# Resonant Directly Coupled Inductors–Capacitors Ladder Network Shows a New, Interesting Property Useful for Application in the Sensor Field, Down to Micrometric Dimensions

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}= 1/√LC is taken equal to 1. In the case of two L–C cells, the normalized ω becomes ω

_{1}= 0.618 … and ω

_{2}= 1.618 … (which do represent the golden section and the golden ratio, respectively) and so on. Another interesting property is represented by the fact that these two frequencies are also present in the case of 7, 12, 17, 22 … and so on cells (i.e., starting from two cells the two solutions are present according to a period of five cells). Furthermore, the transfer function of this kind of L–C L.N. has the property to show all the ω-solutions only in the normalized interval defined by 0 and 2 (as shown in Figure 1). Another property of this L–C L.N. can be seen by looking at this figure while partly shutting one’s eyes. It is possible to see many channels (finite number) that become narrower and narrower by increasing the number of cells. These are the forbidden bands similar to those that we have in a 1-dimensional (1D) array of atoms. This means that whatever the number of cells, the solutions ω

_{i}will never enter these bands.

_{1}= jωL and transversal impedance Z

_{2}= 1/jωC.

^{2}LC, the same changes of either C or L (not simultaneous changes) will produce the same result.

## 2. Materials and Methods

_{b}in each node β of a n-length L.N. where k(s) is the ratio of Z

_{1}/Z

_{2}in the Laplace domain.

## 3. Results

#### 3.1. Fibonacci Relations in the L–C L.N.

^{14}L

^{7}C

^{7}+ 13ω

^{12}L

^{6}C

^{6}− 66ω

^{10}L

^{5}C

^{5}+ 165ω

^{8}L

^{4}C

^{4}− 205ω

^{6}L

^{3}C

^{3}+ 121ω

^{4}L

^{2}C

^{2}− 28ω

^{2}LC + 1

^{(n)}= 2 sin {[(2i − 1)π]/[(2n + 1)2]}

_{1}= 0.20906; ω

_{2}= 0.61903; ω

_{3}= 1; ω

_{4}= 1.33832; ω

_{5}= 1.61803; ω

_{6}= 1.82713; ω

_{7}= 1.9659.

#### 3.2. Sensor Localization Based on Frequency Patterns

^{−6}nF; thus, it could be used in the management of sensors with optimal resolution power, which is crucial when monitoring very small deviations of micrometric sensors’ responses.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Normalized ω solutions in the case of an inductor–capacitor (L–C) ladder network (L.N.) formed by a number of single cells from 1 to 100.

**Figure 4.**Single L–C cell used as a ‘standard’ for the reference condition of the L.N. in this work.

**Figure 5.**Magnitude vs frequency plot of the single-cell L.N. reported in Figure 3 showing the behavior of a low pass filter.

**Figure 8.**Magnitude plots vs frequency of the seven patterns readable by the seven nodes of the seven cells of the L–C L.N.

**Figure 9.**Comparison of the magnitude plots vs frequency of the patterns readable at the seventh node of the L–C L.N. when the capacitance at the 7th cell shifts from 47 nF to 44 nF.

**Figure 10.**Magnitude plot shifting for three different conditions: (

**a**) C shift in cell 1 read by node 1; (

**b**) C shift in cell 2 read by node 2; and (

**c**) C shift in cell 5 read by node 5.

**Figure 11.**Score plot of the first two PCs of the principal component analysis (PCA) model built in the 7 × 13 data array of resonance frequency of the seven-cell L–C L.N. of Figure 5.

**Figure 12.**Representation of the localization action, which can be performed using the PCA model of Figure 11 in finding out the variation that occurred.

Power of Monomials at the Denominator | ||||||||
---|---|---|---|---|---|---|---|---|

