Handbook
of
Electrochemical Impedance Spectroscopy
- Im Z *
2
1
0
0
1
Re Z *
ELECTRICAL CIRCUITS
CONTAINING CPEs
ER@SE/LEPMI
J.-P. Diard, B. Le Gorrec, C. Montella
Hosted by Bio-Logic @ www.bio-logic.info
March 29, 2013
2
Contents
1 Circuits containing one CPE
1.1 Constant Phase Element (CPE), symbol Q .
1.2 Circuit (R+Q) . . . . . . . . . . . . . . . . .
1.2.1 Impedance . . . . . . . . . . . . . . .
1.2.2 Reduced impedance . . . . . . . . . .
1.3 Circuit (R/Q) . . . . . . . . . . . . . . . . . .
1.3.1 Impedance . . . . . . . . . . . . . . .
1.3.2 Reduced impedance . . . . . . . . . .
1.3.3 Pseudocapacitance #1 . . . . . . . . .
1.3.4 Pseudocapacitance #2 . . . . . . . . .
1.4 Circuit (R/Q)+(R/Q)+ .. (Voigt) . . . . . .
1.5 Circuit (R1 +(R2 /Q2 )) . . . . . . . . . . . . .
1.5.1 Impedance . . . . . . . . . . . . . . .
1.5.2 Reduced impedance . . . . . . . . . .
1.6 Circuit (R1 /(R2 +Q2 )) . . . . . . . . . . . . .
1.6.1 Impedance . . . . . . . . . . . . . . .
1.6.2 Reduced impedance . . . . . . . . . .
1.7 Transformation formulae between(R+(R/Q))
and (R/(R+Q)) . . . . . . . . . . . . . . . . .
1.7.1 α21 = α22 . . . . . . . . . . . . . . . .
1.7.2 α21 6= α22 . . . . . . . . . . . . . . . .
2 Circuits made of two CPEs
2.1 Circuit (Q1 +Q2 ) . . . . . .
2.1.1 α1 = α2 = α . . . .
2.1.2 α1 6= α2 . . . . . . .
2.1.3 Reduced impedance
2.2 Circuit (Q1 /Q2 ) . . . . . .
2.2.1 α1 = α2 = α . . . .
2.2.2 α1 6= α2 . . . . . . .
2.2.3 Reduced impedance
3 Circuits made of one R and
3.1 Circuit ((R1 /Q1 ) + Q2 ) .
3.1.1 α1 = α2 = α . . .
3.1.2 α1 6= α2 . . . . . .
3.2 Circuit ((R1 + Q1 )/Q2 ) .
3.2.1 α1 = α2 = α . . .
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13
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14
16
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16
16
CPEs
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4
CONTENTS
3.2.2
α1 6= α2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Circuits made of two Rs and two CPEs
4.1 Circuit ((R1 /Q1 )+(R2 /Q2 )) . . . . . . .
4.2 Circuit ((R1 +(R2 /Q2 ))/Q1 ) . . . . . . .
4.3 Circuit ((Q1 +(R2 /Q2 ))/R1 ) . . . . . . .
4.4 Circuit (((Q2 +R2 )/R1 )/Q1 ) . . . . . . .
A Symbols for CPE
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21
23
23
24
25
26
29
Chapter 1
Circuits containing one
CPE
1.1
Constant Phase Element (CPE), symbol Q
Figure 1.1: Most often used symbol for CPE (see also the Appendix A).
Z=
sα
1
cα
, Im Z = −
α , Re Z =
α
Qω
Q ωα
Q (i ω)
cα = cos(
|Z| =
πα
πα
) , sα = sin(
)
2
2
πα
1
, φZ = −
Q ωα
2
The Q unit (F cm−2 sα−1 ) depends on α (1 ).
1.2
1.2.1
Circuit (R+Q)
Impedance
Z(ω) = R +
1
sα
1
cα
, Im Z = −
α , Re Z = R +
Q ωα
Q ωα
Q (i ω)
Different equations for CPE: Z =
1
Q
[5], Z =
[26].
(i ω)1−α
(Q i ω)α
5
CHAPTER 1. CIRCUITS CONTAINING ONE CPE
Im Y
- Im Z
6
0
Π
- Α €€€€€€
2
0
0
Π
Α €€€€€€
2
0
Re Z
Re Y
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.
