Basics of Measuring the Dielectric Properties of Materials

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Keysight Technologies
Basics of Measuring the
Dielectric Properties of Materials
Application Note
Introduction
A wide variety of industries need a better understanding of the materials they are working with to
shorten design cycles, improve incoming inspection, process monitoring, and quality assurance. Every
material has a unique set of electrical characteristics that are dependent on its dielectric properties. Accurate measurements of these properties can provide scientists and engineers with valuable
information to properly incorporate the material into its intended application for more solid designs or
to monitor a manufacturing process for improved quality control.
A dielectric materials measurement can provide critical design parameter information for many electronics applications. For example, the loss of a cable insulator, the impedance of a substrate, or the
frequency of a dielectric resonator can be related to its dielectric properties. The information is also
useful for improving ferrite, absorber and packaging designs. More recent applications in the area of
aerospace, automotive, food and medical industries have also been found to benefit from knowledge
of dielectric properties.
Keysight Technologies, Inc. offers a variety of instruments, fixtures, and software to measure the
dielectric properties of materials. Keysight measurement instruments, such as network analyzers,
impedance analyzers and LCR meters range in frequency up to 1.1THz. Fixtures to hold the material under test (MUT) are available that are based on coaxial probe, parallel plate, coaxial/waveguide
transmission lines, free space and resonant cavity methods. The table below shows product examples
that can be measured by Keysight’s material test solutions.
Table 1. Materials measurement applications example
Industry
Applications/Products
Electronics
Capacitor, substrates, PCB, PCB antenna, ferrites, magnetic recording heads,
absorbers, SAR phantom materials, sensor
Aerospace/Defense
Stealth, RAM (Radiation Absorbing Materials), radomes
Industrial materials
Ceramics and composites: IC package, aerospace and automotive components,
cement, coatings, bio-implants
Polymers and plastics: fibers, substrates, films, insulation materials
Hydrogel: disposable diaper, soft contact lens
Liquid crystal: displays
Rubber, semiconductors and superconductors
Other products containing these materials: tires, paint, adhesives, etc.
Food & Agriculture
Food preservation (spoilage) research, food development for microwave, packaging,
moisture measurements
Forestry & Mining
Moisture measurements in wood or paper, oil content analysis
Pharmaceutical & Medical Drug research and manufacturing, bio-implants, human tissue characterization,
biomass, chemical concentration, fermentation
3 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Contents
Introduction ........................................................................................2
Dielectric Theory ...............................................................................4
Dielectric constant ......................................................................4
Permeability ...................................................................................7
Electromagnetic Wave Propagation ...........................................8
Dielectric Mechanisms ..................................................................10
Orientation (dipolar) polarization ..........................................11
Electronic and atomic polarization ......................................11
Relaxation time ...........................................................................12
Debye relation .............................................................................12
Cole-Cole diagram ....................................................................13
Ionic conductivity .......................................................................13
Interfacial or space charge polarization ............................14
Measurement Systems .................................................................15
Network analyzers .....................................................................15
Impedance analyzers and LCR meters ..............................16
Fixtures ..........................................................................................16
Software .......................................................................................16
Measurement Techniques ............................................................17
Coaxial probe ..............................................................................17
Transmission line ........................................................................20
Free space ....................................................................................23
Resonant cavity ..........................................................................26
Parallel plate ...............................................................................29
Inductance measurement method .......................................30
Comparison of Methods ...............................................................31
Keysight Solutions ..........................................................................32
References ........................................................................................33
Web Resources ...............................................................................34
4 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Dielectric Theory
The material properties that will be discussed here are permittivity and
permeability. Resistivity is another material property which will not be
discussed here. Information about resistivity and its measurement can be found
in the Keysight Application Note 1369-11. It is important to note that permittivity
and permeability are not constant. They can change with frequency,
temperature, orientation, mixture, pressure, and molecular structure of the
material.
Dielectric constant
A material is classified as “dielectric” if it has the ability to store energy when
an external electric field is applied. If a DC voltage source is placed across a
parallel plate capacitor, more charge is stored when a dielectric material is
between the plates than if no material (a vacuum) is between the plates. The
dielectric material increases the storage capacity of the capacitor by
neutralizing charges at the electrodes, which ordinarily would contribute to the
external field. The capacitance with the dielectric material is related to
dielectric constant. If a DC voltage source V is placed across a parallel plate
capacitor (Figure 1), more charge is stored when a dielectric material is
between the plates than if no material (a vacuum) is between the plates.
A
t
C = C0k'
+
C0 =
k' = er' =
C
C0
V
A
t
+
+
+
+
+
+ +
+
–
Figure 1. Parallel plate capacitor, DC case
Where C and C0 are capacitance with and without dielectric, k’ = e’r is the real
dielectric constant or permittivity, and A and t are the area of the capacitor
plates and the distance between them (Figure 1). The dielectric material
increases the storage capacity of the capacitor by neutralizing charges at the
electrodes, which ordinarily would contribute to the external field. The
capacitance of the dielectric material is related to the dielectric constant as
indicated in the above equations. If an AC sinusoidal voltage source V is placed
across the same capacitor (Figure 2), the resulting current will be made up of a
charging current Ic and a loss current Il that is related to the dielectric constant.
The losses in the material can be represented as a conductance (G) in parallel
with a capacitor (C).
5 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
I
I = IC + II = V (jwC0k' + G)
If G = wC0k'', then
I = V (jwC0)(k'– jk'') = V (jwC0)k
+
A
V
t
w = 2p f
+ - + - +- + - +
+
+
+
C
G
–
Figure 2. Parallel plate capacitor, AC case
The complex dielectric constant k consists of a real part k’ which represents
the storage and an imaginary part k’’ which represents the loss. The following
notations are used for the complex dielectric constant interchangeably
k = k* = er = e*r .
From the point of view of electromagnetic theory, the definition of electric
displacement (electric flux density) Df is:
Df = eE
where e = e* = e0er is the absolute permittivity (or permittivity),
er is the relative permittivity, e0 ≈ 1 x 10−9 F/m is the free space permittivity
36Π
and E is the electric field.
