theoretical impedance of capacitive electrodes

THEORETICAL IMPEDANCES

OF CAPACITIVE ELECTRODES

Jean Jacquelin

Updated edition with the full

mathematical developments.

J.Jacquelin, "Theoretical impedance of capacitive electrodes". [1992-1995, updated : December 2009] - 2 -

THEORETICAL IMPEDANCES OF CAPACITIVE ELECTRODES

Jean Jacquelin

ABSTRACT

Theoretical studies of capacitive electrodes are succinctly presented. In an ideal case,

the electrode is perfectly flat, smooth, uniformly covered with a capacitive layer and subjected

to a uniform electric field. This quite utopian case is equivalent to a resistor coupled in series

with a capacitor.

Many phenomenon are likely to make some disturbances and then modify the

impedance of the electrode. Among a lot of possible causes, three are considered successively

- Non-flat macroscopic shapes (for example a sharp point)

- Non-flat microscopic shapes (roughness for example)

- Non-uniform property of the capacitive layer.

Thick and porous electrodes are outside the scope of the study. Nothing fractal is involved.

The approach is conducted within the Cartesian geometry tradition and no advanced

mathematic is required (But we have to carry out heavy derivations)..

The most interesting result is that, several times, impedances containing the term (iω)ν

with a non-integer exponent (ν ) are obtained. This special impedance is known as Constant

Phase Angle (CPA). In some specific cases, the term Log(iω) also appears.

These examples of CPA seem to confirm again the assumption that this type of special

element may be generated from many different aspects or phenomena. Considering the

structure of the equivalent electrical network found, which is a ladder network, and the

formula which gives the values of the components of this network, a hierarchical character

appears. In the literature, CPA response is generally related to a hierarchical character, and

quite rightly to fractal geometry. However, in our academic examples, there is no fractal

geometry involved and nothing can be suspected to present any fractal aspect. Therefore, a

hierarchical character is also likely to appear in situations where nothing fractal is involved.

The special impedance term Log(iω) can be considered as a borderline case for CPA.

It seems to exist when points or sharp edges are present. Obviously, detecting it while

experimental impedance analysis would be a delicate task because one have to separate it

from the other more important terms of the whole impedance.

KEY WORDS : Impedance, Capacitive Electrode, Modelling, Point/plan, Roughness.


CONTENT

The first paper (pp. 4–25) : "Theoretical impedances of Point / Plane electrode

configuration" was published in Electrochimica Acta, Vol. 38, No. 4, pp. 597-606, 1993.

Point / Plane electrode configuration are studied here analytically and by a numerical

finite element approach, and the results agree. In the formulas obtained for the

admittance and impedance, one remarkable function appears, which is Log(iω). Its

representation in the complex graph is a horizontal line. The mechanism governing the

impedance variation as a function of the frequency is fully described

- The second paper (pp. 26 – 46) : "Theoretical impedance of rough electrodes with

smooth shapes of roughness" was published in Electrochimica Acta, Vol. 39, No. 18, pp.

2673-2684, 1994.

A theoretical study of rough electrodes is succinctly presented, for small-size and

smooth-shape roughness situations. The analytical result is a formula which derives

the resulting impedance from the parameters of the shape of such a roughness. An

equivalent electrical network is then derived with very simple formulae in order to

calculate the component values. The most original part of the results consists in

specific theoretical examples. Several times, impedances with the term (iω)ν with a

non integer exponent (ν) are obtained. This paper shows that from a theoretical point

of view, smooth roughness profiles with some simple shapes may result in a special

type of impedance known as Constant Phase Angle (CPA). In some specific cases,

impedances with the term log(iω) are also obtained.

- The third paper (pp. 47 – 56) : "Theoretical impedances of smooth flat electrodes with

slightly non-uniform double layer capacitance" was written in 1995 (unpublished).

A purely theoretical study of a flat and smooth electrode, covered with a non-uniform

double-layer capacitance, is succinctly presented. The analytical result is a formula

which describes the impedance in terms of parameters of the capacitance distribution.

The whole impedance includes a main resistive term, a main capacitive term and a

special term. Several times, impedances containing the special term (iω)ν with a non-

integer exponent (ν ) are obtained. This special impedance is known as Constant Phase

Angle (CPA). In some specific cases, the term log(iω) also appears. An interesting

result is to have found the same special impedance element in two different situations :

one in the case of some rough electrodes with a uniform double-layer capacitance,

second in the case of some smooth electrodes with non-uniform double-layer

capacitance.

- In addition :

pp. 57-94 : Theoretical impedance of rough electrodes with smooth shapes of

roughness. Complete collection of equations.

pp. 95-106 : Theoretical impedances of smooth flat electrodes with slightly non-

uniform double layer capacitance. Complete collection of equations.


THEORETICAL IMPEDANCES

OF POINT / PLANE ELECTRODE CONFIGURATION

Jean Jacquelin

ABSTRACT

Point / Plane electrode configuration are studied here analytically and by a numerical

finite element approach, and the results agree.

In the formulas obtained for the admittance and impedance, one remarkable function

appears, which is Log(iω). Its representation in the complex graph is a horizontal line.

The mechanism governing the impedance variation as a function of the frequency

is fully described.

KEY WORDS : Impedance, Electrode, Modelling, Point / plane.


(10) INTRODUCTION

It is rare that an analytical solution can be found for an electrode impedance problem,

as the modelling leads to complicated equations of partial derivatives. The search for

analytical solution is usually abandoned, in favour of numerical calculations that are possible

with today's computers. This is fully justified for the current research and development

requirements.

In the context of thought and research farther upstream, we would like to have a set of

analytical solutions that will give us a better understanding of all of the factors of importance,

in parametric form. If we want to make any headway in this direction, we must first attempt to

solve the simplest possible academic cases before trying those cases that are frequently more

arduous. The point / plane configuration, with its very simple geometry, offers some hope of

finding an analytical solution.

The results for the point / plane configuration are given in the second part of this

paper, but first we will consider the even simpler problem of the blade / plane configuration as

represented in figures 1 and 2.

Figure 1 : First blade / plane configuration.

Figure 2 : Second blade / plane configuration.


The space between the electrodes is assumed to be conductive, with a resistivity ρ

(Ω/m). One of the electrodes – either the blade or the plane – is assumed to have a capacitive

surface layer with a capacitance per unit area of γ (F/m2). A sinusoidal difference of potential

V(t)=Vp cos(ω t) of amplitude Vp and angular frequency ω=2πf is applied between the two

electrodes. We want to express Ip appearing in the expression for the current I(t)=Ip cos(ωt-Φ)

analytically, along with the complex admittance Y and the complex impedance Z = l/Y=

(Vp/Ip) exp(iΦ).

These two blade / plane configurations lead to the same mathematical model, shown in

figure 3. Only half is shown, for reason of symmetry. A macroscopic limit (E E') should be

added along with a microscopic limit (ε ε'), for which we will study the influences on the

result.

Figure 3 : Mathematical model for the blade / plane configurations.

The Laplace partial differential equation, 2 2 2 2/ / 0V V x V y∆ = ∂ ∂ + ∂ ∂ = , considered

alone, has an infinite number of solutions. For example, F being an arbitrary function, the

"harmonic" functions F(x+iy) and F(x-iy) are solutions.

The real difficulty appears when we want to find a function that satisfy the boundary

conditions of the modelled domain. In the present case, on the boundary (ε' E') modelling an

electrode, the condition V=Vp (given parameter) is set. This condition is therefore of the

Dirichlet type. The condition ( ) /i V V yω γ ρ = ∂ ∂ is set on the (ε E) boundary modelling the

capacitive electrode. This is a Fourier condition. On the macroscopic and microscopic

boundaries ( E E' and ε ε', respectively), the / 0V n∂ ∂ =

condition ( n

being the normal to the

boundary) is of the Newmann type. This problem of an elliptical type of equation ∆V=0, with

Dirichlet-Fourier-Neumann boundary conditions, is therefore well known, which does not

mean that it has been solved. The well known formula of Green can be used to find the


function V(x,y) when V and /V n∂ ∂

are simultaneously known on the boundary, which is not

the case. In the present circumstances, looking for compatible V and /V n∂ ∂

on the boundary

changes the way the problem is presented without solving the problem itself, and is of no help

to us in finding the analytical solution. A mathematician would say that we must "abandon all

hope of obtaining an explicit solution", even in a simplified geometry, because problems of

this kind remain analytically insoluble.

The results of the theoretical blade / plane problem, and then of the point / plane, have

been obtained rather by two processes that we will be presenting in order: first a conventional

numerical calculation by computer, and then an approximation of the analytical solution by a

variational method, which yields an interesting result in the present case. Finally, the literal

formula obtained is compared with the individual results of the numerical method.

(10) NUMERICAL SOLUTION

The main difficulty encountered results from the very different orders of magnitude of

the complex numbers on which the operations are carried out. Inverting a large dimension

matrix that includes terms that are alternately positive and negative and whose values may

extend over several decades, is a common example. Unless proper care taken, the results can

be highly marred or even completely meaningless. So it is essential to make a careful choice

of algorithm or of processes and means that will ensure maximum precision, and it is also

indispensable to check the validity of numerical results. But these considerations are well

known in the field of numerical computation, and there is no need for further comment.

The finite element method was used, in most cases with a 100 x 100 grid, hence

10,000 nodes. In certain configurations, it is hardly sufficient to carry out the operations to 16

significant decimal places. The grid with equidistant nodes is generated after a change of

reference systems. (x , y) becomes (ζ , θ ) by the transformation ( x=r.cos(θ) ; y=r.sin(θ) ;

r=E.exp(ζ) ). Figure 3 is transformed into figure 4.

The numerical calculation was carried out systematically for different values of the

parameters ( ωγρ, E, ε ) and for different values α, in the domain 0 < α ≤ π/2.

Figure 4: Conform transformation of figure 3.


2. SEARCH FOR AN ANALYTICAL SOLUTION

The search for a solution expressed in the form of a sum of known elementary

solutions generally fails for reasons of non-stability or non-convergence.

For example, in the (ζ , θ ) axis system of figure 4, the function V(ζ , θ ) defined by

the following sum (in which the an are the unknown coefficients) is a solution of the Laplace

equation ∆V=0 and clearly satisfies the condition V(ζ , α )=Vp on the boundary (ε' E') :

( )( , ) exp( )sin ( )p nV V a n nζ θ ζ α θ= + −∑

The coefficients an must be calculated so that the condition exp( ) /i EV Vω γ ρ ζ θ= ∂ ∂ is

satisfied at the (ε E) boundary, i.e. for θ=0. This leads to the recurrence relations:

( )

1

1

cos( )

cos( ) sin ( 1)

p

n n

a i EV

a n i E a n

α ω γ ρ

α ω γ ρ α−

= −

= − −

and thus literally:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )sin ( 1) sin ( 2) ...sin 2 sin( )

! cos cos ( 1) ...cos 2 cos

n

n p

n ni Ea V

n n n

α α α αω γ ρ

α α α α

− −−=

−

So we have in fact obtained a solution V(ζ , θ ) expressed as a sum of elementary solutions

( )exp( )sin ( )na n nζ α θ− , in which all of the coefficients an are known now. But this formal

expression is not viable because we see that, for fixed α, we will always find zero terms or

terms close to zero in the infinite series cos(α),cos(2α), cos(3α), …, which will make some an

very large or indeterminate.

We always encounter this type of difficulty in finding a stable analytical solution,

which was illustrated here in "naïve" form, when solutions are sought in the form of series of

functions of the coordinates (x,y) or (r,θ), or (ζ , θ ), and so on.

A different approach would consist of starting with some "trite" configuration in

which the problem has a known solution, and then trying to deviate slightly from this

configuration and extend the "trite" solution with a slight correction. The search then amounts

to a series expansion as a function of the parameters defining the boundary configuration. In

the present case, the "trite" problem is that of two parallel plane electrodes (α.=0), from

which we deviate slightly by considering small angles α.. The search then consists of series

expansions as a function of α. (and of θ, which is of the same order of magnitude as α. Since

0 ≤ θ ≤ α ). Let f(ζ) and g(ζ) to designate the object functions on the boundary ε E (θ=0) expressed using

a series as a function of α., in wich f0 , f1 , f2 , …g0 , g1 , g2 , … are function of ζ :

20 1 2( ,0) f ( ) ... n

nV f f f fζ ζ α α α= = + + + =∑ (1)

20 1 2

( ,0)

g( ) ... nn

Vg g g f

ζ

ζ α α αθ

∂ = = + + + =

∂ ∑ (2)

The relation between fn and gn is given by the condition set on the boundary ε E :

/ exp( ) ( ,0)V i E Vθ ω γ ρ ζ ζ∂ ∂ = , thus:

exp( )n ng i E fω γ ρ ζ= (3)

The successive derivatives (for θ=0) are found from ∆V=0 :


22 2 2

2 2 2 2

44 4 4

4 4 4 4

dd f

d d

dd f

d d

nn

nn

fV V

fV V

αθ ζ ζ ζ

αθ ζ ζ ζ

∂ ∂= − = − = −

∂ ∂

∂ ∂= + = + = +

∂ ∂

∑

∑

… etc …

23 2 2

3 2 2 2

45 4 4

5 4 4 4

dd g

d d

dd g

d d

nn

nn

gV V

gV V

αθθ ζ ζ ζ

αθθ ζ ζ ζ

∂ ∂ ∂ = − = − = −

∂ ∂ ∂

∂ ∂ ∂ = + = + = +

∂ ∂ ∂

∑

∑

… etc …

The Taylor formula yields V(ζ,θ) expanded in a series of θ :

2 2 2 3 4 4 4 5

2 2 4 4

d f d g d f d g( , ) f g ...

2! 3! 4! 5!d d d dV

θ θ θ θζ θ θ

ζ ζ ζ ζ

= + − − + + +

(4)

2 22 3

2 2

0

4 44 5

4 4

d d( , )

2! 3!d d

d d ...

4! 5!d d

n nn n

n

nn n

f gV f g

f g

θ θζ θ θ

ζ ζ

θ θα

ζ ζ

=

= + − − +

+ + +

∑ (5)

On the boundary ε' E', we have the condition V(ζ ,α)=Vp , therefore :

2 22 3

2 2

0

4 44 5

4 4

d d

2! 3!d d

d d ...

4! 5!d d

n np n n

n

nn n

f gV f g

f g

α αα

ζ ζ

α αα

ζ ζ

=

= + − − +

+ + +

∑ (6)

Grouping the terms in the same power of α together, we get :

2 22 3

1 2 2

0

4 44 5

4 4

d d1 1

2! 3!d d

d d1 1 ...

4! 5!d d

n np n n

n

nn n

f gV f g

f g

ζ ζ

αζ ζ

− −−

=

− −

= + − − +

+ + +

∑ (7)

We can consider a more or less large number of terms in this series in α , α2, α3

, …. This

yields, successively :


0

1 0

20

2 1 2

2201

3 2 2 2

0

d10

2! d

dd1 10

2! 3!d d

pV f

f g

ff g

gff g

ζ

ζ ζ

=

= +

= + −

= + − −

etc …

The successive functions are therefore obtained using (3) :

0 0

21 1

2 32 2

33

( ) exp( )

( ) exp( ) ( ) exp(2 )

( ) exp(2 ) ( ) exp(3 )

1( ) exp(3 ) ( )ex

3

p p

p p

p p

p

f V g V i E

f V i E g V i E

f V i E g V i E

f V i E i E

ω γ ρ ζ

ω γ ρ ζ ω γ ρ ζ

ω γ ρ ζ ω γ ρ ζ

ω γ ρ ζ ω γ ρ

= = −

= − =

= =

= − + p( )ζ

etc …

This leads to a series expansion of V(ζ , 0):

2 2

3 3 4

( ,0) f ( ) 1 ( ) exp( ) ( ) exp(2 )

1 ( ) exp(3 ) ( )exp( ) (...)

3

pV V i E i E

i E i E

ζ ζ ω γ ρ ζ α ω γ ρ ζ α

ω γ ρ ζ ω γ ρ ζ α α

= = − +

− + +

(8)

This formula can be written in the following equivalent form, limiting ourselves the

terms in α2 inclusive:

3( ,0) f ( ) (...)1 exp( )

pVV

i Eζ ζ α

ω γ ρ α ζ= = +

+ (9)

Of course, the series expansion can be taken further, but the more we increase the number of

terms, the closer we come to the infinite series, which would lead to the difficulties pointed

out at the beginning of this section.

This approximate result (9) is substituted in (4) using only the terms in θ2 inclusive (of the

same order as α2, since 0 ≤ θ ≤ α ), and we get :

31 exp( )( ,0) (...)

1 exp( )p

i EV V

i E

ω γ ρ θ ζζ α

ω γ ρ α ζ

+= +

+ (10)

When this formula is expressed in polar coordinates (r,θ), it becomes :

31( , ) (...)

1p

i E rV r V

i E r

ω γ ρ θθ α

ω γ ρ α

+= +

+ (11)

It seems at first sight that the function (10) does not satisfy the equation ∆V=0. That is, we see

that 2 2/ 0V θ∂ ∂ = and 2 2/ 0V ζ∂ ∂ ≠ . But this is normal because, in this method, we are not

looking for an exact equality but rather only looking for the equality for the terms in α of an

order less than the one to which we have limited the series.


The admittance is calculated by :

( 0)

2 d 2 d

1

E EL V r L i E rY

r i E rε εθ

ω γ ρ α

ρ θ ρ ω γ ρ α=

∂ = =

∂ + ∫ ∫ (12)

The integration yields :

( )2

2

1 ( ) 2ln arctan( ) arctan( )

1 ( )

L i E LY i i E i

i

ω γ ρ αω γ ρ α ω γ ρ α ε

ρ α ρ αω γ ρ α ε

+= + −

+ (13)

The expression (13) can be simplified if ω is in the range:

1 1

Eω

γ ρ α γ ρ α ε≪ ≪ (14)

That is, since (ωγραE) is large, unity will first be negligible with respect to it. Then

arctan(ωγραE) ≈ π / 2 . As (ωγραε) is small, we may neglect it with respect to unity

and then arctan(ωγραε) is small and can also be neglected. We get:

( ) ( )2 2

ln lnL L L

Y i E E iπ

ω γ ρ α ω γ ρ αρ α ρ α ρ α

≈ = + (15)

3. COMPARISON OF RESULTS OBTAINED BY THE NUMERICAL METHOD AND

BY ANALYTICAL CALCULATION

Figure 5 gives one example (case α = π / 2) of a comparison of impedance diagrams (Z', Z")

and complex admittance diagrams (Y', Y"). The solid curve corresponds to formula (13) and

the dotted one to the simplified formula (15). The crosses are results obtained by numerical

finite element method.

Figure 5: Impedance and admittance diagrams of blade / plane configuration (α = π/2 ).


A second comparative example of numerical and analytical results is presented in

figure 6, with the following parameters : ( α = π/3 ; ε / E = 10-10

). The voltage on the (ε E)

boundary is plotted as a function of ζ for different values of (ωγρE). The solid line

corresponds to the analytical solution of formula (9). The results of the finite element

calculation are represented by crosses.

The agreement is good, with a slight deviation in the transition zone. This result is all

the more remarkable as the expansion in powers of α by formula (9) has been limited to a

very few terms.

Figure 6: Comparative results. Numerical (crosses) and analytical (solid lines).


Figure 7 compares equipotential plots (in the case ωγρE =105). The dotted lines correspond to

the numerical calculation.

Figure 7: Comparison of numerical (solid lines) and analytical (dotted) results.

Equipotential lines.

To get a more general and more precise view of the comparative results, Table 1 gives

the admittance values (Y=Y'+iY") for a large number of examples, in form of dimensionless

numbers (Y'ρ / 2L) and (Y''ρ / 2L). Three results are compared in each case, from top to

bottom:

- numerical calculation by finite elements

- analytical calculation by formula (13)

- analytical calculation by simplified formula (15), which does not take microscopic or

macroscopic limits into account.

The comparison is very satisfactory. As expected, deviations do appear at low and

high values of ω when condition (14) is not verified. The agreement is very good for all the

configurations calculated.


Table 1: Comparison of numerical and analytical results for blade / plane configuration.

4. RESULTS

4. 1. Impedances and Admittances :

An analytical solution based on a series expansion with respect to α leads to a very

simple formula (11) for the potential distribution in the inter-electrode space. We deduce the

literal formula (13) from this for the admittance Y= 1/Z. Plotted in a complex admittance

graph, Y=Y'+iY" is a very much flattened curve (the solid curve in figure 5).

