*Republished with permission from the Author – Rod Elliott*

## Introduction

Based on the number of emails I receive and the astonishing number of websites that provide erroneous or inaccurate information, power factor must be one of the least well-understood concepts in the electrical field. I have seen weird analogies with horses pulling rail carts, glasses of beer (no, I’m not kidding) and countless vector diagrams, all trying to explain the concept [1].

Most fail, and I have no doubt that I will probably be no more successful than anyone else trying to explain a concept that is impossible to visualise, and only makes complete sense when you understand it. There is a circular reference there – you won’t make sense of it until you understand it, which can’t be done until you can make sense of the concept. I think you see the conundrum.

However, I shall persevere. I can promise no vector diagrams or horse-drawn carts, but you might want to consider the beer – not as an analogy but as a calmative . However, there are two diagrams that I hope will help – at least a little.

In essence, if you have a poor power factor (either from a single appliance or the whole building), you will draw more current from the mains than you actually use. Enough loads with a poor PF place an additional burden on the energy supplier’s equipment, and larger substation transformers and distribution wiring may be needed. Rest assured – they will not pay for that themselves out of the goodness of their hearts ¹, but will recoup their outlay by charging more.

Therefore, it’s up to the users to ensure that all current drawn from the mains is put to work. With simple loads like motors, PFC (power factor correction) is achieved relatively simply, provided the motor loading is consistent. Things get more difficult where the motor load varies, but this is beyond the scope of a simple explanation.

1 – Although ‘hearts’ were mentioned in regard to energy suppliers, this is only an assumption on my part. There appears to be little evidence that any energy supplier is so blessed, so I apologise in advance for any anguish or offence caused

A direct quote from a document I found shows how little regard is paid to non-linear loads in particular, even though some of the examples given are responsible for very non-linear current waveforms … “*Loads on an electrical distribution system can be categorized as resistive, inductive and capacitive. Under normal operating conditions certain electrical loads (e.g. transformers, induction motors, welding equipment, arc furnaces and fluorescent lighting) draw not only active power (kW) from the supply, but also inductive reactive power (kVAr).*” (Source: NHP – Power Factor Correction). Not a mention of any non-linear effect anywhere in the document, other than a passing reference to harmonic distortion!

In its simplest terms, power factor (PF) is a measure of how effectively electrical power is being utilised by a system. It can vary from zero to one, and the higher the number the better. Only *one* component can produce a PF of zero – a capacitor. It absorbs current with each half cycle of the mains, then gives all of it back with no work being done. A high quality capacitor is as close to an ideal reactive component as we can get. Interesting, but not terribly useful.

## 1 – Definition Of Power

First and foremost, we need to define power. Power is the physical work performed by an electrical machine (the load), determined by the voltage across the load and the current through that load. Actual power with DC always follows this simple definition. Work can be a mechanical function (commonly rotary, such as with motors), or the production of heat. It doesn’t matter if heat is the *desired* work or a by-product of inefficiencies in the electro-mechanical system.

Work is measured in Watts (or multiples thereof). A Watt is one Joule per second. Watts represent real energy, and that’s how you are charged by your electricity supplier. Many industrial customers are penalised if the power factor at the connection point is less than a predetermined minimum (typically ~0.9, but it varies). The new smart meters that are being installed worldwide are also (apparently) capable of charging residential customers for a poor power factor if legislation ever allows it.

Unless the reader is familiar with electrical and electronic concepts, defining power is actually harder than it sounds. As noted, power is usually defined as the product of current and voltage, and for DC it works every time. If we have 10V across a 10 Ohm resistor, it will draw 1 Ampere and dissipate 10 Watts. There is no ambiguity – the answer is as accurate as the voltage and current readings will allow.

Where things get complex is when we no longer have a DC power source. Alternating current (AC) is supplied to business and domestic users worldwide, although it’s outside the scope of this article to explain the very good reasons for this. Suffice to say that this is the case, with power delivered at a frequency of either 50Hz or 60Hz at a variety of voltages. Both frequency and voltage are consistent in most countries, and typical combinations are 230V RMS at 50Hz (most of the world, including Europe and Australia), or 120V at 60Hz (the US and Canada, and a few smaller regions).

Provided the load is still a resistor (as above), 10V RMS across a 10 ohm resistor will still cause 1A to flow, and the resistor will dissipate 10W. The frequency and waveform are (surprisingly) irrelevant. Root-Mean-Squared (RMS) voltage measurements mean that the RMS voltage will cause identical heating in a purely resistive load as an equal DC voltage. 10V RMS is exactly equivalent to 10V DC – but *only* with a completely resistive load (e.g. traditional incandescent lamp, radiant heater, toaster, etc.).

