Juni 2021
Resonance frequency, Q-factor, coupling
of a cylindrical cavity and the effect on
graphite from an alternating electric field
Teknisk- naturvetenskaplig fakultet UTH-enheten Bes¨oksadress:
Angstr¨omslaboratoriet˚ L¨aderhyddsv¨agen 1 Hus 4, Plan 0 Postadress:
Box 536
751 21 Uppsala Telefon:
018 - 371 30 03 Telefax:
018 - 471 30 00 Hemsida:
http://www.teknat.uu.se/student
Jakob G¨ol´en, Simon Persson
The purpose of this project was to investigate a cylindrical cav- ity resonator and use microwaves to heat up a material in the cavity. This was done by measuring the Q-factor and the reso- nance frequency of the cavity, both with and without material inside. The chosen material was graphite, and more accurate measurements were done with that specific material. A pro- gram called QZero was used to export the Q-factor and the res- onance frequency from the measurement data received from a VNA and the program also gave error estimations. Then elec- tromagnetic simulations were done using Comsol. Both an empty cavity and a cavity where graphite has been inserted were simulated and the results were compared to the actual measurements. To measure temperatures inside the cavity, a pyrometer was to be used. The cavity resonator has small cir- cular holes through the side, and a frame was designed and produced using a 3D-printer in order to lock the pyrometer in place in front of one of the holes. A power supply was also installed to the pyrometer. In order to send microwaves into the cavity, a signal generator was used. It was connected to an amplifier and the amplification as well as the efficiency was noted. The pyrometer could only measure temperatures above 490◦C. This was not achieved, so a handheld electrical thermometer was used. The temperature of the graphite was measured and then compared to how hot the graphite would be without heat loss.
For the empty cavity, a Q-factor of 3200 for the resonance frequency of around 2.4 GHz was measured, which matched the simulated measurements in Comsol. When graphite was inserted to the cavity, the Q-factor lowered to 300 in the real experiment. A discrepancy was found between the actual mea- surements, and the Comsol simulations in which the graphite only lowered the Q-factor to 2570. The reason for this is be- lieved to be either with an error to how the material was chosen in Comsol, since there were many types of graphite to select with many settings to change. Another reason could be an error with the setup itself due to the sheer complexity of the program.
Teknisk- naturvetenskaplig fakultet UTH-enheten Bes¨oksadress:
Angstr¨omslaboratoriet˚ L¨aderhyddsv¨agen 1 Hus 4, Plan 0 Postadress:
Box 536
751 21 Uppsala Telefon:
018 - 371 30 03 Telefax:
018 - 471 30 00 Hemsida:
http://www.teknat.uu.se/student
sulted a in risk of overheating the amplifier. For this reason only low power could be used. For an output of 2 W, the graphite was heated to 100-150◦C from 30 seconds of expo- sure to the microwaves. This was significantly lower than the theoretical value of 1700◦C calculated from the energy pro- vided to the graphite, which leads to a theory that the temper- ature found an equilibrium at around 100-200◦C or that the resonance frequency changes such that the graphite could no longer absorb the energy.
Handledare: Dragos Dancila Amnesgranskare: Maria Str¨omme¨ Examinator: Martin Sj¨odin
ISSN; 1401-5757, MAT-VET-F-21020 Tryckt av: Uppsala
Popul¨arvetenskaplig sammanfattning
Syftet med detta projekt ¨ar att unders¨oka en cylindrisk kavitet samt att anv¨anda mikrov˚agor f¨or att v¨arma upp ett material i kaviteten. Detta gjordes genom att m¨ata kavitetens Q-faktor och resonansfrekvens med och utan material. Sedan valdes grafit ut som det materialet som skulle testas och mer noggranna m¨atningar gjordes p˚a just det materialet, och programmet QZero anv¨andes f¨or att f˚a ut mer noggrann data samt gav en felmarginal p˚a Q-faktorn.
Efter det gjordes simuleringar i programmet Comsol av kaviteten med och utan grafit och j¨amf¨ordes med de faktiska v¨ardena. F¨or att m¨ata temperatur av materialet i kaviteten desig- nades en h˚allare som en pyrometer skulle f¨astas vid och riktas mot materialet i kaviteten.
