Understanding Dielectrics Electrostriction

7/29/2019 Understanding Dielectrics Electrostriction

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TX to Ae Ch 2. Contents >

2. Components and Materials: Part 5.Contents:

2-16. Understanding Dielectrics.2-17. A realistic capacitor model.2-18. Modelling Polarisation.2-19. Dielectric Absorption.

2-20. Ferroelectricity.

2-21. Electrostriction andPiezoelectricity.2-22. Electric and MagneticCounterparts.2-23. Dielectric Elastomers.

2-16. Understanding dielectrics:The energy stored in a capacitor is given by the expression [6]:

E = C V / 2 Joules

(where C is in Farads and V, the voltage at the terminals, is in Volts). If thedielectric is a vacuum, the energy stored is simply the potential energydue to the physical attraction between the two oppositely charged plates.If a material dielectric is inserted between the plates however, thecapacitance usually increases; which means that there is an increase inthe amount of energy which must be delivered to the capacitor in order toachieve a particular terminal voltage. The only possible conclusion is thatat least one new energy storage mechanism has come into play; and infact there are various mechanisms, which make differing contributions tothe capacitance depending on the nature of the dielectric material and thefrequency of the applied field [8][26].

The two fundamental energy storage mechanisms are known as'electronic polarisation' and 'atomic (or distortion) polarisation'. When adielectric is subjected to an electric field, an electronic re-distributionoccurs within the atoms or molecules which make up the material; themore loosely bound electrons being slightly displaced from their normalorbits to favour regions where they are least shielded from the positivelycharged electrode (to which the are attracted). This is electronicpolarisation. The positively charged core regions (nuclei) of the atomsmeanwhile are displaced slightly towards the negatively chargedelectrode, and this effect is known as atomic polarisation. In molecules,

since the degree of displacement is dependent on the electronicenvironments of the individual atoms, the effect of atomic polarisation isto stretch, compress, or bend chemical bonds. In ionic crystals (materialscomposed of positively and negatively charged ions, such as rock salt Na+Cl-), the effect (also known as ionic polarisation) is to shift the averagepositions of the atoms and so distort the crystal structure. The result, inany case, is a stress in the material which is released upon removal of theelectric field, i.e., the polarisation energy is given back when the capacitoris discharged. The electronic and atomic polarisation processes are almost

NH info: Components and Materials: Part 5. http://www.g3ynh.info/zdocs/comps/par

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perfectly elastic at radio frequencies; i.e., there is very little (but not zero)energy loss as the stored energy is released and the material is allowed toreturn to normal; which means that these processes make a significantcontribution to the capacitance of a capacitor, but only a smallcontribution to the loss resistance. Both of these polarisation processesare also effectively independent of temperature at radio frequencies.

By re-distributing the electrons, the effect of an electric field is to make

an atom or molecule appear slightly positive when viewed from somedirections, and slightly negative when viewed from other directions. Inthis condition, it is said to possess an induced dipole-moment. Manymolecules however possess a permanent dipole-moment, this being due toa naturally non-uniform distribution of electrons within the molecule, andare said to bepolar. In the absence of a field, polar molecules are eitherrandomly orientated, or if packed into crystals, are usually stacked in sucha way that the overall dipole moment is zero. When subjected to anelectric field however, they show a tendency to rotate so that the mostnegative end of the molecule points towards the positive plate and vice

versa. This is orientation polarisation. The process is illustrated below,where the symbol + represents a slight deficit of electron density, and -represents a slight excess of electron density (the deltas in this case arejust a standard notation meaning "a small amount of"; and have nothingto do with the loss angle ). In general, an excess of electron densityoccurs in the vicinity of an electronegative (i.e., electron-hogging) atomsuch as oxygen, nitrogen, fluorine, or chlorine.

The orientation process may involve overall rotation of the molecule aboutits centre of mass; or, especially in long flexible molecules such as thosewhich occur in plastics, it may involve internal rotation. The diagram onthe right (above) illustrates the latter process as it might occur in apolymer molecule with an occasional chlorine atom attached to thecarbon-chain backbone. In this case the chain swivels about several




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carbon-carbon bonds in order to allow the electronegative chlorine atomto move towards the positive electrode. This is arguably an atomicpolarisation mechanism, because it involves the stressing of chemicalbonds; but it also corresponds to a localised orientation within themolecule. Here we will regard internal rotation in large molecules as anorientation process, because it gives rise to a frequency and temperaturedependence of the dielectric constant which is similar to that associated

with overall rotation. Note however, that if every possible substitution sitein the molecule is occupied by the same type of electronegative atom, thenthe molecule will not be polar overall; which means, for example, thatinternal rotation polarisation will occur in partially chlorinated plasticssuch as PVC (polyvinyl chloride), but it will not occur in fully fluorinatedplastics such as PTFE (polytetrafluoroethylene, CF3-CF2-CF2-CF2- . . . . . -CF3).

Although polar molecules show a tendency to become aligned with theincident electric field, near-complete alignment does not occur except inthe presence of extremely strong fields. One reason for this incompletepolarisation is that the orientating effect of the field is in competition with

the randomising effect of thermal motions. It follows, that if the energy putinto orientation is dissipated as a result of thermal randomisation, thenthe real part of the dielectric constant (that which represents realcapacitance) is likely to reduce as the temperature is increased, and theimaginary part (that which represents losses) is likely to increase. Thus, ingeneral, the dielectric constant of a polar material will be temperaturedependent; although, as we shall see, the overall dependence is the sum ofthis and several other effects, and the actual temperature coefficient canbe either positive or negative.

Another factor which mitigates against the alignment of polarmolecules in an electric field is the viscosity of the medium. In effect, amolecule may need to push other things out of the way in order to changeits orientation, and it will take both time and the expenditure of someenergy in order to do so. In general, the molecules in gases are free torotate, the molecules in liquids are somewhat restricted, and the moleculesin solids are highly restricted to a degree which depends on the internalstructure of the material. The fact that it takes time to rearrange themedium means that there will be afrequency dependence of the dielectricconstant, i.e., if the electric field changes too rapidly for the orientationprocess to make any headway, then orientation polarisation will be unableto make a contribution to the real part of the dielectric constant. Put

another way, there will be a cutoff frequency, above which the capacitanceof a capacitor with a polar dielectric will decline [27]. This cutofffrequency incidentally, is likely to vary with temperature; since anincrease in temperature usually involves expansion and a consequentreduction in the restrictions placed on internal motion. It follows also, thatif a polar material is compressed, there will usually be a reduction ininternal freedom; and so the dielectric constant may vary with pressure.

