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1 WAVES
1.1 MICROWAVES
1.1.1 INTRODUCTION
In the laboratory we generally use only low frequency signals that are conveniently routed with
wires or a coaxial cable. However the use of wires becomes intractable in the high frequency
regime where field radiation in the space around the wires results in intolerable energy losses. In
the high frequency realm guided microwaves are more appropriate for conveying electromagnetic
signals. At 10 kHz the attenuation of a signal in a waveguide is ¼ that in an equivalent coaxial
cable. At 1 MHz and above, a coaxial cable has no application.
1.1.2 THEORY
It should be no surprise that electromagnetic radiation may be propagated in a hollow metal tube.
As Feynman facetiously remarks, if the tube is straight, we can see through it! To understand
pictorially how microwaves are propagated in a guide, refer to Figure 1.1 below.
a)
E
b)
E
c)
H
Figure 1.1 Generation of electromagnetic disturbance in a guide
Imagine that an oscillator exists connected to the end of the tube causing a deficiency of
electrons on the bottom wall and a surplus on the top. An electric field exists between the top
and the bottom walls of the conductor causing current to flow as shown in Figure 1.1a). Magnetic
fields trying to encircle these currents are prevented because of the presence of the solid metal
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PART II LABORATORY
wall, so they join up instead, as shown in Figure 1.1b). As the oscillator reverses its polarity, the
magnetic fields generate other currents by induction further along the guide and so the
disturbance is propagated. One cycle of this disturbance is shown in Figure 1.1c).
A more mathematical treatment is required to extract all the interesting observable parameters
from this phenomenon, specifically, what kind of waves can exist in a rectangular pipe? For
electromagnetic radiation we can expect that both electric and magnetic fields will be
perpendicular to the direction of propagation and that any mathematical form expressing these
fields should satisfy Maxwell’s equations. We may construct the coordinate axis as shown in
Figure 1.2.
y
z
a
b
x
Figure 1.2 Co-ordinate axes for a rectangular waveguide
where a is the broad dimension of the waveguide and b is the narrow dimension.
Since we expect no electric field in the z direction we may expect some variation in E across x or
y. This function must satisfy the condition that the electric field goes to zero at the sides of the
conductor since the currents and charges in a conductor conspire so that there is no tangential
component of E at the surface. So we may have Ey varying with x as shown in Figure 1.3.
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Ey
a
x
Figure 1.3 Possible variation of Ey with x
For a rectangular geometry, the waveform is generally simple harmonic so we may guess that the
wave would have some form
E y  E0 sin k x xe j (t k z z )
Eq. 1.1
The field is perpendicular to the top and bottom conductor surfaces and zero at the side walls if
we choose
k x a  n
Eq. 1.2
where n is any integer. The z dependence indicates a wave travelling in the z direction with
frequency  and wave velocity
v

kz
Eq. 1.3
The divergence of E must be zero within the free space inside the conductor. In this case E has
only a y component, and it doesn’t change with y, so
 
.E  0
Eq. 1.4
is satisfied. The other Maxwell equations must also be satisfied, specifically the wave equation
2Ey
x
2

2Ey
y
2

2Ey
z
2
2
1  Ey
 2
c t 2
Eq. 1.5
Unless Ey is zero everywhere this equation specifies that
 k
2
x

kx  kz 
2


Ey  0   kz Ey  
2
2
c
2
2
2
c
2
Ey
Eq. 1.6
Eq. 1.7
0
3
PART II LABORATORY
  2   n 2 2 
k z   2    2 
c   a 
kx is already fixed, so
kz 
Eq. 1.8
2
Eq. 1.9
g
where g is the guide wavelength.
We may define the free space wavelength as:
0 
2c
Eq. 1.10

and so
4 2
g
2

4 2
0
2

n 2 2
a2
Eq. 1.11
n2


2g 20 4a 2
1
1
Eq. 1.12
This is the fundamental relationship between free space and guided wavelengths.
Associated with the electric fields there will be magnetic fields. Since
c 2  B 
y
E
t
Eq. 1.13
E
B
wave
propagation
z
the lines of B will circulate around the regions where E/t is largest, that is between the maxima
and minima of E
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Figure 1.4 Variation of E and B in the waveguide
We have assumed that k z has only one root but it should properly have two, a positive and
negative one for waves travelling in either direction.
  2   n 2 2 
k z    2    2 
c   a 
Eq. 1.14
We have also assumed that kz is real but if

n c
a
Eq. 1.15
then the term in the square root is negative and we apparently have no solution anymore.
However if we define
k z  ik 
Eq. 1.16
with
 n 2 2    2 
k    2    2 
 a  c 
Eq. 1.17
Then for Ey we have
E y  E 0 sin k x xei (t ik z )
 E 0 sin k x xe
 k z
Eq. 1.18
e it
Thus Ey is seen to oscillate with time, but to decay smoothly with z as a real exponential. This
tells us that for
c 
n c
a
Eq. 1.19
or below, waves do not propagate down the guide, but decay from the source exponentially with
k’. It is for this reason that c is called the cut-off frequency and
c 
2a
n
Eq. 1.20
is called the cut-off wavelength.
We may thus recast the relationship between free space and guide wavelength as
1

