Download Experiment 8. Microwaves

Experiment 8.
Microwaves
Updated BWJ August 20, 2015
Online resources
1. David M. Pozar, Microwave Engineering, 4th edition (Wiley, New Jersey, 2012) §6.4
Circular Waveguide Cavities. Click here.
2. S. Ramo, J.R. Winnery & T. van Duzer, Fields & Waves in Communications Electronics, 3rd edition (Wiley, New York, 1994), §10.6 Circular Cylindrical Resonators.
Click here.
3. R.G. Carter, IEEE Trans Microwave Theory Tech. 49, 918 (2001). Click here.
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S ENIOR P HYSICS L ABORATORY
Safety
• This experiment uses microwave radiation. The power is very low (∼ 5 mW) and
the radiation is contained within waveguides and a cavity resonator. Any leakage is
minimal and does not represent a safety hazard.
• If you suspect an item of equipment is not operating correctly, turn it off and, if mains
operated, turn off the power at the mains switch, then consult a tutor.
2
Objective
In this experiment you will investigate:
• a square-law microwave detector
• the concept of matching a cavity resonator to the waveguide
• the Q of the cavity, a parameter that describes the sharpness of the cavity resonance
• the use of the cavity to determine the relative permittivity of nylon at microwave
frequencies
3
Introduction
Microwaves are electromagnetic radiation in the wavelength range of ∼1 mm (300 GHz) to
∼ 30 cm (1 GHz)1 . There are two kinds of microwave generators:
• those based on electron beam excitation of electromagnetic oscillations in cavities,
giving rise to devices such as klystrons, magnetrons (used in microwave ovens) and
gyrotrons, and
• semiconductor devices, such as varactor-tuned oscillators (VTO), Gunn diodes2 and
IMPATT (impact ionisation avalanche transit time) diodes.
4
The experiment
This section provides a brief description of the various components in this experiment.
1
Microwave ovens use radiation at 2.5 GHz, i.e. a wavelength of approximately 12 cm; wi-fi uses frequencies near the lower end of the microwave band; and mobile phones use frequencies that straddle the lower end
of the band: ∼ 850 MHz - 2.3 GHz. Aircraft control radar and weather radar also use microwave frequencies.
2
These diodes exploit the Gunn effect: coherent oscillations generated at microwave frequencies when a
sufficiently large DC electric field is applied to a thin layer of n-type GaAs.
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4.1
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Varactor-tuned oscillator (VTO)
In this experiment the microwaves (at a frequency ∼ 9 GHz) are generated by a varactortuned semiconductor oscillator, which is an LC oscillator based on a high frequency transistor, with a varactor or voltage-controlled capacitor as the tuning element.
A varactor is a special reverse-biased diode in which the depletion layer acts as a capacitor’s
dielectric. The thickness of the depletion layer increases with increasing reverse bias, in
effect widening the gap between the ‘capacitor plates’ and reducing its capacitance. Varying
the voltage applied to the varactor allows its capacitance, and consequently the frequency
of the oscillator, to be changed.
The oscillator has two controls:
• The frequency control which, by adjusting the DC voltage applied to the varactor,
controls the frequency of the oscillator.
• The dispersion control allows the frequency scan range to be varied. It changes the
amplitude of a sawtooth waveform (obtained from the oscilloscope’s timebase waveform) to be varied. This waveform is added to the DC voltage, as shown in Figure 8-1,
such that the scan range starts at the set frequency. If the dispersion is set to zero, the
VTO output is a constant frequency. The frequency meter will give a reading only
when the input frequency is constant, i.e., the dispersion is set to zero.
The output of the VTO is connected, via a coaxial cable, to an antenna that launches microwaves into a rectangular waveguide.
Fig. 8-1 : The voltage waveform applied to the varactor
4.2
Waveguides
Microwaves can, of course, propagate through free space (e.g. mobile phone communications, wi-fi, radar), but the usual way of propagating microwaves over short distances is
through waveguides. A waveguide is a hollow pipe with conducting walls (the cross-section
is usually rectangular, but can also be circular), filled with a dielectric (usually air at atmospheric pressure). Only certain transverse modes will propagate along a waveguide. The
transverse distribution of electric and magnetic fields in a waveguide mode are determined
by the waveguide geometry.