Permittivity describes the interaction of a material with an electric field E and is
a complex quantity.
k=
e
= er = er − j er ''
e0
Dielectric constant (k) is equivalent to relative permittivity (er) or the absolute
permittivity (e) relative to the permittivity of free space (e0). The real part of permittivity (er’) is a measure of how much energy from an external electric field is
stored in a material. The imaginary part of permittivity (er’’) is called the loss
factor and is a measure of how dissipative or lossy a material is to an external
electric field. The imaginary part of permittivity (er”) is always greater than zero
and is usually much smaller than (er’). The loss factor includes the effects of
both dielectric loss and conductivity.
6 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
When complex permittivity is drawn as a simple vector diagram (Figure 3), the
real and imaginary components are 90° out of phase. The vector sum forms an
angle d with the real axis (er’). The relative “lossiness” of a material is the ratio
of the energy lost to the energy stored.
e r''
1
tan d = = D =
Q
er
=
Energy lost per cycle
Energy stored per cycle
Figure 3. Loss tangent vector diagram
The loss tangent or tan d is defined as the ratio of the imaginary part of the
dielectric constant to the real part. D denotes dissipation factor and Q is quality
factor. The loss tangent tan d is called tan delta, tangent loss or dissipation
factor. Sometimes the term “quality factor or Q-factor” is used with respect to
an electronic microwave material, which is the reciprocal of the loss tangent.
For very low loss materials, since tan d ≈ d, the loss tangent can be expressed
in angle units, milliradians or microradians.
7 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Permeability
Permeability (µ) describes the interaction of a material with a magnetic field. A
similar analysis can be performed for permeability using an inductor with
resistance to represent core losses in a magnetic material (Figure 4). If a DC
current source is placed across an inductor, the inductance with the core
material can be related to permeability.
R
L = L0 µ '
µ'=
L
L0
L
Figure 4. Inductor
In the equations L is the inductance with the material, L0 is free space
inductance of the coil and µ’ is the real permeability. If an AC sinusoidal
current source is placed across the same inductor, the resulting voltage will be
made up of an induced voltage and a loss voltage that is related to permeability.
The core loss can be represented by a resistance (R) in series with an inductor
(L). The complex permeability (µ* or µ) consists of a real part (µ’) that
represents the energy storage term and an imaginary part (µ’’) that represents
the energy loss term. Relative permittivity µr is the permittivity relative to free
space:
µr =
m
= µr – jµr ''
m0
µ0 = 4p x 10–7 H/m is the free space permeability
Some materials such as iron (ferrites), cobalt, nickel, and their alloys have
appreciable magnetic properties; however, many materials are nonmagnetic,
making the permeability very close to the permeability of free space (µr = 1). All
materials, on the other hand, have dielectric properties, so the focus of this
discussion will mostly be on permittivity measurements.
8 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Electromagnetic Wave Propagation
In the time-varying case (i.e., a sinusoid), electric fields and magnetic fields
appear together. This electromagnetic wave can propagate through free space
(at the speed of light, c = 3 x 108 m/s) or through materials at slower speed.
Electromagnetic waves of various wavelengths exist. The wavelength l of a
signal is inversely proportional to its frequency f (λ = c/f), such that as the
frequency increases, the wavelength decreases. For example, in free space a
10 MHz signal has a wavelength of 30 m, while at 10 GHz it is just 3 cm.
Many aspects of wave propagation are dependent on the permittivity and
permeability of a material. Let’s use the “optical view” of dielectric behavior.
Consider a flat slab of material (MUT) in space, with a TEM wave incident on
its surface (Figure 5). There will be incident, reflected and transmitted waves.
Since the impedance of the wave in the material Z is different (lower) from the
free space impedance η (or Z0) there will be impedance mismatch and this will
create the reflected wave. Part of the energy will penetrate the sample. Once
in the slab, the wave velocity v, is slower than the speed of light c. The
wavelength λd is shorter than the wavelength λ0 in free space according to the
equations below. Since the material will always have some loss, there will be
attenuation or insertion loss. For simplicity the mismatch on the second
border is not considered.
Z=
ld =
h
e r'
l0
e r'
h = Z0
v=
m0
= 120p
e0
c
e r'
Figure 5. Reflected and transmitted signals
9 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Figure 6 depicts the relation between the dielectric constant of the Material Under Test
(MUT) and the reflection coefficient |G| for an infinitely long sample (no reflection from
the back of the sample is considered). For small values of the dielectric constant
(approximately less than 20), there is a lot of change of the reflection coefficient for a
small change of the dielectric constant. In this range dielectric constant measurement
using the reflection coefficient will be more sensitive and hence precise. Conversely, for
high dielectric constants (for example between 70 and 90) there will be little change of
the reflection coefficient and the measurement will have more uncertainty.
1
0.9
Reflection coefficient
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10
20
30
40
50
60
Dielectric constant
Figure 6. Reflection coefficient versus dielectric constant
70
'
r
80
90
100
10 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Dielectric Mechanisms
A material may have several dielectric mechanisms or polarization effects that
contribute to its overall permittivity (Figure 7). A dielectric material has an
arrangement of electric charge carriers that can be displaced by an electric
field. The charges become polarized to compensate for the electric field such
that the positive and negative charges move in opposite directions.
At the microscopic level, several dielectric mechanisms can contribute to
dielectric behavior. Dipole orientation and ionic conduction interact strongly at
microwave frequencies. Water molecules, for example, are permanent dipoles,
which rotate to follow an alternating electric field. These mechanisms are quite
lossy – which explains why food heats in a microwave oven. Atomic and electronic mechanisms are relatively weak, and usually constant over the microwave region. Each dielectric mechanism has a characteristic “cutoff frequency.”
As frequency increases, the slow mechanisms drop out in turn, leaving the
faster ones to contribute to e’. The loss factor (er’’) will correspondingly peak at
each critical frequency. The magnitude and “cutoff frequency” of each mechanism is unique for different materials. Water has a strong dipolar effect at low
frequencies – but its dielectric constant rolls off dramatically around 22 GHz.
PTFE, on the other hand, has no dipolar mechanisms and its permittivity is
remarkably constant well into the millimeter-wave region.