Q
R
Figure 1.3: Circuit (R+Q).
1.2.2
Reduced impedance
Z ∗ (ω) =
Z(ω)
1
, τ = RQ
=1+
R
τ (i ω)α
The τ unit depends on α: uτ = sα .
Z ∗ (u) = 1 +
1
, u = ω τ 1/α
(i u)α
0
0.5
uc =1
1
0
0
1
Re Z*
uc =1
Im Y*
- Im Z*
2
2
0
1
Re Y*
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)
1.3
7
Circuit (R/Q)
Q
R
Figure 1.5: Circuit (R/Q).
1.3.1
Impedance
Z(ω) =
Re Z(ω) =
1.3.2
R
; τ = RQ
1 + τ (i ω)α
R τ ω α sα
R (1 + τ ω α cα )
;
Im
Z(ω)
=
−
1 + τ 2 ω 2 α + 2 τ ω α cα
1 + τ 2 ω 2 α + 2 τ ω α cα
Reduced impedance
Z ∗ (ω) =
1
Z(ω)
=
α ; τ = RQ
R
1 + τ (i ω)
1 + τ ω α cα
τ ω α sα
; Im Z ∗ (ω) = −
2
α
α
2
1 + ω + 2 τ ω cα
1 + τ ω 2 α + 2 τ ω α cα
α τ ω −1+α −1 + τ 2 ω 2 α sα
dIm Z ∗ (ω)
= 0 ⇒ ωcα = 1/τ [6]
=
dω
(1 + τ 2 ω 2 α + 2 τ ω α cα )2
Re Z ∗ (ω) =
τ2
Re Z ∗ (ωc ) = 1/2 , Im Z ∗ (ωc ) = −
α=
sα
2 (1 + cα )
!
2
2
arccos −1 +
2
π
1 + 4 Im Z ∗ (ωc )
Z ∗ (u) =
1
1/α
α, u = ωτ
1 + (i u)
(Fig. 1.6)
1.3.3
Pseudocapacitance #1
The value of the pseudocapacitance C (C/F cm−2 ) for the (R/C) circuit giving
the same characteristic frequency than that of the (R/Q) circuit (Fig. 1.7) is
obtained from:
1
1
1
1
ωc =
=
⇒ C = Q α R α −1
RC
(R Q)1/α
8
CHAPTER 1. CIRCUITS CONTAINING ONE CPE
0.5
uc =1
- Im Z*
0
Im Y*
2
0
0
1
uc =1
1
0
*
Re Z
1
Re Y*
2
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.
Q
C
R
R
1
Ωc =
0
Ωc = 1HRCL
HRQL
1Α
0
0
0
R
R
Figure 1.7: (R/Q) and (R/C) circuits with the same characteristic frequency at the
apex (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 giving the same impedance for the characteristic frequency of the (RQ /Q) circuit
(Fig. 1.7) is obtained from [2, 9]:
(1/α)−1
C = Q1/α RQ
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
Q
C
RQ
RC
1
- Im Z
Ωc =
0
1Α
JRQ QN
0
RC
RQ
Re Z
Figure 1.8: (RQ /Q) and (RC /C) circuits with the same impedance for the characteristic frequency of the (RQ /Q) circuit.
1.4
Circuit (R/Q)+(R/Q)+ .. (Voigt)
nRQ
Z(ω) =
X
i=1
Ri
; τi = Ri Qi
1 + τi (i ω)αi
nRQ
Re Z(ω) =
X
i=1
Ri (1 + τi ω αi cαi )
1 + τi2 ω 2 αi + 2 τi ω αi cαi )
nRQ
Im Z(ω) = −
X
i=1
1.5
1+
τi2
Ri τi ω αi sαi
ω 2 αi + 2 τi ω αi cαi
Circuit (R1 +(R2/Q2 ))
Q2
R1
R2
Figure 1.9: Circuit (R1 +(R2 /Q2 )).