There are two classes of modes:
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S ENIOR P HYSICS L ABORATORY
• transverse electric (TE) modes for which the transverse magnetic field is zero
• transverse magnetic (TM) modes for which the transverse electric field is zero
A waveguide is a form of transmission line3 . As with coaxial cables, we can define a characteristic impedance and use the same terminology to describe the matching of a waveguide
to a load (in this case the cavity).
4.3
Crystal detector
Crystal detectors (a diode junction formed at the contact of a fine wire - the antenna - with
a semiconductor crystal) are commonly used as microwave detectors. For sufficiently small
signals the output is proportional to the square of the induced voltage, i.e. to the incident
power. When operating under such conditions the detectors are called square-law detectors.
A diode detector is a current source; however, it is the voltage across an internal load resistor
that is usually measured, as shown in Figure 8-2.
Fig. 8-2 : Schematic representation of a crystal detector
4.4
Variable attenuator
The micrometer drive of the attenuator moves an absorbing plate across the waveguide, with
attenuation increasing as the absorbing plate moves into the higher field region in the centre
of the waveguide. See the appendix for the calibration curve for the attenuator: attenuation
in decibels (dB) as a function of micrometer setting.
4.5
4.5.1
Microwave cavity
Cavity fields
In this experiment we study the simplest resonant mode of a cylindrical cavity - the TM010
mode.4 In order to describe the resonant fields in a cylindrical cavity we use cylindrical
3
An optical fibre is another example. In this case the radiation is confined to the dielectric fibre by the radial
variation of refractive index - e.g. a central core region surrounded by a cladding of lower refractive index.
4
Two mode numbers are need to specify a waveguide mode, eg TM01 , where the mode numbers characterise
the structure of the mode in the two orthogonal transverse directions in the plane normal to the direction of
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polar coordinates: the z axis is parallel to the axis of the cylinder; r and φ are polar coordinates in a plane perpendicular to the z axis. In the TM010 mode the only non-vanishing
field components are Ez and Bφ (see Figure 8-3(a)), given by
Ez = E0 J0 (kr)
(1)
Bφ = (E0 /c)J1 (kr)
(2)
where J0,1 are Bessel functions of the zeroth and first order, respectively, and k = 2π/λ
is the wavenumber corresponding to a wavelength of λ. The radial distributions of Ez and
Bφ are shown in Figure 8-3(b). The fact that Ez → 0 as r → a, is a consequence of the
assumption that the cavity walls are perfectly conducting. For our resonator, made of brass,
this is only approximately true. The field will, in fact, penetrate into the metal walls where
it decays exponentially with a characteristic scale length called the skin depth. The skin
depth is a function of both frequency of the radiation and conductivity of the metal.
Fig. 8-3 : (a) Pictorial representation of the fields Ez and Bφ for a TM010 mode in a cylindrical cavity
(credit: Fig. 10.6, p496 of [1]); (b) Ez (r)/Ez (0) (solid) and Bφ /cEz (0) (dashed) as a function of
normalised radius, r/a, where a is the radius of the cavity.
4.5.2
Equivalent circuit
The cavity can be represented by a parallel LC circuit as shown in Figure 8-4(a); the resonance frequency of this parallel circuit is
ω0 = 2πf0 = √
1
LC
(3)
propagation. Accordingly, three mode numbers are required, in general, to describe a cavity resonance - one
for each orthogonal coordinate.
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The resistor R represents losses due to the finite conductivity of the walls of a real cavity.
When connected to a waveguide, the cavity is ‘loaded’ by the impedance of the waveguide
Z0 , as shown by the equivalent circuit in Figure 8-4(b). When the cavity is matched, there is
no power reflected back at resonance. The matching mechanism behaves like a transformer.
From the point of view of the waveguide, R is transformed to Z0 ; from the point of view
of the cavity the load impedance of the waveguide is transformed to R. As a result the
equivalent circuit of the matched cavity is as shown in Figure 8-4(c).
R!
C!
L!
Z0!
(a)!
C!
R!
L!
R!
(b)!
R!
C!
L!
(c)!
