# Consistent and Efficient Modeling of the Nonlinear Properties of Ferroelectric Materials in Ceramic Capacitors for Frugal Electronic Implants

^{*}

## Abstract

**:**

## 1. Introduction

^{2}was produced. The implantable electronics considered in this paper contain neither batteries nor sensors or active electronic components and are therefore not suitable for autonomous operation. Power is supplied by induction at a frequency below 1 MHz using an extracorporeal wearable device. The amount of inductively transferred power directly impacts the induced voltage.

## 2. Methods

_{1}and capacitor C

_{4}. The duration and interval between the individual power pulses corresponds to the stimulation duration and frequency at the electrode impedance R

_{L}.

_{1}and L

_{2}

_{1}

_{2}and its loss resistance R

_{2}

_{1}

_{1}as a function of the voltage ${u}_{D1}\left(t\right)$

_{4}

_{4}

_{L}

#### 2.1. Characterization of the Voltage Dependency of Ceramic Capacitors

_{2}, C

_{2a}, C

_{2b}and C

_{4}was measured using the precision impedance analyzer Agilent 4294A (Agilent Technologies, Inc., Santa Clara, PA, USA, 4294A R1.11 Mar 25 2013) and the test fixture Agilent 16034E (Agilent Technologies, Inc., Santa Clara, PA, USA). The AC component was set to a frequency of 375 kHz for the capacitor C

_{2}, C

_{2a}and C

_{2b}and to 40 Hz (lower limit of the impedance analyzer) for the voltage rectifier capacitor C

_{4}. The amplitude was set to 5 mV and was superimposed with a DC bias voltage varying in the range from −40 V to +40 V with a resolution of 801 points. To determine the hysteresis, the electrical capacitance of C

_{2}, C

_{2a}, C

_{2b}and C

_{4}was measured by varying the bias voltage from −40 V to +40 V and from +40 V to −40 V. The obtained characteristic curves of the capacitors C

_{2}, C

_{2a}, C

_{2b}and C

_{4}were implemented in the simulation model in Mathcad (PTC, Boston, MA, USA) and ANSYS (ANSYS, Inc., Canonsburg, PA, USA) in order to include the voltage-dependent capacitance change in the calculations (Figure 3 and Figure 4). Additional specifications for the capacitors C

_{2}, C

_{2a}, C

_{2b}and C

_{4}can be found in Section 2.4.

#### 2.2. Calculations in Mathcad Prime 3.1

^{−7}and the number of points for a given solution interval was set to 50 k, 500 k and 5 M. The step size was constant or varying within a solution interval, depending on the solver used. Under consideration of the currents i

_{C2}(t) or i

_{C2a}(t) and i

_{C2b}(t), the hysteresis losses can be incorporated into the model. The characteristic curves of the capacitors C

_{2}, C

_{2a}, C

_{2b}and C

_{4}have been interpolated with third order B-spline functions. In order to achieve different modulations of the electrical capacitance resulting from each circuit topology, the amplitude of the sinusoidal excitation ${u}_{1}\left(t,{A}_{mp},\omega \right)$ was varied from 0.1 V to 10 V in 0.1 V steps at a coupling factor k of 1% and 10%

#### 2.3. Calculations in ANSYS 2019 R3 Simplorer

#### 2.4. Model Validation by Means of a Measurement Setup

_{1}and R

_{1}, C

_{1}as well as L

_{2}and R

_{2}and C

_{4}were measured with the precision impedance analyzer Agilent 4294A and the test fixture HP 1604D (Hewlett Packard, Palo Alto, CA, USA). For all circuit topologies in Figure 2, L

_{1}and R

_{1}(14.53 µH, 0.4 Ω, Würth Elektronik), and C

_{1}(12.45 nF, WIMA, FKP1, 2 kV) remain constant.

_{2}(47 nF, 200 V, C0G), a nonlinear capacitor C

_{2}(47 nF, ±20%, 4 V, X5R, 01005) and an inductance L

_{2}and loss resistance R

_{2}(3.76 µH, 0.3 Ω, Würth Elektronik) were used for the circuit topology shown in Figure 2a. The capacitors C

_{2a}and C

_{2b}(47 nF, ±20%, 4 V, X5R, 01005) and the inductance L

_{2}and loss resistance R

_{2}(8.45 µH, 0.86 Ω, Würth Elektronik) were used for the circuit topology shown in Figure 2b. Finally, the capacitors C

_{2a}and C

_{2b}(47 nF, ±20%, 4 V, X5R, 01005) and the inductance L

_{2}and loss resistance R

_{2}(1.75 µH, 0.28 Ω, Würth Elektronik) were used for the circuit topology in Figure 2c.