Aven en str¨omf¨ors¨orjning till pyrometern installerades. F¨or att skicka in mikrov˚agor i¨ kaviteten anv¨andes en signalgenerator som var kopplad till en f¨orst¨arkare. F¨orst¨arknin- gen m¨attes och f¨orst¨arkarens effekt noterades. I slut¨andan n˚addes inte de temperaturer som kr¨avdes f¨or pyrometern, s˚a en elektrisk termometer anv¨andes f¨or att m¨ata tempera- turen p˚a grafiten och detta j¨amf¨ordes sedan mot den energin som absorberades av grafiten, d˚a energin tillf¨ord till grafiten och grafitens specifika v¨armekapacitet var k¨ant. Prestan- dan av experimentuppst¨allningen unders¨oktes ocks˚a. For den tomma kaviteten m¨attes en Q-faktor p˚a 3200 och resonansfrekvensen var 2.4 GHz. Detta st¨amde bra ¨overens med simuleringarna i Comsol. N¨ar grafit f¨ordes in i kaviteten s¨anktes Q-faktorn till 300. En avvikelse uppt¨acktes mellan de faktiska m¨atningarna och simuleringarna. I simuleringen s¨anktes Q-faktorn bara till 2570, en andledning till detta tros vara antingen vara hur ma- terialet valdes, d˚a det fanns olika typer av grafit med olika inst¨allningar att v¨alja mellan i Comsol. En annan felk¨alla kan vara n˚agot fel med uppst¨allningen p˚a grund af hur avancerat Comsol var. I v¨armeexperimentet hade f¨orst¨arkaren en l˚ag verkningsgrad vilket ledde till
¨overhettning om f¨or stor effekt anv¨andes. Experimentet begr¨ansades d¨arf¨or till att anv¨anda upp till tv˚a watt. Vid exponerig under 30 sekunder v¨armdes grafiten upp till 100-150◦C, vilket var avsev¨art l¨agre ¨an den teoretiska uppv¨armingen till 1700◦ ber¨aknad fr˚an ener- gin tillf¨ord till grafiten. Antagandet ¨ar att temperaturen hamnade i ett j¨amnviktsl¨age kring 100-200◦C eller att resonansfrekvensen ¨andrades vilket ledde till en minskning av energi tillf¨ord till grafiten.
1 Introduction 1
1.1 Goal . . . 1
2 Theory 2 2.1 Q-factor, scattering . . . 2
2.2 Coupling coefficient . . . 3
2.3 The cavity and resonance frequencies . . . 3
2.4 Smith diagram . . . 4
2.5 Vector network analyser . . . 5
2.6 Signal generator . . . 5
2.6.1 dBm . . . 5
2.7 Amplifier . . . 6
2.8 Directional coupler . . . 6
2.9 Circulator . . . 7
2.10 Electric heating . . . 7
2.11 Comsol . . . 7
2.12 QZero . . . 8
2.12.1 Purpose . . . 8
2.12.2 Inner workings . . . 8
2.12.3 Error estimation . . . 9
3 Method 11 3.1 Material . . . 11
3.2 Measurements of material . . . 13
3.3 Accurate measurements . . . 13
3.4 The Pyrometer . . . 14
3.5 The heating experiment . . . 15
3.6 Comsol . . . 17
4 Results and Discussion 18 4.1 Results . . . 18
4.1.1 Material study . . . 18
4.1.2 Measurements with the VNA and in Comsol . . . 18
4.1.3 Amplification and efficiency of amplifier . . . 20
4.1.4 Heating of graphite . . . 22
4.2 Discussion . . . 22
4.2.1 Material study . . . 22
4.2.2 VNA and Comsol measurements obtained by QZero . . . 22
4.2.3 Heating experiment . . . 22
5 Conclusion 23 5.1 Whats next? . . . 23
6 References 25
1 Introduction
Since the 1940s, microwave technology for the purpose of heating have developed rapidly and is today commonplace. Microwave heating works by sending energy directly into the material, which then absorbs it. This creates a disturbance in the electron cloud surround- ing the molecule of the material targeted by the microwaves. A dipole moment is induced and the friction from the movement creates heat, which in turn heats the material.
Microwave heating has many benefits over other, traditional heating processes. It is very energy efficient, since the heating occurs within the material itself, minimizing wasted energy that heats the materials surroundings. Microwave components can also be built to a smaller scale than other heating processes and greater control over the heating process is possible.(1) The most interesting case for microwave heating today is the possibility of 3D-printing with dielectric materials. If made possible a lot of energy could be saved and a more even melting procedure could be done with higher temperatures. This would mainly work for metals but could be done on plastic by blending the plastic with dielectric materials. An example of metal sintering can be found in the 47th European Microwave Conference under (Samuel Hefford 1, Nyle Parker 2, Jonathan Lees 3, and Adrian Porch 4, ”Monitoring Changes in Microwave Absorption of Ti64 Powder During Microwave Sintering”).
1.1 Goal
The goal with this project was to investigate a microwave cavity and use it to heat mate- rials, which in this project was graphite powder. This main goal was then divided into a number of smaller subgoals:
• How to measure the Q-factor and resonance frequency of the cavity, two important properties. This included learning about the measurement equipment (VNA) and what the data consists of (Scattering parameters).
• Doing the measurements in goal 1.
