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Application Note 43
11062012
How to fit transmission lines with ZFit
I
Introduction
ZFit is the impedance fitting tool of
R
EC-Lab⃝
. This note will describe how to fit
transmission lines using one equivalent circuit
elements contained in ZFit.
It is well known that the Warburg impedance
is equivalent to that of a semi-infinite large network i.e. a transmission line, as shown in Fig. 1
[1, 2]. Moreover, the transmission lines are often used for modeling porous electrodes, for
example in the field of photovoltaics.
r
c
¥
Fig. 1: The equivalent circuit of the Warburg
impedance.
More recently it has been shown [3] that the
impedance of a L-long transmission line made
of χ and ζ elements and terminated by a ZL
element (Fig. 2) is given by the general expression:
Χ
Χ
Ζ
with three limiting cases
- open-circuited transmission line
( √ )
√
L χ
ZL = ∞ ⇒ Z = ζ χ coth √
ζ
- short-circuited transmission line
( √ )
√
L χ
√
ZL = 0 ⇒ Z = ζ χ th
ζ
(1)
(2)
- semi-infinite transmission line
√
L→∞⇒Z = ζχ
(3)
Hereafter, some transmission lines are described and the corresponding ”simple” equivalent circuit elements are shown. The opencircuited transmission lines will be explained,
followed by short-circuited and semi-infinite
transmission lines.
II
Open-circuited transmission
lines ZL = ∞
II.1
Open-circuited URC (Uniform distributed RC)
Let us consider the open-circuited tranmission
line made of r and c elements (Fig. 3).
Χ
Ζ
Ζ
r
ZL
L
c
L
Fig. 2: Uniform transmission line made of χ and
ζ elements and terminated by ZL [3].
( √ )
(
)
L χ
√
ζ χ − ZL2 sh
ζ
( √ )
( √ ) + ZL
Z=
√
L χ
L χ
√
√
+ ζ χ ch
ZL sh
ζ
ζ
Fig. 3: L-long open uniform distributed RC
(URC) transmission line [4, 5].
Using Eq. (1), the impedance of the URC transmission line is given by (1 )
√
√ coth(L r c j ω)
1
√
χ = r, ζ =
⇒Z= r
jωc
cj ω
1
The transmission lines are named according to the U-χζ format where U means uniform distributed and χ and ζ are
the element of the transmission line.
Bio-Logic Science Instruments, 1 Rue de l’Europe, 38640 Claix, FRANCE
Tel: +33 476 98 68 31 - Fax: +33 476 98 69 09 www.bio-logic.info
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Application Note 43
11062012
with ω = 2 π f . This impedance is similar to
that of the M element of ZFit
√
coth τd j ω
ZM = Rd √
, Rd = L r, τd = L2 r c
τd j ω
II.2 Open-circuited URQ
Replacing c elements by q elements, with Zq =
1/(q (j ω)α ), leads to transmisssion line shown
in Fig. 4.
r
Fig. 5: Nyquist impedance diagram of a battery
Ni-MH 1900 mAh.
q
L
Fig. 4: L-long open uniform distributed RQ
(URQ) transmission line.
II.3
The transmission line impedance is given by
The equivalent circuit of the so-called anomalous diffusion is show in Fig. 6 [6].
1
χ = r, ζ =
⇒
q (j ω)α
√
√ coth(L r q (j ω)α/2)
Z= r
√
q (j ω)α/2
This impedance is similar to that of the Ma element of ZFit
ZMa = R
ω)α/2
coth(τ j
(τ j ω)α/2
with
R = L r, τ = (L2 r q)1/α
As an example a Nyquist impedance diagram
of a battery Ni-MH 1900 mAh is shown in Fig. 5.
The equivalent circuit R1+L1+Q1/(R2+Ma3),
containing a Ma element, is chosen to fit the
data shown in Fig. 5. The values of the parameters, obtained using the ZFit tool of ECLab, are R1 = 0.049 Ω, L1 = 0.154 × 10−6 H,
Q1 = 0.66 F sα−1 , α1 = 0.61, R2 = 0.0236 Ω,
R3 = L r = 0.057 Ω, τ 3 = (L2 r q)1/α = 2.25 s
and α3 = 0.89.
