Energy

Energy is power accumulated over time. Learn joules and kilowatt-hours, electricity bills, battery capacity (Ah vs Wh), energy stored in capacitors and inductors, and how to size a battery bank.

Power tells you how fast energy flows; energy tells you how much has flowed in total. Your electricity bill charges for energy (kWh), a phone battery stores energy (Wh), and a capacitor bank releases stored energy in a flash.

Engineers constantly convert between energy units and use energy balance — energy in = energy out + losses + change in stored energy — to design everything from battery packs to power plants.

An analogy

Power is the speed of a car; energy is the distance travelled. Driving at 60 km/h for 2 hours covers 120 km just as a 60 W lamp running for 2 hours uses 120 Wh.

Joules and kilowatt-hours

The SI unit of energy is the joule (1 W for 1 s). It is tiny for household use, so utilities bill in kilowatt-hours: a 1 kW load running for 1 hour. Cost = energy (kWh) × tariff.

Battery capacity: Ah vs Wh

An amp-hour rating says how much charge a battery holds; multiply by its nominal voltage to get energy. A 12 V 100 Ah battery and a 24 V 50 Ah battery store the same 1.2 kWh. Always compare batteries in Wh.

Usable energy: depth of discharge, efficiency, C-rate

Not all stored energy is usable. Lead-acid batteries should stay above ~50 % state of charge; LiFePO₄ tolerates 80–90 % depth of discharge. Inverter losses, cable losses and capacity loss at high discharge rates (Peukert's effect) all reduce what reaches the load.

• C-rate: a 100 Ah battery discharged at 1C supplies 100 A for about an hour; at 0.2C, 20 A for about 5 hours.
• Round-trip efficiency: Li-ion ≈ 90–95 %, lead-acid ≈ 75–85 %.

Energy in fields

Capacitors store energy in an electric field (½CV²); inductors store it in a magnetic field (½LI²). Because the energy depends on V² or I², doubling the voltage on a capacitor quadruples its stored energy — and its danger.

Energy balance and grid-scale storage

Grids must balance generation and demand every instant. Pumped hydro (~70–80 % round trip), battery energy storage systems, flywheels and hydrogen shift energy in time. Levelised cost of storage (cost per MWh delivered over the system's life) is the metric used to compare them.

Formulas

• Energy from power: E = P · t
• Unit conversion: 1 kWh = 1000 W × 3600 s = 3.6 × 10⁶ J
• Battery energy: E (Wh) = V_nominal × capacity (Ah)
• Stored energy: E_C = ½ C V², E_L = ½ L I²

Worked example: Heater running cost

Given: P = 2 kW; t = 3 h/day for 30 days; Tariff = 0.15 per kWh

Find: Monthly energy and cost.

• Daily energy = 2 kW × 3 h = 6 kWh
• Monthly = 6 × 30 = 180 kWh
• Cost = 180 × 0.15 = 27.00

Answer: 180 kWh ≈ 27 (currency units) per month

Worked example: Energy in a capacitor bank

Given: C = 4700 µF; V = 400 V (DC link)

Find: Stored energy.

• E = ½ C V²
• = 0.5 × 4700 × 10⁻⁶ × 400²
• = 0.5 × 4.7 × 10⁻³ × 160 000 = 376 J

Answer: 376 J — enough to be lethal; bleeder resistors are mandatory

Common mistakes

• Wrong: Comparing batteries by Ah across different voltages. — Right: Convert to Wh (V × Ah) before comparing.
• Wrong: Assuming 100 % of a battery's rating is usable. — Right: Apply depth of discharge, efficiency and temperature derating.

Applications

• Solar + battery homes: Sizing a home battery starts from the evening energy use (kWh) and the depth of discharge the chemistry allows.
• EV range: An EV using 150 Wh/km with a 60 kWh pack has a nominal range of 400 km; cold weather and speed change the Wh/km.
• Camera flash and defibrillators: Capacitors store energy slowly and release it in milliseconds — power far higher than the battery could supply directly.

Key takeaways

• Energy = power × time; 1 kWh = 3.6 MJ.
• Compare batteries in Wh, and apply DoD and efficiency before sizing.
• Capacitors store ½CV², inductors ½LI².
• Energy is the area under the power-vs-time curve.

Going further

• State-of-charge estimation: Coulomb counting drifts; open-circuit-voltage lookup only works at rest. Production BMSs fuse both with a battery model in a Kalman filter to estimate SOC and state of health.
• Ragone plots: Energy storage technologies are compared on a Ragone plot: energy density (Wh/kg) against power density (W/kg). Batteries sit high in energy, supercapacitors high in power, which is why hybrid systems pair them.
• Demand vs energy charges: Commercial tariffs bill both energy (kWh) and peak demand (kW, averaged over 15–30 min). Peak shaving with batteries or load scheduling can cut bills even when total energy stays the same.