OpenEnergyMonitor

AC Power Theory 1

Understanding AC power.

A whole house energy monitor measures the energy used by appliances connected to the house mains. To understand how it does this it is useful to know something about how appliances interact with the electrical system.

Not all appliances interact with the electricity system in the same way. This article will first discuss resistive loads and how the power they use is calculated. It then goes on to discuss reactive loads and a bit about non-linear loads.

Resistive loads

Incandescent light bulbs, kettles, irons, electric water heaters, electric cookers are all quite straight forward they use all the energy given to them. They are resistive loads which means that their current draw is equal to the voltage divided by their resistance (Ohms Law). A purely resistive load gives a voltage and current waveform output similar to the following:

Diagram 1 – Voltage, current, power graph of a resistive load

The yellow line is power at a given time (at any given instant its called instantaneous power) which is equal to the product of the voltage and current at a given time. Notice how the power is always positive in this case, the positive direction is energy flowing to the load.

Partially reactive loads

However things like fridges, washing machines, pillar drills and arc welders are not so straight forward as these appliances take in a certain amount of energy and then release some energy back into the mains supply. These have inductive (usually motors) or capacitive (arc welders) components in addition to the resistive component. A partially inductive load gives a voltage and current waveform output similar to the following:

Diagram 2 – Voltage, current, power graph of a partially reactive load

Notice how the yellow line now goes negative for a period of time, the positive bit is energy flowing to the load and the negative bit is energy flowing back from the load.

The other thing to think about is that the voltage and current waveforms have been shifted apart, imagine charging a fairly large capacitor with a resistor in series (so that it can't charge instantly): To start with the capacitor is empty. The voltage on the supply rises and is higher than the voltage on the capacitor and so current flows into the capacitor (the positive direction on the graph), which leads to the voltage on the capacitor rising. The voltage on the supply now falls, the voltage across the capacitor is now higher than the supply voltage as it is charged and so current starts to flow back in the direction of the supply (the negative direction on the graph). This causes the current waveform to appear as if it is shifted as in the graph. (The shift is known as phase shift).

Real Power, Reactive power and Apparent Power.

Looking at both voltage, current and power graphs above at mains frequency the power draw fluctuates 50/60 times a second, for us humans we cant keep up with change at this speed and so we have a more useful value for power: the average of the instantaneous power which we call real power! (or active power).

Real power is often defined as the power used by a device to produce useful work. Looking at the graph above the positive bits are power going to the load from the supply and the negative bits are power going back from the load to the supply, the power that was actually used by the load the power going to minus the power going back is the real power.

Reactive power is a measure of the power going back from the load to the supply.

Another useful measure of power is Apparent Power which is the product of the Root-Mean-Squared (RMS) of the Voltage and the RMS of the Current. For purely resistive loads real power is equal to apparent power. But for all other loads real power is less than apparent power. Apparent power is a measure of the real and reactive power but it is not a sum of the two, as the sum of the two does not take into account phase differences.

Relationship between real, reactive and apparent power for IDEAL sinusoidal loads:

Real Power = Apparent Power x cos(theta)

Reactive Power = Apparent Power x sin(theta)

cos(theta) is also known as power factor.

However a note about non-linear loads:

power factor =is only valid for linear sinusoidal loads. Most power supplies for DC devices like Laptops exert a non-linear load on the mains, their current draw often look like this:

We can still calculate powerfactor from the following equation:

Power Factor = Real Power / Apparent Power

but power factor = cos(theta) would not be correct, higher order harmonics would need to be added.

The power factor value measures how much the mains efficiency is affected by BOTH phase lag φ AND harmonic content of the input current.

Ref: understanding power factor by L Wuidart

To sum up

There are a lot of things we can measure about energy use in AC systems. Each one with its uses. For the household energy metering application however, real power is likely to be the most useful value as it is tells you how much power all your appliances are actually using and it is what the utility bills you for.

Further reading

http://en.wikipedia.org/wiki/AC_power

Power in resistive and reactive loads: http://www.allaboutcircuits.com/vol_2/chpt_11/1.html

Single phase power systems: http://www.allaboutcircuits.com/vol_2/chpt_10/1.html

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