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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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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