In fact, all polarisation mechanisms have a cutoff frequency, which isrelated to a quantity known as the relaxation time for the process. The




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relaxation time is the time it takes for a displaced particle to return to thevicinity of its normal position after removal of the electric field, and isrelated to the reciprocal of the natural resonant frequency for thedisplacement. In effect, if an internal motion in the material is stimulatedby a field varying at less than its resonant frequency, then the motion cantrack the changing field and energy can be stored in the spring-like forceswhich restore the particles involved to their normal positions when the

field is removed. If the motion is stimulated at greater than its resonantfrequency however, then it cannot keep up, and there will be no effectivedisplacement and hence no capacitive energy storage. If the motion isstimulated at its resonant frequency, then its reactance will disappear (i.e.,the motion will be in phase with the applied field), but its degree ofexcitation will be at a maximum; in which case it will be maximallyefficient at absorbing energy from the field and donating it, throughfrictional losses, to the general thermal energy of the material. Thus, if thefrequency of the applied field is increased, and should happen to sweepthrough the effective resonant frequency of some polarisation process,

there will be a drop in the real part of the dielectric constant r', and acorresponding peak in the dielectric losses in the region where the fall-offis occurring. Thus it transpires, that the real and imaginary parts of thedielectric constant cannot vary independently: they are, as was impliedearlier, different aspects of a single physical phenomenon; and ouridentification of permittivity as a complex quantity is found to befundamental. For those who wish to pursue the matter further, therelationship between ' and " is described by equations known as theKramers-Kronig relations [1][8][9]. These relationships have profoundimplications for circuit theory because they arise from considerations ofcausality; for which reason they also imply constraints on the real andimaginary parts of any impedance function. Ultimately, every electricaland optical phenomenon can be understood in terms of the scattering ofelectromagnetic radiation by the arrays of oscillators of which matter iscomposed; and since scattered radiation cannot overtake incidentradiation, there are natural restrictions on the relationship betweenresistance and reactance. The most obvious restriction is that impedancemust always lie in the right-hand side of the complex number plane, i.e.,resistance is positive; but a more subtle implication, which underpins thetechnique of circuit analysis, is that all impedances can ultimately berepresented by networks of idealised positive resistances, capacitances,

and inductances.The behaviour of the dielectric constant is illustrated below for a

hypothetical material with only three resonance regions in its spectrum,all being somewhat more conveniently distinct than is usual. This is thegeneral form however; and it allows us to make our first properconnection between electricity and general optics, by noting that theimaginary part of the dielectric constant, r", represents the absorptionspectrum of the material.




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In the diagram above, which represents an example dielectric rather thanall dielectrics; the ability of the material to store energy as a result oforientation polarisation is shown to collapse as we sweep through theradio-frequency region of the spectrum. The corresponding peak indielectric losses is shown to be broad because, in reality, particularly insolid and liquid materials, the resonances involved have low Q, and therewill be a wide range of relaxation times due to the effect of thesurrounding medium on the motions of the particles involved. There mayalso be several distinct steps in r', due to the collapse of differentorientation processes at different frequencies (e.g., internal as well asoverall rotation); and correspondingly, several bumps in r", all of whichmay overlap to some extent. If, on the other hand, the material iscompletely non-polar, there will be no capacitance due to orientationpolarisation, and this situation is indicated by the dashed linecorresponding to a value of r' of about 2.5.

What we see, in the idealised spectrum above, is that non-polarmaterials give a modest increase in capacitance when inserted betweenthe plates of a capacitor (typically about 2 - 2 times for solids), and thecontribution remains almost constant until the frequency moves into theinfrared region. Polar materials, on the other hand, give a much greaterincrease, but this may vary with frequency and temperature. What the

diagram does not show, is that the range of frequencies over which anorientation process drops out of the dielectric constant, and thecontribution to the capacitance below this range, may vary enormouslydepending on the material. In general, the drop-out range is in themicrowave region for polar liquids, but may lie anywhere from millihertzto microwave for solid polar materials. The contribution to the capacitancedue to orientation polarisation may also vary from a few percent to afactor of about 100, depending on the size of the dipole moment and theextent to which the orientation process can overcome thermal effects and




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environment. Thus there are many 'good' polar dielectrics, with amoderate contribution from orientation polarisation which is sustainedover a very wide frequency range. In general however, the larger thecontribution, the greater the losses when the useful frequency rangecomes to its end.

The atomic polarisation process generally drops out of the dielectricconstant in the infrared, and the corresponding peak in losses is the

vibrational absorption spectrum of the material. This effect, being outsidethe electrical frequency range, must be measured using optical methods.In an infrared spectrometer, the absorption region of a molecular mediumcan be resolved into separate peaks for the various internalbond-stretching and bending processes. In crystals, the absorption regioncan be resolved into the various lattice vibration modes.

Higher still in frequency: there is another absorption region associatedwith the collapse of the electronic polarisation process. This usually occursin the ultraviolet, but some materials have absorptions which begin in thevisible part of the spectrum. The dropping-out of electronic polarisation

corresponds to the electronic spectrum of the material; and once again,this time with the aid of a UV-visible spectrometer, it may be resolved intoseparate peaks for the various electronic energy transitions which maytake place.

Should any part of the absorption spectrum of a material encroachupon the visible range, then the material will appear to be coloured. Thehuman eye samples the relative intensities of light at three wavelengths,these being approximately 600, 540, and 450 nanometres. The

corresponding frequencies are 51014Hz (red), 5.61014Hz (green), and

6.7&times1014Hz (blue). The eye perceives colour in transparent objects

when one or two of these frequencies is attenuated relative to the ambientillumination. The slight greenish tinge of soda glass, for example, is due tothe tail-end of the vibrational spectrum attenuating some of the red (lowfrequency) content in the transmitted light, and the run-up to theelectronic spectrum absorbing some of the blue (high-frequency) content.

It is no coincidence that James Clerk-Maxwell was also studying thevisible spectrum and the human perception of colour at the time when hedeveloped the electromagnetic theory. By showing that electricity is a formof light, he established a definite relationship between the refractive indexin optics and the dielectric constant obtained from electrical

measurements. We will not go into the details here; but it is necessary tomention the relationship in order explain the use of the term "dispersion"in the discussion of dielectrics. A glass prism (i.e., a block of opticallytransparent dielectric material) will split or "disperse" white light into itscomponent colours. Dispersion occurs because the refractive index isdifferent for different colours (i.e., different frequencies) of light.Refractive index is not synonymous with permittivity, but it is linked to itby simple mathematical function (in non-magnetic materials it is simplythe square-root ofr); and so it is legitimate and also conventional todescribe a region of the spectrum where the dielectric constant is changing




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as a dispersion region. All transparent dielectrics (except vacuum) haveat least a weak dispersion region in the visible part of the spectrum, thisbeing the precursor to the collapse of electronic polarisation which occursin the ultraviolet. Polar solid and liquid dielectrics often have dispersionregions in the RF or microwave parts of the spectrum, this being due to thecollapse of orientation polarisation.

