REMARKS ON THE MEASUREMENT OF STATIC PER- MITTIVITY ... · of supercapacitor [14]. Such studies...

Progress In Electromagnetics Research C, Vol. 33, 95–108, 2012

REMARKS ON THE MEASUREMENT OF STATIC PER-MITTIVITY THROUGH A CLASSICAL DESCRIPTION

B. Tellini1, * and M. Bologna2

1Department of Energy and Systems Engineering, University of Pisa,Largo L. Lazzarino, Pisa I-56122, Italy2Instituto de Alta Investigacion IAI, Universidad de Tarapaca, Casilla7-D, Arica, Chile

Abstract—In this paper, the concept of static electric permittivityand its measurement are discussed. A classical description ofpolarization via a harmonically bound charge model is revisited andthe evolution of the polarization concept in the presence of freeelectrons is shown. Various electrostatic problems are defined underideal conditions. The measurement procedures for characterizing thestatic permittivity of dielectrics and conductors via the measurementof induction-electric field, charge-potential difference and electrostaticenergy variation are discussed. Two basic experiments with a lossydielectric are described. In one case we reach an electrostaticequilibrium with an indeterminate solution. In the other case wedefine a magnetostatic problem. Finally, we comment on the caseof a laboratory experiment remarking on the proper use of the low-frequency limit of dielectric constant and showing experimental resultsperformed on a supercapacitor.

1. INTRODUCTION

Electric permittivity ε is a constitutive parameter relating themacroscopic fields induction D and electric field E. The assumptionthat D depends only on E holds for constant fields, while for variablefields a relationship between D and H is possible. Such an effectremains on the order of a/λ, where a is the atomic dimension and λ isthe field wavelength [1]. This effect is generally negligible, especiallyin slowly varying fields.

Received 16 June 2012, Accepted 18 September 2012, Scheduled 1 October 2012* Corresponding author: Bernardo Tellini ([email protected]).

96 Tellini and Bologna

Under the hypothesis of linearity, D and E generally show afrequency dispersion relationship. This means that ε varies as afunction of the angular frequency ω and D(ω) = ε(ω)E(ω). In thetime representation, D(t) and E(t) are not instantaneously related.The field D(t) depends on the history of the electric field [1, 2], i.e., itholds the convolution integral relationship

D(t) = ε0

{E(t) +

∫ +∞

−∞G(τ)E(t− τ)dτ

}(1)

where the kernel G(τ) depends on the structure of the dielectricmaterial. In more detail, G(τ) is the inverse Fourier transform of theelectric susceptibility χ(ω) = ε(ω)/ε0 − 1.

A full description of the physics of dispersion in dielectric mediaremains, in general, a very complex task and it should involvethe study of electron and ion motion through a combined use ofclassical and quantum mechanic theory. However, a simple model ofharmonically bound charges (electrons and ions) is generally adoptedto illustrate this phenomenon. In terms of classical models of themolecular properties, the relationship between the macroscopicallydefined parameter ε and the molecular properties can be summarizedby the Clausius-Mossotti ’s equation:

γ =3N

ε/ε0 − 1ε/ε0 + 2

(2)

that relates the molecular polarizability γ to ε and

χ =Nγ

1− 13Nγ

(3)

where N is the average number of molecules per unit volume.In 1929 Debye presented a model for the response of electric

dipoles to an alternating electric field [3, 4]. The model showed anexponential relaxation of dipole polarization and this led to the knownexpression

ε(ω) = ε(∞) +ε(0)− ε(∞)

1 + iωτ(4)

where τ is the relaxation time, ω the angular frequency of thealternating electric field, and ε(0) and ε(∞) are the zero- and high-frequency limits. A simple physical consideration shows that whenω → ∞, the function ε(ω) → ε(∞), i.e., ε0. Indeed, for a validmacroscopic description the condition ω ¿ c/a must hold. Since λ ≈c/ω, as the frequency increases λ becomes comparable with the atomicdimension a. This is clearly discussed in [1].

Progress In Electromagnetics Research C, Vol. 33, 2012 97

Development of the Debye model and further investigation of therelaxation response were a consequence of the possibility of applyingthe dielectric spectroscopy over an extremely broad frequency range.Today it is possible to measure from µHz to tens of GHz withthe use of commercially available spectrometers. Measurements atlow frequency, f ≤ 2GHz, are generally performed through the useof dielectric samples prepared in parallel-plate geometry [5]. Athigher frequencies reflectrometric techniques with a sample mountedat the end of the transmission lines and analysis of transmissionproperties are adopted [6, 7]. Note that time domain and frequencydomain measurement techniques are generally used to characterizethe dielectrics in the µHz-MHz frequency range. Reflective coaxial,transmission coaxial and quasi-optical methods are used to cover therange 108–1011 Hz, while higher frequency ranges from 1011–1015 Hzare covered by Fourier-transform methods [8–10]. In the very highfrequency range, ε(ω) is generally expressed as

ε(ω) = ε0

(1− 4πNe2/mω2

).

