MAPUA INSTITUTE OF TECHNOLOGY
School of EE-ECE-CoE
COMMUNICATIONS
LABORATORY 3
Experiment Number 1
Two Wire Line Investigation Demonstrating the Effects of
Frequency, Attenuation, Loading Coils and Determining
Characteristic Impedance
Course/Section: ECE123L/E01
Name: Padilla, Melvin Mark L.
Date of Performance: 06/02/2020
Date of Submission: 06/09/2020
Engr. Jessie Balbin
Grade
Interpretation of Sample Data
In the Table 4, we can see the transmission response of 0.85km twowire line. In open circuit, as the frequency increases the attenuation seems
to be the same all throughout. Also when the frequency increases, the output
voltages also decrease. It is similar also in the case when terminating with
the impedance but the attenuations are different but with minimal
discrepancies. In termination, the line will remain the same or constant once
the load becomes equal with the characteristic impedance. Also the log
attenuation are constant because U1 and U2 have small difference in value.
The case is still the same when the length of the line is increased to
5km. Both the attenuation and log attenuation increase as the frequency
increases which means that they are directly proportional to each other. But
the voltage outputs decrease as the frequency increases which means that
they are inversely proportional to each other. It is evident as shown in the
graph of attenuation versus frequency using open circuit and 600 ohms
termination.
In Table 6, loading coils were not used yet but it can be seen that the
attenuation decreased as the frequency increased which it became inversely
proportional to each other because of the effect of longer two-wire line.
In Table 7, loading coils are used. Both in open circuit and termination
600 ohms, the log attenuations and attenuations are increased. The increase
is directly attributed to the loading coils due its high reactance at high
frequencies. It is the same with results in Table 8.
For Tables 9 and 10, it is seen that in lower frequencies, there will be
higher impedances as shown in the graph. Also, when the diameter and line
length are increasing, the line termination is decreasing.
Sample Problems
1. At a frequency of 4 MHz a parallel wire transmission line has the
following parameters: R = 0.025 /m, L = 2 H/m, G = 0, C = 5.56
pF/m. The line is 100 meters long, terminated in a resistance of 300 .
Find the standing wave ratio and voltage reflection coefficient of the
load.
2. Consider a lossless coaxial transmission line having distributed
parameters L = 245 nH/m and C = 200 pF/m. The line is terminated
with a resistor RL = 100 as shown. The operating frequency is f =1 GHz.
Determine the characteristic impedance and phase velocity of the line.
3. A load with impedance ZL = (25 -j50) is to be connected to a lossless
transmission line with characteristic impedance Zo, with Zo chosen
such that the standing-wave ratio is the smallest possible. What should
Zo be?
4. A transmission line has the propagation constant y = 0.1 + j10 lm, and
characteristic impedance Zo = 50 + j5 . The line is terminated in an
impedance 100 -j30 . Find the impedance at a distance of 1.5m from
the load.
5. A load impedance, (200 + j0)  is to be matched to a 50 lossless
transmission line by using a quarter wave line transformer (QWT). The
characteristic impedance of the QWT required is
Discussion
A transmission line is used for the transmission of electrical power from
generating substation to the various distribution units. It transmits the wave of
voltage and current from one end to another. The transmission line is made up of a
conductor having a uniform cross-section along the line. Air act as an insulating or
dielectric medium between the conductors [1].
Fundamentals of Transmission Lines are considered to be impedancematching circuits designed to deliver power (RF) from the transmitter to
the antenna, and maximum signal from the antenna to the receiver. From
such a broad definition, any system of wires can be considered as forming
one or more transmission lines. If the properties of these lines must be taken
into account, the lines might as well be arranged in some simple, constant
pattern. This will make the properties much easier to calculate, and it will
also make them constant for any type of transmission line. All practical
Fundamentals of Transmission Lines are arranged in some uniform pattern.
This simplifies calculations, reduces costs and increases convenience [2].
There are two types of commonly used Fundamentals of Transmission
Lines. The parallel-wire (balanced) line is shown in Figure 1, and the coaxial
(unbalanced) line.
Figure 1
All transmission lines are characterized by conductivity and dielectric
losses. In some cases losses can be neglected, but not in others. When
losses are small, some simplifications of transmission line parameters can be
made.
The complex propagation constant is
when the
transmission line has low conduction and dielectric losses, we can assume
that R<<wL and G<<wC.
So for
is the
characteristic impedance. This approximation is called low-loss high
frequency approximation and can be used when losses of the transmission
line are low.
It does mean that the different phase parts of the wideband signal will
travel with different phase velocities and arrive at the reciever at different
times. This effect will lead to the dispersion and the distortion of the signal.
So the longer the transmission line, the bigger the phase effect.
