When designing high-power DC systems, understanding the charge on capacitors in series is the difference between a stable inverter and a tripped BMS. The direct answer to how charge behaves in a series capacitor string is absolute: the charge (Q, in Coulombs) on every capacitor in a series string is identical, regardless of their individual capacitance values. The total charge is calculated as $Q_{total} = C_{eq} \times V_{total}$, where $C_{eq}$ is the equivalent series capacitance. While voltage divides across the string, the physical charge stored on each dielectric plate remains perfectly matched.

In off-grid and hybrid power systems, we exploit this principle by wiring low-voltage supercapacitors in series to buffer high-voltage DC buses. This article bridges fundamental capacitor theory with practical energy storage sizing, showing you exactly how to calculate series charge, apply Peukert’s law to your battery bank, and select the right inverter for heavy surge loads.

The Physics: Charge on Capacitors in Series vs. Parallel

To build a reliable energy storage system, you must understand how series and parallel topologies alter voltage, capacity, and charge. The rules for capacitors are the exact inverse of resistors, and they differ fundamentally from chemical batteries.

Topology Capacitors (Supercaps) Chemical Batteries (LiFePO4)
Series Voltage ratings add. Equivalent capacitance drops ($1/C_{eq} = 1/C_1 + 1/C_2$). Charge (Coulombs) is identical on each cell. Voltage adds (e.g., 4x 12V = 48V). Amp-hours (Ah) remain identical. Total energy (Wh) adds.
Parallel Capacitance adds ($C_{eq} = C_1 + C_2$). Voltage rating remains limited by the lowest cell. Charge adds. Amp-hours add (e.g., 2x 100Ah = 200Ah). Voltage remains identical. Total energy (Wh) adds.
CRITICAL WARNING: Never parallel mismatched cells. Whether you are wiring LiFePO4 batteries or supercapacitors in parallel, mismatched internal resistance (ESR) or state-of-charge will cause massive cross-currents. A fully charged cell will dump its energy into a depleted cell at a rate limited only by the wiring resistance, leading to melted lugs, thermal runaway, or venting. Always parallel only identical, age-matched, and voltage-matched cells.

System Architecture: Source to Load Block Description

Let us define a concrete system block for a 48V off-grid well pump application. The power flows through four distinct stages:

  1. Source: Solar MPPT charge controller and grid-tied AC charger.
  2. Primary Storage: 48V nominal LiFePO4 battery bank (high energy density, handles continuous baseline loads).
  3. Surge Buffer (The Capacitor Bank): Supercapacitors wired in series to match the 48V bus, acting as a low-impedance shock absorber for inverter surges.
  4. Conversion & Load: 48V-to-120V Pure Sine Wave Inverter driving a 1.5 HP well pump (requiring a 4000W startup surge for 3 seconds).

Inverter/Charger Sizing: For a 1.5 HP pump (approx. 1200W running, 4000W starting surge), your inverter must be rated for at least 5000W peak. A Victron MultiPlus-II 48/5000 is the benchmark here, providing 5000VA (4000W continuous, 9000W peak for short durations). The built-in 120A charger is sufficient to replenish the bank at roughly C/2 for a 200Ah bank.

Sizing Math: Peukert, Efficiency, and Joule Requirements

Sizing a hybrid battery-capacitor bank requires two different mathematical models: one for the electrochemical battery (which suffers from voltage sag and capacity loss at high currents) and one for the electrostatic capacitor bank.

1. Sizing the Supercapacitor Bank (Joule Math)

We need the capacitor bank to supply 4000W for 3 seconds to clear the motor startup surge without dragging the battery voltage below the inverter's low-voltage disconnect (LVD).

  • Energy required ($E$) = Power $\times$ Time = $4000W \times 3s = 12,000$ Joules.
  • Assume the 48V bus is allowed to sag from 51.2V (fully charged) down to 44.0V during the surge.
  • Capacitor energy formula: $E = 0.5 \times C_{eq} \times (V_{high}^2 - V_{low}^2)$
  • $12,000 = 0.5 \times C_{eq} \times (51.2^2 - 44.0^2)$
  • $12,000 = 0.5 \times C_{eq} \times (2621.44 - 1936)$
  • $12,000 = 342.72 \times C_{eq} \implies C_{eq} = 35.0$ Farads.

If we use standard 16V, 150F supercapacitor cells, we must wire them in series to survive the 51.2V bus. Wiring 4 cells in series gives a maximum voltage rating of 64V. The equivalent capacitance of four 150F caps in series is $150 / 4 = 37.5F$. This satisfies our 35.0F requirement.

Calculating the Charge on Capacitors in Series:
At a nominal 48V, the total charge stored in this string is:
$Q = C_{eq} \times V = 37.5F \times 48V = 1,800$ Coulombs.
Because they are in series, each individual 150F capacitor holds exactly 1,800 Coulombs of charge. The voltage across each capacitor will be $V = Q / C = 1800 / 150 = 12.0V$, safely below the 16V absolute maximum rating.

