# Quantitative Analysis of Insulator Degradation in a Single Layer Solenoid by Renormalization of the Transmission Parameter

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Transmission Line Model Including a Locally Degraded Part Inside

_{s}indicates the equivalent series inductance of the solenoid, R

_{s}indicates the conductor’s resistance, and C

_{p}and R

_{i}are parallel capacitance and parallel resistance, respectively. This model has been used to estimate reactance [10,13,14], and induced torque with a plunger from measured voltage and current [15] in a low-frequency range. However, it is not suitable for estimating the distributed degradation of the insulator inside a solenoid due to the limit of lumped element modeling itself.

_{C}is the characteristic impedance of the transmission line, β is a propagation constant, and Z

_{0}is the reference impedance of the measurement equipment. In the next section, assuming that the permittivity change in a solenoid could be similar to the permittivity change in a transmission line, the S-parameter formulas with a discontinuous parameter in the transmission line, to mimic the deterioration of the insulator inside, is developed.

#### 2.1. Discontinuous Transmission Line Model Mimicking Deterioration of the Insulator

_{1}, characteristic impedance Z

_{C}, and phase constant β) and area 3 (length l

_{3}, characteristic impedance Z

_{C}and phase constant β) are in a healthy state, and area 2 (length l

_{2}, characteristic impedance Z’

_{C}and phase constant β’) indicates the degraded part. That is, the pink area in the diagram indicates area 2, where the insulator locally deteriorates with the prime symbols. The total ABCD parameter with ‘t’ in the superscript can be calculated by multiplying three ABCD matrices in sequence as Equation (1) [17].

#### 2.2. Changes in Permittivity of the Transmission Parameter

_{21}is linearly decreased as frequency increases as in Equation (7a). For the healthy transmission line only, the phase of S

_{21}can be as in Equation (7b). If the characteristic impedance Z

_{C}is not matched to a reference impedance Z

_{0}, the trace of the phase in S

_{21}shows local fluctuations.

#### 2.3. Renormalization Technique of an S-Parameter

_{0}, and Figure 3b shows the renormalized S-parameter when terminated with Z

_{new}. Basically, the S-parameter is one of the network parameters, which is dependent on the reference (termination) impedance, but the Z-parameter is not changed by the termination impedance. The relationship between those two parameters is expressed in Equations (9a) and (9b), where

**G**and

**F**are the termination impedance matrix and the real part of them, respectively, and

**I**is the identity matrix. The star is for the conjugate of complex value. In Equation (9a),

**S**is expressed in terms of

**Z**, and

**Z**is expressed in terms of

**S,**in Equation (9b).

**G**and

**F**as in Equation (10), which comes from Equation (9a). Since the

**Z**-parameter is unique in a given network, Equation (9b) can be plugged into Equation (10), and if we make use of reflection coefficient

**Γ**, Equation (11) can be derived [21].

**Γ**is a diagonal reflection coefficient matrix from Z

_{0}(

**G**) and Z

_{new}(

**G**

_{new}). Z

_{new}can be selected by the designer as needed. Through this technique, the reference impedance can be matched to the characteristic impedance of the device under test.

## 3. Fabrication and Measurement Setup for a Prototype of Single-Layered Solenoid

#### 3.1. Structure of a Prototype Single-Layered Solenoid

^{7}Siemens/m).

#### 3.2. Setup for Experimental Measurement and De-Embedding Interfaces

_{eff}is the effective permittivity in the background, and h is the height from the reference plane. Applying the physical data (r

_{in}= 0.57 mm, r

_{out}= 4.1 mm, h = 12.25 mm, ε

_{eff}

_{,coax}= 1, and ε

_{eff,SW}= 2.3), the characteristic impedance of each part can be calculated. Furthermore, the input impedance toward the VNA, as in Figure 5b [17], can be calculated as in Equation (14).