10
CHAPTER 1. CIRCUITS CONTAINING ONE CPE
1.5.1
Impedance
1
Z(ω) = R1 +
(i ω)α2 Q2 +
Z(ω) =
1.5.2
1
R2
R1 R2 Q2
(R1 + R2 ) (1 + (i ω)α2 τ2 )
, τ1 = R2 Q2 , τ2 =
1 + (i ω)α2 τ1
R1 + R2
Reduced impedance
Z ∗ (u) =
1/α2
u = τ1
Z(u)
1 + T (i u)α2
=
R1 + R2
1 + (i u)α2
(1.1)
ω, T = τ2 /τ1 = R1 /(R1 + R2 ) < 1
Re Z ∗ (u) =
T cα uα2 + cα uα2 + T u2α2 + 1
2 cα uα2 + u2α2 + 1
Im Z ∗ (u) = −
(1 − T ) uα2 sα
2 cα uα2 + u2α2 + 1
1
1
Τ1 1Α
Τ2 1Α
0
0
1
T 1Α
0
R1 +R2
R1
uc = 1
- Im Z *
- Im Z
Ωc =
0
T
1
Re Z *
Re Z
Figure 1.10: Nyquist diagrams of the impedance and reduced impedance for the
(R1 +(R2 /Q2 )) circuit.
1.6
Circuit (R1 /(R2 +Q2 ))
R1
Q2
R2
Figure 1.11: Circuit (R1 /(R2 +Q2 )).
1.7. TRANSFORMATION FORMULAE BETWEEN(R+(R/Q)) AND (R/(R+Q))11
1.6.1
Impedance
R1 (1 + τ2 (i ω)α2 )
, τ1 = (R1 + R2 ) Q2 , τ2 = R2 Q2
1 + τ1 (i ω)α2
2
R1 cos πα
(τ1 + τ2 ) ω α2 + τ1 τ2 ω 2α2 + 1
2
Re Z(ω) =
2
ω α2 + 1
τ1 τ1 ω α2 + 2 cos πα
2
2
ω α2 sin πα
R1 (τ1 − τ2 )
2
Im Z(ω) = −
2
ω α2 + 1
τ1 τ1 ω α2 + 2 cos πα
2
Z(ω) =
1.6.2
Reduced impedance
Z ∗ (u) =
1/α2
u = τ1
Z(u)
1 + T (i u)α2
=
R1
1 + (i u)α2
ω, T = τ2 /τ1 = R2 /(R1 + R2 ) < 1
cf. Eq. (1.1) and Fig. 1.10.
1.7
1.7.1
Transformation formulae between(R+(R/Q))
and (R/(R+Q))
α21 = α22
Q21
R12
R11
Q22
R21
R22
Figure 1.12: The (R+(R/Q)) and (R/(R+Q)) circuits are non-distinguishable for
α21 = α22 [1].
Transformations formulae (R+(R/Q)) → (R/(R+Q))
R12 = R11 + R21 , R22 =
2
2
R11
Q21 R21
+ R11 , Q22 =
R21
(R11 + R21 ) 2
Transformations formulae (R/(R+Q))→(R+(R/Q))
Q21 =
1.7.2
2
R12 R22
R12
Q22 (R12 + R22 ) 2
, R11 =
, R21 =
2
R12
R12 + R22
R12 + R22
α21 6= α22
The (R+(R/Q)) and (R/(R+Q)) circuits (Fig. 1.12) are distinguishable for
α21 6= α22
12
CHAPTER 1. CIRCUITS CONTAINING ONE CPE
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
1
+
Q1
Q2
1
Q1 Q2
1
=
, Q=
(i ω)α
Q (i ω)α
Q1 + Q2
cf. § 1.1.