Power flow across the transmission line helps to find the attenuation
constant, and this method is called perturbation method. When the
reflections at the transmission line can be estimated as absent, the
attenuation constant can be viewed as
Figure 2. L-length transmission line with losses
According to Mechaik [3], Figure 3 shows waveform plots for
conductor, dielectric, and total attenuation losses in dB/m versus frequency
in GHz for a typical coaxial cable characterized by: inner radius of 0.362 mm,
outer radius of 1.524 mm, and the dielectric material is Teflon. The waveform
plots of Figure 3 show several interesting conclusions: The total attenuation
loss is the sum of conductive losses and dielectric losses. Note that the curve
for the dielectric loss is multiplied by a value of 10K in order to show its
variations with respect to the other two curves. All three losses increase with
increasing frequency. For the frequencies of interest (i.e., < 25 GHz), the
total loss is totally dominated by the conductor losses. Dielectric losses are
almost negligible below 2.5 GHz as compared to conductor losses. This is the
case only for the frequencies considered as for higher frequencies the loss
tangent increases with increasing frequency. The conductor losses and hence
the total losses increase almost as the square root of frequency when all
other parameters are held constant. This observation is consistent with the
square-root dependence of resistance with frequency. This is usually referred
to as skin effect dependence. Extrapolation of the three curves in Figure 3
shows that at much higher frequencies, dielectric losses dominate resistive
losses. Digital systems in general operate at much lower frequencies thereby
making resistive losses dominate dielectric losses for most frequencies of
interest. Transmission of data bits over a distance of 10 m can suffer as much
attenuation as 9 dBs at 5 GHz. This loss can even be greater at higher
frequencies. This loss of power is significant considering the amount of
power a typical downstream signal carries since most CMOS transistors can
provide up to less than 5 volts of output voltage. For TTL and ECL transistors,
the power loss can even be greater.
Figure 3: Transmission Losses through a Coaxial Cable
Figure 4 shows three waveform plots for the total
attenuation loss with outer to inner radii ratios of 2, 4, and 8. All three curves
are obtained using the same data used for Figure 3. An interesting
observation is that as the ratio of the outer to inner radius increases, the
total attenuation loss decreases but not by the same proportion. Increasing
the ratio of the radii by a factor of 2 decreases the ratio of the losses by
about a factor between 1.25 and 2. This behavior is related to the complex
dependence of the losses on the inner and outer radii. Increasing the outer
radius with respect to the inner radius of the coaxial cable increases the
thickness of the dielectric. Such increase decreases the dielectric loss but it
also decreases the conductive losses by a smaller amount. This nonlinear
dependence of the losses on the radii of the conductors decreases the losses
in a complex manner.
Attenuation loss in coaxial cables can be minimized by
doping different dielectric materials of different permittivities and dissipation
factors. This doping, being similar to ion implementation in metals, lowers
the dielectric constant and dissipation factors at the expense of slowing
down the signals.
Figure 4: Coaxial cable transmission loss with different ratios of outer to inner
radii
All transmission lines have two ends. The end of a two-wire
transmission line connected to a source is ordinarily called the INPUT END or
the GENERATOR END. Other names given to this end are TRANSMITTER END,
SENDING END, and SOURCE. The other end of the line is called the OUTPUT
END or RECEIVING END. Other names given to the output end are LOAD END
and SINK.
According to Jones [4], one can describe a transmission line in terms of
its impedance. The ratio of voltage to current (Ein/Iin) at the input end is
known as the INPUT IMPEDANCE (Zin). This is the impedance presented to the
transmitter by the transmission line and its load, the antenna. The ratio
of voltage to current at the output (E out/Iout) end is known as the
OUTPUT IMPEDANCE (Zout). This is the impedance presented to the load by
the transmission line and its source. If an infinitely long transmission line
could be used, the ratio of voltage to current at any point on that
transmission line would be some particular value of impedance. This
impedance is known as the CHARACTERISTIC IMPEDANCE.
A transmission line’s characteristic impedance increases as the
conductor spacing increases. If the conductors are moved away from each
other, the distributed capacitance will decrease, and the distributed
inductance will increase. Less parallel capacitance and more series
inductance result in a smaller current drawn by the line for any given amount
of applied voltage, which by definition is a greater impedance. Conversely,
bringing the two conductors closer together increases the parallel
capacitance and decreases the series inductance. Both changes result in a
larger current drawn for a given applied voltage, equating to a lesser
impedance.
[5] Barring any dissipative effects such as dielectric leakage and
conductor resistance, the characteristic impedance of a transmission line is
equal to the square root of the ratio of the line’s inductance per unit length
divided by the line’s capacitance per unit length:
Or:
Note that in its general form, characteristic impedance can be a
complex number. Also note that it only becomes complex if either R' or G' are
non-zero. In practice we try to achieve nearly lossless transmission lines. For
a low-loss transmission line, the following relationships will occur
Then for all practical purposes we can ignore the contributions of R'
and G' from the equation and end up with a nice scalar quantity for
characteristic impedance. For lossless transmission lines the characteristic
impedance equation reduces to:
L' is the tendency of a transmission line to oppose a change in current,
while C' is the tendency of a transmission line to oppose a change in voltage.