Fig. 8-4 : Equivalent circuits for (a) an isolated cavity, (b) the cavity ‘loaded’ by the waveguide, and
(c) the cavity ‘matched’ to the waveguide
4.5.3
Q of the cavity
The Q of the cavity resonance is a figure of merit which characterises the sharpness of the
resonance. The general definition of Q is
Q = 2π
Energy stored in the cavity
Energy lost per oscillation cycle
(4)
It can be shown that this definition is equivalent to
Q=
f0
∆f
(5)
where f0 is the resonant frequency and ∆f is the FWHM of the resonance. Note that the
wider the resonance, the lower the value of Q.
For the parallel resonant circuit of Figure 8-4(a), the Q of the resonance is given by
s
Q = ω0 RC = R
C
L
(6)
In this experiment we use the variation of reflected power as a function of frequency in the
vicinity of the resonance to estimate the Q of the cavity.
4.6
Matching the cavity to the waveguide
At resonance the reflected power from the cavity is a minimum, but in general not zero.
This is the loaded condition shown in Figure 8-4b. Matching eliminates this reflected power
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by inserting a stub into the waveguide in front of the cavity and adjusting its position so that
the reflection from the stub interferes destructively with the reflection from the cavity. The
result is the matched condition shown in Figure 8-4c.5
4.7
Perturbation of the cavity
When a small object is introduced into a microwave cavity the resonant frequency shifts by
a small amount.6 The effects of various perturbations are discussed by Carter [2].
5
Experimental setup
The microwave power radiated from the antenna propagates down the waveguide to a magic
tee where it is spit equally between the two horizontal arms. The short arm to the left has
an absorbing load in it so that entering microwaves are completely absorbed. The right arm
is connected via the matching stub to the cavity where microwaves can enter via a small
aperture in the side wall.
Fig. 8-5 : Power flows through the magic tee
Except when the oscillator is tuned close to a cavity resonance, most of the incident power
is reflected. At the magic tee the reflected wave is coupled to the vertical waveguide where
it propagates to the crystal detector via the variable attenuator - see Figure 8-5.
As the oscillator’s dispersion input is the oscilloscope’s timebase waveform (see Figure 81), the oscilloscope can be used to display reflected power as a function of (uncalibrated)
microwave frequency. If the other oscilloscope input is grounded, it can be used as a reference level.
5
There is similarity between this situation and what happens in the antireflecting coatings applied to optical
components such as camera lenses. The amplitudes of reflection from the front and back surfaces of the coatings
have to be equal and 180◦ out of phase to interfere destructively.
6
At resonance the average stored electric and magnetic energies are equal. When the small object is introduced one of these energies will, in general, be affected more than the other. The resonant frequency then shifts
by a small amount to again equalise the energies.
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Variation of the dispersion potentiometer changes the frequency scan range. In particular,
when it is set to zero there is no frequency scanning, i.e., the oscillator output is a single frequency. In such circumstances the frequency meter can be used to measure the microwave
frequency.7
6
Prework
1. With reference to section 4.5, where do the cavity fields Ez and Bφ
(a) vanish,
(b) have their maximum values
2. Given that the radius of the cavity is approximately 13 mm, estimate the frequency,
f0 , of the TM010 cavity mode.
3. At resonance the fields in the cavity oscillate at frequency f0 , requiring charge to
move between the top and bottom at this frequency.
(a) where do currents flow?
(b) where is power dissipated?
4. This experiment uses a variable attenuator, for which a calibration of attenuation in
decibels is supplied. If the input power is P0 , and the attenuation is ∆dB decibels,
write down
(a) an expression for the output power P in terms of P0 and ∆dB, and
(b) an expression for log P .
5. The microwave detector is called a crystal detector. It is a diode formed at the
contact of a fine wire (which acts as the antenna) and a semiconductor (the crystal). Given a voltage across the diode of V0 sin ωt and a diode characteristic of
I = I0 (exp(−eV /kT ) − 1), where I0 is the reverse bias current, show that for sufficiently small induced signals the time-averaged current is given by
Iav ≈ I0
eV0
2kT
2
.
(7)
As Iav is proportional to V02 , the crystal detector is known as a square-law detector
and its output voltage (Iav RL ) will be proportional to the incident microwave power.8
6. If the Q of the matched cavity is measured to be Qm , what is the Q of the cavity if it
is isolated, i.e. unloaded?
7. By consulting the paper by Carter [2], find an expression for the shift in resonance
frequency of the cavity when a thin dielectric rod (we will use nylon fishing line)
is inserted along the axis of the cavity. Define all quantities in the equation. Note:
Carter refers erroneously to mode TM100 ; it should be mode TM010 .
