The Phasance Concept Jean Jacquelin
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"The Phasance concept : a review" (here, pp. 2-13) was published in Current Topics in Electrochemistry, Vol.4, (1997), Edit.: Research Trends, 695 012 India. This paper is intended for specialists in various fields of electrotechnics (for example electrochemistry), using impedance analysis for materials characterization (ionic conductors, dielectrics, etc.) ABSTRACT : The Phasance concept is introduced using either a practical or a theoretical approach. The potential difference, function of time, V(t), causing a current, I(t), to flow in a circuit element, can be expressed in terms of the fractional derivative of the current, thanks to the Rieman~LiouviIIe transform: V=Pϕ.dνI/dtν, where Pϕ is the Phasance and ν is the degree of fractional derivation. In the sinusoidal case, the relationship between the impedance (Z) and the Phasance is Z = Pϕ (i ω)ν, where ν is related to the phase angle displacement (ϕ). A basic element, so-called Phasor (Pϕ, ν=2ϕ/π), is described in the case of constant Pϕ and ν. The key characteristic of this element is a constant energy efficiency. The consideration of Resistance, Capacitance, Inductance as boundary cases of Phasance, leads to a more general approach in mathematical treatments. Theoretical and practical examples are presented, which result from a review of information, notions, and ideas derived from the literature. The unified concept of Phasance is especially useful in Electrochemistry. Thanks to this theoretical basis, one can understand why some kinds of complex impedance behaviours are so common and are likely to be caused by many different phenomena. This is also the starting point for the derivation of a number of models and equivalent circuits.
"A number of models for CPA of conductors and for relaxation in non-Debye dielectrics" (here, pp.14-18) is more specifically addressed to specialists in the field of dielectrics. It was published in the Journal of Non-Crystalline Solids,131-133, (1991) 1080-1083, Edit. Elsevier.Science Publishers (North-Holland). ABSTRACT: The impedance, admittance, or permittance diagrams of materials, when presented in the complex plane, are often well defined straight lines, or well defined semi-circles with axes depressed below the real axis. Using a general theoretical approach, it is suggested that this behaviour is not characteristic of a single particular phenomenon or theory (hence, not characteristic of fractal geometry for example). The potential difference, V(t), causing a current, I(t), to flow in any type of electrical circuit, can be expressed in terms of the 'fractional derivative' of the current: V=P dνI / drν where P ('phasance', magnitude) and ν (degree of fractional derivation) depend on the physical properties. A particular case corresponds to constant P and ν , to a constant energy efficiency, to a straight line on the complex impedance diagram and to the 'constant phase angle' (CPA) behaviour. As an example, the direct application of the mathematical theory leads to several simple and exact distribution functions of relaxation times and a number of models for non-Debye dielectrics.
"Impedance analysis and modelization of the alternating current properties of ionic conductors" (here, pp.19-32) should interest scientists dealing with ionic conductors, especially in electrochemistry. This paper, written in 1984, was not published. ABSTRACT: The common inclined semi-circular complex impedance diagrams of ionic conductors can be generated by various distribution laws of the physical properties. Four different examples of models are given, suggesting some kinds of physical phenomena likely to cause this behaviour. Adequate structures of networks can be found to modelize them accurately.
The three papers were aimed at specialists while the more general paper "La dérivation fractionnaire" (not translated as of yet) is intended to general public, published in the magazine Quadrature Vol.40, (2000) 10-12, Edit. EDP Sciences (France) and now available on : http://www.scribd.com/JJacquelin/documents All these papers are in close relationship. They are derived from studies made around 1980 reported in a technical document : "Use of Fractional Derivatives to express the Properties of Energy Storage Phenomena in Electrical Networks", (1982, re-print 1984, now out of print). Edit.: Laboratoires de Marcoussis, Route de Nozay, 91460, Marcoussis, France.
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The Phasance concept: a review Jean Jacquelin Alcatel Alsthom Recherche. Route de Nozay. 91460. Marcoussis, France
ABSTRACT The Phasance concept is introduced using either a practical or a theoretical approach. The potential difference, function of time, V(t), causing a current, I(t), to flow in a circuit element, can be expressed in terms of the fractional derivative of the current, thanks to the Rieman~LiouviIIe transform: V=Pϕ.dνI/dtν, where Pϕ is the Phasance and ν is the degree of fractional derivation. In the sinusoidal case, the relationship between the impedance (Z) and the Phasance is Z = Pϕ (i ω)ν, where ν is related to the phase angle displacement (ϕ). A basic element, socalled Phasor (Pϕ, ν=2ϕ/π), is described in the case of constant Pϕ and ν. The key characteristic of this element is a constant energy efficiency. The consideration of Resistance, Capacitance, Inductance as boundary cases of Phasance, leads to a more general approach in mathematical treatments. Theoretical and practical examples are presented, which result from a review of information, notions, and ideas derived from the literature. The unified concept of Phasance is especially useful in Electrochemistry. Thanks to this theoretical basis, one can understand why some kinds of complex impedance behaviours are so common and are likely to be caused by many different phenomena. This is also the starting point for the derivation of a number of models and equivalent circuits.
1. INTRODUCTION The aim of this paper is to make a synthesis of several already known physical and mathematical points. A major consideration is to give a comprehensible overview on notions, ideas and material taken in the literature from various fields: mathematics, physics, electrochemistry, dielectric science, etc.. In many areas of electrical engineering, it is common to study the materials properties, components, networks or electrical systems as a function of frequency. Specialized and sophisticated measurement devices are available, e.g. network analysers or impedance analysers. Electrochemists speak of impedance spectroscopy. One sample system is submitted to a stimulating electrical signal. The response signal is compared to the emitted signal, with the aim of deriving an experimental knowledge of the phenomena inside the studied object. One of the most simple cases is the measurement of the complex impedance, Z=Z'+iZ", at various input frequencies f or ω=2πf. The corresponding experimental results, such as complex impedance tables and diagrams, are displayed with various systems of axes and scales. One of the most common representation is in the complex plane, where the real part is on the x-axis and the imaginary part is on the y-axis, with f or ω as a parameter. Then, it has been observed that the diagrams are often well defined arcs of circles [1-2]. Many authors introduce a term (iω)ν, where (i) is the imaginary unit and (ν) is a constant, to express experimental results [1-2], in the formulas of impedance, Z(ω), or admittance, Y(ω)=1/Z, or complex capacitance C(ω)=l/(iωZ), etc.
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In the field of dielectrics, a considerable amount of experimental data [1] are usually represented by equation (1) :
ε = εp +
εs 1+(iω τ0 )1−α
(1)
The loci of the permittivity (ε), represented in the complex plane, are semi-circles the centres of which are depressed below the real-axis as shown on Figure 1. This behaviour is referred to as Non-Debye.