Cell # | X^{0} | X^{2} | X^{4} | X^{6} | X^{8} | X^{10} | X^{12} | X^{14} |

0 | 1 | - | - | - | - | - | - | - |

1 | 1 | −1 | - | - | - | - | - | - |

2 | 1 | −3 | 1 | - | - | - | - | - |

3 | 1 | −6 | 5 | −1 | - | - | - | - |

4 | 1 | −10 | 15 | −7 | 1 | - | - | - |

5 | 1 | −15 | 35 | −28 | 9 | −1 | - | - |

6 | 1 | −21 | 65 | −84 | 45 | −11 | 1 | - |

7 | 1 | −28 | 121 | −205 | 165 | −66 | 13 | −1 |

Poles | |||||||
---|---|---|---|---|---|---|---|

Cell # | P1 | P2 | P3 | P4 | P5 | P6 | P7 |

1 | 18.61 | 55.01 | 89.02 | 119.15 | 144.04 | 162.66 | 174.18 |

2 | 18.61 | 55.01 | 89.02 | 119.15 | 144.04 | 162.66 | 174.18 |

3 | 18.61 | 55.01 | 0 | 119.15 | 144.04 | 162.66 | 174.18 |

4 | 18.61 | 55.01 | 89.02 | 119.15 | 144.04 | 162.66 | 174.18 |

5 | 18.61 | 0 | 89.02 | 119.15 | 0 | 162.66 | 174.18 |

6 | 18.61 | 55.01 | 0 | 119.15 | 144.04 | 162.66 | 174.18 |

7 | 18.61 | 55.01 | 89.02 | 119.15 | 144.04 | 162.66 | 174.18 |

Zeros | ||||||
---|---|---|---|---|---|---|

Cell # | Z1 | Z2 | Z3 | Z4 | Z5 | Z6 |

1 | 21.46 | 63.13 | 101.13 | 133.26 | 157.65 | 172.86 |

2 | 25.33 | 73.96 | 116.6 | 0 | 149.79 | 170.84 |

3 | 30.91 | 0 | 0 | 136.39 | 0 | 167.3 |

4 | 39.61 | 0 | 111.01 | 0 | 160.43 | 0 |

5 | 0 | 0 | 0 | 0 | 0 | 0 |

6 | 0 | 0 | 0 | 0 | 0 | 0 |

7 | 0 | 0 | 0 | 0 | 0 | 0 |

Poles | |||||||
---|---|---|---|---|---|---|---|

Cell # | P1 | P2 | P3 | P4 | P5 | P6 | P7 |

1 | 18.77 | 55.44 | 89.61 | 119.72 | 144.51 | 162.92 | 174.22 |

2 | 18.77 | 55.44 | 89.61 | 119.72 | 144.51 | 162.92 | 174.22 |

3 | 18.77 | 55.44 | 89.61 | 119.72 | 144.51 | 162.92 | 174.22 |

4 | 18.77 | 55.44 | 89.61 | 119.72 | 144.51 | 162.92 | 174.22 |

5 | 18.77 | 55.44 | 89.61 | 119.72 | 144.51 | 162.92 | 174.22 |

6 | 18.77 | 55.44 | 89.61 | 119.72 | 144.51 | 162.92 | 174.22 |

7 | 18.77 | 55.44 | 89.61 | 119.72 | 144.51 | 162.92 | 174.22 |

Zeros | ||||||
---|---|---|---|---|---|---|

Cell # | Z1 | Z2 | Z3 | Z4 | Z5 | Z6 |

1 | 21.67 | 63.69 | 101.85 | 133.87 | 158.01 | 172.98 |

2 | 25.63 | 74.69 | 117.4 | 0 | 150.34 | 171 |

3 | 31.35 | 0 | 90.01 | 137.24 | 0 | 167.61 |

4 | 40.31 | 0 | 112.3 | 0 | 161.02 | 0 |

5 | 0 | 56.33 | 0 | 0 | 145.41 | 0 |

6 | 0 | 0 | 92 | 0 | 0 | 0 |

7 | 0 | 0 | 0 | 0 | 0 | 0 |

**Table 4.**Root-mean-square error in cross validation (RMSECV) in the identification of the capacitance shifts (from C1 to C7), as obtained from the partial least square discriminant analysis (PLS-DA) model by using the leave-one-out cross-validation criterion. Each row of the table reports the node (the point of observation for monitoring the ‘sensor’ variation) in the first column and the minimum detectable variation in that node, in the second column.

Node | RMSECV [nF] |
---|---|

1 | 0.36 |

2 | 0.19 |

3 | 0.17 |

4 | 0.31 |

5 | 2.08 × 10^{−6} |

6 | 0.16 |

7 | 0.20 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

D’Amico, A.; Santonico, M.; Pennazza, G.; Zompanti, A.; Scipioni, E.; Ferri, G.; Stornelli, V.; Salmeri, M.; Lojacono, R.
Resonant Directly Coupled Inductors–Capacitors Ladder Network Shows a New, Interesting Property Useful for Application in the Sensor Field, Down to Micrometric Dimensions. *Micromachines* **2018**, *9*, 343.
https://doi.org/10.3390/mi9070343

**AMA Style**

D’Amico A, Santonico M, Pennazza G, Zompanti A, Scipioni E, Ferri G, Stornelli V, Salmeri M, Lojacono R.
Resonant Directly Coupled Inductors–Capacitors Ladder Network Shows a New, Interesting Property Useful for Application in the Sensor Field, Down to Micrometric Dimensions. *Micromachines*. 2018; 9(7):343.
https://doi.org/10.3390/mi9070343

**Chicago/Turabian Style**

D’Amico, Arnaldo, Marco Santonico, Giorgio Pennazza, Alessandro Zompanti, Emma Scipioni, Giuseppe Ferri, Vincenzo Stornelli, Marcello Salmeri, and Roberto Lojacono.
2018. "Resonant Directly Coupled Inductors–Capacitors Ladder Network Shows a New, Interesting Property Useful for Application in the Sensor Field, Down to Micrometric Dimensions" *Micromachines* 9, no. 7: 343.
https://doi.org/10.3390/mi9070343