# Charge Estimation of Piezoelectric Actuators: A Comparative Study

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## Abstract

**:**

## 1. Introduction

## 2. Background-Fundamentals of Position Control of Piezoelectric Actuators

_{e}stands for the excitation voltage, the input voltage to the piezoelectric actuator. A precise position sensor (e.g., among the ones listed in [15]) imposes considerable cost and more importantly serious limits in terms of space and calibration on the micro/nanopositioning system. Consequently, sensorless approaches have attracted a lot of attention [22,27,28]. The simplest control architecture for sensorless control is feedforward, as depicted in Figure 1b (similar to [29]); however, this architecture sacrifices accuracy, having no signal representing the actuator’s position.

_{P}, and the voltage across a sensing element, known as sensor voltage, V

_{S}. Figure 2 depicts a common arrangement to find V

_{P}and V

_{S}; evidently, if there is no sensing element in Figure 2, V

_{P}= V

_{e}.

_{P}has mostly been used as the input to a model to estimate position, e.g., in Figure 3. Sensorless control systems using V

_{P}are known as voltage-based sensorless systems. They are based on models mapping V

_{P}to position. A very wide range of research has been carried out to develop such models, due to inherited nonlinearity and the complexity of the V

_{P}and position relationship [30,31,32,33,34,35]. Inverted versions of these models have been developed to receive the desired positions and estimate their corresponding V

_{P}, which is equal to V

_{e}in the absence of sensing elements. Such inverted models can be the controller in Figure 1b [33,36].

_{S}is often used to estimate charge rather than position. Advantageously, charge and position are proportionally related in wide operating areas for many piezoelectric actuators [37]; thus, finding charge from position or vice versa is often not a difficult task. Hence, sensorless control systems using V

_{S}are known as charge-based sensorless systems [37,38]. Figure 4 shows a general schematic of such a control system.

## 3. Different Types of Charge Estimators for Piezoelectric Actuators

- Type I: Charge estimators with a sensing capacitor
- Type II: Charge estimators with a sensing resistor

#### 3.1. Type I—Charge Estimators with a Sensing Capacitor

_{S}. As V

_{S}is not applied on the actuator, it is also known as the voltage drop.

_{S}, the voltage across the sensing capacitor of C

_{S}, in the Laplace domain:

_{S}is the current passing the sensing capacitor. The voltage amplifier (the triangle) is not grounded; thus, only a tiny current passes through it. Therefore, I

_{S}is nearly equal to the current passing the piezoelectric actuator, I

_{P}:

_{S}≈ I

_{P}= qs,

_{S}, amplified by a voltage amplifier with a gain of C

_{S}, can estimate the charge:

_{b}, not included in Equations (1) and (3). I

_{b}is added to I

_{P}and is integrated by the sensing capacitor [39]:

_{b}, the estimated charge presented in (3) and Figure 5 ramps away the real charge of the piezoelectric actuator, q. This phenomenon is named ‘drift’ and has been observed since the emergence of charge estimators for piezoelectric actuators [38,43]. Two drift removal methods have been reported in the literature for type I charge estimators, initialisation circuits, and analogue high-pass filters, resulting in two sub-types of type I charge estimators.

#### 3.1.1. Type IA—Type I with an Initialisation Circuit

#### 3.1.2. Type IB—Type I with an Analogue High-Pass Filter

_{P}[41]. As to Figure 7, the voltage across R is same as the voltage across the actuator, as presented in (6), since they are in parallel.

_{R}is the current passing R. Noting that (i) the current passing the amplifier is negligible, (ii) I

_{P}= qs and (6), both (7) and (8) present I

_{S}; the current passing the sensing capacitor is shown in (7).

_{S}= V

_{S}C

_{S}s.

_{P}R)

^{−1}rad/s restrains the DC (or very low-frequency) current of I

_{b}, presented in (5). As a drawback, the filter suppresses other low-frequency components; consequently, these estimators may not capture low-frequency charge signals.

#### 3.2. Type II—Charge Estimators with a Sensing Resistor

_{P}, the current passing the piezoelectric actuator. As a result, (10) presents the voltage across the sensing resistor:

_{S}≃ I

_{P}R

_{S}.