Figure 1: Complex plane locus of the complex capacity or complex permittivity and equivalent circuit, as required by experimentaI evidence, for dielectrics referred to as "Non-Debye". In the field of ionic conductors, it has long been recognized that many complex impedance loci result in well-defined semi-circles (Figure 2). A common empirical formula (2), as required by experimental evidence, expresses the complex impedance as a function of frequency:
Z = Rs +
Rp
(2)
1 + (iω τ 0 )1− α
Figure 2: Complex plane locus of the complex impedance and equivalent circuit, as required by experimental evidence, for many ionic conductors.
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Linear loci are also commonly observed, generated by other phenomena, including rough electrode surfaces [3], porous electrodes [4], or diffusion [5], etc. A linear locus, Figure 3, corresponds to equation (3), which is a boundary case of equation (2)
Z = Rs +
Rp
(3)
(iωτ0 )1− α
Figure 3: Complex plane locus of the complex impedance and equivalent circuit, as required by experimental evidence, observed in many cases, for example rough electrode surfaces, thick porous electrodes, diffusion, etc.. In all the above equations (1-3), an (iω)ν term is involved, in which the exponent (ν) is no longer an integer. The very large number of experimental observations involving (iω)ν terms draws one to think that such a term is a direct. consequence of fundamental electrical laws and is not specific to some particular phenomenon.
2. THE FRACTIONAL CALCULUS The fractional derivative of degree (ν) of a function f(t) is introduced by the means of Rieman-Liouville's transform (4):
d ν f(t) 1 = ν Γ( − ν) dt
∫
t 0
f(x) dx (t − x) ν +1
(4)
A vast literature deals with the pure mathematics of these subject. The fractional calculus [6] gives an excellent review: background, mathematics and recent physical applications. The use in electrotechnics of the operator of fractional derivation is not new, since Heaviside used it as early as 1892. A number of fractional derivatives of various functions can be found in the tables of RiernanLiouville transforms [7]. For studies involving sinusoidal alternating current, it is essential to know the fractional derivative of the functions cos(ωt) and exp(iωt), respectively (5, 6):
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dν π cos(ωt) ) = ω ν cos ωt + 2 ν − D ( ωt , − ν ) ν( dt 1 where D ( ωt , − ν ) = ( cos(ωt)C ( ωt , − ν ) + sin(ωt)S ( ωt , − ν ) ) Γ ( −ν )
( (
)
)
dν exp(iωt) ) = exp(iωt) (iω) ν − F ( ωt , − ν ) ν( dt ων where F ( ωt , − ν ) = ( C ( ωt , − ν ) − i S ( ωt , − ν ) ) Γ(−ν)
(
)
(5)
(6)
where Γ is the Gamma function, ( [8], p. 255), and C(ωt,-ν) and S(ωt,-ν) are the Generalized Fresnel Integrals from Ref. [8], p. 262 . It can be shown that the Fresnel Integrals tend to zero if (ωt) is large enough. So, in a steady state, or more strictly a periodic state in which the initial switching-on of the signal is so remote in time so that its effect is negligible, equation (5) tends to (7) and equation (6) tends to (8):
dν π cos(ωt) = ω ν cos ωt + 2 ν ( ) ν dt Steady State
(
dν exp(iωt) = (iω) ν exp(iωt) ( ) ν dt Steady State
)
(7)
(8)
It can be observed that, if the lower limit for the integration (4) is t=-∞ instead of t=0, equations (7, 8) are directly obtained, corresponding to the particular case of the Weyl operator [7].
3. THE PHASANCE CONCEPT [9-10] A practical approach consists in considering the characteristic laws of the three fundamental components which are recapitulated on Figure 4.
Figure 4 : Generalization of the three basic laws. The phasance concept and boundary cases. 6
In addition, Figure 4 reveals a generalization of the time relationship between voltage drop and current, thanks to the fractional derivation operator. By etymological analogy, the denomination Phasance [9] has been proposed to designate the behaviour of a passive component characterized by Z=Pϕ.(iω)ν. The principal characteristic of these kind of components is to have a constant Phase displacement, suggesting the prefix Phas-. The suffix -ance is added by analogy with resistance, capacitance, inductance, which are boundary cases of phasances. We may also venture to suggest the use of Pϕ as practical symbol. The special kind of element endowed with the phasance behaviour may be named Phasor. The basic components, ideal or perfect -Resistor, -Capacitor, -Coil, can be considered, from a pure mathematical viewpoint as -boundary cases of Phasors, respectively (R, 0), (C, -1), (L, 1). In electrochemistry, the term "CPA" (Constant Phase Angle) is sometimes used, or alternatively "CPE" (Constant Phase Element). The Phasance concept can be introduced on a more fundamental basis. We can either consider the real current Ip.cos(ωt), the real voltage Vp.cos(ωt+ϕ) and equations (5), (7), or the complex current Ip.exp(iωt), the complex voltage Vp.exp(i(ωt+ϕ) and equations (6), (8). The fractional derivation (degree ν=2ϕ/π ) of the current leads to the relationship (9), which then can be expressed as (10):
d (2ϕ /π) 1 = (iω)(2ϕ /π) V (2ϕ /π) dt Z (2ϕ /π) d V = Pϕ (2ϕ /π) ; Z = Pϕ (iω)(2ϕ /π) dt
(9)
(10)
In most cases, Pϕ and ν depend on the frequency. A particular case relates to constant Pϕ and ν, which corresponds to the basic behaviour appearing in Figure 4. It is an easy exercise to compute the energy transfer in a Phasor. A quantity of energy Ein is absorbed and Eout is returned each cycle (11), (12). The energy efficiency of the electrical storage inside the Phasor is Eout/Ein , equation (13):
Ein =
Vp Ip
E out =
( π − ϕ cos ϕ +sinϕ )
2ω Vp Ip 2ω
(11)
( − ϕ cos ϕ + sin ϕ )
(12)
E out π =1 − Ein π − ϕ + tan ϕ
(13)
We can see that the energy efficiency is constant, if ϕ is constant, which is common knowledge. A constant energy efficiency is a characteristic feature of a Phasor. It is therefore not surprising that similar behaviour is observed in many circumstances, in which different phenomena may show a constant energy efficiency, at least in a limited range of frequencies.
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4. THE SIMPLEST ASSOCIATIONS ELEMENTARY COMPONENT
OF
A
PHASOR
WITH
ANOTHER
In the case of a Phasor (Pϕ , ν=2ϕ/π) coupled in parallel with a resistor (R), the principal characteristics are given in Figure 5. The corresponding impedance diagram appears in Figure 6. The Z-plot is the arc of a circle. The Z-plot corresponding to the Phasor distinctly considered is the straight line.
Figure 5: Main characteristics of the association of a Phasor and a Resistor in parallel.
Figure 6: Complex impedance diagram of a Phasor coupled in parallel with a resistor.