_{0}in Equation (14) is the reference impedance in the VNA (50 Ω), the input impedance Z

_{in}can be calculated, of course, and reversely the Z

_{0}can be calculated for a given input impedance value. If we need to convert Z

_{in}into the solenoid impedance, that is, if we need the input impedance Z

_{in}to be matched to the solenoid impedance, the new Z

_{0}can be calculated using Equation (14). Since the measured S-parameter is based on the reference impedance of 50 Ω, it can be converted into new S-parameter based on the new Z

_{0}, which is calculated using Equation (14). In the equation, l

_{coax}is 18 mm, and l

_{SW}is 14 mm, respectively, and if ${Z}_{C,\mathrm{coax}}={Z}_{C,SW}={Z}_{0}$, then we have ${Z}_{in}={Z}_{0}$, as expected. S

_{sol}in Figure 5b represents the de-embedded S-parameter of the solenoid using Equation (14). Keysight’s E5061B model of the vector network analyzer was used for the two-port measurement, and the measurement was performed with 1601 points from 1 kHz to 250 MHz on a linear scale. When measuring the solenoid’s S-parameter, the solenoid was enclosed by six sides of the aluminum harness plane for shielding, reducing interference from the outer electromagnetic environment.

#### 3.3. Setup for Mimicking Degraded Insulator in the Prototype Solenoid

_{3–4}, r

_{9–10}, r

_{15–16}, r

_{21–22}, r

_{27–28}, r

_{3,29}were used for installation, and so on. Note that the last one (r

_{3,29}) indicates the two rings are separately located at r

_{3}and r

_{29}(the red color in Table 2), while the others are consecutively installed. The number of rings and their locations in Table 2 imply both the level of effective permittivity and the scatteredness of the degradation in the insulator.

## 4. Detection of Insulator Degradation in Solenoid Using S-Parameters

_{21}/S

_{12}.

_{in}, which most effectively transfers the signal. Figure 7 describes how to determine the appropriate Z

_{in}for the analysis.

_{in}is set, and then the measured S-parameters are converted with reference to the new Z

_{0}(Z

_{in}), and the solenoid S-parameters are checked if S

_{sol},

_{12}is high enough in the low-frequency range. If S

_{sol}

_{,12}is high enough, then the final Z

_{0}(Z

_{in}) to be used is determined.

#### Quantitative Detection of Insulator Degradation Using Renormalized S-parameters

_{sol}

_{,12}, of the single-layered solenoid. One can find that the transmission characteristic has been significantly enhanced with 1500 Ω reference impedance as compared with 50 Ω reference impedance; for S

_{sol}

_{,12}with 1500 Ω reference impedance, the magnitude increases significantly in the low-frequency range up to ~100 MHz, and the phase decreases more linearly than S

_{sol}

_{,12}with 50 Ω reference impedance. This could suggest that the characteristic impedance of the single-layered solenoid would be close to 1500 Ω, not to 50 Ω in this specific frequency range up to 100 MHz. After ~100 MHz, the magnitude of S

_{sol}

_{,12}decreases more, but the phase keeps linearity up to ~250 MHz, as seen in Figure 8b. The S

_{sol}

_{,12}signals with degraded insulator (51 1–5 rings) are also shown in Figure 8 with 1500 Ω reference in various colors. Nevertheless, it is observed that the S

_{sol}

_{,12}signals are overlapped and become a thick trace, where it is not easy to identify each line. To identify the traces more clearly, the phase difference in S

_{sol}

_{,12}, between the healthy and degraded solenoids, was calculated using Equation (8). Figure 9a shows the phase of S

_{sol}

_{,12}with 1500 Ω reference impedance, which has been wrapped between −π and π, and the nine points for the multiples of −π are indicated as sky blue colored squares. It can be seen that the curves get thicker as the frequency increases, which means that the differences are getting larger as the frequency increases.

_{p}is phase velocity, and t

_{d}is delay time due to the degraded section in the solenoid. One can see that the phase differences in Figure 9c are fluctuating even though they are normalized with respect to the frequency, especially at the local nine least peak frequencies. The local nine least peaks actually correspond to the nine sky-blue small boxes in Figure 9a and Figure 8b.

