Handbook of Electrochemical Impedance Spectroscopy ELECTRICAL CIRCUITS CONTAINING
Transcript of Handbook of Electrochemical Impedance Spectroscopy ELECTRICAL CIRCUITS CONTAINING
Handbookof
Electrochemical Impedance Spectroscopy
0 10
1
2
Re Z *
-Im
Z*
ELECTRICAL CIRCUITS
CONTAINING CPEs
ER@SE/LEPMIJ.-P. Diard, B. Le Gorrec, C. Montella
Hosted by Bio-Logic @ www.bio-logic.info
March 29, 2013
Contents
1 Circuits containing one CPE 51.1 Constant Phase Element (CPE), symbol Q . . . . . . . . . . . . 51.2 Circuit (R+Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Reduced impedance . . . . . . . . . . . . . . . . . . . . . 6
1.3 Circuit (R/Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.2 Reduced impedance . . . . . . . . . . . . . . . . . . . . . 71.3.3 Pseudocapacitance #1 . . . . . . . . . . . . . . . . . . . . 71.3.4 Pseudocapacitance #2 . . . . . . . . . . . . . . . . . . . . 8
1.4 Circuit (R/Q)+(R/Q)+ .. (Voigt) . . . . . . . . . . . . . . . . . 91.5 Circuit (R1+(R2/Q2)) . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5.1 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5.2 Reduced impedance . . . . . . . . . . . . . . . . . . . . . 10
1.6 Circuit (R1/(R2+Q2)) . . . . . . . . . . . . . . . . . . . . . . . . 101.6.1 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6.2 Reduced impedance . . . . . . . . . . . . . . . . . . . . . 11
1.7 Transformation formulae between(R+(R/Q))and (R/(R+Q)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.7.1 α21 = α22 . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.7.2 α21 6= α22 . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Circuits made of two CPEs 132.1 Circuit (Q1+Q2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 α1 = α2 = α . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 α1 6= α2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.3 Reduced impedance . . . . . . . . . . . . . . . . . . . . . 14
2.2 Circuit (Q1/Q2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 α1 = α2 = α . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 α1 6= α2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.3 Reduced impedance . . . . . . . . . . . . . . . . . . . . . 16
3 Circuits made of one R and two CPEs 193.1 Circuit ((R1/Q1) + Q2) . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 α1 = α2 = α . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.2 α1 6= α2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Circuit ((R1 + Q1)/Q2) . . . . . . . . . . . . . . . . . . . . . . . 213.2.1 α1 = α2 = α . . . . . . . . . . . . . . . . . . . . . . . . . 21
3
4 CONTENTS
3.2.2 α1 6= α2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Circuits made of two Rs and two CPEs 234.1 Circuit ((R1/Q1)+(R2/Q2)) . . . . . . . . . . . . . . . . . . . . . 234.2 Circuit ((R1+(R2/Q2))/Q1) . . . . . . . . . . . . . . . . . . . . . 244.3 Circuit ((Q1+(R2/Q2))/R1) . . . . . . . . . . . . . . . . . . . . . 254.4 Circuit (((Q2+R2)/R1)/Q1) . . . . . . . . . . . . . . . . . . . . . 26
A Symbols for CPE 29
Chapter 1
Circuits containing oneCPE
1.1 Constant Phase Element (CPE), symbol Q
Figure 1.1: Most often used symbol for CPE (see also the Appendix A).
Z =1
Q (i ω)α , Re Z =
cα
Q ωα, Im Z = −
sα
Q ωα
cα = cos(π α
2) , sα = sin(
π α
2)
|Z| =1
Q ωα, φZ = −
π α
2
The Q unit (F cm−2 sα−1) depends on α (1).
1.2 Circuit (R+Q)
1.2.1 Impedance
Z(ω) = R +1
Q (iω)α , Re Z = R +
cα
Q ωα, Im Z = −
sα
Q ωα
1 Different equations for CPE: Z =Q
(i ω)1−α[5], Z =
1
(Q i ω)α[26].
5
6 CHAPTER 1. CIRCUITS CONTAINING ONE CPE
0Re Z
0
-Im
Z
- ΑΠ2
0Re Y
0
ImY
ΑΠ2
Figure 1.2: Nyquist diagram of the impedance and admittance for the CPE element,plotted for α = 0.8. The arrows always indicate the increasing frequency direction.
R
Q
Figure 1.3: Circuit (R+Q).
