From Full to Empty: The Mathematics of Battery Drain

Abstract

Smartphone battery life exhibits substantial variability under seemingly similar usage conditions, creating persistent challenges for both end users who require reliable runtime estimates and manufacturers pursuing energy efficiency optimization. This study develops a comprehensive analytical framework for battery discharge dynamics through multi-component power decomposition, temperature-dependent capacity modeling, and aging-aware state estimation. The total power consumption is decomposed into screen, processor, network, GPS, and background components, with empirical evidence demonstrating that OLED display panels constitute the single largest power consumer in modern smartphone systems, while connectivity modules represent another primary contributor. Temperature effects are characterized through an Arrhenius-type relationship, as established by experimental investigations confirming that lithium-ion battery capacity correlates with absolute temperature following the Arrhenius formulation. Battery aging is modeled via square-root-of-time kinetics, a widely adopted semi-empirical approach grounded in the diffusion-limited growth of the solid electrolyte interphase layer. The framework enables systematic prediction of State of Charge trajectories and Time-to-Empty across diverse usage scenarios. Sensitivity analysis identifies processor-related workload as the dominant contributor to endurance variation, with normalized importance scores reaching approximately 40 percent. Practical recommendations are derived for both end users and operating system developers, prioritizing interventions that yield the greatest improvements in battery longevity. The proposed framework offers a unified analytical approach to battery performance characterization and is generalizable to other portable electronic devices through component parameter adjustment.