_{C}= Z

_{0}in front of the tangent function in Equation (6), the least peak frequencies in Figure 9d could have the same values as Equation (6) predicts. Since we are using the measured S-parameters, which are converted into S-parameters with another reference impedance, the coefficient term in front of the tangent function in Equation (6) cannot be deleted in principle. However, we can try to use the phase information only without the coefficient term in front of the tangent function in Equation (6) if we pick up the phase data at multiples of π (−nπ: n = 1–9). That is, if we make use of the phase data at multiples of π only, the phase of Equation (6) will be zero anyway regardless of the coefficient term in front of the tangent function in Equation (6). It is known that the phase data can be used to calculate the effective permittivity of the insulator, and the effective permittivity in the solenoid can be calculated as Equation (16).

## 5. Conclusions

_{21}), especially the phase of S

_{21}, could be very useful to quantify the degradation in the insulator if the S-parameters are converted into other S-parameters with appropriate reference impedance using a renormalization technique.

_{21}between healthy and degraded (with rings installed) solenoids showed clear classification into five groups (one–five rings), demonstrating the usefulness of the proposed method to quantify the degradation of the insulator. Furthermore, the increment of effective permittivity data (the difference from the healthy solenoid and the solenoid with rings installed) at multiple π clearly showed proportionality as the number of the rings increased.

_{21}). We believe that the proposed method can be used for any solenoid valve to evaluate the degradation in the insulator part, including SOVs in NPP in an overhaul period. For the next step, the proposed method needs to be repeated for physically degraded solenoids. It is concluded that the suggested method would be very promising to quantitatively diagnose future degradation of a solenoid.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AWG | American Wire Gauge |

GW(e) | Gigawatt electrical |

IAEA | International Atomic Energy Agency |

NPP | Nuclear power plant |

RF | Radio frequency |

S-parameters | Scattering parameters |

S_{11} parameters | Input port voltage reflection coefficient in two-port network |

S_{12} parameters | Reverse voltage gain in two-port network |

S_{21} parameters | Forward voltage gain in two-port network |

S_{22} parameters | Output port voltage reflection coefficient in two-port network |