The backward bends at the ends of this curve are the effect of the microscopic and

macroscopic limits. When the frequency is within the range given by (14), the disturbances

due to the microscopic and macroscopic limits become negligible. The admittance formula

then simplifies to (15). The complex admittance diagram is then a perfectly straight horizontal

line close to the real axis (dotted line in figure 5). We see a ln(iω))=ln(ω) + i π/2 term appear.

The analytical solution (15) shows the main effects of the physical parameters ρ and γ , and

the geometric parameter α.. This is in agreement with the results of the numerical method,

which are plotted in figures 8 and 9. The complex total current ( I ) is computed by summing

the elementary currents along the (εE) boundary. We deduce from this the total admittance

(Y = I / Vp ). This was done for various frequencies, and we obtained the results shown in

figure 8. We see that the real part of the admittance, (Y'), is proportional to ln(ω) and that the

imaginary part of admittance, (Y"), is constant.


Figure 8: Admittances as a function of the frequency.


Figure 9: Complex admittance diagrams.

4. 2. Description of phenomena near the point :

Figure 10 gives an example of the equipotential lines obtained in the (ζ , θ ) axis

system and then converts it to the initial (x,y) axis system, in several nlargements.

In the point zone, i.e. the vicinity of the microscopic limit, the voltage is very close to

Vp , so all the difference of potential is concentrated in the capacitive surface layer. This

microscopic area therefore has an admittance (Ym=1/ Zm) very close to that of a capacitor,

Ym=iωγs. The value of this capacitor, cm=γs, depends on the area (s). The admittance of this

zone therefore does not depend on the geometry but only on the area of the capacitive layer

immersed in this zone.

4. 3. Description of phenomena far from the point :

It can also be seen in figure 10 that, in the vicinity of the macroscopic limit far from

the point, all the difference of potential is distributed through the resistive medium. While the

difference of potential through the capacitive layer is very small. This macroscopic zone

therefore has an admittance ( YM=1/ ZM ) very close to that of a pure resistor, ZM=RM. The

value of this pure resistor depends on the resistivity p and on the geometrical configuration of

the macroscopic limit.


Figure 10: Views of various enlargements of the equipotential lines.


Figure 11: Shifts of equipotential lines with frequency.


Figure 12: Schematic description of the cause of admittance variation

as a function of frequency.

4. 4. Description of the phenomena according to the frequency :

Figure 11 shows the effect of the frequency (taking thee examples of three different

values of ω ). It is seen that the map of equipotential lines remains the same, except it is

translated into the (ζ , θ) representation. This corresponds to a scaling in the real ( x , y)


representation. This "transition" zone between the microscopic equivalent capacitor (cm) and

the macroscopic equivalent resistor (RM) shifts with the frequency.

The total admittance therefore depends on the frequency : the higher the frequency, the

closer the "transition" zone comes to the electrode point. This displacement is accompanied

by a replacement of capacitance by conductance. This qualitatively describes the electrical

phenomenon that is the fundamental cause of the impedance variation as a function of the

frequency. Figure 12 gives a schematic representation of all of these phenomena.

4. 5. Description of the limits :

It is obvious that all the above results are valid only if the "transition" zone (which

shifts along the electrode as a function of the frequency) remains well within the microscopic

and macroscopic limits.

If this is not the case, e.g. if the frequency is high enough to bring this transition zone

to the tip of the point, the capacitive layer does not affect the equipotential map, which is then

entirely determined by the conductive medium.

If, on the other hand, the frequency is low enough to place this "transition" zone near

the macroscopic limit, the conductive medium is practically equipotential. All the difference

of potential is then shifted to the entire capacitive layer, the impedance of which completely

determines the impedance of the whole.

The effects of these limits on the impedance and admittance appear clearly in figure 5.

5. POINT / PLANE ELECTRODE CONFIGURATIONS

Figure 13: First point / plane configuration: capacitive layer on the point.


Figure 14: Second point / plane configuration: capacitive layer on the plane.

Figure 15: Third point / plane configuration: capacitive layers on both electrodes


The same method was used to deal with conic point / plane configurations (see figures

13, 14, 15). By symmetry of revolution the problem can be reduced to two dimensions. The

equations are different, but they can be treated like those of the blade / plane configuration.

In (r , θ ) coordinates, the Laplace equation is written :

2

2

1 1cos( )

cos( )

V VV r

r rrθ

θ θ θ

∂ ∂ ∂ ∂ ∆ = +

∂ ∂ ∂ ∂ (16)

Changing this to (ζ , β ) coordinates leads to equation (18) as follows:

1 1 sin( )

; ln2 1 sin( )

E

r

θξ β

θ

+= =

− (17)

2 2

2 2

2 2cosh ( ) 0

V Vξ β

ξ β

∂ ∂+ =

∂ ∂ (18)

The modelling in this system of coordinates is given in figure 16.

.

Figure 16: Mathematical model for the point / plane configurations.


The solution uses the same variational method. The parameter of the series expansion is the

maximum value β, when θ=α , i.e. β=B : 1 1 sin( )

ln2 1 sin( )

Bα

α

+=

− (19)

Considering the different conditions on the boundaries in the three configurations of figures

13 to 15, we arrive at the following equations (20) to (22), respectively :

3cos( )(...)

1 cos( )p

i rV V B

i r B

ω γ ρ α β

ω γ ρ α= +

+ (20)

3(...)1

pi r

V V Bi r B

ω γ ρ β

ω γ ρ= +

+ (21)

( )

( )3cos( ) 1

(...)1 cos( ) 1

p

i rV V B

i r B

α ω γ ρ β

α ω γ ρ

+= +

+ + (22)

In the se formulas, β is given by (17) and B by (19). The admittance is calculated by the

following integration :

( 0)

2d

E VY r

ε θ

π

ρ θ =

∂ =

∂ ∫ (23)

A term Yr appears that corresponds to the admittance of the purely resistive point / plane

configuration, i.e. with no capacitive layer (or with it, but at a very high frequency).

2

rE

YB

π

ρ= (24)

Finally we get:

2 1 1

ln1

ri E B

Y YB i B i B

π ω γ ρ κ

ρ ω γ ρ κ ω γ ρ κ ε

+= −

+ (25)

In this formula, the coefficient (κ) depends on the electrode configuration:

Configuration no. 1: (capacitive point) κ = cos(α) (26)

Configuration no. 2: (capacitive plane) κ = l (27)

Configuration no. 3: (capacitive point and plane) κ = cos(α) / (1+cos(α) ) (28)

When the microscopic and macroscopic limits do not disturb the phenomena, formula (25)

simplifies to the following theoretical formula:

( ) ( )2 2

2ln( ) ln( )ri Y Y E B i

B

πω γ ρ κ ω

γ ρ κ− = − + (29)

The description of the phenomena is therefore similar to what was explained in the blade /

plane case. We also find the ln(iω) term again in the resulting formulas. In these point / plane

cases, the very much flattened curve is visible when (Y-Yr)iω is plotted in a complex graph

(e.g. figure 17 corresponding to the configuration of figure 13).

Table 2 gives the results for a large number of examples in the configuration of figure

13 (case no. 1, point with capacitive layer, plane without ). In each case the three results are

compared :

- result obtained by finite element computation with ε / E = 10-10

.

- analytical result according to the formulae (24), (25) and (26)

- analytical result according to the simplified formula (29), which does not take the

microscopic or macroscopic limits into account.

For all of the cases analyzed, the agreement is very good in the range of validity

determined by the microscopic and macroscopic limits.


Table 2: Comparison of numerical and analytical results for the first point / plane

configuration (capacitive layer only on the point).

From top to bottom: -results obtained by finite element computation

-analytical results according to the formulae (24, 25, 26)

-analytical results according to the simplified formula (29)


Figure 17: Examples of numerical (crosses) and analytical (solid lines) results in

the first point / plane configuration (capacitive layer only on the point).

6. CONCLUSION

In very simple blade / plane and point / plane electrode configurations, the analytical

and finite-element numerical approaches have yielded concordant results.

In the admittance and impedance formulae obtained, one remarkable function appears.

This is ln(iω), which is represented in the complex plane as a horizontal straight line.

The mechanism governing the variation of admittance as a function of the frequency is

fully described, as follow. In the vicinity of the point, the inter-electrode medium is

equipotential. All the difference of potential that exists is applied at the capacitive surface

layer, which is equivalent to a microscopic capacitor. Its admittance does not depend on the

geometry but only on the area of the capacitive layer immersed in this zone.

Far from the point, all the difference of potential is distributed through the resistive

medium, and any difference of potential there is in the capacitive layer is negligible. This

zone is equivalent to a resistor, which value depends on the geometrical configuration of the

macroscopic limit.

There is a "transition" zone between the microscopic capacitive zone and the

macroscopic resistive one. This "transition" zone moves as a function of frequency. The

higher the frequency, the closer it comes to the electrode point. The distribution of

equipotentials is still similar. This shift is accompanied by a replacement of capacitance by

conductance, which determines the admittance variation law.

This theoretical study shows how a mathematical analytical approach may be of use in

understanding a phenomenon. Of course these blade / plane and point / plane configurations

are far from those used in laboratories. That is, in practice, the distance between the electrodes

is always large. Any experiment under conditions representative of those studied theoretically

here would certainly raise major experimental difficulties.

The results of this theoretical approach using the simplest possible configurations may

nonetheless arouse some interest among physicists and mathematicians who wish to make

some headway in solving more complicated configurations.


THEORETICAL IMPEDANCE OF ROUGH ELECTRODES

WITH SMOOTH SHAPES OF ROUGHNESS

Jean Jacquelin

ABSTRACT :

A theoretical study of rough electrodes is succinctly presented, for small-size and

smooth-shape roughness situations. The analytical result is a formula which derives the

resulting impedance from the parameters of the shape of such a roughness. An equivalent

electrical network is then derived with very simple formulae in order to calculate the

component values.

The most original part of the results consists in specific theoretical examples. Several times,

impedances with the term (iω)ν with a non integer exponent (ν) are obtained. This paper

shows that from a theoretical point of view, smooth roughness profiles with some simple

shapes may result in a special type of impedance known as Constant Phase Angle (CPA). In

some specific cases, impedances with the term log(iω) are also obtained.

KEY WORDS : Impedance, Electrode, Modelling, Roughness

LIST OF SYMBOLS :

A (V/m) coefficient in equations (1), (1b)

an , anm (V) coefficients in equation (1), (1b)

Cs , cs (F/m2) capacitance (per unit of front-surface)

D (m) length (on x-axis)

f(y,z) (dimensionless) shape of roughness

H (m) length (on y-axis)

i (dimensionless) imaginary unit

Ip (A) total current

kn , knm (dimensionless) coefficients in Fourier series

L (m) length (on z-axis)

p (dimensionless) exponent

Rs , rs, rn , rnm , Re (Ω.m2) resistances (per unit of front-surface)

s (m2) area

V(x,y,z) (V) 3-D. potential function

Vp (V) total voltage

Z, Zs , zs (Ω.m2) impedances (per unit of front-surface)

α (rad) angle

γ (F/m2) capacity per surface-unit

ε (m) size of roughness

ν (dimensionless) exponent in CPA law

ρ (Ω.m) resistivity

ω , ωe (s-l) pulsation (ω = 2*π*frequency)


(10) INTRODUCTION

It has long been observed that the impedance of an electrode depends on the roughness

of its surface. In the area of electrochemistry, R. De Levie [1] quotes one of the first papers

dated 1950. In spite of very significant experimental works done worldwide over the last

decades (*), it was not possible to establish a precise relationship between the geometrical

characteristics of the roughness and its resulting impedance.

The theoretical studies which tried to establish the analytical relation between the

roughness and the impedance faced great mathematical difficulties. The rigorous formulation

of this kind of problem leads to a partial differential Laplace equation ∆V = 0, with Dirichlet-

Fourier-Neumann boundary conditions [2], whose solution is not known in the general case.

Analytical solutions have been found for a very simplified model in which the second order

partial differential equation is replaced by a simple differential equation [1]. It is then difficult

to know to what degree the result is modified by the model simplification.

In the 1980s, a new research direction emerged with the application of fractal

geometry to electrochemistry. This original approach raised hope. Unfortunately, the

mathematical problem was not simplified. It is indeed " a difficult computing task especially

for fractal geometry " [3]. The results are still not always consistent. Even the simplest

correlations and resulting formulae are controversial [4]. We do not intent to discuss anymore

this subject because, as a matter of fact, fractal roughness is outside of the scope of the present

paper.

The study reported here only deals with smooth and simple shapes of roughness. The

approach is conducted within the Cartesian geometry tradition and no advanced mathematic is

involved : it is no more than a textbook case.

We would forestall an objection which could be raised, concerning some examples

given later : of course, most of them, with smooth shapes and without any angular shape, are

exactly in the scope of the study. But the objection may concern a few examples in which a

point or an edge appears. In fact, these examples are given in the aim of showing the limit

cases for the other examples. We don't claim that the computation method is correct in those

limit cases. Nevertheless, some considerations tend to strengthen the validity of the results

extended to these ambiguous cases: In the case of point/plane electrodes, previously published

[2], it was observed that there is a microscopic quasi-equipotential area at the end of a point or

near the edge of an angular side face. This area is equivalent to a micro-capacitor whose

capacity value is (γs), (s) being the submerged surface in this area. The shape of the point in

this area is therefore not important. A sharp point can thus be fictitiously replaced by a

microscopic rounded end, provided that one stays within the validity limits determined. Thus,

the present study (which is formally limited to smooth shapes) is probably correct in more

complicated cases. This could be a subject to further theoretical work.

(*) Note : Paper written in 1993.


2. STRIPED ELECTRODES : THEORETICAL FRAMEWORK

2.1. Basics equations of the theoretical model

The theoretical study of striped or corrugated electrodes gives rise to a two-

dimensional problem, which is therefore simpler than the case of a three-dimensional rough

electrode which will be studied later. Modelisation is showed in figure 1.

Figure 1 : Model of a striped electrode. The shape is magnified for better visualisation.

It is assumed that the striped electrode, length L, is uniformly covered by a capacitive

layer (capacity per unit surface = γ ). In the case of a non periodical striping, H is the total

height of the electrode. In the case of several equidistant identical stripes, H is the height of

one single shape. Opposite to the capacitive electrode, there is a counter electrode placed at a

great-distance from the electrode (distance D >> H ), in order to avoid modifications in the

distribution of the electric field and of the current in the vicinity of the first electrode. We

assume a conductive medium with a resistivity ρ between the two. In this medium, the

potential V(x,y) complies with the Laplace equation ∆V = 0.

For the boundaries y = 0 and y = H, the boundary condition is 0V

y

∂=

∂

A general solution is given by formula (1 ) which fulfils all the above conditions.

1

( , ) ( ) sinh cosp n

n

x D yV x y V A x D a n n

H Hπ π

∞

=

− = + − +

∑ (1)

In this formula, the coefficients ( A, al, a2, …., an ) values are unknown and must be

determined in order to satisfy condition (2) of the capacitive electrode.

1 V

i Vn

ω γρ

∂=

∂ (2)


The whole current ( Ip ) is computed by integration on the plane conter-electrode ( x=D ). The

result (3) is very simple due to the fact that the integration from 0 to H of cos(nπy/H) gives 0,

and so all an coefficients are deleted.

0 ( )

H

px D

L V A L HI dy

xρ ρ=

∂ = =

∂ ∫ (3)

The resulting impedance (4) for a unit surface in the front of the electrode is :

1p p

p

V VZ

I L H Aρ= = (4)

The whole problem is therefore to calculate the coefficient (A) with respect to the parameters

which define the shape of the striped electrode.

2. 2. Definition of the shape of the roughness.

Of course the shape of the surface must be known. We assume that it is defined by a relation

(5) giving (x) as a function of (y) :

( )x f yε= (5)

A size coefficient (ε) and a shape function f(y) are then part of this function (5). The function

f(y) can be defined by a Fourier series (6) with the given coefficients km.

0

( ) cosm

m

yf y k m

Hπ

∞

=

=

∑ (6)

Note that on figure (1), the position of the axis 0y does not depend on the shape of the actual

surface. We locate this axis at the mean position of the curve x = ε f(y), i.e. determined by the

relation (7). Consequently, the constant term of the Fourier series will be zero ( k0 = 0).

00

( ) 0 0H

f y dy k= ⇒ =∫ (7)

2. 3. Approximate solution by the method of variation of εεεε

The analytical approach is a "perturbation" method. This is a general technic and they

are many ways and manners to apply it. For example, it was already used to compute the

double-layer impedance at a rough surface [5], with a formalism very different from the

formalism applied here.

We start with a "trite" geometry problem which has an obvious solution, then adds

small variations of boundary conditions to the volume studied. In the present case concerning

rough electrodes, we start therefore with a "trite" plane electrode to which small deformations

will be subsequently applied in order to simulate roughness. This lead to series expansions

whose parameters are determined by the definition of the model boundaries. In [2], the series

expansions are limited to only three terms ( the main term and the perturbative terms on the

first and second order) and this method gave results which agree with the simulation results of

the numerical computation. Our approach remains limited to low level roughness profiles

with smooth shapes, because complex cases like fractal roughness should be in formal

contradiction with our main perturbation hypothesis, which is to have only small variations of

the boundary conditions. Especially, we take care at the continuity and the differentiability of

the functions, which means that, in the cases studied here, the vector perpendicular to the


surface is always unique and defined (except in the few ambiguous examples mentioned in

introduction).

According to the method outlined, equations (1) and (2) must be expanded in a series

of ascending powers of ε. Therefore, we start from the smooth surface ( ε = 0) and move

towards deformations which become greater and greater. The correlative development is

given in Appendix 1 (Equations 8 to 25).

Finally, the impedance Z of the whole system (rough electrode, conductive medium,

conter-electrode) appears as the sum of different elements Cs , Rs , rs , Zs . All refer to the unit

surface on the front of the electrode.

1

( )s s ss

Z R r Zi Cω

= − + + (26)

Figure 2 : Equivalent electrical network from the analytical solution

Rs is the mean resistance between electrodes. The term (-rs ) does not mean that a negative

resistance exists. It is a corrective term (27) of resistance Rs from the mean value. Rs and rs

values depend on the choice (7) made for the position of the axis 0y, but ( Rs-rs ) is

independent of this position.

2

2

1

; ( )2

s s n

n

R D r n kH

π ερ ρ

∞

=

= = ∑ (27)

Cs. is the total capacity. The corresponding partial impedance l/(iω Cs) is (28) :

2

2 2 2

2

1

1 11 ( )

4n

sn

n ki C i H

πε

ω ω γ

∞

=

= −

∑ (28)

Zs is a special impedance resulting from the electrode roughness.

2 22

1

( )

2

ns

n

n kZ

HHn i

π ερ

ω γ ρπ

∞

=

=

+

∑ (29)

Formula (29) is therefore the required result. It gives the relation between the parameters (kn)

which define the geometric shape (6) and the impedance (Zs)

(Zs) can be written as a serial parallel network of elementary resistors and capacitors (30-31).


1

1

1s

n nn

Z

i cr

ω

∞

=

=

+∑ (30)

( )

2 2 2

2

( ) 2;

2

nn n

n

n k Hr c

H n k

ρ π ε γ

π ε= = (31)

An electrical equivalent network is then derived (figure 2), with the well-known structure of a

ladder network. Such a result is therefore particularly interesting. We shall now consider some

simple shapes f(y) and the resulting impedances Zs.

3. EXAMPLES OF IMPEDANCES (Zs)

3. 1. The dimensionless graphs

It is easy to see, in equation (29), that they are two dimensionless numbers:

(10) The first is related to the impedance Zs :

s

e

Z

R

where 2

2eR

H

π ρ ε=

(10) The second is related to the frequency ( or to ω):

e

ω

ω

where eH

πω

γ ρ=

Then equation (29) can be written on a dimensionless form :

( )2

1

ns

en

e

n kZ

Rn i

ω

ω

∞

=

=

+

∑

The dimensionless parameters allow us to reduce a lot the number of cases to compute and

display, since we can consider only these two dimensionless variables instead of a lot of

variables and parameters (π, γ, ω, ε, H, Zs ). All the following examples were computed with

dimensionless numbers and the corresponding graphs are drawn with dimensionless scales.