_{b}, not considered in (10) and (11) [45]. In other words, V

_{S}+ V

_{b}enters the digital processor in reality rather than V

_{S}. Therefore,

## 4. Problem Statement

- The high-pass filter distorts low-frequency charge signals.
- The voltage across the sensing element, known as voltage drop, is not exerted on the actuator and is practically wasted.

_{S}with a data-driven model, and the methods depicted in Figure 7 and Figure 8 are used for high-frequency operating areas only, as described in [22,48]. This approach can be employed for both type IB and II charge estimators. Therefore, this drawback will not be further investigated in the paper.

_{S}, is required for charge estimation without loss of accuracy, as detailed in [26]. An important question is how much charge can be estimated with this inevitable voltage drop in two comparable type IB and II charge estimators.

## 5. Analytical Investigation

_{P}[49].

#### 5.1. Analytical Formulation for Type I Charge Estimators

_{P}:

_{S}:

_{S}is proportional to the excitation voltage, V

_{e}; therefore, a bias (time independent component) in V

_{e}leads to a bias in V

_{S}. In addition, considering A

_{e}and A

_{S}as the amplitudes of sinusoidal V

_{e}and V

_{S}, respectively, (15) leads to (16):

#### 5.2. Analytical Formulation for Type II Charge Estimators

_{e}= A

_{e}sin ωt, leads to a sensing voltage (also known as the voltage drop) of

_{S}= A

_{S}[sin ωt + 0.5π − arctan (R

_{S}C

_{P})] ≈ A

_{S}cos ωt.

_{P}is a very small number, as mentioned in Section 6, arctan (R

_{S}C

_{P}) ≈ 0.

_{e}and V

_{S}, A

_{S}, and A

_{e}have the following relationship:

_{e}= A

_{e}sin ωt + B, since (19) is linear, superposition may be used, and the sensing voltage, V

_{S}, can be assumed as the sum of two components influenced by A

_{e}sin ωt and B (bias or time-independent excitation). The final value of the component of V

_{S}influenced by B, V

_{SB}, is shown to be zero in (22):

#### 5.3. Results of Approximate Analytical Investigation

_{S}, for any given sinusoidal excitation voltage.

_{S}, for a given excitation voltage amplitude of A

_{e}, the sensing capacitor should be selected according to (23):

_{S}V

_{S}= C

_{S}A

_{S}sin ωt. Thus,

_{range-I}= 2C

_{S}A

_{S},

_{range-I}is the range of charge with a type I estimator in the case of a voltage drop amplitude of A

_{S}.

_{S}:

_{e}≫ 1 in Equations (16) and (27), Equation (28) can be derived to increase the comparability of type I and II estimators. In fact, the charge range of a type II estimator with resistance calculated with (26) is presented with a formula based on a capacitance calculated with (23):

_{range-II}= 2(C

_{S}+ C

_{P}) A

_{S}.

- Theoretically, (23)/(26) can calculate the sensing capacitance/resistance leading to a sinusoidal voltage drop with an amplitude of A
_{S}with a type I/II charge estimator, respectively, where A_{e}is the amplitude of the sinusoidal excitation voltage. - Based on (23) and (26), the sensing capacitance/resistance of type I/II estimators is dependent on/independent of the excitation frequency. This is a merit for type I estimators.
- A fixed component (bias) in excitation voltage leads to a fixed component voltage drop in type I estimators, as per (15) and superposition. Such a component (bias) has no enduring effect on the voltage drop in type II estimators, per (22).
- For an identical voltage drop, according to (24) and (28), type II estimators estimate a larger charge compared with type I ones, assuming A
_{e}≫ 1. However, considering the values of C_{S}and C_{P}reported in experiments, the difference in estimated charge is insubstantial.