2
g

1

2
0

1

Eq. 1.21
2
c
If operation is below the cut-off frequency the wave is said to be ‘evanescent’.
a)
Modes in the guide
We have been working with the transverse electric mode TEn0 where n denotes the number of full
variations of the electric field along the broad dimension, a and 0 denote the number of full
variations along the narrow dimension. There are many other possible TE modes. These
correspond to other solutions of the wave equations.
Pre-lab Question (a) Draw the TE10, TE01, TE11 and TE20 modes.
If in a mode TEnm, the integers m and n are both non-zero, then the equation for the cut-off
wavelength generalises to
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PART II LABORATORY
cmn 
2
2
n m
   
a  b 
Eq. 1.22
2
All modes up until the cut-off wavelength exist simultaneously in the cavity. Thus the dimensions
of the cavity can be tailored to exclude all modes but one, or one or two etc.
TM modes are also possible. Some simple ones are shown in Figure 1.5.
TM21
TM11
Figure 1.5 Simple TM modes
In TM modes a kind of propagation is produced whereby no component of the magnetic field is in
the direction of propagation, while there is some component of the electric field in the z direction.
1.1.3 APPARATUS
a)
The Gunn Diode
You will be using a solid state source to produce electromagnetic radiation of microwave
frequencies. The central component of a Gunn diode is a piece of Gallium Arsenide crystal.
Generally, in crystals, the repetition of a regular array in space results in allowable energy states
for the electrons, depending on their wave number, k. In the absence of an applied electric field,
the electrons will choose energy levels as low as possible given a certain temperature. The net
electron flow will be zero since there is no preferred direction. If an external voltage is applied,
the electron velocity will increase linearly with the voltage until electron collisions with the lattice
are energetic enough to damage the crystal.
In GaAs, the linear increase of electron velocity with voltage obtains until an internal field of 3
kV/cm is achieved. At this point the electrons gain enough energy to occupy a second energy
level. In this state, the electrons have a higher effective mass and a lower velocity. Since the
velocity is lower, the current is appreciably smaller. Thus even though the voltage is rising the
current is decreasing, meaning that the crystal is behaving as a negative resistance. Oscillations
result.
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The oscillation mechanism is complex and involves many modes. The usual mode involves a
dipole distribution of charge on either side of a non-uniformity in the crystal. This dipole domain
drifts toward the anode and dissipates. The field across the sample rises, causing a new dipole
domain to grow. The frequency of this oscillation depends critically on the domain drift velocity
(or applied voltage) and the lateral dimension of the sample. These parameters may be tuned to
achieve electromagnetic disturbances of a microwave frequency.
b)
The Cavity Wavemeter
The cavity wavemeter is used to measure the wavelength of the standing wave in the waveguide.
The cavity wavemeter consists of a cavity terminated by a fixed short circuit and a moveable
short circuit coupled to the main waveguide as shown.
Figure 1.6 Schematic of a cavity wavemeter
Fixed Short
Circuit
Secondary
Guide
Coupling Hole
Main Guide
Moveable Short
Circuit
Micrometer
Drive
If a short circuit is placed across a cavity, then at that point the voltage must be zero. Thus the
only wave that can exist in the wavemeter
is one circuit
that has a voltage minimum at both ends. Some
short
of the allowable modes are shown below.
Figure 1.7 Allowable modes in the cavity wavemeter
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short circuit
PART II LABORATORY
In this case the wavemeter length is tuned so that only the first mode is possible.
To obtain a reading of the wavelength of the standing wave in the main guide the micrometer is
adjusted until the short circuits are in the configuration shown above. When this is done the
cavity resonates at the frequency of the radiation in the guide, sucking energy from the
propagation in the main guide. This is detected as a resonance dip in the standing wavemeter
(described next). At this point, the wavelength (in centimetres) may be read off directly from the
red scale on the micrometer, corresponding to a previous calibration. The wavemeter should be
taken off resonance (detuned) after measurement to increase the available power in the guide.
c)
The Standing Wavemeter
The standing wavemeter consists of a probe mounted in a slot through the waveguide wall. This
probe is connected to a silicon crystal slab connected directly to a thin tungsten wire. Passing
microwave energy induces a current in the probe. However since the microwave disturbance is