3. 2. Examples: The smoothed rectangular shapes

The smoothed rectangular profiles

are defined by the following Fourier-series

and they are shown in figure 3. The limit-

case p=0 corresponds to the rectangular

profile. For p>0, we obtain a group of

profiles, the shapes becoming smoother

and smoother as (p) increases.

( )

1

0

1 cos (2 1)4

( )(2 1)

m

p

m

ym

Hf y

m

π

π

∞

+=

− +

=+

∑

2 2 1 1

4 ( 1)0 ;

(2 1)

m

m m pk k

mπ+ +

−= =

+

Carrying back this term k2m+ 1 in (29)

Zs is obtained :

20

16 1

2 1 (2 1)

s

e pm

e

Z

Rm i m

π ω

ω

∞

=

=

+ + +

∑

Figure 3. The "smoothed rectangular shapes" :

Example of a group of profiles with increasingly smooth shapes following the parameter p.


Some of the following steps in derivation are given in Appendix 2 . The limit-case p=0 is first

described. It is shown that the impedance includes a term ln(iω). The complex impedance

diagram, Figure 4, is close to a straight horizontal line. This behaviour has already been

mentioned, in different circumstances [2].

Figure 4. The "smoothed rectangular shapes" ( 0<p<0.5 ) : impedances diagrams Zs .


When (p) is increased from 0 to 0.5, a special impedance (iω)ν appears, common to

many other circumstances (identified as CPA. 'Constant Phase Angle' ). The views of the

impedances diagrams, Figure 4, permit us to see that they include an almost straight part,

which is horizontal when p ≈ 0 ; which tends towards a semi-circle when (p) increases.

The magnified views,

Figure 5, clearly shows

the impedance diagrams

as inclined straight lines,

which are characteristic.

The smoother the profile

is (while p increases), the

more the straight line

turns from horizontal to

vertical. With the aim of

comparing with the term

(iω)ν , the corresponding

straight lines are drawn

and we see that the

exponent (ν) varies

roughly as ν ≈ -2p.

This is consistent with

the relationship between

ν and p obtained in

Appendix 2.

Figure 5. The "smoothed rectangular shapes" ( 0<p<0.5 ) :

Magnified views of the impedances diagrams.


When (p) tends towards 0.5

the diagram tends towards a semi-

circle. In fact, this is not exactly a

semi-circle because the impedance

(Zs) contains a residual term (zs), as

expressed in Appendix 2.

The complex diagram of (zs)

is represented on Figure 6.

Again, an horizontal straight line

(case p = 0.5) is visible and the

term ln(iω) appears in (zs).

If we continue to increase (p)

from 0.5 to 1, the term (iω)ν appears

again in (zs). The inclined straight

lines turn and go from vertical to

horizontal (Figure 6).

So, when increasing (p),

each time (2p) is close to an integer,

a new residual term (zs) containing

ln(iω) appears. When (2p) is far

from an integer, the impedance

contains special elements such as

(iω)ν.

Figure 6. The "smoothed rectangular shapes" ( 0.5<p<1)

Complex impedance diagrams of the term (ωzs)


3. 3. Other examples : The smoothed sawtooth profiles

The smoothed sawtooth profiles are defined by the following Fourier-series and they

are shown on figure 7. The limit-case p=0 corresponds to the pure sawtooth profile. For p>0,

we obtain a group of profiles, the shapes becoming smoother and smoother as (p) increases.

Figure 7. The "smoothed sawtooth shapes" :

Example of a group of profiles with increasingly smooth shapes following the parameter (p).

2

0

cos (2 1)4

( )(2 1) p

m

ym

Hf y

m

π

π

∞

+=

+

=+

∑ ; 2 2 1 2

40 ;

(2 1)m m p

k kmπ

+ += =

+

Carrying back this term k2m+ 1 in (29) Zs is obtained :

2 2

0

16 1

(2 1) (2 1)

s

e pm

e

Z

Rm i m

π ω

ω

∞

+=

=

+ + +

∑


The expression (Zs / Re) found in the smoothed sawtooth case and those previously

found in the smoothed rectangular case obviously are identical, but with a shift of (p). If we

replace (p) by (p+1) in the Zs-formula for the smoothed rectangular profile, we obtain exactly

the Zs-formula for the smoothed sawtooth profile. When (p) increases, both smoothed-

rectangular (Figure 3) and smoothed-sawtooth (Figure 7) profiles tend towards a common

sinusoidal profile.

In the smoothed rectangular case we started from p=0 to 0.5. At this point, we took

apart the residue zs. Then, it was necessary to draw (ωzs) instead of (zs) on the complex graph

to make obvious the special impedances such as ln(iω) or (iω)ν . We continued to increase (p)

up to 1. At this point, we are exactly at the starting point for p=0 in the smoothed sawtooth

case. Again we consider the residue (zs) and, we have to draw (ω2zs) instead of (ωzs) on the

complex graph to make obvious the special terms of impedance. The result is represented on

Figure 8. Therefore, the derivation in the smoothed sawtooth case is just the continuation of

the treatment reported in Appendix 2 for the smoothed rectangular case. It is useless to go

again into the details of derivation.

Figure 8. The "smoothed sawtooth shapes" : impedances diagrams Zs and (ω2zs)


The preceeding examples illustrate a behaviour frequently appearing when we

compute the impedance corresponding to various smooth profiles: we obtain impedances

including some main terms, resistances and capacitances, and secondary terms including (iω)ν

But, in specific conditions, which are particular or limit cases of the preceeding ones, the term

ln(iω) is encountered. This surprising behaviour can be explained analytically. When ν tends

towards an integer N, the power function ( )s iνζ ω= expands as a series with a main part

(iω)Ν. The residual part is proportional to (iω)

Ν ln(iω) :

( )( ) 1 ( ) ln( ) ...Ns i N iζ ω ν ω= + − +

If we set apart the "pure" term (iω)Ν and if ν is close to N, the residual part ( )s

Ns iζ ζ ω= −

becomes visible and is almost proportional to (iω)Nln(iω). Then, if we draw on the complex

plane the term ( )( )sNiζ ω − , which is thus almost proportional to ln(iω), we roughly obtain a

straight line on the complex graph. Of course, some deviations appear, at low and high (ω), as

it was explained in [2].

4. ROUGH ELECTRODES.

In the case of a rough electrode, the three-dimensional model (x, y, z) is shown on

figure 9.

Figure 9. 3-D model of a rough electrode.

We will limit ourselves to the starting basis and the final results, without going into the

derivation. In order to facilitate the comparison with the section "Striped Electrodes", the

numbers of the corresponding equations are the same except a "b" index must be added. For

example, the general 2-D solution (1 ) becomes the 3-D. equation (1b).


1/ 22 2

2 2

0 0

( , , ) ( )

sinh ( ) cos cos

p

nm

n m

V x y z V A x D

n m y za x D n m

H LH Lπ π π

∞ ∞

= =

= + − +

+ + − ∑∑

(1b)

The function of roughness of shape f(y,z) is expanded in a double Fourier series (6b) with

coefficients knm. The position of the plane 0yz is fixed by k00 = 0.

0 0

( , ) cos cosnm

n m

y zx f y z k n m

H Lε ε π π

∞ ∞

= =

= =

∑∑ (6b)

The previous condition (2) is applied with the derivative with respect to the

perpendicular to the surface (9b)

1/ 2

2 2

1

V f V f V

V x y y z z

nf f

y z

ε ε

ε ε

∂ ∂ ∂ ∂ ∂− −

∂ ∂ ∂ ∂ ∂ ∂=

∂ ∂ ∂ + + ∂ ∂

(9b)

Once all the developments have been made following the process shown in Appendix

1, we obtain a set of formulae which are the same as (29)-(31), with knm instead of kn and with

(n2/H

2 + m

2/L

2)1/2

instead of (n/H). We find again that the impedance Z, in formula (26), is

written as a sum of partial impedances ( )( ) 1/( )s s s sZ R r i C Zω= − + + .

Again, this impedance may be written as a sum (30b) of elementary resistors and

capacitors, their values being given by (31b). An equivalent electrical network can be derived,

with a structure identical to the one which was already described, figure 2.

0 1

1

1s

n m nmnm

Z

i cr

ω

∞ ∞

= =

=

+∑∑ (30b)

( )

1/ 2 12 2 2 2 2 2

2 2 2 2 2

( ) 2;

2

nmnm n

k n m n mr c

H L H Lk

ρ π ε γ

π ε

−

= + = +

(31b)

The formulae are similar in the case of stripes and roughness profiles, with different

numerical coefficients. Taking into account this similarity, it is clear that the given examples

for different shapes f(y) allow us to foresee similar results with f(y,z) roughness shapes.


5. DISCUSSION

The present study is a theoretical one, very far from the complexity of physical reality.

On the other hand, only relatively simple and basic mathematics are involved (But we have to

carry out heavy derivations).

In reference [2], the impedance of the type ln(iω) appeared and was found again in the

present study. It is even more interesting to have found again CPA, a term which designates

the type (iω)ν. Some smooth roughness shapes provide CPA with a large range of values of

the exponent (ν). It is remarkable that small roughness situations with very simple shapes,

with rather rounded profiles, lead to such a result. Moreover, the given examples (Figures 4

and 8) shows that (ν) varies extensively without the shape, size and dimension changing too

much (Figures 3 and 7 respectively).

These examples of CPA seem to confirm some more the assumption that this type of

impedance may be generated from many different aspects or phenomena. This is not a new

observation [6], [7].

A formula (29 ) was established, between the parameters of the shape of the roughness

and the impedance created by such a roughness. Both geometrical kn and electrical (ω, γ, p)

parameters contribute to this relation. This leads to the idea that a simple universal relation is

quite an utopian view. A somewhat similar point of view has been expressed in the case of

Fractal roughness [5], which is very different from the smooth roughness.

Of course, the results which were obtained here have a limited scope and it is

necessary to avoid giving to them a general scope they do not have. It would be inadequate to

try to apply it to porous electrodes (complex or thick shapes). It should be also an aberration

to draw any conclusion from the comparison of our results with those related to the fractal

shapes.

The study which was presented only achieves a small progress, after a previous step

[2], in some academic cases far from the complexity of the real electrodes. Nevertheless, it is

of interest to make known an example showing that smooth roughness profiles may result in a

special type of impedance known as Constant Phase Angle (CPA).

There is another remarkable point to think about: Considering the structure of the

equivalent network Figure 2, which is a ladder network, and the formula (31) which gives the

values of the components, a hierarchical character appears. In the literature, CPA response is

generally related to a hierarchical character, and the most frequently to fractal geometry [3].

However, in our academic examples, there is no fractal geometry involved and no phenomena

which can be suspected to present any fractal aspect. Does it mean that a hierarchical

character can also appear in situations where nothing fractal is involved ?


REFERENCES

[1] R. de Levie, "The influence of surface roughness of solid electrodes on

electrochemical measurements", Electrochimica Acta, 10, p. 113 (1965).

[2] J. Jacquelin, "Theoretical impedances of point/plane electrode configuration ",

Electrochimica Acta, 38, p. 597 (1993).

[3] P. Meakin, B. Sapoval, "Random-walk simulation of the response of irregular or

fractal interfaces and membranes", Phys. Rev. A 43, p. 2993 (1991).

[4] B. Sapoval, "Letters to the Editor", Solid State Ionics, 25, p. 100 (1987).

[5] T. C. Halsey, "Double-layer impedance at a rough surface: A perturbative approach ",

Phys. Rev., A 36, p. 5877 (1987).

[6] J. Jacquelin, "A number of models for CPA impedances of conductors and for

relaxation in non-Debye dielectrics", Journal of Non Crystalline Solids, 131-133, p.

1080 (1991).

[7] R. de Levie, "On porous electrodes in electrolyte solutions ", Electrochimica Acta, 8,

p. 751 (1963).

[8] J. Jacquelin, "Theoretical impedance of rough electrodes. Complete collection of

equations", Unpublished, v.2, (1993). [ Added to the 2009 edition, pp. 57-94 ]

[9] M. A. Abramowitz, 1. A. Stegun, "Handbook of mathematical functions", Dover

Publications, N.Y. (1970).

APPENDIX

MAIN STEPS IN DERIVATIONS AND COMPUTATIONS

It is unfortunately impossible, in a reasonable space, to provide sufficiently detailed

information on the derivations in the order to lend a full credibility of the results. I am aware

that the present appendix do not contains all the steps and the reader may found a chore to

deduce some intermediate steps. In fact, the full information is available in a document [8]

containing heavy developments in 38 pages, which are summarized hereafter.

APPENDIX 1 : THE TWO-DIMENSIONAL CASE (Model Figure 1).

The basic hypothesis and equations are stated in the main body, equations (1-7). In figure 1,

we see that tan( ) ( ) / 'd f dy fα ε ε= − = , and (8) then (9) are derived from it.

2 2

1 'cos( ) ; sin( )

1 ( ') 1 ( ')

f

f f

εα α

ε ε

−= =

+ + (8)

( )1/ 2

2cos( ) sin( ) 1 ( ') 'V V V V V

f fn x y x y

α α ε ε− ∂ ∂ ∂ ∂ ∂

= + = + − ∂ ∂ ∂ ∂ ∂ (9)

Condition (2) on the electrode may be written as (10), using (9) and (1), as well as the

derivatives of (1) with respect to x and y.


( )1

21

21

sinh cos

1cosh cos

1 ( ')

1' sinh sin

1 ( ')

p

n

n

n

n

n

f D yi V A f D n n

H H

n f D yA a n n

h H Hf

n f D yf a n n

h H Hf

εω γ ρ ε π π

π επ π

ε

π εε π π

ε

∞

=

∞

=

∞

=

− + − + =

− = + + +

− + +

∑

∑

∑

(10)

Equation (10) includes the (y) variable only. Therefore, the unknown constants (A, an) must

hold for all values of y.

The unknown coefficients (A, an) are expanded in series of ε :

20 1 2 ...A A A Aε ε= + + + (11)

20 1 2 ...n n n na a a aε ε= + + + (12)

The functions of ε are expanded in series of ε. For example (13) :

( )1/ 2

2 2 211 ( ') 1 ( ') ...

2f fε ε

−+ = − + (13)

The condition ' 1fε < is easy to fulfil. In any example of shape corresponding to given f(y)

and ' d / df f y= , it suffices to chose ε small enough : max

'fε < . where max

'f is the

maximum value of 'f . But this cannot be fulfilled if max

'f = ∞ .

That is why this study is formally restricted to smooth shapes of roughness. As said in

introduction, the results may hold for more complicated cases which anyway are beyond the

scope of this paper.

The effect of the counter electrode is seen by presence of the terms (D/H). In the

present study, we assume that the distance between the electrodes is great (D >>H).

In the numerical computations, we set for example D = 100 H.

2 21sinh exp 1 ...

2 2

f D D n nn n f f

H H H H

ε π π επ π ε

− ≈ − − + +

(14)

So, the equation (10) fully expanded in series of ε is written as :


( )( )

( )

20 1 2

20 1 2

1

22

...

1 exp ...

2

1 ... cos

p

n n n

n

i V i A A A f D

Di n a a a

H

n f n f yn

H H H

ω γ ρ ω γ ρ ε ε ε

ω γ ρ π ε ε

π πε ε π

∞

=

+ + + + −

− + + +

× − + +

∑

( )

( )

2 2 20 1 2

2 2 20 1 2

1

22

11 ( ') ... ...

2

1 1 ( ') ... exp ...

2

1 ... cos 'sin

n n n

n

f A A A

n Df n a a a

H H

n f n f y yn f n

H H H H

ε ε ε

πε π ε ε

π πε ε π ε π

∞

=

=

= − + + + + +

+ − + + + + ×

× − + + −

∑

(15)

The "trite" solution is given only by the terms (16) which do not contain ε:

00

1

0 0

1

exp cos2

exp cos2

np

n

n

n

a D yi V A D n n

H H

n D yA a n n

H H H

ω γ ρ π π

ππ π

∞

=

∞

=

− − =

= +

∑

∑ (16)

If the different terms of the Fourier series are made equal, A0 and an0 are obtained :

01

pi VA

i D

ω γ ρ

ω γ ρ=

+ (17)

0 0na = (18)

The terms of equation (15) which contain only ε give the equation (19). The function f(y) has

been replaced by the series expansion (6).

0 1

1

11

1

cos

exp cos2

n

n

n

n

yi A k n i A D

H

a n D yA i n n

H H H

ω γ ρ π ω γ ρ

πω γ ρ π π

∞

=

∞

=

− =

= + +

∑

∑ (19)

If the different terms of the Fourier expansion are made equal, Al and an1 are obtained :

1 0A = (20)

1 02 expn ni D

a A k nn H

iH

ω γ ρπ

πω γ ρ

= −

+

(21)

The same procedure is then applied to the terms of the equation (15) which contain ε2,

leading to (22) : When f(y) is replaced by series (6), an equation including cosine products and


sine products is found. These products have to be replaced by sums of cosines, which leads to

the appearance of constant terms with a coefficient ½. If the approximation of the impedance

calculation (4) is limited to the second order (i.e. up to ε2 terms), only A2 has to be calculated.

20

2 1 2

1 1

exp2 4

n n n

n n

An D ni D A i a k n A k

H H H

π πω γ ρ ω γ ρ π

∞ ∞

= =

− + = −

∑ ∑ (22)

Carrying back in (22) the value of anl given by (21), A2 is obtained :

22

202

1

( )1 1

( )1 2 4

n

n

ni

A nHA kni D H

iH

πω γ ρ π

πω γ ρ ω γ ρ

∞

=

= + + +

∑ (23)

The impedance is given by the formula (4) which can be written as a series of ε.

222

0 0220 20

0

1 ......

1 ...

p p pV V V AZ

A AAA AA

A

ρ ρ ρε

εε

= = = − +

+ + + +

(24)

Carrying back in (24) the values of A0 and A2 (17), (23) that were found, after some

conversions, we obtain (25) :

2

2

1

1 1( )

2 2n

n

n

n n HZ D kni H i H

iH

πρε π π

ρ ρπω γ ω γ ω γ ρ

∞

=

= + − + − +

∑ (25)

Then, from (25), we obtain the formulas (26-31) given in the main section of this paper.

APPENDIX 2 : THE SMOOTHED RECTANGULAR PROFILES.

Appendix 2. 1. A "limit" example: The rectangular shape ( specific case p=0 )

( )

0

1 cos (2 1)4

( )(2 1)

m

m

ym

Hf y

m

π

π

∞

=

− +

=+∑ ; 2 2 1

4 ( 1)0 ;

(2 1)

m

m mk kmπ+−

= =+

Carrying back (k2m+ 1 ) in (27) and (29) the result is formula (37) of the impedance Zs altered

with the series resistance rs :

2

0

8 1 1

(2 1)(2 1)

s s s

m

z Z rHH m

m i

ρ ε

π ω γ ρπ

∞

=

= − = − + + +

∑

We can compute this sum thanks to the Polygamma functions ψ , which use is explained in

[9].

28 1 1 3 1 3

12 2 2 2 2

1s

Hz i

HHi

ρ εψ ω γ ρ ψ

π πω γ ρπ

= − + − + +


When the value of ω is great (ωγρΗ >>3π), the Polygamma function tends towards the

logarithmic function:

3

ln ...2 2 2

H Hi iψ ω γ ρ ω γ ρ

π π

+ ≈ +

We obtain an approximated formula which includes ln(iω), in a good agreement with the

complex impedance diagram (z's , z"s ) of figure 4, showing the characteristic horizontal line.

24

ln2

sH

z iH

ρ εω γ ρ

π π

≈

Appendix 2. 2. Smoothed rectangular shapes in the cases 0 < p< 0.5

( )

1

0

1 cos (2 1)4

( )(2 1)

m

p

m

ym

Hf y

m

π

π

∞

+=

− +

=+

∑ ; 2 2 1 1

4 ( 1)0 ;

(2 1)

m

m m pk k

mπ+ +

−= =

+

2 22 2

20 0

( ) 8 1

22 1 (2 1)

ns

pm m

n kZ

H HH Hn i m i m

π ε ερ ρ

πω γ ρ ω γ ρ

π π

∞ ∞

= =

= =

+ + + +

∑ ∑

We obtain an approximation of the sum as :

( ) ( )2 2

1/ 20

1 d

2 1 (2 1) 2 1 (2 1)p p

mm m

µ

λ µ λ µ

∞ ∞

−=

≈+ + + + + +

∑ ∫

where H

iλ ω γ ρπ

=

The solution of this integral is known, leading to :

( ) 2 2

0

1 1( 0 0.5 )

sin(2 )2 1 (2 1) 2p p

m

ppm m

π

πλ λ

∞

=

≈ < <+ + +

∑

2 2

2 1 2 2

4

sin(2 ) ( )

p

s p p pZ

p H i

ρ ε π

π γ ω+≈

We see that Zs contains the term (iω)ν where ν = -2p, in a good agreement with the complex

impedance diagram (Z's , Z"s) of figure 4, showing the characteristic inclined straight lines.