• Simulate the cavity in Comsol and compare the data to the actual data obtained on location.
• Setting up and doing the actual experiment. This included finding a way to measure the temperature in the cavity (pyrometer and how to fit the pyrometer to the cavity), as well as measuring the temperature of the graphite and compare it to what would reasonably happen.
2 Theory
2.1 Q-factor, scattering
To understand the experiment, it is crucial to at least have heard about a couple of im- portant terms. One of which is the quality factor. The Q-factor can be described as how much power in in the form of the input signal is stored inside the system compared to the dissipated energy per wavelength as shown in equation 1. In simpler terms, a low Q-factor means that a lot of the energy is absorbed inside the system while a high Q-factor points towards a low absorption. When changing the heat of a system it may be interesting to analyze the change of Q-factor to understand the material’s properties.(2). To be able to calculate the Q-factor it is first helpful to know about scattering parameters (s-parameters).
Examples of them can be seen in equation 2 and 3 where a system is coupled in two places.
Q= 2π ∗ (energy stored in system)
dissipated energy per cycle (1)
S11 =out1
in1 S21= out2
in1 (2)
S22 =out2
in2 S12= out1
in2 (3)
Of these s-parameters, only S11 is interesting for the Q-factor, though it is good to understand the principle behind all of them. When sending a signal into a system and the signal interacts with the system it is expected that the signal is affected in some way. For a keen eye looking at the equations the reader may understand that scattering is about some kind of relationship between the signal going into and out of a system, and for that the reader is correct. Since a linear system is assumed when scattering is measured the only two interesting relationships to look at are the amplitude and phase of the signals. For S11 a signal is sent into the first coupling of the system and the signal that is reflected back to the coupling is compared to the input signal. S11 is also called the reflection coefficient.
For S21the part of the signal exiting through the second coupling is compared to the input into the first coupling.(3)
From the reflection coefficient for the different frequencies sent into the system a Smith diagram can be made, illustrated in figure 2. From S11 it is possible to finally calculate the Q-factor, but also the coupling coefficient of the system. One way to approximate the Q-factor is by only looking at the amplitude difference of S11 in dB. This can be done by
dividing the resonance frequency with the bandwidth of the resonance seen in equation 4 and illustrated in figure 1.(2)
Qtot= resonance f requency
Bandwidth (4)
Figure 1: Resonance frequency and Bandwidth for S11parameters
2.2 Coupling coefficient
A system can be over, under and critically coupled. The most important thing to under- stand is that when the system is critically coupled all of the input signal is transferred to the system. Critical coupling is reached when the signal source has the same impedance at resonance frequency as the system, making the coupling coefficient equal to one k = 1.
When the impedance of the source is larger than the system, over coupling is reached k > 1 and under coupling is reached when the source has a smaller impedance than the system k< 1.(4)
2.3 The cavity and resonance frequencies
A resonance cavity does not have a defined shape though a cylindrical one is used in this project. As described by the name it is a cavity in which resonance can be achieved and most often it is EM-waves being sent in to the cavity. The Q-factor for cylindrical cavities
Figure 2: Smith Diagram (6)
is affected by the impedance in the walls, the coupling points and the objects inside the cavity.
When thinking about resonance frequencies in a rope it is possible to have different modes of resonances but only in one direction, while using a 3D space makes it possible to have different modes both in overtones and in directions. Not only that but electromagnetic waves have two parts, one electric and one magnetic, both of which affect which kind of resonance is reached at different frequencies. When the electric waves are perpendicular to the direction of the signal it is called transverse electric modes (T Elnm) while transverse magnetic mode (T Mlnm) is for magnetic waves. lnm describes the mode in direction and overtone of the wave. (5). For this experiment TM010 is preferred and used as seen in figures 21 and 22 where the resonance is half a wavelength along the diameter and where the propagation of the wave is always perpendicular to the magnetic field.
2.4 Smith diagram
In figure 2 a Smith diagram is visualised. The Smith diagram is used to help visualizing how a signal interacts with the cavity and how the scattering change with frequency. It is not necessary to understand exactly how the smith diagram works but a couple of details
from it will be discussed. First. The frequency of the signal barely changes the resistance experienced in the system while the reactance will change, resulting in a diagram following the same pattern as the real lines in figure 2. Secondly the resonance frequency is the frequency closest to the point describing where the impedance (z=1) of the cavity would match that of the signal generator. Thirdly, if the circle is larger (closer to RE(z) = 0) there is over-coupling while there is under-coupling if the circle is smaller and does not reach z= 1.(7)
2.5 Vector network analyser
An important tool used for this project is the vector network analyzer, normally called a VNA. The most common VNA function is to measure scattering parameters but can also be used to measure y-,z- and h-parameters. Only the scattering is relevant for this report.
