Zitao Ni

Ph.D. in Materials Science and Engineering

Study Notes · Part 4: Full SPM Implementation & Voltage Coupling

Chapter 4: Full SPM Implementation & Voltage Coupling

Connecting OCV, Diffusion, and BV into a complete SPM engine — understanding step() dataflow, voltage coupling, discharge curves, and C-rate analysis.

Learning Materials

Resource Description
Main Learning Notes Complete Q&A-style study notes covering all 4 sessions
Session Background Course background & learning plan for starting a new conversation
Learning Plan Snapshot Chapter 4 → Chapter 5 progress snapshot & preview

Four-Session Structure

Session Topic Core Activity Output
4.1 step() Four-Step Dataflow Hand-derive current→flux→overpotential→diffusion→voltage step_dataflow.png
4.2 Voltage Coupling & Code Practice Expand voltage formula → write spm_my.py from skeleton → validate spm_my.py (0 nV diff)
4.3 Discharge Curves & C-rate Voltage breakdown chart + C-rate comparison chart spm_discharge.png
4.4 Fully-Implicit Coupling Derive why diffusion is independent of η → intra-step iteration unnecessary Key insight

Key Concepts Covered

  • step() Four-Step Dataflow: $i_{app} \rightarrow j \rightarrow \eta \rightarrow c(r) \rightarrow V_{cell}$
  • Voltage Coupling: $V_{cell} = U_p + \eta_p - U_n - \eta_n = (U_p-U_n) - (|\eta_p|+|\eta_n|)$
  • Sign Convention: $j_n = -i_{app}/(a_{s,n} L_n F)$, $j_p = +i_{app}/(a_{s,p} L_p F)$
  • Cross-Step Coupling: Diffusion does not depend on η within a step; coupling occurs across time steps
  • C-rate Effect: arcsinh is sub-linear; higher C-rate → larger overpotential share of voltage drop
  • Initial Concentration: At t=0, concentration is uniform throughout the particle (diffusion steady state)
Figure Description
step_dataflow.png SPM four-step dataflow diagram
spm_discharge.png 4-subplot: discharge curve + OCV evolution + η evolution + comparison
crate_comparison.png 4-subplot: multi C-rate curves + η contribution + η detail + bar chart

All SPM Equations Assembled

Equation Formula Source
OCV $U(\theta) = f(\mathrm{SOC})$ Ch1
Diffusion $\partial c / \partial t = D \cdot \nabla^2 c$ Ch2
Butler-Volmer $j = j_0 \cdot [e^{\alpha F\eta/RT} - e^{-(1-\alpha)F\eta/RT}]$ Ch3
Voltage Coupling $V_{cell} = U_p + \eta_p - U_n - \eta_n$ Ch4

Python Components Built

Component Purpose Status
SPMModelMy.__init__() Initialize grid, concentrations, diffusion matrices ✅ Self-written
step() Advance dt seconds through four steps ✅ Self-written
simulate() Time loop + result recording ✅ Self-written
build_diffusion_matrix() Diffusion matrix construction (Ch2 component) ✅
solve_one_step() One-step diffusion (Ch2 component) ✅
bv_overpotential() BV inversion (Ch3 component) ✅

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