2. Sizing the LiFePO4 Bank (Peukert & Efficiency)

Chemical batteries lose effective capacity when discharged at high rates. We model this using Peukert’s Law: $t = H \times (C / (I \times H))^k$.

  • Assume a 48V 200Ah LiFePO4 bank. Rated at the 20-hour rate ($H = 20$), so $C = 200Ah$.
  • Peukert exponent ($k$) for high-quality LiFePO4 is roughly 1.05 (compared to 1.3 for lead-acid).
  • If the inverter pulls a continuous 2000W load, accounting for 93% inverter efficiency, the DC draw is $2000 / (48 \times 0.93) = 44.8A$.
  • $t = 20 \times (200 / (44.8 \times 20))^{1.05} = 20 \times (0.223)^{1.05} = 20 \times 0.206 = 4.12$ hours.

Instead of the theoretical 4.46 hours ($200Ah / 44.8A$), Peukert losses reduce your actual runtime to 4.12 hours. To achieve a true 80% Depth-of-Discharge (DoD) usable runtime of 4 hours at this load, you must spec a 280Ah battery bank.

Charge/Discharge Limits, C-Rates, and Safety

Pushing components past their physical limits is where bench prototypes turn into fire hazards. You must respect the charge and discharge limits of both topologies.

Supercapacitor Balancing is Non-Negotiable: Because leakage currents vary slightly between manufacturing batches, the voltage across series capacitors will drift over time. One cell might creep to 17V while another drops to 10V, destroying the 17V cell. You must install passive balancing resistors across each cell. For 16V cells with 1mA leakage, use a 1.5kΩ, 1W resistor across each terminal to force a 10mA bleed current, dominating the leakage variance.
Parameter LiFePO4 Battery Limits Supercapacitor Limits
Max Discharge C-Rate 1C continuous, 3C for <10 seconds. Limited only by ESR and terminal thermal mass (often 50C+).
Max Charge C-Rate 0.5C standard (BMS usually limits to 100A). Limited by series resistance and wiring gauge.
Depth of Discharge (DoD) 80% to 90% for maximum cycle life. 100% (Down to 0V, though inverter LVD cuts it earlier).
Primary Failure Mode Thermal runaway, lithium plating if charged <0°C. Electrolyte venting if cell voltage exceeds 2.7V absolute max.
Lithium Fire-Safety Protocol: LiFePO4 is the safest lithium chemistry, but a short circuit can still cause catastrophic thermal runaway. Your battery bank must include a Class-T fuse or DC breaker rated for the inverter's peak fault current (e.g., 250A minimum for a 5000W 48V inverter) installed within 18 inches of the positive terminal. Furthermore, the BMS must have low-temperature charge cutoff to prevent lithium plating, which causes internal dendrites and subsequent short circuits. Always install LiFePO4 banks in a fire-rated enclosure or away from combustible structural members, adhering to NFPA 855 spacing requirements.

Decision Tree: Selecting Your Storage Topology

Do not guess your storage architecture. Use this decision matrix to determine whether you need pure batteries, pure supercapacitors, or a hybrid system based on your specific load profile.

IF your load profile is... AND your surge duration is... THEN choose this topology... Concrete Part Pick (48V System)
Continuous high draw (e.g., space heating, server rack) No surges > 2x running wattage Pure LiFePO4 Bank. Oversize Ah to handle the continuous C-rate without Peukert sag. 2x SOK 48V 100Ah Server Rack Batteries in parallel.
Micro-pulsing (e.g., spot welders, laser cutters) Milliseconds to 1 second Pure Supercapacitor Bank. Batteries cannot react fast enough; ESR will cause massive voltage sag. 16x Eaton/Vishay 16V 500F Supercaps (4S4P matrix).
Inductive motor starting (e.g., well pumps, compressors, table saws) 1 to 5 seconds at 3x-5x running wattage Hybrid (LiFePO4 + Series Supercaps). Caps absorb the inductive spike, saving the battery BMS from tripping on overcurrent. Victron MultiPlus-II 48/5000 + 4x 16V 150F Caps in series + 48V 200Ah LiFePO4.

The Default Recommendation

For the vast majority of off-grid and backup power builders dealing with standard household AC loads and well pumps, the Hybrid Topology is the definitive choice. Relying solely on a battery bank to handle 4000W inductive surges forces you to oversize the battery bank purely for its peak C-rate capability, wasting money on unused Amp-hours.

Your final bill of materials for a robust 48V surge-buffered system:
Procure a Victron MultiPlus-II 48/5000 inverter/charger. Wire it to a 48V 200Ah LiFePO4 battery bank (protected by a 250A Class-T fuse). On the DC bus terminals of the inverter, install a series string of four Eaton/Vishay 16V 150F supercapacitors, each bypassed by a 1.5kΩ 1W balancing resistor. This setup guarantees that the charge on capacitors in series will absorb the inductive kickback, keeping your battery BMS happy and your lights from flickering when the well pump kicks on.