The typical network analyzer has two ports so it can send in a signal on one side and measure the output signal on the other end but as mentioned earlier we are only interested in the reflected parameter meaning the signal will go in and out of the same port. When measuring s-parameters with the VNA the signal is represented as a complex number with its phase and amplitude.(3)
2.6 Signal generator
A signal generator is quite self explanatory. A machine that generates signals. Now there are multiple different kinds of signal generators, some can only create normal sinusoidal signals, some can almost create any physically possible signal given, for this project only simple sinusoidal signals are relevant. The frequency range, maximal amplitude/power and noise is also different for different signal generators.(8)
2.6.1 dBm
dBm is short for decibel-milliwatts and is used to avoid having to make multiple calcula- tions when changing the strength of a signal in dB. 0 dBm is the same as one milliwatt, from there you can convert any watt to dBm through equation 5.(9)
dBm= 10 ∗ log( PmW
1mW) (5)
2.7 Amplifier
An amplifier recieves a signal and amplifies it with a certain amount of dB. If the amplifi- cation is 20 dB and the input signal is 10 dBm we can express the conversion as in equation 6. For the amplifier to be able to amplify the signal it needs a DC power supply.(10)
10dBm + 20dB = 30dBm = 1W (6)
One common difficulty with amplifiers is that they do not have perfect transmission of electricity and some are lost to heat. The efficiency can be calculated by dividing the output power by the input voltage and ampere provided by the power supply. Equation 7 shows the efficiency of the amplifier.
µ =Pout
Pin = Pout
VinIin (7)
2.8 Directional coupler
Directional couplers are used to redirect a specific part of a signal to the side while letting the rest of the signal through.
Figure 3: Bi-directional coupler
Sending in a signal through port one will transport most of the signal to port two but a defined part of the signal will exit at port three. For this project about -40 dB of the input signal will exit through port three while the rest goes directly to port two. If instead the signal enters through port two most will exit through port one while a small part will be redirected to port four. In reality a non-zero amount of signal will travel from port one to four and from port two to three but is often too small to make a real difference or is canceled out from a mirrored signal.(11)
2.9 Circulator
A circulator as usually has three ports shown in figure 4. Its function is to receive a signal through one port and sen it to the next port clockwise but not anti-clockwise. Sending a signal into port 1 will output it to port 2, using port 2 as input will result in port 3 as output and port 3 will send the signal to port 1.(12)
Figure 4: Clockwise circulator
2.10 Electric heating
In a normal microwave food is heated through the vibration of the dielectric properties of water molecules in a changing electric field.(13) For graphite microplasma is instead created and reacts with the alternating field.(14) To calculate the heating of a material absorbing electromagnetic waves, equations with specific heat can be preformed as in equation 8.
∆T = P∗ t
m∗ cρ (8)
Where ∆T is the temperature difference, P is the power received by the material, t is the amount of time the material has received the power, m is the mass of the material and cρ is the specific heat.(15)
2.11 Comsol
Comsol is a simulation program used for complex physical simulations involving elec- tromagnetism, magnetism, heat distribution, stress and more. It is used in this project to validate the results from the practical tests of the real cavity. Here Comsol is used to find resonant frequencies and values for the Smith diagram.
2.12 QZero
2.12.1 Purpose
QZero is a program designed specifically to extract the Q-factor from the input parameters.
It is a pretty versatile program, capable of taking S11 parameters, impedances and even pure data from certain VNA:s as input and outputs a smith diagram as well as the Q-factor and the resonance frequency. For this project, S11 parameters were used as input. This required the data that was going to be used as input to be written in a table with three columns. The first column contained the frequency at which the measurement took place.
The second column contained the real part of the S11 parameter, and the third column contained the imaginary part of the S11parameter.
The QZero program used in the project was a demo version, called QDEMOW. The demo version was identical to the full version of QZero except that some features (Which were irrelevant for this project, such as measurements of the other S-parameters, i.e. S22, S21and S12) were disabled and the number of data points that could be used as input were limited to 201. This meant that after measurements had been done with the VNA-machine or within Comsol, only the 201 data points closest to the resonance frequency should be used and the rest of the data points had to be discarded. Otherwise, the computational steps and the output from the program was the same as the full version(16).
2.12.2 Inner workings
The QZero software works in the following way.
First, it looks at the S11 parameter from the measurements. It then tries to fit the data to the following mathematical formula:
S11= a1t+ a2
a3t+ 1 (9)
t= 2f− fL
fL (10)
where fL is the loaded resonance frequency, which QZero defines as the frequency with the smallest magnitude of the S11 parameter, and a1, a2 and a3 are three unknown complex coefficients. If there are many measurement points (which there are when QZero is used in this project, since typically around 200 measurement points were used) it leads to an over-determined system, since there are only three unknowns. The three unknown coefficients can then be determined with the least-square method. (7)
When the data has been fitted to equation 9, the S11 parameters displayed in a Smith chart form a circle. The diameter of this circle can be determined by the three found complex coefficients:
d= |a2−a1
a3| (11)
When the diameter of the circle is known, the coupling coefficient is given by κ = d
2 − d (12)
The coupling coefficient is the ratio of the external conductance (i.e. the conductance in the connections to the cavity resonator) and the conductance in the cavity resonator.
