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Previous class • Data Validation • Linear KKT • Equivalent Electrical Circuit (EEC) analysis • Choice of Voigt, Maxwell...

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Previous class • Data Validation • Linear KKT

• Equivalent Electrical Circuit (EEC) analysis • Choice of Voigt, Maxwell, Ladder • Identification of number of (Maxwell or Voigt or Ladder) pairs • Confidence intervals, AIC

Today • EEC fit • Initial values • Distinguishability • Element value  Physical phenomena

• Confidence in the parameter values

50

-Z

Im

/

Identify circuit

• How many elements 0 should we use? 0 • Example: Synthetic data with ‘noise’ Element Till C2

R1

R2 R3 R4

Re

/

Till C3

100

Till C4

R1 () 22.4 (< 3%) 19.9 ( < 1 %) 19.97 ( < 1 %)

C2

1

50 Z

2 C3 C4

R2 () 95.2 ( < 3 %) 100.6 ( < 1 %) 101 (< 4 %) C2 (mF) 16.6 (< 5 %) 10 (3 %)

10 (< 3 %)

R3 () --

52.2 (< 6 %) 52.1 ( < 6 %)

C3 (mF) --

9.8 ( < 3 %)

R4 () -C4 (mF) --

9.8 ( < 3 %) 23370 ( ~ 767 %) 7.5 ( ~ 1704 %)

Identify circuit

Im

/

50

R1

R2 R3

R4

Circuit Till C2 Till C3 Till C4

0 0

50 Z

Re

/

100

AIC  2 k  n ln  RSS 

C2 1

-Z

• How many elements should we use? • Example: Synthetic data with ‘noise’

2 C3

AICc  AIC 

C4

# of params (k) 3 5 7

‘n’ – number of observations ‘k’ – number of parameters

2k  k  1 n  k 1

Residuals

AIC

AICc

1196 76 76

296.6 187.3 191.5

297.2 189.0 194.9

• Residual sum square (RSS) • Akaike Information Criterion • Corrected AIC [ AICc]

Examples • Simple circuit fit (Randel01, Maxwell 01) • With inductors (Maxwell 02) • Complex data set analysis (Maxwell 03, Ex.NegRes.Ind.Loop.00) • Importance of initial guess (Maxwell 04, using LR and CR)

• Initial guess values – • • • •

Rsol = 0 , Cdl = 10-5 F R1 = -299  C2 = 1.3×10-2 F, R2 = 14.7  L3 = 0.63 H, R3 = 63 

R3   R1

Distinguishability • Following 4 circuits are identical • Degenerate R1

 R4 

L1

1

C3

R3 2

1

2

R2

R4

L8

C6 1

1

R5

R7

2

2

R6

R8

Fletcher, S. (1994). Tables of Degenerate Electrical Networks for Use in the Equivalent‐Circuit Analysis of Electrochemical Systems. Journal of the Electrochemical Society, 141(7), 1823-1826.

1

C3  

L1

R12

  R1    R2  1

1

R1   R3

L1  C3 R32

 R2 

1

1

  R3    R4 

1

Distinguishability

C2

• We can replace a part of a circuit

R1

1

C2

2 R6 R5

R2

R1

1

2 C6

R3

C3

C3

R3 1

C6 2

1

R5

2

R6 R4

Remaining elements will NOT be affected Demo