1.2.2 Reduced impedance
Z∗(ω) =Z(ω)
R= 1 +
1
τ (i ω)α, τ = R Q
The τ unit depends on α: uτ = sα.
Z∗(u) = 1 +1
(i u)α, u = ω τ1/α
0 1 2Re Z*
0
1
2
-Im
Z*
uc=1
0 1Re Y*
0
0.5
ImY*
uc=1
Figure 1.4: Nyquist diagram of the reduced impedance and admittance (Y ∗ = R Y )for the (R+Q) circuit, plotted for α = 0.8.
1.3. CIRCUIT (R/Q) 7
1.3 Circuit (R/Q)
R
Q
Figure 1.5: Circuit (R/Q).
1.3.1 Impedance
Z(ω) =R
1 + τ (i ω)α ; τ = R Q
Re Z(ω) =R (1 + τ ωα cα)
1 + τ2 ω2 α + 2 τ ωα cα; Im Z(ω) = −
R τ ωα sα
1 + τ2 ω2 α + 2 τ ωα cα
1.3.2 Reduced impedance
Z∗(ω) =Z(ω)
R=
1
1 + τ (i ω)α ; τ = R Q
Re Z∗(ω) =1 + τ ωα cα
1 + τ2 ω2 α + 2 τ ωα cα; Im Z∗(ω) = −
τ ωα sα
1 + τ2 ω2 α + 2 τ ωα cα
dIm Z∗(ω)
dω=
α τ ω−1+α(
−1 + τ2 ω2 α)
sα
(1 + τ2 ω2 α + 2 τ ωα cα)2= 0 ⇒ ωα
c = 1/τ [6]
Re Z∗(ωc) = 1/2 , Im Z∗(ωc) = −sα
2 (1 + cα)
α =2
πarccos
(
−1 +2
1 + 4 Im Z∗(ωc)2
)
Z∗(u) =1
1 + (i u)α , u = ω τ1/α
(Fig. 1.6)
1.3.3 Pseudocapacitance #1
The value of the pseudocapacitance C (C/F cm−2) for the (R/C) circuit givingthe same characteristic frequency than that of the (R/Q) circuit (Fig. 1.7) isobtained from:
ωc =1
(R Q)1/α=
1
R C⇒ C = Q
1α R
1α−1
8 CHAPTER 1. CIRCUITS CONTAINING ONE CPE
0 1Re Z*
0
0.5
-Im
Z*
uc=1
0 1 2Re Y*
0
1
2
ImY*
uc=1
Figure 1.6: Nyquist diagram of the reduced impedance (depressed semi-circle [22])and admittance (Y ∗ = R Y ) for the (R/Q) circuit, plotted for α = 0.8.
R
Q
R
C
0 R0
Ωc=1
HRQL1Α
0 R0
Ωc= 1HRCL
Figure 1.7: (R/Q) and (R/C) circuits with the same characteristic frequency at theapex (or summit) of impedance arc.
1.3.4 Pseudocapacitance #2
The value of the pseudocapacitance C (C/F cm−2) for the (RC/C) circuit giv-ing the same impedance for the characteristic frequency of the (RQ/Q) circuit(Fig. 1.7) is obtained from [2, 9]:
C = Q1/αR(1/α)−1Q sin(α π/2), RC =
RQ
2 (cos(α π/4))2
with:
τ(RC/C) = (RQ Q)1/α tan(α π/4)
1.4. CIRCUIT (R/Q)+(R/Q)+ .. (VOIGT) 9
RQ
Q
RC
C
0 RC RQ0
Re Z
-Im
Z
Ωc =1
JRQ QN1Α
Figure 1.8: (RQ/Q) and (RC/C) circuits with the same impedance for the character-istic frequency of the (RQ/Q) circuit.
1.4 Circuit (R/Q)+(R/Q)+ .. (Voigt)
Z(ω) =
nRQ∑
i=1
Ri
1 + τi (i ω)αi; τi = Ri Qi
Re Z(ω) =
nRQ∑
i=1
Ri (1 + τi ωαi cαi)
1 + τ2i ω2 αi + 2 τi ωαi cαi)
Im Z(ω) = −
nRQ∑
i=1
Ri τi ωαi sαi
1 + τ2i ω2 αi + 2 τi ωαi cαi
1.5 Circuit (R1+(R2/Q2))
R1
R2
Q2
Figure 1.9: Circuit (R1+(R2/Q2)).