From the a3 coefficient, the loaded Q-factor, which is the Q-factor that includes coupling losses due to the coupling structure to and from the cavity(17), is given (7) by taking the imaginary part of the a3coefficient.(18)
Lastly, the unloaded Q-factor, which is the Q-factor due to losses inside the cavity resonator (17) and which is the Q-factor which is of interest to this project, is given by:
Q0= QL(1 + κ) (13)
2.12.3 Error estimation
While the QZero software determine the three unknown coefficients, since the system is overdetermined the software also calculates the standard deviation σ of the three coeffi- cients. To determine the standard deviation of the unloaded Q-factor, many steps are taken.
Firstly, the standard deviation of the loaded Q is calculated by the software as:
σ (QL) = q
(Re(a3))2+ σ (a3)2 (14)
Re(a3) means the real part of the a3 coefficient and σ (a3) is the standard deviation of a3. Next, the program calculates the standard deviation of the circle in the smith diagram which it tries to fit the data to.
σ (d) = s
σ2(a1)
|a3|2 + σ2(a2) + |a1
a3|2σ2(a3) (15) When the standard deviation of the diameter of the Q circle has been calculated, the program moves on to calculate the standard deviation of the coupling coefficient from
σ (d).
σ (κ ) = 2
(2 − d)2σ (d) (16)
Finally, the program uses σ (QL) and σ (κ) to find the standard deviation of the un- loaded Q-factor.
σ (Q0) = q
(1 + κ)2σ2(QL) + Q2Lσ2(κ) (17) (18)
However, with this method, it is only possible to describe the random error due to fitting the S11 parameter according to equation 9. It is not possible to know any possible systematic error due to the VNA itself.(7) The systematic error is minimized or eliminated if the VNA is calibrated correctly.
3 Method
3.1 Material
A large part of the project was doing different measurement with a cylindrical cavity res- onator. The cavity consisted of a hollow aluminium cylinder with a detachable lid held together with six screws to the main body. The resonator had two ports which could be used to connect a VNA-machine to the cavity in order to do measurements of the scatter- ing parameters. The cavity had one hole in the center of the lid used to insert material into the cavity and three holes were fitted on the side of the main body, one of which could be used to let a pyrometer measure the temperature inside the cavity resonator. Every part of the cavity resonator were measured and a blueprint was created.
Figure 5: The cavity used in the project
Figure 6: Blueprint of the cylindrical cavity, as well as some measurements of the pyrom- eter
3.2 Measurements of material
The first experiment that was done with the cavity resonator was to measure how different materials affected the resonance frequency and the Q-factor inside the cavity. In order to measure the scattering parameters a VNA (FieldFox RF Analyzer N9912A) was used, which can measure scattering parameters for up to 4 GHz. The VNA was set to display the S11 parameter in dB along the y-axis and the frequency domain along the x-axis.
Firstly, an approximation of where the resonance frequency was located was made by visually looking on the VNA display. At the resonance frequency, a drop of dB occur which can be seen as a pit in the graph. A good example of this can be seen in figure 15. The resonance frequency was determined to be around 2.46 GHz, so the VNA was set to measure frequencies from 2 GHz to 3 GHz with 401 measurement points. When the range of the VNA had been set, the VNA was calibrated by attaching an open standard, a short standard and a load standard to the output cable of the VNA. When the VNA had been calibrated, the output cable was connected to the port of the cavity. Extra caution was made to ensure the cable was attached tightly to the cavity, to minimize losses. The measurements then began. The desired material that was to be measured was placed in the cavity, and the lid was screwed on tight. For each material, the resonance frequency was noted. Then the bandwidth was determined by finding where the magnitude of the S11 parameter differed by 3dB from the magnitude at the resonance frequency. The Q- factor could then be calculated using equation 4. The results of these measurements are presented in table 1.
3.3 Accurate measurements
After the measurements had been done, it was found that the calibration of the VNA machine had been unsatisfactory. New measurements therefore had to be done. New, better calibration tools from a calibration kit was used and the calibration was double checked so that the magnitude of the S11-parameter was as close to 0 dB as possible far away from the resonance frequency. The range was also reduced to 2.4-2.5 GHz and the number of measurement points were increased to 1601 points.