A resonant effect is usually associated with electronic or atomic polarization. A
relaxation effect is usually associated with orientation polarization.
'
r
Dipolar
(Rotational)
+
+
+
-
-
Atomic
Ionic
-
Electronic
''
r
10
3
10
6
10
9
MW
10
12
IR
Figure 7. Frequency response of dielectric mechanisms
10
V
15
UV
f, Hz
11 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Orientation (dipolar) polarization
A molecule is formed when atoms combine to share one or more of theirs
electrons. This rearrangement of electrons may cause an imbalance in charge
distribution creating a permanent dipole moment. These moments are oriented
in a random manner in the absence of an electric field so that no polarization
exists. The electric field E will exercise torque T on the electric dipole, and the
dipole will rotate to align with the electric field causing orientation polarization
to occur (Figure 8). If the field changes the direction, the torque will also
change.
T
E
F
–
+
F
Figure 8. Dipole rotation in electric field
The friction accompanying the orientation of the dipole will contribute to the
dielectric losses. The dipole rotation causes a variation in both er’ and er’’ at
the relaxation frequency which usually occurs in the microwave region. As
mentioned, water is an example of a substance that exhibits a strong
orientation polarization.
Electronic and atomic polarization
Electronic polarization occurs in neutral atoms when an electric field displaces
the nucleus with respect to the electrons that surround it. Atomic polarization
occurs when adjacent positive and negative ions “stretch” under an applied
electric field. For many dry solids, these are the dominant polarization
mechanisms at microwave frequencies, although the actual resonance occurs
at a much higher frequency. In the infrared and visible light regions the inertia
of the orbiting electrons must be taken into account. Atoms can be modeled as
oscillators with a damping effect similar to a mechanical spring and mass
system (Figure 7). The amplitude of the oscillations will be small for any
frequency other than the resonant frequency. Far below resonance, the
electronic and atomic mechanisms contribute only a small constant amount to
er’ and are almost lossless. The resonant frequency is identified by a resonant
response in er’ and a peak of maximum absorption in er’’. Above the resonance,
the contribution from these mechanisms
disappears.
12 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Relaxation time
Relaxation time t is a measure of the mobility of the molecules (dipoles) that exist
in a material. It is the time required for a displaced system aligned in an electric
field to return to 1/e of its random equilibrium value (or the time required for
dipoles to become oriented in an electric field). Liquid and solid materials have
molecules that are in a condensed state with limited freedom to move when an
electric field is applied. Constant collisions cause internal friction so that the
molecules turn slowly and exponentially approach the final state of orientation
polarization with relaxation time constant t.
t =
1
1
=
wc 2 p fc
When the field is switched off, the sequence is reversed and random distribution is
restored with the same time constant. The relaxation frequency fc is inversely
related to relaxation time:
At frequencies below relaxation the alternating electric field is slow enough that the
dipoles are able to keep pace with the field variations. Because the polarization is
able to develop fully, the loss (er’’) is directly proportional to the frequency (Figure 9).
As the frequency increases, er’’ continues to increase but the storage (er’) begins to
decrease due to the phase lag between the dipole alignment and the electric field.
Above the relaxation frequency both er’’ and er’ drop off as the electric field is too
fast to influence the dipole rotation and the orientation polarization disappears.
'
r
Debye equation: e(w) = e∞ +
e S – e∞
1 + jwt
For w = 0, e(0) = eS
For w = ∞, e(∞) = e∞
,
''
r
'
r
60
40
20
"
r
0.1
1
10
100
f, GHz
Figure 9. Debye relaxation of water at 30º C
Debye relation
Materials that exhibit a single relaxation time constant can be modeled by the
Debye relation, which appears as a characteristic response in permittivity as a
function of frequency (Figure 9). er’ is constant above and below the relaxation with
the transition occurring near the relaxation frequency (22 GHz). Additionally, er’’ is
small above and below relaxation and peaks in the transition region at the
relaxation frequency.
In calculating the above curves the static (DC) value of the dielectric constant is
es= 76.47, the optical (infinite frequency) value of the dielectric constant is e∞= 4.9
and the relaxation time t = 7.2 ps.
13 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Cole-Cole diagram
The complex permittivity may also be shown on a Cole-Cole diagram by
plotting the imaginary part (er’’) on the vertical axis and the real part (er’) on the
horizontal axis with frequency as the independent parameter (Figure 10). A ColeCole diagram is, to some extent, similar to the Smith chart. A material that has a
single relaxation frequency as exhibited by the Debye relation will appear as a
semicircle with its center lying on the horizontal er’’ = 0 axis and the peak of the
loss factor occurring at 1/τ. A material with multiple relaxation frequencies will
be a semicircle (symmetric distribution) or an arc (nonsymmetrical distribution)
with its center lying below the horizontal er’’= 0 axis.
The curve in Figure 10 is a half circle with its center on the x-axis and its
radius e − e . The maximum imaginary part of the dielectric constant e’rmax
s
2
∞
will be equal to the radius. The frequency moves counter clockwise on the curve.
"
r
30
"
r max
20
=
s
-
2
Increasing
f (GHz)
= 35.8
10
Center
0
10
20
30
40
= 4.9
50
60
70
s
er'
= 76.47
Figure 10. Cole-Cole diagram of Figure 9
Ionic conductivity
The measured loss of material can actually be expressed as a function of
both dielectric loss (erd’’) and conductivity (σ).
er'' = e rd'' +
s
we 0
At low frequencies, the overall conductivity can be made up of many different
conduction mechanisms, but ionic conductivity is the most prevalent in moist
materials. er’’ is dominated by the influence of electrolytic conduction caused by free
ions which exist in the presence of a solvent (usually water). Ionic conductivity only
introduces losses into a material. At low frequencies the effect of ionic conductivity
is inversely proportional to frequency and appears as a 1/f slope of the er’’ curve.
14 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Interfacial or space charge polarization
Electronic, atomic, and orientation polarization occur when charges are locally
bound in atoms, molecules, or structures of solids or liquids. Charge carriers
also exist that can migrate over a distance through the material when a low
frequency electric field is applied. Interfacial or space charge polarization
occurs when the motion of these migrating charges is impeded. The charges
can become trapped within the interfaces of a material. Motion may also be
impeded when charges cannot be freely discharged or replaced at the
electrodes. The field distortion caused by the accumulation of these charges
increases the overall capacitance of a material which appears as an increase in
er’.
