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