This time, the measurements were focused on a single material: graphite. The graphite used was in powder form. It was placed in a glass tube which had an inner diameter of 4 mm. The glass tube was then manually inserted into the hole in the top of the cavity resonator, and held so that the graphite was located close the middle of the empty space inside the cylindrical cavity.
Instead of noting down the resonance frequency and the bandwidth, an USB stick was inserted into the VNA. This made it possible to copy the data points measured by the VNA directly to the USB and save the data as a CSV file. The data consisted of the frequency
measured, accompanied by the real and imaginary parts of the S11 parameter. The number of data points was then reduced to 200 and the ones closest to the resonance frequency were chosen. The data could then be imported to the QZero program, which produced the Q-factor as well as the error for the Q-factor. The program also produced a smith chart.
The results from these measurements are presented in chapter 4.1.2.
3.4 The Pyrometer
To measure the temperature inside the cavity, a pyrometer would be used. In order to get accurate readings a frame to lock the pyrometer in place at one of the holes in the side of the cavity had to be designed. A model was designed in Solidworks, which consisted of a curved edge with two holes, designed to go along the rim of the top of the cylinder and was to be held in place with the screws that were holding the lid and the main body together, effectively squeezing the frame between the body and the lid. The part that was supposed to hold the pyrometer in place went down the side of the cavity and a hole the size of the pyrometer was used to screw the pyrometer into the holder and hold it in place.
The model was printed using a 3D-printer and was made of plastic. The holder can be viewed in figure 7.
Figure 7: The 3D printed holder for the pyrometer
Next, a power supply was needed to power the pyrometer, for this, an extension cable for a 2.1 mm DC plug was used and electric tape was used to squeeze the cable with the help of a screw at the input of the pyrometer, effectively holding it in place. The cable was
then cut, and the ground and live wire was connected to each respective connector. The power supply connection can be seen in 8 and the finished setup in figure 9.
Figure 8: The connection of the power supply to the pyrometer, which can be seen in the left hand side of the picture
Figure 9: The finished setup, with the pyrom- eter connected to the cavity
3.5 The heating experiment
The heating experiment consisted of a signal generator, connected to an amplifier. The amplifier connects to a circulator where the signal goes through port one to two into port one of a bi-directional coupler. Port three of the coupler was then connected to a power meter and the cavity or resistance was connected to port two, the ports being defined as in fig 3 and 4. The amplifier amplified the input signal and the predicted output was specified in the data sheet (19). Since -40dB of the input signal exited through port 3, the power meter was configured such that it displayed +40dBm in relation to its actual measurement, effectively making it approximate the power that exited through port 2.
For the first part of the experiment, the coupler was connected to a load instead of the cavity. The power from the signal generator and the power from the power meter was noted for a variety of different inputs. For a few inputs the current and voltage at the input to the amplifier was measured as well to get the efficiency of the amplifier.
For the second part the cavity was connected to the circuit. A VNA machine measured the resonance frequency and the signal generator was set to oscillate at the resonance frequency. About 50 mg graphite was inserted into the cavity inside a glass tube. Unfor- tunately it became clear that the pyrometer would be of no use since it begins measuring temperatures at 490◦C, which was never reached. Instead a handheld electrical thermome- ter measured the temperature of the graphite as it was pulled out of the cavity. Since
Figure 10: The experiment setup. The important components are a signal generator(1), an amplifier(2), a circulator(3), a directional coupler(4), a power meter(5), drain and gate voltage for the amplifier(6 & 7), VNA(8) and a resistance(9)
the power provided to the graphite as well as the duration, the energy was known, which made it possible to make a prediction of what temperature was expected. This was then compared to the actual temperature obtained in the experiment.
Figure 11: The cavity with the graphite inserted in a glass tube
3.6 Comsol
Three different simulations was made using Comsol to help with analyzing and predicting the behaviour of the Q-factor and resonance of the cavity. All the simulations used one port to send the signal into the cavity with the second port disconnected, making it just a piece of copper inside a cavity. The first two simulations was to investigate if the tests from the VNA on the empty cavity matches up with the simulations and vice versa. For the first simulation all the walls of the cavity had perfect conductance where only the input port was set to have an impedance reflecting the properties of the built in copper material, while the second simulation used impedance properties for every part of the model. The walls of the cavity are set to aluminum, the port is set to copper and the plastic around the port is set as PEEK100. The third simulation also added a cylinder made of graphite in the middle of the cavity to represent the graphite powder in the real experiment and tried a couple of different properties for the graphite material. This is then compared to the test with the VNA on cavity with graphite powder. In figure 12 the cavity walls is 1, where the signal leaves the port is 2, the signal is generated in 3, the plastic surrounding the port is 4 and the graphite is 5.