2.1.2
α1 6= α2
Impedance
1
1
Q1 (i ω)α1 + Q2 (i ω)α2
+
=
α
α
1
2
Q1 (i ω)
Q2 (i ω)
Q1 Q2 (i ω)α1 +α2
−α
−α
1
2
cos πα
cos πα
ω 1
ω 2
2
2
Re Z(ω) =
+
Q1
Q2
−α
2
1
sin πα
ω −α1
ω 2
sin πα
2
2
−
Im Z(ω) = −
Q1
Q2
Z(ω) =
|ZQ1 | = |ZQ2 | ⇒ ω = ωc =
13
Q2
Q1
α
1
1 −α2
14
CHAPTER 2. CIRCUITS MADE OF TWO CPES
• α1 < α2 (Figs. 2.2 and 2.3)
ω → 0 ⇒ Z(ω) ≈
1
1
, ω → ∞ ⇒ Z(ω) ≈
α
2
Q2 (i ω)
Q1 (i ω)α1
• α1 > α2
ω → 0 ⇒ Z(ω) ≈
1
1
, ω → ∞ ⇒ Z(ω) ≈
Q1 (i ω)α1
Q2 (i ω)α2
4
- Im ZW
logH- Im ZWL
200
1
0
0
200
1
Re ZW
4
logHRe ZWL
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 ) .
-Α1 А2
ΦZ °
log ÈZWÈ
-45
- Α2
5
- Α1
0
-Α2 А2
-5
HQ2 Q1 L1HΑ1 -Α2 L
log ِHrd s-1 L
-90
5
-5
HQ2 Q1 L1HΑ1 -Α2 L
5
log ِHrd s-1 L
Figure 2.3: Bode diagrams of the impedance for the (Q1 +Q2 ) circuit. Same values
of parameters as in Fig. 2.2. α1 < α2 .
2.1.3
Reduced impedance
Z ∗ (u) = Q1 ωcα1 Z(ω) =
1
ω
1
+
, u=
α
α
1
2
(i u)
(i u)
ωc
2.1. CIRCUIT (Q1 +Q2 )
15
2
log H- Im Z * L
- Im Z *
3
2
1
0
0
-2
0
1
2
Re Z
-2
3
0
2
log HRe Z 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.
-45
- Α2
-Α1 А2
0
- Α1
ΦZ * °
log ÈZ * È
3
-Α2 А2
-3
-90
-3
0
log u
3
-3
0
3
log u
Figure 2.5: Bode diagrams of the impedance for the (Q1 +Q2 ) circuit. Same values
of parameters as in Fig. 2.4. α1 < α2 .
16
CHAPTER 2. CIRCUITS MADE OF TWO CPES
2.2
Circuit (Q1 /Q2 )
Q2
Q1
Figure 2.6: Circuit (Q1 /Q2 ).
2.2.1
α1 = α2 = α
Z(ω) =
1
1
=
, Q = Q1 + Q2
(Q1 + Q2 ) (i ω)α
Q (i ω)α
cf. § 1.1.
2.2.2
α1 6= α2
Impedance
1
Q1 (i ω)α1 + Q2 (i ω)α2
1
2
cos πα
Q1 ω α1 + cos πα
Q2 ω α2
2
2
Re Z(ω) = 2 2α
1
Q1 ω 1 + Q22 ω 2α2 + 2 cos 2 π (α1 − α2 ) Q1 Q2 ω α1 +α2
1
2
sin πα
Q1 ω α1 + sin πα
Q2 ω α2
2
2
Im Z(ω) = − 2 2α
Q1 ω 1 + Q22 ω 2α2 + 2 cos 12 π (α1 − α2 ) Q1 Q2 ω α1 +α2
Z(ω) =
• α1 < α2 (Figs. 2.7 and 2.8)
ω → 0 ⇒ Z(ω) ≈
1
1
, ω → ∞ ⇒ Z(ω) ≈
α
1
Q1 (i ω)
Q2 (i ω)α2
• α1 > α2
ω → 0 ⇒ Z(ω) ≈
2.2.3
1
1
, ω → ∞ ⇒ Z(ω) ≈
Q2 (i ω)α2
Q1 (i ω)α1
Reduced impedance
Z ∗ (u) = Q1 ωcα1 Z(ω) =
(i u)α1
ω
1
, u=
α
2
+ (i u)
ωc
2.2. CIRCUIT (Q1 /Q2 )
17
logH- Im ZWL
- Im ZW
3
50
0
0
0
50
0
Re ZW
3
logHRe ZWL
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 ) .