Recently, the authors developed various experimental and theoreticalstudies in the field of metamaterials, mainly focusing on themagnetic permeability of resonant repeated structures at industrialfrequencies [11–13] and on the characterization and modelingof supercapacitor [14]. Such studies showed some conceptualand practical problems for the characterization of materials withpermittivity or permeability dispersion at low frequencies. Withreference to the study developed in [14] on a supercapacitor, an evidentdispersion of the equivalent capacitance was observed down to 1mHz.In such a sense, some remarks can be made concerning the definitionof ε in the case of static fields. With reference to the Debye formulawe revisit the meaning and measurement of ε at zero frequency [15]

ε(0) = limω→0

ε(ω). (5)

Referring to non-variable fields, in principle we can distinguish betweenconstant field configurations with and without motion of charges. Forclarity, the steady-state condition is characterized in the first case by adirect motion of charges, thus it is defined as a magnetostatic problem.In the second case there is no charge motion after the transient phaseis extinguished, resulting in an electrostatic problem.

In [1] and [2] the determination of the field due to a static pointcharge at a specific distance from a boundary plane between twodifferent dielectric media 1 and 2 is analyzed. The analysis showsthat for ε2 À ε1 medium 2 behaves much more like a conductorthan a dielectric. In the limit case of ε2 → ∞ the solution leads


to the same results obtained by the method of images for a conductorin place of medium 2. This suggests the same result, ε = ∞, forlossy dielectrics in the presence of constant fields. In fact, a dielectricmedium with zero electrical conductivity, σ = 0, is an ideal assumption,while in practice all the dielectric materials exhibit a σ differentfrom 0. As a consequence, once any transient phase is extinguishedthe dielectric materials behave like a conductor under non-variablefields. In particular, we show how a lossy capacitor connected to adirect current generator defines a magnetostatic problem, and how therelationship V/I = R, with V the potential difference I the currentand R the resistance of the dielectric, is representative of the steadystate condition, providing ε →∞ the same as for a conductor.

This paper is organized as follows: In Section 2, we readdressthe theory of polarization through a classical description based ona harmonically bound charge model, and we discuss how ε → ∞in the limit of a static field if unbounded electrons are present inthe medium. In Section 3, we define various electrostatic problemsand we show how to estimate ε for a conductor via the measurementof, induction-electric field D-E, charge-potential difference Q-V andthrough an energy analysis. In Section 4, we discuss an electrostaticand a magnetostatic problem corresponding to actual measurementconditions and we comment on the measurement of permittivity in apractical case.

2. CLASSICAL MODEL OF POLARIZATION

For the sake of clarity we shall consider a short review of a simplemodel of fields in the matter based on Eq. (1) [1, 2, 16]. Specifically,we neglect the difference between the applied electric field and thefield in the matter and assume that they are uniform. The model istherefore appropriate only for substances of relatively low density. Theequation of motion for an electron of charge e bound by a harmonicforce and under the influence of an electric field E(t) is

m[x + γx + ω2

0x]

= −eE(t) (6)

where γ is the phenomenological friction constant of the damping force.Considering the dipole moment contributed by one electron, p = −ex,and passing to the Fourier representation we obtain

p =e2

m

[ω2

0 − ω2 − ıωγ]−1 E(ω). (7)


If we suppose that there are N/V molecules per unit volume with Zelectrons per molecule, then we may write

P = ε0χE(ω) =N

V

Ze2

m

[ω2

0 − ω2 − ıωγ]−1 E(ω). (8)

We deduce that the dielectric constant isε

ε0= 1 +

N

V

Ze2

mε0

[ω2

0 − ω2 − ıωγ]−1

. (9)

We assume that the electric field is in the form of

E(t) = E0 cos[ωct] (10)

so as to obtain the case of a constant and uniform electric field takingthe limit ωc → 0. Going back to the time representation, from Eq. (8)we obtain

P =e2NZ

(ω2

0 − ω2c

)E0 cos[ωct]

2πmV[γ2ω2

c +(ω2

0 − ω2c

)2] +

e2NZγωcE0 sin[ωct]

2πmV[γ2ω2

c +(ω2

0 − ω2c

)2] . (11)

From Eq. (11), for ωc → 0, the vector P is

P =e2NZ

2πmVω20

E0 ≡ ε0χE0. (12)

Note that ω0 → 0 means that the electrons are somewhat free and themedium approaches the condition of a conductor in the static case. Asexpected, ε →∞.