Figure 12: Cavity from Comsol with graphite in center
Table 1: Q factor, resonant frequency and bandwidth of different materials, measured with VNA
Material Resonant frequency [GHz] Bandwidth [Ghz] Q0
Empty Cavity 2.46428 2.46403 - 2.46453 4917
Cardboard 2.4227 2.41949 - 2.42536 413
Cloth 2.36559 2.33876 - 2.38414 52
Rubber (ball of rubber bands) 2.379272 2.37884 - 2.37970 2756 Plastic lid (from soda bottle) 2.44197 2.4414 - 2.4425 2365 Plastic lid filled with graphite 2.3849 2.3805 - 2.3884 302
Plastic lid filled with water 2.2072 - -
4 Results and Discussion
4.1 Results
4.1.1 Material study
Table 1 presents the results from the measurements of the different materials tested. The lack of data for bandwidth and Q-factor for water is due to the type of calculations done in order to determine the Q-factor. For this experiment, the Q-factor was determined by the ratio between the resonance frequency and the bandwidth. For the measurements of water however, the bandwidth could not be determined since the magnitude of the S11 parameter never differed by 3 dB or more compared to the magnitude at the resonance frequency. A theory for this behaviour is that it is possible that the plastic lid was filled with too much water, which increased the effect it had on the Q-factor to the point that it was too small to be measured.
4.1.2 Measurements with the VNA and in Comsol
Table 2: Measurements with the VNA and in Comsol, obtained with QZero
Measurement Resonant frequency [GHz] Q0 κ
Empty Cavity (VNA) 2.463 3208.1± 82.3 11.987 ± 0.331
Graphite (VNA) 2.436 299.2 ± 40.6 1.006 ± 0.262
Empty Cavity (Comsol without impedance) 2.454 3341.2 ± 255.5 11.842 ± 0.978 Empty Cavity (Comsol with impedance) 2.455 2624.1 ± 153.2 9.587 ± 0.614
Graphite (Comsol without impedance) 2.448 2571.9 ± 86.2 9.427 ± 0.347
The empty cavity is overcoupled which can be seen in the smith chart in figure 14, and is closer to an open circuit than is ideal which leads to more of the signal being reflected.
2.4 2.41 2.42 2.43 2.44 2.45 2.46 2.47 2.48 2.49 2.5
Frequency [Hz] 109
-2 -1.5 -1 -0.5 0 0.5
S11 [dB]
S11 of the empty cavity measured with the VNA
Figure 13: Measured S11(in dB) of the empty cavity with the VNA
Figure 14: Measured S11 of the empty cavity with the VNA, presented in a Smith-chart
2 2.1 2.2 2.3 2.4 2.5 2.6
Frequency [Hz] 109
-30 -25 -20 -15 -10 -5 0
S11 [dB]
S11 of the cavity when graphite was inserted, measured with the VNA
Figure 15: Measured S11(in dB) of the cavity when graphite was inserted, measured with the VNA
Figure 16: Measured S11 of the cavity when graphite was inserted, measured with the VNA and presented in a Smith- chart
This is also supported by the high coupling coefficient from table 3. This changes when the graphite is inserted. The system then gets critically coupled since the system matches the impedance of the signal source, which can be seen in figure 16. This leads to mini- mal losses of reflection at the resonance frequency, which can be seen by comparing the magnitude of figure 13 and 15.
2.42 2.43 2.44 2.45 2.46 2.47 2.48
Frequency [Hz] 109
-3 -2.5 -2 -1.5 -1 -0.5 0
S11 [dB]
S11 (in dB) of the empty cavity, simulated in Comsol
Figure 17: Measured S11(in dB) of the empty cavity, simulated in Comsol
Figure 18: Measured S11 of the empty cavity simulated in Comsol, presented in a Smith-chart
The empty simulated cavity seems to match the actually empty cavity pretty well.
They both are within the same margin of error for both the Q-factor and the coupling coefficient. The simulation with graphite does not seem to match the actual cavity with graphite. Though there is a decrease in the Q-factor and the coupling coefficient is slightly better, it is not nearly on the same level as the actual cavity.
The colors in figure 21 and 22 describe the strength of the electric field inside the cav- ities and its field pattern matches that of the T M010 mode described in the theory section.
The signal is sent from the left coupler in both pictures.
4.1.3 Amplification and efficiency of amplifier
From these values it is possible to calculate the efficiency for the amplifier through equa- tion 7, resulting in µ = 25dBm17.3W ≈ 0.3W17W ≈ 2% for the output of 0.3W and µ = 34dBm48W ≈
2.5W
48W ≈ 5% for the output of 2.5 W.