10 CHAPTER 1. CIRCUITS CONTAINING ONE CPE
1.5.1 Impedance
Z(ω) = R1 +1
(i ω)α2 Q2 +1
R2
Z(ω) =(R1 + R2) (1 + (i ω)α2 τ2)
1 + (i ω)α2 τ1, τ1 = R2 Q2, τ2 =
R1 R2 Q2
R1 + R2
1.5.2 Reduced impedance
Z∗(u) =Z(u)
R1 + R2=
1 + T (i u)α2
1 + (i u)α2(1.1)
u = τ1/α2
1 ω, T = τ2/τ1 = R1/(R1 + R2) < 1
Re Z∗(u) =T cα uα2 + cα uα2 + Tu2α2 + 1
2 cα uα2 + u2α2 + 1
Im Z∗(u) = −(1 − T )uα2 sα
2 cα uα2 + u2α2 + 1
0 R1 R1+R2
0
Re Z
-Im
Z
Ωc=1
Τ11Α
1
Τ21Α
0 T 10
Re Z*
-Im
Z* uc= 1
1
T1Α
Figure 1.10: Nyquist diagrams of the impedance and reduced impedance for the(R1+(R2/Q2)) circuit.
1.6 Circuit (R1/(R2+Q2))
R2
R1
Q2
Figure 1.11: Circuit (R1/(R2+Q2)).
1.7. TRANSFORMATION FORMULAE BETWEEN(R+(R/Q)) AND (R/(R+Q))11
1.6.1 Impedance
Z(ω) =R1 (1 + τ2 (i ω)α2)
1 + τ1 (i ω)α2, τ1 = (R1 + R2)Q2, τ2 = R2 Q2
Re Z(ω) =R1
(
cos(
πα2
2
)
(τ1 + τ2)ωα2 + τ1τ2ω2α2 + 1
)
τ1
(
τ1ωα2 + 2 cos(
πα2
2
))
ωα2 + 1
Im Z(ω) = −ωα2 sin
(
πα2
2
)
R1 (τ1 − τ2)
τ1
(
τ1ωα2 + 2 cos(
πα2
2
))
ωα2 + 1
1.6.2 Reduced impedance
Z∗(u) =Z(u)
R1=
1 + T (i u)α2
1 + (i u)α2
u = τ1/α2
1 ω, T = τ2/τ1 = R2/(R1 + R2) < 1
cf. Eq. (1.1) and Fig. 1.10.
1.7 Transformation formulae between(R+(R/Q))and (R/(R+Q))
1.7.1 α21 = α22
R11
R21
Q21
R22
R12
Q22
Figure 1.12: The (R+(R/Q)) and (R/(R+Q)) circuits are non-distinguishable forα21 = α22 [1].
Transformations formulae (R+(R/Q)) → (R/(R+Q))
R12 = R11 + R21, R22 =R2
11
R21+ R11, Q22 =
Q21R221
(R11 + R21) 2
Transformations formulae (R/(R+Q))→(R+(R/Q))
Q21 =Q22 (R12 + R22)
2
R212
, R11 =R12R22
R12 + R22, R21 =
R212
R12 + R22
1.7.2 α21 6= α22
The (R+(R/Q)) and (R/(R+Q)) circuits (Fig. 1.12) are distinguishable forα21 6= α22
Chapter 2
Circuits made of two CPEs
2.1 Circuit (Q1+Q2)
Q1 Q2
Figure 2.1: Circuit (Q1+Q2).
2.1.1 α1 = α2 = α
Z(ω) =
(
1
Q1+
1
Q2
)
1
(i ω)α=
1
Q (i ω)α, Q =
Q1 Q2
Q1 + Q2
cf. § 1.1.
2.1.2 α1 6= α2
Impedance
Z(ω) =1
Q1 (i ω)α1+
1
Q2 (i ω)α2=
Q1 (i ω)α1 + Q2 (i ω)α2
Q1 Q2 (i ω)α1+α2
Re Z(ω) =cos(
πα1
2
)
ω−α1
Q1+
cos(
πα2
2
)
ω−α2
Q2
Im Z(ω) = −sin(
πα1
2
)
ω−α1
Q1−
sin(
πα2
2
)
ω−α2
Q2
|ZQ1| = |ZQ2
| ⇒ ω = ωc =
(
Q2
Q1
)1
α1−α2
13
14 CHAPTER 2. CIRCUITS MADE OF TWO CPES
• α1 < α2 (Figs. 2.2 and 2.3)
ω → 0 ⇒ Z(ω) ≈1
Q2 (i ω)α2, ω → ∞ ⇒ Z(ω) ≈
1
Q1 (i ω)α1
• α1 > α2
ω → 0 ⇒ Z(ω) ≈1
Q1 (i ω)α1, ω → ∞ ⇒ Z(ω) ≈
1
Q2 (i ω)α2
0 2000
200
Re ZW
-Im
ZW
1 4
1
4
logHRe ZWL
logH-
ImZWL
Figure 2.2: Nyquist and log Nyquist [8] diagrams of the impedance for the (Q1+Q2)circuit, plotted for Q1 = 10−2 F cm−2 sα1−1, Q2 = 10−2 F cm−2 sα2−1, α1 = 0.6, α2 =0.9 (α1 < α2). Dots: ωc = (Q2/Q1)
1/(α1−α2).
