SOME FURTHER NOTES ON THE FIRST PAPER

We start by defining the variables that we will need for the calculations to follow.

imganary operator

Alpha; The Golden Ratio

Beta; The negative root of the Fibonacci quadratic equation

The number of cascaded T sections. Must be an integer.

The Fibonacci numbers of 2n

The Lucas numbers of 2n

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Set series impedance of cascaded T sections in ohms

Set shunt admittance of cascaded T sections in mho's

Choose any number for these values as long as Zt Yt = 1

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Calculate propagation constant, consisting of the attenuation constant and the phase constant of a single T section with and Note that no matter what values are chosen as long as .Zt Yt = 1 the propagation constant will always be the same.

Propagation constant:

Taking the logarithm to arrive at the propagation constant in Napier's

The propagation constant for 1 T section. This is not a function of N.

The propagation constant is the logarithm of the Golden ratio squared

The attenuation constant is real and is less than 1, while the Phase constant is 0.

It is possible for the phase constant to be zero. An example is seen in this purely resistive network. This network will attenuate an applied signal and the input and output signals will be in phase with each other, hence the phase constant is zero.

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Calculate the Characteristic impedance of a single T section with and .

Characteristic impedance or image impedance equation in ohms

The characteristic impedance of 1 T section. Again this is not a function of N. This is equal to

Or the impedance of the T section times the Golden section minus 1/2.

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Now lets review just what the ABCD matrix is:

We start with the loop equations. The voltage and current at one end are known and the voltage and current at the other end are to be found.

where Vs and are the sending end voltage and sending end Current respectively and V.R and IR are the receiving end voltage and current respectively.

We now change to matrix form

and if we matrix multiply rows by columns we have the above equations

From which we have:

With

Forward voltage transfer ratio or the open circuit forward voltage gain

output open

With

Forward voltage to current ratio or short circuit forward impedance.

output shorted

with

Forward current to voltage ratio or open circuit forward admittance.

output open

Forward current transfer ratio or short circuit forward current gain.

with

output shorted

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We will now place the numerical values into the ABCD matrix for 1 T section and then for Sections

The first series branch element of the T

The shunt arm element of the T

Now combine these elements into an unsymmetrical L section

Unsymmetrical L section

Now combine another series branch element for the complete symmetrical T section

Here is the complete ABCD matrix for 1 T section

Find ABCD values for sections by raising the ABCD matrix to the power. This mathematically cascades the T sections together.

Value of the ABDC matrix for

Find characteristic impedance using matrix notation.

The characteristic impedance in ohms

The open circuit impedance in ohms

The short circuit impedance in ohms

The characteristic impedance by using the open and short circuit impedances

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By using the values of VS / VR , with output open, from the ABCD matrix, or the value we can calculate the propagation constant for the values of any N, in this case the value of T sections

This equation is

This overall propagation constant is the summation of the individual propagation constants or N times the first propagation constant which is times the value of = N ln(α2) with the cascaded T sections terminated in ZZO .

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We will now find the A,B,C,D values using Fibonacci and Lucas numbers of 2N and Zt:

Define ABCD values

Define ABCD matrix

We have the same electrical values as above for cascaded T sections.

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We know from many textbooks on transmission engineering that the image impedance and the propagation constant of the line are:

The image impedance of the line

The propagation constant of the line

Solving these two equations simultaneously for the impedance and admittance of the line we have the impedance and admittance as functions of the propagation constant and image impedance of the line.

The impedance of the line

The admittance of the line

We now determine the numerical values by setting the image impedance (ZZot) and the propagation constant (γt) of a single T section equal to the impedance and the admittance of the line and then finding the corresponding numerical values of impedance ( ZZL) and admittance (YYL) of the line.

Impedance of the line using single T section values

Admittance of the line using single T section values

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The differential circuit equations for distributed constants of voltage and current of the line are as follows.

Rate of change of voltage with respect to number of T sections

Rate of change of current with respect to the number of T sections

These expressions say that the rate of change of voltage and current with respect to N are equal to the current and voltage times the impedance and admittance of the line respectively. They are the fundamental transmission line equations for voltage and current.

Second order differential equation for voltage

Second order differential equation for current

From the second order differential equations and from many textbooks we can find a solution of the transmission line equations as:

For voltage.

For current:

These are equations of the infinitesimal line we wrote about in the First Paper.

We will now substitute the values obtained above for ZZL and YYL For the values of ZL and YL . These are the values obtained by using the image impedance and propagation constant of a single T section.

Substitute single T section values for impedance of the line.

Substitute single T section values for admittance of the line.

Check propagation constant for a single T section

Check image impedance

Set up the hyperbolic equations of the line in matrix format.

Set up matrix elements

We have the same ABCD electrical values as the cascaded T sections matrix and the Fibonacci and Lucas matrix values only these values are obtained from the equation of the infinitesimal line using N as an integer distance or N as the number of cascaded T sections.

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So now lets turn our attention to the Alternating Current case.

The differential circuit equations for distributed constants of voltage and current of the line are as follows. Where omega is the radian frequency and L and C are the inductance and capacitance respectively.

Where Z = jωL and Y = jωC

Rate of change of voltage with respect to number of T sections

Rate of change of current with respect to number of T sections

Here we have the rate of change of voltage and current with respect to N as functions of current and voltage times the AC series reactance and shunt susceptance and frequency.

Second order differential equation of the line for voltage

Second order differential equation of the line for current

From the second order differential equations and from many textbooks we find a solution of the AC transmission line equations as:

For voltage

For current

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We ask what value of L and C will be equal to our previously computes ZZL and YYL if the frequency is the same for both reactance and susceptance?

So we set:

and

An imaginary value, equal to a real value.

Setting the radian frequencies equal to each other.

We have:

Or

Any value of capacitance or inductance can be used as a starting value

We set the capacitance to an arbitrary value and rename C as CAP and L as LIND .

Capacitance in Farads

We use the above value of L / C times the capacitance to determine our inductor value

Inductance in Henrys

Sense

Then

Where the radian frequency is:

We have a negative imaginary frequency

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Image impedance is real

Velocity of transmission in units per second is real

Cutoff frequency is real

Series impedance and shunt admittance are real

Propagation constant for 1 T section is real

Propagation constant for T sections which means the phase constant is 0.

Set up ABCD matrix

We have the same ABCD electrical values as the cascaded T sections matrix, the Fibonacci and Lucas matrix and our line of distributed constants values, only these AC values are obtained from the equation of the infinitesimal line using N as an integer distance or N as the number of cascaded T sections.

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REFERENCES:

1. Same as the first paper.

2. Brent Maxfield P. E., Essential Mathcad For Engineering, Science, and Math, Academic

Press, 2009.

3. Ronald W. Larsen, Introduction to Mathcad 15, Prentice Hall Inc., 2011.

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