_{1}(MULTICOMP, 1N4148WS.) and R

_{L}(1 kΩ, ±1%) were taken from the datasheets. The capacitors C

_{2}(47 nF, 200 V, C0G), C

_{2a}and C

_{2b}(47 nF, ±20%, 4 V, X5R, 01005) and C

_{4}(4.7 µF, 50 V) were determined according to Section 2.1 (Figure 3 and Figure 4).

_{2}, C

_{2a}, C

_{2b}and C

_{4}were set by changing the distance between the inductances L

_{1}and L

_{2}on the primary and secondary sides. A loose coupling between the inductances L

_{1}and L

_{2}was ensured, so that the detuning of the resonant circuits on the primary and secondary sides was avoided in order to be able to compare the calculations and the measurements. The voltage Uc2RMS, resulting from the root mean square value over time of the voltage u

_{c2}(t) across the circuit topology consisting of nonlinear capacitors, and the voltage Uc4Mean, resulting from the mean value over time from the voltage u

_{c4}(t) at the load R

_{L}, were measured with the digital oscilloscope RIGOL MSO4054 (RIGOL Technologies, Inc., Suzhou, China). It should be noted that the measurement was performed on the internal memory and not on the graphical memory, otherwise the root mean square value would be wrong, due to insufficient resolution. The internal memory was accessed using the UltraSigma and UltraScope programs (RIGOL Technologies, Inc., Suzhou, China). The measured values refer to a time span of 14 ms, with a sampling rate of 4 GS/s. Furthermore, a pulsed inductive power transfer at a frequency of 375 kHz, a duration of 5 ms and a period of at least 1 s was performed, so that the thermal detuning of the capacitors C

_{2}, C

_{2a}, C

_{2b}and C

_{4}can be neglected.

## 3. Results and Discussion

_{2}(47 nF, 200 V, C0G) and C

_{4}(4.7 µF, 50 V). The capacitors C

_{2}and C

_{4}were defined as constant at 48 nF and 4.56 µF. The deviation S between the calculations with ANSYS/Mathcad and the measurements is shown in Table 1.

_{2}and C

_{4}, most calculation methods in ANSYS and all calculation methods in Mathcad lead to a high consistency between calculations and measurements. As an additional result, the memory consumption and computing time of the calculation methods used in Table 1 are shown in Table 2 and Table 3, respectively. The calculations were performed on a workstation HP Z250 (L8T12AV, Intel Xeon E3-1280 v5 (8M Cache, 3.70 GHz), 32 GB DDR4, 256 GB SSD, Windows 10 Pro 64-bit).

_{2a}and C

_{2b}, or to another cause. Since these discrepancies occur in the case of both, a linear capacitor C

_{2}and two serially connected nonlinear capacitors, C

_{2a}and C

_{2b}, they cannot be assigned to the modeling of the nonlinear capacitors.

_{2}(Figure 2a) and C

_{4}. The coupling factor was varied from 1% to 10% in 1% steps and the amplitude of the sinusoidal voltage source was adjusted so that the voltage Uc2RMS was equal to 2.123 V. Figure 5 shows that an increasing coupling factor directly impacts the amplitude and time constant of the voltage Uc4. By changing the coupling factor between 1% and 4% (Figure 5a), the amplitude and time constant of voltage Uc4 change significantly. For coupling factors above 4%, the impact of the coupling factor on the amplitude and time constant becomes less significant (Figure 5b,c). At a coupling factor between 7% and 10%, the consistency between the calculated and measured time-related voltage curves Uc4 is highest (Figure 5c). Consequently, it should be ensured that the coupling factor is sufficiently high to achieve more accurate results even in the case of loose coupling.

_{4}on the model consistency was determined. The calculations were performed with the circuit in Figure 1 having a linear capacitor C

_{2}(Figure 2a), a nonlinear capacitor C

_{4}(Figure 4) and a coupling factor of 10%. According to Figure 6, the nonlinearity of the voltage rectifier capacitor C

_{4}has no significant impact on the consistency of the model.

_{2a}and C

_{2b}(Figure 7b). The value of Uc2RMS at which the voltage Uc4Mean increases changes from about 23 V to 20 V due to the hysteresis losses. The same behavior can also be observed with a nonlinear capacitor, C

_{2}, and two nonlinear capacitors, C

_{2a}and C

_{2b}, connected in parallel (Figure 7a,c), in a range of Uc2RMS between about 6 V and 9 V. An interesting point in Figure 7 is that depending on the circuit topology used, an approximately constant range of Uc4Mean is achieved within a specific range of Uc2RMS. The range of Uc2RMS in which Uc4Mean is approximately constant and the slope of Uc4Mean within this range are defined by the circuit topology of nonlinear capacitors.

