However, while this formula is also true for purely resistive AC circuits, the situation is slightly more complex in an AC circuits containing reactive components as this volt-amp product can change with frequency.

In an AC circuit, the product of voltage and current is expressed as volt-amperes (VA) or kilo volt-amperes (kVA) and is known as *Apparent power*, symbol S. In a non-inductive purely resistive circuit such as heaters, irons, kettles and filament bulbs etc, their reactance is practically zero, so the impedance of the circuit is composed almost entirely of just resistance.

For an AC resistive circuit, the current and voltage are in-phase and the power at any instant can be found by multiplying the voltage by the current at that instant, and because of this “in-phase” relationship, the rms values can be used to find the equivalent DC power or heating effect.

However, if the circuit contains reactive components, the voltage and current waveforms will be “out-of-phase” by some amount determined by the circuits phase angle. If the phase angle between the voltage and the current is at its maximum of 90^{o}, the volt-amp product will have equal positive and negative values.

In other words, the reactive circuit returns as much power to the supply as it consumes resulting in the average power consumed by the circuit being zero, as the same amount of energy keeps flowing alternately from source to the load and back from load to source.

Since we have a voltage and a current but no power dissipated, the expression of P = IV (rms) is no longer valid and it therefore follows that the volt-amp product in an AC circuit does not necessarily give the power consumed. Then in order to determine the “real power”, also called *Active power*, symbol P consumed by an AC circuit, we need to account for not only the volt-amp product but also the phase angle difference between the voltage and the current waveforms given by the equation: VI.cosΦ.

Then we can write the relationship between the apparent power and active or real power as:

Note that power factor (PF) is defined as the ratio between the active power in watts and the apparent power in volt-amperes and indicates how effectively electrical power is being used. In a non-inductive resistive AC circuit, the active power will be equal to the apparent power as the fraction of P/S becomes equal to one or unity. A circuits power factor can be expressed either as a decimal value or as a percentage.

But as well as the active and apparent powers in AC circuits, there is also another power component that is present whenever there is a phase angle. This component is called **Reactive Power** (sometimes referred to as imaginary power) and is expressed in a unit called “volt-amperes reactive”, (VAr), symbol Q and is given by the equation: VI.sinΦ.

Reactive power, or VAr, is not really power at all but represents the product of volts and amperes that are out-of-phase with each other. Reactive power is the portion of electricity that helps establish and sustain the electric and magnetic fields required by alternating current equipment. The amount of reactive power present in an AC circuit will depend upon the phase shift or phase angle between the voltage and the current and just like active power, reactive power is positive when it is “supplied” and negative when it is “consumed”.

Reactive power is used by most types of electrical equipment that uses a magnetic field, such as motors, generators and transformers. It is also required to supply the reactive losses on overhead power transmission lines.

The relationship of the three elements of power, active power, (watts) apparent power, (VA) and reactive power, (VAr) in an AC circuit can be represented by the three sides of right-angled triangle. This representation is called a **Power Triangle** as shown:

### Power in an AC Circuit

From the above power triangle we can see that AC circuits supply or consume two kinds of power: active power and reactive power. Also, active power is never negative, whereas reactive power can be either positive or negative in value so it is always advantageous to reduce reactive power in order to improve system efficiency.

The main advantage of using AC electrical power distribution is that the supply voltage level can be changed using transformers, but transformers and induction motors of household appliances, air conditioners and industrial equipment all consume reactive power which takes up space on the transmission lines since larger conductors and transformers are required to handle the larger currents which you need to pay for.

Reactive Power Analogy with Beer

In many ways, reactive power can be thought of like the foam head on a pint or glass of beer. You pay the barman for a full glass of beer but only drink the actual liquid beer which is always less than a full glass.

This is because the head (or froth) of the beer takes up additional wasted space in the glass leaving less room for the real beer that you consume, and the same idea is true for reactive power.

But for many industrial power applications, reactive power is often useful for an electrical circuit to have. While the real or active power is the energy supplied to run a motor, heat a home, or illuminate an electric light bulb, reactive power provides the important function of regulating the voltage thereby helping to move power effectively through the utility grid and transmission lines to where it is required by the load.

While reducing reactive power to help improve the power factor and system efficiency is a good thing, one of the disadvantages of reactive power is that a sufficient quantity of it is required to control the voltage and overcome the losses in a transmission network. This is because if the electrical network voltage is not high enough, active power cannot be supplied. But having too much reactive power flowing around in the network can cause excess heating (**I**^{2}R losses) and undesirable voltage drops and loss of power along the transmission lines.

