Electronic – Should I multiply by time to determine Watt-hours

powersolar cell

I have done an experiment with solar panels over the course of the day to determine the total power output in Wh. I measured voltage and current through my solar panel circuit using a multimeter every half an hour. I multiplied these together to find the power being generated each half hour of the day. Assuming the power rises and falls linearly between each measurement, I determined the energy for each half hour is equal to 30 times the power at the mid-mark (either 15 or 45 minutes into the hour). Now, to determine the total energy in Wh the panel produced during the day, do I multiply the result of the previous operation by 0.5 hrs and then add all the interval's power together or can I just add them all together without multiplying by 0.5 hrs?

Best Answer

The term "watt-hour" (Wh) is often not correctly understood. Watt (W) is a unit of power. Energy is measured in Joules (J) and is related to power by $$ E = {P_\text{avg}}\times \Delta t, $$ where \$\Delta t\$ is time in question and \$P_\text{avg}\$ is the average power spent during that time.

With only two measurements, one at the start and one at the end, the best guess is that the average power was $$ P_\text{avg} = \frac{P_\text{start} + P_\text{end}}{2} $$

Notice that "watt-hours" is power multiplied with time, so it is a measure of energy. This is what power companies really sell you, and what they should charge you for; though that's for some reason not always the case.

Joules is the same as a "watt-second", so $$ 1\;\text{Wh} = (1\;\text{W})\times(1\;\text{h}) = (1\;\text{W})\times(3,600\;\text{s}) = 3,600\;\text{J}=3.6\;\text{kJ} $$

Sidenote

If you know calculus, then a more accurate formula for the relationship between power and energy is $$ E = \int_{t_\text{start}}^{t_\text{end}} P(t)\;dt, $$ or even $$ P(t) = \frac{dE(t)}{dt}, $$ which says that power is the rate-of-change of energy consumption. You can make use of this, if you know the sample-rate of your measurements, for instance by using an ADC from a micro-controller. If the time between samples is dt, then the total energy can be calculated by:

loop() {
  current = currentADC.sample();
  voltage = voltageADC.sample();
  time = time_now();
  dt = time - time_last;
  power = current * voltage;
  energy += power * dt;
  time_last = time;
}
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