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Likewise, if a Phasor is mounted in series with a resistor, an arc of a circle is obtained on the complex admittance ( Y = 1/ Z ) diagram. If a Phasor is mounted in series with a capacitor, an arc of a circle is obtained on the complex capacitance ( C=1 / (iωZ) ) diagram. All these arcs of circles are not centred on the axis (Fig. 4). They are commonly referred as inclined or as skewed . The appearance of well defined inclined arcs of circles in various representations (Impedances Z, Admittances Y, Complex Permittivities ε, etc...) and in many different circumstances has been a subject of thought. The critical comments of Dr. A.K. Jonscher [11] may be cited: "Neither skewed Y-plots nor skewed Z-plots are satisfactorily explained to our knowledge, they appear to be accepted as manifestation of a phenomenon loosely referred to as distribution of relaxation times, but it is not made at all clear how this should lead to a well-shaped skewed circular plot in either representation. Distribution of relaxation times is likewise frequently invoked as being responsible for the skewed ε-plots, such as they are, but it is easy to show that skewed circular ε, Y·, and Z- plots are not mutually compatible with one another. Thus any explanation that may be proposed should make clear which of these skewed representations it is intended to interpret". From the preceding arguments an explanation comes to light : First, a common behaviour, called here Phasance behaviour, gives a single type of plot which is compatible in any of the representations Z, Y, ε, or many other, and which are inclined straight lines, as it is obvious from the next equations: Z=Pϕ.(iω)ν , Y= Pϕ−1.(iω)-ν , C= Pϕ−1.(iω)-1-ν , etc. Second, the simplest kind of deformation of these linear plots leads to skewed circular plots: - In the complex Z-plane : Phasor in parallel with a resistor (Fig. 2). . In the complex Y-plane: Phasor in series with a resistor. - In the complex C-plane : Phasor in series with a capacitor (Fig. 1) or a coil. - In the complex l/C-plane: Phasor in parallel with a capacitor or a coil. - Etc. More complicated combinations of course lead to more complicated geometrical forms. However, in limited ranges of frequencies, the predominance of one of these preceding cases is likely to produce a part of an inclined straight line or a part of inclined circle, appearing in one or another kind of graphical representation.
5. THEORETICAL EXAMPLE OF PHASANCE For many continuous media endowed with properties of conduction and electrical storage mixed in a common volume, a phasance type of behaviour will be observed. For examples, if the resistance distribution, ρ(x), and the capacitance distribution, γ(x), are respectively equations (14) with constant coefficients a, b, γ1, ρ1, and in the case of medium properties corresponding respectively to equations (15), then the differential equation (16) is to be solved. This involves the Kelvin functions [8]. This leads to functions I(x) and V(x). Then, it can be shown that the impedance Z is expressed by (17), which is the typical phasance equation, where the coefficients are constant (18).
γ = γ1x a ; ρ = ρ1x b dV dI = ρI ; = iω γ V dx dx
(14) (15)
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d -a dI b x = iωρ1γ1x I dx dx V Z = = Pϕ (iω) ν I x=0
(16) (17)
Γ ( − ν ) (a + 1) ρ1 (ρ1 γ1 ) ν b +1 ν =− ; Pϕ = a+b+2 Γ ( 2 + ν ) (a + b + 2) 2(1+ ν)
(18)
A particular and well-known case in electrochemistry [4] is the semi-infinite homogeneous medium (ρ and γ constant, a=0, b=0 ), in which v=-0.5, and the phase shift is ϕ=-π / 4. This particular case is often loosely referred to as Warburg's impedance, whatever the causal phenomenon might be. Many other examples of theoretical phasance behaviour could be given. A number of distribution functions different than (14) give similar results. Moreover, many media with basic equations different than (15) give also similar results [9]. One may be aware that the phasance behaviour may have many other causes than the particular cause suggested by the preceding theoretical example. 6. EQUIVALENT CIRCUITS AND EXAMPLES OF PHASORS MADE WITH DISCRETE COMPONENTS
Figure 7: Example of four networks (a), (b), (c), (d), convenient for modelling the "phasance" behaviour, and their associations with a resistor in parallel. The complex impedance diagram is common to the four networks
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Coming back to the preceding example of a continuous medium, if we cut it into slices of arbitrary thickness and if each slice is replaced by an equivalent resistor and capacitor, a network with impedance as close as desired to the phasance of the initial medium is obtained (also, clearly, with certain deviations resulting from the replacement of a continuous function by a neighbouring discretized function). For example, this is how the network "Pϕ" of Figure 7(a) was obtained. The above method and formula are not unique : other different ones lead to networks (b), (c), (d) of Figure 7. What is more, many others can be found. One could think, at first sight at Figure 7, that all structures of equivalent networks are endowed with self-similarity. This is not true : an infinity of other networks equivalent to a Phasor, not endowed with selfsimilarity can be found. A few examples are represented in Figure 8.
Figure 8 : Examples of networks, taken among the huge number of possible ones, equivalent to those given in Figure 7, on the frequency band 10 Hz - 100 kHz.
7. SOME PRACTICAL USE OF THE PHASANCE CONCEPT AND PHASORS The phasance concept introduces a somewhat different approach to describing and modelling some electrical phenomenon. For example, in the field of dielectrics, the basic Von Schweidler approach [12], followed by Wagner and many others, involves some distributions of relaxation times. As a reference, a clear review of the Wagner treatment of distribution of relaxation times in dielectrics is given in [13]. The Schweidler's treatment is applied to the entire complex-ε or complex capacitance, whereas, in the present approach, we first separate the Phasors from the pure capacitors or resistors. Then, distinctly considering a Phasor, the distribution law can be very simple and more comprehensive, compared to the complicated distribution laws previously found. 11
In impedance analysis post-treatments, the introduction of Phasors as basic elements in equivalent networks permits synthetic representations and simplifies the network graphs. For example some ladder networks which require a lot of elementary components can simply be represented by a Phasor symbol. On a more applied viewpoint, Phasors can be used for analog models, for standardization of impedance meters in large frequency band, for constructing standard phaseshifting devices, for teaching and training in the field of impedance measurements, for constructing Analog Fractional DifferIntegrators (AFDI), and probably for many other uses. The principle of an AFDI is given in Figure 9, using a linear amplifier, a Phasor and a resistor. Other characteristics can be obtained, using a capacitor, or a coil, or another Phasor, instead of the resistor. For example, Dr. K.B.Oldham has described a particular kind of AFDI to make a so-called semi-integral analyser [14], in the case ν=-1/2 (ϕ=-π/4). Despite this special case of phase-shift, the device is an excellent illustration of the more general thesis given earlier.
Figure 9: Schematic diagrams of Analog Fractional Differ-Integrators, using linear amplifiers and Phasors.