## 4. Conclusions

_{4}does not significantly improve the model, but increases the computing time. Therefore, with regard to consistency and computing time, we recommend neglecting the nonlinear properties of the capacitor C

_{4}for further modeling purposes.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Representation of the inductively coupled system for power transmission: (

**a**) Primary side consisting of an ideal voltage source ${u}_{1}\left(t,{A}_{mp},\omega \right)$ and a series resonant circuit consisting of a capacitor C

_{1}, an inductance L

_{1}and a loss resistor R

_{1}; (

**b**) Secondary side consisting of a parallel resonant circuit, which is composed of the inductance L

_{2}, a circuit topology consisting of nonlinear capacitors f

_{C2}(u

_{C2}(t)) and the loss resistance R

_{2}, a rectifier consisting of the diode D

_{1}and the capacitor C

_{4}and an ohmic load R

_{L}resulting from the biological tissue and electrode properties. The inductive coupling between the primary and secondary inductances is represented by the coupling factor k.

**Figure 2.**Circuit topologies with nonlinear capacitors: (

**a**) one nonlinear capacitor, C

_{2}; (

**b**) two serially connected nonlinear capacitors, C

_{2a}and C

_{2b}; (

**c**) two nonlinear capacitors, C

_{2a}and C

_{2b}, connected in parallel.

**Figure 3.**Measured electrical capacitance C of the capacitor: (

**a**) C

_{2a}; (

**b**) C

_{2b}. The electrical capacitance of C

_{2a}and C

_{2b}for a change in the bias voltage from −40 V to +40 V is represented by C

_{2a+}and C

_{2b+}and for a change in the bias voltage from +40 V to −40 V by C

_{2a−}and C

_{2b−}. C

_{2a}and C

_{2b}corresponds to the averaged capacitances of C

_{2a+}and C

_{2a−}and C

_{2b+}and C

_{2b−}over the entire bias voltage range.

**Figure 4.**Measured electrical capacitance C of the voltage rectifier capacitor C

_{4}: (

**a**) C

_{4+}corresponds to the electrical capacitance of C

_{4}for a change in the bias voltage from −40 V to +40 V and C

_{4−}corresponds to the electrical capacitance of C

_{4}for a change in the bias voltage from +40 V to −40 V; (

**b**) Averaged capacitance of C

_{4+}and C

_{4−}over the entire bias voltage range.

**Figure 5.**Representation of the measured (black) and calculated voltage Uc4 (Mathcad) over time t. Table 2. RMS was set to 2.123 V and the coupling factor k between the two inductances, L1 and L2, was set to: (

**a**) 1% (red), 2% (green), 3% (blue), 4% (pink); (

**b**) 5% (red), 6% (green), 7% (blue); (

**c**) 8% (red), 9% (green), 10% (blue).

**Figure 6.**Representation of the measured (black) and calculated voltage (Mathcad) over time t. The voltage Uc2RMS was set to 2.123 V and the coupling factor, k, between the two inductances, L1 and L2, was set to 10%. Linear capacitor C4 (red), nonlinear capacitor C4 (green).

**Figure 7.**Representation of the measured (blue) and calculated voltage Uc4Mean with hysteresis losses (red) and without hysteresis losses (black) versus the voltage Uc2RMS. The calculations were performed for a circuit topology consisting of: (

**a**) one nonlinear capacitor, C

_{2}(see Figure 2a); (

**b**) two serially connected nonlinear capacitors, C

_{2a}and C

_{2b}, (see Figure 2b); (

**c**) two nonlinear capacitors, C

_{2a}and C

_{2b}, connected in parallel (see Figure 2c). Furthermore, the Adams method was used with a resolution of 50 k points and a coupling factor k of 10%.

**Table 1.**Deviations between the measured and calculated voltage Uc4Mean in a range of Uc2RMS from 0.7 V to 21 V at a coupling factor k of 1%.

Method | 50 k Points | 500 k Points | 5 M Points |
---|---|---|---|

Adams | 0.5 V | 0.5 V | 0.5 V |

Bulirsch–Stoer | 0.3 V | 0.3 V | 0.3 V |

Runge–Kutta ^{1} | 0.5 V | 0.5 V | 0.5 V |

Runge–Kutta ^{2} | 0.3 V | 0.3 V | 0.3 V |

BDF ^{4} | 0.5 V | 0.5 V | 0.5 V |

Radau5 | 0.5 V | 0.5 V | 0.5 V |

Euler ^{1} | 15.5 V | 14.8 V | 0.5 V |

Trapezoid ^{1} | 0.5 V | 0.6 V | 0.6 V |

ATE ^{1,3} | 10.7 V | 0.6 V | 0.6 V |

Euler ^{2} | 15.5 V | ||

Trapezoid ^{2} | 0.6 V | ||

ATE ^{2,3} | 0.5 V |

^{1}With constant step size;

^{2}With variable step size;

^{3}Adaptive Trapezoid-Euler;

^{4}Backward Differentiation Formula.