symmetrical about zero, any meter detection of this current would read zero since the fluctuations
are so frequent. The crystal-wire configuration rectifies this signal. Electron flow through the
silicon crystal is hampered by the scarcity of electrons. Tungsten as a metal, has surplus
electrons, so current can flow readily in this direction only, which rectifies the signal as shown.
The average signal is a measure of the microwave energy in the cavity.
RMS value
a) real signal
b) rectified signal
Figure 1.8 Rectification by the crystal detector
We read off the RMS signal due to the frequency of the fluctuations.
1.1.4 EXPERIMENT
a)
Familiarisation
A block diagram of the microwave bench is shown in Figure 1.9.
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Gunn
diode
cavity
wave
meter
standing
wave
meter
twist
variable
end
Power
supply
Figure 1.9 The microwave bench
You should identify and understand the operation of each component. Your demonstrator will
help you with this if you are unsure.
Connect the DC voltmeter to the standing wave detector. Adjust the cavity wavemeter until a
resonance dip occurs. This corresponds to a reduction in output of 40% from the standing
wavemeter. Record g and calculate the frequency of the oscillation.
b)
Polarisation of Microwaves in the Guide
Attach a microwave horn to the free end of the cavity. Horns simply direct the microwaves to a
narrow beam. Position the other horn with the crystal detector attachment some distance in front
of the first horn. Connect the detector to a multimeter.
Place the wire grille between the two horns and observe the response in the crystal detector for
grille orientations both parallel and perpendicular to the bench surface.
c)
Question (a)
What conclusions may you draw from this exercise?
Question (b)
In what direction do the electric field vectors point in this system?
Relation Between Guide and Free Space Wavelengths
Retaining the set-up from the previous exercise place an aluminium plate 20 cm in front of the
horn. Attach a multimeter to the standing wave detector and observe the standing wave pattern
by moving the carriage along the slot in the top of the waveguide. Refer to the diagram below to
understand how a standing wave is produced.
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PART II LABORATORY
a/2
g/2
main guide
aluminium
plate
horn
Figure 1.10 Standing waves produced using reflection from an aluminium plate
The radiation leaves the main guide via the horn and is reflected back from the plate producing a
standing wave with wavelength a between the plate and the horn. Reflected radiation also
enters the guide, producing a standing wave in the guide with wavelength g. Moving the plate
simply translates the standing wave pattern laterally by the same amount.
Determine the guide wavelength g by measuring the distance between successive minima. Now
measure a, the free space wavelength by keeping the standing wave detector stationary and
moving the aluminium plate. Develop and describe your technique for minimising error.
Question (c)
Predict the value of c using the relationship between a and g.
Compare this to the theoretical value of c calculated assuming that the TE10
mode is being propagated.
d)
Impedance Measurement
Since microwaves are a wave phenomenon we may use an identical formalism as that described
in the sound experiment to represent them. Thus we may assume identical expressions for the
phase change,  = 2kx0, the reflection coefficient, r, and the impedance of a discontinuity, Z T/Z0,
to characterise termination in this case.
Terminate the line with a short circuit, setting its micrometer to zero. Plot the standing wave
pattern for one wavelength, that is two maxima and minima, using the standing wave detector.
You should record the distance from the flange to the first minimum so that you may calculate the
true position of the first minimum. Calculate r,  and ZT/Z0 for the short circuit.
Repeat the impedance measurement for the matched load and open guide.
The short circuit may be described as the perfect mismatch, with almost all the microwave energy
being reflected from the termination. The matched load is the perfect match, containing a
dissipative wedge of iron that absorbs most of the microwave energy.
Question (d)
Do your results reflect these differences?
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1.2 REFERENCES
Note: Some of these books are available in the Laboratory
Cross, A. W., Experimental Microwaves. Marconi Instruments Limited, Sanders Division,
Stevenage Hertforshire, England (1977).
Bowers, Raymond., A Solid State Source of Microwaves. Scientific American.
Corson & Lorrain., Electromagnetic Fields and Waves. Chapter 13. W. H. Freeman and
Company, San Francisco, 2nd Edition (1970).
Feynman, Leighton & Sands., The Feynman Lectures. Volume II, Chapter 23 and 24. AddisonWesley (1964)
Hecht., Optics. Addison-Wesley, 2nd Edition (1987)
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