $t_s$ allows avoiding the superposition of the current responses and thus reduces the losses, and the maximum energy stored in the inductor. In order to avoid any negative current in the underdamped regime, where $\xi < 1$, the following step is applied at $t_0 = \frac{\pi}{\omega_0\sqrt{1-\xi^2}}$ when the current is passing through zero (i.e $t_s = t_0$). When $\xi \approx 1$, $t_0$ diverges, because in the overdamped regime (i.e. $\xi > 1$) the current never vanishes (i.e. is always positive). In this case, to get a finite charging time, $t_s$ is fixed at 99% of the RC circuit full charge time. More precisely, $t_s = t_0$ when $\xi < \xi_{trans}$, otherwise $t_s = -RC \ln(1-0.99)$, with $\xi_{trans} \approx 0.93$ (see Appendix V for more details).

The current amplitude is linear with respect to the voltage step $V_s$. Consequently the Joule loss and the energy stored in the inductor (both related to the square of the current) are linear with respect to the energy transferred to the capacitor (i.e. proportional to $V_s^2$). This makes the energy transferred when the capacitor charges from 0 to $V_s$, $E_C^s = CV_s^2/2$ a natural unit to normalize all the other relevant energies at the single step scale. Indeed, through the analytical study of the RLC circuit, the expressions for the loss $E_J^k/E_C^s$, and for the inductor energy $E_L^k/E_C^s$ of a given step $k$ (see expressions (20) and (19) respectively, in Appendix V) can be worked out.
A quite relevant issue when using SMSC is the possible detrimental effect deriving from transient regimes that can affect the initials, and final steps of the process. Actually, the voltage and current initial conditions for the *k* step, $\Delta V[k] = v_{RLC}(kt_s) - v_C(kt_s)$ and $i(kt_s)$, can change from one step to the other till before stabilizing over a steady state regime, as shown in detail in Appendix V-C). Along the transient regime, the loss $E_J^k/E_C^s$, and the inductance energy $E_L^k/E_C^s$, differ from one step to the other due to the varying initial conditions. On the contrary, within the steady state regime both losses, and inductance energy are constant along the steps, $E_J^{ss}/E_C^s$ and $E_L^{ss}/E_C^s$ respectively. To optimize the overall charge process, the energies (current) overshoot associated with the transient steps must be reduced, making the stationary regime dominant. This goal can be achieved by controlling the transient through pre-charge and post-charge sequences. In what follows, a simple example will show how this problem can be tackled with. Let us consider a three-steps charging process and calculate its voltage and current response. Firstly, when the circuit is overdamped, for $\xi > 0.93$, no differences appear between steps, namely the whole charge process take place in steady state (as apparent from the dash-dotted lines in Fig. 2). Things are quite different when the circuit is underdamped, with $\xi \le 0.93$, where a non-negligible difference between steps appears. In this case, to reduce as much as possible the role of the transient regime, a pre-charge, and post-charge sequences (black continuous line in Fig. 2) are needed. The resulting adapted transient (bold continuous blue and red lines in Fig. 2) partially modifies the shape of the first, and of the last steps so that the non-stationary part of the charge process have energies below the stationary step. A more detailed description of the pre-charge and post-charge protocols can be found in Appendix V-D. The main points we want to make here can be summarized as follows:

* when $\xi > 0.93$, no specific sequence is required as the difference between the current response of the first, and last steps, and the steady state regime are negligible. The RLC voltage in the last step of the sequence still differs from the voltage on the capacitor (because $t_s$ is 99% of the RC charge, 1% remains), requiring an additional negligible time to complete the charge at the final voltage $U$;

* when $\xi < 0.93$, the transient leads to a surge in the energy and loss that can be limited by imposing a pre-charge, and a post-charge protocol that directly drives the system into the steady state regime (i.e. pre-charge) and allows to reach the final voltage $U$ avoiding an oscillatory transient (i.e. post-charge).

