Citation: Boumediene Benyahia, Tewfik Sari. Effect of a new variable integration on steady states of a two-step Anaerobic Digestion Model[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5504-5533. doi: 10.3934/mbe.2020296
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| [2] | Andrés Donoso-Bravo, Pedro Gajardo, Matthieu Sebbah, Diego Vicencio . Comparison of performance in an anaerobic digestion process: one-reactor vs two-reactor configurations. Mathematical Biosciences and Engineering, 2019, 16(4): 2447-2465. doi: 10.3934/mbe.2019122 |
| [3] | Tewfik Sari, Miled El Hajji, Jérôme Harmand . The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat. Mathematical Biosciences and Engineering, 2012, 9(3): 627-645. doi: 10.3934/mbe.2012.9.627 |
| [4] | Amer Hassan Albargi, Miled El Hajji . Mathematical analysis of a two-tiered microbial food-web model for the anaerobic digestion process. Mathematical Biosciences and Engineering, 2023, 20(4): 6591-6611. doi: 10.3934/mbe.2023283 |
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The neuron-electrode interface is crucial for many biomedical applications such as [1], retina implants [2], deep brain stimulation [3], closed loop deep brain stimulation [4] or electrocorticography [5]. For electrodes used for electrically stimulating neural tissue a large charge injection capacity is required to guarantee effective stimulation with parameters that are safe for biological tissue. Recording electrodes on the other hand should have low impedance resulting in low thermal noise and a high signal-to-noise ratio (SNR). In both cases the geometrical surface area of neuroelectrodes should be in the dimension of the cell size to guarantee the detection or stimulation of single cell activity. Small electrodes further allow arranging multiple electrodes in a high density array enabling high spatial resolution. One parameter that influences the SNR is the noise of the electrode which is mainly of thermal origin. Consequently noise amplitudes can be reduced by reducing the electrodesx impedance. Since the impedance is dominantly capacitive this can be achieved by increasing the electrochemical active surface area. A large electrochemical surface area in turn can be achieved by a nanostructured or rough electrode material such as carbon nanotubes [6,7,8], gold nanopillars [9,10], Polypyrrol (PPy) [11] or PEDOT [12]. Although there has been a lot of experimental work on different electrode materials, there is a lack of theoretical models that help to understand and predict the behavior at the interface.
Here we present a surface area based approach to model and simulate the electrode behavior and compare the theoretical expectations with real characteristics of manufactured microelectrodes. As model systems unordered carbon nanotubes and nearly vertically aligned gold nanowires are integrated onto electrodes. In section 2.1 methods of how to integrate the high aspect ratio nano objects onto microelectrodes are presented. In section 2.2 the interface between electrode and neuron is modelled by the well-known point-contact-model. To investigate the influence of biological and electrical parameters on the efficiency of a neuroelectrode, a transfer function describing the interface between neuronal membrane potential and measurable extracellular potential is evolved. From this holistic model a simplified electrode model is derived and the expected properties simulated in MatlabTM (The MathWorks Inc. , Natick, USA). Finally these results are compared to real electrodes behavior manufactured by standard microelectromechanical systems (MEMS) technology. Cyclic voltammetry as well as electrochemical impedance spectroscopy are performed on manufactured microelectrodes to investigate the real characteristics and compare it to the simulated data.
A widely used technique to measure and stimulate neurons in-vitro and in-vivo is an array of microelectrodes (microelectrode array, MEA), suggested in different configurations, e.g. different electrode sizes, distances, arrangements or shapes [13,14,15,16]. Here a microelectrode array with an electrode diameter of 30 µm and distance (center to center) of 200 µm is designed. Metallic microelectrodes and circuit paths are structured lithographically and insulated by a thin layer of polyimide or photoresist. Microelectrodes and contact pads are as well opened lithographically and two types of nano objects are integrated subsequently.
Several approaches have been described to manufacture microelectrodes with integrated unordered carbon nanotubes (CNTs) such as chemical vapor depositioning [17], electroplating CNT/polymer composites [18] or transferring CNTs as dispersion [19]. The manufacturing process applied here is described in detail in [20] and schematically depicted in Figure 1. In short, multiwalled carbon nanotubes (Sigma-Aldrich, Taufkirchen, Germany) are transferred in a 1mg/ml-dispersion in N-Methyl-2-pyrrolidon (NMP), dispensed onto a microelectrode array (Figure 1a-b) and the solvent is allowed to evaporate. After removing excessive CNTs with a squeegee, the nano objects remain in the electrode cavities exclusively as depicted in Figure 1c.
