1. The Waveguide Cutoff Method
A New Method for Measuring the Complex Permittivity of Liquid and Semi-Solid Materials at Microwave Frequencies

2. What is Permittivity?
Permittivity is a measure of the energy stored and dissipated by a material in an electric field
σ = conductivity
ω = 2πf

3. Why is it Important?
Permittivity can be a measure of several different parameters
Density
Temperature
Consistency
Viscosity
Purity

4. Specific Applications
Industrial Applications: Process Monitoring
Polymers and thermoplastics
Steam
Chemical reactions
Mixing and Chemical Composition

5. Specific Applications
Biomedical and Food Applications
Water concentration and detection
Foodstuffs
Soils
Medicines
Fat and Meat quality
Cancer Detection
Blood Glucose Concentration

6. Methods of Permittivity Measurement for Liquids and Semi-Solids
Open-Ended Coaxial Probe technique
Cavity Perturbation Method
Transmission/Reflection Method
Coaxial Line Method
Waveguide Method
Time Domain Spectroscopy

7. Cavity Perturbation
How it works: Uses the Q-factor and the Frequency shift in the resonant frequency to determine the permittivity using ‘Perturbation theory’
Pros
Effective at measuring low-loss materials
Accurate as long as all of the assumptions are met
Cons
Sample size influences effectiveness and accuracy
Small samples only
Narrow Band/ Single Band
Must be precisely machined

10. Transmission/ Reflection
How it works: Uses S-Parameters from a network analyzer and the relates the permittivity to the reflection and transmission of energy through the sample
Pros
Relatively Broadband (One Decade for waveguides up to 20GHz for coaxial)
Excellent for high-loss samples
Cons
Sample size must be corrected
May only use the TE10 mode of propagation for a simple mathematical model
Must be precisely machined

11. Transmission/Reflection

12. Time-Domain Spectroscopy
How it works: uses a frequency domain signal and the FFT to relate the transmission time through an object to the complex permittivity.
Pros
Broadband, but limited by FFT and instrument (10Ghz)
Old method, no surprises
Cons
Very Expensive system
Large system complexity
System and software memory limitations for accuracy from the FFT calculation

14. Open-Ended Coaxial Probe Kit
How it works: Relates the reflection of energy off of the sample to the complex permittivity.
Pros
Commercially available, convenient
Broadband ( 200MHz – 20GHz)
Cons
Some limitations by sample size, temperature dependencies
Air gaps

16. The Waveguide-Cutoff Method
A simple, broadband calibration method for the measurement of liquid and semi-solid materials

17. Semi-Solid Powders Gels Colloids (salt water) Mixtures (Pulp stock)
Malleable solids ( Silly Putty or meat)
Any solid whose dimensions are much smaller than the smallest wavelength in the measurement.

18. Advantages of the Waveguide-Cutoff Method
Broadband (20 GHz)
Calculations are limited to a non-linear curve fit routine
Relatively inexpensive machined parts for having such a large accuracy
Does not suffer from the same restrictions or inaccuracies in calibration as the Probe Kit

19. Waveguide-Cutoff
Minimum frequency within the rectangular chamber that allows electromagnetic energy to travel through it.
Governed by this equation:
fc is the cutoff frequency in Hertz ,
a is the width of the waveguide in m,
b is the height of the waveguide in meters,
m is the number of ½-wavelength variations of fields in the "a" direction,
n is number of ½-wavelength variations of fields in the "b" direction,
µ is the permeability of the material inside the waveguide
ε is the complex permittivity of the material inside of the waveguide

20. Waveguide-Cutoff λ
Chamber
Cutoff!

21. Transmission of Water through the Chamber (S21)

22. Modes of Propagation
Moding occurs when the waveguide is exited with energy which has an integer multiple wavelength smaller than the guide.

23. Modes of Propagation

24. The Chamber

25. Side View of Chamber

26. Excitation
VNA
Output
VNA
Input

27. The Relationship
All the other methods related some measurable aspect to the permittivity, using a model
This model relates the transmission of the wave through the chamber and the subsequent shift in cutoff frequency to the complex permittivity

28. Model Derivation Begin with the propagation vector kz
This is the vector in the direction of wave propagation by crossing the Electric field and Magnetic field by the right hand rule.
Note that this equation contains the chamber dimensions as well as the permittivity

29. Model Derivation (cont.)
Now represent the transmission of energy through the chamber in polar form.
Note the propagation vector and the dependence on frequency and the mode m
Z is the position of the receiving antenna
This is literally the transmission S-parameter from port 1 (input) to port 2 (output)

30. Model Derivation (cont.)
So far, the energy may be calculated for a single mode at a single frequency
However, we would like to have a model which emulates the type of data that can be attained: that which comes from our Vector Network Analyzer (VNA)

31. Model Derivation (cont.)
The VNA outputs the total transmission through the chamber, which includes the cutoff frequency, and all of the existing modes added together
All of these parameters must be included in the model

32. Model Derivation (cont.)
The VNA outputs the logarithmic ratio of the received energy to the amount of energy excited by the input.
So the model must also include a form in values of dB.
Begin by taking the natural log of the transmission of the first few modes added together.