Furthermore, Fig. 3 shows a comparison between the normalized losses, and the inductance energy in the transient regime $E_J^t/E_C^s$, $E_L^t/E_C^s$ respectively (dash-dotted lines), and in the steady state one $E_J^{ss}/E_C^s$, $E_L^{ss}/E_C^s$ respectively (continuous lines). When $\xi \le 0.93$, because the pre-charge and post-charge count as one step, $E_J^t/E_C^s$, $E_L^t/E_C^s$ sum-up the Pre and Post charge (i.e. Pre & Post charge in Fig. 3). Moreover, when $\xi > 0.93$ beside the steady state continuous curves two dotted curves have been traced, one relative to the first step,

<figure><img src="image_2.png" /><figcaption>Fig. 2. Normalized applied current (right y-axis) and voltage (left y-axis) for a three level structure, with $\xi = 0.2 < 0.93$ (underdamped regime, full lines) and with $\xi = 1.01 > 0.93$ (overdamped regime, dash lines). The full thick lines highlight the pre-charge and the post-charge.</figcaption></figure>

the other to the last one. Fig. 3 shows quite unambiguously that: on the one hand, the difference between the first and last steps and the stationary regime are rather small in the overdamped regime; on the other, due to the adapted transient protocol (i.e. the pre-charge and post-charge voltage), the difference between steady state and transient regimes becomes negligible also in underdamped circuits.

It is worth noting that the number of steps *N*, and the damping factor $\xi$, represent the two key parameters to optimize the device. The former is inherent in the SMSC approach, and allows to reduce the maximum energy stored in the inductor, and the losses by decreasing $V_s$. The latter tunes the trade-off between the losses $E_J^{ss}/E_C^s$, and the inductance's energy $E_L^{ss}/E_C^s$, as apparent from Fig. 3 when comparing the blue and the red lines. Actually, when $\xi$ approaches to 1, $E_L^{ss}$ (i.e. the inductance volume) decreases sharply but the energy dissipated becomes nearly equal to the energy transferred per step, $E_J^{ss}/E_C^s \approx 1$, leading to a tiny charge efficiency. Conversely, at low $\xi$, $E_J^{ss}/E_C^s$ decreases while the ratio $E_L^{ss}/E_C^s$ keeps constant around 0.24. From Eq. (2), and Fig. 3, it appears that the Joule losses reduction is mostly related to the time factor reduction ($0.4/N$ when the circuit is overdamped to $0/N$ when it is underdamped) and to the current factor (passes only between $\approx 2.35$ when the circuit is overdamped to $\approx 1.23$ when it is underdamped).

The losses for the overall SMSC process, $E_J$, are the sum of the losses of all the steps (i.e. the steady state and the adapted transient ones). The Joule loss is normalized to the total energy transferred to the capacitor, $E_C = C(NV_s)^2/2$. As shown above, the difference between the transient and the steady state energies ratio are negligible, either because the circuit is overdamped, or because of the adapted transient protocol. Consequently, the sum over Joule loss per step can be approximated to a sum over *N* steady state terms. As, in general, transient losses are lower than steady-state ones, this approximation corresponds to a slight overestimation of the total loss. We can write,
<figure><img src="image_3.png" /><figcaption>Fig. 3. Normalized Joule losses and maximum energy stored in the inductance (i.e. energy - loss ratio). Full lines are used for the steady state, dash-dotted ones for the adapted transient step. The pre and the post charge loss have been summed up. In steady state, yellow dashed and continuous lines on the right axis show respectively the current factor and N times the time factors defined with of Eq. (2)</figcaption></figure>

$$ \frac{E_J}{E_C}(\xi, N) \approx \frac{\sum_{k=0}^{N-1} E_J^{ss}}{C(NV_s)^2/2} = \frac{1}{N} \frac{E_J^{ss}}{E_C^s}(\xi), \quad (3) $$

whence the main advantage of the SMSC strategy appears. Namely, decoupling the current response makes the sum over individual step losses proportional to $NV_s^2$, while $E_C \propto (NV_s)^2$, the normalized loss being eventually proportional to $1/N$. This way, increasing N makes possible a significant reduction of Joule losses. The inductance energy $E_L$ is the maximum energy stored along the steps,

$$ \frac{E_L}{E_C}(\xi, N) \approx \frac{\max(E_J^{ss}, E_J^t)}{C(NV_s)^2/2} = \frac{1}{N^2} \frac{E_L^{ss}}{E_C^s}(\xi). \quad (4) $$

It is worth noting that, as apparent from Fig. 3, and as shown in detail in Appendix V-E, neither $E_J^{ss}/E_C^s$, nor $E_L^{ss}/E_C^s$ depend on the step amplitude $V_s$. Indeed, they are function of $\xi$ only. Thence, Eq. (3) and Eq. (4) can be used to work out $E_J^{ss}/E_C^s$ as a function of $E_L^{ss}/E_C^s$ as plotted in Fig. 4 for different values of N.

