Electrical Equivalence of discrete cascaded T sections, Electrical Transmission Lines, and Fibonacci and Lucas Numbers
Abstract  The Fibonacci numbers, Lucas numbers and the Golden Section are reflected in many types of Electrical Networks. This paper concentrates on identical and symmetrical T networks connected in series. The ladder networks ABCD Constants are described by a matrix of n cascaded T sections which under certain conditions can be shown to be electrically equivalent to the hyperbolic equations of a transmission line and also electrical equivalent to a matrix incorporating the Fibonacci and Lucas numbers.
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1. INTRODUCTION
It is well known that in electrical networks, Fibonacci numbers, Lucas numbers and the Golden section occur. For example, if a series of "n" equal resistances are connected alternately in series and in parallel, the total resistance can be described by the continued fraction expansion of
then for successively larger numbers of resistances, the value of R_{T} / r is:
and if the first resistance in the circuit is in parallel instead of in series then the value of R/r becomes:
where

and
Eq. (1.1) is often called the Golden Section and in some text is represented by f (Greek letter phi) or by some other symbol, we shall continue to use a (Alpha) in this paper. Eq. (1.2) is Beta or the negative reciprocal of Alpha and Eq.(1.3) and (1.4) are the equations for the Fibonacci numbers and Lucas numbers respectively.
In other types of electrical networks consisting of equal series and shunt Z (impedance), it can be shown that the ratio of input impedance (output open circuit) to Z for the L, T and networks are:
for the L network
for the T network
For the network
Where n is the number of cascaded sections (branches).
In this paper, a different approach is taken. It can be shown that under certain conditions the voltage and current distributions of cascaded symmetrical T networks are also electrically equivalent to a matrix consisting of classical transmission line equations which are functions of the image impedance and the propagation constant of a single T section and are electrically equivalent to a matrix consisting of Fibonacci and Lucas numbers.
2. THE SYMMETRICAL T NETWORK
Fig. 1
The value of Z_{O} (image impedance) for a symmetrical network can be easily determined. For the symmetrical T network of Fig. 1, terminated in its image impedance Z_{O}, and if Z_{1 }= Z_{2 }= Z_{T} then from many textbooks:
(2.1) 
(2.2) 

Under Z_{O} termination, input and output voltage and current are:
(2.3) 

If there are n such terminated sections then the input and output voltages and currents, under Z_{O} terminations are:

Where g is the propagation constant for one T section., eg can be evaluated as:
(2.5) 

if Z_{1 }= Z_{2 }= Z_{T} then taking the natural logarithm, the propagation constant of a single T section becomes:
(2.6) 
g _{T} = ln a ^{2} 
3. EQUIVALENT NETWORKS
A typical electrical network is expressed in ABCD Matrix form as:
(3.1) 
Where the ABCD General Circuit constants are given by:
One of the most valuable aspects of the ABCD parameters is that they are readily combined to find the overall parameters when networks are connected in cascade. For a single T section, as shown in Fig. 1, where
Z_{1 }= Z_{2 }= Z_{T} =Z
(3.2) 
And for "n" cascaded T sections each having equal Z. The Electrical characteristics can be expressed in terms of the Fibonacci and Lucas numbers of 2n as:

Expressed in another mathematical form:
(3.4) 

Where I is the unit matrix 
We now make the assumption that a change in voltage along the T section is proportional to the current per section (I) and the product of the characteristic impedance ( ) of the T section, the propagation constant ( ) of the T section and the change in the number of T sections ( ). We then have:
(3.5) 
For Voltage. 

(3.6) 
For Current. 
Forcing the equality and changing to differential notation:
For Voltage. 

For Current. 
differentiating equation (3.7) and (3.8) with respect to n:
For Voltage. 

For Current. 
Combining equations (3.7) and (3.8) with (3.9) and(3.10) we have:
For Voltage. 

For Current. 
Equations (3.11) and (3.12) are equations of the same fo rm as the equations of the infinitesimal transmission line of distributed parameters and will have solutions of the same form.
With solutions of the form: and
From which the solutions by conventional methods leads to the same form of solution, for the lumped circuits, as the classical set of equations do for the distributed circuits of a transmission line, they are shown in hyperbolic form:
and shown in ABCD matrix form
Combining equations (3.3) and (3.15)
Where the image impedance is:
(3.17) 

and the propagation constant is:
(3.18) 
g _{T} = ln a ^{2} 
That is the Voltage and Current distributions of a series of n (integer) discreet cascaded T sections whose
Z_{1 }= Z_{2 }= Z_{T} =Z has the same form as the distributed line equations, n (integer) units of distance from the generator and whose image impedance and propagation constant is the same as a single T section, and whose complete electrical characteristics can be expressed by the Fibonacci and Lucas numbers of 2n.
Of course other parameters of the line can also be expressed in terms of Fn, Ln and Zo as:
The input impedance (output open circuited) 
Also:
The input impedance (output short circuited) 

And as n grows large 
Also:
Sending to receiving voltage and current ratio as in equation (2.4). 

And the input to output power ratio 

4. THE INFINITESIMAL LINE
Consider the infinitesimal transmission line. It is recognized immediately that this line, in the limit may be considered as made up of cascaded infinitesimal T sections. The distribution of Voltage and Current are shown in hyperbolic form:
And shown in matrix form:
Where Z_{L} and Y_{L} are the series impedance and shunt admittance per unit length of line respectively.
Where the image impedance of the line is:
And the Propagation constant of the line is:
And s is the distance to the point of observation, measured from the receiving end of the line.
Equations (4.1) and (4.2) are of the same form as equations (3.13) and (3.14) and are solutions to the wave equation.
Let us define a set of expressions such that:


Where 

Also note that: 
If we now substitute equations (4.6) and (4.7) into equations (4.4) and (4.5), and allowing we have:
And so by choosing and then using equations (4.6) and (4.7) to find , both Real, Imaginary or Complex, then equations (4.3) will be equivalent to equation (3.15) and equation (3.16).
So that the infinitesimal transmission line of distributed parameters, with Z and Y of the line as found from equations (4.6)and (4.7), a distance S from the generator, is now electrically equivalent to a line of N individual T sections whose .
VI. CONCLUSION
After a short introduction to the appearance of Fibonacci and Lucas numbers in electrical networks, the symmetrical T network was discussed. The Voltage and Current distributions of cascaded T networks were shown to be represented by Fibonacci and Lucas numbers as expressed in ABCD matrix form.
A set of differential equations, for the lumped T elements, were developed whose solutions were of the same form as the transmission line equations for distributed circuits and gave us another way of expressing , in hyperbolic form, the electrical characteristics as functions of alpha and beta.
A series of expressions were developed for the series impedance and shunt admittance of the transmission line of distributed elements, which were functions of the propagation constant and the series and shunt impedances of a single T section, which when substituted into the equations of the line gave us hyperbolic equations which could be expressed by Fibonacci and Lucas numbers.
We have shown that a series of cascaded T sections whose ZY=1, is electrically equivalent to the hyperbolic equations of the infinitesimal transmission line and whose electrical characteristics can be completely expressed by the Fibonacci and Lucas numbers of 2n.
We would also find that the fibonacci and Lucas numbers should be embodied in the electrical descriptions of other types of circuit configurations, such as the Pie and L type of networks.
Although these lines are practical and can be realized using lumped discrete components, unfortunately such lines are not as practical to realize using distributed parameters with off the shelf components.
However, these lines can be modeled with any value of series impedance and shunt admittance by the latest mathematics modeling applications or the latest electrical circuit simulation applications.
REFERENCES
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