Application Note 43

11062012

How to fit transmission lines with ZFit

I Introduction with three limiting cases

- open-circuited transmission line
ZFit is the impedance fitting tool of ( √ )
√ L χ
EC-Lab⃝ R
. This note will describe how to fit ZL = ∞ ⇒ Z = ζ χ coth √ (1)
ζ
transmission lines using one equivalent circuit
elements contained in ZFit. - short-circuited transmission line
( √ )
It is well known that the Warburg impedance √ L χ
is equivalent to that of a semi-infinite large net- ZL = 0 ⇒ Z = ζ χ th √ (2)
ζ
work i.e. a transmission line, as shown in Fig. 1
[1, 2]. Moreover, the transmission lines are of- - semi-infinite transmission line
√
ten used for modeling porous electrodes, for L→∞⇒Z = ζχ (3)
example in the field of photovoltaics.
Hereafter, some transmission lines are de-
scribed and the corresponding ”simple” equiv-
r alent circuit elements are shown. The open-
circuited transmission lines will be explained,
c
followed by short-circuited and semi-infinite
¥ transmission lines.

Fig. 1: The equivalent circuit of the Warburg

impedance. II Open-circuited transmission
lines ZL = ∞
More recently it has been shown [3] that the
impedance of a L-long transmission line made II.1 Open-circuited URC (Uniform dis-
of χ and ζ elements and terminated by a ZL tributed RC)
element (Fig. 2) is given by the general expres-
sion: Let us consider the open-circuited tranmission
line made of r and c elements (Fig. 3).
Χ Χ Χ

Ζ Ζ Ζ ZL r

c
L
L

Fig. 2: Uniform transmission line made of χ and

ζ elements and terminated by ZL [3]. Fig. 3: L-long open uniform distributed RC
(URC) transmission line [4, 5].

( √ ) Using Eq. (1), the impedance of the URC trans-

( ) L χ
ζ χ − ZL2 sh √ mission line is given by (1 )
Z= ( √ )
ζ
( √ ) + ZL √
L χ √ L χ 1 √ coth(L r c j ω)
ZL sh √ + ζ χ ch √ χ = r, ζ = ⇒Z= r √
ζ ζ 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.

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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
q Ni-MH 1900 mAh.
L
Fig. 4: L-long open uniform distributed RQ
(URQ) transmission line. II.3 Open-circuited UQC
The equivalent circuit of the so-called anoma-
The transmission line impedance is given by lous diffusion is show in Fig. 6 [6].

q
1
χ = r, ζ = ⇒
q (j ω)α
√ c
√ coth(L r q (j ω)α/2)
Z= r √ L
q (j ω)α/2 Fig. 6: L-long open uniform distributed QC
(UQC) transmission line. Anomalous diffu-
This impedance is similar to that of the Ma ele- sion [6].
ment of ZFit
The anomalous diffusion impedance is given
coth(τ jω)α/2 by
ZMa = R with
(τ j ω)α/2
1 1
R = L r, τ = (L2 r q)1/α χ= α
, ζ= ⇒
q (j ω) cj ω
( √ )
c 1
−α
As an example a Nyquist impedance diagram coth L (j ω) 2 2
q
of a battery Ni-MH 1900 mAh is shown in Fig. 5. Z= √ α 1
The equivalent circuit R1+L1+Q1/(R2+Ma3), c q (j ω) 2 + 2
containing a Ma element, is chosen to fit the This impedance is similar to that of the Mg ele-
data shown in Fig. 5. The values of the pa- ment of ZFit
rameters, obtained using the ZFit tool of EC-
Lab, are R1 = 0.049 Ω, L1 = 0.154 × 10−6 H, coth(τ j ω)γ/2
Q1 = 0.66 F sα−1 , α1 = 0.61, R2 = 0.0236 Ω, ZMg = R with γ = 1 − α,
(τ j ω)1−γ/2
R3 = L r = 0.057 Ω, τ 3 = (L2 r q)1/α = 2.25 s 1
−1 2
−1 −1/γ 1

and α3 = 0.89. R = cγ Lγ q , τ = c γ L2/γ q −1/γ

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 Application Note 43
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III Short-circuited transmission As an example a Nyquist impedance diagram

lines ZL = 0 of a Fe(II)/Fe(III) system is shown in Fig. 8.

III.1 Short-circuited URC

Fig. 7: L-long short-circuited uniform dis-

tributed RC (URC) transmission line.