From this Fig. 4, the two main behaviour of the system can be worked out:

* At fixed N (along full lines), when $\xi$ decreases, the losses $\frac{E_J}{E_C}$ are initially constant and the $\frac{E_L}{E_C}$ increases. Passing $\xi = 1$, losses decrease while $\frac{E_L}{E_C}$ is constant. This constant $\frac{E_L}{E_C}$ is achieved by decreasing the maximum current $i$ at the expense of an increasing of the inductance. Beyond a given value of L the inductor implementation becomes impossible due to the volume issue.

* At fixed $\xi$ (along dash lines), the $\frac{E_L^{ss}}{E_C^s}$ and $\frac{E_L^{ss}}{E_C^s}$ ratio are fixed (Fig. 3). Thus when increasing the number of stages by a factor N, the $\frac{E_L}{E_C}$ ratio scales as $1/N^2$ and the $\frac{E_L}{E_C}$ scales as $1/N$.

As an example to illustrate the mechanism underlying the loss/inductance energy reduction through the SMSC method, let us focus on the 2% loss ratio goal represented in Fig. 4 with a red horizontal line. Without SMSC (i.e. when the charge

<figure><img src="image_4.png" /><figcaption>Fig. 4. Losses versus the maximum energy stored in the inductance normalized with respect to the energy transferred to the capacitor. Each line corresponds to a different number of steps N, and is obtained for a damping ratio spanning an interval from 0 to 100. The red line corresponds to a 2% loss ratio. The dashed lines show the loss ratio at damping factor $\xi$ constant.</figcaption></figure>

process is carried out through a single step, $N = 1$), the loss ratio cross the red line at a rather high inductance energy ratio. Increasing the number of steps N makes possible a relevant inductance energy reduction. Fig. 4 clearly shows the trade-off between $E_J$, and $E_L$ in action. Indeed, targeting 2% losses ratio at charge we can either use a high inductance (i.e. high $\xi$), or increase the number of steps N.

### III. EXPERIMENTS AND DISCUSSION

The analytical results presented in the previous section have been validated against a series of measurements carried using the Solide-State Marx Modulator (S²M²) shown in Fig.5. Using this setup, SMSC can be implemented. A detailed description of the S²M² setup, and of its functioning has been recently published in [14]. The transistors used are 600 V n-channel MOSFET BSS225 from Infineon with an ON-resistance of 45 Ω. In order to fit within the analytical approach, the resistances of the MOSFET have been included into the overall resistance R. Finally, the R resistor takes into account the access resistor of the DEA, the series resistor of the inductor and the ON-state resistors of the MOSFETs (1.15 kΩ). The voltage drops in the L1207C101MPWST D₁,D₂...Dₙ₋₁ Diode (see Fig. 1) is not taken into account.

The RC parameter fit the cardiac assist device [15] with $R=10kΩ$ and $C=1$ nF.

It is noteworthy that a slightly oversized inductor bench, visible in Fig. 5, has been used for the sake of the versatility of the setup. To achieve the optimal size for a specific application, a dedicated inductor needs to be designed.

Fig. 6a, and 6b show the experimental, and theoretical voltage/current profiles for the underdamped, and overdamped cases respectively. The current $i$ and the voltage $V_C$ are measured by a voltage probe Tektronix P3010 in parallel with a shunt resistor and a voltage probe Cal Test CT4026 respectively. Theory and data are on the average in good agreement also thanks to the pre, and post charge protocols used to avoid any current surge. The small deviations between calculated, and measured profiles are easily explained by some
<figure><img src="image_5.png" /><figcaption>Fig. 5. Experimental setup: Solide-State Marx Modulator (the green disc) is connected to a load simulating an EAP, the capacitor is 978 pF (WIMA FKP1), the inductor (10 x B82144B1107J000 EPCOS / TDK) and the resistance (RS-201 resistance substituter from IET) can be tuned to adjust the damping factor. Voltages and the currents are acquired by an oscilloscope through voltage probes (Cal test CT 4026 and Tektronix P3010).</figcaption></figure>

features of the experimental setup. Notably, the slight lack of synchronization between the wave forms is due to the wireless power-transfer device [14] used to drive the stages. Moreover, there are voltage drops in the capacitor and in the MOSFETs. In addition, when switching the MOSFETs, the charging of the parasitic capacitance may require a longer time especially when the initial current is close to zero. This is causing some distortions in the voltage/current steps.