Mixtures of materials with electrically conducting regions that are not in
contact with each other (separated by non-conducting regions) exhibit the
Maxwell-Wagner effect at low frequencies. If the charge layers are thin and
much smaller than the particle dimensions, the charge responds independently
of the charge on nearby particles. At low frequencies the charges have time to
accumulate at the borders of the conducting regions causing er’ to increase. At
higher frequencies the charges do not have time to accumulate and polarization
does not occur since the charge displacement is small compared to the
dimensions of the conducting region. As the frequency increases, er’ decreases
and the losses exhibit the same 1/f slope as normal ionic conductivity.
Many other dielectric mechanisms can occur in this low frequency region
causing a significant variation in permittivity. For example, colloidal suspension
occurs if the charge layer is on the same order of thickness or larger than the
particle dimensions. The Maxwell-Wagner effect is no longer applicable since
the response is now affected by the charge distribution of adjacent particles.
15 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Measurement System
Network analyzers
A measurement of the reflection from and/or transmission through a material along
with knowledge of its physical dimensions provides the information to characterize
the permittivity and permeability of the material. Vector network analyzers such as
the PNA family, ENA series and FieldFox make swept high frequency stimulusresponse measurements from 9 kHz to 1.1 THz. (Figure 12). A vector network
analyzer consists of a signal source, a receiver and a display (Figure 11). The source
launches a signal at a single frequency to the material under test. The receiver is
tuned to that frequency to detect the reflected and transmitted signals from the
material. The measured response produces the magnitude and phase data at that
frequency. The source is then stepped to the next frequency and the measurement
is repeated to display the reflection and transmission measurement response as a
function of frequency. More information on the network analyzer functioning and
architecture is available in the application notes 1287-12 and 1287-23.
Simple components and connecting wires that perform well at low frequencies
behave differently at high frequencies. At microwave frequencies wavelengths
become small compared to the physical dimensions of the devices such that two
closely spaced points can have a significant phase difference. Low frequency
lumped-circuit element techniques must be replaced by transmission line theory to
analyze the behavior of devices at higher frequencies. Additional high frequency
effects such as radiation loss, dielectric loss and capacitive coupling make
microwave circuits more complex and expensive. It is time consuming and costly to
try to design a perfect microwave network analyzer.
Fixture
Incident
Transmitted
MUT
Source
Reflected
Signal
separation
Incident
(R)
Reflected
(A)
Transmitted
(B)
Receiver/detector
Processor/display
Figure 11. Network analyzer
Instead, a measurement calibration is used to eliminate the systematic (stable and
repeatable) measurement errors caused by the imperfections of the system.
Random errors due to noise, drift, or the environment (temperature, humidity,
pressure) cannot be removed with a measurement calibration. This makes a
microwave measurement susceptible to errors from small changes in the
measurement system. These errors can be minimized by adopting good
measurement practices, such as visually inspecting all connectors for dirt or
damage and by minimizing any physical movement of the test port cables after a
calibration. More information on the network analyzer calibration is available in the
Application Note 1287-34.
16 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Impedance analyzers and LCR meters
Impedance analyzers and LCR meters such as the ones listed in Figure 12 are
used to measure the material properties at lower frequencies. The material is
stimulated with an AC source and the actual voltage across the material is
monitored. Material test parameters are derived by knowing the dimensions of
the material and by measuring its capacitance and dissipation factor.
PNA family
Network
analyzers
ENA series
FieldFox Handheld VNA
Impedance/Material
Analyzer
E4991B
Impedance Analyzer
4294A
E4980A , 4285A
DC
10 1
10 2
10 3
10 4
10 5
LCR meters
10 6
10 7
10 8
10 9
10 10
10 11
10 12
f (Hz)
Figure 12. Frequency coverage of Keysight's instruments used for dielectric
measurements
Fixtures
Before the dielectric properties of a material can be measured with network
analyzer, impedance analyzer, or LCR meter, a measurement fixture (or sample
holder) is required to apply the electromagnetic fields in a predictable way and
to allow connection to the measurement instrument. The type of fixture
required will depend on the chosen measurement technique and the physical
properties of the material (solid, liquid, powder, gas).
Software
The measured data from the instrument is not always presented in the most
convenient terminology or format. In this case, software is required to convert
the measured data to permittivity or permeability. Software may also be
required to model any interaction between the fixture and MUT to allow the
extraction of the bulk material properties.
17 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Measurement Techniques
Coaxial probe
Method features
––Broadband
––Simple and convenient (non-destructive)
––Limited er accuracy and tan d low loss resolution
––Best for liquids or semi-solids
Material assumptions
––“Semi-infinite” thickness
––Non-magnetic
––Isotropic and homogeneous
––Flat surface
––No air gaps
The open-ended coaxial probe is a cut off section of transmission line. The
material is measured by immersing the probe into a liquid or touching it to the
flat face of a solid (or powder) material. The fields at the probe end “fringe”
into the material and change as they come into contact with the MUT
(Figure 13). The reflected signal5 (S11) can be measured and related to er*.
A typical measurement system using a coaxial probe method consists of a
network analyzer or impedance analyzer, software to calculate permittivity, and a
coaxial probe, probe stand and cable. The coaxial probe, probe stand and cable
are available in the 85070E dielectric probe kit. The software is now available in
N1500A materials measurement suite. The software can be installed on an
external PC and interfaced over GPIB, LAN or USB, depending on the analyzer.
Or, with ENA or PNA series network analyzers, the software can be installed
directly on the analyzer, eliminating the need for an external PC.
Solids
Semisolids (powder)
Reflection
(S 11)
Liquids
S11
Figure 13. Coaxial probe method
r
18 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Figure 14 shows the three probes that are available in the 85070E kit; the high
temperature probe (a), the slim form probe (b), and the performance probe (c).
The high temperature probe (a) is shown with the shorting block to the right.
Three slim probes are shown at the bottom of (b) with the short on the top and
a couple of other accessories. The performance probe (c) is shown with the
shorting block to the top.
