The Waveguide Cutoff Method

7/29/2019 The Waveguide Cutoff Method

1/64

The Waveguide Cutoff Method

A New Method for Measuring theComplex Permittivity of Liquid and

Semi-Solid Materials at MicrowaveFrequencies


2/64

What is Permittivity?

Permittivity is a

measure of the

energy stored and

dissipated by amaterial in an

electric field

= conductivity

= 2f

''' j

''


3/64

Why is it Important?

Permittivity can be a measure of

several different parameters

Density

Temperature

Consistency

Viscosity

Purity


4/64

Specific Applications

Industrial Applications: Process

Monitoring

Polymers and thermoplastics

Steam

Chemical reactions

Mixing and Chemical Composition


5/64

Specific Applications

Biomedical and Food Applications

Water concentration and detection

Foodstuffs

Soils

Medicines

Fat and Meat quality

Cancer Detection Blood Glucose Concentration


6/64

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


7/64

Cavity Perturbation

How it works: Uses the Q-factor and the Frequencyshift in the resonant frequency to determine thepermittivity 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


8/64


9/64


10/64

Transmission/ Reflection

How it works: Uses S-Parameters from a networkanalyzer and the relates the permittivity to thereflection and transmission of energy through thesample

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 simplemathematical model

Must be precisely machined


11/64

Transmission/Reflection

Reflection

Reflection

Transmission Transmission


12/64

Time-Domain

Spectroscopy How it works: uses a frequency domain signal and

the FFT to relate the transmission time through anobject 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 fromthe FFT calculation


13/64


14/64

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


15/64


16/64

The Waveguide-Cutoff

Method

A simple, broadband calibration

method for the measurement ofliquid and semi-solid materials


17/64

Semi-Solid

Powders

Gels

Colloids (salt water) Mixtures (Pulp stock)

Malleable solids ( Silly Putty or meat)

Any solid whose dimensions are muchsmaller than the smallest wavelength

in the measurement.

Ad t f th


18/64

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 samerestrictions or inaccuracies incalibration as the Probe Kit


19/64

Waveguide-Cutoff

Minimum frequency within the rectangular

chamber that allows electromagnetic

energy to travel through it.

Governed by this equation:22

2

1

b

n

a

mfc

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 -wavelength

variations of fields in the "a" direction,

n is number of -wavelength variations

of fields in the "b" direction,

is the permeability of the material inside

the waveguide

is the complex permittivity of thematerial inside of the waveguide


20/64

Waveguide-Cutoff

ChamberCutoff!

T i i f W t


21/64

Transmission of Water

through the Chamber

(S21)


22/64

Modes of Propagation

Moding occurs when the waveguide is

exited with energy which has an

integer multiple wavelength smaller

than the guide.


23/64

Modes of Propagation


24/64

The Chamber


25/64

Side View of Chamber


26/64

Excitation

VNA

Input

VNA

Output


27/64

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


28/64

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

b

n

a

mfkz

2222 )()()2(


29/64

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)

zfKj zemfS )(12 ),(


30/64


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)


31/64


The VNA outputs the total

transmission through the chamber,

which includes the cutoff frequency,

and all of the existing modes addedtogether

All of these parameters must be

included in the model


32/64


The VNA outputs the logarithmic ratio of thereceived energy to the amount of energyexcited 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 addedtogether.

))7,()6,()5,()4,()3,()2,()1,(ln()( 12121212121212 fSfSfSfSfSfSfSfX


33/64


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 bereceived.


34/64

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

LfKjLfKjLfKj

LfKjLfKjLfKjLfKj

zzz

zzzz

eee

eeeefX

)6,()4,()2,(

)7,()5,()3,()1,()(

ln686.8)(


35/64


36/64

Model Results

Here are the uncalibrated results for water


37/64

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.


38/64

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


39/64

How PSO works

Imagine a surface,where you arelooking for thelowest point on the

curve In this case, you

are solving for 3variables, which

would be the 3Dmidpoint on thesurface


40/64

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, willfall into the hole


41/64

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 moreabstract


42/64

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 solutionand a bit of randomness


43/64

Each particle has a velocity and position

and these are calculated for each iteration

of the PSO

The update equations for the velocity andposition are below:

Next_v[ ] = v[ ] + c1

* rand * (pbest[ ] -

present[ ]) + c2 * rand * (gbest[ ] - present[ ])

Next_present[ ] = present[ ] + v[ ]


44/64

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.


45/64

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


46/64

Mode Calibration

We want to make thefinal transmission asclose to the actual dataas possible, so scalingfactors are added tothe final transmissionequation

Water, Air, ethanol andmethanol were used as

calibration materialssince they have knownpermittivity values


47/64

Mode Calibration

LfKjLfKjLfKj

LfKjLfKjLfKjLfKj

zzz

zzzz

eee

eeeefX

)6,()4,()2,(

)7,()5,()3,()1,()(

ln686.8)(

LfKjLfKjLfKj

LfKjLfKjLfKjLfKj

zzz

zzzz

ececec

ecececec

fX )6,(7

)4,(

6

)2,(

5

)7,(

4

)5,(

3

)3,(

2

)1,(1)(

ln686.8)(


48/64

Electrical Length

Calibration Temperature and humidity can change the

size of the chamber and the effectiveelectrical length of the chamber

Recall that the propagation vector kz,

depends upon the dimensions of thechamber

A second calibration is used to determinethese values before any data is to be taken,in an attempt to remove these effects.

This calibration utilizes temperaturecontrolled water and air and the previouscalibration to fine tune the model

Th D b M d l f


49/64

The Debye Model of

Permittivity

Complex

permittivity changes

over frequency

This is sometimesmodeled using a

Debye Relaxation

Model

22)2(1

)()('

ff

fi

f

0

22 2)2(1

2)()(''

ff

ff

fi


50/64

Debye Relaxation Model


51/64

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


52/64

Debye Swarm Results


53/64

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 callsExplosion

After a certain number of iterations, the

particles will scatter at high velocity in a

random direction, but retain their personalbest solution


54/64

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 theend

The swarm also determines the average

and standard deviation for all the iterations

to attempt to remove any outliers where theswarm has fallen into a false minimum


55/64

Results

All results are for substances at 20

degrees

All of the calibration substances had

known and recorded DebyeParameters

Comparisons were made between this

system and an Open Coaxial-LineDielectric Probe Kit


56/64

Ethanol


57/64

Methanol


58/64

Air


59/64

Water


60/64

Oil


61/64

70% Isopropyl Alcohol


62/64

Acetone


63/64

Error Analysis

With , there is a 5% error between this

system and the reference instrument

With , 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.


64/64

Conclusions

This system has similar accuracy and

uncertainty with that of the Open Coaxial-

Line Dielectric Probe Kit

However, this system does not share thesame 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 withthe Waveguide-Cutoff method.

Date post:	04-Apr-2018
Category:	Documents
Upload:	-
View:	226 times
Download:	0 times

The Waveguide Cutoff Method

Documents