Signal Propagation in NEF LC Ladder Network Using Fibonacci Wave Functions

DOI: http://dx.doi.org/10.24018/ejece.2019.3.6.155 1 Abstract—In this paper, new general model for an infinite LC ladder network using Fibonacci wave functions (FWF) is introduced. This general model is derived from a first order resistive-capacitive (RC) or resistive-inductive (RL) circuit. The nth order Fibonacci wave function of an LC ladder denominator and numerator coefficients are determined from Pascal’s triangle new general form. The coefficients follow specific distribution pattern with respect to the golden ratio. The LC ladder network model can be developed to any order for each inductor current or flux and for each capacitor voltage or charge. Based on this new proposed method, nth order FWF general models were created and their signal propagation behaviors were compared with nth order RC and LC electrical circuits modeled with Matlab-Simulink. These models can be used to represent and analyze lossless transmission lines and other applications such as particles interaction behavior in quantum mechanics, sound propagation model.


I. INTRODUCTION
Fibonacci wave functions (FWFs) are transfer functions with high degree that are irreducible. Their characteristic behavior is unique. Their response to a step input signal gives multiple intermediate stationary regimes before reaching the final steady state which presents oscillations with low amplitudes. The FWFs can be created theoretically from a first-order origin wave function [1]. Fibonacci wave functions have multiple resonance and anti-resonance frequencies organized in a perfect way with respect to each other. Moreover, they have two well defined Fibonacci boundary systems using Pascal's triangle [2]. In this paper, a step by step development methodology of new electrical circuit application of FWFs called Fibonacci electrical circuits (FECs) is introduced to model perfectly the recurrent LC ladder network. These FECs can be used to model transmission cables [3], [4], the behavior and interaction of the infinitely small particles using the infinite LC networks [5] in quantum mechanics, the neural dynamic in biology [6], etc.
The paper is organized as follow. Section II describes a general model of resistive-capacitive Fibonacci electrical circuit (RC-FEC). Section III presents a comparative study of Fibonacci wave functions (FWFs) model and its equivalent Matlab-Simulink RC-FEC circuit. Section IV describes a general model of resistive-inductive Fibonacci electrical circuit (RL-FEC). Section V gives a comparative study of Fibonacci wave functions (FWFs) and its equivalent Matlab-Simulink RL-FEC circuit. N th order RC-FEC and RL-FEC FWF general models are presented in section VI. Section VII presents an application of FWF to transmission lines and are compared with particular case of short circuit found in [3].

II. RC FIBONACCI ELECTRICAL CIRCUIT (RC-FEC)
The original Fibonacci wave function has the following form. The first order electrical circuit resistive-capacitive Fibonacci electrical circuit (RC-FEC) is presented in figure  1.
The second order RC-FEC circuit diagram is shown in figure  2 and its wave function in (3).
The wave function of the third order RC-FEC (4) is derived from circuit diagram presented in figure 3 One can see that an even JK order RC-FEC (figure 4) will have voltage as input and current as output.
n N is the total number of capacitors in the circuit. n O is the total number of inductance in the circuit. The FWF of this circuit is.
For JK odd order, the wave function is.

III. SIMULATION OF RC-FEC AND FWFs
Simulation studies were conducted to compare the previous electrical circuits with Fibonacci wave functions and Matlab-Simulink electrical circuit model. The studies confirm that these circuits follow the logic of a recurrent Fibonacci sequence and can be modelled by FWFs.

A. Case 1: R=1W; L=1H; C=1F
In this case ( , 1 ) = (1,1). Pascal's Triangle in table II will be used to determine all FWFs. Order 14 FWF taken as example is an even function, using its numerator and denominator coefficients are expressed in (8) using table II.      RL-FEC is determined in the same way as RC-FEC. The first order circuit in figure 8 and its FWF is presented in (9). The 2 nd order RL-FEC will be defined with current input and voltage output (figure 9) and its FWF expressed in (10).
The third order RL-FEC will be defined with an input voltage and output current ( Figure 10) and its FWF in (11).
In general, RL-FEC with an even JK order in figure 11 has current as input and voltage as output.
n N is the total number of capacitors in the circuit. n O is the total number of inductance in the circuit.
A RL-FEC with JK odd order ( Figure 12) has voltage as input and current as output and its wave function expressed in (14).   In the same way, the RL-FEC model is illustrated in figure  19 below for = 40.

VII. FIBONACCI WAVE FUNCTIONS APPLIED TO TRANSMISSION LINES
In the literature, many papers have studied and modeled the transmission lines [3] with different analytical methods but very few have noticed that Fibonacci numbers and especially Pascal's triangle can be used [3], [4]. It is well known that the communications lines can be modeled with a recurrent LC depending on the length of the cable. There have been studies to analyze the input impedance as well as the load impedance to better understand the reflection phenomena when the input impedance is different from the load. In [3], it was shown that in a transmission cable with a short-circuit (R=0W), the input impedance or admittance can be found using Pascal's triangle for the case L=1H and C=1F. This is a particular case of Pascal's triangle general form detailed in table II with =1 and 1 = ¥.
Below is an example of %[ ( ) using Pascal's triangle above.       The LC ladder general model for energy transfer between L and C in each section can be used for many other applications that use lossless LC recurrent circuit as model for more research and analysis especially, in quantum mechanics, biology and communication.