Experiment 8. Microwaves

Experiment 8. Microwaves

Updated BWJ March 4, 2015

Online resources

1. David M. Pozar, Microwave Engineering, 4th edition (Wiley, New Jersey, 2012) §6.4Circular Waveguide Cavities. Click here.

2. S. Ramo, J.R. Winnery & T. van Duzer, Fields & Waves in Communications Elec-tronics, 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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1 Safety

• This experiment uses microwave radiation. The power is very low (∼ 5 mW) andthe radiation is contained within waveguides and a cavity resonator. Any leakage isminimal and does not represent a safety hazard.

• If you suspect an item of equipment is not operating correctly, turn it off and, if mainsoperated, 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 microwavefrequencies

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) andgyrotrons, and

• semiconductor devices, such as varactor-tuned oscillators (VTO), Gunn diodes2 andIMPATT (impact ionisation avalanche transit time) diodes.

4 The experiment

This section provides a brief description of the various components in this experiment.

1Microwave ovens use radiation at 2.5 GHz, i.e. a wavelength of approximately 12 cm; wi-fi uses frequen-cies near the lower end of the microwave band; and mobile phones use frequencies that straddle the lower endof the band: ∼ 850 MHz - 2.3 GHz. Aircraft control radar and weather radar also use microwave frequencies.

2These diodes exploit the Gunn effect: coherent oscillations generated at microwave frequencies when asufficiently large DC electric field is applied to a thin layer of n-type GaAs.

MICROWAVES 8–3

4.1 Varactor-tuned oscillator (VTO)

In this experiment the microwaves (at a frequency ∼ 9 GHz) are generated by a varactor-tuned semiconductor oscillator, which is an LC oscillator based on a high frequency tran-sistor, 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’sdielectric. The thickness of the depletion layer increases with increasing reverse bias, ineffect widening the gap between the ‘capacitor plates’ and reducing its capacitance. Varyingthe voltage applied to the varactor allows its capacitance, and consequently the frequencyof 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 theamplitude of a sawtooth waveform (obtained from the oscilloscope’s timebase wave-form) to be varied. This waveform is added to the DC voltage, as shown in Figure ??,such that the scan range starts at the set frequency. If the dispersion is set to zero, theVTO output is a constant frequency. The frequency meter will give a reading onlywhen 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 mi-crowaves 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 communi-cations, wi-fi, radar), but the usual way of propagating microwaves over short distances isthrough waveguides. A waveguide is a hollow pipe with conducting walls (the cross-sectionis usually rectangular, but can also be circular), filled with a dielectric (usually air at atmo-spheric pressure). Only certain transverse modes will propagate along a waveguide. Thetransverse distribution of electric and magnetic fields in a waveguide mode are determinedby the waveguide geometry.

There are two classes of modes:

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• 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 char-acteristic impedance and use the same terminology to describe the matching of a waveguideto 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 - witha semiconductor crystal) are commonly used as microwave detectors. For sufficiently smallsignals the output is proportional to the square of the induced voltage, i.e. to the incidentpower. 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 resistorthat is usually measured, as shown in Figure ??.

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, withattenuation increasing as the absorbing plate moves into the higher field region in the centreof the waveguide. See the appendix for the calibration curve for the attenuator: attenuationin decibels (dB) as a function of micrometer setting.

4.5 Microwave cavity

4.5.1 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

3An optical fibre is another example. In this case the radiation is confined to the dielectric fibre by the radialvariation of refractive index - e.g. a central core region surrounded by a cladding of lower refractive index.

4Two mode numbers are need to specify a waveguide mode, eg TM01, where the mode numbers characterisethe structure of the mode in the two orthogonal transverse directions in the plane normal to the direction of

MICROWAVES 8–5

polar coordinates: the z axis is parallel to the axis of the cylinder; r and φ are polar coor-dinates in a plane perpendicular to the z axis. In the TM010 mode the only non-vanishingfield components are Ez and Bφ (see Figure ??(a)), given by

Ez = E0J0(kr) (1)

Bφ = (E0/c)J1(kr) (2)

where J0,1 are Bessel functions of the zeroth and first order, respectively, and k = 2π/λis the wavenumber corresponding to a wavelength of λ. The radial distributions of Ez andBφ are shown in Figure ??(b). The fact that Ez → 0 as r → 0, is a consequence of theassumption 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 whereit decays exponentially with a characteristic scale length called the skin depth. The skindepth is a function of both frequency of the radiation and conductivity of the metal.

Fig. 8-3 : (a) Pictorial representation of the fieldsEz andBφ for a TM010 mode in a cylindrical cavity(credit: Fig. 10.6, p496 of [?]); (b) Ez(r)/Ez(0) (solid) and Bφ/cEz(0) (dashed) as a function ofnormalised 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 ??(a); the reso-nance 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 - onefor 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 waveguideZ0, as shown by the equivalent circuit in Figure ??(b). When the cavity is matched, there isno 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 viewof the cavity the load impedance of the waveguide is transformed to R. As a result theequivalent circuit of the matched cavity is as shown in Figure ??(c).

C! L!Z0! R! C! L!R! R!C! L!R!

(a)! (b)! (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 theresonance. The general definition of Q is

Q = 2πEnergy 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 thewider the resonance, the lower the value of Q.