8. CONCLUSION The potential difference, function of time, V(t), causing a current, I(t), to flow in a circuit element, can be expressed in terms of the fractional derivative of the current, thanks to the Rieman-Liouville transform:
d νI V = Pϕ dt ν where Pϕ is the phasance and ν is the degree of fractional derivation. In the sinusoidal case, the relationship between phasance Pϕ and complex impedance Z is:
Z = Pϕ (iω)(2ϕ /π)
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Cole, K. S. and Cole, R. H. [1] pointed out the importance of considering the notion of stored energy and suggest that "the problem may profitably be considered from this viewpoint". It is clear that the behaviour of the energy storages is the key aspect. It is common knowledge that phase displacement is directly related to energy storage efficiency. The paper is focused on one of the most typical case of basic element, named Phasor, characterized by a constant Pϕ and ν . A key characteristic of a Phasor is to have a constant efficiency in a large frequency range. Pure -Resistors (Pϕ=R, ϕ=0), -Capacitors (Pϕ=1/C, ϕ=-π/2), -Coils (Pϕ =L, ϕ=π/2) are boundary cases of Phasors, with energy efficiencies which are constant and respectively equal to 0, 1 and 1. The phasance property is the starting point for understanding why this kind of behaviour is encountered in many practical cases and for very different kinds of phenomena. The conclusion is clear: if the complex impedance analysis leads to inclined straight lines or inclined semi-circles (as on Fig. 1-3), this features mean only that the system includes some part(s) transferring energy with a constant efficiency in a range of frequencies. Any affirmation of the nature and mechanisms of the phenomena involved would be pure hypothesis if it is only based on impedance analysis. The phasance concept permits simpler interpretations of the complex diagrams for Impedances, Admittances, complex Capacitances and many others. We saw that there are many manners to make Phasors and they can be used in many practical applications.
REFERENCES [1] : [2] : [3] : [4] : [5] : [6] : [7] : [8] : [9] :
[10] : [11] : [12] : [13] : [14] :
Cole, KS. and Cole, R.H. 1941, J. Chem. Phys.,341. Jonsher, A.K 1978, J. Mat. Sci., 553. De Levie, R. 1965, Electrochim. Acta, 113. De Levie, R. 1964, Electrochirn. Acta, 1231. De Levie, R. and Pospisil, L. 1969. J. Electroanal. Chem., 277. Oldham, K B. and Spanier, J. 1974. The Fractional Calculus, Acad. Press, New York Bateman, H. 1954. Tables of Integral Transforms, McGraw-Hill, New York. Abramowitz, M. and Stegun, I. 1970, A. Handbook of Mathematical Functions, 9th. print, Dover Publ. New York. Jacquelin, J. 1982, Document: Use of Fractional Derivatives to express the Properties of Energy Storage Phenomena in Electrical Networks, (re-print 1984), Laboratoires de Marcoussis, Route de Nozay, 91460, Marcoussis, France. Jacquelin, J. 1991. J. of Non-Crystalline Solids, 1080. Jonsher, A. K 1975. Phys. Stat. Sol. 665. Von Schweidler, E. 1907, Ann. d. Physik, 711. Yager, W. A. 1936, Physics, 434. Oldham, K B. 1973, J. Analytical. Chem., 39.
Nota bene (Nov. 2011) : The paper "The Phasance concept : a review" published in 1997 is the result of studies made around 1980. Since these almost ancient times, many relevant documents where published which, of course, do not appear in the short bibliography above.
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A number of models for CPA impedances of conductors and for relaxation in non-Debye dielectrics Jean Jacquelin Alcatel Alsthom Recherche, route de Nozay. 91460, Marcoussis, France The impedance, admittance, or permittance diagrams of materials, when presented in the complex plane, are often well defined straight lines, or well defined semi-circles with axes depressed below the real axis. Using a general theoretical approach, it is suggested that this behavior is not characteristic of a single particular phenomenon or theory (hence, not characteristic of fractal geometry for example). The potential difference, V(t), causing a current, I(t), to flow in any type of electrical circuit, can be expressed in terms of the 'fractional derivative' of the current: V=P dνI / drν where P ('phasance', magnitude) and ν (degree of fractional derivation) depend on the physical properties. A particular case corresponds to constant P and ν , to a constant energy efficiency, to a straight line on the complex impedance diagram and to the 'constant phase angle' (CPA) behavior. As an example, the direct application of the mathematical theory leads to several simple and exact distribution functions of relaxation times and a number of models for non-Debye dielectrics.
1. Introduction The impedance, admittance, or permittance diagrams of materials, when presented in the complex plane, are often well defined straight lines or semi-circles, with axes depressed below the real axis. In the field of dielectrics, a considerable amount of experimental data [1] can be represented by the equation
ε =εp +
εs 1 + (i ωτ 0 )1−α
The loci of the complex permittivity are semi-circles the centres of which are depressed below the real-axis. This behavior is referred to as 'non-Debye' . In the field of ionic conductors, it has long been recognized that many complex impedance loci result in well-defined semi-circles. An empirical formula, as required by experimental evidence, expresses the complex impedance as a function of frequency:
Z = Rs +
Rp 1 + (i ωτ 0 )1−α
In all the above, an (iω)k term is involved. This common feature is related to a constant ratio between the energy lost per cycle and the energy stored per cycle [1,2].
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2. Use of fractional derivatives to express the properties of phenomena in electrical networks The fractional derivative of degree, ν, of a function, f(t), is introduced by the means of Rieman-Liouville's transform:
dν f (t ) 1 = Γ(−ν ) dtν
∫
t 0
f ( x) dx (t − x)ν +1
The 'fractional calculus' [3] gives an excellent review of the subject background, mathematics and recent physical applications. Use in electrotechnics of the operator of fractional derivation is not new, since Heaviside used it as early as 1892. The characteristic laws of the three fundamental components are recapitulated in table 1. In addition, table 1 reveals a generalization of the time relationship between voltage drop and current, thank to the fractional derivation operator. By etymological analogy, the denomination 'phasance' has been proposed to designate a dipolar passive component characterized by Z = Pϕ (iω)ν.. The elements L, R, C are then represented as limit cases of phasances [4]. The laws are obviously valid whatever the electrical function applied, whether cyclic or not. Table 1: Generalization of the three classical laws
3. Sinusoidal case In electrotechnics, since the sinusoidal function occupies a first-order importance, the special terminology used (phase shift angle, impedance, etc. . .) is recapitulated in addition in the right hand columns of table 1. In most cases, Pϕ and ν depend on the frequency. A particular case relates to constant Pϕ and ν, constant energy efficiency and straight lines on the complex impedance and complex capacitance diagrams:
Z = Pϕ (i ω ) 2ϕ / π
; C=
1
iω Z
=
1 Pϕ (i ω )1+ 2ϕ / π
If such a component (' phasor') is mounted in parallel across a resistor, an impedance diagram in the form of an arc of a circle is obtained. Likewise, if a phasor is mounted in series with a capacitor, an arc of a circle is obtained in the complex capacitance diagram.