## Power Factor Correction of Reactive Power

One way to avoid reactive power charges, is to install power factor correction capacitors. Normally residential customers are charged only for the active power consumed in kilo-watt hours (kWhr) because nearly all residential and single phase power factor values are essentially the same due to power factor correction capacitors being built into most domestic appliances by the manufacturer.

Industrial customers, on the other hand, which use 3-phase supplies have widely different power factors, and for this reason, the electrical utility may have to take the power factors of these industrial customers into account paying a penalty if their power factor drops below a prescribed value because it costs the utility companies more to supply industrial customers since larger conductors, larger transformers, larger switchgear, etc, is required to handle the larger currents.

Generally, for a load with a power factor of less than 0.95 more reactive power is required. For a load with a power factor value higher than 0.95 is considered good as the power is being consumed more effectively, and a load with a power factor of 1.0 or unity is considered perfect and does not use any reactive power.

Then we have seen that “apparent power” is a combination of both “reactive power” and “active power”. Active or real power is a result of a circuit containing resistive components only, while reactive power results from a circuit containing either capacitive and inductive components. Almost all AC circuits will contain a combination of these R, L and C components.

Since reactive power takes away from the active power, it must be considered in an electrical system to ensure that the apparent power supplied is sufficient to supply the load. This is a critical aspect of understanding AC power sources because the power source must be capable of supplying the necessary volt-amp (VA) power for any given load.

i have V=141.4sin(wt+30) and i=11.31cos(wt-30) and i want to calculate the power factor to calculate the active and reactive power

P = v x i where: v = 141.4 sin(wt +30) and i = 11.31 cos(wt -30). The power factor angle (phi) can be found by phase angle subtraction, but only if v and i have the same sinusoidal form which clearly here they do not. Then the cosine term of the current, (i) must be converted to the same sine form as the voltage, (v). This is done using the cosine identity of: cos x = sin(x +90).

Thus i = 11.31 cos(wt -30) = 11.31 sin(wt -30 +90) = 11.31 sin(wt +60)

So the power factor angle becomes +30 – (+60) = -30 thus the power factor, pf is: cos(-30) = 0.866 leading because phi is negative.

v and i are peak values, so:

v = 141.4*0.7071 = 100 volts rms

i = 11.31*0.7071 = 8 amps rms

P = v*i*cos(phi) = 100*8*0.866 = 692.8 watts (rms)

I am having 2.2kw, 240/415v , .8 pf and I want it to be used as self exited generator and the prime mover is 0.75kw 230/415, .82 pf.

What sizes of capacitors should I use to generate 240v in both generator and prime mover motor ?

Thanks in advance.

Please explain little more how capacitor can be used in power factor correction ?

Thank you

example taken is very Precise& long lasting

It is possible that there is power factor correction (in the form of a capacitor) built into LED and CFL bulbs, in which case .90 that I measured would have been worse without the power factor correction. But if there is power factor correction (in the form of a capacitor) built into LED and CFL bulbs, then why didn’t the manufacturer add just a slighter greater amount of capacitance so that power factor would be raised all the way to unity (1) ? Hence, I doubt that power factor correction is built into LED and CFL bulbs.

Compact Fluorescent Lamps and LED lamps use rectifiers to produce a DC supply to drive the lamps, therefore they do not require power factor correction, but they do have transient suppresion (snubber) capacitors.

“… nearly all residential and single phase power factor values are essentially the same due to power factor correction capacitors being built into most domestic appliances by the manufacturer.”

I don’t think that is correct. I know of no consumer appliances that include power factor correction capacitors. Perhaps there is one somewhere, but I haven’t seen it.

In the vast majority of residences, there are a few motors, which are highly inductive. There are also a large number of light bulbs and perhaps an electric range and a toaster, all of which exhibit minimal inductance. Therefore, the total power factor at a residence is usually around 0.85 to 0.90, which is not enough to cause issues for the public power supplier.

Even LED and CFL bulbs have a power factor not much below 1. I measured a few LED table lamp bulbs and found that the PF was 0.86 to 0.87. I also measured some CFLs, and the power factor was about 0.90.

In domestic motor run appliances (white goods), electric discharge lamps and fluorescent fittings, reactive power compensation is provided at all voltage levels by fixed capacitors. Obviously, standard light bulbs, electric fires and toasters use resistive elements operating at or around unity power factor.

Thanks.alot for valuable information and good idea for electrical.

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very helpfull

Very good explaination. Quite useful