Power Electronics, Ultracapacitor Energy Storage & Pulse-Power Sizing
EDLC Supercapacitor 75% Usable Energy & ESR Drop Formula
Calculate Electric Double-Layer Supercapacitor (EDLC) Usable Extractable Energy (Wh) Between V_max and Converter Cutoff V_min, Usable Energy Fraction (75% at 0.5*V_max), Instantaneous ESR Voltage Sag (V), and Matched-Impedance Peak Power (kW).
Never size supercapacitor banks with 0.5*C*V²—apply the 50%-voltage 75%-usable-energy rule!
Unlike Li-ion batteries whose terminal voltage stays nearly flat across discharge, an Electric Double-Layer Supercapacitor (EDLC) voltage drops linearly with charge (V = Q / C)! Because a downstream DC/DC converter’s input current I = P / V doubles when capacitor voltage falls to V_min = 0.5 * V_max, power-electronic converters cut off at half rated voltage—leaving (0.5)^2 = 25% of total stored energy stranded and delivering Usable Energy E_use (Wh) = (0.5 * C * (V_max^2 - V_min^2)) / 3600 (exactly 75% of 0.5*C*V_max^2)! Engineers must also check Instantaneous Step-Current Voltage Sag ΔV_ESR = I_pulse * ESR and Theoretical Matched-Load Peak Power P_max (kW) = V_max^2 / (4 * ESR * 1000) (note USABC 95%-efficiency pulse power is 0.12 * V_max^2 / ESR)!
Verify on a 48 V Hybrid-Bus Regenerative Braking Module—Bank_Capacitance = 500 F (B2), Rated_Max_Voltage = 48.0 V (total stored energy = 160.00 Wh, B3), Converter_Cutoff_Vmin = 24.0 V (B4 -> Usable Extractable Energy = 120.00 Wh in E2, or 0.7500 = 75.00% in E3), Module_ESR = 0.012 Ω (12 mΩ, B5), and Peak_Pulse_Current = 250 A (B6): Instantaneous ESR Voltage Sag is 250 * 0.012 = 3.00 V (E4), and Matched-Impedance Peak Power reaches 48^2 / (4 * 0.012 * 1000) = 48.00 kW (E5)!
| EDLC Supercapacitor Metric (500 F, 48 V -> 24 V Cutoff, 12 mΩ ESR, 250 A Pulse) | Formula | Verified Output |
|---|---|---|
| Usable Extractable Energy (Wh) | (0.5 * C * (V_max^2 - V_min^2)) / 3600 | 120.00 Wh (81.60 Wh @ 33.6 V) |
| Usable Energy Fraction | 1 - (V_min / V_max)^2 | 0.7500 (0.5100 @ 33.6 V Min) |
| Instantaneous ESR Sag (V) | Peak_Pulse_Current_A * ESR_Ohm | 3.00 V |
| Matched-Load Max Power (kW) | (V_max^2) / (4 * ESR_Ohm * 1000) | 48.00 kW |
Supercapacitor Usable Energy Window Formula (Wh)
Validate Capacitance > 0, 0 <= V_min < V_max, ESR > 0, and Pulse_Current >= 0!
=LET(cp_f, B2, vm_x, B3, vm_n, B4, es_r, B5, ip_a, B6, IF(OR(cp_f <= 0, vm_x <= 0, vm_n < 0, vm_n >= vm_x, es_r <= 0, ip_a < 0), "Invalid EDLC Bank Input", ROUND((0.5 * cp_f * ((vm_x ^ 2) - (vm_n ^ 2))) / 3600, 2)))Narrow 70% Voltage Floor Check (Set Converter Cutoff V_min = 33.6 V)
If a direct-coupled inverter trips at 70% of V_max (B4 = 33.6 V), Usable Energy Fraction E3 drops to 1 - 0.70^2 = 0.5100 (51.00%), cutting Usable Energy E2 from 120.00 Wh down to 81.60 Wh!
=IF(ISTEXT(E2), E2, ROUND(1 - ((B4 / B3) ^ 2), 4))Try the example
Use the original values or change them to test your understanding.