Open-circuited UQC
q
c
L
Fig. 6: L-long open uniform distributed QC
(UQC) transmission line. Anomalous diffusion [6].
The anomalous diffusion impedance is given
by
χ=
1
1
, ζ=
⇒
α
q (j ω)
cj ω
( √
)
1
c
−α
2
2
coth L
(j ω)
q
Z=
1
α
√
c q (j ω) 2 + 2
This impedance is similar to that of the Mg element of ZFit
ZMg = R
coth(τ j ω)γ/2
with γ = 1 − α,
(τ j ω)1−γ/2
1
R = cγ
−1
2
Lγ
−1 −1/γ
q
1
, τ = c γ L2/γ q −1/γ
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Tel: +33 476 98 68 31 - Fax: +33 476 98 69 09 www.bio-logic.info
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Application Note 43
11062012
III
Short-circuited transmission
lines ZL = 0
As an example a Nyquist impedance diagram
of a Fe(II)/Fe(III) system is shown in Fig. 8.
III.1 Short-circuited URC
r
c
L
Fig. 7: L-long short-circuited uniform distributed RC (URC) transmission line.
Using Eq. (2), the impedance of the shortcircuited transmission line made of r and c elements (Fig. 7) is given by
χ = r, ζ =
1
⇒
cj ω
√
th (L r c j ω)
√
Z=r
rcj ω
(4)
This impedance is similar to that of the Wd element of ZFit
√
th τd j ω
, Rd = L r, τd = L2 r c
ZWd = Rd √
τd j ω
IV Semi-infinite
lines L → ∞
transmission
Fig. 8: Nyquist impedance diagram of a
Fe(II)/Fe(III) system in basic medium.
The Randles circuit R1+Q2/(R2+W2), containing a Warburg element, is chosen to fit the data
shown in Fig. 8. The values of the parameters for equivalent circuit are R1 = 47.57 Ω,
Q2 = 17.09 × 10−6 F sα−1 , α = √
0.885, R2 =
−1/2
70.94 Ω and σ2 = 85.33 Ω s
⇒ r/c = 42.7
Ω s−1/2 .
IV.2 Semi-infinite URRC
First of all, let us calculate the impedance of
the L-long URRC transmission line (Fig. 9) corresponding to diffusion-reaction and diffusiontrapping impedance [7]:
IV.1 Semi-infinite URC
The impedance of the semi-infinite transmission line shown in Fig. 1 is obtained making
L → ∞ in Eq. (4).
√
√
th (L r c j ω)
r
√
L→∞⇒Z =r
≈ √
rcj ω
cj ω
This expression is similar to that of the Warburg
(W) element of ZFit
√
2σ
r
ZW = √
with σ = √
2 c
jω
r1
r2
c
L
Fig. 9: L-long short-circuited uniform distributed RRC (URRC) transmission line.
χ = r1 , ζ =
r2
⇒
1 + r2 c j ω
Bio-Logic Science Instruments, 1 Rue de l’Europe, 38640 Claix, FRANCE
Tel: +33 476 98 68 31 - Fax: +33 476 98 69 09 www.bio-logic.info
3
Application Note 43
11062012
Z
( √
)
r1
th L
(1 + r2 c j ω)
r2
√
√
r 1 r2
1 + r2 c j ω
=
r1
q
r2
r1
¥
c
r2
Fig. 12: Semi-infinite short-circuited uniform
distributed RRQ (URRQ) transmission line.
¥
and
√
r1 r 2
L→∞⇒Z≈ √
1 + r2 q (j ω)α
Fig. 10: Semi-infinite short-circuited uniform
distributed RRC (URRC) transmission line.