Figure 1. Integration of carbon nanotubes onto microelectrodes. a) An array of metallic microelectrodes is manufactured by standard MEMS-technology. b) The CNT-dispersion is applied and the solvent allowed to evaporate. c) After the removal of exessive nanotubes, CNTs can only be found in the electrodes cavity.
Gold nanowires can be integrated by either seed mediated growth [21] or electrodepositioning gold into a template such as aluminum oxide [9] or track etched polycarbonate (PC) membranes [22]. The manufacturing process applied here is described in detail in [10] and depicted in Figure 2. In short a track etched PC-membrane is thermally laminated onto a previously structured photoresist and thus integrated into the microsystem (Figure 2a). Gold is subsequently electrodepositioned into the pores of the membrane (Figure 2b), which is finally removed in a solvent (Figure 2c).
Figure 2. Integration of gold nanowires. a) A track etched PC-membrane is thermally laminated onto a structured thin layer of photoresist. b) the pores are subsequently filled with gold. After removal of the template free standing nanowires are integrated onto the microelectroes.
The interface between a neuron and a microelectrode is modeled by a point-contact model [23] as depicted in Figure 3, where the interface between electrode and electrolyte is modeled by a parallel circuit of the capacitor Cel and resistor Rel. Similarly, the cell membrane is described by Cm and Rm. Membrane segments that do not participate in the interface are indexed with nm (non-coupling membrane). These Cnm and Rnm are neglected for further investigations. The ohmic resistance of the electrolyte is represented by Rspread and the resistance of the metallic leads by Rmet. The quality of the cell adhesion can be modeled by the seal-resistance Rseal. The closer the coupling the higher is its value. The insulation layer causes a capacity Cshunt in the circuit. The voltage is measured between the microelectrode and a reference electrode, which is represented by a Rref - Cref parallel circuit. This impedance however can be neglected because it is very small due to the large size of the reference electrode. Finally the used amplifier is represented by its input impedance Zamp, which is a R-C parallel circuit (not shown). Since Cshunt is in parallel to the amplifier capacity these two capacities can be combined. Um is the intracellular and Uout the measurable voltage.
Figure 3. Derivation of the transfer function from the point-contactmodel. a) Point-contact- model of the neuron-electrode interface. All parameters are explained in detail in the text. b)Equivalent circuit derived from the point-contact model of the neuron-electrode interface after simplification. The impedances Zamp, Zm and Zel are given in Equations 1-3. For clarification the two meshes M1 and M2 and the knot K1 are marked, as well as some specific currents and voltages.
With these considerations and the impedances Zamp, Zm and Zel given in Equations 1-3 a simplified equivalent circuit (see Figure 3b) is evolved from the complex point-contact-model in Figure 3a.
| Zamp(jω)=Rop1+jωCshuntRop |
(1) |
| Zm(jω)=Rm1+jωCmRm |
(2) |
| Zel(jω)=Rel1+jωCelRel+Rspread+Rmet |
(3) |
From this simplified equivalent circuit the following equations can be derived [24,25,26]. Here Equations 4 and 5 describe the relationship between voltage and current applying the respective impedance. Equation 6 and 7 are extracted from mesh M1 and mesh M2 respectively. Finally Equation 8 is the knot equation in K1.
| Useal(jω)=Rseal⋅Iseal(jω) |
(4) |
| Uout(jw) = Zamp(jw)·Iel(jw) | (5) |
| Um(jω)−Useal(jω)=Zm(jω)⋅Im(jω) |
(6) |
| Umseal(jω)−Uout(jω)=Zel(jω)⋅Iel(jω) |
(7) |
| Im(jω)=Iseal(jω)+Iel(jω) |
(8) |
| H(jω)=ZampRseal(Rseal+Zm)(Zamp+Zel)+RsealZm |
(9) |
The transfer function H(jω) generally is defined as ratio between input and output voltage, in our case @{\rm{H}}\left( {{\rm{j\omega }}} \right) = \frac{{{{\rm{U}}_{{\rm{out}}}}}}{{{{\rm{U}}_{\rm{m}}}}}@. By inserting Equations 1-3 and 4-8, the transfer function can be written as below.