-5 5HQ2Q1L1HΑ1-Α2L
0
5
log ΩHrd s-1L
logÈZWÈ
- Α2
- Α1
-5 5HQ2Q1L1HΑ1-Α2L
-45
-90
log ΩHrd s-1L
ΦZ°
-Α1 Π2
-Α2 Π2
Figure 2.3: Bode diagrams of the impedance for the (Q1+Q2) circuit. Same valuesof parameters as in Fig. 2.2. α1 < α2.
2.1.3 Reduced impedance
Z∗(u) = Q1 ωα1
c Z(ω) =1
(i u)α1+
1
(i u)α2, u =
ω
ωc
2.1. CIRCUIT (Q1+Q2) 15
0 1 2 30
1
2
3
Re Z*
-Im
Z*
-2 0 2-2
0
2
log HRe Z*L
logH-
ImZ*L
Figure 2.4: Nyquist and log Nyquist [8] diagrams of the reduced impedance for the(Q1+Q2) circuit, plotted for α1 = 0.6, α2 = 0.9 (α1 < α2). Dots: uc = 1.
-3 30
-3
0
3
log u
logÈZ*È
- Α2
- Α1
-3 30
-45
-90
log u
ΦZ*°
-Α1 Π2
-Α2 Π2
Figure 2.5: Bode diagrams of the impedance for the (Q1+Q2) circuit. Same valuesof parameters as in Fig. 2.4. α1 < α2.
16 CHAPTER 2. CIRCUITS MADE OF TWO CPES
2.2 Circuit (Q1/Q2)
Q1
Q2
Figure 2.6: Circuit (Q1/Q2).
2.2.1 α1 = α2 = α
Z(ω) =1
(Q1 + Q2) (i ω)α=
1
Q (i ω)α, Q = Q1 + Q2
cf. § 1.1.
2.2.2 α1 6= α2
Impedance
Z(ω) =1
Q1 (i ω)α1 + Q2 (i ω)α2
Re Z(ω) =cos(
πα1
2
)
Q1ωα1 + cos
(
πα2
2
)
Q2ωα2
Q21ω
2α1 + Q22ω
2α2 + 2 cos(
12π (α1 − α2)
)
Q1Q2ωα1+α2
Im Z(ω) = −sin(
πα1
2
)
Q1ωα1 + sin
(
πα2
2
)
Q2ωα2
Q21ω
2α1 + Q22ω
2α2 + 2 cos(
12π (α1 − α2)
)
Q1Q2ωα1+α2
• α1 < α2 (Figs. 2.7 and 2.8)
ω → 0 ⇒ Z(ω) ≈1
Q1 (i ω)α1, ω → ∞ ⇒ Z(ω) ≈
1
Q2 (i ω)α2
• α1 > α2
ω → 0 ⇒ Z(ω) ≈1
Q2 (i ω)α2, ω → ∞ ⇒ Z(ω) ≈
1
Q1 (i ω)α1
2.2.3 Reduced impedance
Z∗(u) = Q1 ωα1
c Z(ω) =1
(i u)α1 + (i u)α2, u =
ω
ωc
2.2. CIRCUIT (Q1/Q2) 17
0 500
50
Re ZW
-Im
ZW
0 3
0
3
logHRe ZWL
logH-
ImZWL
Figure 2.7: Nyquist and log Nyquist [8] diagrams of the impedance for the (Q1/Q2)circuit plotted for Q1 = 10−2 F cm−2 sα1−1, Q2 = 10−2 F cm−2 sα2−1, α1 = 0.6, α2 =0.9 (α1 < α2). Dots: ωc = (Q2/Q1)
1/(α1−α2).
