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This is
the first experiment, and it is done to introduce the basic microwave principles
and measurements. A waveguide trainer is used, and a cavity wavemeter is set up
as part of the equipment. This cavity wavemeter is used with relation to the
frequency of the microwave through the waveguide. Also, a probe detector
assembly is used to measure the wavelength of a standing wave.
It is
therefore crucial to have good understanding of the relationship of the cavity
length during resonance and frequency of the microwave. In addition to this, in
depth knowledge of standing waves would also be of help.
1.
Description of the waveguide
The definition of a waveguide is “a material medium that confines and guides a propagating electromagnetic wave” as quoted from the website <http://www.its.bldrdoc.gov/fs-1037/dir-040/_5860.htm>
The electromagnetic waves (for example, microwave, which it used in this experiment) can travel in three dimensions in free space. For transmission purposes, it is desirable to have the wave propagate in a single direction. For that reason, waveguides are used.
Waves traveling through waveguides usually don’t undergo much attenuation even when the waveguide itself is bent. The only dielectric in the waveguide is air, which introduces very low dielectric losses. The contact between the waveguide structure and the wave itself does not lose contact when routed, and does not generate reflections.
For measurement purposes, different coordinate systems are used depending on the cross-sectional shape of the waveguide. In other words, if a rectangular waveguide is used, rectangular Cartesian coordinates are used. This is the same with circular and cylindrical coordinates.
The setbeck with using waveguides is that it only works for waves above a certain frequency. Below the cut-off frequency, waveguides cannot transmit the wave. Furthermore, waveguides are difficult to install due to their hollow-pipe structure.
A rectangular
waveguide, as used in this lab, cannot support TEM waves because an exclusive
voltage cannot be identified since there is only one conductor in a rectangular
waveguide.
The picture below shows a TE wave (H10) and
a TM wave (E11) propagating in a rectangular waveguide.
This
picture is originally from “The Services Textbook of Radio", vol
5, 1958 HMSO”. The magnetic field lines are green in color, the electric field
lines are red, and the currents in the guide walls are colored blue.
2.
The relationship between cavity length during resonance and frequency of the
wave
The cavity wavemeter is made with a sliding short, and has the precision and potential accuracy of 1/Q of the cavity, whereby Q refers to the quality factor (usually in the range of 1,000-1,0000). The cavity Q factor sets the width of the resonance and the accuracy is mainly dependent on the calibration. The cavity length is changed through alteration of the sliding short. When no resonance occurs, the cavity absorbs very little power from the transmission system. But at resonance, the cavity starts to absorb a lot of power, resulting in a huge power lost in the main transmission system. When this loss is noticed, the length at which the sliding short had moved is noted for calculations based on the equations below:
Frequency, f = c * Ö[1/(4*L²) + 1/lc²]
Whereby L = length of cavity at resonance
c = speed of light
lc = wavelength of the cut-off frequency
3.
Standing waves
A standing wave is formed when a wave in a medium is reflected from one end of the medium to interfere with incident waves from the source, causing some specific points of the wave along the medium to appear stationary. Standing waves can only be created in the medium at specific frequencies of vibrations, known as harmonics. Otherwise, the interference between the incident wave and the reflected wave would result in disturbance in the medium with is non-recurring.
To make things clearer, consider an incident wave traveling
in a medium, formed by a source at one end of the medium. When the incident wave reaches the end, it is reflected and undergoes
inversion (see the picture below).
At the same time, another
incident wave is propagated through, and interferes with the reflected wave. If
the timing is precise, a standing wave is formed.
In short, a standing wave is the result of the perfectly timed interference of two waves passing through the same medium. Standing waves are not actually waves, but rather, patterns due to the interference of waves traveling in opposite directions.
As was mentioned earlier, some points on the wave pattern along the medium will appear stationary and does not have any displacement. These points are known as nodes. Conversely, there are points along the medium that undergoes maximum displacement (vibrating between the positive and negative amplitudes). The points are known as antinodes. The standing wave pattern will always have nodes and antinodes in an alternating fashion. The nodes and the antinodes of the standing wave are always located at the same position along the medium.
The nodes are produced at locations where destructive interference occurs. This is the position where the amplitude of one wave is met with the wave of opposite amplitude traveling in the opposite direction. In contrast, antinodes are produced at locations where constructive interference occurs. Antinodes are produced when the wave of a certain amplitude meets with a wave of the same amplitude traveling in the opposite direction. Antinodes are always vibrating back and forth between positive and negative amplitudes since at this point, 2 positive amplitudes can meet, and half a cycle later, two negative amplitude will meet. The picture below shows the standing wave pattern.
Now, back to harmonics in a wave, it was mentioned earlier that the standing wave patterns can only be obtained at specific frequencies known as harmonics. The first harmonic, or fundamental harmonic is the harmonic with the longest wavelength and the lowest frequency. Below is the first harmonic wave:
There are only 2 nodes in the first harmonic of a medium with both ends closed. Looking at the second and third harmonics, it can be noted that all three wave patterns have one thing in common, that is, the distance between a node and the next consecutive node is exactly half a wavelength.
