Structural Circuits

4.2 Basic Equivalent Circuits

4.2.3 Structural Circuits

168 X-Z. Yuan, C. Song, H. Wang and J. Zhang

a=2R₀Qωⁿcos(π 2 n), b=2R₀Qωⁿsin(π

2n), c=(e^acosb−1) cos(−π

2 n)−e^asinbsin(−π 2 n), d=Qωⁿe^acosb+Qωⁿ,

f =(e^acosb−1) sin(−π

2n)+e^asinbcos(−π

2 n), and g=Qωⁿe^asinb.

Assuming s=d²+g², u=cd+ fg, and v=cg− fd: Z(ω⁾=R_el+ [(R_cts+u)²+v²](s²R_ct+su)

(s²R_ct+su)²+[ωC_dl(R_cts+u)²+ωC_dlv²+sv]²

−i[(R_cts+u)²+v²][ω^Cdl(R_cts+u)²+ω^Cdlv²+sv]

(s²R_ct+su)²+[ω^Cdl(R_cts+u)²+ω^Cdlv²+sv]² (4.72) The complex-plane impedance diagram of the modified bounded Randles cell is given in Figure 4.20b. If distortions in the double-layer capacitance are assumed, a CPE can be used to replace C_dl. More examples of this modified bounded Randles cell can be found in Appendix D (Model D19).

4.2.3 Structural Circuits

Figure 4.21. Maxwell’s structure

A simple example of Maxwell’s structure with two parallel processes is shown in Figure 4.22a.

The total impedance can be calculated by

Z(ω)=[iωC₁+R₁⁻¹+[(iωC₂)⁻¹+R₂]⁻¹+[(iωC₃)⁻¹+R₃]⁻¹]⁻¹ (4.75) Thus, it can be separated into two parts:

Z(ω⁾= R₁ab[ab+R₁bω^2C22R₂+R₁aω^2C32R₃]

[ab+R₁bω^2C22R₂+R₁aω^2C32R₃]²+[R₁abω^C1+R₁bω^C2+R₁aω^C3]²

−i R₁ab[R₁abωC₁+R₁bωC₂+R₁aωC₃]

[ab+R₁bω²C₂²R₂+R₁aω²C₃²R₃]²+[R₁abωC₁+R₁bωC₂+R₁aωC₃]² (4.76) where a=1+(ωC₂R₂)² and b=1+(ωC₃R₃)².

The complex-plane impedance diagram of this simple Maxwell structure is presented in Figure 4.22b. More examples of this model are provided in Appendix D (Model D20).

170 X-Z. Yuan, C. Song, H. Wang and J. Zhang

a

b

Figure 4.22. a Simple example of Maxwell’s structure with two parallel processes (Model D20); b Nyquist plot of a simple Maxwell structure over the frequency range 1 MHz to 1 mHz (Model D20: R1 = 200 Ω, C1 = 0.000001 F, R2 = 400 Ω, C2 = 0.0001 F, R3 = 800 Ω, C3

= 0.01 F)

4.2.3.2 Voigt’s Structure

Voigt’s structure consists of a number of RC circuits in series, as shown in Figure 4.23. This structure is often applied to the description of solid electrochemical cells. It is known that the electrode/electrolyte interface of a solid electrochemical cell may not have perfect contact between two smooth and clean solid surfaces, and the electrode reaction involves species not only from the electrode and electrolyte but also from the gas phase. Sometimes, the electrolyte resistance produces two arcs in the very high frequency range, corresponding to charge transport through the bulk phase and at the grain boundaries. For polycrystalline

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Zre (Ω) -Zim (Ω)

materials with uniform composition of the grains and grain boundaries, only one semicircle will show.

Figure 4.23. Voigt’s structure

The overall impedance of elements in series can be calculated by the sum of the individual impedances of the branch, i.e.,

Z(ω)=

∑

Z_k(ω) ^(4.77)

Thus, the impedance with Voigt’s structure, which is used to describe the impedance of solid-state bulk samples, is expressed as

Z(ω)= (R_k⁻¹+iωCk)⁻¹

k=1

∑

n ^(4.78)

Two-RC Circuits

The bulk and grain boundary behaviour of a polycrystalline electroceramic material can be described by a Voigt structure consisting of two RC circuits. This simple Voigt structure is shown in Figure 4.24a. The parameters of this model all have direct physical meanings: R₁=R_b, C₁=C_b, R₂=R_gb, and C₂=C_gb, where b refers to bulk and gb refers to grain boundary.