-5 5HQ2Q1L1HΑ1-Α2L
0
5
log ΩHrd s-1L
logÈZWÈ
- Α1
- Α2
-5 5HQ2Q1L1HΑ1-Α2L
-45
-90
log ΩHrd s-1L
ΦZ°
-Α1 Π2
-Α2 Π2
Figure 2.8: Bode diagrams of the impedance for the (Q1/Q2) circuit. Same values ofparameters as in Fig. 2.7. α1 < α2.
0 10
1
Re Z*
-Im
Z*
-2 0 2-2
0
2
log HRe Z*L
logH-
ImZ*L
Figure 2.9: Nyquist and log Nyquist [8] diagrams of the reduced impedance for the(Q1/Q2) circuit, plotted for α1 = 0.6, α2 = 0.9 (α1 < α2). Dots: uc = 1.
18 CHAPTER 2. CIRCUITS MADE OF TWO CPES
-3 30
-3
0
3
log u
logÈZ*È
- Α1
- Α2
-3 30
-45
-90
log u
ΦZ*°
-Α1 Π2
-Α2 Π2
Figure 2.10: Bode diagrams of the impedance for the (Q1/Q2) circuit. Same valuesof parameters as in Fig. 2.9. α1 < α2.
Chapter 3
Circuits made of one R andtwo CPEs
3.1 Circuit ((R1/Q1) + Q2)
Q2
Q1
R1
Figure 3.1: Circuit ((R1/Q1)+Q2).
3.1.1 α1 = α2 = α
Impedance
Z(ω) =1
1
R1+ Q1 (i ω)α
+1
Q2 (i ω)α
Z(ω) =1 + (i ω)α τ2
(i ω)α Q2 (1 + (i ω)α τ1), τ1 = R1 Q1, τ2 = (Q1 + Q2)R1, τ1 < τ2
Re Z(ω) = −cos(
πα2
) (
τ1τ2ω2α + 1
)
ω−α + cos(πα)τ1 + τ2
Q2(
τ1
(
τ1ωα + 2 cos(
πα2
))
ωα + 1)
Im Z(ω) = −sin(
πα2
) (
τ1τ2ω2α + 1
)
ω−α + sin(πα)τ1
Q2(
τ1
(
τ1ωα + 2 cos(
πα2
))
ωα + 1)
19
20 CHAPTER 3. CIRCUITS MADE OF ONE R AND TWO CPES
Reduced impedance
Z∗(u) =Z(u)
R1=
1
T − 1
1 + T (i u)α
(i u)α (1 + (i u)α)(3.1)
u = ω τ1/α, T = τ2/τ1 = 1 + Q2/Q1 > 1
Re Z∗(u) =u−α
(
(T + cos(απ))uα +(
Tu2α + 1)
cos(
απ2
))
(T − 1)(
2 cos(
απ2
)
uα + u2α + 1)
Im Z∗(u) = u−α
(
1
1 − T−
u2α
2 cos(
απ2
)
uα + u2α + 1
)
sin(απ
2
)
0 10
1
2
Re Z *
-Im
Z*
Figure 3.2: Nyquist diagram of the reduced impedance for the ((R1/Q1)+Q2) circuit(Fig. 3.1, Eq. (3.1)), plotted for T = 4, 9, 90 and α = 0.85. The line thickness increaseswith increasing T . Dots: reduced characteristic angular frequency uc1 = 1; circles:reduced characteristic angular frequency uc2 = 1/T 1/α (φuc1 = φuc2).
3.1.2 α1 6= α2
Impedance
Z(ω) =1
1
R1+ Q1 (i ω)α1
+1
Q2 (i ω)α2
Re Z(ω) =cos(
πα2
2
)
ω−α2
Q2+
R1
(
cos(
πα1
2
)
Q1R1ωα1 + 1
)
Q1R1
(
Q1R1ωα1 + 2 cos(
πα1
2
))
ωα1 + 1
Im Z(ω) = −sin(
πα1
2
)
Q1R21ω
α1
Q1R1
(
Q1R1ωα1 + 2 cos(
πα1
2
))
ωα1 + 1−
sin(
πα2
2
)
ω−α2
Q2
3.2. CIRCUIT ((R1 + Q1)/Q2) 21
R1
Q1
Q2
Figure 3.3: Circuit ((R1+Q2)/Q1).