If typical values from literature are inserted into these equations the transfer function can be depicted in the form of a bode plot as shown in Figure 4. Here the following values were used [25,26,27]: Cm = 293 pF, Rm = 34 MΩ, Cel = 1, 14 nF, Rel = 140 kΩ, Rms = 12 kΩ, Rseal = 1 MΩ, Cshunt = 20 pF and Rop = 100 GΩ.
Figure 4. Bode plot of the electrode-neuron transfer function H(jω) according to Equation 6 with typical values taken from literature: Cm =293 pF, Rm = 34 MΩ, Cel = 1,14 nF, Rel = 140 kΩ, Rms =12kΩ, Rseal =1 MΩ, Cshunt = 20 pF and Rop =100 GΩ.
Because parameters like the electrode properties Cel and Rel, the amplifier impedance Zamp or the capacity Cshunt only have a minor effect on the transfer function or can hardly be adjusted in reality like the membrane properties Cm and Rm simulation results are not plotted here. However the transfer function is dramatically influenced by the coupling resistance Rseal which is shown in Figure 5.
Figure 5. Bode plot of the electrode-neuron transfer function with different values for the coupling resistance Rseal. The higher the value the lower is the damping of the signal. This leads to a high measurable signal amplitude at the amplifier.
Obviously the coupling resistance Rseal has a great impact on the transfer function as emphasized in Table 1, where magnitude and phase is summarized at the relevant frequency of f = 1 kHz (ω = 6,283 rad/s). This frequency is typically used to evaluate and compare electrode performances because it is within the frequency spectrum of neural activity (800-3, 000 Hz) [28].
| Rseal | 1 kΩ | 100 kΩ | 1 MΩ | 10 MΩ | 1 GΩ |
| Magnitude |H(jω)| in dB at f = 1 kHz | −54.77 | −14.96 | −1.7 | −0.66 | −0.65 |
| Phase ∠H(jω) in ° at f = 1 kHz | 88.39 | 77.41 | 26.17 | 2.31 | −0.58 |
DownLoad: CSVFrom these results it can be concluded that the measurable signal amplitude can be highly increased by improving the coupling between neuron and electrode. Although the electrode properties itself have only a minor influence on the transfer function, the electrode surface can in fact influence the coupling of a neuron onto the electrode. It has been shown that neurons show a very good coupling to nanostructured surfaces [29,30]. Thus integrating nano objects onto microelectrodes will not only decrease the impedance (see Section 4) and lower the thermal noise but might also increase the signal amplitude by improving the coupling between cell and electrode.
To simulate and predict the behavior of an electrode it is sufficient to focus on the electrodes impedance Zel (see Equation 3). The charge transfer resistance of an electrode, Rel, is typically rather large and can be omitted. Thus if only the electrode characteristics are simulated, an even further simplified equivalent circuit of the electrodes capacity Cel in series with the two resistors Rspread and Rmet, which are combined as Rms = Rspread + Rmet, can be used. Since the impedance determines the thermal noise of the electrode, the effects of integrated carbon nanotubes and gold nanowires shall be studied in the following section. Using the simplified equivalent circuit an electrodes impedance is given by Equation 10 which is a simplification of Equation 3.
| Zel=Rms+1jωCel |
(10) |
Since the capacity is linearly increasing with surface area, the impedance is reduced by the integration of nano objects.
Carbon nanotubes are tubes of two-dimensional graphene with extraordinary electrical, mechanical and chemical properties. By integrating these nano objects onto microelectrodes the electrochemically active surface area is expected to increase dramatically. To model the properties of electrodes with integrated nanotubes, the expected surface area has to be calculated. Assuming a perfectly cylindrical form the gain in surface area RA,CNT can be calculated with Equation 11. It is assumed that the inside and the outside walls of carbon nanotubes are wetted by electrolyte. If this is not the case the inner diameter rin can be set to 0 and the nanotubes be treated as nanowires.
