Sodium-Ion Battery: Hard-Carbon Anode Voltage Curve

Cell Operation

Idle at 50% state of charge

C-rate (charge/discharge current) 1.0C

Cycles aged (capacity fade) 1

Live Readouts

Terminal voltage

2.87 V

State of charge

50%

Capacity retention

99.9%

Anode potential

0.34 V vs Na

How it works

The cell voltage is the difference between two independent open-circuit potentials, each a function of the shared state of charge x (0 = empty, 1 = full), minus the resistive drop from the applied current:

V_cathode(x) = 3.00 + 0.90(x-0.5) - 0.05 ln[(x+0.02)/(1.02-x)]
V_anode(x)   = 0.08 + 0.50 e^(-6x) + 0.03 ln[(1.03-x)/(x+0.03)]
V_cell       = V_cathode(x) - V_anode(x) - I_eff * R_int

The layered-oxide cathode term is a smooth, mildly sloping Nernst-like curve. The hard-carbon anode term is deliberately two-part: an exponential "sloping" region at low Na content that collapses toward a near-flat, near-zero-volt plateau as x → 1 — the real, well-documented signature of hard carbon and the reason Na-ion cells must be charged carefully near full state of charge to avoid metallic Na plating on that plateau.

Charge / Discharge — sets the current direction; Na⁺ ions visibly shuttle from the cathode lattice into the hard-carbon anode on charge, and back on discharge.
C-rate — current magnitude. Higher C-rate moves the sweep front faster and increases the I·R_int overpotential, pulling the terminal voltage away from the open-circuit curve (up on charge, down on discharge).
Cycles aged — capacity retention follows SOH = 100 e^(-cycles/1400), a simple exponential fade; as SOH drops, the accessible state-of-charge window shrinks so the cell can no longer be pushed to a full 100%.

This is a simplified, illustrative two-electrode intercalation model (the same style of construction used in porous-electrode battery models), not a fit to a specific manufacturer's cell.

Na⁺ ion (mobile) Hard-carbon anode Layered-oxide cathode

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