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

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

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

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

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

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

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

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

Transmission/Reflection

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

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

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

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.

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

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

Waveguide-Cutoff λ Chamber Cutoff!

Transmission of Water through the Chamber (S21)

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

Modes of Propagation

The Chamber

Side View of Chamber

Excitation VNA Output VNA Input

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

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

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)

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)

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

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.

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.

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

Model Results Here are the uncalibrated results for water

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.

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

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

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’

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

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

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[ ]

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.

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

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

Mode Calibration

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

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

Debye Relaxation Model

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

Debye Swarm Results

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

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

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

Ethanol

Methanol

Air

Water

Oil

70% Isopropyl Alcohol

Acetone

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.

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.