2-17. A realistic capacitor model:In formulating component models, it is important to bear in mind thatresistance, capacitance, and inductance are not human inventions. Theyare manifestations of the basic laws of nature which join together in theelectromagnetic theory; and the fact that we can make resistors,capacitors, and inductors, which very nearly isolate these three properties,tells us that nature has not tried very hard to prevent us from discoveringthe underlying rules. As we try to push our passive electronic componentsto work at ever higher frequencies however, we find that the simpledescriptions break down; and ultimately we are forced to recognise all

electrical phenomena as the interaction between fields and matter. Thequestion which then arises is that of; "how do we translate the physics intoa practical circuit theory?" The answer is no secret, and it has been statedin various ways in the preceding sections, but having taken a look at whatgoes on inside matter, we are in a position to state it in its mostfundamental form. The pure component descriptions, which turn out to betoo simple to describe real components, fail because there is a minimumunit of circuitry; this minimum being best described as a scatteringelement.

One of the fundamental tenets of the electromagnetic theory is that

charges lose energy (i.e., radiate) when they are accelerated. Deceleration,incidentally, is just negative acceleration, and so the term 'acceleration'should be taken to include it. Time-varying electromagnetic fields causecharged particles to accelerate, and so every particle affected by a fieldabsorbs energy from the field and re-radiates it. This absorption andre-radiation process is known as scattering. The result is that radiationcannot pass through matter unhindered, and the resulting field is the sumor 'superposition' of all of the fields due to the incident and scatteredradiation. Later on, we will see how scattering accounts for the apparentreduction in the speed of light which occurs when radiation travels in adielectric material, but in the present context, we are interested in the factthat the charged particles are bound together into structures (atoms,molecules, crystals, etc.) which undergo natural oscillatory motion. Thismeans that natural scattering elements contained within matter areresonators.

In order to make the connection between basic physics and circuittheory, we need to recognise that scattering works on any scale; i.e., anassembly or collection of scattering elements is also a scattering element.The most graphic illustration of this point is given by the Yagi antenna,which can be understood as a giant form of dielectric. In this case, there is




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usually one driven element or wave launcher, and a collection ofscattering elements which absorb energy from the incident field andre-radiate it. The principal difference between this and a microscopicscattering array is that we can alter the resonant frequencies of theelements by altering their lengths, thereby tailoring the phaserelationships between the incident and scattered waves so that we mayachieve addition in a particular direction, and a reasonable degree of

cancellation in all other directions. This is just an obvious example ofscattering however, because the concept of scattering provides us with theonly reasonable way in which we can explain how the device works.Ultimately we must recognise that all electrical circuits are arrays ofscattering elements which, by appropriate choice of phase relationships,allow us to direct energy from one place to another. This allows us betterto understand that energy is not confined by the circuit, but merely guidedby it; and while this might seem like an alarming extension to simplecircuit theory, it will not lead us into any great theoretical difficulty. It isjust a reiteration of a point made earlier, which is that there are

constraints on the impedance versus frequency characteristics of anyphysically realisable device. The overall constraint is: that if a proposeddevice doesn't either scatter or completely absorb radiation, it can't bemade (the only possible non-scattering non-absorbing 'device' is avacuum). We can therefore deduce that the minimum circuit element musthave at least one resonant frequency, and it will in general have somelosses; which means that it can only be modelled by a network containingat least one pure resistance, one pure capacitance, and one pureinductance. We knew that already of course, but what is less obvious isthat we can chop the circuit down into smaller pieces, and the same basicrules will continue to apply. This means that the impedance-relatedconsequences of scattering on a microscopic scale can still be modelled byidealised LCR networks.

In the previous section we showed how the real and imaginary parts ofthe dielectric constant vary with frequency, and explained this behaviourin purely physical terms. Now however, we may presume that all of thepolarisation processes can be represented by LCR networks, orcombinations thereof; and so we may deduce an equivalent circuit which,as it will turn out, behaves in almost exactly the same way as a practicalcapacitor (see below). The resulting model is still not completely generalfor two reasons: firstly, it neglects non-linear effects, e.g., it ignores the fact

that components can be destroyed by exceeding their power or voltageratings; and secondly, it assumes that there is no time difference betweenevents occurring in different parts of the component, i.e., it does notinclude transmission line effects. It is however, good enough to modelphysically small components (less than 0.1 in any dimension); and it willsuffice for its purpose, which is to show how the processes occurringwithin the capacitor contribute to its impedance.




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The elements on the left hand side of the diagram, L S, Rts, and C0, are ofcourse the impedance contribution due to the circuitry and the capacitorplates. This contribution is not properly modelled, because LS and Rts varywith frequency and so are not pure. We will set this limitation asidehowever, because we are primarily interested in the properties of thedielectric, which are divided into conduction and polarisation processes.

Several conduction processes are shown, because several conduction

processes are possible. Ordinary conduction involves electrons, but it isalso possible for a current to be carried by ions (i.e., atoms or moleculeswith one or more missing or additional electrons), and in semiconductingmaterials, by holes. Different current carriers have different mobilities;and limited mobility has the effect of reducing the conductivity due to aparticular carrier as the frequency is increased. This effect is modelled bya series inductance; which causes the impedance of a particularconduction process to rise with frequency, and its contribution to thecapacitor impedance to fall. This inductance, of course, represents themagnetic field due to the conduction process. The mandatory capacitance

associated with conduction is provided by C0, i.e., it is associated with theelectric field which drives the process.