Seeking an experimental validation of the theoretical results, measurements of losses as a function of *N* have been carried adjusting the resistance in order to keep the damping factor *ξ* constant, (i.e. iso-*ξ* measurements), and keeping the loss ratio constant (i.e. iso-loss measurements). The measurements were carried out for *N*=8,12,18,19,24 on the 24-stages experimental prototype by closing the lowside MOSFETs of the unused stages. From the measurement of *i* and *V*<sub>C</sub> the energy transfer to the capacitor *E*<sub>C</sub>, the maximum energy stored in the inductor *E*<sub>L</sub> and the Joules Losses *E*<sub>J</sub> are computed, using the value of R, L and C.

The measurements are shown in Fig. 7 and in Fig. 8 (red squares). In both figures, the agreement between theoretical calculations and experimental data is apparent. It is worth noting that the iso-loss line (2.1% losses) stops at *N* = 12. This is due to the fact that below 12 steps an inductance *L* > 1 Henry would be needed, far beyond the possibility of the experimental setup used here, and definitely too cumbersome to be used in an actual device. This confirms the need of a SMSC approach to answer to the needs of actual devices deploying EAP, in terms of efficiency, and volume.

Furthermore, Fig. 7 shows that the energy loss, and the inductance energy scale, as a function of *N* as in Eq. (4) and Eq. (3), namely:

<figure><img src="image_6.png" /><figcaption>Fig. 6. Voltage (left axis) and current (right axis) on the EAP during the charge. Dashed lines show the theoretical calculations, full lines the measured data. The charge has been carried using a 24 steps SMSC with 66.7 V of input voltage.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 7. Energy ratio $\frac{E_L}{E_C}$ (in blue) and loss ratio $\frac{E_J}{E_C}$ (in red) with markers for the measurements and lines for the model. Two configurations are plotted, one with a constant $\xi$ ($\xi$=2.74 and $L$ = 3.3 mH), and one kept at constant loss (2.1%), star and the square markers respectively.</figcaption></figure>

* at constant $\xi$, $E_L/E_C \propto N^{-2}$, and $E_J/E_C \propto N^{-1}$;

* at constant loss, a trade-off between the number of stages *N* and the value of *ξ* (i.e. the value of the inductor) as to be taken into account for defining the optimal working
<figure><img src="image_8.png" /><figcaption>Fig. 8. Losses versus the maximum energy stored in the inductance normalized with respect to the energy transferred to the capacitor. Each line corresponds to a different number of steps N, and is obtained for a damping ratio spanning an interval from 0 to 100. The red line corresponds to a 2.1% loss ratio. The dashed lines show the loss ratio at damping factor ξ constant. The experimental points from Fig. 7 are represented by the red squares. (N=24, 19, 18, 12, 8 for iso-ξ and N=24, 12, 8 for iso-losses)</figcaption></figure>

point.

These are the main findings of this work as they show that SMSC technique represents a reliable answer to the need for reducing loss ratio keeping, at the same time, a reduced inductance volume. For example, the charging time needed to achieve 2% losses (i.e. 84% overall EAP efficiency) must be greater or equal to 1 ms, as can be deduced from Eq. (2). This can be done by slowing down the current thanks to a strong inductive effect (ξ ≪ 1) [3] or by splitting the voltage increase over N steps through the SMSC method, or adopting a wise balance between the two approaches based on the aforementioned trade-off between N and ξ.

This model takes into account the Joule losses within the MOSFET, the inductor and the access resistance of the EAP neglecting the switching losses. Actually, compared to the energy that is transferred to the load in the literature, these losses are rather small indeed. Considering our design, the output capacitance of the BSS225 MOSFET is 8pF. The number of switches per cycle of charge/discharge is equal to twice the number of stages. Each switch is achieved when the current through the transistor is close to zero, leading to a maximum switching loss of 80 µJ for a 14 kV load with 24 stages a rather negligible value indeed. The losses in the Marx generator for specific cases have been considered in detail elsewhere [14].