| RA,CNT=nCNTACNTA+1 |
(11) |
| RA,CNT=nCNT2π(rout+rin)lCNTA+1, |
where nCNT is the number of carbon nanotubes inside an electrode cavity, ACNT is the total surface area of a single carbon nanotube and A is the geometrical surface of the microelectrode. The number of CNTs inside a cavity can be estimated by the cavities and the CNTs volumes as done in Equation 12. Since not the entire cavity volume is filled with nanotubes, a factor pfill is introduced, representing the ratio of filled volume
| nCNT=pfillVcavityVCNT=pfillAhisoπr2outlCNT, |
(12) |
with Vcavity being the volume of the electrode cavity, VCNT the volume of a single CNT, hiso the height of the insulation layer and lCNT the length of a single nanotube. Inserting Equation 12 into Equation 11 an easy formula to calculate the win of surface area as depicted in Equation 13 is derived
| RA,CNT=2pfillhiso(rout+rin)r2out+1, |
(13) |
with a typical length of lCNT= 1.5 µm, and radii of rout = 4.25 nm and rin = 3.1 nm [31], a height of hiso = 5 µm and a filling ratio of pfill = 1/3 RA,CNT equals 1,357. Figure 6 depicts the bode plots for a planar gold electrode with Cel = 0.1 nF and Rms = 10 kΩ [9] and a CNT electrode with increased capacity. The value for the resistor Rms is kept constant.
Figure 6. Bode plot for planar gold electrodes and electrodes with integrated carbon nanotubes. Obviously the nanostructures lead to a highly decreased impedance over the entire frequency range. For frequencies higher than approximatly 100 Hz when phase is larger than −45° the resistor dominates the electrodes behavior of CNT electrodes.
Clearly the integration of carbon nanotubes leads to a decrease in electrode impedance over the entire frequency range. For higher frequencies larger than approximately 100 Hz, when phase is larger than −45° the resistor dominates the impedance of CNT electrodes, resulting in a magnitude of approximately 10 kΩ. A phase angle of 0° is approached for frequencies in excess of 103 to 104 kHz.
The noble metal gold is known for its biocompatibility and high corrosion resistivity and thus ideal for this application. Although the material itself has no impact on the simulation, measurements will be done with gold. In contrast to carbon nanotubes metallic nanowires are bulk material and thus only the (outer) surface of the nanowires add to the new overall electrode area. By integrating nanowires into the microsystem the capacity can be increased by increasing the active surface area. The gain in surface area RA, AuNW in this case depends on the height of the wires
| RA,AuNW=2πrAuNWhAuNWρAuNW+1, |
(14) |
where rAuNW is the radius of a gold nanowire, hAuNW is its height and ρAuNW the nano wire density. Assuming a density of ρAuNW = 6 µm−2, a radius of rAuNW = 100 nm and a planar capacity of C0 the electrodes capacity can be calculated according to C = (3.77 hAuNW/µm + 1) C0. If the planar capacity is again set to C0 = 0.1 nF and the resistor to Rms = 10 kΩ the height depended electrodes properties summarized in Table 2 and depicted in Figure 7 are obtained.
| hAuNW | 0 µm | 5 µm | 10 µm | 15 µm | 25 µm |
| Magnitude |Z(jω)| in kΩ at f = 1 kHz | 1592 | 81 | 42 | 29 | 19 |
| Phase ∠Z(jω) in ° at f = 1 kHz | −90 | −83 | −76 | −70 | −59 |
DownLoad: CSV
Figure 7. Bode plot for a planar gold electrode and electrodes with integrated gold nanowires with different heights. By integrating gold nanowires the impedance can be decreased over the entire frequency range. As expected the heights hAuNW has a strong influence on the impedance spectra.
As expected the higher the nanowires are, the lower the impedance is. By integrating gold nanowires the impedance can be decreased for the entire frequency range.
To evaluate the simulated electrode properties with experimental findings microelectrodes are manufactured and characterized by cyclic voltammetry and electrochemical impedance spectroscopy. All measurements are done with electrolyte phosphate buffered saline (PBS, Sigma Aldrich). Cyclic voltammetry is performed in a three-electrode set-up, using the microelectrodes as working electrodes, a platinum mesh as counter electrode and an Ag/AgCl electrode as reference. From the resulting U-I-diagrams (not shown) the DC-capacity is evaluated [20]. Impedance spectroscopy is performed in a frequency range of 1 Hz-100 kHz to cover the entire neuronal frequency spectrum and beyond. The manufactured microelectrodes serve as working electrodes and an Ag/AgCl pellet electrode (Multichannel Systems, Reutlingen, Germany) as combined counter/reference electrode. Furthermore the impedance data can be fitted with an equivalent circuit to extract the AC-properties of the electrode such as capacity and resistance. Due to ion idleness in AC-conditions not the entire surface is effective at the interface and the AC-capacity is expected to be smaller than the DC-capacity.