(a)
(b)
Short
(c)
Short
Flange
Aperture
High temperature probe
Short
Performance Probe
Slim probes
Figure 14. Three dielectric probe configurations
Rugged in design, the high temperature probe (a) features a hermetic glass-tometal seal, which makes it resistant to corrosive or abrasive chemicals. The
probe withstands a wide –40 to +200 °C temperature range, which allows
measurements versus frequency and temperature. The large flange allows
measurements of flat surfaced solid materials, in addition to liquids and semisolids. The slim form probe (b) features a slim design, which allows it to fit
easily in fermentation tanks, chemical reaction chambers, or other equipment
with small apertures. The slim design also allows it to be used with smaller
sample sizes. This probe is best used for liquids and soft semi-solids. For
castable solids, the probe is economical enough to be cast into the material
and left in place. Because of the consumable nature of this design, these
probes are offered in sets of three. The slim form probe kit comes with a sealed
slim form holder that adapts a 2.2 mm outer diameter to 10 mm inner diameter
bracket included in the kit as well as commercially available “Midi” sized
adapters and bushings. The performance probe (c) combines rugged, high
temperature and frequency performance in a slim design, perfect for your most
demanding applications. The probe is sealed on both the probe tip and the
connector end, which makes it our most rugged probe. The probe withstands a
wide –40 ºC to +200 ºC temperature range, which allows measurements versus
frequency and temperature. The probe can be autoclaved, so it is perfect for
applications in the food, medical, and chemical industries where sterilization is
a must. The slim design allows it to fit easily in fermentation tanks, chemical
reaction chambers, or other equipment with small apertures. It is useful for
measuring liquid, semi-solid, as well as flat surfaced solid materials. Additional
detailed information is available in the Dielectric Probe Technical Overview6 and
Software Online Help7.
19 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
The dielectric probes are compatible with the Keysight network analyzers and
the E4991A impedance analyzer. With the impedance analyzer the high
temperature probe is specified from 10 MHz.
Before measuring, calibration at the tip of the probe must be performed. A
three-term calibration corrects for the directivity, tracking, and source match
errors that can be present in a reflection measurement. In order to solve for
these three error terms, three well-known standards are measured. The
difference between the predicted and actual values is used to remove the
systematic (repeatable) errors from the measurement. The three known
standards are air, a short circuit, and distillate and de-ionized water. Even after
calibrating the probe, there are additional sources of error that can affect the
accuracy of a measurement. There are three main sources of errors:
–– Cable stability
–– Air Gaps
–– Sample thickness
It is important to allow enough time for the cable (that connects the probe to
the network analyzer) to stabilize before making a measurement and to be sure
that the cable is not flexed between calibration and measurement. The
automated Electronic Calibration Refresh feature recalibrates the system
automatically, in seconds, just before each measurement is made. This virtually
eliminates cable instability and system drift errors.
For solid materials, an air gap between the probe and sample can be a
significant source of error unless the sample face is machined to be at least as
flat as the probe face. For liquid samples air bubbles on the tip of the probe can
act in the same way as an air gap on a solid sample.
The sample must also be thick enough to appear “infinite” to the probe. There
is a simple equation6 to calculate the approximate thickness of the sample for
the high temperature probe sample and suggested thickness for the slim probe
sample. A simple practical approach is to put a short behind the sample and
check to see if it affects the measurement results.
Figure 15 shows a comparison of measurements of dielectric constant and loss
factor of methanol at room temperature (25 °C) using the high temperature
probe, with theoretical calculations using the Cole-Cole model. The following
parameters are used in the Cole-Cole calculations:
eS = 33.7, e∞ = 4.45, t = 4.95 x 10–11, a = 0.036
20 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
"
r
'
r
30
Theory
Measurement
10
20
5
10
0.1
(a)
1
10
f, GHz
Theory
0.1
Measurement
1
10
f, GHz
(b)
Figure 15. Measured dielectric constant (a) and loss factor (b) of methanol at 25
ºC compared with Cole-Cole model
A disadvantage of the coaxial probe method is limited accuracy under some
conditions when compared to the transmission line, free space, or resonant
cavity methods.
Transmission line
Transmission line methods involve placing the material inside a portion of an
enclosed transmission line. The line is usually a section of rectangular
waveguide or coaxial airline (Figure 16). er* and µr* are computed from the
measurement of the reflected signal (S11) and transmitted signal (S21).
Material assumptions
––Sample fills fixture cross section
––No air gaps at fixture walls
––Smooth, flat faces, perpendicular to long axis
––Homogeneous
Method features
––Broadband – low end limited by practical sample length
––Limited low loss resolution (depends on sample length)
––Measures magnetic materials
––Anisotropic materials can be measured in waveguide
21 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Waveguide
l
Reflection
(S11)
Coax
S11
S 21
Transmission
(S21 )
r
µr
Figure 16. Transmission line method; waveguide and coaxial
line case
Coaxial transmission lines cover a broad frequency range, but a toroid shaped
sample is more difficult to manufacture (Figure 17(a)). Waveguide fixtures
extend to the mm-wave frequencies and the samples are simpler to machine,
but their frequency coverage is banded (Figure 17(b)). A typical measurement
system using the transmission line method consists of a vector network
analyzer, a coaxial or waveguide transmission line, and software to calculate
permittiivy and permeability. The software is available in N1500A materials
measurement suite. The software can be installed on an external PC and
interfaced over GPIB, LAN or USB, depending on the analyzer. Or, with ENA or
PNA series network analyzers, the software can be installed directly on the
analyzer, eliminating the need for an external PC. Additional information about
N1500A materials measurement suite can be found in the Technical Overview8
and Software Online Help9.
(a)
(b)
Figure 17. Coaxial 7 mm air line with samples (a) and X-band waveguide
straight section with samples (b)
The 50 Ohm airline from Keysight verification kits (Figure 17(a)) is the
recommended coaxial sample holder. Every waveguide calibration kit in the
11644A family contains a precision waveguide section (Figure 17(b)),
recommended for a waveguide sample holder.