SOLT | Short-open-load-thru |

SOV | Solenoid operated valve |

VNA | Vector network analyzer |

Z_{C} | Characteristic impedance of the transmission line |

Z_{0} | Reference impedance of the ports |

Z_{in} | Input impedance from the end of solenoid to port |

β | Propagation constant of the transmission line |

ε_{eff} | Effective permittivity |

Γ | Matrix of the reflection coefficient from each port |

t_{d} | Delay time |

v_{p} | Phase velocity |

## References

- World Nuclear Association. World Nuclear Performance Report; World Nuclear Association: London, UK, 2020. [Google Scholar]
- International Atomic Energy Agency. Operating Experience with Nuclear Power Stations in Member States in 2019; International Atomic Energy Agency: Vienna, Austria, 2019. [Google Scholar]
- Varga, I.; Bartha, T.; Szabó, G.; Kiss, B. Status and Actual Risk Monitoring in a NPP Reactor Protection System. In Probabilistic Safety Assessment and Management; Springer: Berlin/Heidelberg, Germany, 2004; pp. 2654–2659. [Google Scholar]
- Upadhyaya, B.R.; Eryurek, E. Application of neural networks for sensor validation and plant monitoring. Nucl. Technol.
**1992**, 97, 170–176. [Google Scholar] [CrossRef] - Ma, J.; Jiang, J. Applications of fault detection and diagnosis methods in nuclear power plants: A review. Prog. Nucl. Energy
**2011**, 53, 255–266. [Google Scholar] [CrossRef] - Gong, Y.; Su, X.; Qian, H.; Yang, N. Research on fault diagnosis methods for the reactor coolant system of nuclear power plant based on D-S evidence theory. Ann. Nucl. Energy
**2018**, 112, 395–399. [Google Scholar] [CrossRef] - Kwon, K.-C.; Kim, J.-H. A Stochastic Approach with Hidden Markov Model for Accident Diagnosis in Nuclear Power Plants. In Industrial and Engineering Applications of Artificial Intelligence and Expert Systems: Proceedings of the Tenth International Conference; CRC Press: Boca Raton, FL, USA, 2020. [Google Scholar]
- Wang, H.; Peng, M.-J.; Hines, J.W.; Zheng, G.-Y.; Liu, Y.-K.; Upadhyaya, B.R. A hybrid fault diagnosis methodology with support vector machine and improved particle swarm optimization for nuclear power plants. ISA Trans.
**2019**, 95, 358–371. [Google Scholar] [CrossRef] [PubMed] - Ikonomopoulos, A.; Tsoukalas, L.H.; Uhrig, R.E.; Mullens, J.A. Monitoring nuclear reactor systems using neural networks and fuzzy logic. In Proceedings of the American Nuclear Society (ANS) Topical Meeting on Advances in Reactor Physics, Charleston, SC, USA, 8–11 March 1992. [Google Scholar]
- Jameson, N.J.; Azarian, M.H.; Pecht, M.; Wang, K.; Aidong, X. Electromagnetic coil equivalent circuit model sensitivity analysis for impedance-based insulation health monitoring. In Proceedings of the 2017 Prognostics and System Health Management Conference (PHM-Harbin), Harbin, China, 9–12 July 2017; pp. 1–6. [Google Scholar]
- Jameson, N.J.; Azarian, M.H.; Pecht, M. Improved electromagnetic coil insulation health monitoring using equivalent circuit model analysis. Int. J. Electr. Power Energy Syst.
**2020**, 119, 105829. [Google Scholar] [CrossRef] - Jameson, N.J.; Azarian, M.H.; Pecht, M. Fault diagnostic opportunities for solenoid operated valves using physics-of-failure analysis. In Proceedings of the 2014 International Conference on Prognostics and Health Management, Cheney, WA, USA, 22–25 June 2014; pp. 1–6. [Google Scholar]
- Rahman, M.; Cheung, N.; Lim, K.W. A sensorless position estimator for a nonlinear solenoid actuator. In Proceedings of the IECON ’95—21st Annual Conference on IEEE Industrial Electronics, Orlando, FL, USA, 6–10 November 1995; p. 2. [Google Scholar]
- König, N.; Nienhaus, M. A Solution to Ambiguities in Position Estimation for Solenoid Actuators by Exploiting Eddy Current Variations. Sensors
**2020**, 20, 3441. [Google Scholar] [CrossRef] [PubMed] - Wu, S.; Zhao, X.; Li, C.; Jiao, Z.; Qu, F. Multiobjective Optimization of a Hollow Plunger Type Solenoid for High Speed On/Off Valve. IEEE Trans. Ind. Electron.
**2017**, 65, 3115–3124. [Google Scholar] [CrossRef] - Zhijian, J.; Minglin, Z.; Zishu, Z. Fault location of transformer winding deformation using frequency response analysis. In Proceedings of the 2001 International Symposium on Electrical Insulating Materials (ISEIM 2001). 2001 Asian Conference on Electrical Insulating Diagnosis (ACEID 2001). 33rd Symposium on Electrical and Electronic Insulating Materials and Applications in System, Himeji, Japan, 22–22 November 2001; pp. 841–844. [Google Scholar]
- Pozar, D.M. Microwave Engineering; John Wiley & Sons: Hoboken, NJ, USA, 2009. [Google Scholar]
- Frei, J.; Cai, X.-D.; Muller, S. Multiport S-Parameter and T-Parameter Conversion with Symmetry Extension. IEEE Trans. Microw. Theory Tech.
**2008**, 56, 2493–2504. [Google Scholar] [CrossRef] - Lu, H.-C.; Chu, T.-H. Multiport scattering matrix measurement using a reduced-port network analyzer. IEEE Trans. Microw. Theory Tech.
**2003**, 51, 1525–1533. [Google Scholar] [CrossRef] - Chen, C.-J.; Chu, T.-H. Accuracy Criterion for S-Matrix Reconstruction Transforms on Multiport Networks. IEEE Trans. Microw. Theory Tech.
**2011**, 59, 2331–2339. [Google Scholar] [CrossRef] - Kurokawa, K. Power Waves and the Scattering Matrix. IEEE Trans. Microw. Theory Tech.
**1965**, 13, 194–202. [Google Scholar] [CrossRef] [Green Version] - Diaham, S.; Locatelli, M.-L. Dielectric properties of polyamide-imide. J. Phys. D Appl. Phys.
**2013**, 46, 185302. [Google Scholar] [CrossRef] - Frankel, S. Characteristic Impedance of Parallel Wires in Rectangular Troughs. Proc. IRE
**1942**, 30, 182–190. [Google Scholar] [CrossRef] - Arvia, E.M.; Sheldon, R.T.; Bowler, N. A capacitive test method for cable insulation degradation assessment. In Proceedings of the 2014 IEEE Conference on Electrical Insulation and Dielectric Phenomena (CEIDP), Des Moines, IA, USA, 19–22 October 2014; pp. 514–517. [Google Scholar]
- Tanaka, T. Aging of polymeric and composite insulating materials. Aspects of interfacial performance in aging. IEEE Trans. Dielectr. Electr. Insul.
**2002**, 9, 704–716. [Google Scholar] [CrossRef] - Kim, K.; Shin, J.; Kim, B.-S.; Nah, W.; Lim, C.; Chai, J. Electrical and mechanical diagnosis of aging 600 V rated STP cables in a nuclear power plant. IEEE Trans. Dielectr. Electr. Insul.
**2017**, 24, 1574–1581. [Google Scholar] [CrossRef] - Asipuela, A.; Mustafa, E.; Afia, R.S.A.; Adam, T.Z.; Khan, M.Y.A. Electrical Condition Monitoring of Low Voltage Nuclear Power Plant Cables: Tanδ and Capacitance. In Proceedings of the 2018 International Conference on Power Generation Systems and Renewable Energy Technologies (PGSRET), Islamabad, Pakistan, 10–12 September 2018; pp. 1–4. [Google Scholar]