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4. Equivalent networks For a given impedance diagram or a given table of impedances and frequencies, there exists an infinity of dipolar equivalent networks [5]. Accordingly, it is normal to find many network structures equivalent to phasors. Simple methods are indicated in refs. [4] and [6]. Figure 1 shows four different networks resulting in the same complex capacitance diagram. An infinity of other configurations giving the same phasance can be found. The reader will doubtless perceive that the structures of the networks in figure 1 are endowed with selfsimilarity. However, this is not a necessity: an infinity of other equivalent networks not endowed with self-similarity can be found [6].
Fig. 1. Example of four networks convenient for modelling 'Phasance' behaviour, associated with a capacitor in series.
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5. Comments The prime object of this paper is to point out the fact that a term such as (iω)k is a direct consequence of fundamental electrical laws and is not specific to some particular phenomenon. This term is a general characteristic of any electrical system in which energy efficiency does not depend on the frequency. Then, it is not surprising that similar behavior is observed in many fields of electrotechnic, in which different phenomena can display a constant energy efficiency, at least within a limited range of frequencies. This general theoretical approach suggests that this behavior is not characteristic of a single particular phenomenon or theory (hence, not characteristic for example of fractal geometry). There should be no confusion between 'fractional operator' and 'fractal geometry'. Fractals are restrictive cases of application: One cannot exclude the possibility that constant efficient phenomena develop in geometrical structures other than fractal structures, or omit the cases where the influence of geometry, whatever it may be, would be negligible.
Appendix. Example of calculus of an electric network equivalent to a phasor and comments For many continuous media endowed with properties of conduction and electrical storage mixed in a common volume, a phasance type of behavior will be observed. For example, if the resistance distribution, ρ(x), and the capacitance distribution, γ(x), are respectively γ = γ1xa and ρ = ρ1xb (simplified case f = x in next equations), or more generally, selecting arbitrarily a function, f(x), which allowed the open choice to numerous very simple laws of distribution (linear, or square root, or power function, exponential, logarithmic, etc... ),
γ = γ1 f a
dx df
; ρ = ρ1 f b
dx df
with constant coefficients a, b, ρ1 ,γ1 and in the case of dissipation and storage of electrical energy in the medium correspond respectively to the following equations:
dV 1 d Istor. = ρ I disip. ; V = dx γ i ω dx I = I disip. + Istor. It can be shown that the impedance is expressed by:
Z = Pϕ (i ω ) 2ϕ / π where the coefficients are constant:
ϕ =−
π b +1 2 a+b
; Pϕ =
π ρ1 (γ 1ρ1 )−2ϕ / π . (a + b)sin(−2ϕ )
If we cut the sample into slices of arbitrary thickness and if each is replaced by an equivalent resistor and capacitor, a network with an impedance as close as desired to the phasance of the initial continuous medium is obtained (also, clearly, with certain deviations resulting from the replacement of a continuous function by a neighboring discretized function). For example, this is how the network ' Pϕ ' of fig. 1(a) was obtained. The above method and formula are not unique: other different ones lead to networks (b), (c), (d) of fig. 1. What is more, many others can be found. These structures are endowed with selfsimilarity. However, this is not a necessity: an infinity of other equivalent networks not endowed with self-similarity can be found [6].
17
In dielectrics, as a starting point in examining non-Debye effects, it would seem to be more advantageous to consider the distributions of elementary capacitances and resistances instead of proposing a theoretical distribution of relaxation times, which will afterwards be difficult to understand from a practical viewpoint. It is not necessary for the distributions to tend to zero in the lower and higher frequencies. It is also realistic to consider some distributions that do not tend to zero at the limits, but which are incomplete (i.e. abruptly cut off, thus defining a range of validity). Under these conditions, very simple laws of distribution lead to non-Debye behavior, over wide ranges of validity. The concept of 'distribution of relaxation times' is adequate in some cases, for example those corresponding to figs. 1(a) and (b), where the various relaxation times are well defined and clearly visible and comprehensible. A majority of cases, for example the models represented in figs. 1(c) and (d), are somewhat different, since a consideration of distribution of relaxation times only does not give a complete picture. In attempting to explain a given experimental capacitance diagram, it is therefore generally not difficult to find several different models (with or without self-similarity) which fit perfectly and hence present several different hypotheses, of which the truly applicable one cannot be determined. It would be presumptuous to assert that the models constructed with the components, C0, and a network of the kind 'Pϕ,' in series may represent all aspects of the problem. It is however certain that the method roughly mentioned here will allow us to obtain equivalent circuits giving perfect inclined semi-circles in the complex capacitance plane, characteristic of a widespread behavior of many dielectrics, referred to as non-Debye. To choose the situation corresponding to reality among these models, or amongst models obtained by other possible methods, is a different matter, going beyond the scope of this paper. In the more complicated cases of non-linear or non-semi-circular complex impedance or capacitance diagrams, in the cases of any shape of the complex plots, many equivalents networks can be obtained. A direct and reliable mathematical method and the related software exist. The principle and principal steps of the theory are presented in ref. [5].
References [1] [2] [3] [4]
[5] [6]
KS. Cole and R.H. Cole, J. Chem. Phys. 9 (1941) 341. A.K Jonsher, J. Mater. Sci. 13 (1978) 553. KB. Oldham and J. Spanier, The Fractional Calculus (Academie Press, New York, 1974). J. Jacquelin, 'Use of fractional derivatives to express the properties of energy storage phenomena in electrical networks', Technical report, Laboratoires de Marcoussis (1st Ed. 1982, re-edit. 1984). J. Jacquelin, in: 3ème Forum sur les Impédances Electrochimiques, Montrouge, France, Nov. 1988, p. 91. J. Jacquelin, in: 2ème Forum sur les Impédances Electrochimiques, Montrouge, France, Oct. 1987, p. 279.
18
Impedance analysis and modelization of the alternating current properties of ionic conductors Jean Jacquelin
ABSTRACT: The common inclined semi-circular complex impedance diagrams of ionic conductors can be generated by various distribution laws of the physical properties. Four different examples of models are given, suggesting some kinds of physical phenomena likely to cause this behaviour. Adequate structures of networks can be found to modelize them accurately.
