Institut für Mikro- und Nanomaterialien

Praktikum zur Vorlesung „Werkstoffe der Elektrotechnik“

Versuch: IMPEDANZSPEKTROSKOPIE

Update: 15. Januar 2010 Autor: Benjamin Riedmüller

Contents

1 Introduction 2

2 Dielectric Behaviour 3

2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Dielectric Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Frequency behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.1 Dipole Polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.2 Atomic Polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.3 Electronic Polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Material Behaviour 7

3.1 Ferroelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Ion Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Glass-Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Interpreting the Impedance 13

4.1 Equivalent electrical curcuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Impedance and complex permitivity . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Measurements 15

5.1 Measurement Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.2 Measurement Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.3 Software Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.4 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.4.1 Reference Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.4.2 PVAC Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.4.3 BaTiO Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6 Preparation 19

6.1 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.2 Reference literature for this lab course . . . . . . . . . . . . . . . . . . . . . . . 20

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Chapter 1

Introduction

Impedance Spectroscopy is a very powerful and also easy tool to study the physics of dielectricmaterials. With a simple measurement of the electrical impedance Z = U

Iin dependance of

frequency and/or temperature macroscopic information like permittivity ε and the conductivityσ become available. Today one is able to measure in a very frequency range from 10−4 Hz up to1010 Hz and temperature regions from 100 K up to 800 K. This broad range of operation makeimpedance spectroscopy one of the most important research methods for dielectric materials.Scanning a wide frequency range gives detailed information about molecular or atomic pro-cesses like the movement and time behaviour of permament dipole moments or phenomena likeion conductivity or ferroelectricity. With these results even structural information like phasetransitions or grain size can be studied. One can say that impedance spectroscopy has becomeone of the most important research tools for dielectrics in the last decades.

During this lab experiment we will study the basic principle of this method with differentmaterials and will also give a first interpretation of the assembled values.

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Chapter 2

Dielectric Behaviour

2.1 General

Impedance spectroscopy means measuring the electrical complex resistance (Impedance Z)while changing the electrical frequency (spectroscopy). Since a lot of physical effects are stronglydependant on temperature one can also vary the materials temperature in order to study thetime and temperature behaviour at the same time. The dielectrical behaviour can be regardedas a response of a material to an externally applied electrical field at a certain frequency. Thefield can either interact with permament dipoles in the material or induce them itself. Allthese interactions are summarized in the complex permittivity εr which defines the strengthof this interactions. While measuring, all the different interactions occur at the same time, soone cannot seperate them from each other with one single response. With either changing thefrequency or the temperature one is able to seperate all the different effects on polarisation andattribute them to their physical background.

2.2 Dielectric Theory

Dielectric materials are normally described by the so called polarisation which is their responseto an externally applied electrical field. This is described by the basic material equation:

~P = ε0χ~E (2.1)

with ~D the electrical displacement field, ~P as the polarisation, ~E the external electric field,ε0 the vacuum permittivity and χ the susceptibility which depends on the strength of thematerial’s response. Please note, that the following is only valid for a linear material that canbe described by equation 2.1.

With the elementary material equation

~D = ε0 ~E + ~P (2.2)

Combining these two equations we get

~D = ε0(1 + χ) ~E = ε0εr ~E (2.3)

with ε0 as vacuum permittivity and εr as permittivity of the material.

It should be noted that this equation is only valid for the macroscopical description of amaterial. For describing the field at a certain position ~r inside the material one would also haveto take the influence of next neighbour atoms into account.

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2.3 Frequency behaviour

There are three different types of internal polarisation which are normally described in lit-erature. They all are different in their frequency response. This effect has its origin in thedifferent masses that try to follow the electrical field. In order to get a brief overview thedifferent contributions to polarisation will be explained in the following.

Figure 1: Different sources of polarisation [1]

2.3.1 Dipole Polarisation

Some materials have molecules which have a permanent dipole moment ~p (like the watermolecule). These molecules will feel a certain force when an external field is applied. Thisforce will try to align them with the external field. It is easy to imagine that whole moleculescan follow only at low frequencies because of their huge masses compared to single atoms orelectrons. Dipole Polarisation can be described by the so called Debeye behaviour. Hencethis will be important for the lab course you will have to get a closer look on this description.

2.3.2 Atomic Polarisation

When there are ion bounds in the material (like Na+Cl−) the electrical field will increaseor decrase the distance between these ions due to the Coulomb force. With a change in thedistance their dipolar moment and so the polarisation will be either increased or decreased.These atomic or ion polarisations can respond to higher frequencies than the molecular ones.Atomic polarisation can be described as a damped harmonic osziallatior with a certain resonancefrequency.

