Electrochemical Energy Storage & EV Powertrains
Battery Ragone Specific Energy & Peak Power Formula
Calculate Matched-Load Peak Specific Power P_max = V_oc^2 / (4 * R_int * Mass), Ragone Voltage Efficiency (1 + SQRT(1 - P_load / P_max)) / 2, Delivered Specific Energy, and High-Power Runtime.
Derive the exact Ragone energy-vs-power knee from open-circuit voltage and internal resistance!
David Ragone (1968) introduced the double-logarithmic Specific Energy (Wh/kg) vs Specific Power (W/kg) chart comparing EV cells, fuel cells, and ultracapacitors, and T. Christen & M.W. Carlen (2000, Journal of Power Sources) solved its exact Thevenin equivalent circuit! By Jacobi’s maximum power transfer theorem, a cell of mass M_kg, open-circuit voltage V_oc, and internal DC resistance R_int delivers Peak Matched Specific Power P_max = V_oc^2 / (4 * R_int * M_kg) (at V_term = 0.5 * V_oc, where 50% of stored energy burns internally as I^2 * R_int heat!). At any requested gravimetric power P_load <= P_max, solving P_load * M_kg = (V_oc - I * R_int) * I yields Ragone Discharge Efficiency eta_Ragone = (1 + SQRT(1 - P_load / P_max)) / 2 and Delivered Specific Energy E_del = E_0 * eta_Ragone!
Verify on High-Power EV Pouch Cell—Open_Circuit_Voltage V_oc = 4.00 V (B2), Internal_DCIR R_int = 0.0040 Ohm (4.0 mOhm, B3), Cell_Mass M_kg = 0.40 kg (B4 -> Peak Specific Power P_max = 16 / (4 * 0.004 * 0.40) = 2,500.00 W/kg in E2), Intrinsic_Energy E_0 = 250.0 Wh/kg (B5), and Requested_Track_Power P_load = 900.0 W/kg (P_load / P_max = 0.36, B6): Ragone Efficiency is (1 + SQRT(1 - 0.36)) / 2 = (1 + 0.80) / 2 = 0.9000 (90.00% in E3); Delivered Specific Energy is 250 * 0.90 = 225.00 Wh/kg (E4); and Full-Power Discharge Runtime is 60 * (225 / 900) = 15.00 minutes (E5)!
| Ragone Cell Metric (Voc=4.0V, Rint=4mOhm, Mass=0.40kg, E0=250Wh/kg, Pload=900W/kg) | Formula | Verified Output |
|---|---|---|
| Peak Matched Specific Power P_max | V_oc^2 / (4 * R_int * Mass_kg) | 2,500.00 W/kg |
| Ragone Voltage Efficiency | (1 + SQRT(1 - P_load / P_max)) / 2 | 0.9000 (0.5000 @ P_load = P_max) |
| Delivered Specific Energy (Wh/kg) | E_0_Whkg * Ragone_Efficiency | 225.00 Wh/kg (125.00 @ P_max) |
| Constant-Power Runtime (min) | 60 * (Delivered_Whkg / P_load_Wkg) | 15.00 minutes (3.00 min @ P_max) |
Christen-Carlen Ragone Delivered Specific Energy Formula
Guard P_load > P_max ("Voltage Collapse (P_load > P_max)"—requesting power above V_oc^2 / (4*R_int) drives the quadratic discriminant negative and collapses terminal voltage!)!
=LET(v_oc, B2, r_in, B3, ms_k, B4, e_zr, B5, p_ld, B6, IF(OR(v_oc <= 0, r_in <= 0, ms_k <= 0, e_zr <= 0, p_ld <= 0), "Invalid Cell Spec", LET(p_mx, (v_oc ^ 2) / (4 * r_in * ms_k), IF(p_ld > p_mx, "Voltage Collapse (P_load > P_max)", LET(et_r, (1 + SQRT(1 - p_ld / p_mx)) / 2, ROUND(e_zr * et_r, 2))))))Matched-Impedance Knee Test (P_load = P_max = 2,500 W/kg)
Set Requested_Power B6 = 2,500 W/kg: Ragone Efficiency drops to 0.5000, Delivered Energy halves to 125.00 Wh/kg, and Runtime hits 3.00 minutes!
=IF(ISTEXT(E4), E4, ROUND(60 * (E4 / B6), 2))Try the example
Use the original values or change them to test your understanding.