In the previous section, we stated that polarisation gives rise to abehaviour of the dielectric constant whereby a drop in capacitance isalways associated with a peak in dielectric losses. This behaviour isperfectly modelled by a series LCR network in parallel with a capacitance.The properties of an LCR network are always that, at low frequencies, thecapacitive reactance dominates to the extent that the resistive andinductive contributions are negligible, and the network is to all intents andpurposes a pure capacitor. As the resonant frequency is approached

however, the impedance of the network falls as the inductive reactancestarts to cancel the capacitive reactance. This causes the current to rise fora given applied voltage; and hence causes the power dissipated in the lossresistance to increase, reaching a peak at the effective resonant frequency.Above the resonant frequency, the impedance of the network increasesagain as inductive reactance takes over, and the power dissipated in theloss resistance declines. In modelling terms, the low frequencycontribution to r' is set by choosing the capacitance value Cpn (forpolarisation process n), the peak contribution to r" is set by choosing theresistance Rpn, and the resonant frequency is set by choosing the




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inductance Lpn. We will demonstrate all of this in the next section; byconverting the model into an expression for the dielectric constant, andthen plotting it as a set of graphs.

2-18. Modelling Polarisation:In the capacitor model given above, the elements representing conductionand polarisation constitute an equivalent circuit for the dielectric. What

they really represent however, is a mathematical function describing thefrequency dependence of the dielectric constant. Here we will explore thisfunction for the case of a simplified version of the model; which has noconduction processes (infinite resistivity), and only one polarisationresonance in the frequency range of interest. This model, despite thesimplification, can give a perfectly adequate description of a dielectricwhich is a good insulator, and which has a polarisation process with areasonably narrow range of relaxation times. It therefore applies to asurprisingly large variety of capacitive devices provided that thefrequency range chosen is not too wide. The technique required for the

analysis however, is perfectly general; and can easily be extended to coveran arbitrarily wide frequency range because it resolves the dielectricconstant into a series of additive terms.

In the simplified model (right), C0 is, as usual,the capacitance of the capacitor without adielectric, and the series LCR networkrepresents the polarisation process to besimulated. Other polarisation processes (e.g.,atomic and electronic) are assumed to havedispersion regions at frequencies well above

the range of interest, and so (as we willdemonstrate) can be represented by a singlecapacitance C.

In order to find an expression for the dielectric constant, we mustequate the impedance of a capacitor expressed in terms of the dielectricconstant to the impedance of the model network. The impedance of acapacitor ZC is equal tojXC, where Xc=-1/(2fC), and C=C0(r'-jr"). Hence:ZC = -j / [2fC0 (r' -jr") ]and impedance of the model is:ZC =jXC0 // [Rp +j(XLp + XCp)] //jXC

(where " // " means "in parallel with").We could, of course, simply add C0 to C and treat them as a singlecapacitor; but we will not do so because, by keeping them separate we willend up with an arithmetic series for r in which the first term is 1 (anarithmetic series is an equation in the form A = A0 + A1 + A2 + A3 + ...... + An,where A is an arbitrarily chosen symbol). The first term will thereforerepresent the capacitance in the absence of a dielectric ( r=1 for vacuum),and all of the other terms will then be associated with the dielectric only.

We now need to extract r' and r" from the expression:-j / [2fC0 (r' -jr") ] =jXC0 // [Rp +j (XLp + XCp)] //jXC


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and the obvious approach would be to put both sides into the form 'a+jb'and equate the real and imaginary parts. It does not take muchinvestigation however, to verify that if the problem is attacked in that waythe algebra will soon become very messy. We will therefore redefine theproblem in terms of admittances (i.e., reciprocal impedances. See [ACTheory, 44] ).The impedance of the model network is:

ZC =jXC0 // [Rp +j(XLp + XCp)] //jXCand so the admittance is:YC = 1/ZC = (1/jXC0) + (1/[Rp +j(XLp + XCp)]) + (1/jXC)and recalling that 1/j = -j we have:YC = (-j/XC0) + (1/[Rp +j(XLp + XCp)]) + (-j/XC)and since XC=-1/2fC, we obtain:YC =j2fC0 + (1/[Rp +j(XLp + XCp)]) +j2fC

The impedance expressed in terms of the dielectric constant is:ZC = -j / [2fC0 (r' -jr") ]

which, upon inversion gives:YC =j2fC0(r' -jr")Equating this to the model gives:j2fC0(r' -jr") =j2fC0 + (1/[Rp +j(XLp + XCp)]) +j2fCand dividing throughout byj2fC0 gives:

r' -jr" = 1 +1

j2fC0 [ Rp +j(XLp + XCp) ]+

C

C0

All we have to do now is re-arrange the middle term, a process which wemay begin by using the substitutions 1/j = -j and 1/2fC0 = -XC0, hence:

r' -jr" = 1 +jXC0

[ Rp +j(XLp + XCp) ]+

C

C0

We then multiply the numerator and denominator of the middle term bythe complex conjugate of its denominator:

r' -jr" = 1 +jXC0 [ Rp -j(XLp + XCp) ]

[ Rp + (XLp + XCp) ]+

C

C0

Finally we put the middle term into the form A+jB:

r' -jr" = 1 +XC0(XLp + XCp) +jXC0Rp

[ Rp + (XLp + XCp) ]

+C

C0

Although we will explore it as it stands, note that this representation forthe dielectric constant can easily be extended to cover an arbitrarynumber or polarisation processes having dispersion regions in thefrequency range of interest. All that happens when another series-LCRnetwork is added to the model; is that another additive term, identical inform to the middle term in the expression above, appears in the equation.For this simplified example we have not bothered to number thepolarisation parameters, but assuming that we replace Lp, Cp, and Rp with




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Lpn Cpn, and Rpn (where n=1, 2, 3, ... etc.), and use a different n for eachterm, we can go on adding terms and adjusting the parameters to modeldielectric behaviour of any complexity. We can even model conductionprocesses by adding terms in which XCpn has been set to zero. The onlything we cannot do is set C0 to zero, since that would make |XC0| and hence|r| infinitely large; but setting C0 to zero is physically impossible becauseit leaves us with no mechanism for applying an electric field to the

dielectric. This incidentally, is not a failure of the model, it is simply alimiting condition: it tells us that when the impedance of the capacitorbecomes infinite, then Tan=r"/r' tends towards /, which means thatthe loss angle is can take on any value between the limits 0 and 90 (i.e.,the device is still notionally a capacitor, but we can't say what the lossangle is).

Equating real and imaginary parts of the expression above to separate r'and r" we obtain:

r' = 1 +XC0(XLp + XCp)

[ Rp + (XLp + XCp) ]+

C

C0

and

r" =-XC0Rp

[ Rp + (XLp + XCp) ]

Notice in these expressions, that at low frequencies, the quantity XCp willbecome very large and so will dominate the denominator of thepolarisation term. The result will be that r" will tend towards zero; andthe middle term in the expression for r' will tend towardsXC0XCp/XCp=XC0/XCp. Since XC0=-1/2fC0 and XCp=-1/2fCp, then XC0/XCp=Cp/C0.