**Figure 1.**(

**a**) Lumped element model for a solenoid in low frequency range (under the second resonance frequency); (

**b**) Finite length transmission line model emphasizing distribution elements.

**Figure 2.**Discontinuous transmission line model. The pink part is introduced for the degraded insulator, and the two sky blue parts are in healthy state.

**Figure 3.**(

**a**) Two-port S-parameter terminated by Z

_{0}; (

**b**) The renormalized two-port S-parameter terminated by Z

_{new}.

**Figure 4.**Structure of prototype 30-turn single-layered solenoid. (

**a**) Drawing in front view; (

**b**) Drawing in port side view; (

**c**) A picture from above with radio frequency (RF) cables connected and with upper housing removed.

**Figure 5.**Measurement for prototype solenoid. (

**a**) A picture of the two-port measurement setup with the vector network analyzer (VNA); (

**b**) Interfacing parts to the VNA and its cross-sectional views.

**Figure 6.**Installation of ring-type insulators on the solenoid. (

**a**) 30 conductor turns and 31-ring insulators are numbered from left and to right. (

**b**) A picture of the solenoid when the 21st ring is installed, (

**c**) A picture of the solenoid when three rings from the 20th to 22nd ring are installed, consecutively.

**Figure 7.**Flow chart for determining Z

_{0}(Z

_{in}) to effectively renormalize scattering S-parameter for the diagnosis of insulator degradation.

**Figure 8.**Transmission parameter of one-layered solenoid, S

_{sol}

_{,12}, referenced to 1500 Ω and 50 Ω. (

**a**) Magnitude; (

**b**) Phase characteristics. Colored traces are for 55 S

_{sol}

_{,12}data with 1–5 rings installed in the solenoid.

**Figure 9.**(

**a**) Phase of S

_{sol,12}with 1500 Ω reference impedance which has been wrapped between −π and π; (

**b**) An enlargement of the red-dotted box in (a); (

**c**) Normalized phase difference with respect to angular frequency using Equation (15); (

**d**) An enlargement of the red-dotted box in (c).

**Figure 10.**(

**a**) Calculated average effective permittivity for healthy and 1–5 rings installed using Equation (16); (

**b**) Calculated average effective permittivity at the multiples of π; (

**c**) Incremental (difference) average effective permittivity at the multiples of π; (

**d**) Incremental (difference) average effective permittivity as the number of rings increases.

Design Parameter | Length (mm) | Description | Design Parameter | Length (mm) | Description |
---|---|---|---|---|---|

f_{h} | 64.5 | flange height | h_{h} | 68.5 | housing height |

f_{w} | 64.5 | flange width | h_{w} | 68.5 | housing width |

f_{t} | 10 | flange thickness | h_{l} | 132 | housing length |

b_{l} | 64 | bobbin length | h_{t} | 2 | housing thickness |

b_{d} | 40 | bobbin diameter | i_{l} | 20 | interface length |

p | 2 | pitch of winding | - | - | - |

**Table 2.**Number and location of sheath rubber rings installed on the solenoid. Rings are separated on the bobbin for the red-colored rings.

