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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
'
'
' 

 j




 
'
'
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
The Waveguide Cutoff Method to performance charaterization.ppt
The Waveguide Cutoff Method to performance charaterization.ppt
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
Reflection
Reflection
Transmission Transmission
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
The Waveguide Cutoff Method to performance charaterization.ppt
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 to performance charaterization.ppt
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:
2
2
2
1














b
n
a
m
fc




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
Chamber
Cutoff!
λ
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
Input
VNA
Output
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
b
n
a
m
f
kz
2
2
2
2 )
(
)
(
)
2
(














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)
z
f
K
j z
e
m
f
S 


 )
(
12 )
,
(
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.
))
7
,
(
)
6
,
(
)
5
,
(
)
4
,
(
)
3
,
(
)
2
,
(
)
1
,
(
ln(
)
( 12
12
12
12
12
12
12 f
S
f
S
f
S
f
S
f
S
f
S
f
S
f
X 






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






















 




















L
f
K
j
L
f
K
j
L
f
K
j
L
f
K
j
L
f
K
j
L
f
K
j
L
f
K
j
z
z
z
z
z
z
z
e
e
e
e
e
e
e
f
X )
6
,
(
)
4
,
(
)
2
,
(
)
7
,
(
)
5
,
(
)
3
,
(
)
1
,
(
)
(
ln
686
.
8
)
(
The Waveguide Cutoff Method to performance charaterization.ppt
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






















 




















L
f
K
j
L
f
K
j
L
f
K
j
L
f
K
j
L
f
K
j
L
f
K
j
L
f
K
j
z
z
z
z
z
z
z
e
e
e
e
e
e
e
f
X )
6
,
(
)
4
,
(
)
2
,
(
)
7
,
(
)
5
,
(
)
3
,
(
)
1
,
(
)
(
ln
686
.
8
)
(






















 




















L
f
K
j
L
f
K
j
L
f
K
j
L
f
K
j
L
f
K
j
L
f
K
j
L
f
K
j
z
z
z
z
z
z
z
e
c
e
c
e
c
e
c
e
c
e
c
e
c
f
X )
6
,
(
7
)
4
,
(
6
)
2
,
(
5
)
7
,
(
4
)
5
,
(
3
)
3
,
(
2
)
1
,
(
1
)
(
ln
686
.
8
)
(
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
2
2
)
2
(
1
)
(
)
(
'













f
f
f
i
f
0
2
2
2
)
2
(
1
2
)
(
)
(
'
'
























f
f
f
f
f
i
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.

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The Waveguide Cutoff Method to performance charaterization.ppt

  • 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 ' ' '    j       ' '
  • 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
  • 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: 2 2 2 1               b n a m fc     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
  • 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.
  • 25. Side View of Chamber
  • 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 b n a m f kz 2 2 2 2 ) ( ) ( ) 2 (              
  • 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) z f K j z e m f S     ) ( 12 ) , (
  • 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. )) 7 , ( ) 6 , ( ) 5 , ( ) 4 , ( ) 3 , ( ) 2 , ( ) 1 , ( ln( ) ( 12 12 12 12 12 12 12 f S f S f S f S f S f S f S f X       
  • 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                                             L f K j L f K j L f K j L f K j L f K j L f K j L f K j z z z z z z z e e e e e e e f X ) 6 , ( ) 4 , ( ) 2 , ( ) 7 , ( ) 5 , ( ) 3 , ( ) 1 , ( ) ( ln 686 . 8 ) (
  • 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                                             L f K j L f K j L f K j L f K j L f K j L f K j L f K j z z z z z z z e e e e e e e f X ) 6 , ( ) 4 , ( ) 2 , ( ) 7 , ( ) 5 , ( ) 3 , ( ) 1 , ( ) ( ln 686 . 8 ) (                                             L f K j L f K j L f K j L f K j L f K j L f K j L f K j z z z z z z z e c e c e c e c e c e c e c f X ) 6 , ( 7 ) 4 , ( 6 ) 2 , ( 5 ) 7 , ( 4 ) 5 , ( 3 ) 3 , ( 2 ) 1 , ( 1 ) ( ln 686 . 8 ) (
  • 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 2 2 ) 2 ( 1 ) ( ) ( '              f f f i f 0 2 2 2 ) 2 ( 1 2 ) ( ) ( ' '                         f f f f f i
  • 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
  • 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
  • 58. Air
  • 59. Water
  • 60. Oil
  • 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.