22 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Figure 18 shows measurement results of permittivity (a) and loss tangent (b) of
two Plexiglas samples with lengths of 25 mm and 31 mm respectively, in an
X-band waveguide. The sample holder is the precise waveguide section of
140 mm length that is provided with the X11644A calibration kit (Figure 17(b)).
The network analyzer is a PNA, the calibration type is TRL and the precision
NIST algorithm9 is used for calculation. In both graphs below there are two
pairs of traces for two different measurements of the same samples. The top
two measurements of each graph are performed for the case when the sample
holder is not calibrated out.
'
r
tan
25 mm
25 mm
2.58
(a)
31 mm
31 mm
25 mm
2.56
2.54
0.005
31 mm
9
10
0.004
calibrated out
sample holder
11
12
25 mm
31 mm
f, GHz
0.003
9
10
calibrated out
sample holder
11
12
f, GHz
(b)
Figure 18. Measurement of two Plexiglas samples, 25 mm and 31 mm long in a
X-band waveguide
In this case based on the sample length and sample holder length, the N1500A
materials measurement suite will rotate the calibration plane correctly to the
sample face, but will not compensate for the losses of the waveguide. The
bottom two measurements of the same samples are performed for the case
when the sample holder is part of the calibration and the waveguide losses and
electrical length are calibrated out. As expected, the loss tangent curves (b)
show lower values when the sample holder is calibrated out and they are more
constant with respect to frequency. This is due to the fact that the waveguide
losses are no longer added to the sample’s losses. If a sample cannot stand up
by itself, for example because it is too thin or in some form other than a rigid
solid, it is possible to back it on one or both sides with a dielectric material
with known permittivity and thickness. In this case, de-embedding can be used
to remove the effects of the dielectric backing from the measurement.
23 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Free space
Material assumptions
––Large, flat, parallel-faced samples
––Homogeneous
Method features
––Non-contacting, non-destructive
––High frequency – low end limited by practical sample size
––Useful for high temperature
––Antenna polarization may be varied for anisotropic materials
––Measures magnetic materials
Free space methods use antennas to focus microwave energy at or through a
slab or sheet of material (Figure 19). This method is non-contacting and can be
applied to materials to be tested under high temperatures and hostile
environments. Figure 19 shows two typical setups: an S-parameter free space
transmission configuration (upper) and an NRL arch reflectivity configuration
(lower). A typical system consists of a vector network analyzer, appropriate free
space fixture and software to calculate permittivity and permeability for the free
space transmission method, or reflectivity with the arch method. The software
is available in N1500A materials measurement suite. It can be installed on an
external PC and interfaced over GPIB, LAN or USB, depending on the analyzer.
Or, with ENA or PNA series network analyzers, the software can be installed
directly on the analyzer, eliminating the need for an external PC.
Material Sample
To Port 1
of network analyzer
To Port 1 of
network
analyzer
Figure 19. Free space measurement setups
To Port 2
of network analyzer
To Port 2 of
network
analyzer
24 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
High temperature measurements are easy to perform in free space since the
sample is never touched or contacted (Figure 20). The sample can be heated by
placing it within a furnace that has “windows” of insulation material that are
transparent to microwaves. Keysight does not provide the furnace needed for
such a type of measurement. Figure 20 illustrates the basic set up.
Heating panels
Furnace
Thermal
insulation
Sample
Thermocouple
Figure 20. High temperature measurement in free space
Calibrating the network analyzer for a free space measurement is challenging.
Free space calibration standards present special problems since they are
“connector-less”. A calibration can be as simple as a response calibration or as
complex as a full two-port calibration depending on the convenience and
accuracy desired.
The N1500A materials measurement suite offers free space calibration method
called GRL (Gated match, Reflect, Line). This calibration routine increases the
ease of use and reduces the costs associated with some other calibration
methods, such as TRM (Thru, Reflect, Match) and TRL (Thru, Reflect, Line). Use
of this option requires a network analyzer with the time domain option, an
appropriate free space fixture, and a metal calibration plate. This option also
includes a gated isolation/response calibration, which reduces errors from
diffraction effects at the sample edges and multiple residual reflections
between the antennas. The N1500A materials measurement suite automatically
sets up all the free space calibration definitions and network analyzer
parameters, saving engineering time. A guided calibration wizard steps the user
through the easy calibration process.
25 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
2.6
'
r
2.5
2.4
45
50
55
f, GHz
Figure 21. Measurement of Rexolite sample in a U-band (40 – 60 GHz)
Figure 21 depicts the result of a GRL calibration measuring Rexolite material in U-band
(40-60 GHz) with a PNA network analyzer and N1500A materials measurement suite.
The fixture was made with standard gain horns and a readily available, domestic use,
shelving unit to demonstrate that when doing a GRL calibration, even with the
simplest set up, it is still possible to perform reasonable measurements. For precise
measurements, more rigid fixtures with focused horns are recommended.
Figure 22. 330-500 GHz Thomas Keating Ltd. Quasi-Optical Table with Gaussian
beam horns, focusing mirrors and sample holder.
At mm-wave and submm-wave frequencies, Quasi-Optical Tables are ideal. They can
be purchased from Thomas Keating Ltd, or through Keysight Special Handling
Engineering. Keysight model numbers:
60 to 90 GHz
75 to 110 Ghz
90 to 140 GHz
140 to 220 GHz
220 to 325 GHz
325 to 500 GHz
Quasi-Optical Table
Quasi-Optical Table
Additional Set of Horns
Additional Set of Horns
Additional Set of Horns
Additional Set of Horns
85071E E02
85071E E01
85071E E22
85071E E23
85071E E18
85071E E24
Additional frequencies, as well as tables covering multiple frequency bands may be
available on request.
26 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Resonant Cavity
Resonant versus broadband techniques
Resonant techniques
––High impedance environment
––Reasonable measurements possible with small samples
––Measurements at only one or a few frequencies
––Well suited for low loss materials
Broadband techniques
––Low impedance environment
––Requires larger samples to obtain reasonable measurements
––Measurement at “any” frequency
Resonant cavities are high Q structures that resonate at specific frequencies.
A piece of sample material inserted into the cavity affects the resonant
frequency (f) and quality factor (Q) of the cavity. From these parameters, the
complex permittivity of the material can be calculated at a single frequency. A
typical measurement system consists of a network analyzer, a resonant cavity
fixture and software to make the calculations.