## IV. CONCLUSION

In the two extremes cases, the single step charge reduces the Joule losses to the cost of a bulky inductance of energy of the order of $E_C$ (sometimes unfeasible), whereas SMSC do it without any inductor but by increasing the number of stages.

Concluding, this work argue for the need of studying EAP efficiency jointly with the functioning of the converter used for charging. An integrated converter-EAP approach is presented where the key parameters have been quantified providing some useful guidelines to the design of the incoming generation of

EAP-based devices. Analytical, as well as experimental results are used to show that an N steps charge process (SMSC) entails an inductor energy (i.e. volume) scaling as $1/N^2$, and a Joule loss scaling a $1/N$. To go beyond these compact and key results, the analytical approach in equations 3 and 4 can be extended to consider the evolution of the R and C at each step, helping to take into account the non-linearity of the load (EAP, electrodes...)

Further investigations will be devoted to optimize the number of levels of the Solid-State Marx Modulator presented in [14] and to carry a detailed comparison between the SMSC method presented here and single-pulse approaches such as the pulse frequency modulation method reported in [16].

## V. APPENDIX

The following Appendixes detail the SMSC analytical and numerical solutions, they are organized as follows: Part A, describes how the time step is chosen taking into account the other time constants; Part B shows the analytical solutions for the current, and the voltage; Part C describes the evolution of the boundary conditions between each step; Part D describes how the energies are obtained; Part E describes how the pre and the post charge are obtained. Simulation files are available on request.

### A. Choice of the time step

Fig. 9 shows different RLC characteristic times, in particular the time step, $t_s$, intercalated between voltage steps. The other characteristic times are:

* $t_0$ — the time when the current pass at 0 when $\xi < 1$,

$$t_0 = \frac{\pi}{\omega_0 \sqrt{1 - \xi^2}} \quad (5)$$

* $t_{RC99}$ — the RC time constant at 99% of the full charge,

$$t_{RC99} = -RC\log(1 - 0.99) = -\frac{2\xi}{\omega_0}\log(1 - 0.99) \quad (6)$$

* $t_{RLC99}$ — the RLC time constant to reach 99% of the maximum voltage is extrapolated numerically;

Moreover, $\xi_{trans} \approx 0.93$ has been defined as the damping factor where $t_0 = t_{RC99}$.

### B. Step response of a RLC circuit

To ease the writing of the solution, the following variables are introduced:

$$\begin{aligned} c^+ &= (\xi + \sqrt{\xi^2 - 1}), \\ c^- &= (\xi - \sqrt{\xi^2 - 1}), \end{aligned} \quad (7)$$

whence the following useful relations,

$$\begin{aligned} c^+ c^- &= 1, \\ c^- - c^+ &= -2\sqrt{\xi^2 - 1}, \\ RC\omega_0 &= 2\xi. \end{aligned} \quad (8)$$

In the multi-steps case, as the voltage step is absorbed by the capacitor, it is more convenient to write the solution using the voltage difference $(k + 1)V_s - v_C(t)$. In addition, the
<figure><img src="image_9.png" /><figcaption>Fig. 9. Different characteristic times: the time when the current is zero in the underdamped regime, $t_0$, the numerical extrapolation of the time at 99% of the final voltage value, $t_{99}$ and the time step used in this work, $t_{step}$ to separate the voltage steps.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 10. Example of a charge process, on an underdamped circuit, without adapted transient. The values are: $\xi = 0.13$, $L = 152$ mH, $V_s = 1$ kV, $i_{max} = 127$ mA in steady state and 211 mA in transient.</figcaption></figure>

voltage response is normalized to the voltage step $V_s$, so that the response for step $k$ when $(k+1)t_s \le t \le kt_s$, writes:

$$ \frac{(k+1)V_s - v_C(t)}{V_s} = \frac{(k+1)V_s - v_C(kt_s)}{V_s} \frac{c^+e^{-c^+\theta} - c^-e^{-c^-\theta}}{2\sqrt{\xi^2 - 1}} + \frac{Ri(kt_s)}{V_s} \frac{e^{-c^+\theta} - e^{-c^-\theta}}{4\xi\sqrt{\xi^2 - 1}}, \quad (9) $$

and,

$$ \frac{Ri(t)}{V_s} = \frac{(k+1)V_s - v_C(kt_s)}{V_s} \frac{\xi c^+ c^- (e^{-c^+\theta} - e^{-c^-\theta})}{\sqrt{\xi^2 - 1}} + \frac{Ri(kt_s)}{V_s} \frac{c^+ e^{-c^+\theta} - c^- e^{-c^-\theta}}{2\sqrt{\xi^2 - 1}}, \quad (10) $$

where $\theta = \omega_0(t - kt_s)$. Figure 10 shows an example of a solution for a transient that has not been adapted through a pre- and post-charge protocol.

