Passive Self-Homogenization in Series-Connected Battery Packs: A Structural Theorem, a First-Order Saturation Law, and Linearized Stability

Problem. Cell-to-cell internal-resistance dispersion is amplified at sub-zero temperatures and is widely regarded as the root cause of pack-level capacity loss and lifetime divergence. Countermeasures converge on one idea: eliminate the dispersion—active balancing, external preheating, and stringent dual-parameter sorting (capacity and resistance). A reversed proposition has been largely overlooked: in a series string, where current is common, resistance dispersion itself drives a passive self-homogenization mechanismthe highest-resistance cell dissipates the most Joule heat, warms the most, and its resistance drops the most, so it “catches up” with its lower-resistance neighbours. Challenge. All existing evidence for this mechanism is phenomenological simulation. Three fundamental questions lack analytical answers: Is the observed compression a physical necessity or a parameter coincidence? How large is it, and is there a simple scaling law? Does it converge in a real pack with inter-cell thermal diffusion? Contributions. Under three explicit assumptions we establish: (i) a self-homogenization theorem—the operating relative resistance dispersion is compressed by 1/(1+Λ) ∈ (0,1], whose sign is locked solely by 𝜕R/𝜕T<0 and is therefore independent of parameter values and implementation; (ii) a first-order saturation law flat% = Λ/(1+Λ)—exact in the pure-resistance channel (zero capacity dispersion), predicting the equalization magnitude from the single parameter Λ = 𝛽ꞏΔT̄ (15.2% predicted vs 16.2% simulated at the baseline cell, Δ = 1.0 pp), together with a two-channel superposition formula for the general case and an honest disclosure of the quantitative gap between the first-order estimate and simulation (8.6% vs 16.2%, attributed to the convexity of the cut-off map); (iii) asymptotic stability within the linearized modelwith inter-cell thermal diffusion the coupled thermo-electric ODE has a diagonally dominant Jacobian with all-negative eigenvalues, so self-equalization is an eigenstate of the linear system. The theory further yields an analytic left-boundary criterion for the operating sweet spot (ΔT̄≥ 0.176/𝛽) and explains why sodium-ion cells are the largest beneficiaries.