There are many different methods and types of fixtures.
Keysight N1500A materials measurement suite automates three methods: Split
Cylinder method, Split Post Dielectric Resonator method and ASTM D252010
Cavity Perturbation method. An external computer can be used to control the
network analyzer, interfacing over LAN, USB or through GPIB. For the PNA
family and the ENA series of network analyzers, the software can be installed
directly in the analyzer and there is no need for an external computer. Keysight
also offers high Q resonant cavity fixtures for the Split Cylinder13 and Split
Post14 methods.
Split cylinder resonator
Figure 23. Keysight 85072A 10 GHz split cylinder resonator
27 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
The split cylinder resonator is a cylindrical resonant cavity separated into two
halves. The sample is loaded in a gap between the two cylinder halves. One
cylinder half is fixed, and the other adjusts allowing the gap to accommodate
varying sample thicknesses. The real part of permittivity, e’, and loss tangent or
tan delta, tand, are calculated from the sample thickness, cylinder length, and
S-parameter measurements of the split cylinder resonator, both empty and
loaded with the sample. Using a mode matching model developed at NIST in
Boulder, Colorado14 permittivity and loss tangent can be calculated at the
10 GHz TE011 mode. It may also be possible to measure at some higher order
TE0np modes2 where no interfering modes exist. This method was adopted by
the IPC as TM-650 2.5.5.13 standard test method.15
Split post dielectric resonator
Figure 24. QWED 5 GHz split post dielectric resonator, available from Keysight as
85071E-E04
Split Post Dielectric Resonators from QWED, use low loss dielectric materials
which make it possible to build resonators having higher Q-factors and better
thermal stability than traditional all-metal cavities. This method is one of the
easiest and highest accuracy methods for measuring complex permittivity and
loss tangent of low loss and thin sheet materials16. The relatively inexpensive
fixtures can be purchased from QWED or through Keysight Special Handling
Engineering in single frequencies from 1 to 15 GHz.
Keysight model numbers:
1.1 GHz
2.5 GHz
5 GHz
15 GHz
85071E E19
85071E E03
85071E E04
85071E E15
Additional frequencies may be available on request.
28 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Cavity perturbation (ASTM D2520)
e'r =
VC (fC – fS)
+1
2VSfS
e"r =
VC ". 1
1 "0
–
4VS "/ QS QS "1
Iris-coupled end plates
Q0
QS
f
Sample
fS
V is the volume
index c is for the empty cavity,
index s is for the sample loaded
f
r
fC
or µ r
Q
Figure 25. Resonant cavity measurement
The ASTM 252010 cavity perturbation method uses a rectangular waveguide
with iris-coupled end plates, operating in TE10n mode (Figure 25). For a
dielectric measurement, the sample should be placed in a maximum electric
field. Although Keysight does not provide a ready-made resonator fixture for the
cavity perturbation method, it is not difficult to adapt a precision waveguide
straight section, such as those available in the11644A series waveguide
calibration Kits. A hole needs to be drilled exactly in the middle of the
waveguide length and the two iris-coupled end plates need to be manufactured.
The dimension of the iris hole is b/2.2, where b is the narrow dimension of the
waveguide cross section. If the sample is inserted through a hole in the middle
of the waveguide length, then an odd number of half wavelengths will bring the
maximum electric field to the sample location, so that the dielectric properties
of the sample can be measured. (An even number of half wavelengths will
bring the maximum magnetic field to the sample location so that magnetic
properties of the sample can also be measured.)
The cavity perturbation method requires a very small sample such that the
fields in the cavity are only slightly disturbed to shift the measured resonant
frequency and cavity Q. This assumption allows simplifying the theory to use
the equations above to calculate the dielectric properties of the material.
29 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Parallel plate
The parallel plate method, also called the three terminal method in ASTM
standard D15012, involves sandwiching a thin sheet of material or liquid
between two electrodes to form a capacitor. The measured capacitance is then
used to calculate permittivity. In an actual test setup, two electrodes are
configured with a test fixture sandwiching dielectric material. The impedancemeasuring instrument would measure vector components of capacitance (C)
and dissipation (D) and a software program would calculate permittivity and
loss tangent. The method works best for accurate, low frequency
measurements of thin sheets or liquids. A typical measurement system using
the parallel plate method consists of an impedance analyzer or LCR meter and a
fixture such as the 16451B and 16453A dielectric test fixture, which operates
up to 1 GHz. The 16452A test fixture is offered for measuring liquids. More
information about the parallel plate method and other Keysight low frequency
materials measurement solutions are available in
Application Note 1369-1 (P/N 5980-2862EN)1 and 380-111.
Y =G + j Cp
Electrodes (Area=A)
Equivalent Circuit
⎛C
G ⎞
= j C 0 ⎜⎜ p − j
⎟
C0 ⎠
⎝ C0
Co : Air Capacitance
Cp
G
*
r
=
′ =
r
Solid
Thickness = t
Liquid
′′ =
r
Cp
−j
C0
G
C0
t ⋅C p
A⋅
0
t
⋅ Rp ⋅ A⋅
0
Figure 26. Parallel plate method
Figure 27. Keysight 16451B and 16453A dielectric test fixture with impedance
analyzer
30 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Inductance measurement method
Relative permeability of magnetic material derived from the self-inductance of a
cored inductor that has a closed loop (such as the toroidal core) is often called
effective permeability. The conventional method of measuring effective
permeability is to wind some wire around the core and evaluate the inductance
with respect to the ends of the wire. This type of measurement is usually
performed with an impedance analyzer. Effective permeability is derived from
the inductance measurement result. The Keysight 16454A magnetic material
test fixture provides an ideal structure for single-turn inductor, with no flux
leakage when a toroidal core is inserted in it. More information about the
inductance measurement method is available in the Application Note 1369-1
(P/N 5980-2862EN)1.