With cyclic voltammetry a capacity gain between a planar gold electrode and an electrode with integrated carbon nanotubes is measured to 1,800. This value is somewhat higher than the expected factor of 1,357. The discrepancy can be explained by slight deviations in electrode size due to the manufacturing process.
When observing the AC-properties of the electrode it becomes obvious that under these circumstances not the entire surface contributes to the electrochemical interface, which results in a reduced capacity. Ions are not able to follow the alternating electric field into the pores, eliminating some of the effective capacity. This effect causes the impedance to decrease only by a factor of 62 and not as expected by a factor of 158. Measured impedance spectra for a planar gold electrode and an electrode with carbon nanotubes are depicted in Figure 8. The fundamental shape is in good agreement with the theoretical properties (see Figure 6). It can be seen that the resistor is dominating at a frequency of about 2000 Hz since the capacity is nearly short circuited at this frequency. From simulations this was expected to happen already at about 200Hz. Reason for this discrepancy could be that not the entire surface area contributes to the electrochemical interface in AC-conditions.
Figure 8. Meassured impedance spectra of manufacutred 30 µm-microelectrodes with planar gold and integrated carbon nanotubes in comparison to simulated data from Figure 6. The measurements (blue lines) are generally in good agreement with the simulated data (black lines).
For microelectrodes with integrated gold nanowires a DC-capacity linearly dependent on the height is revealed in cyclic voltammetry as shown in Table 3. This table also summarizes the AC-properties revealed by electrochemical impedance spectroscopy which are also shown in Figure 9. The measured impedance spectra are in good agreement with the simulated electrode properties. If the ratios are calculated for 20.4 µm nanowires the DC-capacity increases by a factor of 90 and the impedance is decreased by a factor of 109.
| hAuNW | 0 µm | 5.2 µm | 10.7 µm | 20.4 µm |
| DC-capacity CDC in mF/cm2 | 0.035± 0.028 | 0.66 ± 0.24 | 1.61 ± 0.83 | 3.17 ± 1.04 |
| Magnitude |Z(jω)| in dB at f = 1kHz | 1585 ± 141 | 121 ± 46 | 53.83 ± 9.03 | 14.5 ± 2.6 |
| Phase ∠Z(jω) in ° at f = 1kHz | −84.32 ± 0.58 | −79.44 ± 2.03 | −75.57 ± 2.03 | −66.60 ± 2.00 |
DownLoad: CSV
Figure 9. Meassured impedance spectra of manufacutred 30µm-microelectrodes with planar gold and integrated gold nanowires with several heights. These measurements are in very good agreement with the simulated data (see Figure 7).
Because the DC-capacity and impedance at 1 kHz are influenced by similar factors it can be concluded that the interspaces between gold nanowires are wide enough that ions are able to follow even an alternating field. Thus the entire surface area is contributing to the electrochemical interface.
Figure 10 shows the simulated and measured magnitude and phase at 1 kHz plotted against the height of gold nanowires. From about 15 µm onwards the impedance is not reduced much further. This is due to the large capacity that it is nearly short-circuited at 1 kHz. Consequently the impedance is dominated by the resistor Rms. The measured results are in very good agreement to the simulated data.
Figure 10. Height dependence of the impedances magnitude and phase at 1 kHz simulated (dotted lines) and measured (line) with manufactured 30 µm-microelectrodes. Simulation and measurements are in good agreement.
For recording electrodes in the nervous system a high signal-to-noise ratio is desired. In this work we present a transfer function defined as ratio between the membrane potential of neuron and an extracellular microelectrode. The influence of relevant parameters was studied to find that the coupling resistance Rseal has a great effect on the function. This resistance can be increased by an improved adhesion between cell and electrode, resulting in low losses and thus an increase in signal amplitude. To further increase the SNR not only the recorded signal amplitude should be increased but also the noise should be decreased at the same time. Since noise is dominantly thermal, this can be achieved by decreasing the electrodes impedance by increasing the electrochemical active surface area through the integration of different nano structures. The influence of the integration of carbon nanotubes and gold nanowires onto the electrodes behavior (impedance) was simulated and measured with manufactured electrodes.
