The Physics of Charge in Capacitors in Series

When designing a DC energy storage system using ultracapacitors (EDLCs), you rarely use a single cell. Because individual supercapacitors are typically rated between 2.5V and 2.85V, you must wire them in series to reach usable system voltages like 12V, 24V, or 48V. This creates a unique electrical environment where the rules of energy storage diverge sharply from chemical batteries.

The fundamental rule of charge in capacitors in series is one of the most elegant symmetries in circuit theory: the total charge ($Q$, measured in Coulombs) stored in the entire series string is exactly equal to the charge stored on any single capacitor in that string.

💡 The Workbench Rule: If you wire six 350F, 2.85V capacitors in series, the total capacitance drops to 58.3F, and the total voltage rating rises to 17.1V. The total charge is $Q = C_{total} \times V_{total} = 58.3F \times 17.1V = 997$ Coulombs. Notice that a single cell holds $350F \times 2.85V = 997.5$ Coulombs. The charge doesn't multiply in series; it remains constant across the chain.

System Block Description: Source to Load

To integrate a series capacitor bank into a practical power system, you must buffer the wide voltage swing of the capacitors. A typical off-grid or backup topology looks like this:

  • Source: Solar array or wind turbine rectifier.
  • Charge Controller: MPPT or buck converter limiting max output to the bank's absolute max series voltage (e.g., 16.5V for a 6-cell string).
  • Storage Medium: Series ultracapacitor bank equipped with an active cell-balancing PCB.
  • Conditioning: Wide-input DC-DC buck/boost converter to stabilize the bus.
  • Inversion: Pure sine wave inverter converting the stabilized DC bus to 120V/240V AC for the load.

Sizing Math: Capacitor ESR vs. Battery Peukert Effect

When sizing chemical batteries for high-surge loads, engineers must apply Peukert’s Law, which dictates that a battery's effective capacity shrinks as the discharge current increases. The formula is $t = H \cdot (C/I)^k$, where $k$ (the Peukert exponent) is typically 1.1 to 1.3 for lead-acid and around 1.05 for lithium.

Capacitors do not suffer from the Peukert effect; their Peukert exponent is effectively 1.0. A 58.3F bank will deliver the exact same Coulombs at 10A as it will at 100A. However, capacitors are constrained by Equivalent Series Resistance (ESR) and efficiency factors.

The usable energy in a capacitor is governed by $E = \frac{1}{2}CV^2$. Because energy scales with the square of the voltage, you lose 75% of your stored energy by the time the bank discharges to half its maximum voltage. Furthermore, high-current discharge causes instantaneous voltage sag dictated by Ohm's Law: $\Delta V = I \times ESR_{total}$.

Worked Sizing Example:
Suppose you need to support a 1500W load at a 14V nominal bus for 10 seconds during a grid transfer switch delay.
1. Required current: $1500W / (0.85 \text{ inverter efficiency} \times 12V \text{ min}) = 147A$.
2. Allowable voltage sag: Assuming a 14V start and an 11V inverter Low Voltage Disconnect (LVD), max sag is 3V.
3. Max allowable ESR: $R = V / I = 3V / 147A = 0.020\Omega$ (20mΩ).
If you use six 350F cells (each with a 3.2mΩ ESR) in series, your total ESR is $19.2m\Omega$. This barely passes the threshold, highlighting why high-power capacitor banks require massive cells or parallel-series matrices.

Series vs. Parallel: Voltage, Capacity, and Balancing

Understanding the consequence of series vs parallel wiring is critical for both capacitors and batteries, though the units of measurement differ.

Configuration Capacitors (EDLC) Lithium Batteries (LiFePO4) Primary Failure Risk
Series Voltage adds ($V_t = V_1 + V_2$).
Capacitance drops ($1/C_t = 1/C_1 + 1/C_2$).
Coulombs (Charge) remain constant.		 Voltage adds.
Ah capacity remains constant.
Energy (Wh) adds.		 Overvoltage on the weakest cell due to leakage current mismatch.
Parallel Voltage remains constant.
Capacitance adds ($C_t = C_1 + C_2$).
Total charge (Coulombs) adds.		 Voltage remains constant.
Ah capacity adds.
Energy (Wh) adds.		 Cross-currents and thermal runaway if cells are mismatched in voltage or internal resistance.