1. INTRODUCTION It has long been recognized that the complex impedance loci of many ionic conductors give well-defined semi-circles having their axes depressed below the real axis, (figure 1). In fact, in practice, these semi-circular diagrams are generally valid only in a limited range of frequencies: often, considerable deviations appear in higher or lower frequencies, but this matter will not be discussed here. An empirical formula, required by experimental evidence, express the complex impedance ( Z ) as a function of the frequency ( f ):
Z = R∞ +
R0 1+(i ω τ 0 )1− α
(1-1)
where: ω = 2 π f R∞ denotes the high frequency value of the impedance. (R∞+R0) denotes the low frequency value of the impedance. τ0 and α are empirical coefficients. A commonly accepted equivalent circuit is represented in figure 1. I would cite a pertinent critical comment from A.K.Jonscher in [1-2]: «The inclined semi-circular impedance plots are commonly, if rather loosely interpreted in terms of so-called distribution of relaxation times..., by analogy with the well-known empirical complex permittivity formula of the "non-debye" dielectrics:
ε = ε∞ +
ε0 1 + (i ω τ 0 )1− α
(1-2)
But it is easy to show that inclined circular ε- and Z-plots are not mutually compatible with one another ». In fact, in the two Eqs.(1-1) and (1-2), α has not the same signification: in Eq.(1-1), α corresponds to a phase displacement -(1-α)π/2 and on the other hand in (1-2), the phase displacement is -απ/2. Another amazing point has been emphasized: a term like (iω)1-α , such as appearing in Eq.(1-1), implies the existence of energy storage phenomena acting with a constant efficiency (i.e. an efficiency which is not function of the frequency), [1], [3]. Moreover, constant efficient energy storages exist in various other cases than ionic conductors. For example, in dielectrics and in electrochemistry in some cases of porous
19
electrodes, rough surfaces, interfaces, diffusion, etc...Inclined semi-circle loci are common in various representations in the complex impedance plane or in the complex admittance plane, or in the complex permittivity plane, etc..., caused by many kind of phenomena. The general thesis developed in [3] sheds light on this matter. Another way of writing the forgoing relationship (1-1) is the impedance z of an imperfect energy storage is considered, [3]:
z = Pϕ (iω)2ϕ/π
(1-3)
where ϕ is the constant phase displacement. In the imperfect storage ( -π/2 < ϕ < 0 ) and Pϕ is a constant coefficient. The relationships between R0, τ0, α and ϕ, Pϕ are:
π ϕ = (1 − α) 2
and
Pϕ = R 0 τ0α −1
(1–4)
If we will not take account of (1-3), we consider that Eqs.(1-4) are the definitions of ϕ and Pϕ from R0, τ0, α. Hence, the Eq.(1-1) can be writ ten:
Z = R∞ +
1
(1–5)
1 1 + R 0 Pϕ (iω)2ϕ/π
This relationship (1-5) is another manner to express this kind of behaviour and is strictly equivalent to (1-1). This is only a matter of notations, which is important to make clear, but it does not change the root of the problem. It is not our aim to make a list of the many kind of phenomena which could be the causes of this behaviour: It should be a masterpiece, out of our possibilities and beyond our knowledge, to sift all the experiments in which inclined semi-circles appear, and to list the phenomena more or less involved. Our concern is only to make available several models perfectly fitting to the Eq.(1-1) or (1-5). This material will probably be useful in modelizing various kinds of phenomena, as it is suggested at the end of this paper. All the theoretical parts and the practical examples of models given here stem from some more general ideas, already known and collected in [3] . But it is not necessary to have read [3], to understand the following pages and to use the practical results.
2. FIRST EXAMPLE, WITH CONTINUOUS DISTRIBUTIONS: In the field of dielectrics, the formula (1-2) has been extensively studied and discussed. The concept of a distribution of relaxation times was first introduced by E.Von Schweidler (1907), in trying to justify the empirical relaxation function proposed first by J.Hopkinson. Several laws of distribution have been proposed. One well-known is the Gaussian law of distribution proposed by K.W.Wagner (1913), leading to results close to the experiments, but not perfectly fitting, [4]. In the case of the ionic conductors, the phenomena involved are different, but, taking account of the similitude of the Eqs.(1-1) and (1-2), a similar mathematical treatment could be applied. The problem remains to assume appropriate distribution laws of relaxation times and, afterwards, to make them understandable on a physical point of view. Interesting results were obtained in some particular cases. For example, in [5], J.F.McCann and S.P.S.Badwal assumed an exponential law as a function of a distributed "activation energy" , acting on some physical properties. But, it is interesting to remark that the exponential law is not the unique law leading to a correct result. In this paper, and to give another view on the matter, we will chose arbitrarily another distribution law. The approach to the basic statements is somewhat
20
different. As a matter of fact, if a physical property is distributed as a function of a variable, in a limited range of validity, you have a large choice of various mathematical functions likely to be adjusted close to the actual physical distribution. Under these conditions, and in the range of validity, any one of these adjusted functions will, of course, lead to almost identical correct results. Now, suppose that some physical properties depend on a parameter (χ ), in a given range, from x to X. ( x<χ
γ 1 1 = + iω dz (ρ dχ) dχ
hence:
with the distribution of relaxation times:
z=
The total impedance is::
∫
X x
ρ dχ 1 + iω τ τ(χ) = ρ(χ) γ(χ) dz =
ρ dχ 1 + iω τ
(2-1 ) ( 2-2) (2-3)
The real and imaginary parts of z = z'+i z" are: X
ρ dχ 2 2 x 1+ ω τ X ωρ τ dχ z'' = 2 2 x 1+ ω τ z' =
∫ ∫
(2-4) (2-5)
If the distributions γ(χ) and ρ(χ) , hence τ(χ) , are known or presumed, it is possible to compute z'(ω) and z"(ω), then z(ω) and finally, with R0 in parallel and R∞ in series, you obtain Z(ω). Now, an example of law of distribution will be chosen. Remember that it is not unique: other laws lead to similar results. For example:
γ = γ1x a
(if a = 0 , γ = γ1 = constant)
(2-6)
ρ = ρ1x b
(2-7)
τ = τ1x a
(2-8)
(if b = 0 , ρ = ρ1 = constant) (with a,b constant). Hence: with τ1 = γ1 ρ1 The impedance z= z' + i z'' is obtain from (2-4) and (2-5):
z' = ρ1
∫
X x
x b dχ 1 + ω2 τ12 x 2(a+b)
z'' = ρ1τ1ω
∫
X x
(2-9)
x a+2b dχ 1 + ω2 τ12 x 2(a+b)
(2-10)
In a first step, we will consider complete distributions, i.e. distributions including all the relaxation times from 0 to ∞. This is the case if 0<χ<∞. Then, there is no major problem 21
in integrating (2-9) and (2-10): use the transformation ξ = ω2 τ12 x2(a+b) and the definite integral is known (see [6] p.256, line 6.1.17). The result is:
πρ1 (ω τ1 ) −(b +1)/(a + b) πρ1 (ω τ1 ) −(b +1)/(a + b) z' = and z'' = − π(b + 1) π(b + 1) 2(a + b) sin 2(a + b) cos 2(a + b) 2(a + b)
(2-11)
It is easy to deduce the complex impedance z = z'+iz" :
πρ1τ1− (b +1)/(a + b) z= (iω)− (b +1)/(a + b) π(b + 1) (a + b) sin 2(a + b) Finally, we obtain:
(2-12)
z = Pϕ (iω)2ϕ/π
(2-13)
πρ1τ1− (b +1)/(a + b) π(b + 1) (a + b) sin 2(a + b) π b +1 ϕ=− 2 a+b Pϕ =
where:
(2-14)
(2-15)
At this point, we already see that the chosen distribution leads to a correct expression (2-13), convenient to be introduced in (1-5). As a consequence, if we put in parallel R0 and in series R∞, we obtain the right impedance Z, given by (1-5), and the characteristic inclined semi-circle in the complex Impedance plane. One could object that a complete distribution,( i.e. from 0 to ∞ ), is not realistic: so, we must limit the distribution in a given range. This will be done in the next §. : The preceding results will be applied in practice, to make an actual equivalent circuit, which will give an effective inclined semi-circular plot in the complex impedance plane.
