Capacity after n cycles: 1000 × (0.98)^n - High Altitude Science

Understanding the Context

What Does This Function Represent?

The formula 1000 × (0.98)ⁿ describes a 1000-unit initial capacity that decays by 2% per cycle. Because 0.98 is equivalent to 1 minus 0.02, this exponential function captures how system performance diminishes gradually but steadily over time.

Why Use Exponential Decay for Capacity?

Key Insights

Real-world components often experience slow degradation due to physical, chemical, or mechanical wear. For example:

• Lithium-ion batteries lose capacity over repeated charging cycles, typically around 2–3% per cycle initially.
  
• Hard disk drives and electronic memory degrade gradually under reading/writing stress.
  
• Software/RDBMS systems may lose efficiency or data retention accuracy over time due to entropy and maintenance lag.

The exponential model reflects a natural assumption: the rate of loss depends on the current capacity, not a fixed amount—meaning older components retain more than new ones, aligning with observed behavior.

How Capacity Diminishes: A Closer Look

Final Thoughts

Let’s analyze this mathematically.

• Starting at n = 0:
  Capacity = 1000 × (0.98)⁰ = 1000 units — full original performance.
  
• After 1 cycle (n = 1):
  Capacity = 1000 × 0.98 = 980 units — a 2% drop.
  
• After 10 cycles (n = 10):
  Capacity = 1000 × (0.98)¹⁰ ≈ 817.07 units.
  
• After 100 cycles (n = 100):
  Capacity = 1000 × (0.98)¹⁰⁰ ≈ 133.63 units — over 25% lost.
  
• After 500 cycles (n = 500):
  Capacity ≈ 1000 × (0.98)⁵⁰⁰ ≈ 3.17 units—almost depleted.

This trajectory illustrates aggressive yet realistic degradation, appropriate for long-term planning.

Practical Applications

1. Battery Life Forecasting
   Engineers use this formula to estimate battery health after repeated cycles, enabling accurate lifespan predictions and warranty assessments.