The impedance is calculated as follows Z₁(ω)= R₁

1+ω²τ1

2−i ωR₁τ1

1+ω²τ1

2 (4.79)

Z₂(ω)= R₂ 1+ω²τ2

2−i ωR2τ2

1+ω²τ2

2 (4.80)

Z(ω)=Z₁(ω)+Z₂(ω)= R₁ 1+ω²τ1

2+ R₂

1+ω²τ2 2

−i( ωR1τ1

1+ω²τ1

2+ ωR₂τ2

1+ω²τ2

2) (4.81)

where τ1=R₁C₁ and τ2=R₂C₂.

The complex-plane impedance diagram of the two-RC Voigt structure is depicted in Figure 4.24b. It is characterized by two time constants, τ1 and τ2. The

172 X-Z. Yuan, C. Song, H. Wang and J. Zhang

mixing of the two semicircles caused by the ratio of the two capacitors for this model can be found in Appendix D (Model D21).

a

b

Figure 4.24. a Voigt structure with two RC in series (Model D21); b Nyquist plot of Voigt structure with two RC in series, over the frequency range 1 MHz to 1 mHz (Model D21: R1

= 50 Ω, R2 = 100 Ω, C1 = 0.00001 F, C2 = 0.01 F) Three-RC Circuits

Figure 4.25a shows the Voigt structure with three RC circuits in series. The physical meanings of the parameters in Figure 4.25a are: R₁=R_b, C₁=C_b,

R₂=R_gb,R₃=R_ct, C₂=C_gb, and C₃=C_dl. The total impedance is calculated as

Z(ω)=Z₁(ω)+Z₂(ω)+Z₃(ω) (4.82) Z₁(ω)= R₁

1+ω²τ1

2−i ωR₁τ 1+ω²τ1

2 (4.83)

Z₂(ω)= R₂ 1+ω²τ2

2−i ωR2τ 1+ω²τ2

2 (4.84)

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Z

_re

( Ω )

-Z

im

( Ω )

Z₃(ω)= R₃ 1+ω²τ3

2−i ωR₃τ 1+ω²τ3

2 (4.85)

where τ1=R₁C₁, τ2=R₂C₂, andτ3=R₃C₃.

The complex-plane impedance diagram of the Voigt structure with three RC circuits is depicted in Figure 4.25b. It is characterized by three time constants, τ1, τ2, and τ3. More examples of this model can be found in Appendix D (Model D22).

a

b

Figure 4.25. a Voigt structure with three RC in series (Model D22); b Nyquist plot of Voigt structure with three RC in series, over the frequency range 1 MHz to 1 mHz (Model D22: R₁

= 150 Ω, R2 = 200 Ω, R3 = 100 Ω, C1 = 0.00001 F, C2 = 0.001 F, C3 = 0.1 F) 4.2.3.3 Ladder Structure

The ladder structure is presented in Figure 4.26. This structure consists of a number of kernels, which occur sequentially. The impedance of the ladder structure is

Z(ω)=Z₁(ω)+ Z₂⁻¹(ω)+^⎡_⎣⎢Z₃(ω)+

(

Z₄⁻¹(ω)+...

)

⁻¹⎤

⎦⎥

⎧ −1

⎨⎩

⎫⎬

⎭

−1

(4.86) where Z_k(ω) is the impedance of the ladder elements. Then the total impedance of

this structure can be expressed as

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Zre (Ω) -Zim (Ω)

174 X-Z. Yuan, C. Song, H. Wang and J. Zhang

Z(ω)=R₁+ 1 iωC₁+ 1

R₂+ 1 iωC2+...

(4.87)

Figure 4.26. Ladder structure

Faradaic Reaction Involving One Adsorbed Species

Figure 4.27a shows the equivalent circuit of a simple example of the ladder structure, for electrochemical systems that are known as Faradaic reactions involving one adsorbed species. This heterogeneous reaction occurs in two steps in the absence of diffusion limitation:

A⎯ → ⎯ ^k1 B+e⁻¹ (4.88)

B⎯ → ⎯ ^k2 C+e⁻¹ (4.89) Species A is transported to the electrode surface where it adsorbs and reacts, producing species C. B is the adsorbed species.