For the parallel resonant circuit of Figure ??(a), the Q of the resonance is given by

Q = ω0RC = R

√C

L(6)

In this experiment we use the variation of reflected power as a function of frequency in thevicinity 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 ??b. Matching eliminates this reflected power

MICROWAVES 8–7

by inserting a stub into the waveguide in front of the cavity and adjusting its position so thatthe reflection from the stub interferes destructively with the reflection from the cavity. Theresult is the matched condition shown in Figure ??c.5

4.7 Perturbation of the cavity

When a small object is introduced into a microwave cavity the resonant frequency shifts bya small amount.6 The effects of various perturbations are discussed by Carter [?].

5 Experimental setup

The microwave power radiated from the antenna propagates down the waveguide to a magictee where it is spit equally between the two horizontal arms. The short arm to the left hasan absorbing load in it so that entering microwaves are completely absorbed. The right armis connected via the matching stub to the cavity where microwaves can enter via a smallaperture 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 poweris reflected. At the magic tee the reflected wave is coupled to the vertical waveguide whereit propagates to the crystal detector via the variable attenuator - see Figure ??.

As the oscillator’s dispersion input is the oscilloscope’s timebase waveform (see Figure ??),the oscilloscope can be used to display reflected power as a function of (uncalibrated) mi-crowave frequency. If the other oscilloscope input is grounded, it can be used as a referencelevel.

5There is similarity between this situation and what happens in the antireflecting coatings applied to opticalcomponents such as camera lenses. The amplitudes of reflection from the front and back surfaces of the coatingshave to be equal and 180 out of phase to interfere destructively.

6At resonance the average stored electric and magnetic energies are equal. When the small object is intro-duced one of these energies will, in general, be affected more than the other. The resonant frequency then shiftsby 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 fre-quency. In such circumstances the frequency meter can be used to measure the microwavefrequency.7

6 Prework

1. With reference to section ??, 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 tomove 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 indecibels 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 logP .

5. The microwave detector is called a crystal detector. It is a diode formed at thecontact of a fine wire (which acts as the antenna) and a semiconductor (the crys-tal). Given a voltage across the diode of V0 sinωt and a diode characteristic ofI = I0(exp(−eV/kT )− 1), where I0 is the reverse bias current, show that for suffi-ciently small induced signals the time-averaged current is given by

Iav ≈ I0(eV0

2kT

)2

. (7)

As Iav is proportional to V 20 , the crystal detector is known as a square-law detector

and its output voltage (IavRL) 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 itis isolated, i.e. unloaded?

7. By consulting the paper by Carter [?], find an expression for the shift in resonancefrequency 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.

7The frequency meter will register a value only when the dispersion is set to zero, i.e., when the oscillatoris not repeatedly scanning through a range of frequencies.

8Crystal detector load resistors usually have a value of 50 Ω.

MICROWAVES 8–9

7 Procedure

7.1 The TM010 resonance

1. Ensure that the movable matching stub is retracted and set the dispersion control tozero, so that there is no frequency scanning.

2. With the aid of the frequency meter, set the oscillator frequency close to the value forthe TM010 cavity resonance calculated in the pre-work.

3. Observe the reflected power from the cavity on the oscilloscope, and adjust the fre-quency and dispersion controls until you see clearly a frequency scan through theTM010 resonance. The displayed trace should look like the one shown in Figure ??.

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 thedetuning rod (a wire bent into an ‘L’ shape that can be inserted through the hole inthe 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 andmeasure its frequency and estimate an uncertainty.

6. The reliability of the resonance frequency measurement is also dependent upon yourability to set the oscillator precisely on the resonant frequency. Set the oscillatoron resonance again and make a measurement. Devise a way of obtaining a reliablemeasurement of the resonance frequency and its uncertainty.

7. From the measured resonant frequency calculate a value for the internal radius of thecavity a, including an uncertainty. Unscrew the top of the cavity and check that yourresult for a is consistent with a direct measurement (a vernier caliper is available forthis measurement).

If you do not know how to use vernier calipers, an instructional video is availablehere.

C1 .

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7.2 Detector response

1. With the aid of the attenuator determine the dependence of the detector output onmicrowave power. Ensure that you use a frequency that leads to a significant reflectedpower from the cavity (i.e. avoid the cavity resonance!).

2. For an ideal square-law detector V ∝ P . Assuming V ∝ Pn, what value of n isindicated by your data? Do your measurements fit a power law over the completerange of powers measured?

C2 .

7.3 Measurement of the cavity Q

1. Using the oscilloscope to observe the passage through resonance, insert the matchingstub 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, determinethe FWHM of the reflected power profile. If the cavity is well matched, the reflectedpower at the resonance frequency should be very close to zero. Remember that thedetuning 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 estimatean uncertainty.

4. Deduce the value of the Q of the isolated cavity (see Figure ??).

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 stretchedtaut along the axis of the cavity, determine the relative permittivity of nylon. A micrometeris provided for measuring the diameter of the nylon line.

Compare your result with accepted values (see, for example, the CRC Handbook of Chem-istry and Physics).

Question 3: Slacken the tension on the nylon line so that it bows away from the cavity’saxis and observe the change in resonant frequency. Explain your observation.

C4 .

MICROWAVES 8–11

References

[1] S. Ramo, J.R. Whinnery and T. Van Duzer, Fields and Waves in Communication Elec-tronics, 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 ?? is the calibration curve for the precision attenuator.

Fig. 8-7 : Calibration curve for the precision attenuator: attenuation in dB as a function of microm-eter 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± 0.11b = −53.1± 2.2c = 319± 11.