1. Introduction
Passive ladder networks (L.N.s) made by a number of single cells showing both longitudinal and transversal impedances have been studied for a long time [
1,
2,
3,
4,
5,
6] due to their versatility in representing a good model for mechanical, chemical, thermal, and electronic systems and also because they have been frequently employed in passive filters [
7,
8]. In fact, the representation of complex dynamic systems by using the analogy of electrical networks has proven to be very useful in the realization of interfaces for monitoring integrated sensors, sensor systems, and microstructures, especially when the sensors work at the micro-dimensional level. In this analogy, the role of sensor activity is played by those electronic components, which present measurable time variability, such as inductors or capacitors. These kinds of electrical networks are called inductor–capacitor (L–C) networks and are often used in modeling a particular kind of information transmission. When the inductor (L) and capacitor (C) elements represent sensors operating in a real scenario, the content of each variation is a part of global information conveyed by the whole network of electro-mechanical sensors. The interest in L.N.s today is still alive due to new applications, for example, in analog neural networks. Furthermore, we cannot exclude in the near future their possible implications in the study of the electric behavior of both DNA and RNA structures [
9] and related aspects of epigenomics.
This paper, on the other hand, also considers these kinds of networks from another viewpoint: the search of the presence of links with Fibonacci numbers. In many cases these famous numbers are present as expression of impedances, voltages, and currents and may facilitate the rapid calculation of their amplitudes [
4].
The particular structure of the directly coupled L–C single cell L.N. shows very strong peculiarities, which are shown in
Figure 1, reported below, and taken from D’Amico et al. [
5].
In the vertical axes we have, for example, the number of L–C cells from 1 to 100. In the horizontal axes, we have the normalized frequencies, which means that in the case of only one cell ω
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.
The novelty in this paper concerns the identification of a new property related to the L–C L.N. that can be of a certain utility in the field of sensor network. In this case we have imagined dealing with capacitive sensors for either mechanical of chemical quantities. In fact, measuring the frequencies vector in only one of the randomly selected nodes of a given L–C L.N., we have found that it is possible to determine the capacitor that has changed its value due to a given sensing action. Of course, the same property is evident if we consider the inductors as sensors. The result is the same due to the fact that the transfer function of this network is related to the ratio (K) between the longitudinal Z1 = jωL and transversal impedance Z2 = 1/jωC.
In fact, being that K = ω2LC, the same changes of either C or L (not simultaneous changes) will produce the same result.
In this paper, we have investigated, as an example, the transfer function of seven L–C cells directly coupled forming a discrete L.N. and found a new interesting property useful for applications in the field of sensors. In fact, surprisingly, each node brings the information of each cell, and when one capacitor of a cell is changed, then the voltages in the seven nodes change also. Of course, each one changes in a different way, but the identification of the modified cell can be performed whatever the testing node. This new property is found to be of interest for the control of sensorial multi-points: this attitude is strategic when applied to the monitoring of networks developed at the micrometric level, when the ‘interrogation’ and the localization of the ‘sensor-points’ could be not so easy. We have estimated the resolution of this system as a final contribution to the knowledge of this peculiar discrete network. Definitively, the purpose of this work is the study of useful electrical properties of directly coupled L–C cells forming a discrete ladder network (L–C L.N.) to be applied to the sensor field.
2. Materials and Methods
A typical L–C L.N. is shown in
Figure 2.
As shown in the literature [
6], the transfer function of this L–C L.N., whatever the number of cell, can be easily determined by the use of the DFF triangle [
4], which gives the modules of the coefficients of the polynomial at the denominator of the transfer function.
In the same paper, it is shown the following expression, which gives the voltage
Vb 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.
This triangle is here reported in
Table 1 and is framed to give the coefficients for a seven-cell L.N. (
Figure 3), which is the number taken into account in this paper for the demonstration of the new property of this L.N.
In order to obtain a general model to be used for any kind of L–C network with n-cell, a bottom-up strategy has been used, starting from the analysis of a single L–C cell. The starting condition (assumed as a reference) for this cell is given by the following values: L = 68 uH and C = 47 nF (see
Figure 4).
Studying the transfer function of this cell and simulating the relative electronic circuit in MultiSim (National Instruments, Austin, TX, USA), we obtain the magnitude curve (
Figure 5), which confirms the following theoretical resonance frequency:
This frequency is characteristic of the elementary cell (
Figure 5), and the calculus is here implemented for a n-cell ladder network.
The pattern of frequencies relative to the
n poles and to the
m zeros, related to the voltages at internal nodes, is typical of the ladder network. The
n cells of the ladder network and the
n + m frequencies registered form a multidimensional
n(
n +
m) array: this array provides a dynamic picture of the network, and its elaboration via multivariate data analysis techniques gives important information on the network condition. The study of this array has been here performed with both a qualitative and a quantitative approach. Principal component analysis (PCA) has been used to explore the array variation with the aim of dynamically identifying whether a specific pattern is able to identify the point of observation on the network or the element (C or L) whose value has changed or both. Partial least square discriminant analysis (cross-validated via the leave-one-out criterion) has been used here in order to quantify the occurring variation of capacitance and/or inductance. Let us consider the seven-cell L–C ladder network with L = 68 uH and C = 47 nF as reported in
Figure 6. This is the electronic circuit analyzed in the following of the work.
4. Conclusions
Ladder networks have been employed in many applications in the engineering context, and in this paper, we have shown that another important property can be attributed to them in the special case when the L.N. is formed by longitudinal inductors and transversal capacitors.
The results obtained in this work demonstrate that one of the transversal elements, capacitors in this example, can be univocally identified by using the arrays of the resonance frequencies, which can be easily determined by the transfer function of the L.N. itself. Moreover, by analyzing these arrays with multivariate data analysis technique, it is possible to localize the element where in the L.N. the variation is occurring and to quantify this variation independently from the observation node. The model here presented has a general validity and can be used for any kind of network as far as the number of cells is concerned.
This method could be of particular utility in cases of complex network configurations, when the identification of the variation of single sensorial element is not easy, such as in the case of sensor networks operating at the micrometric dimensional level. Of course, there is an important aspect to be considered when dimensions of sensors are reduced, which deals with the possibility of keeping the overall performance analysis when operating frequencies become higher and higher. The identification of limits in this direction is a challenge for near-future technological work. Moreover, whenever the L.N. represents a sensor network, an automated localization system based on such model could be useful to speed up the identification process and the potential warning activation, especially in case of hazardous or extended areas.