Characteristic impedance is a measure of the balance between the two.
The characteristic impedance of coaxial cable or any type of
transmission line is constant, regardless of its length. This metric is
expressed in ohms but cannot be measured by an ohmmeter. The
measurement takes a time domain reflectometer, some models costing
thousands of dollars. An oscilloscope can also be used to ascertain this value.
But it’s usually unnecessary to make this measurement on short lengths of
coaxial cable; coax is manufactured to exacting specifications and labeled
accordingly.
To understand characteristic impedance, we must visualize a
transmission line of infinite length. As apparent in the accompanying
diagram, the transmission line may be modeled as consisting of an infinite
number of capacitances. This is entirely realistic because in coaxial cable the
two conductors are the plates of a capacitor and the dielectric layer is the
insulating material separating them. Similarly, conductors have a certain
specific inductance per unit length. In this thought experiment we shall
disregard the dc resistance of the wires, imagining they are cooled to close
to absolute zero and have become superconductors.
Figure 5: L-C model of an infinite transmission line
When voltage is applied at the input of this infinitely long transmission
line, the capacitors charge, a process that progresses down the line close to
the speed of light. Each parallel-connected capacitor charges, dropping the
applied voltage by a slight amount during the charging process. On an
infinitely long cable, there are an infinite number of capacitors to charge
down the line. Simultaneously, the series-connected inductors representing
the cable diminish the current as they establish magnetic fields about them.
As each magnetic field becomes fully established, the inductance no longer
opposes the flow of current, but there are always more inductors
downstream on an infinitely long cable.
The characteristic impedance of a waveguide is very important in
many areas of their use. Like other forms of feeder, waveguides have a
characteristic impedance. By matching the waveguide impedance to the
source and load, the maximum power transfer occurs on each occasion.
Methods of determining the waveguide characteristic impedance tend
to provide results that are within a factor of two of the free space impedance
of 377 ohms, i.e. most results for the waveguide impedance fall between
about 190 and 750.
To obtain the optimum power transfer between a waveguide and its
source or load, the impedance of both items at the junction should be the
same. When the impedance of the waveguide is not accurately matched to
the load, standing waves result, and not all the power is transferred.
Similarly when a source is providing power to the waveguide and there is an
impedance mismatch, then it is not possible for all the available power to be
transferred. To overcome the mismatch it is necessary to use impedance
matching techniques.
For Stripline Transmission Lines:
[6] The characteristic Impedance is therefore:
However, there are no exact analytic solutions for the capacitance and
inductance of stripline. They must be numerically analyzed. However, we can
use those results to form an analytic approximation of characteristic
impedance:
Where We is a value describing the effective width of the center
conductor:
Loading coils are a simple lump series inductance that produce an
effect called loading. Loading increases the series inductance of the loop and
effectively makes the loop a low pass filter, increasing the impedance of the
line which drops signal attenuation. A typical 26 gauge local loop pair is
loaded with a 26H88 loading coil. The letter H designates a coil that is added
every 6000 feet, 26 represents 26 gauge wire and 88 indicates the
inductance of the coil is 88 mH. This loading makes the loop perform as a low
pass filter and cuts the frequency off sharply at around 3.4KHz. Loading coils
work great for the low bandwidth requirements of voice but causes problems
when you want to transmit data at higher bandwidths over these same wires.
[7]
At voice frequencies, the cutoff frequency (fC) for a transmission line
can be approximated as follows:
L = Loading Coil Inductance
D = Distance in miles btwn loading coils
C = capacitance per mile
References
[1] (June 2020) Transmission Lines. Retrieved from
https://circuitglobe.com/transmission-lines.html
[2] (November 12, 2018) Fundamentals of Transmission Lines. Retrieved from
https://www.eeeguide.com/fundamentals-of-transmission-lines/
[3] M.M Mechaik (2001) Signal Attenuation in Transmission Lines. IEEE.
DOI: 10.1109/ISQED.2001.915226
[4] Jones (2010) Manuals Combined: U.S. Navy ELECTRONICS TECHNICIAN,
VOLUMES 01 - 08
[5] www.allaboutcircuits.com (2013) Characteristic Impedance. Retrieved
from: https://www.allaboutcircuits.com/textbook/alternating-current/chpt14/characteristic-impedance/#:~:text=Characteristic%20impedance%20is
%20also%20known,equal%20to%20some%20large%20fraction
[6] Stiles, J (02/02/2009). Transmission Lines and Waveguides.
http://www.ittc.ku.edu/~jstiles/723/handouts/chapter_3_Transmission_Lines_a
nd_Waveguides_package.pdf
[7] Gordon (07/14/2011). Loading Coils- More On the Local Loop.
http://www.gordostuff.com/2011/07/loading-coils-more-on-local-loop.html