3. ELECTROSTATIC PROBLEM — IDEALCONDITIONS

Let us imagine a charged ideal parallel-plate capacitor as that shownin Fig. 1.

If the charge density is σ, the electric field inside the capacitor isuniform and its magnitude is E = σ.

Now consider an indefinite slab of a non-charged conductor andput it into the capacitor as shown in Fig. 2. Assuming the thickness ofthe conductor is small when compared with the other dimensions, wecan deduce that, by induction, the two surfaces will be opposite thecharge of the capacitor. This configuration has the following electricalfield distribution in the three regions VI , VII , VIII

EI = σk, EII = 0, EIII = σk


Figure 1. Charged ideal parallel-plate capacitor. Absolute vacuumis between the electrodes. The terms V0 and h stand for the per unitarea volume and separation distance between the electrodes.

Figure 2. Charged ideal parallel-plate capacitor with a conductingslab inserted into it. Absolute vacuum is between the electrodes andthe slab. The terms VI , VII , and VIII stand for the volume per unitarea between the upper electrode and the slab, of the slab and betweenthe lower electrode and the slab.

where k is the unit vector normal to the capacitor plates. Since theconductor did not carry any charge, the correspondent induction vectorD is continuous

DI = DII = DIII .

In region II the field is in a medium and it holds the relationship DII =εEII while in the other regions we have DI = DIII = σk. Since EII = 0,the continuity of the field D is preserved if ε →∞.

The same result is obtained approaching the estimation of ε viathe measurement of the integral quantities Q and V . With reference toFig. 1, the potential difference V0 between the electrodes is V0 = E0hand the capacitance of the capacitor is C0 = Q/V0. Inserting theconducting slab into the capacitor as shown in Fig. 2, the potentialdifference between the electrodes at the steady-state condition is V =VI + VIII = E0(hI + hIII ) since EII = 0 and VII = 0. The capacitanceof the system is C = Q/V and its inverse 1/C = VI/Q + VIII /Q.This is equivalent to a dielectric plate II with infinite permittivity


inserted into the capacitor. In such a case, a zero potential differenceis maintained between the upper and lower boundary surfaces of thedielectric plate as well as for the conducting slab.

3.1. Electrostatic Configuration: Dielectric Slab

Now we consider a further example to better clarify how to defineelectrostatic properties, such as the dielectric constant, through theanalysis of the variation of the electrostatic energy. For this purposewe consider an ideal capacitor with the two plates charged with anelectrical charge ±Q0, surface S, and separated by a distance h, as forFig. 1. For the sake of simplicity, in this section only we will set ε0

as unit. As previously mentioned, the uniform electric field inside hasmagnitude E0 = σ0 = Q0/S. Let us charge a dielectric slab on the twosurfaces with charge Q1. The slab generates a uniform electric fieldoutside the slab

E1 = ±Q1

Sk = ±σ1k (13)

where the minus sign refers to the semi-space underneath the slab.Inside the slab the electric field vanishes. The electrostatic energyof the system capacitor and dielectric slab shown in Fig. 3 is simplythe sum of the single electrostatic energies given by the formula U =12

∫V E ·DdV, namely

U1 =12E2

0V0 +12E2

1Vu +12E2

1Vd (14)

where V0 = Sh, Vu and Vd are the volumes of the two semi-spacesoutside the slab.

Inserting the slab into the capacitor as shown in Fig. 4 we have

U2 =12(E0 + E1)2VI +

12ε

E20VII +

12(E0 − E1)2VIII

+12E2

1(Vu − VI) +12E2

1(Vd − VIII ). (15)

Figure 3. Charged ideal parallel-plate capacitor and charged dielectricslab.


Evaluating the change of the electrostatic energy of the system we have

∆U = −12E2

0

(1− 1

ε

)VII + E1E0(VI − VIII ). (16)

The electrical field is E1 = σ1 = (DI − DII ) · n, where n is theouter-pointing normal from the slab. Measuring the variation of theelectrostatic energy we have a measure of the induction vector D.

