ANALYSIS OF A METHOD FOR MEASURING DEPOSIT IMPEDANCE PARAMETERS USING CHARGE AMPLIFIER AND LOCK-IN VOLTMETER

Methods for measuring deposit parameters are often based on a capacitance or conductivity measurement aimed at estimating, e.g. deposit moisture content. In practice, these methods fail for materials with a low degree of homogeneity, a diverse porous structure or high conductivity, e.g. due to a high water content. This article demonstrates an approach that enables a more precise estimation of the parameters of any deposit. The presented method involves the use of a measuring system in a charge amplifier configuration and the application of a technique using lock-in detection or a lock-in voltmeter to determine resistance and capacitance parameters of a deposit based on signals received from the measuring system. This method can be successfully used wherever the test deposit material is highly heterogeneous and contains both dielectric and conductive materials. The article presents an example of a solution to a measuring system using two planar electrodes that can be dimensioned depending on the deposit dimensions. It is followed by a presentation of a method for converting the signal from the measuring system into impedance parameters of the deposit using a lock-in voltmeter. The analysis of the operation of the entire measuring system was modelled in Matlab/Simulink, and the operation results were presented. Method for measuring deposit parameters In this paper, methods for measuring deposit parameters are to be understood as electrical quantities that characterise a deposit, such as conductivity and capacitance. These quantities are often used for indirect measurement of other parameters, e.g. deposit material moisture content or bulk density or for spatial imaging. A study by (Anton Fuchs et al., 2008) (Yuesheng Tan et al., 2017) presents a measurement of solid material deposit capacitance parameters taken to determine the moisture content. The measuring element applied was a sensor in the form of two planar electrodes. The test materials

method involves the use of a measuring system in a charge amplifier configuration and the application of a technique using lock-in detection or a lock-in voltmeter to determine resistance and capacitance parameters of a deposit based on signals received from the measuring system. This method can be successfully used wherever the test deposit material is highly heterogeneous and contains both dielectric and conductive materials. The article presents an example of a solution to a measuring system using two planar electrodes that can be dimensioned depending on the deposit dimensions. It is followed by a presentation of a method for converting the signal from the measuring system into impedance parameters of the deposit using a lock-in voltmeter. The analysis of the operation of the entire measuring system was modelled in Matlab/Simulink, and the operation results were presented.

Method for measuring deposit parameters
In this paper, methods for measuring deposit parameters are to be understood as electrical quantities that characterise a deposit, such as conductivity and capacitance. These quantities are often used for indirect measurement of other parameters, e.g. deposit material moisture content or bulk density or for spatial imaging.
A study by (Anton Fuchs et al., 2008) (Yuesheng Tan et al., 2017) presents a measurement of solid material deposit capacitance parameters taken to determine the moisture content. The measuring element applied was a sensor in the form of two planar electrodes. The test materials 3 were pellets, homogeneously structured powders and plant materials. A different approach is presented in a study by (Yuesheng Tan et al., 2017), which examined the possibility for the use of deposit capacitance parameters to approximately determine the bulk density of miscanthus.
In addition, the effects of the moisture content and particle size on the measurement results were investigated. An interesting application of capacitance parameters of a deposit is presented in a study by (Aulen and Shipley, 2012), where deposit capacitance parameters within the nanofarad range were used to assess the root system of crop plants.
Methods for measuring the impedance parameters of a deposit are also used in imaging techniques, e.g. electrical capacitance tomography (ECT) or electrical impedance tomography (EIT). Work in this area was performed in a study by Wegleiter (2006). Capacitance tomography is also used in medical sciences. Ambika M. et al. (2019) used capacitance tomography to analyse bone density. Capacitance tomography systems using methods which also enable impedance measurements were also demonstrated in a study by Smolik W. (2017). The methods presented in the study is based on the application of a lock-in amplifier as a method for measuring very low capacitances of the fF fraction order as well as impedance through the application of an additional reference signal. This paper will focus on deposit material in the form of a thick, porous layer with a heterogeneous structure characterised by a significant gradient of moisture content changes in space. An example of such a deposit is presented in Fig. 1.