2.3.3 Electronic Polarisation

Every atom consists of a positivly charged core and a negativly charged electron cloud whichsurrounds the core. When these electrons feel an electrical field the electron cloud is moved ashort distance due to the coulomb force and so it leaves its center position. This movementwill lead to a polarisation which can follow higher frequencies than the other polarisationsbecause of the very small mass of the electrons. Like the atomic polarisation the electronic oneis also always described as a harmonic oscillatin with a certain damping factor. The measuredpermittivity now consists of all these contributions to the polarisation. When now measuringthe polarisation or the permittivity of a material the value will depend on the applied frequency.As described above, both will decrease with increasing the frequency.

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Taking all these effects into account the polarisation and so the permittivity will have thefrequency behaviour shown in figure 2.

One can see a difference in the materials’ response from dipole, atomic and electronic polarisa-tion as described above.

Figure 2: Frequency behaviour of complex permittivity [1]

As the figure shows the permittivity is always described by real and an imaginary part. Thereal part of the permittivity is directly connected with the polarisation effects as describedabove and shows the proposed behaviour. In order to describe the meaning of the imaginarypart, a short calculation has to be made:

We define our charging current density as:

i =d ~D

dt= ε0εr ~E (2.4)

If we assume a harmonic oszillation of the electrical field ~E = ~E0ejωt we get

i =d ~D

dt=

d

dt

(ε0εr ~E0e

jωt)

(2.5)

With the assumption εr = ε′ − jε′′r we finally get

i = ωε0ε′′r~E + jωε0ε

′r~E (2.6)

As shown in equation (2.6) the imaginary part of the material directly is related to the elec-trical losses that occur in the material. This loss has its’ origin in the phase shift between theexternal electrical field ~E and the dielectric response ~D. ( ~D = ~Eejφ) At very low frequenciesthe dipoles follow the field whithout any time delay so that there is no phase shift. The higherthe frequency gets the more phase shift occurs between field and reponse to the field and so theloss due to this phase shift increases. When the frequency reaches very high values the dipolescan’t follow the field any longer which decreases the loss in the system. One measures the lossin the so called loss factor.

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It is defined as:

tan(δ) =IcIr

(2.7)

Figure 3 shows the vector model of the two currents in a dielectric system.

Re

Im

Ir

Ic

Figure 3: Current model in a dielectric

Here Ic is the displacement current and Ir is the loss current.

At low frequencies there also can occur DC loss due to leakage currents at the surface or thebulk material. This loss is not shown figure 2 and would lead to a higher ε′′ value in the lowfrequency range.

As one can see in figure 2 a drop in the real part of permittivity always leads to a loss peak inthe imaginary part. In our case ε′ and ε′′ are measured automatically at the same time. Butthese two values are not independent. If one of them is known over the whole frequency rangethe other part can be directly calculated with the Kramers-Kronig relation.

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Chapter 3

Material Behaviour

In this Lab Course we are going to study three different material properties. The first oneis the ferroelectricity, which is an effect that leads to very high permittivity (so called high-kmaterials). The second one is the glass transition that can occur at a certain temperatureTg. The last effect is the ion conductivity which will be studied in the ZrO2 : Y2O3 materialsystem which is therefore used as a oxygen pressure sensor in cars. After a short introductionto each of these effects it will be shown how these effects can be measured with the methods ofimpedance spectroscopy.

3.1 Ferroelectricity

Ferroelectricity can only occur in materials which have a permanent dipole moment. These ma-terials do have a remaining polarisation ~P after applying an electrical field, even when the fieldis removed after a certain time. One calls it ferroelectricity for the similarity to ferromagnetismwhich leads to a magnetisation ~M (similar to ~P ) without any externally applied magnetic field.One well-known ferroelectrical material is BaTiO3 (barium titanate). Its crystallographic or-der allows different positions for the Ti ions depending on the temperature. Below 120 CBaTiO3 is ferroelectric with a very high permittity. When heating up over 120 C it becomesparaelectric and the permittivity follows the Curie-Law.

ε′ ∝ C

T −Θ(3.1)

with C as a constant and Θ as the ferroelectric Curie temperature.

According to this the temperature between the two states is called the ferroelectrical CurieTemperature Tc.

The charging curve (P-E) of a ferroelectrical material is very similar to a ferromagnetic material.These materials will always show some hysteresis loop and a remanence polarisation (P(E=0)6=0). Both effects are shown in figure 4. With the high permittivity BaTiO3 is able to store a lotof energy when subjected to an electrical field. So it is often used for very high capacitances atlow frequencies (higher frequencies would lead to higher losses in the marial). BaTiO3 is alsooften used as a piezoelectric transducer which allows one to translate mechanical into electricalsignals. In this lab course we will only deal with its ferroelectric behaviour.