Note that if VIII = VI then ∆U does not depend on the externalcharge Q1. Finally, taking the limit for ε → ∞ we rediscover thevariation of energy for a conductor

∆U = −12E2

0VII + E1E0(VI − VIII ). (17)

3.2. Electrostatic Configuration: Dielectric Sphere

Next we study an analogous case considering a dielectric sphere ofradius a with a charge distribution on the surface given by σ(θ) =σex cos θ. The function σ(θ) has been selected in such a way as tofind a simple analytical solution. Solving the Laplace equation for thepotential, ∇2φ = 0, we obtain for the potential the following

φin =σex

ε + 2r cos θ (18)

φout =ε− 1ε + 2

E0a3 cos θ

r2+

σexa3

ε + 2cos θ

r2. (19)

As before, we imagine inserting the sphere into the capacitor as shownin Fig. 5. The field of the sphere is assumed to vanish at the electrodes.Combining the electric field given by Eqs. (18) and (19) with the well

Figure 4. Charged idealparallel-plate capacitor withcharged dielectric slab insertedinto it.

Figure 5. Charged dielec-tric sphere inserted into an idealparallel-plate capacitor. Absolutevacuum is between the electrodesand the sphere.


known solution of a dielectric sphere in a uniform electric field weobtain for the field inside the sphere

Ein =(

3E0

ε + 2− σex

ε + 2

)k, (20)

where k is the unit vector in the z direction. In terms of polar unitvector

k = cos θr− sin θθ.

Outside of the sphere we have

Eout = E0k + EP , (21)

where the components of the vector EP are

EPr = 2ε− 1ε + 2

E0a3

r3cos θ +

2a3σex

ε + 2cos θ

r3(22)

EPθ =a3E0(ε− 1) sin θ

r3(ε + 2)+

a3σex

(ε + 2)sin θ

r3. (23)

With respect to the vector D, inside the sphere it holds

D = εEin (24)

while outside the sphere it coincides with the field Eout. The energyvariation of the system is

∆U =12

∫

VE ·DdV − 1

2E2

0VC − 2a3πσ2ex

3(ε + 2)(25)

where VC = Sh is the volume enclosed by the capacitor. If a ¿ hwe can perform the integration considering a vanishing electrical fieldoutside of the capacitor. We may rewrite Eq. (25) as

∆U =4π

3(ε + 2)a3

[12(ε− 1)E2

0 − E0σex

]. (26)

Note that for ε → ∞ the contribution of the external charge, σex,vanishes. The same result is obtained for the case of a chargedconducting sphere, although the considered charge distribution is anideal assumption since a charge on a conducting sphere tends todistribute uniformly.

We shall end this section considering a dielectric sphere with auniform charge distribution on its surface. Using the same notation ofthe previous example we can write, for the field inside the sphere

Ein =3E0

ε + 2k, (27)


and for the field outside the sphere

Eout = E0k + EP (28)

EPr =q

4πr2+ E0 cos θ +

2a3E0(ε− 1) cos θ

r3(ε + 2)(29)

EPθ =a3E0(ε− 1) sin θ

r3(ε + 2). (30)

The variation of electrostatic energy is

∆U =12

∫

VE ·DdV − 1

2E2

0VC − q

8aπ=

43πa3 ε− 1

ε + 2E2

0

2(31)

that is independent from the external charge q. The case of a conductoris obtained evaluating the limit for ε →∞.

4. ELECTROSTATIC AND MAGNETOSTATICPROBLEM — ACTUAL CONDITIONS

Dielectrics and conductors exhibit different values of the electricalconductivity σ. Such a difference is evident over certain time scalesor frequency ranges within which low and high values of σ are morecharacteristic of an insulating and conducting behavior. On the otherhand, it is common for a material to change its behavior showing aproper frequency dispersion law for its parameters.

In particular, apart from the absolute vacuum any other materialbehaves as a conductor in static conditions. This poses some questionson the possibility of measuring the static ε under “actual conditions”,i.e., with a lossy dielectric. Is it possible to reach an electrostaticsteady-state or could we reach other conditions from which it wouldbe possible to derive the correct estimation of static ε?

Such considerations find an experimental evidence and apractical application with new dielectric materials, such as those forsupercapacitors that show a permittivity dispersion at frequency wellbelow 1 Hz [14]. In Table 1, we report some experimental resultsperformed on a 120 F Maxwell c© double layer supercapacitor. Theresults are expressed in terms of real part of the capacitance of aG (conductance)-C (capacitance) parallel equivalent circuit. The dataclearly show a dispersion of the estimated parameter down to 1mHz.