The Balancing Imperative

In a series string, manufacturing tolerances mean no two capacitors have the exact same leakage current. When charging to the absolute maximum voltage (e.g., 17.1V for six 2.85V cells), the cell with the highest leakage current will see a disproportionately high voltage share, potentially exceeding its 2.85V dielectric breakdown limit. You must use an active or passive balancing circuit. Passive balancers bleed excess voltage as heat via resistors; active balancers shuttle charge between cells using switched-capacitor ICs, which is mandatory for high-efficiency power storage systems.

Lithium vs. Ultracapacitor Banks: Safety and Limits

Many builders pivot from ultracapacitors to lithium chemistry when they realize that a 58.3F capacitor bank at 17V stores only ~8.4 kilojoules (2.3 Wh) of usable energy—enough for surge buffering, but useless for sustained off-grid runtime. If your application requires sustained runtime, you will likely transition to lithium, which introduces strict safety and operational limits.

⚠️ Lithium Fire-Safety Mandate: If your DC bus design pivots from ultracapacitors to LiFePO4 or NMC lithium cells to increase energy density, you must use a certified BMS with cell-level over-voltage/under-voltage cutoff and thermal runaway protection. Never parallel mismatched lithium cells, and always charge with a lithium-specific profile. Ultracapacitors do not suffer thermal runaway, making them inherently safer for high-surge, unmonitored bench setups.

When sizing a lithium bank for the same 150A surge, you must respect C-rates and Depth of Discharge (DoD). A 100Ah LiFePO4 battery rated at 1C can safely deliver 100A continuously, but pulling 150A (1.5C) will trigger the BMS over-current protection or degrade the cell chemistry. Furthermore, while lithium can technically discharge to 0%, practical DoD is limited to 80-90% to preserve cycle life. Capacitors, conversely, can be cycled 100% to their minimum voltage millions of times with zero degradation.

Inverter Sizing and System Integration

Integrating a capacitive DC source with a commercial inverter requires careful attention to the inverter's input capacitors and Low Voltage Disconnect (LVD) thresholds.

Charge/Discharge Limits:
Ultracapacitor charge limits are strictly voltage-based. Never exceed the manufacturer's max cell voltage (usually 2.7V or 2.85V) at any temperature. Discharge limits are thermal. Continuous RMS current is limited by the $I^2R$ heating of the ESR. For a 350F cell with a 3.2mΩ ESR, a continuous 50A draw generates 8W of heat per cell. In a tightly packed series module without forced air, this will push the core temperature past the 65°C derating threshold, halving the capacitor's operational lifespan.

Inverter Selection:
Standard off-grid inverters assume a stiff, low-impedance battery source. When connected directly to a series capacitor bank, the inverter's internal switching transients can cause high-frequency ringing. You must size the inverter's internal DC bus capacitance to handle the ripple current, or insert a DC-DC converter between the capacitor bank and the inverter to present a stable, regulated voltage to the inverter's LVD circuit.

Decision Tree: Selecting Your Energy Storage Medium

Choosing between a series capacitor bank, a lithium battery, or a hybrid approach depends entirely on your load profile's time-domain characteristics. Use the decision matrix below to terminate your design process with a concrete component selection.

Load Profile Characteristic Time Duration System Requirement Concrete Component Pick
High-surge engine cranking, motor starting, or grid-transfer bridging. 1 to 30 seconds Maximum power density, zero Peukert loss, high cycle life. Voltage sag is the only limit. Eaton (Maxwell) ESHSR-0350C0-2R85 (2.85V, 350F, 3.2mΩ ESR). Wire 6 in series with an active balancer.
Sustained off-grid runtime, solar smoothing, or backup power for electronics. 10 minutes to 12 hours Maximum energy density (Wh), stable voltage output, BMS protected. LiTime 12V 100Ah LiFePO4 (Built-in 100A BMS, 1.28kWh capacity). DoD limited to 80% for longevity.
Hybrid: High-surge buffering combined with sustained baseline runtime. Seconds + Hours Capacitor bank absorbs high-frequency transients; battery provides bulk energy. Hybrid Bus: 6x Eaton 350F caps in series, paralleled to a LiFePO4 bank via a high-current Schottky diode or ideal diode controller to prevent reverse charging.

For pure power storage applications requiring rapid charge/discharge cycling without chemical degradation, the Eaton ESHSR-0350C0-2R85 wired in a 6-cell series string is the definitive benchmark. It provides the 17.1V headroom necessary for 12V-nominal DC-DC conversion while maintaining the sub-20mΩ ESR required to prevent inverter LVD trips during high-amperage surges. Always pair this series string with a dedicated active balancing module (such as those based on the TI BQ33100 IC) to ensure equal charge distribution and prevent catastrophic dielectric overvoltage on individual cells.