3.2 Circuit ((R1 + Q1)/Q2)
3.2.1 α1 = α2 = α
Impedance
Z(ω) =1
(i ω)α Q1 +1
R1 +1
(i ω)α Q2
=1 + Q2 R1(i ω)α
(i ω)α (Q1 + Q2)
(
1 +(i ω)α Q1 Q2 R1
Q1 + Q2
)
Z(ω) =1 + τ2(i ω)α
(i ω)α (Q1 + Q2) (1 + (i ω)α τ1), τ1 =
Q1 Q2 R1
Q1 + Q2, τ2 = Q2 R1
Re Z(ω) =ω−α
(
cos(πα)ωα + τ2ωα + cos
(
πα2
) (
τ2ω2α + 1
))
(
2 cos(
πα2
)
ωα + ω2α + 1)
(Q1 + Q2) τ1
Im Z(ω) = −ω−α sin
(
πα2
) (
2 cos(
πα2
)
ωα + τ2ω2α + 1
)
(
2 cos(
πα2
)
ωα + ω2α + 1)
(Q1 + Q2) τ1
Reduced impedance
Z∗(u) =Z(u)
R1=
T − 1
T 2
1 + T (i u)α
(i u)α (1 + (i u)α)(3.2)
u = ω τ1/α, T = τ2/τ1 = 1 + Q2/Q1 > 1
Re Z∗(u) =(T − 1)u−α
(
(T + cos(απ))uα +(
Tu2α + 1)
cos(
απ2
))
T 2(
2 cos(
απ2
)
uα + u2α + 1)
Im Z∗(u) = −(T − 1)u−α
(
2 cos(
απ2
)
uα + Tu2α + 1)
sin(
απ2
)
T 2(
2 cos(
απ2
)
uα + u2α + 1)
3.2.2 α1 6= α2
Z(ω) =
1
(i ω)α2 Q2+ R1
(i ω)α1 Q1
(
1
(i ω)α1 Q1+
1
(i ω)α2 Q2+ R1
)
22 CHAPTER 3. CIRCUITS MADE OF ONE R AND TWO CPES
0 10
1
2
Re Z *
-Im
Z*
Figure 3.4: Nyquist diagram of the reduced impedance for the ((R1+Q1)/Q2) circuit(Fig. 3.3, Eq. (3.2)), plotted for T = 4, 9, 90 and α = 0.85. The line thickness increaseswith increasing T . Dots: reduced characteristic angular frequency uc1 = 1; circles:reduced characteristic angular frequency uc2 = 1/T 1/α (φuc1 = φuc2).
Z(ω) =1 + τ (i ω)
α2
(i ω)α1 Q1 + (i ω)α2 Q2 + τ (i ω)α1+α2 Q1
, τ = R1 Q2
Re Z(ω) =(
ωα1 cα1
(
1 + τ2 ω2 α2 + 2 τ ωα2 cα2
)
Q1 + ωα2 (τ ωα2 + cα2) Q2
)
/(
ω2 α1(
1 + τ2 ω2 α2 + 2 τ ωα2 cα2
)
Q12 + 2 ωα1+α2 (τ ωα2 cα1 + cα1mα2) Q1 Q2 + ω2 α2 Q2
2)
cα1mα2 = cos
(
π (α1 − α2)
2
)
Im Z(ω) =(
−ωα1(
1 + τ2 ω2 α2 + 2 τ ωα2 cα2
)
Q1 sα1 − ωα2 Q2 sα2
)
/(
ω2 α1(
1 + τ2 ω2 α2 + 2 τ ωα2 α2)
Q12 + 2 ωα1+α2 (τ ωα2 cα1 + cα1mα2) Q1 Q2 + ω2 α2 Q2
2)
Chapter 4
Circuits made of two Rsand two CPEs
4.1 Circuit ((R1/Q1)+(R2/Q2))
R1
Q1
R2
Q2
Figure 4.1: Circuit ((R1/Q1)+(R2/Q2)).
Z(ω) =1
(i ω)α1 Q1 +1
R1
+1
(i ω)α2 Q2 +1
R2
Z(ω) =R1
1 + (i ω)α1 τ1
+R2
1 + (iω)α2 τ2
, τ1 = R1 Q1 , τ2 = R2 Q2
Z(ω) =R1 + R2 + (i ω)
α1 R2 τ1 + (i ω)α2 R1 τ2
(1 + (i ω)α1 τ1) (1 + (i ω)
α2 τ2)
Re Z(ω) =R1 (1 + ωα1 cα1 τ1)
1 + ωα1 τ1 (2 cα1 + ωα1 τ1)+
R2 (1 + ωα2 cα2 τ2)
1 + ωα2 τ2 (2 cα2 + ωα2 τ2)
Im Z(ω) = −ωα1 R1 sα1 τ1
1 + ωα1 τ1 (2 cα1 + ωα1 τ1)−
ωα2 R2 sα2 τ2
1 + ωα2 τ2 (2 cα2 + ωα2 τ2)
23
24 CHAPTER 4. CIRCUITS MADE OF TWO RS AND TWO CPES
0 0.5 10
0.3
Re Z *
-Im
Z* uc1 uc2
Figure 4.2: Nyquist diagrams of the reduced impedance for the ((R1/Q1)+(R2/Q2))circuit (Fig. 4.1). R1 = R2, α1 = α2, Q2 ≫ Q1.