Using Eq. (2), the impedance of the short- Fig. 8: Nyquist impedance diagram of a
circuited transmission line made of r and c ele- Fe(II)/Fe(III) system in basic medium.
ments (Fig. 7) is given by
The Randles circuit R1+Q2/(R2+W2), contain-
1
χ = r, ζ = ⇒ ing a Warburg element, is chosen to fit the data
cj ω
√ shown in Fig. 8. The values of the parame-
th (L r c j ω) ters for equivalent circuit are R1 = 47.57 Ω,
Z=r √ (4)
rcj ω Q2 = 17.09 × 10−6 F sα−1 , α = √ 0.885, R2 =
70.94 Ω and σ2 = 85.33 Ω s −1/2 ⇒ r/c = 42.7
This impedance is similar to that of the Wd ele-
Ω s−1/2 .
ment of ZFit
√
th τd j ω
ZWd = Rd √ , Rd = L r, τd = L2 r c IV.2 Semi-infinite URRC
τd j ω

First of all, let us calculate the impedance of

IV Semi-infinite transmission the L-long URRC transmission line (Fig. 9) cor-
lines L → ∞ responding to diffusion-reaction and diffusion-
trapping impedance [7]:
IV.1 Semi-infinite URC
r1
The impedance of the semi-infinite transmis-
sion line shown in Fig. 1 is obtained making c
r2
L → ∞ in Eq. (4).
√ √ L
th (L r c j ω) r
L→∞⇒Z =r √ ≈ √
rcj ω cj ω
Fig. 9: L-long short-circuited uniform dis-
tributed RRC (URRC) transmission line.
This expression is similar to that of the Warburg
(W) element of ZFit
√
2σ r r2
ZW = √ with σ = √ χ = r1 , ζ = ⇒
jω 2 c 1 + r2 c j ω

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 Application Note 43
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( √ )
r1
th L (1 + r2 c j ω)
√ r2 r1
Z = r 1 r2 √
1 + r2 c j ω
q
r2

r1 ¥

c
r2 Fig. 12: Semi-infinite short-circuited uniform
distributed RRQ (URRQ) transmission line.

¥
and
√
Fig. 10: Semi-infinite short-circuited uniform r1 r 2
L→∞⇒Z≈ √
distributed RRC (URRC) transmission line. 1 + r2 q (j ω)α

With L → ∞ it is obtained [8]: This expression is similar to that of the Ga ele-

ment of ZFit
√
r1 r2
L→∞⇒Z≈ √ R √
1 + r2 c j ω ZGa = √ , R= r1 r2 , τ = r2 q
1 + τ (j ω)α
This expression is similar to that of the
Gerischer element G of ZFit [9]:
V Conclusion
RG √
ZG = √ , R G = r1 r2 , τ G = r2 c Seven elements, W, Wd, M, Ma, Mg, G and
1 + τG j ω
Ga, available in ZFit, can be used to represent
the impedance of seven different transmission
IV.3 Semi-infinite URRQ lines, as summarized in the table below (Tabs.
1 (p. 4), 2 (p. 6)).
r1

q
r2
Table 1: Summary table.

L Transmission line ZFit Element

URC M
Fig. 11: L-long short-circuited uniform dis- Open Circuited URQ Ma
tributed RRQ (URRQ) transmission line. UQC Mg
Short circuited URC Wd
Replacing c elements by q elements URC W
Semi-∞ URRC G
r2
χ = r1 , ζ = ⇒ URRQ Ga
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

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 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). [7] J.-P. D IARD and C. M ONTELLA, J. Electroanal.
Chem. 557, 19 (2003).
[2] M. S LUYTERS -R EHBACH, Pure & Appl. Chem.
66, 1831 (1994). [8] B. A. B OUKAMP and H. J.-M. B OUWMEESTER,
[3] J. B ISQUERT, Phys. Chem. Chem. Phys. 2, Solid State Ionics 157, 29 (2003).
4185 (2000). [9] H. G ERISCHER, Z. Physik. Chem. (Leipzig) 198,
[4] G. C. T EMES and J. W. L A PATRA, Introduction 286 (1951).
to Circuits Synthesis and Design, McGraw-Hill,
New-York, 1977. Nicolas Murer, Ph. D.,
[5] J.-P. D IARD, B. L E G ORREC, and C. M ON - Aymeric Pellissier, Ph. D.,
TELLA , J. Electroanal. Chem. 471, 126 (1999). Jean-Paul Diard, Pr.

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 Application Note 43
11062012

Table 2: ZFit elements vs. transmission lines.

ZFit element Equations Transmission line

√
coth τd j ω r
Rd √
M τd j ω c
Rd = L r, τd = L2 r c
L

coth(τ j ω)α/2 r
R
Ma (τ j ω)α/2
R = Lr q

τ = (L2 r q)1/α L

q
coth(τ j ω)γ/2
R
(τ j ω)1−γ/2
Mg 1
−1 2 −1
R = c γ L γ q −1/γ c
1
τ = c γ L2/γ q −1/γ L

√
th τ j ω
Rd √ d r
Wd τd j ω
Rd = L r c

τ d = L2 r c L

2σ
√ r
j√
ω
W r c
σ= √
2 c ¥

r1
RG
√
1 +√
τG j ω c
G R G = r1 r2 r2

τG = r 2 c
¥

r1
RG
√
Ga 1 + τG√(j ω)α q
R G = r1 r2 r2

τ G = r2 q
¥

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