Theoretical introduction
The method for measuring impedance parameters is based on the measurement of electrical conductivity and the capacitance of a deposit. It is assumed that the deposit will be placed in a reactor equipped with two parallelly arranged planar electrodes with a thin galvanic isolation layer. The system obtained in this way will be characterised by a certain capacitance of the order of single pF in the absence of a deposit and of the order of a maximum of several tens of pF in the presence of a very moist material. Conductivity between the electrodes will be strongly determined by the moisture content of the material placed between the electrodes. It was also assumed that the planar electrode surfaces had dimensions ensuring that electric field lines would penetrate through the largest possible cross-section of the deposit, which is aimed at ensuring the measurement of parameters for the largest possible volume of the test sample. Based on assumptions thus defined, the test system can be presented in the form of a planar loss capacitor 4 with electrode surfaces S and the distance between them d. The system described in this way can be presented schematically, as in Fig. 4.
If there is air between the plates, the capacitor capacitance is described by the following relationship: where: − 8.854 • 10 −12 / − ; The relative dielectric constant for the air is 1.0006. Typical dielectric materials, e.g. plastic or oil, are characterised by a time-constant ranging from 3 to 10, while for polar fluids, e.g.
water, the time-constant is 50 and more, depending on the temperature. Fig. 2 presents the distribution of field force lines for a planar capacitor. (Baxter L., 1997) Where the distances d for the electrodes are significant in relation to each other, in relation to plane A, the field lines at the planar capacitor electrode edges are arranged in semi-circles. This is such a strong phenomenon that formula (1) is only valid for small distances d between electrodes, such that the field lines are perpendicular to the electrode surfaces. For example,

Fig. 2. Planar capacitor's field force lines
where the plates with a surface S=1 m 2 are separated by a distance d=1 mm, the planar capacitor capacitance is 88.54 pF, and is consistent with expression (1). Where the distance between electrodes is considerable, the so-called edge fields (which are not perpendicular in relation to the electrodes' internal surface) are formed at the electrode edges. Therefore, the capacitance between the electrodes can be considerably greater than that determined using formula (1). Fig.   3 presents a distorted arrangement of the field lines at the electrode edges and a way to counteract this phenomenon. -guard electrode (Larry B., 1997) The introduction of guard electrodes counteracts this phenomenon and brings the distorted field lines at the edges to the ground potential. Thanks to this solution, the distorted fields at the edges are insignificant for the measurement, and only the field lines that are perpendicular to the electrode plane play a role. Because of the application of guard electrodes, the distorted field lines generate a capacitance that is connected to the ground potential. Fig. 4 presents an electrical diagram of connections that eliminates the impact of field distortions at the electrode edges on the measurement result.

Fig. 4. A diagram of electrical connections of guard and measurement electrodes that neutralise field distortions
The use of guard electrodes enables increasing the distance d of guard electrodes to a size allowing a porous deposit to be placed between the electrodes.

Measuring system
The system for measuring impedance parameters of a deposit needs to exhibit characteristics that will ensure the measurement of small values of the physical parameters between the measurement electrodes. It should be stressed that parasitic capacitances of the measuring circuits can be greater than the capacitance between the electrodes by an order of magnitude. The study adopted an electrode model using a loss capacitor with capacitance C and parallel resistance Rp. The electrode circuit model is shown in Fig. 5.

Fig. 5. A model of the measurement electrode circuit as a loss capacitor with capacitance C and loss resistance Rp. Phasor diagram of currents for the circuit
For dielectrics such as plastics or ceramics, the loss factor D is relatively constant with an increase in frequency. On the other hand, for water, the loss factor D changes 4-fold with a change in frequency from 100 kHz to 1 MHz at a constant temperature of 25 ºC. Measurement of parameters for water below 100 kHz is difficult as the loss angle δ is almost 90°. At 100 kHz, the loss factor D amounts to 4, which means that the impedance angle is 76°, and the resistive element has a much lower impedance than that of the reactance element. 6 Fig. 9 shows that for materials containing water, the loss factor D is a more sensitive indicator than the relative permittivity εr. A change in water temperature from 15 to 75 ºC results in a 3-fold increase in the loss factor and a 2.5-fold decrease in the dielectric constant.
The design of the measuring system assumes the use of a two-electrode measuring element that will be subjected to sinusoidal excitations in order to obtain the best possible signal-to-noise ratio (Rofee J., 1997). A basic diagram of the measuring transducer circuit is presented in Fig. 7.
while electrode #2 serves as a detection electrode. The signal in the form of an electric charge Q will be received from the detection probe and converted into a value proportional to charge Qx transferred between the electrodes. The presented configuration is known in the literature as a charge amplifier and is widely used in applications in which the output signal is proportional to the electric charge. The relationship between the excitation signal and the output signal from the charge amplifier is described by the following relationship: where: It follows from expression (3) that it is possible to easily determine the unknown capacitance Cx based on the parameters of the input signal, output signal and the known feedback capacitance Cf.
The solution presented above is suitable for materials for which the conductivity between electrodes is negligibly low, i.e. of the order of μS. For conductive material, e.g. contaminated water, the situation changes, and conductivity has a greater influence on changes in the deposit impedance. Thus, a measuring transducer circuit in the form presented in Fig. 8 is obtained.