## 6. Experimentation

^{3}piezoelectric stack, with epoxy coating and the code of SA070742 detailed in [51]. The actuator had a stiffness of 51 N/μm and capacitance, C

_{P}, of 6.23 µF, measured with an LCR meter at the amplitude of 1 V and the frequency of 1 kHz. V

_{e}, the excitation voltage in Figure 8 and Figure 9, was originally generated in Simulink then transferred to an AETECHRON 7114 liner power amplifier through the Simulink Real-Time Desktop Toolbox and the PCIe-6323 card.

_{e}= A

_{e}sin ωt; excitation frequency in Hz is defined as

_{e}= ω/2π.

_{S}= 1 V, as mentioned in the problem statement, based on approximate formulae derived in Section 5. In type I estimators, with the use of (16), for the sensing capacitors of 20, 40, and 80 µF, the theoretical A

_{e}values would be 4.21, 7.42, and 13.84 V, respectively. A sinusoidal excitation voltage with each of these values of amplitude was applied on a setup with its respective sensing capacitor. Excitation frequency, f

_{e}, had the values of 20, 30, 40, 50, 60, and 70 Hz for every pair of capacitors and A

_{e}. This means 18 experiments were performed to assess type I estimators. Similar experiments, with the same values of A

_{e}, were carried out for type II estimators; however, for each excitation frequency, the sensing resistance was calculated based on (26). In all experiments, the ranges of charge, q

_{range}, and A

_{S}(practically half of V

_{S}range) were measured. A sample time of 10

^{−4}s was used in all experiments. As an implementation point, the employed capacitors, e.g., the one shown in Figure 10, were bulkier than the resistors/potentiometer used.

## 7. Experimental Results and Discussion

_{S}= 1 V, (23) and (26) lead to R

_{S}and C

_{S}, shown in (30) and (31).

_{S}) of 1 V and the range of charge presented in (32) and (33) for any given sinusoidal excitation voltage with the amplitude of A

_{e}. (32) and (33) are the results from A

_{S}= 1 and (24) and (28). In summary, with R

_{S}and C

_{S}determined with (30) and (31), the following approximate theoretical outcomes are expected:

_{S}= 1 V

_{range-I}= 2C

_{S}.

_{range-II}= 2(C

_{S}+ C

_{P}).

#### 7.1. Observation 1: Discrepancy between Estimator Types in Frequency Dependency

_{e}presented in Table 1, Table 2 and Table 3, the voltage drop amplitude for type I estimators, A

_{S-I}, and their range of charge, q

_{range-I}, are nearly fixed across different excitation frequencies, f

_{e}. This is in agreement with finding (2) in Section 5, presented in (16), and is a substantial advantage. On the other hand, in type II estimators, for the higher values of A

_{e}presented in Table 2 and Table 3, A

_{S-II}and q

_{range-II}decrease meaningly with an increase in f

_{e}; however, the resistor changes with frequency according to (31) to maintain A

_{S}at 1 V.

**Table 1.**Experimental results for the excitation voltage amplitude of 4.21 V. Indices I and II refer to type I and II estimators.

A_{e} = 4.21 V, C_{S} = 20 μFq _{range-I-analytical} = 40 μC, q_{range-II-analytical} = 52.46 μC | |||||
---|---|---|---|---|---|

f_{e}(Hz) | R (Ω) | A_{S-I}(V) | A_{S-II}(V) | q_{range-I} (μC) | q_{range-II} (μC) |

20 | 303 | 1.10 | 0.99 | 44.14 | 53.07 |

30 | 202 | 1.10 | 1.00 | 44.08 | 54.62 |

40 | 152 | 1.10 | 1.00 | 44.01 | 54.35 |

50 | 121 | 1.10 | 1.02 | 43.88 | 54.19 |

60 | 101 | 1.10 | 0.97 | 43.94 | 52.77 |

70 | 87 | 1.10 | 0.96 | 43.94 | 53.64 |

**Table 2.**Experimental results for the excitation voltage amplitude of 7.42 V. Indices I and II refer to type I and II estimators.

