Signal Propagation in Transmission Lines with Losses Using Fibonacci Wave Functions

DOI: http://dx.doi.org/10.24018/ejece.2021.5.5.360 Vol 5 | Issue 5 | October 2021 32 Abstract — In this paper, the general model for an infinite LC ladder network using Fibonacci wave functions that were applied to lossless transmission lines will be extended to transmission lines including losses. The general model that was derived from a first order system transfer function representing a simple RC or RL circuit will be used to model and analyze transmission lines presenting losses. The LC ladder network model can be applied to any order for each inductor current with its parasitic rL resistor and for each capacitor voltage with its parasitic rC resistor. The extension of the proposed general model to transmission lines with losses is subject to Heaviside condition for both resistors rCand rL.


I. INTRODUCTION
In the previous papers [1] and [2], Fibonacci wave functions (FWFs) were introduced to model an infinite LC ladder network without losses. A general model for any LC ladder network order was also introduced.
In the literature, very few papers analyzed either LC ladder or transmission cable including losses with exhaustive mathematical calculations [4]. Most other papers used only lossless modeling and analysis. Introducing losses in such systems leads to another level of difficulties in modelling and analyzing these systems [3], [5] and [6]. Fibonacci wave functions may help simplify and extend the analysis to another horizon.
In this paper, the extension of FWFs with its general model to LC ladder network with losses to model a transmission lines with resistive losses will be modeled and analyzed for different orders and will be compared to MATLAB-Simulink LC model presenting the same losses. The paper is organized as follow. Sections II and III are dedicated to the general model extension to resistive-capacitive including losses Fibonacci electrical circuit (RC-FEC-Losses) and resistive inductive including losses Fibonacci electrical circuit (RL-FEC-Losses. Section IV describes a comparative study of FWFs and RC-FEC-Losses using MATLAB-Simulink model. Section V shows a comparative study of FWFs and RC-FEC-Losses of transmission line with short and open terminations.

II. RC-FEC-LOSSES FIBONACCI ELECTRICAL CIRCUIT
The original Fibonacci transfer function has the following form [1]. The first order electrical circuit RC-FEC-Losses is presented in Fig. 1. The equivalent impedance between the capacitor C and its parasitic resistor can be easily calculated as follows. The second order RC-FEC-Losses circuit diagram shown in Fig. 2   The equivalent impedance between the inductor L and its parasitic resistor can be easily calculated as follows: and changing Laplace variable to the new reference = + with the condition = which is exactly the known Heaviside condition for no signal distortion through the transmission lines. This condition does not eliminate losses through the lines. This condition is necessary to extend the Fibonacci wave functions to transmission lines that present losses.
The impedance becomes = . Furthermore, the circuit in figure 2 can easily be written in the new Laplace reference defined by Laplace variable .
The transfer function of the third order RC-FEC-Losses (5) is derived from circuit diagram presented in Fig. 3.
One can see that an even ℎ order RC-FEC-Losses ( Fig.  4) will have voltage as input and current as output.
n c total number of capacitors in the circuit.
n L total number of inductances in the circuit. The FWF of the ℎ even order of the circuit in Fig. 4 is: For the ℎ odd order in the circuit in Fig. 5, the FWF is:

III. RL-FEC-LOSSES FIBONACCI ELECTRICAL CIRCUIT
As RC-FEC-Losses Fibonacci wave functions FWFs, RL-FEC-Losses can be calculated easily. The first order circuit in Fig. 8 and its FWF is presented in (9). For the 2 nd order RL-FEC-Losses the FWF is expressed in (10).
The third order RL-FEC-Losses will be defined with an input voltage and output current (Fig. 10) and its FTF in (11).
an even ℎ order RL-FEC-Losses in figure 11 has its FWF expressed in (13).

= + (12)
n c is the total number of capacitors in the circuit. n L is the total number of inductance in the circuit.
An odd ℎ order RL-FEC-Losses in Fig. 12 has its FWF expressed in (14).

IV. COMPARATIVE SIMULATION OF RC-FEC-LOSSES AND THEIR FWFS
Comparative simulation studies were conducted between the electrical circuits using MATLAB-Simulink model with Fibonacci wave functions for each order in Table I. The studies confirm that these electrical LC ladder circuits for different orders representing different transmission lines length with losses defined by and follow the same recurrent Fibonacci sequence modelled by FWFs [1].

Case 1: R=1; L=1H; C=1F; = 0.02 = = 50
In this case ( , , ) = (1,1,0.2). Pascal's Triangle in Table II presented in [1] will be used to determine all FWFs. 14 ( , , ) ( ) taken as example is an even function, using its numerator and denominator coefficients are expressed in (15) using  Using input voltage ( ) = for all even orders FWFs and input current ( ) = for all odd orders FWFs. Fig. 15 shows the block diagram in both references ℒ( ) and ℒ( , ) for voltage as input, the figure is also applicable when the input is current. Simulation will be used in ℒ( , ) since all FWFs are perfectly known. This block diagram is also applicable for all odd orders FWFs with current as input and voltage as output.
Simulations of ( ) 40  Steady states values (SSV) can be easily found using Table  IV below    The input impedance or admittance can be found using Pascal's triangle for the case R=1; L=1H; C=1F; = 0.02 = = 50.
This is a particular case of Pascal's triangle general form detailed in [1].
Below is an example of 13 ( ) and 14 ( ) using Pascal's triangle in Table III. For 13 ℎ and 13 ℎ order RL-FEC-Losses circuit in Fig. 19 is used to determine the corresponding FWFs.

= 1
For RC-FEC-Losses or RL-FEC-Losses, the input impedance or admittance for any order and for both shortcircuit and open-circuit is as follow.

VI. CONCLUSION
In this paper, a complete Fibonacci wave functions FWFs model that was first applied to an LC ladder network representing lossless transmission line [1] and [2] is extended to LC ladder network presenting losses ( , ) to model a transmission line with no distortion subject to Heaviside condition. The general model with losses proposed in this paper helps find automatically all transfer functions, any input impedance or admittance for any recurrent LC network order.