This tells us that the assumption that all polarisation processes whichresonate above the range of interest can be represented by a term C /C0 isperfectly valid, because all polarisation terms take on this form when thefrequency is well below the resonant frequency.

The model is now ready for use, except that the parameters L p, Cp, andRp are somewhat unfriendly if we wish to think about dielectrics in anatural way. We will therefore define a more meaningful set ofparameters, from which Lp, Cp, and Rp can be obtained. These are: Cp/C0,which is the contribution the polarisation process makes to thepermittivity at low frequencies; fp, which is the resonant frequency of the

polarisation process; and Qp, which is the Q at resonance of thepolarisation process. fp is defined by the resonance formula, thus:fp = 1 / [ 2(LpCp) ]and Qp is the magnitude at resonance of either XCp or XLp divided by Rp,thus:Qp = 1 / (2fpCpRp )orQp = 2fpLp / Rp

In order to see how the model behaves, we must choose some parameter




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values. For the purpose of illustration, we will choose C0=10pF, Cp=20pF,C=10pF, and fp=10MHz. The choice of Cp and fp fixes Lp at 12.665H (aremarkably large inductance); and 2fpLp=795.77. Rp is obtained from theexpression Rp=2fpLp/Qp. Entering the expressions for r' and r" into aspreadsheet program, using the chosen parameters, and varying Rp to varyQp, yields the following graphs:

What we find is that when Qp is substantially less than 1, the modelproduces classic orientation polarisation behaviour. The only point to note

about this is that the peak in r" is skewed to the low-frequency side of fp,and those who care to experiment with the model will discover that, whilethe overall appearance of the graph remains unchanged, the peak movesto lower frequency as Qp is reduced. This Q-dependent movability of thedispersion region is one possible source of temperature variability of thedielectric constant (although temperature variation must also be expectedto alter the relaxation frequency and the L/C ratio, giving rise to a range ofpossible behaviours). The model is therefore successful, and is capable ofdescribing a wide variety of ordinary polar dielectrics; but what isinteresting is the behaviour it exhibits when Qp is allowed to become




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greater than 1.One of the intriguing properties of the series LCR model for a

polarisation process is that it predicts effects which we might not haveexpected, but which actually occur. If the Q of the network is high enough,there will be a region just above the resonant frequency where thenetwork places a very low inductive reactance in parallel with C 0. This canhave the effect of making r' less than 1, or even negative, provided that

the inductance can overcome the capacitance due to other polarisationprocesses. This effect does not show up at radio frequencies in normaldielectrics, because the polarisation drop-out regions are very broad; but itcan be seen very clearly at optical frequencies as a refractive index of lessthan unity on the high-frequency side of sharp vibrational and electronicresonances. Thus a humble engineer's model intended to describe radiocomponents predicts bizarre optical effects which really exist; whichhopefully underlines the point that the association we have made betweencircuit theory and polarisation is genuine. The model goes even furtherhowever, because when the Q is allowed to become very high, it describes

the mechanical resonances which occur in piezoelectric materials (whichwill be discussed in a later section). Thus all of the behaviours predicted bythe model turn out to have natural counterparts, which leads us to aninteresting philosophical point: The test of any theory lies not in itsdescriptive power, but in itspredictive power; and the theory in this case isthat the world of matter is composed of resonant scattering objects.

Returning now to the behaviour of ordinary dielectrics: recall thatmatter is composed of huge numbers of microscopic resonators; whereas,provided that all of the resonators vibrate at roughly the same frequency,the model somehow manages to describe them as a single resonator. Itmay seem strange that this is possible, but we can understand why byconsidering what happens when several identical series LCR networks areplaced in parallel. Each network can be regarded as an impedance:Zp=Rp+j(XLp+XCp), with an admittance Yp=1/Zp; and we can write anexpression for n such admittances in parallel as:Y = 1/Z = (1/Zp) + (1/Zp) + (1/Zp) + ........+ (1/Zp)hence:1/Z = n / Zpwhich, upon inversion, gives:Z = Zp / ni.e.,

Z = [ Rp +j(XLp+XCp) ] / n

which gives us two very important pieces of information: firstly, thefrequency at which (XLp+XCp) goes to zero is unchanged, and so theresonant frequency of a large number of identical series resonatorsconnected in parallel is the same as that of the individual resonators; andsecondly, the ratio of reactance to resistance of the individual resonators isthe same as the ratio for the array of resonators (i.e.,Xp/Rp=[Xp/n]/[Rp/n],because the n's cancel). This means that the Q of the resonance is the sameregardless of how many resonators are connected in parallel (provided




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that those resonators do not interact). Consequently (since the Q of aresonance dictates its bandwidth); connecting a number of identicalnon-interacting series-resonators in parallel can only change themagnitude of the resonance behaviour, it cannot change the bandwidth ofthe resonance. This brings us to the point, that the measurement ofquantities relating to the dielectric constant can be worked back to obtainresonance parameters for the individual scattering elements in the

dielectric. This is the basis of the various techniques of atomic andmolecular spectroscopy (which are our principal source of information onthe microscopic universe); where at radio frequencies the relevantmeasurements are the frequency dependencies of r' and r", and at opticalfrequencies the relevant measurements are the frequency dependencies ofthe refractive index and the optical absorption coefficient (i.e., the amountby which light of a particular frequency is attenuated on passing through asubstance).

There are considerably complexities involved in obtaining atomic andmolecular data from spectroscopic measurements, and readers will need

to delve deeply into the subjects of chemistry and physics if they wish toappreciate all of the issues. Here however, we can at least come to anunderstanding of what electricity is, and explain why a realistic model fora dielectric requires a startlingly large amount of inductance. What hasjust been stated is: that we can represent a polarisation process as an LCRnetwork, and if we can work out how many units undergoing that processare contained within a sample of material, then we can work out an LCRnetwork for the individual units. The units in question are often atoms ormolecules, but there are exceptions: In the case of a material such as PVC,for example, we may be interested not in the number of moleculespresent, but in the number of carbon-chlorine bonds. In the case of acrystal, we may need to consider something called the "unit cell", which isthe smallest group of atoms which can be regarded as a complete crystal,the crystal itself being a stack or array of these cells. The unit cell maycontain several molecules, or it may be a collection of ions, and it may bedifficult to decide whether or not multiple processes can occur within thecell. Crystals also have a habit of behaving as giant molecules, and so thereare polarisation processes such as piezoelectricity, which involve thecrystal as a whole. The list of exceptions could go on and on, but the pointis in the principle: which is that we can notionally divide matter downuntil we reach a single unit which, if we divide it further, will suddenly

cease to exhibit the behaviour we are studying. That unit will always be acollection of charged particles, and its behaviour in response to an electricfield is as though a number of LCR networks, at least one for eachresonance, has been placed across the plates of a capacitor. There is thennothing left to choose between the oscillating electric and magnetic fieldsassociated with the equivalent circuit, and the oscillating electric andmagnetic fields associated with the charged particles as they vibrate orrotate. The large inductance is therefore associated with charged particlesin motion, which, of course, constitute an electrical current even if all the