Because of the temperature dependance of this effect we will study the permittivity at differenttemperatures. According to the transition from ferroelectric to paraelectric at T = Tc weshould see a big peak in the permittivty near this temperature. At higher temperatures thepermittivity should follow the Curie Law as in equation (3.1)

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Figure 4: Crystallite order of BaTiO3 in the ferroelectric phase [1]

Figure 5: Charging curve of a ferroelectric material [1]

3.2 Ion Conductivity

In dielectric materials are no free or semi-free electrons like in metals or semiconductors. Theonly way to transport electronic charge is to move ions through the material. In a perfectcrystal one could not move any ions through the lattice; defects are required to obtain ionconductivity. They can be generated by thermal energy kT (Frenkel or Shottky defects) orexternally induced by doping. A very well known material system is ZrO2 stabilized with Y2O3

(Yttrium stabilized Zirconia). The Yttrium will sit at the lattice sites of Zirconia and suppliesthree outer electrons instead of the four outer electrons from Zr for binding to the oxygen. Forcharge neutrality there have to be some empty oxygen places which allows oxygen ions (O2−)to be transported through the crystal.

Ion Conductivity is described as:σ = eNµ (3.2)

where e is the elementary charge, N the ion density and µ the mobility of the ions.

The mobility depends on how easily the ions can diffuse through the material and is given as

µ =D

kT(3.3)

where D is the Diffusion constant, and kT the thermal Energy. An ion always needs a certainamount of energy for the diffusion process to change its position in the crystal. This energy iscalled activation energy Ea. Measurements have shown that the diffusion constant follows an

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Arrhenius law which is illustrated in figure 5.

D = Doe−EakT (3.4)

Combining these equations we get for the ion conductivity:

σ =eN

kTD0e

−EakT (3.5)

Figure 6: Ion conductivity as function of temperature

This oxygen ion transport is used to measure the oxygen pressure in the so called λ sensorwhich is used in the control system of catalytic converters

3.3 Glass-Transition

When a liquid is cooled so fast that it cannot reach any crystalline state it may undergo a glasstransition. Glasses are liquids that remain amorphous when cooled to the solid state. Whencooling throuh the so called glass temperature the viscosity of the material doesn’t change in adiscontinous way but continously. One way to measure viscosity in a qualitative way is to applyan external electrical field with a varying frequency. The lower the viscosity the higher the isthe frequency the molecules are able to follow. This time dependance is described by a timecalled the relaxation time τr which contains information on how fast the molecules get back toequilibrium state after being distorted. One defines the glass temperatur Tg as following:

µ(T = Tg) = 1012Pa · s (3.6)

with µ as viscosity of the material

Hence µ and τr are related to the same phenomena one could also define the glass temperatureas:

τr(T = Tg) = 100 s (3.7)

It should be noted that these values don’t have a well-defined physical meaning. But in thisregion especially the time constant τ can be measured without performing long time measure-ments.

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This relaxation time normally has a strong temperature dependance. In the simplest way itcan be described as

τr = τ0e− E

kT (3.8)

With this behaviour τr can be easily extracted by extrapolation of a τ(T ) diagram.

Especially in complex molecules like polymers the relaxation time shows a different temperaturebehaviour that can not be described by a thermally activated Arrhenius behaviour like inequation 3.9. For these molecules there is an equation called the Vogel-Fulcher equationwhich describes the temperature behaviour of the relaxation time by:

τr,V F = τ0e− E

k(T−TV F ) (3.9)

with TV F as the Vogel-Fulcher temperature

Both relations for the relaxation time are illustrated in figure 7

Figure 7: Arrhenius and Vogel-Fulcher description of the relaxation time [4]

In this lab course we will measure the dielectric behaviour of PVAC (poly venil acetate) whichis an organic polymer and which has a glass temperature that is near to room temperature.According to this we will see a change in the frequency response of our sample with increasingtemperature.

Figure 8: Chemical structure of PVAC (CH2CHOCOCH3) [3]

The dielectric behaviour near the glass transition can be described by the so called Debeyerelaxator. It is the easiest way to describe the frequency response of the orientation polarisationto an externally applied AC field.

In this description one gets:

εr = ε∞ +∆ε

1 + jωτr(3.10)

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Here ε∞ is the response at very high frequencies that are too fast for the permanent dipolesand ∆ε = ε0 - ε∞ where ε0 is the response to a DC field (frequency = 0).