-45
-Α1 А2
- Α1
ΦZ °
log ÈZWÈ
5
0
- Α2
-5
HQ2 Q1 L1HΑ1 -Α2 L
-Α2 А2
-90
-5
5
log ِHrd s-1 L
HQ2 Q1 L1HΑ1 -Α2 L
5
log ِHrd s-1 L
Figure 2.8: Bode diagrams of the impedance for the (Q1 /Q2 ) circuit. Same values of
parameters as in Fig. 2.7. α1 < α2 .
2
- Im Z *
log H- Im Z * L
1
0
0
-2
0
1
Re Z *
-2
0
2
log HRe Z * 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
-45
3
0
- Α1
ΦZ * °
log ÈZ * È
-Α1 А2
- Α2
-3
-Α2 А2
-90
-3
0
log u
3
-3
0
3
log u
Figure 2.10: Bode diagrams of the impedance for the (Q1 /Q2 ) circuit. Same values
of parameters as in Fig. 2.9. α1 < α2 .
Chapter 3
Circuits made of one R and
two CPEs
3.1
Circuit ((R1/Q1) + Q2 )
Q1
Q2
R1
Figure 3.1: Circuit ((R1 /Q1 )+Q2 ).
3.1.1
α1 = α2 = α
Impedance
Z(ω) =
Z(ω) =
1
1
+
α
1
Q
(i
2 ω)
+ Q1 (i ω)α
R1
1 + (i ω)α τ2
, τ1 = R1 Q1 , τ2 = (Q1 + Q2 ) R1 , τ1 < τ2
Q2 (1 + (i ω)α τ1 )
(i ω)α
Re Z(ω) = −
cos
πα
2
τ1 τ2 ω 2α + 1 ω −α + cos(πα)τ1 + τ2
ωα + 1
Q2 τ1 τ1 ω α + 2 cos πα
2
τ1 τ2 ω 2α + 1 ω −α + sin(πα)τ1
sin πα
2
Im Z(ω) = −
ωα + 1
Q2 τ1 τ1 ω α + 2 cos πα
2
19
20
CHAPTER 3. CIRCUITS MADE OF ONE R AND TWO CPES
Reduced impedance
Z ∗ (u) =
1 + T (i u)α
1
Z(u)
=
R1
T − 1 (i u)α (1 + (i u)α )
(3.1)
u = ω τ 1/α , T = τ2 /τ1 = 1 + Q2 /Q1 > 1
u−α (T + cos(απ))uα + T u2α + 1 cos απ
2
Re Z (u) =
uα + u2α + 1
(T − 1) 2 cos απ
2
!
απ 1
u2α
∗
−α
Im Z (u) = u
−
sin
1−T
2
2 cos απ
uα + u2α + 1
2
∗
- Im Z *
2
1
0
0
1
Re 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 increases
with 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
1
1
+
1
Q2 (i ω)α2
+ Q1 (i ω)α1
R1
−α
πα2
1
R1 cos πα
cos 2 ω 2
Q1 R1 ω α1 + 1
2
+
Re Z(ω) =
1
Q2
ω α1 + 1
Q1 R1 Q1 R1 ω α1 + 2 cos πα
2
−α
1
2
sin πα
sin πα
Q1 R12 ω α1
ω 2
2
2
Im Z(ω) = −
−
1
Q2
ω α1 + 1
Q1 R1 Q1 R1 ω α1 + 2 cos πα
2
Z(ω) =
3.2. CIRCUIT ((R1 + Q1 )/Q2 )
21
Q1
Q2
R1
Figure 3.3: Circuit ((R1 +Q2 )/Q1 ).