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particles do is move backwards and forwards. The capacitance, as wealready know, is associated with the displacement of charges from theirequilibrium positions, and the resistance is associated with the leakage ofenergy away from the process which is being stimulated. Thus we have thecomplete connection between electrical (and optical) phenomena andmechanical processes involving charged particles. In any mechanicalsystem, the total extractable energy is the sum of the potential energy and

the kinetic energy at any instant (kinetic energy is the energy due tomotion). When a system oscillates, kinetic energy is at its maximum whenthe particles are moving fastest (as they pass through and overshoot theirequilibrium positions), and potential energy is at its maximum when theparticles come to a momentary standstill at maximum displacement (thinkof a weight on a spring). If such a system is represented by an electricalnetwork; the instantaneous potential energy term is the energy stored inthe capacitance: Epotential=CVC/2, and the instantaneous kinetic energy is theenergy stored in the inductance Ekinetic=LIL/2. In any system involvingalternating electric and magnetic fields, there must be constant trading

between potential and kinetic energy, and so inductance and capacitanceare inseparable on a fundamental level. This is, of course, why we cannotrepresent a dielectric correctly without including inductance, and wecannot create a circuit which does not include both inductance andcapacitance.

Before we move on to the more practical aspects of the polarisationtheory, it is perhaps appropriate to offer some kind of conclusion to thediscussion so far; and the point is this: people routinely use componentmodels which omit either inductance or capacitance, and then presumethat the resulting simulations imply realistic behaviour. The problem isthat such models violate the Kramers-Krnig relations and thereby violatethe principle of causality. This does not mean that models which fail toreproduce the system resonances are no good, far from it; but they areonly applicable to narrow frequency ranges, and it is important to beaware of their limitations.

2-19. Dielectric absorption:An article on the subject of dielectrics cannot be complete without somemention of dielectric absorption. This is the propensity of certain types ofcapacitor to recharge spontaneously after having been discharged brieflyand then left open-circuit. The necessary condition for dielectricabsorption to occur is that the capacitor should have been maintained in acharged condition for a 'long' period prior to discharging.

The best demonstration of dielectric absorption is given by ordinaryaluminium electrolytic capacitors, i.e., capacitors in which the dielectric isa thin layer of aluminium oxide (Al2O3) formed on the surface of analuminium foil. If such a capacitor is charged via resistor from a DCsupply, the current will initially fall exponentially according to idealcapacitor behaviour, but there will remain a small residual current whichdwindles extremely slowly over a period of several tens of minutes. If the




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capacitor is allowed to charge for half-an-hour or so, then removed fromthe supply and discharged rapidly through a low-value resistor, theterminal voltage will typically rise to about 5% of the original chargevoltage after the discharging resistor has been removed.

Dielectric absorption is primarily a hazard for those who work withhigh-voltage power supplies, and necessitates the use of bleed resistors toensure that the equipment is safe after it has been switched off. It can also

be troublesome in accurate measuring circuits involving capacitors, suchas sample-and-hold amplifiers. In general, vacuum and gas capacitors canbe assumed to be free from the effect, but all solid-dielectric capacitorsmust be assumed to exhibit it to some extent until proven innocent.Polystyrene, for example, a nominally non-polar dielectric with r'=2.6,shows a voltage recovery of about 0.05%; whereas polyethylene-terephthalate (PET or Mylar), a slightly polar dielectric with r' of about 3,shows a voltage recovery of about 0.25% [29].

In view of the discussion in the preceding sections, it should be obviousthat dielectric absorption can be explained as a polarisation process with a

long relaxation time. We must therefore expect it to be evident in highlypolar materials with limited internal freedom (i.e., polar solids), and thisturns out to be correct. It is however, also present to a small extent innon-polar materials, and this warrants some additional explanation. Theclue, in this case, is that it varies according to the age, purity, opticalclarity, and water content of the sample. We may therefore attribute it towhat are known as "interfacial effects", which can be understood asfollows:

Imagine a material as a compressed collection of irregularly shapedgranules, with diffuse patches of resistive material trapped in the cracksbetween them. If the material is made into a capacitor and charged; therewill be an initial polarisation due to the overall electric field between theplates but, by leakage of charges through cracks and pores, it will bepossible for isolated pockets of dielectric to be subjected to electric fieldsstronger than the average. In effect, resistive inclusions will shunt parts ofthe dielectric, reducing the effective plate-spacing and thereby increasingthe capacitance if the charging period is long enough. Thus it will appearthat there is an additional capacitance, with a large series resistance, inparallel with the main capacitance. It is in this additional capacitance thatthe dielectric absorption charge is stored, and it will leak back into themain capacitance if the capacitor is briefly discharged and then left

open-circuit. This effect is indistinguishable from slow orientationpolarisation, except that it also occurs in non-polar materials.

2-20. Ferroelectricity:In some materials, there is a giant polarisation effect calledferroelectricity, which can give rise to an r' of up to 15000.Ferroelectricity is a property of crystals which have a permanent dipolemoment in the unit cell, and in which the conditions are such that thedipole moment can flip to a new orientation with minimal rearrangement




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of the atoms [26][30]. Extreme alignment with the field is possible, hencethe large r', but it is also possible to saturate the alignment process, i.e., toalign all of the dipoles which can be aligned; the result being that thecapacitance of a capacitor with such a dielectric will diminish as the field-strength is increased (i.e., the polarisability will diminish as the number ofdipoles available to be aligned is reduced). The ferroelectric effect istherefore non-linear, in addition to being temperature dependent.