Appendix 2. 3. Smoothed rectangular shapes in the limit case p=0.5

2

0

8 1

2 1 (2 1)s

m

ZHH

m i m

ερ

πω γ ρ

π

∞

=

=

+ + +

∑

2

0

8 1 1 1

(2 1)2 1

s

m

ZH HH m

i m i

ερ

π ω γ ρ ω γ ρπ π

∞

=

= −

+ + +

∑


The same process, with the polygamma functions when ωγρH is large, leads to

2

2

4 1ln

2s

HZ i

iH

εω γ ρ

ω γ π

≈ −

On the complex diagram, we draw (ωZs), which is proportion al to lng(iω). We obtain a

vertical straight line instead of an horizontal one (Figure 6). In fact, the -π/2 rotation is due to

the term (1/i )=-i.

Appendix 2. 4. Smoothed rectangular shapes in the cases 0.5 < p< 1

Let p' = p-0.5 , hence 0 < p' < 0.5 , and we obtain :

2

1 2 '0

8 1

2 1 (2 1)s

pm

ZHH

m i m

ερ

πω γ ρ

π

∞

+=

=

+ + +

∑

The next series is converging, which permit to define a capacitor (c) corresponding to the

partial impedance Z1 = 1/iωc :

2

1 1 2 '

0

1 8 1

(2 1) p

m

Zi c H i H m

ε πρ

ω π ω γ ρ

∞

+=

= =+

∑

This "pure" element (c) is taken apart and we consider the residue (zs) :

2

22 '

0

1 1 8 1

2 1 (2 1)s s

pm

z ZHi c i H m i m

ε

ω ω γω γ ρ

π

∞

=

= − = −

+ + +

∑

If we compare zs with Zs previously obtained in the case 0<p<0.5, we observe that both

contains the same term Σ . The two differences are: p' instead of p and the appearance of the

factor (π / iωγρH). Accordingly, the comparison of the complex graphs (Figures 4 and 6

respectively) reveals that both are quite identical , except the following differences : first,

(ωzs) is drawn instead of (Zs); second, (p) is shifted to (p'=p-0.5); third, we observe the -π/2

rotation due to the factor (1/i). Then, from the preceeding result in the case 0<p<0.5, it is easy

to deduce the next relationship for large (ω) in the case 0.5<p<1 :

2 2 ' 2 2

2 ' 1 2 ' 2 ' 2 1 2 2

4 4

sin(2 ' ) ( ) sin(2 ) ( )

p p

s p p p p p pZ

i Hp H i p H i

ρ ε π π ρ ε π

ω γ ρπ γ ω π γ ω+ +≈ = −

Hence, the impedance Zs contains the term (iω)ν where ν=-2p.

This process can be repeated for p≥1. Let p"=p-l so that: 0<p"<0.5 .The "pure"

element (c) already defined is taken apart and the next "pure" element as well. Then, we draw

the term (ω2zs) where zs is the residual impedance obtained.

22

1 2 1 2 1 2 ''

0

1 8 1; ;

(2 1)s s p

m

z Z z z z zi c H i H m

ρ ε π

ω π ω γ ρ

∞

+=

= − − = =

+ ∑

This leads to Figure 8. In fact, Figure 8 corresponds to the "smoothed sawtooth" case. But

Figure 8 also corresponds to the "smoothed rectangular" case with a shift of (p) of one unit.

This is obvious when we compare the Zs-equations (stated in the main paper) of both

rectangular and sawtooth cases.


THEORETICAL IMPEDANCES OF SMOOTH FLAT ELECTRODES WITH

SLIGHTLY NON-UNIFORM DOUBLE LAYER CAPACITANCE

Jean Jacquelin

ABSTRACT

A purely theoretical study of a flat and smooth electrode, covered with a non-uniform

double-layer capacitance, is succinctly presented. The analytical result is a formula which

describes the impedance in terms of parameters of the capacitance distribution. The whole

impedance includes a main resistive term, a main capacitive term and a special term.

Several times, impedances containing the special term (iω)ν with a non-integer

exponent (ν ) are obtained. This special impedance is known as Constant Phase Angle (CPA).

In some specific cases, the term log(iω) also appears.

An interesting result is to have found the same special impedance element in two

different situations : one in the case of some rough electrodes with a uniform double-layer

capacitance, second in the case of some smooth electrodes with non-uniform double-layer

capacitance.

KEY WORDS: Impedance, electrode, modelling.

LIST OF SYMBOLS :

A (V/m) coefficient in equations (2)

an , anm (V) coefficients in equation (2)

Css (F/m2) capacitance (per unit of front-surface)

D (m) length (on x-axis)

f(y,z) (dimensionless) "pattern" function

H (m) length (on y-axis)

i (dimensionless) imaginary unit

Ip (A) total current

kn , knm (dimensionless) coefficients in Fourier series (f)

L (m) length (on z-axis)

p (dimensionless) exponent

Rs , Re (Ω.m2) resistances (per unit of front-surface)

V(x,y,z) (V) 3-D. potential function

Vp (V) total voltage

Z, Zs (Ω.m2) impedances (per unit of front-surface)

γ (y,z) (F/m2) capacity distribution (per surface-unit)

γ 0 (F/m2) mean capacity (per surface-unit)

ε (m) magnitude coefficient

ν (dimensionless) exponent in CPA law

ρ (Ω.m) resistivity

ω , ωe (s-l) pulsation (ω = 2*π*frequency)


(10) INTRODUCTION

In the present study, we consider flat and smooth electrodes (without any roughness

and without any porosity). If we observe at a microscopic scale a polished electrode, we can

see random or periodic crystal arrangements and grain boundaries. Considering the diversity

of the surface appearance, it may be asked whether the impedance of the electrode could be

appreciably modified if the double-layer capacitance was not uniformly distributed over the

whole surface.

In a previous paper [1], the case of rough electrode covered with a uniform

capacitance double-layer was considered. We will use exactly the same model, with the same

assumptions. We only replace the rough electrode of the previous model (Figure 9 in [1] ) by

a smooth electrode (Figure 1 in the present paper) and the previous constant capacitance (γ ) is

now function γ (y,z) on the location over the electrode.

Figure 1. "Spotty" and "streaky" electrode models :

(a): "Clear-cut spotty" pattern ; (b): "Graduated spotty" pattern

(c): "Clear-cut streaky" pattern ; (d): "Graduated streaky" pattern.

The graduated grey colour is supposed to represent a smooth variation

of the double-layer capacitance as a function of y and z.


An aim of this study is to compare the both different cases:

- The previous results in case of uniform capacitance on a rough electrode

- The results in case on non-uniform capacitance on a smooth electrode.

On Figure 1, the so-called "spotty electrode" model gives rise to a three-dimensional

problem. A simpler model (two-dimensional), called "streaky electrode", will first be

considered. H and L are respectively the height and the length of the capacitive electrode. In

the case of periodical distribution of capacitance, H and L are the length of one single pattern.

Opposite to the spotty or streaky electrode, there is a counter electrode placed at a great

distance from the electrode (distance D >>H). We assume a purely conductive medium with a

resistivity ρ between the two.

In the large space between the electrodes, the potential V(x,y,z) complies with the

Laplace equation ∆V = 0. Of course, the Laplace law is only valid in the volume phase of the

conductive medium. In the vicinity of the capacitive electrode, the term "double-layer

capacitance" understand more complicated phenomena. A vast literature deals with this

subject [2] which is beyond the scope of the present paper. We doesn't intend to describe the

internal structure of the double-layer, which overall properties are globally modelled by the

capacitance term (γ ) and next equation (1). This simple capacitive model [3] for double-layer

is convenient (except in the very low frequency range) in the cases where more complicated

phenomena (diffusion, etc…) are not considered. For this reason, we consider only the case of

a. c. currents of low amplitude, without d. c. current.

For the boundaries y = 0 and y = H, the boundary condition is / 0V y∂ ∂ = . For the

boundaries z = a and z = H, the boundary condition is / 0V z∂ ∂ =

On the capacitive electrode, where the double-layer capacitance is γ(y,z) in the spotty-

case, or is γ(y) in the streaky-case, the boundary condition is (1) :

1 V

i Vx

ω γρ

∂=

∂ (1)

The problem is to find the impedance per unit of front-surface corresponding to a given

distribution function γ(y,z).

The form of Eq. (1) slightly differs from Eq.(2) in [1]. Clearly, in these equations, we

previously had n

instead of x because we had to take account of the shape of roughness ( n

was the normal vector on the surface in the case of Fig.9 in [1] ), and (γ ) was constant whilst

it is now function on y, z. These are the only (and basic) differences between the present and

the previous models.

2. STREAKY ELECTRODES: THEORETICAL FRAMEWORK

2. 1. Basic equations of the theoretical model

The general solution already used in [1], is given by formula (2) which fulfils the

Laplace equation and the boundary conditions.

1


n


H Hπ π

∞

=

− = + − +

∑ (2)

In this formula, the coefficients (A , al , a2 , …., an , …) values are unknown and must be

determined in order to satisfy condition (1) of the capacitive electrode.


The whole current (Ip), which is common an along the entire circuit, is computed by

integration on the section of the conductive medium, at any distance (x), in 0 < x < D. It can

be also computed on the counter-electrode x=D. The result (3) is very simple due to the fact

that the integration from 0 to H of cos (nπy / H) gives 0, and so all coefficients (an) disappear

in (3).

0

1d

H

pV A L H

I yxρ ρ

∂= =

∂∫ (3)

The resulting impedance (per surface unit) of the electrode is :

p p

p

V VZ L H

I Aρ= = (4)

The whole problem is therefore to calculate the coefficient (A) with respect to the parameters

which define the distribution of capacitance on the streaky electrode.

2. 2. Boundary condition on the electrode.

The double-layer capacitance distribution, γ(y) in the streaky case, can be defined by a

Fourier series (5) with given coefficients γ 0 , γ 1 , …, γ m, …, etc.

0

( ) cosm

m

yy m

Hγ γ π

∞

=

=

∑ (5)

Note that the first term γ 0 is the mean value of γ(y). We consider a slightly non-uniform

distribution. Hence, γ(y) differs only slightly from γ 0 . This means that the other coefficients

γ1 , …, γm, …, are relatively small. To make this more obvious, we will use a different

notation. Let γm / γ 0 = ε km (m≥1). Then, the Fourier series can be written as (6-7).

( )0γ( ) 1 f ( )y yγ ε= + (6)

1

f ( ) cosm

m

yy k m

Hπ

∞

=

=

∑ (7)

We call "pattern function" the function f(y). The capacitance distribution γ(y) is then defined

by means of γ 0 and a small non-uniform term ( ε γ 0 f(y) ). A magnitude coefficient (ε) and a

pattern function f(y) are then part of this term.

The case ε=0 is the "trite" case of uniform distribution ( γ =γ 0 ) which solution is

obvious. For a given pattern f(y) of capacitance distribution, we can go from zero towards a

given magnitude of the non-uniform distribution by increasing the magnitude coefficient (ε ). This method leads to the analytical solution expressed as a series of ascending powers of (ε ).

2. 3. Approximate solution by the method of variation of εεεε

We use a so-called "variational" method. The main steps are reported in Appendix, Equations

(8) to (17). Finally, the impedance Z of the whole system (streaky electrode, conductive

medium, counter-electrode) appears as the sum of different elements Rs, Cs, Zs . All refer to

the unit surface on the front of the electrode.

1

s ss

Z R Zi Cω

= + + (18)


Rs = ρ D is the pure resistance (per unit surface) between electrodes.

Cs = γ 0 is the mean capacitance (per unit surface) corresponding to the mean value of

the double-layer capacitance. Note that, if we had supposed a capacitive counter-electrode

(instead of an ideal counter-electrode with zero impedance ), we would have obtain this

additional capacitance in series.

Zs is a special impedance resulting from the non-uniform double layer capacitance

distribution (from Eq.17 in Appendix).

22

01

2

ns

n

kHZ

n i H

ρ ε

π ω ρ γ

∞

=

=+∑ (19)

Formula (19) is therefore the required result. It gives the relation between the impedance (Zs)

and the parameters (kn) which define the pattern of the capacitance distribution (6-7). We see

that (Zs) is an impedance defined by distributed constants. An example of equivalent electrical

circuit was derived and shown in [1]. Comparing Eq.(19) with previous Eq.(29) in [1], we

observe a same structure, but with different values of coefficients.

As usual the variational method which was used, raises the problem of the range of

validity, in relation with the expansion coefficient (ε). This has theoretical and practical

consequences. We propose to come back to this point later, in Section "Discussion".

3. EXAMPLES OF IMPEDANCES (Zs)

3. 1. The dimensionless form of results

The following dimensionless numbers were stated in [1] :

s

e

Z

R

where 2

2e

HR

ρ ε

π= (20)

e

ω

ω

where 0

eH

πω

γ ρ= (21)

Then the equation (19) can be profitably written on a dimensionless form :

( )2

1

ns

en

e

n kZ

Rn i

ω

ω

∞

=

=

+

∑ (22)

3. 2. Example: The "dear-cut streaky" pattern

In this case, we suppose that the electrode surface is divided in a series of bands. This

kind of non-uniform distribution of capacitance is represented in Figure 2 (in the case p=0).

The Fourier series for this pattern is (23). The result is formula (24) which gives the

impedance Zs

( )

0

1 cos (2 1)4

( )(2 1)

m

m

ym

Hf y

m

π

π

∞

=

− +

=+∑ ; 2 2 1

4 ( 1)0 ;

(2 1)

m

m mk kmπ+−

= =+

(23)


2

0

16 1

(2 1) (2 1)

s

em

e

Z

Rm m i

π ω

ω

∞

=

=

+ + +

∑ (24)

The sum can be obtain by numerical computation. Alternatively, we can compute this sum

thanks to the Polygamma functions [4].

3. 3. Other examples : The "graduated streaky" patterns

In this cases, we suppose that the local value γ(y) of double-layer capacitance is a

smooth function. A group of graduated patterns is represented in Figure 2 corresponding to

equation (25). At each value p corresponds a different pattern. The case p=0 is the preceding

"clear cut pattern". The larger is p, the smoother is the function γ(y) and the more graduated

the pattern appears.

Figure 2 : Different "graduated streaky patterns", depending on p.

The result is formula (26) which gives the impedance Zs corresponding to a pattern with a

given value p.

( )

1

0

1 cos (2 1)4

( )(2 1)

m

p

m

ym

Hf y

m

π

π

∞

+=

− +

=+

∑ ; 2 2 1 1

4 ( 1)0 ;

(2 1)

m

m m pk k

mπ+ +

−= =

+ (25)

2 20

16 1

(2 1) (2 1)

s

e pm

e

Z

Rm m i

π ω

ω

∞

+=

=

+ + +

∑ (26)


3. 4. Results :

It is remarkable to observe that equation (26) is exactly the same as the equation

already found in a very different case" published in [1], the case of rough electrodes with a so-

called "smoothed saw-tooth profile". This common equation appears in [1], page 2677. The

development and solution are detailed in [1], Appendix 2. It is not of use to repeat here the

already published mathematical development. The corresponding impedance diagrams are

also published in [1], Fig. 8. It has been shown the appearance of quite linear loci. This

behaviour is due to a term (iω)ν with non-integer (ν ), and is refereed to as CPA (Constant

Phase Angle). In some particular conditions, for example in the case (p=0), the special term

ln(iω) is obtained, corresponding to an horizontal straight line in the complex graph (Fig.8 in

[1] ).

These results lead one to think that, in some cases, identical complex impedance

response as a function of frequency may be obtained in two different situations : one in the

case of some rough electrodes with a uniform double-layer capacitance, second in the case of

some smooth electrodes with non-uniform double-layer capacitance.

4. SPOTTY ELECTRODES

The three-dimensional model (x, y, z) of a spotty electrode is shown on Figure 1. We

will limit ourselves to the starting basis and the final results, without going into the derivation.

The general two-dimensional solution (2) becomes the next equation, in the three dimensional

case:

1/ 22 2

2 2

0 0

( , , ) ( )

sinh ( ) cos cos

p

nm

n m

V x y z V A x D

n m y za x D n m

H LH Lπ π π

∞ ∞

= =

= + − +

+ + − ∑∑

The pattern function f(y,z) is expanded in a double Fourier series with coefficients knm.

0 0

f ( , ) cos cosnm

n m

y zy z k n m

H Lπ π

∞ ∞

= =

=

∑∑

Once all the developments have been made following the same process as in the streaky case,

we obtain a set of formula which are the same as (18-19) with knm instead of kn and with

(n2/H

2 + m

2/L

2)1/2

instead of (n/H). The formulae are similar in the case of spotty and streaky

patters, with different numerical coefficients. Taking into account this similarity, it is clear

that the given examples for different streaky patters f(y) allow to foresee similar results with

f(y,z) spotty patters.

5. DISCUSSION

In reference [1], the impedance of the types (iω)ν and ln(iω) appeared and was found

again in the present study. These examples of CPA seem to confirm again the assumption that

this type of special element may be generated from many different aspects or phenomena.

It is even more interesting to have found the same complex impedance formulae in

two different situations: one in a case of rough electrode with a uniform double-layer


capacitance [1], second in a case of smooth electrode with non-uniform double-layer

capacitance .

A formula (19) was established, between the parameters of the double-layer

capacitance non-uniform distribution and the impedance created by such a non-uniformity.

The impedance Z includes a pure resistive part (ρ D), a pure capacitive part (l/iωCs) and a

special impedance zs resulting from the non-uniform double-layer capacitance distribution.

The variational method supposes that, starting from a "trite" configuration (ε=0), we increase

ε without going too far from this configuration. We found that the expansion coefficient (ε)

must remain lower than a value which, as expected, corresponds to 1/s sz D i Cρ ω< + (in

Appendix). This is the reason why we consider, in this study, only the case of slightly non-

uniform capacitance distribution. More precisely, we consider only the cases where the

magnitude of the special impedance sz is lower than the magnitude of the main resistive and

capacitive parts of the impedance.

Experimental considerations are outside the scope of the present study which is a

purely theoretical one. Nevertheless, the preceding theoretical limitation, and other

contingencies, draw one to think that any experiment under conditions representative of those

studied theoretically here would certainly raise experimental difficulties. These difficulties are

the same order than those in the different case of "saw-tooth shapes of roughness" published

in [1]. Even if R- and/or C-corrected impedance data are nowadays obtained using modern

impedance spectrometers, the zs contribution to Z is certainly difficult to be accurately

resolved. The results of this theoretical approach may nonetheless arouse some interest among

physicists and mathematicians who wish to find solutions valid in larger range of parameters,

or in more complicated configurations.

In fact, we do not expect from this kind of theoretical studies an immediate benefit for

the patricians requiring concrete predictive or explanatory results. Nevertheless, in the context

of thought and research in the field of modelling and methods to solve the very complicated

problems raised, we must first attempt to solve the simplest possible academic cases before

trying those cases that are more representative of reality, but more difficult to approach. The

present paper is an attempt on this long way.

REFERENCES

[1] : J. Jacquelin, Electrochim. Acta 39, 2673 (1994)

.

[2]: J. OlMo Bockris, A. K. N. Reddy, "Modern Electrochemistry", Plenum Press, New-

York (1973).

[3] : R. De Levie, Electrochim. Acta 8, 751 (1963).

[4]: M. A. Abramowitz, 1. A. Stegun, "Handbook of mathematical functions ", Dover

Publications, New-York (1970).

[5]: J. Jacquelin, " Theoretical Impedances of Smooth Flat Electrodes with non-uniform

double-layer capacitance: Complete collection of equations ",

unpublished (1995). [ Added to the 2009 edition, pp. 95-106 ]

[6] : J. Jacquelin, Electrochim. Acta 38, 597 (1993).