Fig. 9. Block diagram of the lock-in voltmeter circuit. X -real component, Y -imaginary component, Z -module, a -phase shift
In accordance with the linear system theory, a signal with the same frequency but with a changed amplitude and the angular shift of will be obtained on detection electrode #2.
On the other hand, signals representing the real component , imaginary component , module | |, and phase shift will be obtained at the output of the lock-in voltmeter: = | |cos ( ) = | |sin ( ) (8) Based on the quantities obtained from the lock-in voltmeter and relationship (4), it is possible to determine the parameters of the sought impedance Zx. Following the substitutions and conversions, the sought Zx and its components are described with the following relationships: where: Matlab/Simulink is presented in Fig. 10.

Fig. 10. A simulation model of the deposit impedance parameter measuring system
The simulation model comprises three independent blocks. The first block is responsible for simulations of the behaviour of measurement electrodes along with measuring transducer in accordance with equation (4) and the electrode model in the form of a loss capacitor with impedance Zx. The second block simulates the operation of a lock-in voltmeter. In this block, synchronous detection and low-pass filtration occur, and the signal parameters in relation to the reference signal are then determined. The sinusoidal signal described by equation (5), which, at the same time, is the excitation signal of measurement electrode #1, was used as the reference signal. In a system with a single reference signal, it is possible to determine the amplitude Uao of the test signal, which is received from measurement electrode #2. The determination of the phase shift between the reference signal and the test signal from detection electrode #2 requires the use of an additional reference signal phase-shifted by 90°. As a result of lock-in detection and the separation of the constant component using a third-order Butterworth low-pass filter with the corner frequency of 10 kHz, the parameters , , | | and are obtained. The third block is used to determine impedance parameters of the test deposit according to equations (12)-(16).
Simulation testing was conducted for feedback parameters = 22 and = 220 . In practice, the feedback parameters should be set to the expected variability of the deposit parameters in order to obtain flat amplification characteristics over a wide frequency range. Fig. 11 shows an example of the characteristics of the shift of a measuring transducer with ideal WO for the parameters shown in Table 1. It follows from the characteristics provided in Fig. 11 that for the assumed feedback parameters, it is possible to obtain a 3 transfer band from a frequency of 200 krad/s.   The results of simulation testing of the system for measuring deposit parameters are presented in Tables 2 and 3. The simulation testing was conducted by setting capacitance Cx and resistance Rx that model the measurement electrodes, followed by simulations of the system with the parameters set. Cf. This is due to the fact that charge Qx accumulated on measurement electrodes #1 and #2 is presented by the following expression: and, from relationships (3) and (17) Obviously, the signal from the measuring transducer requires further amplification in real measuring systems, although this is not required for simulation purposes

Conclusions
The method for measuring impedance parameters of a deposit using a charge amplifier and a lock-in voltmeter allows very high accuracy to be achieved. The results of the simulation model testing lead to the conclusion that accuracy is limited only and exclusively by the accuracy of numerical methods of simulation programs. This is due to the fact that the process of determining impedance parameters in a numerical manner results directly from algebraic equations (7)-(16). A properly selected low-pass filter, which separates the constant component from the variable component, which has a frequency twice as high as that of the excitation signal, guarantees accuracy arising directly from the numerical accuracy of algebraic transformations.
The presented method ensures the certainty of the results obtained through simulation, but the situation is different as regards practical implementations. In practice, the process of detection 11 from the detection electrode is affected by the influence of many interfering quantities, mainly in the form of parasitic capacitances of connections. These quantities are often higher by an order of magnitude than those being measured. Proper design of measuring circuits, shielding from external fields and proper design of guard electrodes will eliminate or significantly reduce their impact on the measurement process. As demonstrated using the examples of measuring transducer parameter values from relationship (19), it is possible to detect changes in capacitance on measurement electrodes of the femtofarad order. Methods using lock-in voltmeters are, due to their properties, often employed in many technical solutions, inter alia in capacitance tomography systems. The use of this method for measuring deposit impedance parameters will allow high accuracy to be achieved over a wide range of parameter changes. The practical implementation of a measuring system requires the preparation of measurement electrodes and their shielding and the selection of suitable electronic systems that ensure appropriate operation frequencies. The practical implementation can be carried out entirely using an analogue technique or partly using a digital technique.   Figure 1.