The fundamental electrical charge formula is Q = I × t (Charge = Current × Time). In physics, this yields Coulombs (Amperes × seconds). In practical battery sizing and power system design, we translate this into Amp-hours (Ah) by dividing by 3,600, or more commonly, we calculate required capacity using Watt-hours: Ah = (Watts / Volts) × Hours. However, applying this raw formula to a real-world battery bank without adjusting for Depth of Discharge (DoD), inverter efficiency, and Peukert’s Law will leave you with a severely undersized system that suffers from premature voltage sag.
System Block Description: From Source to Load
Before running the math, you must map the energy flow. Every conversion stage introduces losses that the basic electrical charge formula ignores. A standard off-grid or backup power system follows this block sequence:
- Source (Solar Array or Grid): Generates raw DC or AC power.
- Charge Controller / Rectifier: Regulates voltage and current to safely charge the bank (MPPT controllers operate at 92–98% efficiency).
- Battery Bank (Storage): Stores energy chemically. Internal resistance and Peukert effects reduce usable capacity under high loads.
- Inverter: Converts DC bank voltage to AC load voltage. High-frequency inverters run at 85–90% efficiency; low-frequency transformer-based units hit 90–93%.
- AC Panel (Load): The final destination. Wire gauge and distance introduce minor voltage drops here.
When sizing your storage, you must work backward from the Load to the Source, compounding the efficiency losses at each step. According to All About Circuits, ignoring these conversion losses is the most common reason DIY solar systems fail to meet expected runtimes.
The Core Electrical Charge Formula and Sizing Math
Let’s size a battery bank for a 1,200W continuous AC load that needs to run for 3 hours on a 12V nominal system.
Step 1: Base Capacity Calculation
First, find the DC current draw. Assuming a 90% efficient inverter, the DC load is 1,200W / 0.90 = 1,333W.
Current (I) = 1,333W / 12V = 111 Amps.
Using the electrical charge formula (Q = I × t): 111A × 3 hours = 333 Ah.
Step 2: Adjusting for Depth of Discharge (DoD)
You cannot drain a battery to 0V without destroying it. If you are using Lithium Iron Phosphate (LiFePO4) cells, a safe DoD is 80%. For Lead-Acid (AGM/Gel), it is 50%.
Required Ah (LiFePO4) = 333 Ah / 0.80 = 416 Ah.
Step 3: Peukert’s Law and Efficiency Factors
The basic formula assumes capacity is linear. It is not. Peukert’s Law states that as discharge current increases, the effective capacity of a battery decreases. The formula is t = H × (C / (I × H))^k, where k is the Peukert exponent.
| Battery Chemistry | Peukert Exponent (k) | Impact at 111A Draw | Max Recommended C-Rate |
|---|---|---|---|
| Flooded Lead-Acid | 1.25 - 1.30 | Effective capacity drops by ~35% | C/10 (0.1C) |
| AGM / Gel | 1.10 - 1.20 | Effective capacity drops by ~18% | C/5 (0.2C) |
| LiFePO4 (Lithium) | 1.02 - 1.05 | Effective capacity drops by < 3% | 1C continuous |
If you attempted this 111A draw on a 416Ah AGM bank, Peukert’s law dictates you would actually experience a much shorter runtime. You would need to oversize the AGM bank to roughly 600Ah to compensate. With LiFePO4, the 416Ah calculation holds true because the exponent is nearly 1.0.
Series vs. Parallel Consequences for V and Ah
To achieve 416Ah, you might use four 12V 100Ah batteries. How you wire them changes the system architecture:
- Series: Voltages add, Ah remains the same. Four 12V 100Ah batteries in series yield 48V at 100Ah. (Total energy: 4,800Wh).
- Parallel: Ah adds, Voltage remains the same. Four 12V 100Ah batteries in parallel yield 12V at 400Ah. (Total energy: 4,800Wh).