Similarly,
the distance between one antinode to the next consecutive antinode is also half
a wavelength.
Applying
the understanding of the connection between nodes and wavelength of the wave
travelling in a medium, the slotted waveguide section and probe detector are
used in the laboratory experiment to determine the wavelength and frequency of
wave in the waveguide.
1.
Part A: Measurement of Source Frequency Using a Cavity Wavemeter
The cavity resonator was set to 21.77mm (the maximum length) and was adjusted until a sharp dip in output power was observed in the meter. This occurred at a cavity gauge length of 18.70mm. Using the equation as discussed in the introduction, which is:
Frequency, f = c * Ö[1/(4*L²) + 1/lc²]
Whereby L = length of cavity at resonance
c = speed of light
lc = wavelength of the cut-off frequency
The values are known to be as below:
c
= 3 x 108 m/s²
L
=18.7mm
lc = 2a
a = 22.86mm
Therefore, frequency f
= 3 x 108 x Ö[1/(4*(18.7 x 10-3)²) + 1/(2 x 22.86 x 10-3)²]
= 10.36 GHz
Modifying the equipment with a short termination, the wave transmitted was reflected back. Using the slotted wavelength section and probe detector, the node of the standing wave pattern obtained were found by observing the consecutive points along the wave at which the power falls drastically. The length of these points of occurrence was measured from a reference point.
X1 = 28 mm
X2 =
46 mm
X3 =
65 mm
lg1 = 2(X2 – X1 )
= 2(46 – 28)
= 36 mm
lg2 = 2(X3 – X2 )
= 2(65 – 46)
= 38 mm
lg3 = X3 – X1
= 65 – 28
= 37 mm
The average wavelength as determined from this experiment is:
lg = (lg1 + lg2 + lg3)/3
= (36 + 38 + 37)/3
= 37 mm
Given the equation:
lg = l0 lc /Ö(lc ² - l0²)
Whereby
l0 = wavelength of the frequency without reflection
lc = wavelength of the cut-off frequency
lg = wavelength of the standing wave frequency
The equation was manipulated to read:
l0 = lg lc /Ö(lg ² + lc²)
Substituting the lg found in the experiment and given lc is 45.72mm,
l0
=
(37 x 10-3 x 45.72
x 10-3)/ Ö((37
x 10-3)²+ (45.72 x 10-3)²)
= 28.76 mm
Since c = fl,
f0 = c/l0
= 3 x 108/28.76 x 10-3
= 10.43 GHz
|
Given
frequency of source (GHz) |
Frequency
of source from Part A (GHz) |
Frequency
of source from Part B (GHz) |
Error
from Part A |
Error
from Part B |
|
10.425 |
10.36 |
10.43 |
0.624% |
0.048% |
Table 1: Summary of the results and calculations
The
frequency of the wave in the waveguide could be estimated using the cavity
resonator while incurring very small errors (refer to Table 1). The error arises
from losses due to “holes” in the waveguide introduced by the two
attenuators and joints from one part of the waveguide to another. The accuracy
of this method is also very much affected by the calibration of the cavity
resonator. A small calibration error may have gone unnoticed when carrying out
the experiment, and this error may have been brought forward to the calculation
of wave frequency, causing the error seen.
Human
errors (such as inaccurate reading of the micrometer) can also cause minor
errors in the experiment. Another possibility is that the source used n the
experiment may not have produced waves of exactly 10.425 GHz as promised.
However
in Part B, the frequency of the source calculated from the experiment is much
closer to the one expected (error is only 0.048%). This shows that the main
error probably occurred in Part A, whereby calibration and human errors were
probably the dominant factors in the resulting error.
Note
that errors may also have occurred in Part B, whereby small losses in the
attenuators and joints would affect the final reading. Human error may also
occur, although not so obvious, since reading the scale of the slotted waveguide
section is less tricky than reading the micrometer on the cavity resonator.
Calibration error would not occur in Part B, since all the readings were taken
from the same reference point, thus eliminating one potential source of error.
To
sum it up, the errors and losses that occur in waveguides are very small and
almost negligible (see Table 1), making it highly suitable for transmission of
electromagnetic waves. Using the standing wave measurements appear to be more
accurate than using the cavity resonator. Nevertheless for practical
applications, it would be easier to insert a cavity resonator to check the
frequency rather than to force a standing wave and measure the distance of the
nodes.
This
experiment also shows that waveguides are very useful and effective mediums for
transmission. However, it would be good to bear in mind that waveguides cannot
work below a cut-off frequency, and therefore its applications are limited.
Waveguides
and Cavity Resonators <http://www.ee.surrey.ac.uk/Personal/D.Jefferies/wguide.html>
Introduction
to transmission lines and waveguides
Introduction
to rectangular waveguides
Standing
waves