A_{e} = 7.42 V, C_{S} = 40 μFq _{range-I-analytical} = 80 μC, q_{range-II-analytical} = 92.45 μC | |||||
---|---|---|---|---|---|

f_{e}(Hz) | R (Ω) | A_{S-I}(V) | A_{S-II}(V) | q_{range-I} (μC) | q_{range-II} (μC) |

20 | 172 | 1.16 | 1.05 | 92.49 | 99.30 |

30 | 115 | 1.16 | 1.04 | 92.49 | 99.88 |

40 | 86 | 1.15 | 0.99 | 92.36 | 92.68 |

50 | 69 | 1.16 | 0.94 | 92.62 | 88.30 |

60 | 57 | 1.15 | 0.92 | 92.36 | 87.25 |

70 | 49 | 1.14 | 0.83 | 91.05 | 82.96 |

**Table 3.**Experimental results for the excitation voltage amplitude of 13.84 V. Indices I and II refer to type I and II estimators.

A_{e} = 13.84 V, C_{S} = 80 μFq _{range-I-analytical} = 160 μC, q_{range-II-analytical} = 172 μC | |||||
---|---|---|---|---|---|

f_{e}(Hz) | R (Ω) | A_{S-I}(V) | A_{S-II}(V) | q_{range-I} (μC) | q_{range-II} (μC) |

20 | 92 | 1.30 | 1.14 | 207.6 | 204.2 |

30 | 62 | 1.29 | 1.07 | 206.3 | 1 97.8 |

40 | 46 | 1.29 | 1.00 | 206.0 | 193.0 |

50 | 37 | 1.28 | 0.99 | 205.3 | 194.8 |

60 | 31 | 1.28 | 0.87 | 204.7 | 177.7 |

70 | 26 | 1.28 | 0.81 | 204.5 | 168.8 |

#### 7.2. Observation II: Discrepancy between Theoretically Expected and Experimental Values of A_{S}

_{S}, in most experiments, are not equal to 1 V, despite theoretical expectations. This discrepancy simply means if (23) and (26) are trusted to find the capacitance/resistance of sensing components, an unexpected value of A

_{S}may happen to exist. Particularly, overly high values of A

_{S}may lead to serious issues as detailed in subsection VII.C of [26].

_{e}, a sinusoidal sensing voltage, V

_{S}, will be observed. Figure 11 depicts V

_{S}for two type II estimators both designed per (31) and excited with a sinusoidal voltage to assess this expectation. For the estimator designed for a higher amplitude of excitation voltage (A

_{e}), V

_{S}is not sinusoidal, as shown by the dashed curve in Figure 11, relevant to the shaded cell in Table 3; this demonstrates nonlinearity, which has been overlooked in the analytical formulation. However, V

_{S}for the other type II estimator, shown by a solid curve in Figure 11, relevant to the shaded cell of Table 1, is nearly sinusoidal. This indicates that nonlinearity is inapparent in some operating areas.

_{S}for type II estimators in Table 1, Table 2 and Table 3 are 0.0167 V, 0.0683 V, and 0.09 V, respectively. These values are obviously larger for type I estimators, 0.1 V, 0.1533 V, and 0.2867 V. However, the discrepancy can be avoided more simply in type I, as it is almost independent of frequency. By the way, the aforementioned discrepancy is no longer a major issue in terms of design, i.e., finding R

_{S}and C

_{S}values to realise a certain value of A

_{S}. Data-driven methods, based on artificial intelligence and statistics, are available to replace (23) and (26) in the design of estimators to assure A

_{S}remains in a safe zone with no or minor loss to the accuracy of charge estimation [26]. Those data-driven methods, nevertheless, provide no physical insight compared with analytical formulation.

#### 7.3. Observation III: Type II Estimators Are Capable of Estimating Slightly Higher Values of Charge

_{S}are different for type I and II estimators, particularly in Table 3. Table 4, alternatively, eases the comparison. This table shows that for the same amplitude of voltage drop (A

_{S}), a type II estimator estimates a 12% to 40% higher amount of charge than a type I estimator for the setup and conditions detailed in Section 6. However, this discrepancy is not as substantial as presented in the experimental results reported in [49]. For instance, for an excitation frequency of 10 Hz, the result in Table 2 of [49] can be reasonably interpreted as showing that type II estimators can estimate 892% more charge than type I ones with the same A

_{S}. Such results have been used as a ground for the superiority of type II estimators. The point is that the method of design (choice of sensing resistance/capacitance) in [49] is not backed by an analytical formulation; hence, the experiments were practically carried out with intuitively chosen values of R

_{S}and C

_{S}.