- Series-Parallel (2S2P): Yields 24V at 200Ah. This is the sweet spot for 2,000W–4,000W inverters, keeping DC current manageable without requiring massive 4/0 AWG cabling.
Inverter and Charger Sizing for Your Load
Once the battery bank is sized via the electrical charge formula, you must match the inverter and the charger. Undersizing the charger leads to chronic undercharging and sulfation (in lead-acid) or BMS faults (in lithium).
| System Parameter | Sizing Rule | Example for 1200W / 416Ah System |
|---|---|---|
| Inverter Continuous | 1.25× the maximum continuous AC load. | 1200W × 1.25 = 1500W minimum |
| Inverter Surge | Must handle motor startup (LRA) for 3-5 seconds. | Ensure 3000W+ surge rating for compressors. |
| Charger Output (Lead-Acid) | 10% to 15% of total bank Ah (0.1C - 0.15C). | 416Ah × 0.15 = 62A DC charger |
| Charger Output (LiFePO4) | 20% to 50% of total bank Ah (0.2C - 0.5C). | 416Ah × 0.3 = 125A DC charger |
For the LiFePO4 example, a 125A charger will replenish the 333Ah drawn in roughly 2.7 hours (accounting for the constant-voltage absorption taper at the end of the charge cycle). If you are using solar as the source, your MPPT charge controller must be rated for at least 125A output at the battery voltage, which typically requires a 150V/100A controller paired with a secondary 50A unit, or a single high-amperage commercial unit.
Frequently Asked Questions
How do you calculate electrical charge in Coulombs vs Amp-hours?
The standard SI unit for electrical charge is the Coulomb (C), defined as one Ampere flowing for one second (1C = 1A × 1s). Because battery capacities involve hours rather than seconds, we use Amp-hours (Ah). To convert between them, multiply the Amp-hours by 3,600 (the number of seconds in an hour). For example, a 100Ah battery holds 360,000 Coulombs of charge. In practical system design, we almost exclusively use Ah or Watt-hours (Wh), reserving Coulombs for capacitor and transient surge calculations.
What is the electrical charge formula for a capacitor vs a battery?
While both store electrical energy, their charge formulas differ fundamentally. For a battery, charge is calculated chemically via Q = I × t, providing a relatively stable voltage over time. For a capacitor, the formula is Q = C × V, where Q is charge in Coulombs, C is capacitance in Farads, and V is voltage. A capacitor's voltage drops linearly as it discharges, whereas a battery maintains a plateau voltage until its chemical reactants are depleted. Capacitors are used for microsecond power conditioning; batteries are used for macro-scale energy storage.
How does temperature affect the electrical charge formula calculations?
The basic Q = I × t formula assumes a standard ambient temperature of 25°C (77°F). In reality, temperature severely impacts chemical reaction rates inside a battery. At 0°C (32°F), a lead-acid battery's usable capacity drops to roughly 80% of its rated Ah, and its internal resistance spikes, exacerbating Peukert losses. LiFePO4 batteries maintain capacity well in the cold but cannot be charged below 0°C without causing permanent lithium plating and internal short circuits. When sizing a bank for winter use, apply a 1.25x temperature derating factor to your final Ah calculation unless the batteries are housed in a climate-controlled enclosure.
Why does my battery drain faster than the electrical charge formula predicts?
If your runtime falls short of the math, you are likely falling victim to three hidden factors:
1. Peukert’s Effect: High current draws reduce effective capacity in lead-acid batteries.
2. Voltage Sag and Cutoffs: Under heavy load, internal resistance causes the terminal voltage to drop. Your inverter’s Low Voltage Disconnect (LVD) might trigger at 11.5V, shutting the system down even if the battery still has 30% chemical capacity left.
3. Phantom Loads and Inverter Quiescent Draw: Inverters consume 15W to 40W just being turned on. Over a 24-hour period, a 30W quiescent draw consumes 720Wh, which the basic load calculation often forgets to include. Always add a 10% buffer to your total daily Watt-hour requirement to account for system overhead.