Looking at the capacitance values reported in Table 1, we cannotice how they are still far from converging to a constant capacitancevalue at lower frequencies. This leads to the question on how tomeasure and to define the static capacitance or, alternatively, onhow to properly read the obtained experimental results. The static


Table 1. Estimated real part of the capacitance value of an equivalentG-C parallel circuit of the supercapacitor.

frequency (Hz) 0.001 0.005 0.01 0.05 0.1 0.5 1capacitance (F) 119.29 104.10 100.80 86.30 63.40 8.10 2.55

capacitance value is usually provided by the manufacturer, but itsmeaning should be properly taken into account. As written inthe introduction, the characterization of static permittivity becomesmore critical for materials showing permittivity dispersion at lowfrequencies. If for example they are used as energy storage devices,it is important to know as accurately as possible the effective storedand suppliable energy.

Let us first find the time scale to reach the static conditionfor lossy dielectrics. For simplicity, imagine inserting an externalcharge density ρ(x, y, z) inside a lossy dielectric sphere. The continuityequation is written as:

∂

∂tρ(r, t) + ∇ · J(r, t) = 0. (32)

Using the phenomenological relation J = σE and Maxwell’sequation ∇ ·E = ρ/ε we obtain for the charge density

∂

∂tρ(r, t) +

σ

ερ(r, t) = 0 (33)

that has as solution

ρ(r, t) = ρ(r, 0) exp[−σ

εt]. (34)

The time constant through which the dynamic system evolves can beestimated via an experimental observation. For example, we can tryto measure the field inside the sphere taking into account that for anyconductor the charge will finally distribute on the external surface ofthe body vanishing the electric field inside the material. Using thesemi-empirical argument given above, at first approximation, we canassume an exponential decay of the field and we can write the timeevolution expression for the charge density as

ρ(x, y, z, t) = ρ(x, y, z, 0) exp[− t

τeff

], (35)

where τeff isτeff =

εeff

σ(36)


and where we can define εeff as the equivalent permittivity betterfitting the time evolution for the charge density. However, onceelectrostatic equilibrium is reached we have defined an indeterminateproblem. Both D and E are zero at equilibrium inside the medium.Outside of the sphere D and E are the same field (except for theuniversal constant ε0) and their magnitude at the sphere surface is thesuperficial charge density. Since the discontinuity of D is the value ofthe superficial charge density, this implies that inside the sphere thefields vanish.

Let us now consider the problem defined in Fig. 6 that consistsof a direct current generator connected to a lossy capacitor. Thesteady-state condition is driven by Ohm’s law V/I = R where Vis the potential difference I is the current and R is the resistanceof the dielectric. Strictly speaking, we have defined a magnetostaticproblem. In principle, from the geometric parameters of the medium, itis possible to estimate the value of σ via the constitutive equation J =σE. On the other hand, we will never reach electrostatic equilibrium.Consequently we are not able to measure the static permittivity.However, as for the previous case of the sphere, we have a conductorbehavior that is in agreement with previous sections and we canattribute this to the infinite value of the static ε.

Referring still to the problem defined in Fig. 6, let us comment onthe case of a common laboratory experiment with low lossy dielectrics.Once the DC generator is closed on the capacitor system a first chargetransient of the capacitor follows when the circuit is switched on. Forsimplicity, we can assume that the charge transient time constant isnegligible, for example when compared to τeff . In practice the materialwould respond to a voltage step input, while the estimated ε would bethe value assumed by the permittivity during the “first instances” ofthe main transient, i.e., for t ¿ τeff .

We believe that the present analysis may provide a contribute tothe characterization of static permittivity and to possible developmentof capacitance standards [17–19].

Figure 6. Magnetostatic field problem. Capacitor setup.


5. CONCLUSIONS

In this paper, we discussed the concept and measurement of staticelectric permittivity. A formal description and measurement definitionof such a parameter was provided via a classical model of polarization,as well as an analysis of various electrostatic configurations. Twobasic experiments with a lossy dielectric were described. In onecase, the charge evolution inside a sphere previously charged by theinsertion of an external charge density was treated. Some remarkson the effective time constant were provided. In the other case amagnetostatic problem was defined by supplying a capacitor filledwith a lossy dielectric via a DC generator. In this case, it wasstressed how the system never reached electrostatic equilibrium. Inboth cases, after the transient phases were extinguished we observeda conductor behavior. As a consequence, we could attribute to thestatic ε of the investigated material an infinite value. The case of acommon laboratory experiment and experimental results performed ona commercial supercapacitor were also commented on to point out themeaning of the value attributed to ε in actual measurement conditions.The presented analysis may help to discuss the dielectric behavior ofactual materials as well as to reassess the concept and measurementof permittivity under static conditions. Finally, this can find practicalapplications with energy storage devices for which dielectric materialsshow permittivity dispersion at low frequencies.

ACKNOWLEDGMENT

MB acknowledges financial support from FONDECYT projectNo. 1110231. We thank C. Beeker for her editorial contribution.

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