The total impedance of this ladder structure can be calculated as Z(ω)=R_el+ 1

iωCdl+ 1 R_ct+ 1

iωC₂+ 1 R₃

(4.90)

Thus,

Z(ω)=R_el+a(1−ω²bC_dl)+ω²b(aC_dl+C₂R₃) (1−ω²C_dlb)²+(ωC_dla+ωC₂R₃)²

−iaω(aC_dl+C₂R₃)−ωb(1−ω²C_dlb)

(1−ω²C_dlb)²+(ωC_dla+ωC₂R₃)² (4.91)

or

Z(ω)=R_el+ (a²+ω²b²)(a+ω²C₂R₃b)

(a+ω²C₂R₃b)²+(ωC_dla²+ω³C_dlb²+ωC₂R₃a−ωb)²

−i (a²+ω²b²)(ωCdla²+ω³C_dlb²+ωC2R₃a−ωb)

(a+ω²C₂R₃b)²+(ωCdla²+ω³C_dlb²+ωC2R₃a−ωb)² (4.92) where a=R₃+R_ct and b=C₂R₃R_ct.

a

b

Figure 4.27. a Ladder structure for electrochemical systems known as Faradaic reactions involving one adsorbed species (Model D23); b Nyquist plot of a ladder structure for the Faradaic reaction involving one adsorbed species, over the frequency range 1 MHz to 1 mHz (Model D23: Rel = 200 Ω, Rct = 400 Ω, R3 = 600 Ω, Cdl = 0.0001 F, C2 = 0.01 F) The simulated complex-plane impedance diagram is shown in Figure 4.27b. As can be seen in the figure, this ladder structure is characterized by two semicircles with two time constants, τ1=R_ctC_dl and τ2=R₃C₂, accounting for the two-step reaction. The element C₂ symbolizes the adsorption capacitance, and τ2 represents the relaxation of the adsorbing process.

The two semicircles in Figure 4.27b may be well pronounced or mixed, depending on the ratio of the two time constants. Note that this characteristic is

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0 200 400 600 800 1000 1200 1400

Zre (Ω) -Zim (Ω)

176 X-Z. Yuan, C. Song, H. Wang and J. Zhang

quite similar to that of the two-time-constant models with the Voigt structure, described earlier. However, the parameter values can be different. More examples of this ladder structure are presented in Appendix D (Model D23).

Faradaic Reaction Involving Two Adsorbed Species

The electrochemical reaction involving two adsorbed species in the absence of diffusion limitations can also be represented with a ladder type of equivalent circuit. The electrochemical reaction from A to C occurs in three steps involving two adsorbed species, B and C.

A⎯ → ⎯ ^k1 B+e⁻¹ (4.93)

B⎯ → ⎯ ^k2 C+e⁻¹ (4.94) C⎯ → ⎯ ^k3 D+e⁻¹ (4.95) This ladder type of equivalent circuit for two adsorbed species is depicted in Figure 4.28a.

The total impedance of this ladder structure includes the Faradaic impedance, the double-layer capacitance, and the electrolyte resistance, which can be calculated as

Z(ω⁾=R_el+ 1 iωC_dl+ 1

R_ct+ 1

iω^C2+ 1 R₃+ 1

iωC₃+ 1 R₄

(4.96)

Assuming a=R₃+R₄, b=C₃R₃R₄, l=a²+ω²b², m=a+ω²C₃R₄b, n=ω^C2a²+ω^3C2b²+ω^C3R₄a−ω^b, and q=m²+n², then the total impedance is given by

Z(ω)=R_el+(R_ctq+lm)(q+ω^Cdlnl)−nlω^Cdl(R_ctq+lm) (q+ω^Cdlnl)²+[ω^Cdl(R_ctq+lm)]²

−inl(q+ω^Cdlnl)+(R_ctq+lm)ω^Cdl(R_ctq+lm)

(q+ωC_dlnl)²+[ωC_dl(R_ctq+lm)]² (4.97) The complex-plane impedance diagram is displayed in Figure 4.28b. The effects of the model parameters on the shape of the spectra can be found in Appendix D (Model D24).

a

b

Figure 4.28. a Ladder structure with two adsorbed species in the absence of diffusion limitation (Model D24); b Nyquist plot of ladder structure with two adsorbed species, over the frequency range 1 MHz to 1 mHz (Model D24: Rel = 50 Ω, Rct = 100 Ω, R3 = 150 Ω, R4 = 50 Ω, Cdl = 0.00001 F, C2 = 0.001 F, C3 = 0.1 F)