The simple loads referred to above (plus many others of course) are never a problem. Power is calculated as the product of voltage and current …

Power = Volts * Amps

There is no ambiguity, and the answer is always right.

Naturally, there are countless machines and appliances that are not a simple resistor. Motors (large and small) and discharge lighting (fluorescent, metal halide, mercury vapour, etc.) have been the traditional ‘problem’ loads in the past, but there are now many loads – some extremely large – that are non-linear. The applied voltage is a (nominal) sinewave from the mains, but the current is not. This causes problems that cannot be calculated by the ‘traditional’ formula and cannot be corrected using otherwise tried and true methods.

Herein lies the problem!

## 2 – Power Factor

I don’t know why, but it seems that perhaps 75% or so of electrical engineers are still stuck firmly in the past, and fail to understand power factor as it applies to non-linear loads. Most rely on an old short-cut formula that considers only one thing – phase angle. The *correct*, and ideally the *only* way to determine power factor is to use the right formula …

Power Factor (PF) = Real Power / Apparent Power

Real power is that which is measured by a wattmeter, such as the one in your meter box. Real power is always measured in watts, and was previously considered to be that part of the supplied mains that performs work. The reactive part of the current waveform is “returned” to the grid when a motor (for example) is running, and you are only billed for that fraction of the current that is used to perform work. ‘Apparent’ power is the product of RMS voltage and RMS current – volts * amps (VA). Unlike the DC condition, VA is no longer the same as watts with AC mains!

Modern switchmode power supplies have changed the definition, because *they are not reactive*, and nothing is returned to the grid. There is no significant out-of-phase component in the current waveform, but the current waveform is often highly distorted and rich in harmonics. Countless websites and/or engineers will try to claim that these supplies are somehow reactive, and any explanation that claims any significant (measurable) reactance is quite simply wrong. Not just woefully inaccurate, *wrong!*

In reality, there actually *is* a tiny amount of reactance caused by the EMI (electromagnetic interference) filter. However, its influence is very small indeed, the mains waveform is not materially affected, and any ‘phase angle’ that may be introduced has absolutely no bearing on the overall power factor. The power factor is changed by such a tiny amount by the addition of the filter components that it’s not worth considering. The difference can be measured on a simulator, but is unlikely to even register on typical test instruments. I mention this purely in the interests of completeness, and to save people the trouble of complaining that I had failed to address it.

There is a simple thought experiment that I hope will give you and idea of both reactive and non-linear loads. I’ve not seen this one used before, but I’m hopeful that it will convey the idea better than some of the more traditional explanations. If not, I apologise in advance .

The diagram shows the general principle of reactive power (A) and non-linear power (B). Imagine that the surface is slippery, and there’s no ‘stiction’ (static friction, where it takes additional force to get something to start moving [3]) With the reactive case, your task is to push the brick along the plank of wood, but you have to do it with little pulses of energy, all exactly equal and preferably at 100 or 120 times a second. When you push, the spring compresses and the brick moves forward a little (work is performed). As you release (this is AC, remember), the spring will return some of the energy you used. It’s not useful and performed no work, but you had to expend the energy to compress the spring, *and* absorb the energy it returns to you. This is a reactive load, and the power factor is determined by how much energy is used to perform work, versus how much is just absorbed and returned by the spring.

So, when the load is reactive (with the spring) not all of the energy you expend (incoming power) is converted to work – moving the brick. This system has a poor power factor. If you remove the spring, all of the energy you put in will move the brick, as long as it’s enough to overcome friction (resistance). In this case, there is no reaction (energy return), and thus approximates unity power factor. In an electrical machine, you don’t have the option of simply removing the spring if you don’t like it, because it’s part of a circuit that won’t function without it.

Should you decide to use a different circuit, you can move the brick with a hammer (again, swing it back and forth 100 or 120 times a second), there will only be a power transfer in the brief period when the hammer strikes the brick at the peak of the swing (please ignore any rebound – that’s a physical phenomenon that’s not part of this thought experiment). The remainder of the swing is basically ‘idle-time’ when no useful work is performed. So it is with the mains – non-linear loads may only draw current for a few milliseconds each half cycle. The rest of the time the mains voltage still swings back and forth much like the hammer, but converts zero energy into work – namely moving the brick.

To see the electrical waveforms, have a look at Figure 2.

Apparent (aka ‘reactive’) power is only possible with AC, and used to be the result of reactive loads – most commonly motors, but also includes traditional iron-core ballasted discharge lamps. More recently, we have had more and more *non-linear* loads, such as computer power supplies, compact fluorescent lamps (CFLs) and DC plug-pack power supplies. There are also much larger non-linear loads, such as inverter technology air-conditioners and microwave ovens, induction cooktops and many other products (TV sets, home theatre systems, etc.). Indeed, there is an almost infinite number of appliances small and large using switchmode power supplies. A large majority of these switchmode supplies draw a highly non-linear mains current, commonly so different from the voltage waveform that it’s sometimes hard to imagine how the supply network survives.

