Werner Strunz, Zahner-elektrik 7. Constant Phase Element [ CPE ] W. Strunz, IWIS 2015: Check of...

Werner Strunz, Zahner-elektrik

www.zahner.de

Outline

W. Strunz, IWIS 2015: Check of Causality of Measured EIS and Modeling …

1. Resistor [ R ]


0 1 2 3 4 5 6

2

Spannung

Strom

2. Inductance [ L ]


0 1 2 3 4 5 6

2

Spannung

Strom

3. Capacitor [ C ]


0 1 2 3 4 5 6

2

Spannung

Strom

4. Warburg Impedance [ W ]


0 1 2 3 4 5 6

2

Spannung

Strom

4. Warburg Impedance [ C ] - Example


?

5. Nernst-Diffusion[ N ]


5. Nernst-Diffusion[ N ] - Example (FC)


6. Finite Diffusion [ FD ]


6. Finite Diffusion [ FD ] - Example


7. Constant Phase Element [ CPE ]


jYZCPE

11

0

1 100 10K

10

100

1m

10m

100m

|Z| /

0

15

30

45

60

75

90

|phase| /

o

frequency / Hz

-0.05 0 0.05 0.1 0.15

0

-0.2

-0.25

-0.1

-0.15

-0.05

Z' /

Z''

/

CPE – Flexible Element


Impedance of an Electrolyte Capacitor

HF : Z ~ (like L)

MF : Z ~ const (like R)

LF : Z ~ 1 / (like C) j > -90 ° => CPE

100m 1 3 10 30 100 1K 3K 10K 100K

100m

1

10

100

1K

|Z| /

0

30

60

90

|phase| / o

frequency / Hz

Constant Phase Element [ CPE ]


Constant Phase Element [ CPE ]


a = - 1

Z w( ) = 1

C⋅

1

j ⋅w

a = - 0,5

Z w( ) = "W"⋅1

j ⋅w( )

a = 0

Z w( ) = R

a = + 1

Z w( ) = L⋅ j ⋅w

Capacitor Diffusion (Warburg) Ohmic Inductance

0 > a ≥ - 1 : Z w( ) = 1

Yo

⋅1

j⋅w( )a

[YO] = f ()

CPE – a frequency dependent capacity

1

0 logloglog YC0YC

norm

norm

Without precaution

f norm = (2)-1

(natural = mathematical

normalization)


Normalization of CPE (R || CPE !)

K.S. Cole, R.H. Cole; J. Chem. Phys. 9 (1941) 341–352

K.S. Cole, R.H. Cole; J. Chem. Phys. 10 (1942) 98-105

G. J. Brug, A. L. G. van den Eeden, M. Sluyters-Rehbach, J. H. Sluyters; Journal of Electroanalytic Chemistry 176 (1984) 275-295

C.H. Hsu, F. Mansfeld; Corrosion 57/ No. 9 (2001) 747-748

M.R. Shoar Abouzari, F. Berkemeier, G. Schmitz, D. Wilmer; Solid State Ionics 180 (2009) 922–927

B. Hirschorn, M. Orazem, B. Tribollet, V. Vivier, I. Frateur, M. Musiani; El. Acta 55 (2010) 6218–6227


Constant Phase Element - Normalization


CPEC ZYC

Z

1111

0

0YC

0YCnorm

norm

OR

Normalization of CPE (R || CPE) ! - simplified derivation

Normalized capacity is independent of the exponent

11

0||

||

1

1

YRC

YRwithj

RZ

jCR

RZ

CPER

CR


Constant Phase Element [ R||CPE ]


Z w( ) = R⋅1

1 + RY0 ⋅ jw( ) a = R⋅ ∫

- ∞

∞1

1 + RC⋅ jw⋅G t( )dt

with ∫- ∞

∞

G t( )dt = 1

Constant Phase Element [ CPE ] Distribution Function (G())


G t( ) =1

2p⋅

sin p⋅ 1 - a( )( )

cosh a⋅x( ) - cos p⋅ 1 - a( )( )

with x = lnt

t0

Constant Phase Element [ CPE ] Distribution Function (G())


G t( ) =1

2p⋅

sin p⋅ 1 - a( )( )

cosh a⋅x( ) - cos p⋅ 1 - a( )( )

with x = lnt

t0





Rn: equidistand spacing in log(/0)

Validation of Spectra


Problems of Daily Life


EIS-Principle at a Single Frequency

How to validate EIS-spectra ?