### C. Boundary conditions

The boundary conditions can change from one step to the other, this is what define the transient state, indeed. Boundary conditions can be written in terms of voltage difference and current (i.e $\Delta V[k] = [(k+1)V_s - v_C(kt_s)]$ and $i[k] = i(kt_s)$ respectively) for each step. It follows,

$$ \frac{(k+1)V_s - v_C(t)}{V_s} = \frac{\Delta V[k]}{V_s} \frac{c^+ e^{-c^+\theta} - c^- e^{-c^-\theta}}{2\sqrt{\xi^2 - 1}} + \frac{Ri[k]}{V_s} \frac{e^{-c^+\theta} - e^{-c^-\theta}}{4\xi\sqrt{\xi^2 - 1}}, \quad (11) $$

$$ \frac{Ri(t)}{V_s} = - \frac{\Delta V[k]}{V_s} \frac{\xi c^+ c^- (e^{-c^+\theta} - e^{-c^-\theta})}{\sqrt{\xi^2 - 1}} + \frac{Ri[k]}{V_s} \frac{c^+ e^{-c^+\theta} - c^- e^{-c^-\theta}}{2\sqrt{\xi^2 - 1}}. \quad (12) $$

The final voltage and current of a step (i.e $\theta = \theta_s = \omega_0 t_s$), define the boundary condition of the following one. For the current, the link is direct

$$ i_n[k+1] = i_n((k+1)t_s) \quad (13) $$

while for the voltage difference we have:

$$ \frac{(k+1)V_s - v_C((k+1)t_s)}{V_s} = \frac{(k+2)V_s - v_C((k+1)t_s)}{V_s} - 1 = \frac{\Delta V[k+1]}{V_s} - 1 \quad (14) $$

Coefficients VV, VI, IV, II, are introduced to define the boundary condition recursive relation,

$$ \begin{aligned} \frac{\Delta V[k+1]}{V_s} &= 1 - VV \frac{\Delta V[k]}{V_s} - VI \frac{Ri[k]}{V_s}, \\ \frac{Ri[k+1]}{V_s} &= IV \frac{\Delta V[k]}{V_s} + II \frac{Ri[k]}{V_s}. \end{aligned} \quad (15) $$

It is worth noting that those coefficients depend just on $\xi$,

$$ \begin{aligned} VV &= \frac{(\xi - \sqrt{\xi^2 - 1})e^{-c^+\theta_s} - (\xi + \sqrt{\xi^2 - 1})e^{-c^-\theta_s}}{2\sqrt{\xi^2 - 1}}, \\ VI &= -\frac{e^{-c^+\theta_s} - e^{-c^-\theta_s}}{4\xi\sqrt{\xi^2 - 1}}, \\ IV &= -\xi \frac{e^{-c^+\theta_s} - e^{-c^-\theta_s}}{\sqrt{\xi^2 - 1}}, \\ II &= \frac{(\xi + \sqrt{\xi^2 - 1})e^{-c^+\theta_s} - (\xi - \sqrt{\xi^2 - 1})e^{-c^-\theta_s}}{2\sqrt{\xi^2 - 1}}. \end{aligned} \quad (16) $$
<figure><img src="image_11.png" /><figcaption>Fig. 11. Steady state boundary conditions as a function of $\xi$. When $\xi < \xi_{trans}$, $\frac{\Delta V[\infty]}{V_s} = 1/(1 + e^{\frac{-\xi\pi}{\sqrt{1-\xi^2}}})$ and $i[\infty] = 0$</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 12. Example of normalized voltage and current for a pre- and a post-charge process with $\xi = 0.4$. Square dots show the limit between the pre- and the post-charge, which are connected to achieve a step of $V_s$</figcaption></figure>

Considering that the initial conditions of the recursive relations are $\Delta V[0]/V_s = 1$ and $R i[0]/V_s = 0$,

In steady state, the boundary conditions become

$$ \frac{\Delta V[\infty]}{V_s} = \frac{1 - II}{IV VI + (1 + VV)(1 - II)} \\ \frac{R i[\infty]}{V_s} = \frac{IV}{IV VI + (1 + VV)(1 - II)} \quad (17) $$

as shown in Fig. 11.

