Efficiency Formula
Calculate how effectively a system converts input energy to useful output.
Applies to engines, motors, and any energy system.
The Formula
η = (Useful Output / Total Input) × 100%
Efficiency measures the percentage of input energy that becomes useful output. No real system is 100% efficient — some energy is always lost to heat, friction, or other waste.
Variables
| Symbol | Meaning |
|---|---|
| η | Efficiency (percentage, 0-100%) |
| Useful Output | Energy or power that does the intended work (joules or watts) |
| Total Input | Total energy or power supplied to the system (joules or watts) |
Example 1
A motor consumes 500 W and produces 400 W of mechanical power
η = (400 / 500) × 100%
η = 80% (100 W lost as heat)
Example 2
A car engine burns fuel with 30,000 J of energy and delivers 9,000 J to the wheels
η = (9,000 / 30,000) × 100%
η = 30% (typical for internal combustion engines)
When to Use It
Use the efficiency formula when:
- Comparing the performance of different machines or engines
- Calculating energy waste and operating costs
- Evaluating renewable energy systems (solar panels, wind turbines)
- Identifying where energy improvements will have the biggest impact
Key Notes
- The second law of thermodynamics guarantees no heat engine can reach 100% efficiency — even a theoretically perfect Carnot engine is limited to η = 1 − (T_cold / T_hot), where temperatures are in Kelvin
- For systems in series (generator → transmission line → motor), total efficiency = η₁ × η₂ × η₃ — small inefficiencies multiply, so a chain of 90% stages gives only 73% end-to-end efficiency
- Efficiency can be calculated in energy (joules in vs out) or power (watts in vs out) — both give the same percentage because time cancels
- LED lighting reaches ~35–50% efficiency vs 5% for incandescent bulbs; the difference is that LEDs convert electricity directly to light rather than routing through heat first
Key Notes
- Formula: η = (useful output / total input) × 100%: Efficiency is always ≤ 100% — the First Law of Thermodynamics (conservation of energy) ensures no more energy comes out than goes in. The "lost" energy doesn't disappear; it converts to heat, sound, or other non-useful forms.
- Cascaded efficiency: η_total = η₁ × η₂ × η₃ × …: When energy passes through multiple stages, multiply (don't add) the efficiencies. A motor at 90% driving a gearbox at 95% driving a pump at 80%: η_total = 0.90 × 0.95 × 0.80 = 68.4%. This explains why transmission losses compound quickly in long energy chains.
- Carnot efficiency — the theoretical maximum for heat engines: η_Carnot = 1 − T_cold/T_hot (temperatures in Kelvin). No real heat engine can exceed Carnot efficiency; real engines achieve 35–45% of the theoretical maximum. This limit applies to power plants, car engines, and refrigerators (coefficient of performance has a Carnot analog).
- Electrical efficiency: For motors and generators: η = P_out / P_in. Electric motors reach 90–97% efficiency; internal combustion engines 25–40%; combined-cycle gas turbines up to ~60%. The difference is why electrification reduces total energy use even when accounting for generation losses.
- Applications: Efficiency formulas guide motor and drive selection in industry, building energy audits (HVAC COP, window U-values), power plant performance benchmarking, supply chain energy analysis, and engineering design — where improving efficiency by even a few percent has significant economic and environmental impact at scale.