APPENDIX: MAIN STEPS IN DERIVATIONS AND COMPUTATIONS

The basic hypothesis and equations are stated in the main body, Eqs.(1-7). The reader

may also profitably refer to [1]. It is unfortunately impossible, in a reasonable space, to

provide sufficiently detailed information on the derivation. 1 am aware that the present

appendix do not contain all steps and the reader may find it a chore to deduce some

intermediate steps. In fact, the full information is available in a document [5] containing

heavy developments which are summarised hereafter.

We start with a "trite" problem (ε=0), then adds small variations of the boundary conditions to

the volume studied. Consequently, equations (1) and (2) must be expanded in a series of

ascending powers of ε. The unknown coefficients (A, an) are expanded in a series of ε

20 1 2 ...A A A Aε ε= + + + (11)

20 1 2 ...n n n na a a aε ε= + + + (12)

Condition (1) on the electrode (x=0) may be written as (10), where V and /V x∂ ∂ comes from

(2), and using (6), (7), (8) and (9). Equation (10) is fully expanded in series of ε :

( )

( )

( )

( )

20 1 2

20 1 2

1

20 0 1 2

0

20 1 2

1...

1 ... cosh cos

1 cos ...

... sinh

n n n

n

m p

m

n n n

A A A

n D ya a a n n

H H H

yi k n V D A A A

H

Da a a n

H

ε ερ

πε ε π π

ρ

ω γ ε π ε ε

ε ε π

∞

=

∞

=

+ + + +

+ + + + =

= + − + + + +

+ + + + −

∑

∑

1

cos

n

yn

Hπ

∞

=

∑

(10)

Equation (10) includes the (y) variable only. Therefore, the unknown constants (A0 , A1 , A2 ,

…, an0 , an1 , an2 , …) must hold for all values of y.

The "trite" solution is given by the terms (11) which do not contain ε :

0 0

1

0 0 0

1

cosh cos

sinh cos

n

n

p n

n

n D yA a n n

H H H

D yi V DA a n n

H H

ππ π

ω ρ γ π π

∞

=

∞

=

+ =

= − + −

∑

∑ (11)

The constants (A0, an0 ) must hold for all values of y. This is possible only if :


0

0 00

and 01

pn

i VA a

i D

ω ρ γ

ω ρ γ= =

+ (12)

The terms of equation (10) which contain only ε lead to equation (13) :

( )1 0 1

1

0 1 0

1 1

1 cosh cos

sinh cos cos

n

n

n m

n m

n D yA i D a n n

H H H

D y yi a n n A k m

H H H

πω ρ γ π π

ω ρ γ π π π

∞

=

∞ ∞

= =

+ + =

= − +

∑

∑ ∑ (13)

If the different terms of the Fourier expansion are made equal, the constants (Al, anl) are

obtained :

01 1

0

0 and

cosh sinh

nn

A kA a

n D Dn i n

H H H

ππ ω ρ γ π

= =

+

(14)

The same procedure is then applied to the terms of equation (10) which contain ε2, leading to

(15). Remark that coth(nπD/H) is very close to 1 because D >>H : the distance between the

electrode and the counter-electrode is large. For example, we set D=100H in numerical

computations.

2 20

20

1 02 1

coth

p n

n

V i kA

n Di Dn i

H H

ω ρ γ

πω ρ γπ ω ρ γ

∞

=

= −

+ +

∑

2 20

20

1 02 1

p n

n

V i kA

ni Di

H

ω ρ γπω ρ γ ω ρ γ

∞

=

≈ −

+ +∑ (15)

We could continue the procedure to express the following terms in the series expansion. In

[6], the series expansions are limited to only three terms (the main term and the corrective

terms on the first and second order) and this method gave results which agree with the

simulation results of the numerical computation. The relationship (4) allows to express the

impedance (16) :

2

0 1 2 ...

pVZ

A A A

ρ

ε ε=

+ + + (16)

Carrying back in (16) the values of A0 , A1 and A2 that were found, after some conversions, we

obtain (17).

22

4

01 0

1(...)

2

n

n

kZ D

nii

H

ρ ερ ε

πω γ ω ρ γ

∞

=

= + + +

+∑ (17)

Then, from (17), we obtain the formulas (18-19) given in the main section of this paper.

Since Al =0, the conversions made to go from (16) to (17) are possible if 22 0/ 1A Aε < .

Carrying back the values A0 and A2 from (12) and (15), and after some conversions, the

preceding condition can be expressed as 01/sZ D iρ ω γ< + , where Zs is defined by (19).


THEORETICAL IMPEDANCE OF ROUGH ELECTRODES

COMPLETE COLLECTION OF EQUATIONS

This "Complete Collection of Equations" was written in order to make easier the

understanding and the verification of the mathematical developments in the main paper :

"Theoretical Impedances of Rough Electrodes"


STRIPED ELECTRODES

1


n


H Hπ π

∞

=

− = + − +

∑ (1)

(1) → (1-2) :

1

cosh cosn

n

V n x D yA a n n

x H H H

ππ π

∞

=

∂ − = +

∂ ∑ (1-2)

(10) → (1-3) :

1

sinh sinn

n

V n x D ya n n

y H H H

ππ π

∞

=

∂ − = −

∂ ∑ (1-3)

1 Vi V

nω γ

ρ

∂=

∂ (2)

( any , in f < < )

0

H

x x DL V

I dyx

ερ

∂ =

∂ ∫ (3-1)

(3-1) , (1-2) → (3-2) :

01

cosh cosH

n

n

L n x D yI A a n n dy

H H H

ππ π

ρ

∞

=

− = +

∑∫ (3-2)

(3-2) → (3-3) :

0 0cos 0

H Hy L L H An dy I A dy

Hπ

ρ ρ

= → = =

∫ ∫ (3-3)

(3-3) → (3-4) :

( ) p p

I AY

L H V Vρ= = (3-4)


0 ( )

H

px D

L V A L HI dy

xρ ρ=

∂ = =

∂ ∫ (3)

(10) (3) → (4) :

1p p

p

V VZ

I L H Aρ= = (4)

( )x f yε= (5)

0

( ) cosm

m

yf y k m

Hπ

∞

=

=

∑ (6)

(6) → (6-2) :

0

' cosm

m

m yf k m

H H

ππ

∞

=

= −

∑ (6-2)

(6) → (7-1) :

0 00

0 001

( ) cos

sin

H H

m

m

y H

mym

yf y dy k m

H

H yk H k m k H

m H

π

ππ

∞

=

∞ =

==

= =

= − =

∑∫ ∫

∑ (7-1)

(7-1) → (7) :

00

( ) 0 0H

f y dy k= ⇒ =∫ (7)

d( f )tan( ) f '

d y

εα ε= − = − (8-1)


(8-1) → (8-2) :

( )2 2

1 1cos( )

1 ( f ' )1 tan( )

αεα

= =++

(8-2)

(8-1) → (8-3) :

( )2 2

tan( ) f 'cos( )

1 ( f ' )1 tan( )

α εα

εα

−= =

++

(8-3)

(8-2) , (8-3) → (8) :

2 2

1 f 'cos( ) ; sin( )

1 ( f ') 1 ( f ' )

εα α

ε ε

−= =

+ + (8)

(8) → (9) :

( )1/ 2

2cos( ) sin( ) 1 ( f ' ) f 'V V V V V

n x y x yα α ε ε

− ∂ ∂ ∂ ∂ ∂= + = + −

∂ ∂ ∂ ∂ ∂ (9)

(2) (2) , (9) → (10-1) :

( )1/ 2

21 ( f ') f 'V V V

i Vn x y

ω γ ρ ε ε− ∂ ∂ ∂

= = + − ∂ ∂ ∂ (10-1)

(10-1) , (1) → (10-2) :

( )1

1/ 22

( ) sinh cos

1 ( f ') f '

p n

n

x D yi V A x D a n n

H H

V V

x y

ω γ ρ π π

ε ε

∞

=

−

− + − + =

∂ ∂= + −

∂ ∂

∑ (10-2)

(10-2) , (1-2) , (1-3) → (10-3) :

: ( )

( ) ( )

1

1/ 22

1

1/ 22

( ) sinh cos

1 ( f ' ) cosh cos

1 ( f ') f ' sinh sin

p n

n

n

n

n

x D yi V A x D a n n

H H

n x D yA a n n

H H H

n x Da n n

H H

ω γ ρ π π

πε π π

πε ε π π

∞

=

∞−

=

−

− + − + =

− = + + +

− + + − −

∑

∑

1n

y

H

∞

=

∑

(10-3)


(10-3) , (5) → (10) :

( )1

21

21

ff sinh cos

1 fcosh cos

1 ( f ')

1 ff ' sinh sin

1 ( f ')

p n

n

n

n

n

n

D yi V A D a n n

H H

n D yA a n n

h H H

n D ya n n

h H H

εω γ ρ ε π π

π επ π

ε

π εε π π

ε

∞

=

∞

=

∞

=

− + − + =

− = + + +

− + +

∑

∑

∑

(10)

2

0 1 2 ...A A A Aε ε= + + + (11)

20 1 2 ...n n n na a a aε ε= + + + (12)

( )1/ 2

2 2 21 11 ( f ') 1 (f ') ...

2 f 'ε ε ε

− + = − + ≤

(13)

f 1 f fsinh exp exp

2

D D Dn n n

H H H

ε ε επ π π

− − − = − −

(14-1)

(14-1) → (14-2) :

f 1 f fsinh exp exp exp exp

2

D D Dn n n n n

H H H H H

ε ε επ π π π π

− − = − −

(14-2)

(14-2) → (14-3) :

f 1 fsinh exp exp

2

D Dn n n

H H H

ε επ π π

− ≈ − −

(14-3)

(14-3) → (14) :

2 2f 1sinh exp 1 f f ...

2 2

D D n nn n

H H H H

ε π π επ π ε

− ≈ − − + +

(14)


f 1 f fcosh exp exp

2

D D Dn n n

H H H

ε ε επ π π

− − − = + −

(14-5)

(14-5) → (14-6) :

f 1 f fcosh exp exp exp exp

2

D D Dn n n n n

H H H H H

ε ε επ π π π π

− − = + −

(14-6)

(14-6) → (14-7) :

f 1 fcosh exp exp

2

D Dn n n

H H H

ε επ π π

− ≈ −

(14-7)

(14-7) → (14-8) :

2 2f 1cosh exp 1 f f ...

2 2

D D n nn n

H H H H

ε π π επ π ε

− ≈ − + +

(14-8)

(10) , (13) → (15-1) :

( )1

2 2

1

2 2

ff sinh cos

1 f1 ( f ' ) ... cosh cos

2

1 f 1 ( f ') ... f ' sinh sin

2

p n

n

n

n

n

n

D yi V A D a n n

H H

n D yA a n n

h H H

n D ya n n

h H H

εω γ ρ ε π π

π εε π π

π εε ε π π

∞

=

∞

=

− + − + =

− = − + + +

− + − +

∑

∑

1

∞

=

∑

(15-1)

(15-1) → (15-2) :

( )1

2 2

2 2

1

ff sinh cos

11 ( f ') ...

2

1 f 1 ( f ' ) ... cosh cos

2

p n

n

n

n

D yi V A D a n n

H H

A

n D ya n n

h H H

εω γ ρ ε π π

ε

π εε π π

∞

=

∞

=

− + − + =

= − + +

− + − + +

∑

∑f

f 'sinh sinD y

n nH H

εε π π

− +

(15-2)


(15-2) , (14) , (14-8) → (15-3) :

( )

2 2

1

2 2 2 2

1

f

1exp 1 f f ... cos

2 2

1 1 1 ( f ' ) ... 1 ( f ' ) ...

2 2

1 exp

2

p

n

n

n

n

i V i A D

D n n yi a n n

H H H H

nA a

h

Dn

H

ω γ ρ ω γ ρ ε

π π εω γ ρ π ε π

πε ε

π

∞

=

∞

=

+ − +

− − + + =

= − + + − +

∑

∑2 2

2 2

1 f f ... cos2

1 f ' exp 1 f f ... sin

2 2

n n yn

H H H

D n n yn n

H H H H

π π εε π

π π εε π ε π

− + + +

+ − − + +

(15-3)

(15-3) → (15-4) :

( )

2 2

1

2 2 2 2

1

f

1exp 1 f f ... cos

2 2

1 1 1 1 ( f ' ) ... 1 ( f ' ) ... exp

2 2 2

1

p

n

n

n

n

i V i A D

D n n yi a n n

H H H H

n DA a n

h H


π π εω γ ρ π ε π

πε ε π

∞

=

∞

=

+ − +

− − + + =

= − + + − + ×

×

∑

∑2 2

f f ... cos f 'sin2

n n y yn n

H H H H

π π εε π ε π

− + + −

(15-4)

(15-4) , (11) , (12) → (15-5) :

( )( )

( )

20 1 2

20 1 2

1

2 2

... f

1 ... exp

2

1 f f ... cos2

p

n n n

n

i V i A A A D

Di a a a n

H

n n yn

H H H


ω γ ρ ε ε π

π π εε π

∞

=

+ + + + − +

+ + + + − ×

× − + +

∑

( )

( )

2 2 20 1 2

2 2 20 1 2

1

2 2

1 1 ( f ' ) ... ...

2

1 1 1 ( f ') ... ... exp

2 2

1 f f ... cos f 'sin2

n n n

n

A A A

n Da a a n

h H

n n yn

H H H

ε ε ε

πε ε ε π

π π εε π ε

∞

=

=

= − + + + + +

+ − + + + + ×

× − + + −

∑

yn

Hπ

(15-5)


(15-5) → (15) :

( ) ( )

( )

20 1 2

20 1 2

1

22

... f

1 exp ...

2

f f 1 ... cos

p

n n n

n

i V i A A A D

Di n a a a

H

n n yn

H H H


ω γ ρ π ε ε

π πε ε π

∞

=

+ + + + −

− + + + ×

× − + +

∑

( )

( )

2 2 20 1 2

2 2 20 1 2

1

22

11 (f ') ... ...

2

1 1 (f ' ) ... exp ...

2

f f 1 ... cos f 'sin

n n n

n

A A A

n Dn a a a

H H

n n y yn n

H H H H

ε ε ε

πε π ε ε

π πε ε π ε π

∞

=

=

= − + + + + +

+ − + + + + ×

× − + + −

∑

(15)

(15) → (16-1) :

( )( ) ( )

( ) ( )

0 0

1

0 0

1

1exp cos

2

exp cos

p n

n

n

n

D yi V i A D i n a n

H H

n D yA n a n

H H H

ω γ ρ ω γ ρ ω γ ρ π π

ππ π

∞

=

∞

=

+ − − =

= +

∑

∑ (16-1)

(16-1) → (16) :

00

1

0 0

1

exp cos2

exp cos2

np

n

n

n

a D yi V A D n n

H H

n D yA a n n

H H H

ω γ ρ π π

ππ π

∞

=

∞

=

− − =

= +

∑

∑ (16)

(16) → (17-1) :

( )0 0pi V A D Aω γ ρ − = (17-1)

(17-1) → (17) :

01

pi VA

i D

ω γ ρ

ω γ ρ=

+ (17)


(16) → (18-1) :

00exp cos exp cos

2 2

nn

a D y n D yi n n a n n

H H H H H

πω γ ρ π π π π

=

(18-1)

(18-1) → (18) :

0 0na = (18)

(15), (18), (16-1) → (19-1) :

( ) ( ) ( )( )

( )

2 20 1 2 1 2

21 2

1

22

... f ...

1 exp ...

2

f f 1 ... cos

n n

n

i A A A i A A D

Di n a a

H

n n yn

H H H

ω γ ρ ε ε ε ω γ ρ ε ε

ω γ ρ π ε ε

π πε ε π

∞

=

+ + + + + + + −

− + + ×

× − + +

∑

( ) ( )

( )

2 2 2 21 2 0 1 2

2 2 21 2

1

22

1... (f ' ) ... ...

2

1 1 (f ') ... exp ...

2

f f 1 ... cos f 'sin

n n

n

A A A A A

n Dn a a

H H

n n yn

H H H

ε ε ε ε ε

πε π ε ε

π πε ε π ε

∞

=

=

= + + + − + + + +

+ − + + + ×

× − + + −

∑

yn

Hπ

(19-1)

(19-1) → (19-2) :

( )( ) ( )( )

( )

( )

0 1 1 2

1 2

1

1 2

... f ...

1 exp ...

2

f 1 ... cos

1... (f

2

n n

n

i A A i A A D

Di n a a

H

n yn

H H

A A

ω γ ρ ε ω γ ρ ε

ω γ ρ π ε

πε π

ε

∞

=

+ + + + + + −

− + + ×

× − + =

= + + + −

∑

( )

( )

20

1 2

1

' ) ... ...

exp ...

f 1 ... cos f 'sin

n n

n

A

n Dn a a

H H

n y yn n

H H H

ε

ππ ε

πε π ε π

∞

=

+ +

+ + + ×

× − + −

∑

(19-2)


(19-2) → (19-3) :

( )

( )

0 1 1

1

1 1

1

1f exp cos

2

exp cos

n

n

n

n

D yi A i A D i n a n

H H

n D yA n a n

H H H


ππ π

∞

=

∞

=

− − =

= +

∑

∑ (19-3)

(19-3), (6) → (19) :

0 1

1

11

1

cos

exp cos2

n

n

n

n

yi A k n i A D

H

a n D yA i n n

H H H


πω γ ρ π π

∞

=

∞

=

− =

= + +

∑

∑ (19)

(19) → (20-1) :

1 1i A D Aω γ ρ− = (20-1)

(19) (20-1) → (20) :

1 0A = (20)

(19) → (21-1) :

10 exp

2

nn

a n Di A k i n

H H

πω γ ρ ω γ ρ π

= +

(20-1)

(21-1) → (21) :

1 02 expn ni D

a A k nn H

iH

ω γ ρπ

πω γ ρ

= −

+

(21)


(19-2) , (19-3) → (22-1) :

( )( ) ( )( )

( )

( )

1 2

2

1

1

1

2

... f ...

1 exp ... cos

2

1 f exp ... ... cos

2

.

n

n

n

n

i A i A D

D yi n a n

H H

D n yi n a n

H H H

A


ω γ ρ π ε π

πω γ ρ π ε ε π

ε

∞

=

∞

=

+ + + −

− +

− + − + =

= +

∑

∑

( ) ( )

( )

( )

20

2

1

1

1

1.. (f ' ) ... ...