3.2
Circuit ((R1 + Q1 )/Q2 )
3.2.1
α1 = α2 = α
Impedance
1
Z(ω) =
1
(i ω)α Q1 +
R1 +
1
(i ω)α Q2
=
1 + Q2 R1 (i ω)α
(i ω)α Q1 Q2 R1
(i ω)α (Q1 + Q2 ) 1 +
Q1 + Q2
1 + τ2 (i ω)α
Q1 Q2 R1
, τ2 = Q2 R1
, τ1 =
(i ω)α (Q1 + Q2 ) (1 + (i ω)α τ1 )
Q1 + Q2
τ2 ω 2α + 1
ω −α cos(πα)ω α + τ2 ω α + cos πα
2
Re Z(ω) =
α
2α + 1 (Q + Q ) τ
2 cos πα
1
2 1
2 ω +ω
πα
ω −α sin πα
ω α + τ2 ω 2α + 1
2 2 cos 2
Im Z(ω) = −
α
2α + 1 (Q + Q ) τ
2 cos πα
1
2 1
2 ω +ω
Z(ω) =
Reduced impedance
Z ∗ (u) =
T −1
1 + T (i u)α
Z(u)
=
2
R1
T
(i u)α (1 + (i u)α )
u = ω τ 1/α , T = τ2 /τ1 = 1 + Q2 /Q1 > 1
(T − 1)u−α (T + cos(απ))uα + T u2α + 1 cos απ
2
Re Z (u) =
α
2α + 1
T 2 2 cos απ
2 u +u
α
2α
+ 1 sin απ
(T − 1)u−α 2 cos απ
2 u + T u
2
Im Z ∗ (u) = −
uα + u2α + 1
T 2 2 cos απ
2
∗
3.2.2
α1 6= α2
1
+ R1
(i ω)α2 Q2
Z(ω) =
1
1
α
1
+
+ R1
(i ω) Q1
(i ω)α1 Q1
(i ω)α2 Q2
(3.2)
22
CHAPTER 3. CIRCUITS MADE OF ONE R AND TWO CPES
- Im Z *
2
1
0
0
1
Re 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 increases
with increasing T . Dots: reduced characteristic angular frequency uc1 = 1; circles:
reduced characteristic angular frequency uc2 = 1/T 1/α (φuc1 = φuc2 ).
α
Z(ω) =
(i ω)α1
1 + τ (i ω) 2
, τ = R1 Q2
Q1 + (i ω)α2 Q2 + τ (i ω)α1 +α2 Q1
Re Z(ω) =
ω 2 α1
ω α1 cα1 1 + τ 2 ω 2 α2 + 2 τ ω α2 cα2 Q1 + ω α2 (τ ω α2 + cα2 ) Q2 /
1 + τ 2 ω 2 α2 + 2 τ ω α2 cα2 Q1 2 + 2 ω α1 +α2 (τ ω α2 cα1 + cα1mα2 ) Q1 Q2 + ω 2 α2 Q2 2
π (α1 − α2 )
cα1mα2 = cos
2
Im Z(ω) =
ω 2 α1
−ω α1 1 + τ 2 ω 2 α2 + 2 τ ω α2 cα2 Q1 sα1 − ω α2 Q2 sα2 /
1 + τ 2 ω 2 α2 + 2 τ ω α2 α2 Q1 2 + 2 ω α1 +α2 (τ ω α2 cα1 + cα1mα2 ) Q1 Q2 + ω 2 α2 Q2 2
Chapter 4
Circuits made of two Rs
and two CPEs
4.1
Circuit ((R1 /Q1 )+(R2 /Q2 ))
Q1
Q2
R1
R2
Figure 4.1: Circuit ((R1 /Q1 )+(R2 /Q2 )).
Z(ω) =
1
α1
(i ω)
Z(ω) =
1
Q1 +
R1
+
1
α2
(i ω)
Q2 +
1
R2
R1
R2
+
, τ1 = R1 Q1 , τ2 = R2 Q2
α1
α
1 + (i ω) τ1
1 + (i ω) 2 τ2
α
Z(ω) =
Re Z(ω) =
α
R1 + R2 + (i ω) 1 R2 τ1 + (i ω) 2 R1 τ2
α
α
(1 + (i ω) 1 τ1 ) (1 + (i ω) 2 τ2 )
R1 (1 + ω α1 cα1 τ1 )
R2 (1 + ω α2 cα2 τ2 )
+
α
α
1 + ω 1 τ1 (2 cα1 + ω 1 τ1 ) 1 + ω α2 τ2 (2 cα2 + ω α2 τ2 )
Im Z(ω) = −
ω α1 R1 sα1 τ1
ω α2 R2 sα2 τ2
−
1 + ω α1 τ1 (2 cα1 + ω α1 τ1 ) 1 + ω α2 τ2 (2 cα2 + ω α2 τ2 )
23
24
CHAPTER 4. CIRCUITS MADE OF TWO RS AND TWO CPES
0.3
- Im Z *
uc1
0
uc2
0
0.5
1
Re Z *
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 .