Once the steady state is reached, the underdamped regime benefits of a significant lowering of the voltage step while in the overdamped one, the initial conditions are nearly unchanged, and $t_s$ effectively decouple the response.

### D. Adapted transient

Underdamped circuits need an adapted transient in order to avoid current peaks that can be detrimental in terms of Joule losses and magnetic energy. In that case precharge/postcharge protocols are imposed.

The precharge voltage sequence is shown in Fig. 12. A constant voltage $V_s$ is applied for a duration $t_1$, followed by 0 voltage lasting a time $t_2$. Times $t_1$ and $t_2$ are tuned to force the system to reach the steady state at the following step.

The postcharge voltage sequence is shown in Fig. 12. Here, the voltage $V_s$ last a time $t_3$ then followed by 0 voltage during a time $t_4$. Times $t_3$ and $t_4$ aims to reach the maximum voltage $NV_s$ without overshoots. Times $t_1, t_2, t_3$ and $t_4$ are found numerically through the steepest descent method to minimize the Joules losses. The result of this numerical resolution are shown in figure 13.

### E. Energy stored in the inductance and Joule losses

The maximum current along a given step $k$ is reached when:

$$ \theta_{imax}[k] = \frac{\log\left(\left(\xi + \sqrt{\xi^2 - 1}\right)^2 \frac{2 \frac{\Delta V[k]}{V_s} \xi + \frac{R i[k]}{V_s} (\xi + \sqrt{\xi^2 - 1})}{2 \frac{\Delta V[k]}{V_s} \xi + \frac{R i[k]}{V_s} (\xi - \sqrt{\xi^2 - 1})}\right)}{2\sqrt{\xi^2 - 1}}, \quad (18) $$

<figure><img src="image_13.png" /><figcaption>Fig. 13. Values of $\omega_0 t_1, \omega_0 t_2, \omega_0 t_3$ and $\omega_0 t_4$ as a function of the damping factor.</figcaption></figure>

giving the following expression for the maximum energy stored in the inductance:

$$ \frac{E_L^k}{E_C^s} = \frac{1}{4\xi^2} \left( \frac{Ri(\theta_{imax}[k])}{Vs} \right)^2. \quad (19) $$

For the step $k$, the Joule loss writes,

$$ \frac{E_J^k}{E_C^s} = \frac{1}{\xi} \int_{k\omega_0 t_s}^{(k+1)\omega_0 t_s} \left(\frac{Ri(\theta)}{V_s}\right)^2 d\theta \quad (20) $$

$$ \frac{E_J^k}{E_C^s} = \frac{c^+ A^2 (1 - e^{-2c^+ \theta})}{2\xi(\xi^2 - 1)} + \frac{c^- B^2 (1 - e^{-2c^- \theta})}{2\xi(\xi^2 - 1)} + \frac{AB(1 - e^{-2\xi\theta})}{\xi^2(\xi^2 - 1)} \quad (21) $$

with $A = -\frac{\Delta V[k]}{V_s}\xi c^- + \frac{R i[k]}{V_s}/2$, $B = \frac{\Delta V[k]}{V_s}\xi c^+ - \frac{R i[k]}{V_s}/2$ and $\theta = \omega_0 t_s$.

Remarkably, the Joule loss and the inductance energy ratios depends only on the damping factor. Pre- / post-charge in case of underdamped circuits, and the first and the last step in case of overdamped ones, as well as the steady states, are plotted in Fig. 3.
# VI. ACKNOWLEDGEMENT

This work has been supported by France's national research agency on the ANR-20-CE05-0044, ANR-23-CE05-0012, Werner Siemens foundation.

## REFERENCES

[1] M. Almanza, F. Clavica, J. Chavanne, D. Moser, D. Obrist, T. Carrel, Y. Civet, and Y. Perriard, "Feasibility of a dielectric elastomer augmented aorta," *Advanced Science*, vol. n/a, p. 2001974, 2021.

[2] R. Ma, Z. Zhang, K. Tong, D. Huber, R. Kornbluh, Y. S. Ju, and Q. Pei, "Highly efficient electrocaloric cooling with electrostatic actuation," *Science*, vol. 357, no. 6356, pp. 1130-1134, 2017.