## 8. Conclusions

- C1.
- Type II estimators have a slightly higher ratio of estimated charge to voltage drop, 12% to 40%, according to experiments.
- C2.
- Both behaviour and the choice of the sensing element are independent of/dependent on frequency in type I/II estimators, as a major advantage of type I estimators.
- C3.
- Bias (a time-independent component) in the excitation voltage has an/no enduring effect on the behaviour of type I/II estimators.
- C4.
- In type I estimators, in order to obtain low voltage drops, high-capacitance sensing capacitors need to be employed. These capacitors are bulkier than the resistors used in type II estimators. Such a capacitor is shown in Figure 10.
- C5.
- Type I estimators can be implemented as an analogue circuit, e.g., Figure 7; while type II estimators need digital processors to be implemented.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**A schematic of (

**a**) a conventional feedback voltage-based system and (

**b**) a feedforward control system for a piezo-actuated nano/micro positioner.

**Figure 3.**A schematic of a voltage-based sensorless control system for a nano/micro positioner with a grounded piezoelectric actuator, where V

_{P}= V

_{e}.

**Figure 7.**Schematic of a charge estimator with a sensing capacitor and an analogue high-pass filter.

**Figure 9.**Proposed implementation of a charge estimator with a sensing capacitor and a digital high-pass filter and gain.

**Figure 10.**Implementation of Figure 9, excluding the computer and the amplifier.

**Figure 11.**Sensing voltage versus time for two type II estimators, with R = 92 Ω and R = 303 Ω excited with A

_{e}= 13.84 V and A

_{e}= 4.21 V, respectively, both with f

_{e}= 20 Hz; i.e., the excitation voltages for each are 13.84sin (20 × 2πt) and 4.21sin (20 × 2πt), respectively. Relevant cells in Table 1 and Table 3 are shaded.

**Table 4.**Experimental results for three amplitudes of excitation voltage. Indices I and II refer to type I and II estimators.

A_{e} = 4.21 VC _{S} = 20 μF | A_{e} = 7.42 VC _{S} = 40 μF | A_{e} = 13.84 VC _{S} = 80 μF | ||||
---|---|---|---|---|---|---|

f_{e} (Hz) | q_{range-I}A _{S-I} | q_{range-II}A _{S-II} | q_{range-I}A _{S-I} | q_{range-II}A _{S-II} | q_{range-I}A _{S-I} | q_{range-II}A _{S-II} |

20 | 40 | 53.60 | 80 | 94.20 | 160 | 179.4 |

30 | 40 | 54.53 | 80 | 96.40 | 160 | 185.3 |

40 | 40 | 54.53 | 80 | 93.77 | 160 | 193.0 |

50 | 40 | 53.23 | 80 | 93.87 | 160 | 196.1 |

60 | 40 | 54.20 | 80 | 94.40 | 160 | 203.5 |

70 | 40 | 56.04 | 80 | 99.69 | 160 | 209.1 |

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## Share and Cite

**MDPI and ACS Style**

Mohammadzaheri, M.; Al-Sulti, S.; Ghodsi, M.; Soltani, P.
Charge Estimation of Piezoelectric Actuators: A Comparative Study. *Energies* **2023**, *16*, 3982.
https://doi.org/10.3390/en16103982

**AMA Style**

Mohammadzaheri M, Al-Sulti S, Ghodsi M, Soltani P.
Charge Estimation of Piezoelectric Actuators: A Comparative Study. *Energies*. 2023; 16(10):3982.
https://doi.org/10.3390/en16103982

**Chicago/Turabian Style**

Mohammadzaheri, Morteza, Sami Al-Sulti, Mojtaba Ghodsi, and Payam Soltani.
2023. "Charge Estimation of Piezoelectric Actuators: A Comparative Study" *Energies* 16, no. 10: 3982.
https://doi.org/10.3390/en16103982