3. FIRST EXAMPLE, MODEL MADE WITH DISCRETE COMPONENTS. Instead of considering a continuous medium with the distributions γ(χ) and ρ(χ) , we will consider a succession of values of (χ): χ0 < χ1 < χ2 < ... < χn < ... < χN-1 < χN A capacitor and a resistor correspond to each interval:
Rn =
∫
χn
ρ(χ)dχ and
χ n-1
1 = Cn
∫
χn
χ n-1
dχ γ(χ)
(3-1)
a
with 1 ≤ n ≤ N (See figure 2). For example with the law of distribution: γ = γ1 χ , ρ = ρ1 χ and with a repartition of the values of (χ) defined by
χ n = βn χ 0
( β is constant > 1 ) it is easy to compute each value of Cn and Rn . Then, let:
h = β n −1
and
k = β n +1
22
(3-2) (3-3)
b
This leads to the recurrence formulas:
Cn = C1 h n-1
and and the Eqs. (2-14) and (2-15) lead to:
ϕ=−
R n = R1 k n-1
(3-4 )
π 1 2 1 + ln(h) ln(k)
h −1 Pϕ = C1 h (2ϕ + π)
(3-5)
2ϕ/π
1+ (2ϕ/π)
−2ϕ R1 k −1
1 sin( − 2ϕ)
(3-6)
In practice, you have the choice of C1, R1, h, k and N. Then, compute the values of the capacitors, resistors (3-4), and the values of ϕ (3-5) and Pϕ (3-6) . If they are connected as represented in figure 2 , the resulting theoretical complex impedance z of this network will be:
z ≃ Pϕ (i ω) 2ϕ/π
(3-7)
The actual result is very close to the theory, as the following actual example will show: Coefficients: C1 = 0.01µF ; R1 = 0.1Ω ; h = 1.78 ; k = 2.15 ; N = 20 . In practice, the actual values of the capacitors and resistors are not exactly the computed values (3-4) but approximate values taken in series of components available on the market. They are given in the table 1 (this table contains also the values corresponding to the other examples, see in §4 ). The result, corresponding to the network represented in figure 2, is very good: compare the theoretical straight line "Pϕ" , drawn from Eq. (3-7), to the actual points given in figure 2. In fact, the range of validity is large: from 0.1Hz to 10MHz in this example. It could be extended by increasing the number of components. But, it is generally not necessary to have a so large range of validity to modelize accurately the actual experiments. lf you put in parallel a resistor, for example R0=150Ω , with this network, you obtain an almost perfect inclined semi-circle. The Eqs.(3-5) and (3-6) permit to compute ϕ and Pϕ. They are close to the empirical values from figure 3 : ϕ ≈ -0.57 π/2 ≈ 51° and Pϕ ≈ 33,000. Then, from (1-4), we obtain: α ≈ 0.43 and τ0 ≈ 80µs. These values can be used in the Eqs (1-5) or (1-9). As a consequence, this shows that the model is perfect to represent this kind of behaviour. In the next §, we will see that many other models are convenient as well. Remark : In the case of the models made with discrete components, the distributions are incomplete. As a consequence, we will observe some deviations at low frequencies and at high frequencies. But the resistor R0 in series makes the deviation negligible at low frequency. The resistor R∞ in series makes the deviation negligible at high frequency. An other point is: When you replace the continuous medium by a succession of discrete components, you introduce another kind of deviations (appearing as undulations around the theoretical locus). All this points: range of validity and deviations must be taken in account in the choice of k, h, N but it would involve too long mathematical development to be studied here.
23
4. OTHER MODELS We have already point out that several different laws of relaxation times can be used, leading to a common kind of model represented in figure 2. Moreover, they are different structures of models convenient as well. To be concise enough, we will only give a sketch of the main results. The method is the same: you define the physical properties of a continuous medium, as we did in §2, but with different properties and, so, different basic physical laws. Then, you choose some appropriate distribution laws acting on the physical coefficients, as we did in §2, but the distribution laws could be different. You solve the mathematical problem of integrating the equations. Then, you replace the continuous medium by a succession of discrete components in a limited range as we did in §3. In some cases, this method leads to perfect inclined semi-circular plots in the complex plane. After the preceding first example, three other kinds of mediums lead to the three different models represented in figures 3, 4 and 5. To simplify, we will always use the same following recurrence formulas (which, in practice, have the advantage to lead to series of resistors and capacitors currently available on the market):
Cn = C1 h n-1
n-1
and R n = R1 k (4-1) (with 1 ≤ n ≤ N ) . So, you have the choice of C1 , R1 , h, k and N. But do not think that this recurrence formulas are unique, others could be chosen, leading to similar results. Not too difficult mathematical treatments permit to show that the following formula is common to these examples:
ϕ=−
π 1 2 1 + ln(h) ln(k)
(4-2)
The complex impedances formulas are also common:
z ≃ Pϕ (i ω) 2ϕ/π
(4-3)
But the values of (Pϕ) are not the same: - In the case represented in figure 2, we already obtained:
h −1 Pϕ = C1 h (2ϕ + π)
2ϕ/π
1+ (2ϕ/π)
−2ϕ R1 k −1
1 sin( − 2ϕ)
(4-4)
sin( − 2ϕ)
(4-5)
- In the case of the model represented in figure 3:
2ϕ + π Pϕ = C1 h −1
2ϕ/π
1+ (2ϕ/π)
k −1 R 1 (−2ϕ) k
- In the case of the ladder network represented in figure 4: ( the main steps of the demonstration are given in [3]. Note: Γ = gamma function, [6] p.255-293)
1 + (2ϕ/π) Pϕ = C1 h −1
2ϕ/π
1+ (2ϕ/π)
−2ϕ/π R1 k −1
Γ ( −2ϕ/π ) Γ (1 + (2ϕ/π) )
(4-6)
- In the case of the ladder network represented in figure 5:
h −1 Pϕ = C1 h (1 + (2ϕ/π) )
2ϕ/π
1+ (2ϕ/π)
k −1 k ( − 2ϕ/π) R1
24
Γ (1 + (2ϕ/π) ) Γ ( −2ϕ/π )
(4-7)
The complex capacitance loci of all these networks are straight lines (lines "Pϕ" in figures 2, 3, 4, 5). lf you put each network in parallel with a resistor R0 and a resistor R∞ in series, you obtain a semicircle locus corresponding to the formula:
1
Z = R∞ +
(4-8)
1 1 + R 0 Pϕ (iω)2ϕ/π
or to the formula:
Z = R∞ + where:
R0 1+(i ω τ 0 )1− α
2ϕ α =1+ π
and
(4-9)
R τ0 = 0 Pϕ
− π /(2 ϕ)
(4-10)
Remark: The Eqs.(4-4) , (4-5) , (4-6) and (4-7) are so much accurate as k, h are not far to 1 and N is large. Some practical examples of complex impedance plots are given in figures 2, 3, 4 and 5. To simplify, we choose the same set of capacitors in all the examples. The values are given in table 1 . They, roughly, correspond to the coefficients: h = 1.78 k = 2.15 N = 20 and C1 = 0.01µF The values of the resistors are not the same in each example. They are given in table 1. The complex impedance of the four networks are represented in figures 2, 3, 4 and 5. (i.e. the straight lines "Pϕ") .With a resistor R0 = 150Ω in parallel with each network, the complex impedance plots become inclined semi-circles. Note that the values of the components have intentionally been chosen so that we obtain appreciatively the same values of ϕ and Pϕ in the four examples:
ϕ ≃ −0.57 π / 2 ≃ −51°
and
Pϕ ≃ 33000 ( unit : s -0.57Ω )
(measured values, from the impedance diagrams). In the Eq.(4-9), these values correspond to: α = 0.43 and τ0 = 80µs As a consequence, the actual complex impedance diagrams of these four different networks are almost identical, as it is obvious in figures 2, 3, 4 and 5. This is to show that completely different structures of networks lead possibly to the same complex impedance in large range of frequencies.