Ferroelectricity, being the property of extremepermittivity due to flippingof electric dipoles, is so called by analogy with its cousin ferromagnetism,which is the property of extremepermeability due to flipping ofmagneticdipoles. Like ferromagnetic materials, ferroelectric materials have a Curiepoint, i.e., a temperature above (and sometimes below) which the effectdisappears. Known ferroelectric materials include: titanates, particularlybarium titanate, BaTiO3; tartarates (salts of tartaric acid) such as Rochellesalt; and potassium dihydrogen phosphate (KDP). The ferroelectricity ofbarium titanate is due to to the titanium ion Ti4+ having more room in theunit cell than it needs. This gives the ion six possible equilibrium positions

with low energy barriers between them, each resulting in a differentorientation of the dipole moment [30]. The Curie point (115C) is thetemperature above which the ion has more energy than it needs to hop thebarriers between between the energy minima, in which case it is no longerlocalised by the barriers and ferroelectricity disappears. Theferroelectricity of tartarates and KDP is due to the rearrangement ofhydrogen bonds (weak inter-atomic bonds) under the influence of anelectric field.

Barium titanate is the principal ingredient of dielectrics used in ceramiccapacitors; its large dielectric constant being of particular value in radio-frequency applications. Dielectrics with a high BaTiO3 content (known as'high k' dielectrics) enable a large amount of capacitance to be obtained ina very small volume, this being highly advantageous in fast-switching andradio-frequency applications because small size permits minimisation ofparasitic inductance. High k ceramic capacitors are therefore thepreferred types for RF coupling and decoupling applications, where lowinductance is more important than the actual capacitance value. Onefurther important property of BaTiO3 however, is that it can be used tomodify rather than dominate the characteristics of a ceramic, allowing theproduction of dielectrics with nearly-perfect RF properties. This is thebasis of the so-called 'Class 1' ultra-stable ceramic capacitor, which

combines low inductance with a very small Tan, minimal dielectricabsorption, and a very low temperature coefficient, making it the idealchoice for critical applications such as filters and oscillators. Modern class1 capacitors generally have RF characteristics which are better than thoseof polystyrene and traditional mica capacitors [29].

2-21. Electrostriction and piezoelectricity:It is a property of all dielectrics that, under the influence of a polarisingforce (i.e., an electric field), their physical dimensions change. This




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behaviour may be broken down into two classes, these beingelectrostriction andpiezoelectricity. The difference between the two effectsis that an electrostrictive material will either expand or contract under theinfluence of a field, regardless of the polarity of the field; whereas apiezoelectric material will alternate between expansion and contraction asthe field alternates in polarity. In either case, an alternating field willlaunch acoustic waves (i.e., pressure waves) in the material. Normally

these vibrations will be ultrasonic, but if the frequency is within the audiorange, and the effect is large enough, there will be an associated sound.The electrical consequences of the two phenomena are however,somewhat different.

Since the electrostrictive effect is dependent only on the magnitude ofthe field, an electrostrictive material can convert electrical energy intoacoustic energy, but it cannot convert acoustic energy directly intoelectricity. From an electrical point of view therefore, electrostriction onlygives rise to losses; and these losses depend on the mechanical resonancesof the dielectric and the degree of acoustic coupling to the surrounding

media. It might seem therefore, that electrostriction should give rise to apeak in r" without a corresponding drop in r', in which case the Kramers-Kronig relations will be violated. Causality however, must prevail; and sonature closes this loophole by ensuring that a dielectric can store moreenergy if it is allowed to change its dimensions than if it is prevented fromdoing so. A material therefore has a different dielectric constantdepending on whether it is clamped orfree. It is extremely difficult toclamp a material in such a way that it cannot expand or contract, but thereis no need. By the same argument as was used in the general discussion ofpolarisation; a material is 'free' in respect of a particular mechanicalresonance if it is stimulated at a frequency below that resonance, and it is'clamped' if it is stimulated at a frequency above that resonance (i.e., itcannot change its dimensions if the field is changing too rapidly for it tofollow). Thus there is a drop in r' on passing through an electrostrictionresonance, and the Kramers-Kronig relations are obeyed.

While electrostriction is, in one sense, just another type of polarisationprocess, it belongs to a special class because it has implications regardingthe linearity of capacitors. The point is that if the dielectric constantchanges due to a change in dielectric dimensions under the influence of afield, then the capacitance of a capacitor will vary according to the voltageappearing across its terminals. Thus electrostriction is a source of signal

distortion; but fortunately, in most dielectrics, the electrostriction effect isnegligibly small, and the amount of distortion produced is only of interestto extreme audiophiles (capacitor distortion effects in sensibly designedaudio amplifiers lie below the compact-disc noise floor, so you needspecially evolved ears to hear them). A large electrostriction effect occurshowever in ferroelectric materials; and such is the association, thatferroelectricity and electrostriction are sometimes taken to besynonymous. In this case, non-linearity due to electrostriction is not suchan issue, because ferroelectrics are already non-linear on account of the




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saturation effect mentioned in the preceding section.Piezoelectricity (pressure electricity) is a property of crystals which lack

a centre of symmetry in the unit cell; i.e., the unit cell is such that if it isrotated in any plane, it will not look the same again until it is returned toits original orientation. The effect of this asymmetry is that if the crystal issubjected to a stress, there will be a separation of the centres of mass forthe positive and negative charges, and the crystal will develop an overall

electric field. Piezoelectricity therefore differs from electrostriction in thatit is a reciprocal effect, i.e., piezoelectric materials can convert electricalenergy into acoustic energy and vice versa. Piezoelectricity is exhibited bynaturally occurring minerals such as quartz and tourmaline, and byartificially grown crystals such as Rochelle salt, lead niobate, lithiumsulphate, and ammonium dihidrogen phosphate (quartz crystals arenowadays also artificially grown). It can also be induced in ferroelectricmaterials which do not already exhibit it, by heating the material to atemperature above the Curie point and and allowing the material to coolwhile under the influence of a strong electric field (the required field

strength is about 50KV/cm for BaTiO3). Materials so treated are known aspoled ferroelectrics, and are used in a large variety of devices such as:ceramic transducers (microphones, sounders, seismographic sensors,sonar transducers, accelerometers, ultrasonic cleaning and weldingequipment), resonators, filters, and spark generators (self-lighting gasappliances, flintless cigarette lighters). In addition to barium titanate, theother commonly used ferroelectrics are: lead metaniobate (PbNb2O6), andlead zirconate titanate (PZT) [31][32][33].

When Piezoelectric devices are basedon naturally piezoelectric materials such a

quartz, they must generally beconstructed using a slab of material cutfrom a single large crystal; and theproperties obtained depend critically onthe orientation of the applied electric fieldwith respect to the orientation of the unitcell (which is by definition asymmetric)[34]. When poled ferroelectrics are usedhowever, the material can bepolycrystalline (i.e., an aggregate ofrandomly orientated microcrystals),because the piezoelectric property isinduced after fabrication. This means thatthe ceramic devices have ease ofmanufacture on their side, and havetherefore largely taken over from naturalpiezoelectrics; except where extremelyhigh Q is required . The maindisadvantages of poled devices are: thatthey may lose their properties with time

1950s vintage quartz resonators.Left: 100KHz. Right: 435KHz.