No. of Rings | Ring Location | No. of Rings | Ring Location |
---|---|---|---|

0 | None (healthy) | 3 | r_{3–5}, r_{9–11}, r_{15–17}, r_{21–23}, r_{27–29}, r_{3,16,29} |

1 | r_{1}, r_{2},…, r_{30}, r_{31} | 4 | r_{2–5}, r_{8–11}, r_{14–17}, r_{20–23}, r_{26–29}, r_{3,11,20,29} |

2 | r_{3–4}, r_{9–10}, r_{15–16}, r_{21–22}, r_{27–28}, r_{3,29} | 5 | r_{1–5}, r_{7–11}, r_{13–17}, s_{19–23}, r_{25–29}, r_{3,8,16,23,29} |

No. of Rings | 0 (Healthy) | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

ε_{eff} at π, f (MHz) | 2.503, (25.16) | 2.512, (25.16) | 2.517, (25.00) | 2.519, (25.00) | 2.528, (25.00) | 2.530, (25.00) |

ε_{eff} at 2π, f (MHz) | 2.484, (50.47) | 2.506, (50.47) | 2.505, (50.31) | 2.527, (50.31) | 2.523, (50.16) | 2.544, (50.16) |

ε_{eff} at 3π, f (MHz) | 2.608, (73.91) | 2.609, (73.75) | 2.623, (73.59) | 2.630, (73.44) | 2.641, (73.28) | 2.662, (73.28) |

ε_{eff} at 4π, f (MHz) | 2.625, (98.13) | 2.636, (98.13) | 2.649, (97.81) | 2.661, (97.66) | 2.672, (97.34) | 2.680, (97.19) |

ε_{eff} at 5π, f (MHz) | 2.643, (122.3) | 2.649, (122.2) | 2.666, (121.9) | 2.678, (121.6) | 2.692, (121.3) | 2.701, (121.1) |

ε_{eff} at 6π, f (MHz) | 2.654, (146.6) | 2.663, (146.3) | 2.676, (145.9) | 2.688, (145.6) | 2.704, (145.2) | 2.716, (144.8) |

ε_{eff} at 7π, f (MHz) | 2.673, (170.3) | 2.679, (170.2) | 2.694, (169.7) | 2.709, (169.2) | 2.724, (168.8) | 2.738, (168.3) |

ε_{eff} at 8π, f (MHz) | 2.622, (196.6) | 2.630, (196.3) | 2.650, (195.5) | 2.658, (195.0) | 2.681, (194.4) | 2.694, (193.9) |

ε_{eff} at 9π, f (MHz) | 2.590, (222.5) | 2.598, (222.2) | 2.619, (221.3) | 2.634, (220.6) | 2.653, (219.8) | 2.665, (219.4) |

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**MDPI and ACS Style**

Kim, K.; Han, J.; Chai, J.; Nah, W.
Quantitative Analysis of Insulator Degradation in a Single Layer Solenoid by Renormalization of the Transmission Parameter. *Electronics* **2020**, *9*, 1984.
https://doi.org/10.3390/electronics9111984

**AMA Style**

Kim K, Han J, Chai J, Nah W.
Quantitative Analysis of Insulator Degradation in a Single Layer Solenoid by Renormalization of the Transmission Parameter. *Electronics*. 2020; 9(11):1984.
https://doi.org/10.3390/electronics9111984

**Chicago/Turabian Style**

Kim, Kwangho, JunHee Han, Jangbom Chai, and Wansoo Nah.
2020. "Quantitative Analysis of Insulator Degradation in a Single Layer Solenoid by Renormalization of the Transmission Parameter" *Electronics* 9, no. 11: 1984.
https://doi.org/10.3390/electronics9111984