Many industrial machines also draw non-linear current, and these machines can be very large – massively so. Even more traditional high-power loads such as 3-phase rectifier systems (used to power electric trains and other large DC motors, electroplating tanks, etc.) present a very unfriendly distorted current waveform back to the grid. Being distorted, the current waveform is rich with harmonics ready to cause havoc. Again, the standard phase-based (cosφ) power factor calculations cannot be used with these non-linear loads.

With AC, apparent power is defined as the product of voltage and current. This gives us a figure that’s called VA (volt-amps), and it may or may not be the same as the power measured by a wattmeter. VA can *never* be lower than the power in watts, but with resistive loads they can be exactly the same. This is a power factor of unity – the best that can be achieved. This means that the voltage and current are not only in phase, but have the same waveform (nominally a sinewave). There is no ‘excess’ or ‘wasted’ current.

Traditionally, the favourite formula has always been that PF = cosφ – the cosine of the phase angle difference between voltage and current. It’s a nice short-cut that only works with *undistorted* sinewaves and linear reactive loads. The simple fact is that a vast number of electrical loads are non-linear, and the formula doesn’t apply – it can’t apply, and it is nonsense to imagine that a formula that applies only to clean sinewaves is functional with any non-linear load.

Above, we can see the two main kinds of load – reactive (top) and non-linear (bottom). The reactive load’s current is not in phase with the voltage, and just like the spring in Figure 1 (A) fails to keep the pressure (voltage) and displacement (current) working at the same moment in time. The same happens with a reactive load. The current lags the voltage by a number of degrees, and we can work out the power factor by taking the cosine of the phase difference. As noted though, this *only* works with reactive loads that have no non-linear function. We can also have capacitive reactive loads, but they are extremely uncommon and won’t be covered here.

In particular, look at the upper waveforms for the reactive load. There is a point during a cycle where the *voltage is positive* while the *current is negative* (and vice versa). For any load to be considered reactive, this condition *must* exist during each cycle. If it doesn’t, then the load is not reactive. This is an important concept to understand. Knowing this instantly allows us to look at the lower non-linear current waveform and note that these prerequisite conditions for a reactive load do not exist!

The lower diagram shows a non-linear load – the voltage causes no current flow at all until it almost reaches the maximum. There’s a short burst of current and then nothing until the next half cycle. The traditional formula does not work with a load such as this. Not even a little bit! A vast number of loads combine both reactive and non-linear functions. A perfect example is a traditional fluorescent tube and ballast, which can never be corrected for a unity power factor. One can get close, but the non-linear current prevents total correction. The typical best case power factor for a fluorescent lamp and ballast is around 0.9 – all other discharge lighting is similarly afflicted.

Until such time as the electrical engineers of the world all wake up to reality and understand that their favourite formula does not work with non-linear loads, we will continue to see non-sensible replies to forum posts, completely inaccurate claims being made and expensive reports that completely fail to address the real issues. There are many otherwise excellent papers about power factor and the benefits of power factor correction (PFC), but some fail to even mention the ‘elephant in the room’ – non-linear loads. Some of these reports (they are in the minority) will mention harmonic currents – these are the *direct result* of non-linear loads, and cannot be created by any truly linear load.

## 3 – What Poor Power Factor Does

In a nutshell, poor power factor causes a load to draw more current than it needs to convert into work. It doesn’t matter why the power factor is bad – the result is that the grid has to supply more current than the equipment actually uses to perform the work. Some poor power factor loads can be corrected – either near the machine responsible or elsewhere, and others cannot be effectively corrected at all. Motors and iron-core ballasted lighting equipment are easily corrected, although with discharge lighting systems there are actually *two* reasons for the bad power factor. As mentioned above, all discharge lighting causes a lagging (inductive) power factor, but the current is also non-linear. This means that it is impossible to get unity power factor, because only the inductive component can be corrected.

Non-linear loads cannot be corrected with any conventional PFC system. The current drawn from the mains is distorted, and there is very little that can be done externally to remove the distortion. While it *can* be done, it is a difficult and expensive exercise, and very few external correction systems currently exist. The best way to solve the problem is to use more sophisticated power supplies that incorporate active PFC (see active power factor correction for a much more complete analysis of power factor correction techniques).

If a load has a power factor of 0.5, that means it will draw twice as much current from the mains, compared to an equivalent load with unity power factor. A 230 watt appliance with a PF of 0.5 will draw 2A from the mains (460VA), while only performing useful work amounting to 230 watts. In most cases, householders are not billed for the extra current, only for the *power* used – 230W in this case. Most large industrial users are charged for poor power factor, and there is a financial incentive to make it as good as it can be.