What‘s the specific property ?

0 500µ 1m 1.5m 2m 2.5m 3m-500

-400

-300

-200

-100

0

100

200

300

400

500Excitation at constant frequency

time /s

fun

ctio

ns

E(t

) &

I(t)

Z

j


Motivation

Fuel cells

Batteries

Rechargeable batteries

Solar cells

Coatings

Development and/or improvement of important technical products

Under load

Under illumination

Water uptake

NON-STATIONARY

CONDITIONS

(may) result in

NON-STATIONARY

SPECTRA


10m 100m 1 3 10 100 1K 10K 100K

10

20

15

30

25

|Z| / m

-30

0

30

60

phase / o

frequency / Hz

artifact

Reliable detection of artifacts


Motivation : what we need

The Kramers-Kronig Relations

0

2

0

20

)(Im

2 - (0)Re=)(Re

d

HPVHH

0

2

0

200

)(Re2 =)(Im

d

HPVH

BUT WHERE ARE THE PROBLEMS ?


The Limited Bandwidth Problem

• Simulation of a coating during water up-take

• Measured frequency range 100 KHz – 50 mHz

• 0 : ?

• : ?


The measurement model


-2 -1 0 1 2 3 4

-2.6

-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2

0

15

30

45

60

75

90

Simulation

log frequency / Hz

log

im

pe

da

nc

e Z

/ O

hm

|Ph

as

e a

ng

le| / °

Measurement Model


Measurement Model

- Drawback: “RC“ is not linear! Z RC ( ) =

R

1+RC⋅

Z RC ( ) =R

1+RC⋅= Z real ( ) + Z imag ( ) =

R

1+ RC⋅( )2− j⋅

R2C⋅

1+ RC⋅( )2

Z RC ( ) =R

1+⋅= Z real ( ) + Z imag ( ) =

R

1+ ⋅( )2− j⋅

R⋅⋅

1+ ⋅( )2

Complex number ⇒ Z = a + j⋅b

Complex Matrix Entry ⇒Zreal

Z imag

=a −b

b a


Measurement Model

- Solution: “RC“ Replacement :“RC=“


Z RC ( ) =R

1+RC⋅

Z RC ( ) =R

1+RC⋅= Z real ( ) + Z imag ( ) =

R

1+ RC⋅( )2− j⋅

R2C⋅

1+ RC⋅( )2

Z RC ( ) =R

1+⋅= Z real ( ) + Z imag ( ) =

R

1+ ⋅( )2− j⋅

R⋅⋅

1+ ⋅( )2



Z imag

=a −b

b a

Linear-KK Check


Z RC ( ) =R

1+RC⋅

Z RC ( ) =R

1+RC⋅= Z real ( ) + Z imag ( ) =

R

1+ RC⋅( )2− j⋅

R2C⋅

1+ RC⋅( )2

Z RC ( ) =R

1+⋅= Z real ( ) + Z imag ( ) =

R

1+ ⋅( )2− j⋅

R⋅⋅

1+ ⋅( )2



Z imag

=a −b

b a


Linear-KK Tool from KIT Karlsruhe

Linear-KK – Battery (I)


Linear-KK – Supercap (Sub m-Range)


Linear-KK – Coating (Huge-Z-Range)


Linear-KK – Battery (II)


Drift in Batteries


2,0 2,2 2,4 2,6 2,8 3,0 3,2 3,4 3,6 3,8 4,0 4,23,20

3,21

3,22

3,23

3,24

3,25

3,26 Discharge 4A (0.1 C)

Relaxation 0A

Battery

voltage / V

time of experiment / h

Measurement time ~ 6 h

The Z-HIT Approximation (evaluation of impedance modulus from the phase angle

jj

ln

)(ln)(

2 + onst. )(ln

0

0d

ddcH

O

S

Detection of artifacts

Detection of instationarities (drift)