2

exp ... cos

f exp ... ... co

n

n

n

n

A

n D yn a n

H H H

n D nn a

H H H

ε

ππ ε π

π ππ ε

∞

=

∞

=

+ − + +

+ +

+ + − +

∑

∑

( )( )1

1

s

exp ... f ' ... sin2

n

n

yn

H

n D yn a n

H H H

π

ππ ε π

∞

=

+ + − +

∑

(22-1)

(22-1) → (22-2) :

( )

( )

( ) ( ) ( )

1 2 2

1

1

1

22 0 2

1

1f exp cos

2

1 exp f cos

2

1 (f ' ) exp cos

2 2

n

n

n

n

n

n

D yi A i A D i n a n

H H

D n yi n a n

H H H

n D yA A n a n

H H H


πω γ ρ π π

ππ π

∞

=

∞

=

=

− −

− − =

= + − +

∑

∑

( )

( )( )

1

1

1

1

f exp cos

2

exp f ' sin2

n

n

n

n

n D n yn a n

H H H H

n D yn a n

H H H

π ππ π

ππ π

∞

∞

=

∞

=

+ −

+ −

∑

∑

∑

(22-2)


(22-2) → (22-3) :

( )

1 2 2

1

1

1

22 0 2

1

1f exp cos

2

exp f cos2

1 (f ' ) exp cos

2 2

n

n

n

n

n

n

D yi A i D A i a n n

H H

n D yi a n n

H H H

n D yA A a n n

H H H


πω γ ρ π π

ππ π

∞

=

∞

=

∞

=

− −

+ =

= − +

∑

∑

∑

( )

( )

2

1

1

1

1

1 exp f cos

2

exp f ' sin2

n

n

n

n

n D ya n n

H H H

n D ya n n

H H H

ππ π

ππ π

∞

=

∞

=

−

−

∑

∑

(22-3)

(22-3) , (6) , (6-2) → (22-4) :

1 2

0

2

1

1

0

cos

1 exp cos

2

exp cos cos2

m

m

n

n

n m

m

yi A k m i D A

H

D yi a n n

H H

n D y yi a n k m n

H H H H


ω γ ρ π π

πω γ ρ π π π

∞

=

∞

=

∞

=

−

−

+

∑

∑

∑1

2

2 0

0

2

1

2

1

1 cos

2

exp cos2

1 exp

2

n

m

m

n

n

n

m yA k m A

H H

n D ya n n

H H H

n Da n k

H H

ππ

ππ π

ππ

∞

=

∞

=

∞

=

=

= − −

+

−

∑

∑

∑

1 0

1

1 0

cos cos

exp cos sin2

m

n m

n m

n m

y ym n

H H

n D m y ya n k m n

H H H H H

π π

π ππ π π

∞ ∞

= =

∞ ∞

= =

− −

∑ ∑

∑ ∑ (22-4)


(22-4) → (22-5) :

1 2

0

2

1

1

0

cos

1 exp cos

2

exp cos cos2

m

m

n

n

m n

m

yi A k m i D A

H

D yi a n n

H H

n D y yi k a n m n

H H H H


ω γ ρ π π

πω γ ρ π π π

∞

=

∞

=

∞

=

−

−

+

∑

∑

∑1

2

2 0

0

2

1

2

1

1 cos

2

exp cos2

exp c2

n

m

m

n

n

mn

m yA k m A

H H

n D ya n n

H H H

k n Da n

H H

ππ

ππ π

ππ

∞

=

∞

=

∞

=

=

= −

+

−

∑

∑

∑

1 0

1

1 0

os cos

exp cos sin2

n m

m n

n m

y ym n

H H

m n D y yk a n m n

H H H H H

π π

π ππ π π

∞ ∞

= =

∞ ∞

= =

+

∑∑

∑∑ (22-5)

1cos cos cos ( ) cos ( )

2

y y y ym n m n m n

H H H Hπ π π π

= − + +

(22-6)

1cos cos 1 cos 2

2

y y yn n n

H H Hπ π π

= +

(22-7)

1sin sin cos ( ) cos ( )

2

y y y ym n m n m n

H H H Hπ π π π

= − − +

(22-8)

1sin sin 1 cos 2

2

y y yn n n

H H Hπ π π

= −

(22-9)


(22-4) , (22-6) , (22-8) → (22-10) :

1 2

0

2

1

1

0

cos

1 exp cos

2

exp cos ( ) cos ( )4

m

m

n

n

m n

m

yi A k m i D A

H

D yi a n n

H H

n D y yi k a n m n m n

H H H H


ω γ ρ π π

πω γ ρ π π π

∞

=

∞

=

∞

=

−

−

+ − + +

∑

∑

1

02

0 0

2

1

2

1

cos ( ) cos ( )4

exp cos2

exp4

n

m n

n m

n

n

mn

A m n y yA k k m n m n

H H H H

n D ya n n

H H H

k n Da n

H H

π ππ π

ππ π

ππ

∞

=

∞ ∞

= =

∞

=

=

= − − − +

+

−

∑∑

∑∑

∑

1 0

1

1 0

cos ( ) cos ( )

exp cos ( ) cos ( )2 2

n m

m n

n m

y ym n m n

H H

m n D y yk a n m n m n

H H H H H

π π

π ππ π π

∞ ∞

= =

∞ ∞

= =

− + +

+ − − +

∑∑

∑∑ (22-10)

(22-10) → (22-11) :

( )

2 1

1

220

2

0

22

1 1

1 1

exp4

4

exp exp4 2

n n

n

n

n

nn n n

n n

n Di D A i k a n

H H

A nA k

H

k n D n Da n k a n

H H H H

πω γ ρ ω γ ρ π

π

π ππ π

∞

=

∞

=

∞ ∞

= =

− + =

= −

− +

∑

∑

∑ ∑

(22-11)

(22-11) → (22) :

20

2 1 2

1 1

exp4 4

n n n

n n

An D ni D A i a k n A k

H H H


∞ ∞

= =

− + = −

∑ ∑ (22)


(22) → (23-1) :

( )2

02 1

1 1

1 exp4 4

n n n

n n

An D ni D A i a k n k

H H H


∞ ∞

= =

+ = +

∑ ∑ (23-1)

(23-1) , (21) → (23-2) :

( ) ( )2

2 02 0

1 1

12 4

n n

n n

An i ni D A i A k k

nH Hi

H

π ω γ ρ πω γ ρ ω γ ρ

πω γ ρ

∞ ∞

= =

+ = +

+∑ ∑ (23-2)

(23-2) → (23) :

22

202

1

( )1 1

( )1 2 4

n

n

ni

A nHA kni D H

iH

πω γ ρ π

πω γ ρ ω γ ρ

∞

=

= + + +

∑ (23)

(4) , (11) → (24) :

222

0 0220 20

0

1 ......

1 ...

p p pV V V AZ

A AAA AA

A

ρ ρ ρε

εε

= = = − +

+ + + +

(24)

(24) , (17) , (23) → (25-1)

222

2

1

( )1 1

1 ( )1 2 4

1

pn

pn

niV nHZ k

i V ni D Hi

Hi D

πω γ ρρ ε π

ω γ ρ πω γ ρ ω γ ρω γ ρ

∞

=

= − + + + +

∑ (25-1)

(25-1) → (25-2) :

222

2

1

1 1( )

2 2n

n

ni

i D nHZ kni i H

iH

πω γ ρω γ ρ ε π

πω γ ω γω γ ρ

∞

=

+

= − + +

∑ (25-2)


(25-2) → (25-3) :

2 22

1

1 1( )

2 2n

n

n i nZ D k

ni H i Hi

H

ε π ω γ ρ πρ

πω γ ω γω γ ρ

∞

=

= + − + +

∑ (25-3)

(25-3) → (25) :

22

1

1 1( )

2 2n

n

n

n n HZ D kni H i H

iH

πρε π π

ρ ρπω γ ω γ ω γ ρ

∞

=

= + − + − +

∑ (25)

1( )s s s

s

Z R r Zi Cω

= − + + (26)

(25-3) , (26) → (27-1) :

22

1

( )2

s n

n

nr k

H

ε πρ

∞

=

= ∑ (27-1)

(25-3) , (26) , (27-1) → (27) :

22

1

; ( )2

s s n

n

R D r n kH

π ερ ρ

∞

=

= = ∑ (27)

(25-3) , (26) → (28-1) :

22

1

1 1 1( )

2 2n

sn

n nk

i C i H i H

ε π π

ω ω γ ω γ

∞

=

= −

∑ (28-1)

(28-1) → (28) :

22 2 2

2

1

1 11 ( )

4n

sn

n ki C i H

πε

ω ω γ

∞

=

= −

∑ (28)


(25-3) , (26) → (29-1) :

22

1

( )2

s n

n

n

n HZ knH

iH

πρε π

πω γ ρ

∞

=

= − − +

∑ (29-1)

(29-1) → (29) :

2 22

1

( )

2

ns

n

n kZ

HHn i

π ερ

ω γ ρπ

∞

=

=

+

∑ (29)

(29) → (30-1) :

1 2 22 2 2

1

1

( ) ( )2 2

s

n

n n

ZH

i

n k n kH H

ω γ ρπ ε π ε

ρ π ρ

∞

=

=

+∑ (30-1)

1

1

1s

n nn

Z

i cr

ω

∞

=

=

+∑ (30)

(30-1) , (30) → (31) :

( )

2 2 2

2

( ) 2;

2

nn n

n

n k Hr c

H n k

ρ π ε γ

π ε= = (31)


EXAMPLES OF IMPEDANCES (Zs)

The "Sawtooth" shape :

2 2 12 2

1,3,5,...

cos4 4

f( ) ; 0 ;(2 1)

m m

n

yn

Hy k k

n m

π

π π

∞

+

=

= = =

+∑ (32)

(29) , (32) → (33-1) :

2 22 2

21 0

( ) 8 1

2(2 1) (2 1)

ns

n m

n kZ

H HH Hn i m i m

π ε ερ ρ

πω γ ρ ω γ ρ

π π

∞ ∞

= =

= =

+ + + +

∑ ∑ (33-1)

2 2 2 2

1 1 1 1; 2 1

( ) ( )

Hi n m

n n n n nκ ω γ ρ

πκ κ κ κ κ

= − + = = +

+ + (33-2)

(33-1) , (33-2) → (33-3) :

1 2 2

2

2

0

8

(2 1)(2 1) (2 1)s

m

H H Hi i i

ZHH mm m i

ω γ ρ ω γ ρ ω γ ρε π π π

ρπ

ω γ ρπ

− − −

∞

=

= − + + + + +

∑ (33-3)

(33-3) → (33-4) :

2 2

2 2 2

0 0

1 8 1 8 1 1 1

2 1(2 1) (2 1)s

m m

ZHi H mH m H m ii

ε ερ

ω γ π ω γ ρω γ ρ ππ

∞ ∞

= =

= − − ++ + +

∑ ∑ (33-4)

2

2 2 2

0

1 1 11 ...

8(2 1) 3 5m

m

π∞

=

= + + + =+

∑ (33-5)

(33-4) , (33-5) → (33-6) :

2 2 2

2 2 2

0 0

1 1 8 1 1

2 1(2 1) (2 1)s

m m

ZHi mH m H m iH i

π ε ρ ε

ω γ ω γ ρπ ω γ ρ ππ

∞ ∞

= =

= + − ++ + +

∑ ∑ (33-6)

2 2

2

1 1;

' 's s

s s

Z zi c i c i H

π ε

ω ω ω γ= + = (33)

2

2

0

8 1 1

2 1(2 1)

s

m

zH mH m iH i

ρ ε

ω γ ρπ ω γ ρ ππ

∞

=

= − + + +

∑ (34)


0

1 1 1 3 1 3 1 11

2 1 2 2 2 2 2(2 1) 1m

Hi

H Hmm i i

ω γ ρπω γ ρ ω γ ρ

π π

∞

=

− = Ψ − Ψ + − + + + + +

∑

(35-1)

[ Summation of rational series by means of Polygamma Functions ψ ,

Handbook of Mathematical Functions, 9th

. Print, Dover Publ., N-Y, 1970, pp.264-265 ]

30.0365

2

Ψ ≃ (35-2)

2

1 1( ) ln( ) ... 1

2 12ξ ξ ξ

ξ ξ Ψ = − − + >> (35-3)

(35-3) → (35-4) :

3 1 3 1 1ln ... 1

2 2 2 22

2

H H Hi i

Hi

ω γ ρ ω γ ρ ω γ ρπ π πω γ ρ

π

Ψ + = + − + >>

(35-4)

3 1 3ln ln 1

2 2 2 32

2

H H Hi i

Hi

ω γ ρ ω γ ρ ω γ ρπ π πω γ ρ

π

+ = + >>

(35-5)

(35-4) , (35-5) → (35-6) :

3 1 1ln ...

2 2 2

2

H Hi i

Hi

ω γ ρ ω γ ρπ π ω γ ρ

π

Ψ + = + +

(35-6)

(35-1) , (35-2) , (35-6) → (35-7) :

0

1 1 1ln 0.982..

2 1 2 2(2 1)m

Hi

H mm i

ω γ ρπω γ ρ

π

∞

=

− − − + + +

∑ ≃ (35-7)

(34) , (35-76) → (35-8) : 2

2

8 1ln 0.982..

2 2s

Hz i

HH i

ρ εω γ ρ

ππ ω γ ρ

π

− −

≃ (35-8)

2

2

2 2 3

4ln 2

2s

Hz i H e

H

π εω γ ρ ω γ ρ π

πρ γ ω

≈ >> (35)

2

2

2 2 3

4ln 2

2 2s

Hz i H e

H

π ε πω γ ρ ω γ ρ π

πρ γ ω

≈ + >> (35-10)


The rectangular shape :

( )2 2 1

0

1 cos (2 1)4 4 ( 1)

f( ) ; 0 ;(2 1) (2 1)

mm

m m

m

ym

Hy k k

m m

π

π π

∞

+

=

− + − = = =

+ +∑ (36)

(29) , (36) → (37-1) :

2 22 2

1 0

( ) 8 1

2(2 1)

ns

n m

n kZ

H HH Hn i m i

π ε ερ ρ

πω γ ρ ω γ ρ

π π

∞ ∞

= =

= =

+ + +

∑ ∑ (37-1)

[ formal expression, valid only in (Zs-rs) ]

(27) , (36) → (37-2) :

22

1

( )2

s n

n

r n kH

π ερ

∞

=

= ∑ (37-2)

(37-1) , (37-2) , (36) → (37) :

2

0

8 1 1

2 12 1

s s s

m

z Z rHH m

m i

ρ ερ

π ω γ ρπ

∞

=

= − = − + + +

∑ (37)

(37) , (35-7) → (38-1) :

28 1

ln 0.982.. 12 2 3

s s sH H

z Z r iH

ρ εω γ ρ ω γ ρ

π π π

= − − − >>

≃ (38-1)

(38-1) → (38) :

2

24ln 2

2s

Hz i H e

H

ρ εω γ ρ ω γ ρ π

π π

≈ − >> (38)

(38) → (38-3) :

2

24ln 2

2 2s

Hz i H e

H

ρ ε πω γ ρ ω γ ρ π

π π

≈ − + >> (38-3)


The smoothed rectangular shapes :

( )2 2 11 1

0

1 cos (2 1)4 4 ( 1)

f( ) ; 0 ;(2 1) (2 1)

mm

m mp p

m

ym

Hy k k

m m

π

π π

∞

++ +=

− + − = = =

+ +∑ (39)

(29) , (39) → (40-1) :

2 22 2

11 1

( ) 1

2 2(2 1) (2 1)

ns

pn n

n kZ

H HH Hn i m i m

π ε π ερ ρ

ω γ ρ ω γ ρπ π

∞ ∞

+= =

= =

+ + + +

∑ ∑ (40-1)

2

20

8 1

(2 1) (2 1)s

pm

ZHH

m i m

ρ ε

πω γ ρ

π

∞

=

=

+ + +

∑ (40)

( ) 12 2

0 2

1 d

(2 1) (2 1) (2 1 )(2 1)p p

mm m

µ

λ µ λ µ

∞ ∞

−=

≈+ + + + + +

∑ ∫ (41-1)

(41-1) → (41-2) :

( ) 2 20

0

1 1 1 d2 1 ; d d ;

2 2(2 1) (2 1) ( )p p

mm m

ββ µ µ β

λ β λ β

∞ ∞

=

= + = ≈+ + + +

∑ ∫ (41-2)

2 2 20 0

d 1 d 1 d; d ;

2 ( ) 2 ( 1)p p p

β β β χχ χ

λ λ β λ β λ χ χ

∞ ∞= = =

+ +∫ ∫ (41-3)

20

d 10

sin(2 ) 2( 1) pp

p

χ π

πχ χ

∞ = < < +∫ (41-4)

(41-2) , (41-3) , (41-4) → (41-5) :

( ) 2 2

0

1 1 10

sin(2 ) 2(2 1) (2 1) 2p p

m

ppm m

π

πλ λ

∞

=

≈ < < + + +

∑ (41-5)


(40) , (41-5) → (41-6) :

2 2

22

0

8 1 8

(2 1) (2 1) 2 sin(2 )

s pp

m

ZHH H Hm i m i p

ρ ε ρ ε π

π πω γ ρ ω γ ρ ππ π

∞

=

= ≈ + + +

∑ (41-6)

(41-6) → (41) :

( )[ ]

2 2

22 2 2 1

40 0.5

sin(2 )

p

s pp p pZ p

p H i

ρ ε π

π γ ρ ω+≈ < < (41)

Case p=0.5 ( verification, case "rectangular shape" ) :

(40) , ( p=0.5) → (42-1) :

2

0

8 1

(2 1) (2 1)s

m

ZHH

m i m

ρ ε

πω γ ρ

π

∞

=

=

+ + +

∑ (42-1)

(42-1) → (42-2) :

2

0

8 1 1 1

2 1(2 1)

s

m

ZH HH m

i m i

ρ ε

π ω γ ρ ω γ ρπ π

∞

=

= − + + +

∑ (42-2)

(42-2) , (35-7) → (42-3) :

28 1 1

ln 0.9822 2

sH

Z iHH

i

ρ εω γ ρ

π πω γ ρπ

− −

≃ (42-3)

(42-3) → (42) :

2

2

2

4 1ln 0.5 ; 2

2s

HZ i p H e

iH

εω γ ρ ω γ ρ π

ω γ π

≈ − = >> (42)


ROUGH ELECTRODES

1/ 2

2 2

2 2n mn m

H Lµ

= +

(0b)

1/ 22 2

2 2

0 0

( , , ) ( )

sinh ( ) cos cos

p

nm

n m

V x y z V A x D

n m y za x D n m

H LH Lπ π π

∞ ∞

= =

= + − +

+ + − ∑∑

(1b)

(1b) , (0b) → (1b-2) :

( )0 0

cosh ( ) cos cosnm nm nm

n m

V y zA a x D n m

x H Lµ π µ π π π

∞ ∞

= =

∂ = + −

∂ ∑∑ (1b-2)

(1b) , (0b) → (1b-3) :

( )0 0

sinh ( ) sin cosnm nm

n m

V n y za x D n m

y H H L

πµ π π π

∞ ∞

= =

∂ = − −

∂ ∑∑ (1b-3)

(1b) , (0b) → (1b-4) :

( )0 0

sinh ( ) cos sinnm nm

n m

V m y za x D n m

z L H L

πµ π π π

∞ ∞

= =

∂ = − −

∂ ∑∑ (1b-4)

1 Vi V

nω γ

ρ

∂=

∂ (2b)

0 0

1 1

( )

y H z L

y z p p

V AY dy dz

x LH V Vρ ρ

= =

= =

∂= = ∂ ∫ ∫ (3b)

(3b) → (4b) :

1 pVZ

Y Aρ= = (4b)


f( , )x y zε= (5b)

0 0

f( , ) cos cosNM

N M

y zx y z k N M

H Lε ε π π

∞ ∞

= =

= =

∑∑ (6b)

(6b) → (6b-2) :

0 0

fsin cosNM

N M

N y zk N M

y H H L

ππ π

∞ ∞

= =

∂ = −

∂ ∑∑ (6b-2)

(6b) → (6b-3) :

0 0

fcos sinNM

N M

M y zk N M

z L H L

ππ π

∞ ∞

= =

∂ = −

∂ ∑∑ (6b-3)

(6b) → (7b-1) :

0 0

000 0

0 0

f ( , )

cos cos

y H z L

y z

y H z L

NMy z

N M

y z dy dz

y zk N M dy dz k H L

H Lπ π

= =

= =

∞ ∞ = =

= == =

=

= =

∫ ∫

∑∑ ∫ ∫ (7b-1)

(7b-1) → (7b) :

000 0

f ( , ) 0 0y H z L

y z

y z dy dz k= =

= == ⇒ =∫ ∫ (7b)

1/ 22 2

f f

d

df f

1

V V V

V x y y z z

n

y y

ε ε

ε ε

∂ ∂ ∂ ∂ ∂− −

∂ ∂ ∂ ∂ ∂= ∂ ∂ + +

∂ ∂

(9b)


(2b) , (9b) → (10b-1) :

1/ 22 2

f f

d

df f

1

V V V

V x y y z zi V

n

y y

ε ε

ω γ ρ

ε ε

∂ ∂ ∂ ∂ ∂− −

∂ ∂ ∂ ∂ ∂= =

∂ ∂ + + ∂ ∂

(10b-1)

(10b-1) , (1b) → (10b-2) :

( )0 0

1/ 22 2

( )

sinh ( ) cos cos

f f f f 1

p

nm nm

n m

i V i A x D

y zi a x D n m

H L

V V V

y y x y y z z

ω γ ρ ω γ ρ

ω γ ρ µ π π π

ε ε ε ε

∞ ∞

= =

−

+ − +

+ − =

∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + − − ∂ ∂ ∂ ∂ ∂ ∂ ∂

∑∑ (10b-2)

(10b-21) , (1b-2) , (1b-3) , (1b-4) → (10b-3) :

( )0 0

1/ 22 2

1/ 22 2

( )

sinh ( ) cos cos

f f 1

f f 1

p

nm nm

n m

nm

i V i A x D

y zi a x D n m

H L

Ay y

ay y

ω γ ρ ω γ ρ


ε ε

ε ε

∞ ∞

= =

−

−

+ − +

+ − =

∂ ∂ = + + + ∂ ∂

∂ ∂ + + + ∂ ∂

∑∑

( )

( )

0 0

cosh ( ) cos cos

f sinh ( ) sin cos

n m

nm nm

nm

y zx D n m

H L

n y zx D n m

y H H L

µ π µ π π π

πε µ π π π

∞ ∞

= =

− +

∂ + − +

∂

∑∑

( )f

sinh ( ) cos sin

nmm y z

x D n mz L H L

πε µ π π π

∂ + − +

∂ (10b-3)


(1b-3) , (5b) → (10b) :

( )0 0

1/ 22 2

1/ 22 2

( f )

sinh ( f ) cos cos

f f 1

f f 1

p

nm nm

n m

nm

i V i A D

y zi a D n m

H L

Ay y

ay y


ω γ ρ µ π ε π π

ε ε

ε ε

∞ ∞

= =

−

−

+ − +

+ − =

∂ ∂ = + + + ∂ ∂

∂ ∂ + + + ∂ ∂

∑∑

( )

( )

0 0

cosh ( f ) cos cos

f sinh ( f ) sin cos

n m

nm nm

nm

y zD n m

H L

n y zD n m

y H H L

µ π µ π ε π π

πε µ π ε π π

∞ ∞

= =

− +

∂ + − +

∂

∑∑

( )f

sinh ( f ) cos sin

nmm y z

D n mz L H L

πε µ π ε π π

∂ + − +

∂ (10b)

20 1 2 ...A A A Aε ε= + + + (11b)

20 1 2 ...nm nm nm nma a a aε ε= + + + (12b)

1/ 2

2 2 2 22

1/ 22 2

f f 1 f f1 1 ...