2.43 2.435 2.44 2.445 2.45 2.455 2.46 2.465 2.47 2.475 Frequency [Hz]
-3 -2.5 -2 -1.5 -1 -0.5 0
S11 [dB]
S11 (in dB) of the cavity when graphite was inserted, simulated in Comsol
Figure 19: Measured S11 of the cavity when graphite was inserted, simulated in Comsol
Figure 20: Measured S11 of the cavity when graphite was inserted, simulated in Comsol and presented in a smith chart
Figure 21: Simulation without graphite Figure 22: Simulations with graphite Table 3: amplification and power input to amplifier
input signal [dBm] output signal [dBm] input power [W]
-20 -23 –
-5 -14 –
0 -6 –
5 9 –
10 25 17.3
13 34 48
4.1.4 Heating of graphite
Using the specific heat of graphite from equation 8 the temperature difference at 33dBm or 2 watt for half a minute, where no energy is dissipated from the graphite is 1700◦C.
This is compared to the actual temperature measured 30 seconds after the signal had been shut down at 100◦C.
4.2 Discussion
4.2.1 Material study
The measurements done in a material study gave a hint of how much microwaves affect the material. Plastic and rubber was affected very little, but the microwave had a huge impact of the water, which is to be expected due to the dipole nature of water.
4.2.2 VNA and Comsol measurements obtained by QZero
The accurate measurements done on the actual cavity and in Comsol are interesting. The first simulation with mostly perfect conductivity gave a result which almost perfectly re- flects the results from the VNA on the empty cavity, when adding impedance to the rest of the cavity the result of the Q-factor is still close to that of the real experiment but further from perfect. Why this is the case is not clear but may have to do with how the materials in Comsol does not represent the real materials correctly. The simulations of graphite gave unexpected results, only slightly lowering the Q-factor while the real experiment had a significant change of quality factor and resonance frequency. The leading theory for the discrepancy is that there was some error when choosing the material. There were many different kinds of graphite to choose from and there may be some important properties of graphite that is not set in the basic materials already part of Comsol. Even if multiple dif- ferent properties was change and added no noticeable difference where observed. Another likely error with the simulation is simply that there was an error with the setup. Comsol is a very complex program and it took many hours to even get a decent result with the empty cavity, so it’s not out of the question that some setting or boundary condition was wrongly defined.
4.2.3 Heating experiment
When the output power from the amplifier goes over one Watt the temperature of the amplifier rise up to 200◦C with the simple setup of a small fan and heat sink. When looking at the data sheet for the used amplifier we can see that the resulting efficiency of
2-5% is not too far off since the amplifier is built for much higher powers than those used here. This can be seen in figure 3 in the datasheet, which shows an efficiency of 30% for an output of 50 watts, and the output is around 2 watts for this experiment, leading to an even worse efficiency(19). The heat resulting in the graphite is significantly lower than the 1700◦C that it would have been if there was no emission from the tube with graphite, making us believe that the temperature found an equilibrium at around 100-150◦C or that the resonance frequency had such a significant change that most of the signal was reflected back to the circulator and out to thin air. As the setup was now it was impossible to measure the heat of the cavity as it was heated up, since the graphite had to be placed on the bottom om the cavity instead of being held in the middle, since the holes for measuring the temperature is placed at the middle of the cavity walls.
5 Conclusion
The four goals have been fulfilled. Knowledge about Q-factors and resonance frequencies have been obtained and the empty cavity has been measured to have a Q-factor of around 3200 and a resonance frequency at 2.46 GHz. Measurements in Comsol seemed to match the real cavity when it was empty with perfect conductivity, but there was a discrepancy when it was simulated with graphite and when proper impedances was set to the walls, which is discussed in the report. The experiment was also done, and the pyrometer was fitted and ready to do measurements, unfortunately it was not used. The efficiency of the amplifier was calculated and the microwaves managed to heat the graphite to 100-150◦C.
This was much lower than the theoretical temperature that could have been used with the energy provided, and was also mentioned in the discussion.
5.1 Whats next?
The end goal with the experimental setup is, one, to be able to measure the change of Q-factor, coupling and resonance frequency, two, to change the frequency sent from the signal generator as the resonance frequency is changing with the heating. To do this the signal generator is supposed to send frequency sweeps every couple of seconds and then reflection parameters can be measured through the third end of the circulator as the signal is reflected from the cavity and compared to the signal coming out from the third port of the directional coupler. With this it is possible to automatically change the frequency of the signal generator to match the changed resonance of the cavity.
To avoid overheating with higher power it may be good to change the amplifier for one that has a better efficiency for lower power levels. Quartz wool has been bought and can
be used to make sure the graphite is placed in the middle of the cavity by filling half of the glass/quartz tube along the center axis of the cavity. The tube is shown in figure 11.
When the graphite has been placed in the middle more tests with higher power needs to be conducted to see if it is possible to reach higher temperatures without needing to adapt the frequency of the input signal. It is still interesting to look at the changing reflection parameter to understand the interaction of graphite or other materials with high intensity microwaves.
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