Since so many loads today are non-linear, the old cosφ formula is useless – it can only ever create wrong answers. I’ve seen laboratory reports claiming that a LED lamp has a power factor of (for example) 0.95 … leading! What rot! The measured power factor is due to non-linearity, not phase shift. It’s neither significantly leading nor lagging – it’s non-linear, and the sooner certified test labs and official standards are brought into the 21st century the better. The traditional formula doesn’t even work properly with many ‘legacy’ lighting systems. As already noted, all discharge lighting creates a non-linear current waveform that cannot be corrected – even if the voltage and current are *perfectly* in phase!

So, having a power factor of less than unity simply means that the appliance (whatever it might be) draws more current from the mains than it can put to effective use. It really doesn’t matter why (when only the PF is being quoted), but by knowing about it an electrician can take the extra current into account when running cables for an installation. It’s no longer possible to just look at the power rating – just because a lamp (for example) is rated for 23W doesn’t mean it only draws 100mA from the 230V mains.

If the power factor is 0.5, each lamp will draw 200mA, so an 8A lighting circuit can only handle 40 of them – not 80 as might be imagined. The same applies for any electrical device with a PF of less than 1. Note that for a variety of reasons, it may only be possible to use perhaps 10 of the hypothetical lamps mentioned above on an 8A lighting circuit, especially if all lamps are to be turned on at the same time (with a single switch for example). This is an issue that has already caused grief in some modern lighting installations. CFLs (compact fluorescent lamps) have really lowered the standards, with many struggling to manage a PF of 0.5 – and some are worse. These are non-linear loads that have a current waveform that looks remarkably similar to the one shown in Figure 2.

## Conclusion

This short article is intended as a primer into the mysterious world of power factor. It is quite deliberately devoid of vector diagrams, complex formulae or other distractions, and will hopefully give the reader at least a basic understanding of the topic. The most important message is to forget the cosφ formula, because as long as people keep using it they will most often get a nonsense answer.

There are innumerable sites on the Net that will also describe power factor, but the vast majority completely omit any reference to non-linear current waveforms and naively assume the only sources of a poor PF are motors and other inductive loads. Energy suppliers (who should know better) are just as guilty of this as anyone else. Whole reports [2] have been written (no doubt at great expense) that have totally ignored non-linear loads. Not a mention. Nothing!

The article Active Power Factor Correction has a great deal more information, and there you will find detailed graphs, waveforms and a detailed explanation of PFC in general and active PFC systems in particular.

In case anyone is wondering why I have used so many examples of lighting products, that’s because lighting is generally considered to be the largest single user of energy [4] of all. Inefficient lighting also has other flow-on effects, because additional cooling is often needed to remove excess heat generated by lighting systems, adding to the overall impact.

Overall, there is a net gain when (for example) incandescent lamps are replaced by CFLs, because although the power factor is much worse, their current draw is far less and the power consumed is also lower. The latest LED lighting products are now more likely to use active PFC, so not only do they draw far less current, but also present a friendly current waveform back to the supply grid.

There are a great many companies who sell (expensive) automatic PFC equipment, but only a few will mention non-linear loads. There is definitely a benefit to installing a PFC cabinet if your business operates many motor-driven machines, but if those machines are fitted with VFD (variable frequency drive) then the mains loading is likely to be non-linear unless the VFD utilises active PFC in its circuitry. Adding a PFC system will achieve nothing useful in either case.

Transformers are regularly cited as a contributor to poor power factor, with their magnetising current taking most of the blame. This only applies when the transformer is lightly loaded. At any reasonable load, power factor is usually excellent, provided the load on the secondary of the transformer is linear! If the secondary is used to obtain DC (as with amplifiers and countless pieces of industrial equipment), the load is not linear, and low power factor is again the result of the non-linear load. Traditional PFC systems won’t work, and nor will the outdated and pointless cosφ formula.

It is extremely important that novice readers in particular read the article The Great “Power Saver” Fraud, lest they think that there might be some (financial) benefit if they improve the power factor of their home’s electricals. The devices offered are almost all total frauds, and will not save you a cent. Indeed, if enough people installed these units (which contain only a small capacitor in almost all cases), the grid could be faced with a *leading* power factor that can’t be corrected by substation equipment.

*Republished with permission from the Author – Rod Elliott*

**References**

- PFC as a ‘green’ approach
- Power Factor Correction Evaluation – Prepared for the Australian Building Codes Board
- Wikipedia – Stiction
- Energy Star, Chapter 6 – Lighting
- Active Power Factor Correction