Nov 6, 2012 - This note will describe how to fit transmission lines using one equivalent circuit elements contained in ZFit. It is well known that the Warburg ...
Application Note 43 11062012
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.
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Application Note 43 11062012
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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Application Note 43 11062012
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
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Application Note 43 11062012
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
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ZFit Element M Ma Mg Wd W G Ga
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Application Note 43 11062012
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.
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Application Note 43 11062012
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
¥
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