0.5
0.5
uc1 = uc2
- Im Z *
- Im Z *
uc1 = uc2
0
0
0.5
Re Z
0
1
- А4
А4
0
0.5
*
Re Z
1
*
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 )
1
Z(ω) =
α1
(i ω)
1
Q1 +
1
R1 +
α2
(i ω)
Q2 +
1
R2
α
Z(ω) =
R1 + R2 + (i ω) 2 Q2 R1 R2
1 + (i ω)α1 Q1 (R1 + R2 ) + (i ω)α2 Q2 R2 + (i ω)α1 +α2 Q1 Q2 R1 R2
Q1
Q2
R1
R2
Figure 4.4: Circuit ((R1 +(R2 /Q2 ))/Q1 ).
4.3. CIRCUIT ((Q1 +(R2 /Q2 ))/R1 )
25
Re Z(ω) = R1 + R2 + ω 2 α2 Q2 2 R1 (1 + ω α1 Cα1 Q1 R1 ) R2 2 +
2
ω α1 Cα1 Q1 (R1 + R2 ) + ω α2 Cα2 Q2 R2 (R2 + 2 R1 (1 + ω α1 Cα1 Q1 (R1 + R2 ))) /
1 + ω 2 α2 Q2 2 (1 + ω α1 Q1 R1 (2 Cα1 + ω α1 Q1 R1 )) R2 2 +
ω α1 Q1 (R1 + R2 ) (2 Cα1 + ω α1 Q1 (R1 + R2 )) + 2 ω α2 Q2 R2
× (Cα2 + ω α1 Q1 (Cα1mα2 R2 + Cα2 R1 (2 Cα1 + ω α1 Q1 (R1 + R2 )))))
cα1mα2 = cos
π (α1 − α2 )
2
Im Z(ω) = ω α1 Q1 −ω 2 α2 Q2 2 R1 2 R2 2 − 2 ω α2 Cα2 Q2 R1 R2 (R1 + R2 ) −
2
(R1 + R2 ) Sα1 − ω α2 Q2 R2 2 Sα2 /
1 + ω 2 α2 Q2 2 (1 + ω α1 Q1 R1 (2 Cα1 + ω α1 Q1 R1 )) R2 2 +
ω α1 Q1 (R1 + R2 ) (2 Cα1 + ω α1 Q1 (R1 + R2 )) + 2 ω α2 Q2 R2
× (Cα2 + ω α1 Q1 (Cα1mα2 R2 + Cα2 R1 (2 Cα1 + ω α1 Q1 (R1 + R2 )))))
4.3
Circuit ((Q1 +(R2/Q2 ))/R1 )
R1
Q2
Q1
R2
Figure 4.5: Circuit ((Q1 +(R2 /Q2 ))/R1 ).
Z(ω) =
Z(ω) =
1
1
+
R1
1
1
+
α
(i ω) 1 Q1
1
α2
(i ω)
1
R2
Q2 +
R1 (1 + (i ω)α1 Q1 R2 + (i ω)α2 Q2 R2 )
α1
1 + (i ω)
α2
Q1 (R1 + R2 ) + (i ω)
α1 +α2
Q2 R2 + (i ω)
Q1 Q2 R1 R2
26
CHAPTER 4. CIRCUITS MADE OF TWO RS AND TWO CPES
Re Z(ω) = R1 1 + ω α2 Q2 R2 (2 Cα2 + ω α2 Q2 R2 ) + ω 2 α1 Q1 2 R2
× (R2 + R1 (1 + ω α2 Cα2 Q2 R2 )) + ω α1 Q1 (2 R2 (Cα1 + ω α2 Cα1mα2 Q2 R2 ) +
Cα1 R1 (1 + ω α2 Q2 R2 (2 Cα2 + ω α2 Q2 R2 ))))) /
1 + ω 2 α2 Q2 2 (1 + ω α1 Q1 R1 (2 Cα1 + ω α1 Q1 R1 )) R2 2 +
2ω
α2
ω α1 Q1 (R1 + R2 ) (2 Cα1 + ω α1 Q1 (R1 + R2 )) +
Q2 R2 (Cα2 + ω α1 Q1 (Cα1mα2 R2 + Cα2 R1 (2 Cα1 + ω α1 Q1 (R1 + R2 )))))