Glass tube diam: 17mm. B7G base.




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(the aligned dipoles eventually becomerandomised), albeit very slowly; and theycan be denatured by accidentaloverheating (such as during soldering).

It is at this point that we should deduce the equivalent circuits fordevices exhibiting electrostrictive or piezoelectric effects. The problem ofhow to do so will not detain us for very long however, because we have

already solved it. Piezoelectricity and electrostriction are polarisationeffects and so, in the case of two-terminal devices, can be describedaccording to our normal polarisation model as series LCR networks placedin parallel with a capacitor [34][35][36]. The big difference betweenpiezoelectrics and normal dielectrics is that Q of the polarisation

resonances can be very high (as much as 106 for quartz), which gives riseto immense excursions for r' in the resulting dispersion regions. Weshould also note that these resonances correspond to the various ways inwhich sound waves can travel in the material and arrive in phase back atthe starting point; and are therefore associated with bulk properties of the

material such as the physical dimensions, density, and stiffness (elasticmodulus). It is also impossible to obtain only a single mechanicalresonance: there will always be a series of overtone resonances ofprogressively diminishing Q associated with each fundamental resonance;and there may be numerous weak fundamental resonances due to thevarious ways in which acoustic waves can propagate in directions otherthan perpendicular to the electric field. An accurate model, of course,requires an LCR network for every one of these resonance modes. Thevarious wanted and spurious resonances of a piezoelectric device can beinvestigated most easily (at radio frequencies) by connecting it to an

antenna analyser (although careful tuning of the VFO is required, becausesuch resonances can be very sharp) [37].

Piezoelectricity and electrostriction, although often desirable, can alsoshow up a spurious circuit effects; particularly in association with devicesconstructed around ceramics and solid minerals. Such effects include:anomalous dispersion behaviour linked to dielectric dimensions (andtherefore not reproducible between different test-capacitor setups),microphony, and audible noise from dielectrics operating at audio andpower-line frequencies. Note that although electrostrictive materialscannot convert acoustic energy directly into electrical energy, they can still

exhibit microphony because pressure waves will modulate the dielectricconstant, causing terminal-voltage variation in charged capacitors. Notealso that, since electrostriction depends on field magnitude, there will betwo expansions for every cycle of the applied field, causing electrostrictivematerials to buzz at twice the excitation frequency. Electrostriction (aproperty of all ferroelectric materials) has a magnetic counterpart calledmagnetostriction, which is a property of all ferromagnetic materials, whichis one of the reasons why mains transformers buzz at twice the power-linefrequency [31].




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See also: Luminous Quartz Resonators.

2-22. Electric and Magnetic Counterparts:

Electric Magnetic

Capacitance C (Farads) Inductance L (Henrys)

Stored Energy E = C V / 2 (Joules) Stored Energy E = L I / 2 (Joules)

Capacitive reactanceXC=-1/(2fC) []

Inductive reactance XL=2fL []

Permittivity [Farads / metre]Vacuum permittivity 0=8.85418782pF/m

Permeability [Henrys / metre]Vacuum permeability 0=400 nH/m

Ferroelectricity Ferromagnetism, Ferrimagnetism

Electret Magnet (permanent)

Electrostriction Magnetostriction

Piezoelectricity Magnetostriction (Piezomagnetism?)

Note 1. An electret (permanently charged dielectric) is formed by melting a polar polymerand allowing it to cool under the influence of a strong electric field. Electrets are used tomake capacitor microphones (eliminating the need for a DC bias supply).Note 2. Piezomagnetism exists, but the term is not used. The effect is instead classified as atype of magnetostriction.

2-23. Dielectric Elastomers:When a capacitor is charged, a force of attraction exists between theplates. If a dielectric material is sandwiched between the plates, the plateswill squeeze the material. That much should be obvious; and so we haveanother mechanical property of capacitors which can, in principle, giverise either to spurious or to exploitable effects. What is perhaps lessobvious, is that a dielectric material, in increasing the capacitance, alsoincreases the force. Indeed, it is not so much that the plates squeeze thedielectric, but that the dielectric squeezes itself. The stress due to thiseffect, which is distinct from electrostriction or piezoelectricity, is knownasMaxwell stress.

In most cases the strain (dimensional change) due to Maxwell stress isvery small, and is subsumed by the much greater (but also usuallynegligible) strain due to electrostriction. In elastomers (rubbery materials)however; the dimensional change can be substantial if the field strength islarge enough, and the force in the event that such change is resisted maybe sufficient to cause mechanical problems.

In general, an engineer should simpy be aware of Maxwell stress whenproposing to subject elastomers (e.g., silicone and acrylic architecturalfillers and sealants) to strong electric fields. In recent times however, therehas been interest in the possibility of using dielectric elastomers to makeartificial muscles (linear motors), both as a source of novel devices and as




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a potential replacement for many of the bulky magnetic motor devices incommon use (e.g., loudspeakers and actuator solenoids). Crucial to thedevelopment electrostatic motors has been the development of elasticcapacitor electrodes, the issue being that as a block of elastomer shrinks inlength under the influence of a field, it also increases in area, and anelectrode is needed which can expand in area accordingly in order tomaintain a reasonably uniform field [38]. The resulting length change and

pulling force can then be substantial: approaching that of biologicalmuscle [39], using drive voltages in the order of 1 to 5KV. Mechanicalpower output, of course, is represented (as always) by the powerdissipated in a resistive component of the capacitor's impedance.

We should also note, of course, that when an elastic capacitor isstretched or compressed, its capacitance will change. This is a possiblesource of electrical instability (microphony, oscillator drift etc.), but canalso be used as the basis for the design of pressure sensors. It must also beadmitted at this point; that although capacitance change is not a basis fordirect conversion of mechanical energy into electrical energy, electrical

energy can nevertheless be extracted by the use of charge-pumpingcircuitry. This latter technique has, for example, been used to extractenergy from elastic capacitors fitted in boot-heels, the act of walkinghaving the potential to produce about 1 Watt of continuous power(sufficient to operate a piece of equipment such as a cellphone) [38].

Part 6 >>

D W Knight 2007.David Knight asserts the right to be recognised as the author of this work.

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