**Table 2.**Memory consumption of the selected calculation methods in Mathcad and ANSYS summed up over 100 independent runs.

Method | 50 k Points | 500 k Points | 5 M Points |
---|---|---|---|

Adams | 0.23 GB | 2.23 GB | 22.3 GB |

Bulirsch–Stoer | 0.23 GB | 2.23 GB | 22.3 GB |

Runge–Kutta ^{1} | 0.23 GB | 2.23 GB | 22.3 GB |

Runge–Kutta ^{2} | 0.23 GB | 2.23 GB | 22.3 GB |

BDF ^{4} | 0.23 GB | 2.23 GB | 22.3 GB |

Radau5 | 0.23 GB | 2.23 GB | 22.3 GB |

Euler ^{1} | 1.18 GB | 11.7 GB | 118 GB |

Trapezoid ^{1} | 1.18 GB | 11.7 GB | 117 GB |

ATE ^{1,3} | 1.18 GB | 11.7 GB | 117 GB |

Euler ^{2} | 2.84 GB | ||

Trapezoid ^{2} | 34.9 GB | ||

ATE ^{2,3} | 2.33 GB |

^{1}With constant step size;

^{2}With variable step size;

^{3}Adaptive Trapezoid-Euler;

^{4}Backward-Differentiation-Formula.

**Table 3.**Computing time (hh:mm:ss) of the selected calculation methods in Mathcad summed up over 80 independent runs.

Method | 50 k Points | 500 k Points | 5 M Points |
---|---|---|---|

Adams | 00:03:17 | 00:03:49 | 00:10:03 |

Bulirsch–Stoer | 00:16:17 | 01:12:57 | 08:56:55 |

Runge–Kutta ^{1} | 00:01:48 | 00:15:41 | 02:42:50 |

Runge–Kutta ^{2} | 00:07:58 | 00:42:35 | 06:39:06 |

BDF ^{4} | 00:12:33 | 00:12:48 | 00:19:13 |

Radau5 | 00:07:42 | 00:08:33 | 00:13:11 |

Euler ^{1} | 00:01:51 | 00:06:29 | 00:44:27 |

Trapezoid ^{1} | 00:01:45 | 00:06:17 | 00:52:25 |

ATE ^{1,3} | 00:04:43 | 00:14:30 | 00:43:16 |

Euler ^{2} | 00:02:49 | ||

Trapezoid ^{2} | 00:21:13 | ||

ATE ^{2,3} | 00:02:04 |

^{1}With constant step size;

^{2}With variable step size;

^{3}Adaptive Trapezoid-Euler;

^{4}Backward-Differentiation-Formula.

**Table 4.**Deviations between the measured and calculated voltage Uc4Mean in a range of Uc2RMS from 0.7 V to 21 V at a coupling factor k of 10%.

Method | 50 k Points | 500 k Points | 5 M Points |
---|---|---|---|

Adams | 0.3 V | 0.3 V | 0.3 V |

Bulirsch–Stoer | 0.3 V | 0.3 V | 0.3 V |

Runge–Kutta ^{1} | 0.4 V | 0.3 V | 0.3 V |

Runge–Kutta ^{2} | 0.3 V | 0.3 V | 0.3 V |

BDF ^{4} | 0.3 V | 0.3 V | 0.3 V |

Radau5 | 0.3 V | 0.3 V | 0.3 V |

Euler ^{1} | 15.5 V | 0.3 V | 0.4 V |

Trapezoid ^{1} | 0.4 V | 0.4 V | 0.4 V |

ATE ^{1,3} | 0.3 V | 0.4 V | 0.4 V |

Euler ^{2} | 15.1 V | ||

Trapezoid ^{2} | 0.4 V | ||

ATE ^{2,3} | 0.4 V |

^{1}With constant step size;

^{2}With variable step size;

^{3}Adaptive Trapezoid-Euler;

^{4}Backward Differentiation Formula.

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**MDPI and ACS Style**

Olsommer, Y.; Ihmig, F.R.
Consistent and Efficient Modeling of the Nonlinear Properties of Ferroelectric Materials in Ceramic Capacitors for Frugal Electronic Implants. *Sensors* **2020**, *20*, 4206.
https://doi.org/10.3390/s20154206

**AMA Style**

Olsommer Y, Ihmig FR.
Consistent and Efficient Modeling of the Nonlinear Properties of Ferroelectric Materials in Ceramic Capacitors for Frugal Electronic Implants. *Sensors*. 2020; 20(15):4206.
https://doi.org/10.3390/s20154206

**Chicago/Turabian Style**

Olsommer, Yves, and Frank R. Ihmig.
2020. "Consistent and Efficient Modeling of the Nonlinear Properties of Ferroelectric Materials in Ceramic Capacitors for Frugal Electronic Implants" *Sensors* 20, no. 15: 4206.
https://doi.org/10.3390/s20154206