0 0.5 10
0.5
Re Z *
-Im
Z*
uc1 = uc2
0 0.5 10
0.5
Re Z *
-Im
Z*
uc1 = uc2
- Π4 Π4
Figure 4.3: Unusual Nyquist diagrams of the reduced impedance for the((R1/Q1)+(R2/Q2)) circuit (Fig. 4.1). R1 = R2, Q2 = Q1, α1 = 1. Left: α2 = 0.3,right: α2 = 0.5.
4.2 Circuit ((R1+(R2/Q2))/Q1)
Z(ω) =1
(i ω)α1 Q1 +
1
R1 +1
(i ω)α2 Q2 +
1
R2
Z(ω) =R1 + R2 + (i ω)
α2 Q2 R1 R2
1 + (i ω)α1 Q1 (R1 + R2) + (i ω)α2 Q2 R2 + (i ω)α1+α2 Q1 Q2 R1 R2
R1
Q1
Q2
R2
Figure 4.4: Circuit ((R1+(R2/Q2))/Q1).
4.3. CIRCUIT ((Q1+(R2/Q2))/R1) 25
Re Z(ω) =(
R1 + R2 + ω2 α2 Q22 R1 (1 + ωα1 Cα1 Q1 R1) R2
2+
ωα1 Cα1 Q1 (R1 + R2)2
+ ωα2 Cα2 Q2 R2 (R2 + 2 R1 (1 + ωα1 Cα1 Q1 (R1 + R2))))
/(
1 + ω2 α2 Q22 (1 + ωα1 Q1 R1 (2 Cα1 + ωα1 Q1 R1)) R2
2+
ωα1 Q1 (R1 + R2) (2 Cα1 + ωα1 Q1 (R1 + R2)) + 2 ωα2 Q2 R2
× (Cα2 + ωα1 Q1 (Cα1mα2 R2 + Cα2 R1 (2 Cα1 + ωα1 Q1 (R1 + R2)))))
cα1mα2 = cos
(
π (α1 − α2)
2
)
Im Z(ω) =(
ωα1 Q1
(
−ω2 α2 Q22 R1
2 R22 − 2 ωα2 Cα2 Q2 R1 R2 (R1 + R2)−
(R1 + R2)2)
Sα1 − ωα2 Q2 R22 Sα2
)
/(
1 + ω2 α2 Q22 (1 + ωα1 Q1 R1 (2 Cα1 + ωα1 Q1 R1)) R2
2+
ωα1 Q1 (R1 + R2) (2 Cα1 + ωα1 Q1 (R1 + R2)) + 2 ωα2 Q2 R2
× (Cα2 + ωα1 Q1 (Cα1mα2 R2 + Cα2 R1 (2 Cα1 + ωα1 Q1 (R1 + R2)))))
4.3 Circuit ((Q1+(R2/Q2))/R1)
Q1
R1
Q2
R2
Figure 4.5: Circuit ((Q1+(R2/Q2))/R1).