Im Z(ω) = −ω α1 Q1 R2 2 (Sα1 + ω α2 Q2 R2 ((2 Cα2 + ω α2 Q2 R2 ) Sα1 + ω α1 Q1 R2 Sα2 )) /
1 + ω 2 α2 Q2 2 (1 + ω α1 Q1 R1 (2 Cα1 + ω α1 Q1 R1 )) R2 2 +
ω α1 Q1 (R1 + R2 ) (2 Cα1 + ω α1 Q1 (R1 + R2 )) +
2 ω α2 Q2 R2 (Cα2 + ω α1 Q1 (Cα1mα2 R2 + Cα2 R1 (2 Cα1 + ω α1 Q1 (R1 + R2 )))))
Z(ω) =
R1 (1 + τ1 (i ω)α1 + τ2 (i ω)α2 )
α1
1 + (1 + R1 /R2 ) τ1 (i ω)
α2
+ τ2 (i ω)
α1 +α2
+ τ1 τ2 (R1 /R2 ) (i ω)
τ1 = Q1 R2 , τ2 = Q2 R2
4.4
Circuit (((Q2 +R2 )/R1 )/Q1 )
Q1
R1
Q2
R2
Figure 4.6: Circuit (((Q2 +R2 )/R1 )/Q1 ).
Z(ω) =
Z(ω) =
1
1
(i ω)α1 Q1 +
+
R1
1
1
+ R2
α
(i ω) 2 Q2
R1 (1 + (i ω)α2 Q2 R2 )
α1
1 + (i ω)
α2
Q1 R1 + (i ω)
α2
Q2 R1 + (i ω)
α1 +α2
Q2 R2 + (i ω)
Q1 Q2 R1 R2
4.4. CIRCUIT (((Q2 +R2 )/R1 )/Q1 )
27
Re Z(ω) = (R1 (1 + ω α2 Q2 (ω α2 Q2 R2 (R1 + R2 ) + Cα2 (R1 + 2 R2 )) +
ω α1 Cα1 Q1 R1 (1 + ω α2 Q2 R2 (2 Cα2 + ω α2 Q2 R2 )))) /
(1 + ω α2 Q2 (R1 + R2 ) (2 Cα2 + ω α2 Q2 (R1 + R2 )) +
ω 2 α1 Q1 2 R1 2 (1 + ω α2 Q2 R2 (2 Cα2 + ω α2 Q2 R2 )) + 2 ω α1 Q1 R1
× (Cα1 + ω α2 Q2 (Cα1mα2 R1 + 2 Cα1 Cα2 R2 + ω α2 Cα1 Q2 R2 (R1 + R2 ))))
Im Z(ω) = R1 2 (− (ω α1 Q1 (1 + ω α2 Q2 R2 (2 Cα2 + ω α2 Q2 R2 )) Sα1 ) − ω α2 Q2 Sα2 ) /
(1 + ω α2 Q2 (R1 + R2 ) (2 Cα2 + ω α2 Q2 (R1 + R2 )) +
ω 2 α1 Q1 2 R1 2 (1 + ω α2 Q2 R2 (2 Cα2 + ω α2 Q2 R2 )) + 2 ω α1 Q1 R1
× (Cα1 + ω α2 Q2 (Cα1mα2 R1 + 2 Cα1 Cα2 R2 + ω α2 Cα1 Q2 R2 (R1 + R2 ))))
α
Z(ω) =
α1
1 + (i ω)
R1 (1 + (i ω) 2 τ2 )
τ1 + (1 + R1 /R2 ) (i ω)α2 τ2 + (i ω)α1 +α2 τ1 τ2
τ1 = Q1 R1 , τ2 = Q2 R2
28
CHAPTER 4. CIRCUITS MADE OF TWO RS AND TWO CPES
Appendix A
Symbols for CPE
29
30
APPENDIX A. SYMBOLS FOR CPE
B
A
C
CPE
D
F
E
G
H
Q
J
K
M
N
P
S
ÙÙ
Q
I
L
O
R
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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