By seperating real and imaginary part one gets:

ε′r = ε∞ +∆ε

1 + (ωτr)2(3.11)

ε′′r =∆ε(ωτr)

1 + (ωτr)2(3.12)

Equation 3.11 and 3.11 describe the frequency behaviour you can see in figure 2 as responseof the orientation polarisation This descrption is only valid for one single relaxation timeτr. When structures get more complicated (like polymers) this will lead to a peak broadeningthat can not be described by one single relaxation time. If measuring the dielectric responseof polymers one will always see a broadening in the curves which only can be explained bya distribution of relaxation times. Figure 9 shows the difference between ideal Debeyerelaxation and real measurements.

Figure 9: Ideal Debeye Behaviour and real measurement [2]

The real behaviour is then normally described by the empirical equations of Havriliak-Negami. They found out that one has to make a modification of the Debeye model.

They describe the complex permittivity as:

εrHN = ε∞ +∆ε

(1 + (jωτr)α)β(3.13)

with the two fitting parameters α and β in the range from 0 to 1.

(Setting α = 1 and β = 1 the Haviliak-Negami equation would lead to the Debeye model asdescribed by equations 3.11 and 3.11

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Figure 10 shows the influence of the two fitting parameters on the Havrillak-Negami functions:

Figure 10: Havrillak Negami Equations with different fitting parameters [2]

With changing α and β one can adjust the peak broadening and even the peak position so thatit fits ideally to the measured data.

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Chapter 4

Interpreting the Impedance

4.1 Equivalent electrical curcuit

As previously mentioned the electrical behaviour of such a material can be described with acomplex permittivity εr. In the simplest curcuit model description one could assume a resistanceand a capacitor in parallel. While the capacitor describes the displacement current the resistorstands for the losses that occur in our sample.

U

Ir Ic

Figure 11: Equivalent parallel curcuit for a dielectric material

Taken the sample dimensions into account one could easily get the permittivity behaviour fromthe measured impedance values.

The measured impedance will be a compex value of Z = ReZ + j Im(Z).A parallel curcuit is easily described with a complex admittance: Y = ReY + j Im(Y ) = 1

Z

The real part of the measured admittance now corresponds to the resistance (or conductance)as R = 1

ReYand the capacitance is C = - ImY

ω

4.2 Impedance and complex permitivity

Assuming a parallel circuit model one is now able to calculate the complex permitivity fromthe measured impedance values.

An ideal parallel plate capacitor can be described by:

C =ε0ε′rA

d(4.1)

⇒ ε′r =C

Cowith Co =

ε0A

d(4.2)

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For this calculation one has to know the thickness d and the electrical contact area A of thedielectric. They both are known within this lab so this calculation can be done.

In the same way one will find for the imaginary part of εr

⇒ ε′′r =G

ωCo(4.3)

One should remark that this is only one possibility for describing a dielectric response with anequivalent electrical curcuit. It is used in this lab course for simplifying the results. Usuallya more complex model of electrical curcuits for describing materials’ behaviour is applied likeseveral parallel curcuits in a row. With such a detailed description one is able to connect theimpedance behaviour to its’ real physical background like grain size or interlayer effects betweenbulk and contact material. For such research a large amount of temperature and frequency datahas to be recorded over a wide temperature and frequency range which can not be done withinthis lab course.

bulk grain boundaries

bulk-contacttransition

R bulkC R gbC R trC

Figure 12: Combination of RC circuits

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Chapter 5

Measurements

5.1 Measurement Hardware

For measuring the electrical impedance of our samples the ”HP 4192A Impedance Spectro-scope” is used. You can see it in figure 13

Figure 13: HP4192A Impedance Sepctroscope

It is able to measure the impedance in a range of 5 Hz to 13 MHz. One has to know thatmeasurements in the very high and very low frequency region always bring systematic errors bythe measurement setup itself. Namely the cables’ stray capacitances and the serial resistanceof them has to be taken into account at these frequencies. Our measurements will be donebetween 1 kHz and 1 MHz in order to avoid these effects. The measurement itself is run in theso called ”4 wire setup” which is illustrated in figure 1. With this technique one ensures thatmeasurements are not influenced by the cable and contact resistances which always occur in a”2 wire setup”

Figure 14: Four wire measurements

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The sample is mounted into a home made sample holder which is illustrated in figure 15. Formeasuring capacitance, values of at least several pF are needed which requires a large contactarea and a small sample thickness. The holder primarily consits of an upper electrode for theHIGH contacts (current and voltage) and a lower electrode for the LOW contacts (currentand voltage). The insulating upper part is made of ceramics. To measure the sample withonly two wires instead of four a connection kit from HP is used which provides a four pointmeasurement with two cables. The temperature of the sample can be controlled with a heatingplate from IKA which is able to reach temperatures up to 250 C. The temperature has tobe controlled during the experiment so that its influence on the measurements can be takeninto account. This is done by a K-type thermocouple which has a well known resistance vs.temperature behaviour. The resistance (and so the temperature) is directly measured by aKeithley multimeter.