History (time) preserving

Reconstruction of causal spectra

=> Reliable interpretation of spectra


Validation of Spectra – Z-HIT


0 2 4 6 8 10 12 1411

12

13

14

15

16

ln Z

ln

-1,4

-1,2

-1,0

-0,8

-0,6

-0,4

-0,2

0,0(O)

E

S

Phase / rad

0 2 4 6 8 10 12 1411

12

13

14

15

16

Integral

(shifted)

ln Z

0 2 4 6 8 10 12 14-0,7

-0,6

-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

[C] ln Z - Integral (j)

dj / dln

0 2 4 6 8 10 12 14-0,7

-0,6

-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

[D]

[B][A]

ln Z - Integral (j)

/6 * dj / d ln

Deduction of the Z-HIT

Randle

circuit


The Sensitivity of Objects (Z & j) - Excellent Examples: Sensors !

- Temperature Dependent Resistor (NTC, PTC)

Pt 100, Pt 1000, KTY 81, …

- Light Dependent Resistor (LDR)

- Magnetic Dependent Resistor (MDR)

- Humidity Dependent Capacity

- …..


The Course of Phase and Impedance when Heating NTC/PTC

Z & j : Phase j is more stable than impedance Z

1 2 3 4 53,4

3,5

3,6

3,7

3,8

3,9

4,0

0

15

30

45

60

75

90

KTY 10k (NTC)

LO

G Im

ped

ance

/

LOG frequency / Hz

|Ph

ase| / °

0 1 2 3 4 5

3,05

3,10

3,15

0

15

30

45

60

75

90

Pt 1000 (PTC)

LO

G Im

ped

ance

/

LOG frequency / Hz

|Ph

ase| / °


onst.ln

)(ln)(

2 )(ln

0

0 cd

ddH

O

S

jj

Implementation of the Z-HIT

1) The experimental data are filtered by a smoothing algorithm. The result is a set of continuous samples equidistant in log f.

2) The integral term is calculated by numerical integration.

3) The first derivate is taken from the smoothing function.

4) The integration constant is determined by a least squares fit.


Battery under Load - Mutual Inductance & Drift

High-frequency Data (inductance) With kind permission of R. Gross, bno-consult, Dettelbach


Z-HIT Examples


Battery (I)

Z-HIT Examples


Supercap

Z-HIT Examples


Battery (II)

Restriction (2-Gate) Z-HIT

What is History (Time) Preserving?

Considering Kramers Kronig relations

0

2

0

200

)(Re2 =)(Im

d

HPVH

jj

ln

)(ln)(

2 + onst. )(ln

0

0d

ddcH

O

S


Integral-Term preserved

integration along the frequency axis leads

to “weighting“ (measuring time)

History (Time) Preserving


Randle circuit with NTC as Charge Transfer Resistance




Only Smoothing Z-HIT refinement




Only Smoothing Z-HIT refinement

Dangerous: expanding the model

without physical justification

Series measurement

Water Uptake - Waterborne Coating

With kind permission of U. Christ, Fraunhofer-Institut für Produktionstechnik und Automatisierung IPA, Stuttgart

1st..4th spectrum

5th spectrum 1 3 10 30 100 1K 3K 10K 100K

1K

10K

100K

1M

10M

100M

1G

|Z| /

0

30

60

90

|phase| / o

frequency / Hz

a a a aaaaaaaaaaaaaaa

aaaaaa

a

a

a

a

a

a

a

a

a

a

a

a

aaaaaaaaaaaaa a

a a


Water Uptake - Waterborne Coating

With kind permission of U. Christ, Fraunhofer-Institut für Produktionstechnik und Automatisierung IPA, Stuttgart


Z-HIT: Estimate of Accuracy (I)

= 1 = 0.5


Z-HIT: Estimate of Accuracy (II)

100.

SIM

HITZSIM

Z

ZZErrorrel


= 1 = 0.5


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Werner Strunz, Zahner-elektrik 7. Constant Phase Element [ CPE ] W. Strunz, IWIS 2015: Check of...

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