6. CONCLUDING REMARKS Very simple laws of distributions of the physical properties lead to the characteristic inclined semi-circle in the complex impedance plane, in large ranges of validity and perfectly consistent with the actual experiments which are, of necessity, limited to some ranges of frequencies. The concept of "distribution of relaxation times" is adequate in some cases, for example corresponding to the figures 2 and 3 : the various relaxation times are well defined, obviously visible and comprehensible. It is somewhat different in the cases of the models represented in figures 4 and 5 : invoking only a distribution of relaxation times does not give a complete and true image. The four examples given in this paper suggest some possible causes for this characteristic kind of behaviour:
25
The first model , represented in figure 2, suggests the case of ionic conductors in which a part is purely resistive (R0) and in which some of the electric charge carriers has to pass through some higher resistive paths distributed in the bulk where the charge carriers remain more or less stored. The second model , represented in figure 3, suggests the case of ionic conductors with imperfect terminal electrodes. Some parts act as elementary blocking electrodes, distributed on the interface. An alternative interpretation could be: some ionic conductive paths are interrupted in the bulk, in places where some charge carriers are stored, constituting the equivalent of distributed capacitances. The third model , represented in figure 4, has often been used in the cases of ionic conductors in porous electrodes.[7], [8]. This kind of model is also used in the cases of ionic conductors in contact with rough electrodes [9]. The fourth model , represented in figure 5, has no evident application in the field of ionic conductors, except if we consider the cases of preponderant electrostatic energy storage phenomena in the bulk: In the different field of dielectrics, a theory involving the penetration of the electromagnetic field in materials and through interfaces, leads to a model of the kind represented in figure 5 , in which the resistors are not actual conductive paths, but are a logical and comprehensible manner to represent the energy losses. It is not necessary to develop this point here, which should be far too long and probably out of subject. Anyhow, we intend to limit this paper to the few following remarks: (i). They are many perfect models, equivalent circuits and laws of distributions of the physical properties, hence of the relaxation times, leading exactly to inclined semi-circular loci in the complex impedance plane. (ii). This strongly suggests that many different phenomena are likely to cause this kind of behaviour. It is not surprising: the literature is extensive concerning various kinds of phenomena more ore less involved in the manifestations of inclined semi-circles not only in the field of ionic conductors, but in completely different fields, for example in dielectrics. (iii). Different structures of models give exactly the same result, i.e. the same complex impedance as a function of the frequency in wide ranges of validity. In this paper we give four examples of networks with different values of components and with different structures, and these four networks give the same response to impedance analysis, i.e. identical inclined semi-circles in the complex impedance plane. (iiii). There is a choice of several different models to represent accurately one experiment. The problem is how to determine, among them, which model is the best representation of the actual physical cause. Many phenomena are likely to give similar impedance diagrams. Many different interpretations can be given concerning one impedance diagram. We could imagine other interpretations than the preceding ones. So, in trying to explain a given experimental impedance diagram, it is generally not difficult to find several different models perfectly fitting and to put forward several hypothesis. But, you do not know which one is the true one. I would say, to conclude, that it should be pretentious to assert that the models made with the components R0, R∞ and a network of the kind "Pϕ" given here, are the bests in all viewpoints. One thing is sure: this method permit to obtain some equivalent circuits which give perfect inclined semi-circles in the complex impedance plane, characteristic of a widespread behaviour of many ionic conductors. It is another thing, to choose, among these models or among other models obtained by other methods if possible, which model is the most appropriate on a physical viewpoint to give the true image of each mechanism of the physical phenomena. This is a different matter, far beyond the content of this paper.
26
REFERENCES [1] [2] [3]
[4] [5] [6] [7] [8J 191
A.K.Jonscher. J.of Mat. Sci. 13 , (1978) , 553. A.K.Jonscher. Phys. Stat. Sol. (a)32 , (1975), 665. J.Jacquelin."Use of fractional derivatives to express the properties of energy storage phenomena". Unpublished. LABORATOIRES DE MARCOUSSIS, Route de Nozay, 91460, Marcoussis.(France) W.A.Yager. Physics. 7 , (1936), 434. J.F.McCann, S.P.S.Badwal, .J. Elect. Chem. Soc. 129 , (1982), 551. M.Abramowitz, I.A.Stegun, "Handbook of Mathematical functions" Dover Publ. N-Y. 9th print (1970). R. de Levie .Electrochimica Acta. 8 , (1963), 751. R. de Levie. Electrochimica Acta. 9 , (1964), 1231. R. de Levie. Electrochimica Acta. 10 , (1965), 113.
TABLE 1 : Values of components of the four examples of equivalent circuits.
27
Figure 1 Complex plane locus of the complex impedance and equivalent circuit, as required by experimental evidence, for many ionic conductors.
28
Figure 2 First example of network convenient to obtain inclined linear or inclined semi-circular plots in the complex impedance plane. (Values of components in table 1) 29
Figure 3 Second example of network convenient to obtain inclined linear or inclined semi-circular plots in the complex impedance plane. (Values of components in table 1)
30
Figure 4 Third example of network convenient to obtain inclined linear or inclined semi-circular plots in the complex impedance plane. (Values of components in table 1)
31
Figure 5 Fourth example of network convenient to obtain inclined linear or inclined semi-circular plots in the complex impedance plane. (Values of components in table 1)
32