Z(ω) =1
1
R1+
11
(i ω)α1 Q1
+1
(i ω)α2 Q2 +
1
R2
Z(ω) =R1 (1 + (i ω)α1 Q1 R2 + (i ω)α2 Q2 R2)
1 + (i ω)α1 Q1 (R1 + R2) + (i ω)
α2 Q2 R2 + (i ω)α1+α2 Q1 Q2 R1 R2
26 CHAPTER 4. CIRCUITS MADE OF TWO RS AND TWO CPES
Re Z(ω) =(
R1
(
1 + ωα2 Q2 R2 (2 Cα2 + ωα2 Q2 R2) + ω2 α1 Q12 R2
× (R2 + R1 (1 + ωα2 Cα2 Q2 R2)) + ωα1 Q1 (2 R2 (Cα1 + ωα2 Cα1mα2 Q2 R2)+
Cα1 R1 (1 + ωα2 Q2 R2 (2 Cα2 + ωα2 Q2 R2))))) /(
1 + ω2 α2 Q22 (1 + ωα1 Q1 R1 (2 Cα1 + ωα1 Q1 R1)) R2
2+
ωα1 Q1 (R1 + R2) (2 Cα1 + ωα1 Q1 (R1 + R2))+
2 ωα2 Q2 R2 (Cα2 + ωα1 Q1 (Cα1mα2 R2 + Cα2 R1 (2 Cα1 + ωα1 Q1 (R1 + R2)))))
Im Z(ω) = −ωα1 Q1 R22 (Sα1 + ωα2 Q2 R2 ((2 Cα2 + ωα2 Q2 R2) Sα1 + ωα1 Q1 R2 Sα2)) /
(
1 + ω2 α2 Q22 (1 + ωα1 Q1 R1 (2 Cα1 + ωα1 Q1 R1)) R2
2+
ωα1 Q1 (R1 + R2) (2 Cα1 + ωα1 Q1 (R1 + R2))+
2 ωα2 Q2 R2 (Cα2 + ωα1 Q1 (Cα1mα2 R2 + Cα2 R1 (2 Cα1 + ωα1 Q1 (R1 + R2)))))
Z(ω) =R1 (1 + τ1 (i ω)α1 + τ2 (i ω)α2)
1 + (1 + R1/R2) τ1 (iω)α1 + τ2 (i ω)
α2 + τ1 τ2 (R1/R2) (i ω)α1+α2
τ1 = Q1 R2 , τ2 = Q2 R2
4.4 Circuit (((Q2+R2)/R1)/Q1)
Q2
R2
R1
Q1
Figure 4.6: Circuit (((Q2+R2)/R1)/Q1).
Z(ω) =1
(i ω)α1 Q1 +1
R1+
11
(i ω)α2 Q2
+ R2
Z(ω) =R1 (1 + (iω)α2 Q2 R2)
1 + (i ω)α1 Q1 R1 + (i ω)
α2 Q2 R1 + (i ω)α2 Q2 R2 + (i ω)
α1+α2 Q1 Q2 R1 R2
4.4. CIRCUIT (((Q2+R2)/R1)/Q1) 27
Re Z(ω) = (R1 (1 + ωα2 Q2 (ωα2 Q2 R2 (R1 + R2) + Cα2 (R1 + 2 R2))+
ωα1 Cα1 Q1 R1 (1 + ωα2 Q2 R2 (2 Cα2 + ωα2 Q2 R2)))) /
(1 + ωα2 Q2 (R1 + R2) (2 Cα2 + ωα2 Q2 (R1 + R2)) +
ω2 α1 Q12 R1
2 (1 + ωα2 Q2 R2 (2 Cα2 + ωα2 Q2 R2)) + 2 ωα1 Q1 R1
× (Cα1 + ωα2 Q2 (Cα1mα2 R1 + 2 Cα1 Cα2 R2 + ωα2 Cα1 Q2 R2 (R1 + R2))))
Im Z(ω) =(
R12 (− (ωα1 Q1 (1 + ωα2 Q2 R2 (2 Cα2 + ωα2 Q2 R2)) Sα1) − ωα2 Q2 Sα2)
)
/
(1 + ωα2 Q2 (R1 + R2) (2 Cα2 + ωα2 Q2 (R1 + R2)) +
ω2 α1 Q12 R1
2 (1 + ωα2 Q2 R2 (2 Cα2 + ωα2 Q2 R2)) + 2 ωα1 Q1 R1
× (Cα1 + ωα2 Q2 (Cα1mα2 R1 + 2 Cα1 Cα2 R2 + ωα2 Cα1 Q2 R2 (R1 + R2))))
Z(ω) =R1 (1 + (i ω)
α2 τ2)
1 + (i ω)α1 τ1 + (1 + R1/R2) (i ω)α2 τ2 + (i ω)α1+α2 τ1 τ2
τ1 = Q1 R1 , τ2 = Q2 R2
30 APPENDIX A. SYMBOLS FOR CPE
A B CCPE
DE
F
G HQ
I
J K L
M N O
ÙÙP Q R
S T
Figure A.1: Some CPE symbols, taken from A: [13], B: [19], C: [23], D: [5], E: [10],F: [17], G: [18], H: [21], I: [12], J: [15], K: [20], L: [2, 9], M: [11], N: [3, 4], O: [14], P:[24], Q, R [25], S [7], T [16].
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