2

f f

y y y y

y y

ε ε ε

ε

−

−

∂ ∂ ∂ ∂ + + = − + + ∂ ∂ ∂ ∂

∂ ∂ ≤ + ∂ ∂

(13b)

( ) ( )2

21sinh ( f ) exp 1 ( f ) ( f ) ...

2 2nm nm nm nmD D

εµ π ε µ π µ π ε µ π

− ≈ − − + +

(14b)

( ) ( )2

21cosh ( f ) exp 1 ( f ) ( f ) ...

2 2nm nm nm nmD D

εµ π ε µ π µ π ε µ π

− ≈ − + +

(14b-8)


(10b) , (13b) → (15b-1) :

( )0 0

2 22

2 22

( f )

sinh ( f ) cos cos

1 f f 1 ...

2

1 f f 1 ...

2

p

nm nm

n m

i V i A D

y zi a D n m

H L

Ay y

y y


ω γ ρ µ π ε π π

ε

ε

∞ ∞

= =

+ − +

+ − =

∂ ∂ = − + + + ∂ ∂

∂ ∂ + − + + ∂ ∂

∑∑

( )

( )

0 0

cosh ( f ) cos cos

f sinh ( f ) sin cos

nm

n m

nm nm

nm

a

y zD n m

H L

n y zD n m

y H H L


πε µ π ε π π

∞ ∞

= =

− +

∂ + − +

∂

∑∑

( )f

sinh ( f ) cos sin

nmm y z

D n mz L H L

πε µ π ε π π

∂ + − +

∂ (15b-1)

(15b-1) , (14b) , (14b-8) → (15b-3) :

( )0 0

22

2 22

1( f ) exp

2

1 ( f ) ( f ) ... cos cos2

1 f f 1 ...

2

p nm nm

n m

nm nm

i V i A D i a D

y zn m

H L

Ay y

ω γ ρ ω γ ρ ε ω γ ρ µ π

εµ π ε µ π π π

ε

∞ ∞

= =

+ − −

− + + =

∂ ∂ = − + + + ∂ ∂

∑∑

( )2 2

2

0 0

22

22

1 f f 1 1 ... exp

2 2

1 ( f ) ( f ) ... cos cos2

f 1 ( f ) ( f ) ...

2

nm nm

n m

nm nm nm

nm nm

a Dy y

y zn m

H L

n

y

ε µ π

εµ π ε µ π µ π π π

ε πε µ π ε µ π

∞ ∞

= =

∂ ∂ + − + + ∂ ∂

− + +

∂− − + +

∂

∑∑

22

sin cos

f 1 ( f ) ( f ) ... cos sin

2

nm nm

y zn m

H H L

m y zn m

z L H L

π π

ε πε µ π ε µ π π π

∂ − − + + ∂

(15b-3)


(15b-3) , (11b) , (12b) → (15b) :

( )

( ) ( )

( )

20 1 2

20 1 2

0 0

22

22

0 1 2

... ( f )

1... exp

2

1 ( f ) ( f ) ... cos cos2

1 f f ... 1

2

p

nm nm nm nm

n m

nm nm

i V i A A A D

i a a a D

y zn m

H L

A A Ay


ω γ ρ ε ε µ π


ε ε

∞ ∞

= =

+ + + + −

− + + +

× − + + =

∂ ∂= + + + − +

∂ ∂

∑∑

( )

( )

22

2 22 2

0 1 2

0 0

22

...

1 f f1 ... ...

2

1 exp 1 ( f ) ( f ) ...

2 2

nm nm nm

n m

nm nm nm

y

a a ay y

D

ε

ε ε ε

εµ π µ π ε µ π

∞ ∞

= =

+ +

∂ ∂ + − + + + + + ∂ ∂

× − + +

∑∑

cos cos

f sin cos

nmy z

n mH L

n y zn m

y H H L

µ π π π

πε π π

×

∂ −

∂

f cos sin

m y zn m

z L H L

πε π π

∂ − ∂

(15b)

(15b) → (16b) :

( )

( )

0

0

0 0

0 0

0 0

1 exp cos cos

2

1 exp cos cos

2

p

nm nm

n m

nm nm nm

n m

i V i A D

y zi a D n m

H L

y zA a D n m

H L

ω γ ρ ω γ ρ


µ π µ π π π

∞ ∞

= =

∞ ∞

= =

−

− =

= +

∑∑

∑∑ (16b)

(16b) → (17b-1) :

( )0 0pi V A D Aω γ ρ − = (17b-1)

(17b-1) → (17b) :

01

pi VA

i D

ω γ ρ

ω γ ρ=

+ (17)


(16b) → (18b-1) :

0

0

cos cos

cos cos

nm

nm nm

y zi a n m

H L

y za n m

H L

ω γ ρ π π

µ π π π

− =

=

(18b-1)

(18b-1) → (18b) :

0 0nma = (18b)

(15b) , (18b) , (16b-1) → (19b-1) :

( ) ( )

( ) ( )

2 21 2 0 1 2

21 2

0 0

22

2

0

... ( ) ... ( f)

1 ... exp

2

1 ( f ) ( f ) ... cos cos2

1 f

2

nm nm nm

n m

nm nm

i A A D i A A A

i a a D

y zn m

H L

Ay

ω γ ρ ε ε ω γ ρ ε ε ε

ω γ ρ ε ε µ π


∞ ∞

= =

+ + − + + + +

− + +

× − + + =

∂ ∂= − +

∂

∑∑

( )

22

2 22 2

1 2

2 22 2

1 2

f...

1 f f ... 1 ...

2

1 f f 1 ...

2nm nm

y

A Ay y

a ay y

ε

ε ε ε

ε ε ε

+ ∂

∂ ∂ + + + − + + ∂ ∂

∂ ∂ + − + + + ∂ ∂ ( )

( )

0 0

22

...

1 exp 1 ( f ) ( f ) ...

2 2

cos cos

n m

nm nm nm

nm

D

y zn m

H L

εµ π µ π ε µ π

µ π π π

∞ ∞

= =

+

× − + +

×

∑∑

f sin cos

f cos sin

n y zn m

y H H L

m y zn m

z L H L

πε π π

πε π π

∂ −

∂

∂ − ∂

(19b-1)


(19b-1) → (19b-2) :

( ) ( )

( ) ( )

21 2 0 1 2

1 2

0 0

22

2 2

0

... ( ) ... ( f )

1... exp

2

1 ( f ) ( f ) ... cos cos2

1 f f

2

nm nm nm

n m

nm nm

i A A D i A A A

i a a D

y zn m

H L

Ay y

ω γ ρ ε ω γ ρ ε ε

ω γ ρ ε µ π


∞ ∞

= =

+ + − + + + +

− + +

× − + + =

∂ ∂= − + ∂ ∂

∑∑

( )

( ) ( )

2 22

1 2

2 22

1 2

0 0

22

...

1 f f... 1 ...

2

1 f f 11 ... ... exp

2 2

1 ( f ) ( f ) .2

nm nm nm

n m

nm nm

A Ay y

a a Dy y

ε

ε ε

ε ε µ π

εµ π ε µ π

∞ ∞

= =

+

∂ ∂ + + + − + + + ∂ ∂

∂ ∂ + − + + + + ∂ ∂

× − + +

∑∑

.. cos cos

f sin cos

f

nmy z

n mH L

n y zn m

y H H L

µ π π π

πε π π

ε

∂ −

∂

∂−

∂cos sin

m y zn m

z L H L

ππ π

(19b-2)

(19b-2) → (19b-3) :

( )

( )

1 0

1

0 0

1 1

0 0

f

1 exp cos cos

2

1 exp cos cos

2

nm nm

n m

nm nm nm

n m

i A D i A

y zi a D n m

H L

y zA a D n m

H L

ω γ ρ ω γ ρ


µ π µ π π π

∞ ∞

= =

∞ ∞

= =

− +

− =

= +

∑∑

∑∑

(19b-3)

(19b-3) , (6b) → (19b) :

:

( ) ( )

1 0

0 0

1 1

0 0

cos cos

1exp cos cos

2

NM

N M

nm nm nm

n m

y zi A D i A k N M

H L

y zA a D i n m

H L

ω γ ρ ω γ ρ π π

µ π µ π ω γ ρ π π

∞ ∞

= =

∞ ∞

= =

− + =

= + +

∑∑

∑∑ (19b)


(19b) → (20b-1) :

1 1i A D Aω γ ρ− = (20b-1)

(20b-1) → (20b) :

1 0A = (20b)

(19b) → (21b-1) :

( ) ( )10 exp

2

nmnm nm nm

ai A k i Dω γ ρ µ π ω γ ρ µ π= + (21b-1)

(21b-1) → (21b) :

( )1 02 expnm nm nmnm

ia A k D

i

ω γ ρµ π

µ π ω γ ρ= −

+ (21b)

(19b-2) , (19b-3) → (22b-1) :

( )

( )

2 1

2

0 0

1

0 0

2 2

0 2

f

1 exp cos cos

2

1exp f cos cos

2

1 f f

2

nm nm

n m

nm nm nm

n m

i A D i A

y zi a D n m

H L

y zi a D n m

H L

A Ay y


ω γ ρ ε µ π π π

ω γ ρ µ π µ π ε π π

ε ε

∞ ∞

= =

∞ ∞

= =

− +

−

+ =

∂ ∂ = − + + ∂ ∂

∑∑

∑∑

( )1

0 0

...

1 f exp sin cos

2

nm nm

n m

n y za D n m

y H H L

πµ π ε π π

∞ ∞

= =

+ +

∂ + − ∂

−

∑∑

( )

( )

2

0 0

21

0 0

fcos sin

1 exp cos cos

2

1 exp ( ) f cos cos

2

nm nm nm

n m

nm nm nm

n m

m y zn m

z L H L

y za D n m

H L

y za D n m

H L

πε π π

ε µ π µ π π π


∞ ∞

= =

∞ ∞

= =

∂ ∂

+

−

∑∑

∑∑ (22b-1)


(22b-1) → (22b3) :

( )

( )

2 1

2

0 0

1

0 0

2 2

0 2

f

1 exp cos cos

2

1exp f cos cos

2

1 f f

2

nm nm

n m

nm nm nm

n m

i A D i A

y zi a D n m

H L

y zi a D n m

H L

A Ay y

ω γ ρ ω γ ρ


ω γ ρ µ π µ π π π

∞ ∞

= =

∞ ∞

= =

− +

−

+ =

∂ ∂ = − + + ∂ ∂

∑∑

∑∑

( )1

0 0

1 f exp sin cos

2

f cos

nm nm

n m

n y za D n m

y H H L

mn

z L

πµ π π π

ππ

∞ ∞

= =

∂ + − ∂

∂−

∂

∑∑

( )

( )

2

0 0

21

0 0

sin

1 exp cos cos

2

1 exp ( ) f cos cos

2

nm nm nm

n m

nm nm nm

n m

y zm

H L

y za D n m

H L

y za D n m

H L

π

µ π µ π π π

µ π µ π π π

∞ ∞

= =

∞ ∞

= =

+

−

∑∑

∑∑ (22b-3)


(22b-3) , (6b) , (6b-2) (6b-31) → (22b-4) :

( )

( )

2 1

0 0

2

0 0

1

0 0 0 0

cos cos

1 exp cos cos

2

1 exp

2

NM

N M

nm nm

n m

nm nm nm

n m N M

y zi A D i A k N M

H L

y zi a D n m

H L

i a D



ω γ ρ µ π µ π

∞ ∞

= =

∞ ∞

= =

∞ ∞ ∞ ∞

= = = =

− +

−

+

∑∑

∑∑

∑∑ ∑∑

2

2 0

0 0

0

0 0

cos cos cos cos

1 sin cos

2

1 cos sin

2

NM

NM

N M

NM

N M

y z y zk N M n m

H L H L

N y zA A k N M

H H L

M y zA k N M

L H L

π π π π

ππ π

ππ π

∞ ∞

= =

∞ ∞

= =

=

= −

−

∑∑

∑∑

( )

( )

2

1

0 0 0 0

1

0 0 0 0

1 exp

2

sin cos sin cos

1 exp

2

nm nm

n m N M

NM

nm nm

n m N M

na D

H

N y z y zk N M n m

H H L H L

ma D

L

πµ π

ππ π π π

πµ π

∞ ∞ ∞ ∞

= = = =

∞ ∞ ∞ ∞

= = = =

+

+

∑∑ ∑∑

∑∑ ∑∑

( )

( )

2

0 0

21

0 0 0 0

cos sin cos sin

1 exp cos cos

2

1 exp ( )

2

NM

nm nm nm

n m

nm nm nm

n m N M

M y z y zk N M n m

L H L H L

y za D n m

H L

a D

ππ π π π

µ π µ π π π

µ π µ π

∞ ∞

= =

∞ ∞ ∞ ∞

= = = =

+

−

∑∑

∑∑ ∑∑

cos cos cos cosNMy z y z

k N M n mH L H L

π π π π

(22b-4)


(22b-4) → (22b-5) :

( )

( )

2 1

0 0

2

0 0

1

0 0 0 0

cos cos

1 exp cos cos

2

exp2

NM

N M

nm nm

n m

nm nm nm

n m N M

y zi A D i A k N M

H L

y zi a D n m

H L

i a D



πω γ ρ µ µ π

∞ ∞

= =

∞ ∞

= =

∞ ∞ ∞ ∞

= = = =

− +

−

+

∑∑

∑∑

∑∑∑∑

2

2 0

0 0

2

0

0 0

cos cos cos cos

1 sin cos

2

1 cos sin

2

NM

NM

N M

NM

N M

y z y zk N M n m

H L H L

N y zA A k N M

H H L

M y zA k N M

L H L

π π π π

ππ π

ππ π

∞ ∞

= =

∞ ∞

= =

=

= −

−

∑∑

∑∑

( )

( )

1

0 0 0 0

1

0 0 0 0

exp2

sin cos sin cos

exp2

nm nm

n m N M

NM

nm nm

n m N M

na D

H

N y z y zk N M n m

H H L H L

ma D

L

πµ π

ππ π π π

πµ π

∞ ∞ ∞ ∞

= = = =

∞ ∞ ∞ ∞

= = = =

+

×

+

∑∑ ∑∑

∑∑ ∑∑

( )

( )

2

0 0

2

1

0 0 0 0

cos sin cos sin

exp cos cos2

( ) exp

2

NM

nm nm nm

n m

nmnm nm

n m N M

M y z y zk N M n m

L H L H L

y za D n m

H L

a D

ππ π π π

πµ µ π π π

µ πµ π

∞ ∞

= =

∞ ∞ ∞ ∞

= = = =

×

+

−

∑∑

∑∑ ∑∑

cos cos cos cosNMy z y z

k N M n mH L H L

π π π π

×

(22b-5)


2

y y y yn N n N n N

H H H Hπ π π π

= − + +

(22b-6)


2

z z z zm M m M m M

L L L Lπ π π π

= − + +

(22b-7)



2

y y y yn N n N n N

H H H Hπ π π π

= − − +

(22b-8)


2

z z z zm M m M m M

L L L Lπ π π π

= − − +

(22b-9)

(22b-5) , (22b-6) , (22b-7) , (22b-8) , (22b-9) → (22b-10) :

( )

( )

2 1

0 0

2

0 0

1

0 0 0 0

cos cos

1 exp cos cos

2

exp8

cos (

NM

N M

nm nm

n m

nm NM nm nm

n m N M

y zi A D i A k N M

H L

y zi a D n m

H L

i a k D

n



πω γ ρ µ µ π

∞ ∞

= =

∞ ∞

= =

∞ ∞ ∞ ∞

= = = =

− +

−

+

× −

∑∑

∑∑

∑∑∑∑

2

2 0

0 0

0

0 0

) cos ( ) cos ( ) cos ( )

1 sin cos

2

1 cos sin

2

NM

N M

NM

N M

y y z zN n N m M m M

H H L L

N y zA A k N M

H H L

M y zA k N M

L H L

π π π π

ππ π

ππ π

∞ ∞

= =

∞ ∞

= =

+ + − + + =

= −

−

∑∑

∑∑

( )

( )

2

2

1 2

0 0 0 0

2

1 2

0 0 0

exp8

cos ( ) cos ( ) cos ( ) cos ( )

exp8

nm NM nm

n m N M

nm NM nm

m N M

n Na k D

H

y y z zn N n N m M m M

H H L L

m Ma k D

L

πµ π

π π π π

πµ π

∞ ∞ ∞ ∞

= = = =

∞ ∞ ∞

= = =

+

× − − + − + +

+

∑∑ ∑∑

∑∑

( )

0

2

0 0

2

1

0 0

cos ( ) cos ( ) cos ( ) cos ( )

exp cos cos2

( ) exp

8

n

nm nm nm

n m

nmnm NM nm

N M


H H L L

y za D n m

H L

a k

π π π π

πµ µ π π π

µ πµ

∞

=

∞ ∞

= =

∞ ∞

= =

× − + + − − +

+

−

∑∑

∑∑

∑∑ ( )0 0

cos ( ) cos ( ) cos ( ) cos ( )

n m

D


H H L L

π

π π π π

∞ ∞

= =

× − + + − + +

∑∑

(22b-10)


(22b-10) → (22b-11) :

( )

( )

2 1

0 0

2 2 2 22 2

2 0 02 2

0 0 0 0

2 2

1 2

0 0

2

1

exp8

( ) ( )8 8

exp8

nm nm nm nm

n m

nm nm

n m n m

nm nm nm

n m

nm nm

i A D i a k D

n mA A k A k

H L

na k D

H

ma k

πω γ ρ ω γ ρ µ µ π

π π

πµ π

∞ ∞

= =

∞ ∞ ∞ ∞

= = = =

∞ ∞

= =

− + =

= − −

+

+

∑∑

∑∑ ∑∑

∑∑

( )

( )

2

2

0 0

2

1

0 0

exp8

( ) exp

8

nm

n m

nmnm nm nm

n m

DL

a k D

πµ π

µ πµ π

∞ ∞

= =

∞ ∞

= =

−

∑∑

∑∑

(22b-11)

(22b-11) , (0b) → (22b) :

( )2 1

0 0

22 2

2 0

0 0

exp8

( ) ( )8

nm nm nm nm

n m

nm nm

n m

i A D i a k D

A A k

πω γ ρ ω γ ρ µ µ π

πµ

∞ ∞

= =

∞ ∞

= =

− + =

= −

∑∑

∑∑ (22b)

(22b) , (21b) → (23b-1) :

( ) 02

0 0

22 2

0

0 0

21

8

( ) ( )8

nmnm nm

nmn m

nm nm

n m

i A ki D A i k

i

A k

ω γ ρ πω γ ρ ω γ ρ µ

µ π ω γ ρ

πµ

∞ ∞

= =

∞ ∞

= =

+ =+

+

∑∑

∑∑ (23b-1)

(23b-1) → (23b-2) :

( )( )

2 222

2 0

0 0

( )( )1 ( )

84

nmnmnm

nmn m

ii D A A k

i

µ ππ µ ω γ ρω γ ρ

µ π ω γ ρ

∞ ∞

= =

+ = + +

∑∑ (23b-2)


(23b-2) → (23b) :

( ) ( )

2 2220

2

0 0

( )( )( )

1 84

nmnmnm

nmn m

A iA k

i D i

µ ππ µ ω γ ρ

ω γ ρ µ π ω γ ρ

∞ ∞

= =

= + + +

∑∑ (23b-2)

(4b) , (11b) → (24b) :

222

0 0220 20

0

1 ......