This expression is similar to that of the Ga element of ZFit
With L → ∞ it is obtained [8]:
√
r1 r2
L→∞⇒Z≈ √
1 + r2 c j ω
ZGa = √
This expression is similar to that of the
Gerischer element G of ZFit [9]:
√
RG
, R G = r1 r2 , τ G = r2 c
ZG = √
1 + τG j ω
IV.3 Semi-infinite URRQ
V
R
1 + τ (j
ω)α
, R=
√
r1 r2 , τ = r2 q
Conclusion
Seven elements, W, Wd, M, Ma, Mg, G and
Ga, available in ZFit, can be used to represent
the impedance of seven different transmission
lines, as summarized in the table below (Tabs.
1 (p. 4), 2 (p. 6)).
r1
q
r2
Table 1: Summary table.
L
Fig. 11: L-long short-circuited uniform distributed RRQ (URRQ) transmission line.
Replacing c elements by q elements
r2
⇒
1 + r2 q (j ω)α
( √
)
r1
α
th L
(1 + r2 q (j ω) )
r2
√
√
Z = r1 r2
1 + r2 q (j ω)α
χ = r1 , ζ =
Transmission line
URC
Open Circuited URQ
UQC
Short circuited
URC
URC
Semi-∞
URRC
URRQ
Bio-Logic Science Instruments, 1 Rue de l’Europe, 38640 Claix, FRANCE
Tel: +33 476 98 68 31 - Fax: +33 476 98 69 09 www.bio-logic.info
ZFit Element
M
Ma
Mg
Wd
W
G
Ga
4
Application Note 43
11062012
References
[6] J. B ISQUERT and A. C OMPTE, J. Electroanal.
Chem. 499, 112 (2001).
[1] J. C. WANG, J. Electrochem. Soc. 134, 1915
(1987).
[2] M. S LUYTERS -R EHBACH, Pure & Appl. Chem.
66, 1831 (1994).
[3] J. B ISQUERT,
4185 (2000).
Phys. Chem. Chem. Phys. 2,
[4] G. C. T EMES and J. W. L A PATRA, Introduction
to Circuits Synthesis and Design, McGraw-Hill,
New-York, 1977.
[5] J.-P. D IARD, B. L E G ORREC, and C. M ON TELLA , J. Electroanal. Chem. 471, 126 (1999).
[7] J.-P. D IARD and C. M ONTELLA, J. Electroanal.
Chem. 557, 19 (2003).
[8] B. A. B OUKAMP and H. J.-M. B OUWMEESTER,
Solid State Ionics 157, 29 (2003).
[9] H. G ERISCHER, Z. Physik. Chem. (Leipzig) 198,
286 (1951).
Nicolas Murer, Ph. D.,
Aymeric Pellissier, Ph. D.,
Jean-Paul Diard, Pr.
Bio-Logic Science Instruments, 1 Rue de l’Europe, 38640 Claix, FRANCE
Tel: +33 476 98 68 31 - Fax: +33 476 98 69 09 www.bio-logic.info
5
Application Note 43
11062012
Table 2: ZFit elements vs. transmission lines.
ZFit element
Equations
M
√
coth τd j ω
Rd √
τd j ω
Rd = L r, τd = L2 r c
coth(τ j ω)α/2
(τ j ω)α/2
R = Lr
τ = (L2 r q)1/α
R
Ma
coth(τ j ω)γ/2
(τ j ω)1−γ/2
1
−1 2 −1
R = c γ L γ q −1/γ
1
τ = c γ L2/γ q −1/γ
√
th τ j ω
Rd √ d
τd j ω
Rd = L r
τ d = L2 r c
Transmission line
r
c
L
r
q
L
q
R
Mg
Wd
2σ
√
j√
ω
W
σ=
L
r
c
L
r
r
√
2 c
RG
1 +√
τG j ω
R G = r1 r2
τG = r 2 c
√
G
c
c
¥
r1
r2
c
¥
√
Ga
RG
1 + τG√(j ω)α
R G = r1 r2
τ G = r2 q
r1
q
r2
¥
Bio-Logic Science Instruments, 1 Rue de l’Europe, 38640 Claix, FRANCE
Tel: +33 476 98 68 31 - Fax: +33 476 98 69 09 www.bio-logic.info
6