LOW

HIGH

Figure 15: Sample Holder

Figure 16: Open Sample Holder

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5.2 Measurement Software

Because measurements are dependant on frequency and temperature, a lot of values have tobe recorded simultaneously. Due to this, a measuring software for this lab was programmedusing ”Labview” which controls the impedance spectrometer and records all needed values overa ”GPIB Bus”. There are also plots included which will show the results directly after themeasurement so that one is able to interpret the measurements. After the measurements aredone the results are saved in a text file so that the students are able to work with them athome for the lab report.

Figure 17: Labview Frontpanel (only example)

5.3 Software Options

• Impedance and Permittivity: Measuring impedance or permittivity at different fre-quencies and at a constant temperature. Frequency range can be either set to linear orlogarithmic scale.Input parameters: Start Frequency, End Frequency, Stray Capacitance, Sample thicknessand diameter, Frequencies per decade (log), Frequency step (lin)

• Timed Measurement: Measuring permittivity at a contant frequency for different tem-peraturesInput parameters: Start Frequency, End Frequency, Stray Capacitance, Sample thicknessand diameter, Frequencies per decade (log), Frequency step (lin)

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5.4 Measurements

5.4.1 Reference Measurement

As first step you will determine the stray capacitance of the measurement setup. Think of howyou could measure this capacitance with the given setup. (Hint: preperation question 1).

You will measure a parallel curcuit consisting of a capacitance with C = 100 pF and R =100 kΩ in the impedance mode. For your protocol these data have to be compared to thetheoretical curve you calculated for the lab.

5.4.2 PVAC Measurement

In the second step you will have to produce a PVAC sample on your own. Since PVAC is athermoplastic material you will be able to form it temperatures of 60 C - 70 C. Measure thediamater and the thickness of the sample. After preparing the sample you will measure it in atemperature range from 40 C - 70 C and in a frequency range of 1 kHz to 1 MHz.You will have to fit these data for your protocol in order to calculate the relaxation timeover temperature behaviour. With this you will be able to determine the glass temperature ofPVAC afterwards.

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Chapter 6

Preparation

These questions have to be worked out in written form by every student before the labcourse will take place. You are allowed to preparate them together but in the end every singleparticipant should be able to answer them. If some of the group members should be not wellpreparated the lab course will not take place for the entire group and you will have to comeagain at another day. If there are any problems with answering the questions please contactyour supervisor before the lab takes place.

6.1 Questions

1. Calculate the capacitance of a cylindrical capacitor (shown in figure 12) filled with airwith the following dimensions:

• Diameter D = 1,8 cm

• Height h = 1 mm

2. Calculate the complex impedance of a resistance R = 100 kΩ in parallel to C = 100pF.How does the Cole-Cole plot (imaginary part over real part) of this curcuit look like?

3. Why can we measure the dielectric behaviour only up to a frequency of ≈ 1MHz and notat higher frequencies?

4. What is the Kramers-Kronig relation and what it can be used for?

5. What is the so called ”Debeye behaviour” of dielectric materials? How can it be describedmathematically? What is the normal behaviour of ε′(f) and ε′′(f) and how can it beexplained? Where can you see the relaxation time in these characteristic curves? (assumea single relaxation time!) Why is the Debeye behaviour not seen in real measurements(especially in polymers)?

6. What is the glass transition? How can it be defined and measured with impedancespectroscopy? What will happen with the dielectric behaviour according to figure 2 whenincreasing the temperatur?

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Figure 18: Parallel Plate Capacitor

6.2 Reference literature for this lab course

•

Following literature has been used for creating this document:

• [1] Gerhard Fasching: ”Werkstoffe der Elektrotechnik” (Springer, Wien; Auflage: 3(2005))

• [2] Versuchsanleitung ”Dieleketrische Relaxationsspektroskopie” des Polymerpraktikumder Universitt Marburg (2005)

• [3] http://de.wikipedia.org/wiki/Polyvinylacetat

• [4] ”Versuch 19 im Fortgeschrittenenpraktikum”, Institut der Physik der Universitt Augs-burg

All pictures have been taken taken from these sources.

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