Quantifying energy input, useful energy output, and energy losses is essential for evaluating system performance, comparing design alternatives, and identifying efficiency improvements. VCE Systems Engineering requires both the conceptual understanding of where energy goes and the ability to perform numerical calculations.
KEY TAKEAWAY: Energy input always equals the sum of useful energy output and losses. Calculating all three confirms the energy balance and reveals how much is wasted at each stage.
Conservation of energy:
$$E_{input} = E_{output} + E_{losses}$$
$$\Rightarrow \quad E_{losses} = E_{input} - E_{output}$$
Efficiency:
$$\eta = \frac{E_{output}}{E_{input}} \times 100\%$$
$$\Rightarrow \quad E_{output} = \eta \times E_{input}$$
$$\Rightarrow \quad E_{losses} = E_{input}(1 - \eta)$$
Power versions (same relationships, per unit time):
$$P_{input} = P_{output} + P_{losses}$$
$$\eta = \frac{P_{output}}{P_{input}} \times 100\%$$
Electrical energy consumed:
$$W = Pt = VIt$$
| Symbol | Quantity | Unit |
|---|---|---|
| $W$ | Energy | Joule (J) |
| $P$ | Power | Watt (W) |
| $t$ | Time | second (s) |
Worked example 1 — Heater:
A 1200 W electric heater operates for 45 minutes. The heater is 100% efficient at converting electrical energy to heat (all electrical input becomes heat output — no mechanical losses).
$$W = P \times t = 1200 \times (45 \times 60) = 1200 \times 2700 = 3{,}240{,}000 \text{ J} = 3.24 \text{ MJ}$$
Worked example 2 — Motor with losses:
A motor draws 500 W and has an efficiency of 80%. Find the useful mechanical power output and the power lost as heat.
$$P_{output} = 0.80 \times 500 = 400 \text{ W}$$
$$P_{losses} = 500 - 400 = 100 \text{ W (as heat in windings and friction)}$$
Over 10 minutes of operation:
$$W_{input} = 500 \times 600 = 300{,}000 \text{ J} = 300 \text{ kJ}$$
$$W_{output} = 400 \times 600 = 240 \text{ kJ}$$
$$W_{losses} = 60 \text{ kJ}$$
Work done:
$$W = F \times d$$
Kinetic energy:
$$E_k = \frac{1}{2}mv^2$$
Gravitational potential energy:
$$E_p = mgh$$
where $m$ = mass (kg), $v$ = velocity (m/s), $g$ = 9.8 m/s², $h$ = height (m).
Worked example — Lifting a load:
A motor lifts a 50 kg load 3 m in 10 s. Calculate: (a) work done on load, (b) power of load, (c) motor input power if 75% efficient.
(a) Work done on load:
$$W = mgh = 50 \times 9.8 \times 3 = 1470 \text{ J}$$
(b) Power of load (output power):
$$P_{output} = \frac{W}{t} = \frac{1470}{10} = 147 \text{ W}$$
(c) Motor input power:
$$P_{input} = \frac{P_{output}}{\eta} = \frac{147}{0.75} = 196 \text{ W}$$
(d) Power lost as heat:
$$P_{losses} = 196 - 147 = 49 \text{ W}$$
VCAA FOCUS: Multi-step problems like the above are common in VCAA exam questions. Show all working, including units at every step. Partial marks are awarded for correct method even if arithmetic errors occur.
For a system with multiple conversion stages, losses accumulate at each stage.
Worked example — Solar → Battery → Motor → Load:
| Stage | Efficiency | Input power | Output power | Losses |
|---|---|---|---|---|
| Solar panel | 18% | 1000 W (sunlight) | 180 W | 820 W |
| Battery charge | 90% | 180 W | 162 W | 18 W |
| Motor | 85% | 162 W | 137.7 W | 24.3 W |
| Total system | 13.77% | 1000 W | 137.7 W | 862.3 W |
$$\eta_{system} = 0.18 \times 0.90 \times 0.85 = 0.1377 = 13.77\%$$
COMMON MISTAKE: When calculating multi-stage efficiency, multiply the individual efficiencies as decimals, not percentages. Using 18 × 90 × 85 gives a nonsensical result; use 0.18 × 0.90 × 0.85 = 0.1377.
Always verify calculations using the energy balance equation:
$$E_{input} = E_{output} + E_{losses}$$
If this does not balance, an error has been made. Also check:
- Units are consistent (all in Joules, or all in Watts — not mixed)
- Efficiency is expressed as a decimal (0.75) when multiplying, percentage (75%) only in the final answer
- Time is in seconds when using SI units
APPLICATION: In a project folio, present an energy analysis table showing input, output, and losses for each subsystem. This demonstrates systematic engineering thinking and is a key criterion in VCAA assessment.