By increasing the coupling resistance Rseal and decreasing the impedance, the integration of nano objects onto microelectrodes has the potential to significantly improve the signal-to-noise ratio of neuroelectrodes. Here simulations of electrode properties are compared with experimental findings. Electrodes with integrated carbon nanotubes show a DC-capacity that is 1, 800-fold higher compared to a planar gold electrode which is in good agreement to simulations. The AC-impedance however at 1 kHz is only reduced by a factor of 62 which is less than the expected value. Ions are not able to follow the alternating electric field into the pores, reducing the effective capacity. Electrodes with integrated gold nanowires show height dependent DC- and AC-properties that are in good agreement with simulation results.
C.N. expresses his gratitude for the Studienstiftung des Deutschen Volkes for supporting his research.
The author declares no conflicts of interest in this paper.
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Figures(17) / Tables(5)
Boumediene Benyahia, Tewfik Sari. Effect of a new variable integration on steady states of a two-step Anaerobic Digestion Model[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5504-5533. doi: 10.3934/mbe.2020296
| Rseal | 1 kΩ | 100 kΩ | 1 MΩ | 10 MΩ | 1 GΩ |
| Magnitude |H(jω)| in dB at f = 1 kHz | −54.77 | −14.96 | −1.7 | −0.66 | −0.65 |
| Phase ∠H(jω) in ° at f = 1 kHz | 88.39 | 77.41 | 26.17 | 2.31 | −0.58 |
DownLoad: CSV| hAuNW | 0 µm | 5 µm | 10 µm | 15 µm | 25 µm |
| Magnitude |Z(jω)| in kΩ at f = 1 kHz | 1592 | 81 | 42 | 29 | 19 |
| Phase ∠Z(jω) in ° at f = 1 kHz | −90 | −83 | −76 | −70 | −59 |
DownLoad: CSV| hAuNW | 0 µm | 5.2 µm | 10.7 µm | 20.4 µm |
| DC-capacity CDC in mF/cm2 | 0.035± 0.028 | 0.66 ± 0.24 | 1.61 ± 0.83 | 3.17 ± 1.04 |
| Magnitude |Z(jω)| in dB at f = 1kHz | 1585 ± 141 | 121 ± 46 | 53.83 ± 9.03 | 14.5 ± 2.6 |
| Phase ∠Z(jω) in ° at f = 1kHz | −84.32 ± 0.58 | −79.44 ± 2.03 | −75.57 ± 2.03 | −66.60 ± 2.00 |
DownLoad: CSV| Rseal | 1 kΩ | 100 kΩ | 1 MΩ | 10 MΩ | 1 GΩ |
| Magnitude |H(jω)| in dB at f = 1 kHz | −54.77 | −14.96 | −1.7 | −0.66 | −0.65 |
| Phase ∠H(jω) in ° at f = 1 kHz | 88.39 | 77.41 | 26.17 | 2.31 | −0.58 |
| hAuNW | 0 µm | 5 µm | 10 µm | 15 µm | 25 µm |
| Magnitude |Z(jω)| in kΩ at f = 1 kHz | 1592 | 81 | 42 | 29 | 19 |
| Phase ∠Z(jω) in ° at f = 1 kHz | −90 | −83 | −76 | −70 | −59 |
| hAuNW | 0 µm | 5.2 µm | 10.7 µm | 20.4 µm |
| DC-capacity CDC in mF/cm2 | 0.035± 0.028 | 0.66 ± 0.24 | 1.61 ± 0.83 | 3.17 ± 1.04 |
| Magnitude |Z(jω)| in dB at f = 1kHz | 1585 ± 141 | 121 ± 46 | 53.83 ± 9.03 | 14.5 ± 2.6 |
| Phase ∠Z(jω) in ° at f = 1kHz | −84.32 ± 0.58 | −79.44 ± 2.03 | −75.57 ± 2.03 | −66.60 ± 2.00 |