[3] Y. Meng, Z. Zhang, H. Wu, R. Wu, J. Wu, H. Wang, and Q. Pei, "A cascade electrocaloric cooling device for large temperature lift," *Nature*, 2020.

[4] B. Rechenbach, M. Willatzen, and B. Lassen, "Theoretical study of the electromechanical efficiency of a loaded tubular dielectric elastomer actuator," *Applied Mathematical Modelling*, vol. 40, pp. 1232-1246, Jan. 2016.

[5] E. Defay, R. Faye, G. Despesse, H. Strozyk, D. Sette, S. Crossley, X. Moya, and N. D. Mathur, "Enhanced electrocaloric efficiency via energy recovery," *Nature Communications*, vol. 9, no. 1, pp. 1-9, 2018.

[6] S. Paul, A. Schlaffer, and J. Nossek, "Optimal charging of capacitors," *IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications*, vol. 47, pp. 1009-1016, July 2000.

[7] Y. Perrin, A. Galisultanov, H. Fanet, and G. Pillonnet, "Optimal Charging of Nonlinear Capacitors," *IEEE Transactions on Power Electronics*, vol. 34, pp. 5023-5026, June 2019.

[8] V. Ravi, S. Satpathy, and N. Lakshminarasamma, "An Energy-Based Analysis for High Voltage Low Power Flyback Converter Feeding Capacitive Load," *IEEE Transactions on Power Electronics*, vol. 35, pp. 546-564, Jan. 2020.

[9] P. Thummala, H. Schneider, Z. Zhang, Z. Ouyang, A. Knott, and M. A. E. Andersen, "Efficiency Optimization by Considering the High-Voltage Flyback Transformer Parasitics Using an Automatic Winding Layout Technique," *IEEE Transactions on Power Electronics*, vol. 30, pp. 5755-5768, Oct. 2015.

[10] R. Mottet, M. Almanza, L. Pniak, A. Boegli, and Y. Perriard, "Ultra-high voltage (7kv) bi-directional flyback converter used to drive capacitive actuators," *IEEE Transactions on Industry Applications*, pp. 1-1, 2021.

[11] L. Pniak, M. Almanza, Y. Civet, and Y. Perriard, "Ultrahigh-Voltage Switch for Bidirectional DC-DC Converter Driving Dielectric Elastomer Actuator," *IEEE Transactions on Power Electronics*, vol. 35, pp. 13172-13181, Dec. 2020. Conference Name: IEEE Transactions on Power Electronics.

[12] T. Todorcevic, R. van Kessel, P. Bauer, and J. A. Ferreira, "A Modulation Strategy for Wide Voltage Output in DAB-Based DC-DC Modular Multilevel Converter for DEAP Wave Energy Conversion," *IEEE Journal of Emerging and Selected Topics in Power Electronics*, vol. 3, pp. 1171-1181, Dec. 2015.

[13] S. Park, L. Gu, and J. Rivas-Davila, "60 V-to-35 kV input-parallel output-series DC-DC converter using multi-level class-DE rectifiers," in *2018 IEEE Applied Power Electronics Conference and Exposition (APEC)*, Mar. 2018. ISSN: 2470-6647.

[14] M. Almanza, T. Martinez, M. Petit, Y. Civet, Y. Perriard, and M. LoBue, "Adaptation of a Solid-State Marx Modulator for Electroactive Polymer," *IEEE Transactions on Power Electronics*, vol. 37, pp. 13014-13021, Nov. 2022.

[15] M. Almanza, F. Clavica, J. Chavanne, D. Moser, D. Obrist, T. Carrel, Y. Civet, and Y. Perriard, "Feasibility of a Dielectric Elastomer Augmented Aorta," *Advanced Science*, vol. 8, p. 2001974, Mar. 2021.

[16] S. Mönch, R. Reiner, K. Mansour, P. Walteroit, M. Basler, R. Quay, C. Molin, S. Gebhardt, D. Bach, R. Binninger, and K. Bartholomé, "A 99.74% Efficient Capacitor-Charging Converter Using Partial Power Processing for Electrocalorics," *IEEE Journal of Emerging and Selected Topics in Power Electronics*, vol. 11, pp. 4491-4507, Aug. 2023.