16454A
h
where,
c
b
No magnetic flux leakage
Figure 28. Inductance measurement method
relative permeability
measured inductance with MUT
measured inductance without MUT
permeability of free space
height of MUT (Material Under Test)
outer diameter of MUT
inner diameter of MUT
31 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Comparison of Methods
Many factors such as accuracy, convenience, and the material shape and form
are important in selecting the most appropriate measurement technique. Some
of the significant factors to consider are summarized here:
–– Frequency range
–– Expected values of er and μr
–– Required measurement accuracy
–– Material properties (i.e., homogeneous, isotropic)
–– Form of material (i.e., liquid, powder, solid, sheet)
–– Sample size restrictions
–– Destructive or nondestructive
–– Contacting or non-contacting
–– Temperature
–– Cost
Figure 29 provides a quick comparison between the measurement methods that
have been discussed already.
Coaxial Probe
r
Transmission Line
r
and µ r
Free Space
r
and µ r
Resonant Cavity
r
Parallel Plate
r
Inductance measurement
µr
Broadband, convenient, non-destructive
Best for lossy MUTs; liquids and semi-solids
Broadband
Best fpr lossy to low loss MUTs;
machineable solids
Broadband; Non-contacting
Best for flats sheets, powders, high temperatures
Single frequency; Accurate
Best for low loss MUTs; small samples
Accurate
Best for low frequencies; thin, flat sheets
Accurate, simple measurement, a toroidal core
sturcture is required
Figure 29. Summary of the measurement techniques
32 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Keysight Solutions
Keysight offers a wide variety of test fixtures to measure the dielectric
properties of materials which covers most material types. Figure 30 shows the
coverage of Keysight test fixtures depending on material types and frequency
ranges.
Materials measurement suite
Material
types
N1500A
Liquid
16452A
Gel
Liquid test fixture
Dielectric probe
85070E
Semi-solids
(Powder)
Solid
16453A
16451B
Substrate
Dielectric test fixture
Toroidal core
DC
16454A
1 kHz
1 MHz
10 GHz split
cylinder resonator
85072A
85071E-Exx
Split post dielectric resonators (SPDR)
Magnetic material test fixture
1 GHz
10 GHz
20 GHz
50 GHz
100 GHz
Frequency
Figure 30. Materials measurement fixtures
Keysight also offers powerful software to help customers automate complex
permittivity and permeability measurement analysis. The N1500A materials
measurement suite streamlines the process of measuring complex permittivity
and permeability with an Keysight network analyzer. The easy-to-use software
guides the user through setup and measurement, instantly converting
S-parameter network analyzer data into the data format of your choice and
displaying the results within seconds. Results can be charted in a variety of
formats:
er’, er”, tan δ, μr’, μr”, tan δm and Cole-Cole
A variety of measurement methods and mathematical models are provided to
meet most application needs.
A free space calibration option provides Keysight’s exclusive gated reflect line
(GRL) calibration for measuring materials in free space. The arch reflectivity
option automates popular NRL arch method for measuring reflections off the
surface of a sample. The resonant cavity option offers the highest loss tangent
accuracy and resolution. The coaxial probe option automates the dielectric
probe kit measurements.
Figure 31 summarizes Keysight fixtures and compatible measurement
instruments.
●
85071E Exx Split post dielectric resonators (SPDR)
●
●
●
85072A
10 GHz split cylinder resonator
●
16451B
Dielectric material test fixture
Method
E4991B
●
4285A
FieldFox
●
Dielectric probe kit
E4980A
ENA
●
85070E1
4294A
PNA
33 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
Coaxial probe
Resonant cavity
Resonant cavity
16452A
Liquid test fixture
16453A
Dielectric material test fixture
●
16454A
Magnetic material test fixture
●
●
●
●
Parallel plate
●
●
●
Parallel plate
Parallel plate
●
Inductance
Figure 31. Keysight Instruments and fixtures
References
1. 85070E with E4991B requires N1500A-004
software revision v2014-20150115 and
above.
1. Solutions for Measuring Permittivity and Permeability with LCR Meters and Impedance Analyzers, Keysight Literature Number 5980-2862EN, October 28, 2008
2. Understanding the Fundamental Principles of Vector Network Analysis, Keysight literature number 5965-7707E, December 12, 2012
3. Exploring the Architectures of Network Analyzers, Keysight literature number
5965-7708E, December 6, 2000
4. Applying Error Correction to Network Analyzer Measurements,
Keysight literature number 5965-7709E, March 27, 2002
5.D. V. Blackham, R. D. Pollard, An Improved Technique for Permittivity
Measurements Using a Coaxial Probe, IEEE Trans. on Instr. Meas., vol. 46, No 5,
Oct. 1997, pp. 1093-1099
6.Technical Overview, Keysight N1500A Materials Measurement Suite, Keysight
literature number 5992-0263EN
7. Technical Overview, Keysight 85070E Dielectric Probe Kit, Keysight literature number 5989-0222EN
8. Online Help for N1500A Materials Measurement Suite, http://na.tm.keysight.
com/materials/downloads.html
9. ASTM Test methods for complex permittivity (Dielectric Constant) of solid
electrical insulating materials at microwave frequencies and temperatures to
1650°, ASTM Standard D2520, American Society for Testing and Materials
10. Dielectric constant measurement of solid materials using the 16451B dielectric
test fixture, Keysight literature number 5950-2390, September 1998
11. ASTM, “Test methods for A-C loss characteristics and permittivity (dielectric
constant) of solid electrical insulating materials,” ASTM Standard D 150,
American Society for Testing and Materials
12. Technical Overview, Keysight 85072A 10GHz Split Cylinder Resonator.
Keysight literature number 5989-6182EN, May 8, 2012.
13. M.D. Janezic, ‘‘Nondestructive Relative Permittivity and Loss Tangent
Measurements using a Split-Cylinder Resonator,’’ Ph.D. Thesis, University of
Colorado at Boulder, 2003.
14. IPC TM-650 2.5.5.13 Relative Permittivity and Loss Tangent Using a SplitCylinder Resonator
15. Split Post Dielectric Resonators for Dielectric Measurements of Substrates.
Keysight literature number 5989-5384EN, July 19, 2006
34 | Keysight | Basics of Measuring the Dielectric Properties of Materials - Application Note
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© Keysight Technologies, 2013-2015
Published in USA, January 27, 2015
5989-2589EN
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