33. Model Derivation (cont.)
Note that the even modes are subtracted from the transmitted energy and the odd modes are added
While the presence of the even modes within the waveguide is not seen by the receiving antenna, their existence still removes energy from what will be received.

34. Currently, the equation X(f) is simply a logarithmic ratio, which is in units of Nepers.
Now a conversion factor from Nepers to DB is required to accurately predict the output of the VNA

36. Model Results
Here are the uncalibrated results for water

37. Model Results
Now that we can accurately predict the behavior of the transmission through the chamber, we will need to calibrate the model.
This is where Particle Swarm Optimization and the non-linear curve fit come in.

38. Particle Swarm Optimization
PSO is a generic non-linear stochastic method for curve-fitting in multiple dimensions.
This method is used to calibrate the instrument for the mode coefficients as well as the electrical length of the chamber

39. How PSO works
Imagine a surface, where you are looking for the lowest point on the curve
In this case, you are solving for 3 variables, which would be the 3D midpoint on the surface

40. How PSO works
Now imagine dropping a number of marbles onto the surface.
Keep track of their positions and their velocities as they roll around the surface
Eventually, most of the marbles, regardless of their initial positions, will fall into the ‘hole’

41. Paradigm shift
Now, instead of a 3D surface, make the solution space in n-dimensions.
Each ‘dimension’ of the space represents one of the parameters that will be changed in the PSO
The particles still have ‘positions’ and ‘velocities’ but they are much more abstract

42. How PSO ‘really’ works
Each particle represents one particular ‘solution’ to the problem within the solution space
The particles ‘move’ around this space, and the movements are based upon three things: their own personal best solution, the global best solution and a bit of randomness

43. Each particle has a velocity and position and these are calculated for each iteration of the PSO
The update equations for the velocity and position are below:
Next_v[ ] = v[ ] + c1 * rand * (pbest[ ] - present[ ]) + c2 * rand * (gbest[ ] - present[ ])
Next_present[ ] = present[ ] + v[ ]

44. Why do we care?
PSO was used in two different calibrations and in the final curve fit to find the model parameters for the complex permittivity.

45. Total Calibration Procedure
Calibrate for the Addition of different Modes
Calibrate for the effective electrical length of the chamber
Perform the Swarm several times to determine the model parameters for the complex permittivity

46. Mode Calibration
We want to make the final transmission as close to the actual data as possible, so scaling factors are added to the final transmission equation
Water, Air, ethanol and methanol were used as calibration materials since they have known permittivity values

47. Mode Calibration

48. Electrical Length Calibration
Temperature and humidity can change the size of the chamber and the effective electrical length of the chamber
Recall that the propagation vector kz, depends upon the dimensions of the chamber
A second calibration is used to determine these values before any data is to be taken, in an attempt to remove these effects.
This calibration utilizes temperature controlled water and air and the previous calibration to fine tune the model

49. The Debye Model of Permittivity
Complex permittivity changes over frequency
This is sometimes modeled using a Debye Relaxation Model

50. Debye Relaxation Model

51. Debye Swarm Process
Each particle is given a randomly selected starting position in the solution space
The solution is represented by the four numbers of the Debye relaxation model
The swarm then changes these four values to minimize the error between the model of the chamber and the actual data from the chamber

52. Debye Swarm Results

53. Preventative Measures
Sometimes a swarm will fall within a local minimum instead of the global minimum
This can be solved through a method of noise injection that Matt Trumbo calls “Explosion”
After a certain number of iterations, the particles will scatter at high velocity in a random direction, but retain their personal best solution

54. Statistical Methods
Most Swarm solutions are close to the actual solution, but not exact.
To reduce random error, the swarm is run for several iterations and averaged at the end
The swarm also determines the average and standard deviation for all the iterations to attempt to remove any outliers where the swarm has fallen into a false minimum

55. Results All results are for substances at 20 degrees
All of the calibration substances had known and recorded Debye Parameters
Comparisons were made between this system and an Open Coaxial-Line Dielectric Probe Kit

56. Ethanol

57. Methanol

58. Air

59. Water

60. Oil

61. 70% Isopropyl Alcohol

62. Acetone

63. Error Analysis
With ε’, there is a 5% error between this system and the reference instrument
With ε’’, however there is up to a 20% maximum error between the Waveguide-Cutoff method and the reference
*BUT* the uncertainty is only ±3
That is, the large 20% error only occurred for low-loss materials.

64. Conclusions
This system has similar accuracy and uncertainty with that of the Open Coaxial-Line Dielectric Probe Kit
However, this system does not share the same problems with sample depth, and air gaps between the sample and the probe
While sample size is larger, much of the uncertainty of measurement is removed with the Waveguide-Cutoff method.