1 ...

p p pV V V AZ

A AAA AA

A

ρ ρ ρε

εε

= = = − +

+ + + +

(24b)

(4b) , (17b) , (23b) → (25b-1) :

( ) ( )

2 2222

0 0

( )( )1 ( )

1 84

1

nmnmp nm

nmn m

p

iV k

i D iZ

i V

i D

µ ππ µ ω γ ρερ

ω γ ρ µ π ω γ ρ

ω γ ρ

ω γ ρ

∞ ∞

= =

− + + +

=

+

∑∑

(25b-1)

(25b-1) → (25b-3) :

( )

2 2222

0 0

( )1( )

4 2

nmnmnm

nmn m

iZ D k

i ii

µ ππ µ ω γ ρερ

ω γ ω γµ π ω γ ρ

∞ ∞

= =

= + − + +

∑∑ (25b-3)

(25b-3) → (25b) :

( )

22

0 0

1( )

4 2

nm nmnm nm

nmn m

Z D ki i i

µ π ρ µ περ π µ ρ

ω γ ω γ µ π ω γ ρ

∞ ∞

= =

= + − + − +

∑∑ (25b)

1( )s s s

s

Z R r Zi Cω

= − + + (26b)

(25b-3) , (26b) → (27b-1) :

22

0 0

( )4

s n nm

n m

r kπ ρ ε

µ

∞ ∞

= =

= ∑∑ (27b-1)


(25b-3) , (26b) , (27b-1) → (27b) :

22

0 0

; ( )4

s s n nm

n m

R D r kπ ρ ε

ρ µ

∞ ∞

= =

= = ∑∑ (27b)

(25b-3) , (26b) → (28b-1) :

22

0 0

1 1( )

4 2

nmn nm

sn m

ki C i i

µ πεπ µ

ω ω γ ω γ

∞ ∞

= =

= −

∑∑ (28b-1)

(28b-1) → (28b) :

22 2 2

0 0

1 11 ( ) ( )

8n nm

sn m

ki C i

πε µ

ω ω γ

∞ ∞

= =

= −

∑∑ (28b)

(25b-3) , (26b) → (29b-1) :

( )

2 22 2

0 0

( ) ( )4

s nm nmnmn m

Z ki

ε π ρµ

µ π ω γ ρ

∞ ∞

= =

=+∑∑ (29b-1)

(29b-1) → (29b) :

2 2 2 22

2 2 1/ 22 2

0 0

2 2

( )4

s nm

n m

n mZ k

H ln m

iH l

ε π ρ

π ω γ ρ

∞ ∞

= =

= +

+ +

∑∑ (29b)

0 1

1

1s

n m nmnm

Z

i cr

ω

∞ ∞

= =

=

+∑∑ (30b)

(30b-1) , (30b) → (31b) :

( )

1/ 2 12 2 2 2 2 2

2 2 2 2 2

( ) 2;

2

nmnm n

k n m n mr c

H L H Lk

ρ π ε γ

π ε

−

= + = +

(31b)


THEORETICAL IMPEDANCE OF SMOOTH FLAT ELECTRODES

WITH SLIGHTLY NON-UNIFORM DOUBLE-LAYER CAPACITANCE

COMPLETE COLLECTION OF EQUATIONS

This "Complete Collection of Equations" was written in order to make easier the

understanding and the verification of the mathematical developments in the main paper :

"Theoretical Impedances of Smooth Flat Electrodes

with non-uniform double-layer capacitance"


1 Vi V

xω γ

ρ

∂=

∂ [ in x=0 ] (1)

( ) ( )( 0) ( )

0 , / 0 , / 0 , ( , ) (2)py y HV V y V y V x D y V

= =∆ = ∂ ∂ = ∂ ∂ = = = →

1


n


H Hπ π

∞

=

− = + − +

∑ (2)

(2) → (2-1) :

1

(0, ) sinh cosp n

n

D yV y V A D a n n

H Hπ π

∞

=

= − −

∑ (2-1)

(2) → (2-2) :

1

cosh cosn

n

V n x D yA a n n

x H H H

ππ π

∞

=

∂ − = +

∂ ∑ (2-2)

(2-2) → (2-3) :

( 0)1

cosh cosnx

n

V n D yA a n n

x H H H

ππ π

∞

= =

∂ = +

∂ ∑ (2-3)

(2-2) → (3) :

0

1d

H

pV A L H

I yxρ ρ

∂= =

∂∫ (3)

(2) → (4) :

p p

p

V VZ L H

I Aρ= = (4)


0

( ) cosm

m

yy m

Hγ γ π

∞

=

=

∑ (5)

( )0γ( ) 1 f ( )y yγ ε= + (6)

1

f ( ) cosm

m

yy k m

Hπ

∞

=

=

∑ (7)

2

0 1 2 ...A A A Aε ε= + + + (8)

20 1 2 ...n n n na a a aε ε= + + + (9)

(1) , (2-1) , (2-3) → (10-0) :

1

1

1sinh cos

sinh cos

p n

n

p n

n

D yV A D a n n

H H

D yi V A D a n n

H H

π πρ

ω γ π π

∞

=

∞

=

− − =

= − −

∑

∑ (10-0)

(10-0) , (8) , (9) → (10) :

( )

( )

( )

( )

20 1 2

20 1 2

1

20 0 1 2

0

20 1 2

1...

1 ... cosh cos

1 cos ...

... sinh

n n n

n

m p

m

n n n

A A A

n D ya a a n n

H H H

yi k m V D A A A

H

Da a a n

H

ε ερ

πε ε π π

ρ

ω γ ε π ε ε

ε ε π

∞

=

∞

=

+ + + +

+ + + + =

= + − + + + +

+ + + + −

∑

∑

1

cos

n

yn

Hπ

∞

=

∑

(10)


(10) , (in ε→0) → (11)

0 0

1

0 0 0

1

cosh cos

sinh cos

n

n

p n

n

n D yA a n n

H H H

D yi V DA a n n

H H

ππ π

ω ρ γ π π

∞

=

∞

=

+ =

= − + −

∑

∑ (11)

(11) → (11-1) :

( )0 0 0pA i V D Aω ρ γ= − (11-1)

(11) → (11-1) :

0 0 0cosh cos sinh cosn nn D y D y

a n n i a n nH H H H H

ππ π ω ρ γ π π

= −

(11-1) , (11-2) → (12) :

:

0

0 00

and 01

pn

i VA a

i D

ω ρ γ

ω ρ γ= =

+ (12)

(10) , (12) → (13-0) :

( )

( )

( )

( )

21 2

21 2

1

20 1 2

21 2

1

0

...

... cosh cos

...

... sinh cos

cos

n n

n

n n

n

m

A A

n D ya a n n

H H H

i D A A

D ya a n n

H H

yi k m

H

ε ε

πε ε π π

ω ρ γ ε ε

ε ε π π

ω ρ γ ε π

∞

=

∞

=

+ + +

+ + + =

= − + + +

+ + + −

+

∑

∑

( )

( )

0 1

0

1

1

...

... sinh cos

p

m

n

n

V D A A

D ya n n

H H

ε

ε π π

∞

=

∞

=

− + + +

+ + −

∑

∑

(13-0)


(13-0) → (13-1) :

( )

( )

( )

( )

1 2

1 2

1

0 1 2

1 2

1

0

0

...

... cosh cos

...

... sinh cos

cos

n n

n

n n

n

m

m

A A

n D ya a n n

H H H

i D A A

D ya a n n

H H

yi k m

H

ε

πε π π

ω ρ γ ε

ε π π

ω ρ γ π

∞

=

∞

=

=

+ + +

+ + + =

= − + + +

+ + + −

+

∑

∑

( )

( )

0 1

1

1

...

... sinh cos

p

n

n

V D A A

D ya n n

H H

ε

ε π π

∞

∞

=

− + + +

+ + −

∑

∑

(13-1)

(13-1) , (in ε→0) → (13-2) :

1 1

1

0 1 1

1

0

0

cosh cos

sinh cos

cos

n

n

n

n

m

m

n D yA a n n

H H H

D yi DA a n n

H H

yi k m

H

ππ π

ω ρ γ π π

ω ρ γ π

∞

=

∞

=

∞

=

+ =

= − + −

+

∑

∑

∑ ( )0pV D A

−

(13-2)

(13-2) , (11-1) → (13) :

( )1 0 1

1

0 1 0 0

1 1

1 cosh cos

sinh cos cos

n

n

n m

n m

n D yA i D a n n

H H H

D y yi a n n i A k m

H H H

πω ρ γ π π

ω ρ γ π π ω ρ γ π

∞

=

∞ ∞

= =

+ + =

= − +

∑

∑ ∑ (13)

(13) → (14) :

01 1

0

0 ;

cosh sinh

nn

A kA a

n D Dn i n

H H H

ππ ω ρ γ π

= =

+

(14)


(13-1) , (14) → (15-0) :

2 2

1

0 2 2

1

0 1

0 1

cosh cos

sinh cos

cos sinh cos

n

n

n

n

m n

m n

n D yA a n n

H H H

D yi D A a n n

H H

y D yi k m a n n

H H H

ππ π

ω ρ γ π π

ω ρ γ π π π

∞

=

∞

=

∞ ∞

= =

+ =

= − + −

+ −

∑

∑

∑ ∑

(15-0)

(15-0) → (15-1), terms not function on y, including the n=m terms :

( ) ( ) ( ) ( )cos / cos / 0.5cos ( ) / 0.5cos ( ) /m y H n y H m n y H m n y Hπ π π π= + + −

In m=n : ( ) ( ) ( )cos / cos / 0.5 0.5cos 2 /n y H n y H y Hπ π π= +

[ ]2 0 2 0 1

1

1sinh

2n n

n

DA i D A i k a n

Hω ρ γ ω ρ γ π

∞

=

= − + −

∑ (15-1)

(15-1) → (15-2) :

( )0

2 10

1

sinh2 1

n n

n

i DA k a n

i D H

ω ρ γπ

ω ρ γ

∞

=

= −

+ ∑ (15-2)

(15-2) , (14) → (15-3) :

( )

20 0

20

1 0

( )

2 1coth

n

n

i A kA

n Di Dn i

H H

ω ρ γ


∞

=

= −+

+

∑ (15-3)

(15-3) , (12) → (15) :

2 2

02

01 0

( )

2 1coth

p n

n

V i kA

n Di Dn i

H H

ω ρ γ


∞

=

= −

+ +

∑ (15)

D >> H → coth(nπD/H) ≈ 1

2 2

02

01 0

( )

2 1

p n

n

V i kA

ni Di

H

ω ρ γπω ρ γ ω ρ γ

∞

=

= −

+ +∑ ( 15 , in D>>H )


(4) , (8) → (16) :

:

20 1 2 ...

pVZ

A A A

ρ

ε ε=

+ + + (16)

(16) , (14) → (16-1) :

:

220

0

1 ...

pVZ

AA

A

ρ

ε

=

+ +

(16-1)

in 22 0/ 1A A ε < then (16-1) → (16-2) :

:

22

0 0

1 ...pV A

ZA A

ρε

= − +

(16-2)

(16-2) , (12) , (15) → (17-1) :

( )( )

2 20 40

0 01 0

1 ( )1 ...( )

2 1

n

n

i D i kZ

ni i Di

H

ρ ω ρ γ ω ρ γ εε

πω ρ γ ω ρ γ ω ρ γ

∞

=

+

= + + + +

∑ (17-1)

(17-1) → (17) :

224

01 0

1(...)

2

n

n

kZ D

nii

H

ρ ερ ε

πω γ ω ρ γ

∞

=

= + + +

+∑ (17)

(17) → (18) :

1s s

s

Z R Zi Cω

= + + (18)

Where Rs=ρD , Cs=γ0 and (19) → Zs :

22

01

2

ns

n

kHZ

n i H

ρ ε

π ω ρ γ

∞

=

=+∑ (19)

(19) → (19-1) :

2 20

2 2

1

21where and

1 2

ns n n

nn nn

H kZ r c

n ki cr

ρ ε γ

π εω

∞

=

= = =

+∑ (19-1)


Condition :

22 0/ 1A A ε < from (16-2) , then (12) , (15) lead to :

220

01 0

( )11

2 1

n

n

i k

ni Di

H

ω ρ γε

πω ρ γ ω ρ γ

∞

=

<+ +

∑

(19) → 202

0

211

2 1

sH Zi

i D H

ω ρ γε

ω ρ γ ρ ε<

+ → 0

0

11

si

Zi D

ω γ

ω ρ γ<

+

0

0 0

1 1s

i DZ D

i i

ω ρ γρ

ω γ ω γ

+< = +

Dimensionless form of results

s

e

Z

R

where 2

2e

HR

ρ ε

π= (20)

e

ω

ω

where 0

eH

πω

γ ρ= (21)

( )2

1

ns

en

e

n kZ

Rn i

ω

ω

∞

=

=

+

∑ (22)


"Clear-cut streaky" pattern :

( )

0

1 cos (2 1)4

f( )(2 1)

m

m

ym

Hy

m

π

π

∞

=

− +

=+∑ ; 2 2 1

4 ( 1)0 ;

(2 1)

m

m mk kmπ+−

= =+

(23)

(22) , (23) → (24) :

22

0

16 1

(2 1) (2 1)

s

em

e

Z

Rm m i

ωπω

∞

=

=

+ + +

∑ (24)

2 2 2 2 2

1 1 1 1; 2 1

( ) ( ) e

i n mn n n n n

ωκ

ωκ κ κ κ κ

= − + = = +

+ + (24-1)

(24) , (24-1) → (24-2) :

2 2

2 2

0

16

(2 1)(2 1)(2 1)

e e e

s

em

e

i i iZ

R mmm i

ω ω ω

ω ω ω

ωπω

∞

=

= − + + + + +

∑ (24-2)

(24-2) → (24-3) : 2

2 2 2

0 0

16 1 16 1 1

(2 1)(2 1)(2 1)

s e e

em m

e

Z

R i i mmm i

ω ω

ω ω ωπ πω

∞ ∞

= =

= − − + + + +

∑ ∑ (24-3)

2

2 2 2

0

1 1 11 ...

8(2 1) 3 5m

m

π∞

=

= + + + = +

∑ (24-4)

(24-3) , (24-4) → (24-5) : 2

2

0

16 1 12

(2 1)(2 1)

s e e

em

e

Z

R i i mm i

ω ω

ω ω ωπω

∞

=

= + −

+ + +

∑ (24-5)

2

0

1 1; 2

' '

es s e

s s

Z z Ri c i c i i

ω ε

ω ω ω ω γ= + = = (24-6)

(24-5) , (24-6) → (24-7) : 2

2

0

16 1 1

(2 1)(2 1)

s e

em

e

z

R i mim

ω

ω ωπω

∞

=

= −

+ + +

∑ (24-7)


[ Summation of rational series by means of Polygamma Functions ψ ,

Handbook of Mathematical Functions, 9th

. Print, Dover Publ., N-Y, 1970, pp.264-265 ]

0

1 1 1 3 1 3 1 11

(2 1) 2 2 2 2 21(2 1)

em

ee

i

imm i

ωωωω

ωω

∞

=

− = Ψ − Ψ + − +

+ ++ +

∑ (24-8)

30.0365

2

Ψ ≃ (24-9)

2

1 1( ) ln( ) ... 1

2 12ξ ξ ξ

ξ ξ Ψ = − − + >> (24-10)

(24-10) → (24-11) :

3 1 3 1 1 3 1ln ... ln ... 1

12 2 2 2 2 22

2

e

e e e e

e

i i i

i i

ωω ω ω ωωω ω ω ω ω

ω

Ψ + = + − + = + − + >>

(24-11)

3 1 1 3 1ln ln ... ln 3 ... 1

12 2 2 2 32

2

e

e e e e

e

i i i

i i

ωω ω ω ωωω ω ω ω ω

ω

+ = − + = + + >>

(24-12)

(24-11) , (24-12) → (24-13) :

3 1 1ln 2 ... 1

2 2 2 3

e

e e e

i i

i

ωω ω ω

ω ω ω ω

Ψ + = + + >>

(24-13)

(24-8) , (24-9) , (24-13) → (24-14) :

0

1 1 1 1ln 0.982.. ...

2 1 2 2(2 1) e

me

i

i mm

ωω ω

ω

∞

=

− = − − +

+ + +

∑ (24-14)

(24-7) , (24-14) → (24-15) : 2

2

16 1 1ln 0.982.. ...

2 2

s e

e e

z i

R i

ω ω

ω ωπ

= − − +

(24-15)

(24-15) → (24-16) → (24-17) , in ω large : 2

2

8 1ln

2

s e

e e

z i

R i

ω ω

ω ωπ

≈ −

(24-16)

2

2

8 1ln

2 2

s e

e e

zi

R i

ω ω π

ω ωπ

≈ − +

(24-17)


Streaky patterns :

( )

1

0

1 cos (2 1)4

f ( )(2 1)

m

p

m

ym

Hy

m

π

π

∞

+=

− +

=+

∑ ; 2 2 1 1

4 ( 1)0 ;

(2 1)

m

m m pk k

mπ+ +

−= =

+ (25)

(22) , (25) →(26) :

22 2

0

16 1

(2 1) (2 1)

s

e pm

e

Z

Rm m i

ωπω

∞

+=

=

+ + +

∑ (26)

(26) , (24-1) → (26-1) :

2 2

2 2 2 1 22

0

16

(2 1) (2 1)(2 1) (2 1)

e e e

sp p

e pm

e

i i iZ

R m mm m i

ω ω ω

ω ω ω

ωπω

∞

+ +=

= − + + + + + +

∑ (26-1)

(26-1) → (26-2) , (26-3) , (26-4) , (26-5) :

2

1 2s e e s

e e

Z zK K

R i i R

ω ω

ω ω

= + +

(26-2)

where :

1 2 2 2

0

16 1

(2 1) p

m

Kmπ

∞

+=

=+

∑ (26-3)

2 2 1 2

0

16 1

(2 1) p

m

Kmπ

∞

+=

= −+

∑ (26-4)

2

22

0

16 1

(2 1) (2 1)

s e

e pm

e

z

R im m i

ω

ω ωπω

∞

=

=

+ + +

∑ (26-5)

Approximate of the sum :

( ) 12 2

0 2

1 d

(2 1) (2 1) (2 1 )(2 1)p p

mm m

µ

λ µ λ µ

∞ ∞

−=

≈+ + + + + +

∑ ∫ (26-6)


(26-6) → (26-7) :

( ) 2 20

0

1 1 1 d2 1 ; d d ;

2 2(2 1) (2 1) ( )p p

mm m

ββ µ µ β

λ β λ β

∞ ∞

=

= + = ≈+ + + +

∑ ∫ (26-7)

(26-7) → (26-8) :

2 2 20 0

d 1 d 1 d; d ;

2 ( ) 2 ( 1)p p p

β β β χχ χ

λ λ β λ β λ χ χ

∞ ∞= = =

+ +∫ ∫ (26-8)

(26-8) → (26-9) :

20

d 10

sin(2 ) 2( 1) pp

p

χ π

πχ χ

∞ = < < +∫ (26-9)

(26-7) , (26-8) , (26-9) → (26-10) :

( ) 2 2

0

1 1 10

sin(2 ) 2(2 1) (2 1) 2p p

m

ppm m

π

πλ λ

∞

=

≈ < < + + +

∑ (26-10)

(λ = iω /ωe ) , (26-5) , (26-10) → (26-11) :

[ ]2

2 2

16 10 0.5

sin(2 )2

s ep

e

e

zp

R i pi

ω π

ω ππ ω

ω

≈ < <

(26-11)

(26-11) → (26-12) :

[ ]2 2

80 0.5

sin(2 )

ps e

e

zp

R p i

ω

π π ω

+

≈ < <

(26-12)

theoretical impedance of capacitive electrodes

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