7
The frequency meter will register a value only when the dispersion is set to zero, i.e., when the oscillator
is not repeatedly scanning through a range of frequencies.
8
Crystal detector load resistors usually have a value of 50 Ω.
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Procedure
7.1
The TM010 resonance
1. Ensure that the movable matching stub is retracted and set the dispersion control to
zero, so that there is no frequency scanning.
2. With the aid of the frequency meter, set the oscillator frequency close to the value for
the TM010 cavity resonance calculated in the pre-work.
3. Observe the reflected power from the cavity on the oscilloscope, and adjust the frequency and dispersion controls until you see clearly a frequency scan through the
TM010 resonance. The displayed trace should look like the one shown in Figure 8-6.
Voltage Proportional
To Incident Power
CHANNEL 2
Voltage Proportional to
Reflected Power at Resonance
CHANNEL 1 ON "GROUND"
CENTRAL RESONANT
FREQUENCY
Fig. 8-6 : Profile of reflected power versus frequency near resonance.
4. Check that that you are indeed seeing a scan through the resonance by inserting the
detuning rod (a wire bent into an ‘L’ shape that can be inserted through the hole in
the top of the cavity, along its axis and out the hole in the bottom).
Question 1: Why does the insertion of the detuning rod kill the resonance?
5. With the dispersion again set to zero, set the oscillator on the TM010 resonance and
measure its frequency and estimate an uncertainty.
6. The reliability of the resonance frequency measurement is also dependent upon your
ability to set the oscillator precisely on the resonant frequency. Set the oscillator
on resonance again and make a measurement. Devise a way of obtaining a reliable
measurement of the resonance frequency and its uncertainty.
7. From the measured resonant frequency calculate a value for the internal radius of the
cavity a, including an uncertainty. Unscrew the top of the cavity and check that your
result for a is consistent with a direct measurement (a vernier caliper is available for
this measurement).
If you do not know how to use vernier calipers, an instructional video is available
here.
C1 .
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Detector response
1. With the aid of the attenuator determine the dependence of the detector output on
microwave power. Ensure that you use a frequency that leads to a significant reflected
power from the cavity (i.e. avoid the cavity resonance!).
2. For an ideal square-law detector V ∝ P . Assuming V ∝ P n , what value of n is
indicated by your data? Do your measurements fit a power law over the complete
range of powers measured?
C2 .
7.3
Measurement of the cavity Q
1. Using the oscilloscope to observe the passage through resonance, insert the matching
stub and adjust its position in order to match the cavity to the waveguide.
2. With dispersion set to zero, and using the DVM to measure reflected power, determine
the FWHM of the reflected power profile. If the cavity is well matched, the reflected
power at the resonance frequency should be very close to zero. Remember that the
detuning rod will be useful for determining reflected power well away from the TM010
resonance.
3. Obtain a value for Qm , the Q of the resonance for the matched cavity, and estimate
an uncertainty.
4. Deduce the value of the Q of the isolated cavity (see Figure 8-4).
Question 2: If there is no power reflected back up the waveguide in the matched case,
where is the incident power going?
C3 .
7.4
Measurement of relative permittivity
By measuring the shift in resonance frequency when a piece of nylon fishing line is stretched
taut along the axis of the cavity, determine the relative permittivity of nylon. A micrometer
is provided for measuring the diameter of the nylon line.
Compare your result with accepted values (see, for example, the CRC Handbook of Chemistry and Physics).
Question 3: Slacken the tension on the nylon line so that it bows away from the cavity’s
axis and observe the change in resonant frequency. Explain your observation.
C4 .
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References
[1] S. Ramo, J.R. Whinnery and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd edition, (Wiley, New York, 1994)
[2] R.G. Carter, IEEE Trans Microwave Theory Tech 49, 918 (2001)
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Appendix: Attenuator calibration
Figure 8-7 is the calibration curve for the precision attenuator.
Fig. 8-7 : Calibration curve for the precision attenuator: attenuation in dB as a function of micrometer reading.
The calibration is fitted very well by the following polynomial.
y = ax2 + bx + c
(8)
where y is the attenuation in decibels (dB), x is the micrometer reading in millimetres, and
a = 2.20 ± 0.11
b = −53.1 ± 2.2
c = 319 ± 11.