The classic definition of power factor involves the phase-relationship between sinusoidal voltage and current waveforms in a load. If the load is purely resistive, the voltage and current will be in phase, as you can see at the top of Figure 1, and the power, shown in red, is always positive. However, if the load is purely capacitive or inductive the voltage an current will be 90˚ out of phase and the power will swing positive and negative throughout each cycle, but average power will be zero. This is shown in the middle and lower part of Figure 1. Note that in these cases, current flows into the load even though there is no net power delivered!
In a purely resistive load (top) the voltage and current are in phase and the power delivered to the load is always positive. In the case of a capacitive load (middle) or inductive load (bottom) the voltage and current are 90˚ out of phase and the average power delivered over each cycle is zero.
In most real-life loads there will be both a resistive element and a reactive (capacitive of inductive) element. This means there are two components of power – a “real” part dissipated in the resistor and an “reactive” part related to the inductance or capacitance. This is often illustrated in a “power triangle” like that shown in Figure 2 for an RL load The real power (that can do useful work) and the reactive power (that can’t) form the adjacent and opposite sides of a right-angled triangle.
The hypotenuse of the triangle represents the total, or “apparent” power that flows into the load, and the angle ø represents the phase angle between voltage and current. The ratio of real power to apparent power is the power factor. Basic trigonometry tells us that this ratio is the cosine of the angle ø. Power factor can vary between 1 for purely resistive loads and zero for purely reactive loads.
In a mixed load (like most real-life loads) the apparent power is the vector sum of the real power and reactive power. The real power can do useful work, but the reactive power is useless although it nevertheless contributes to the apparent power drawn from the source.
Only real power can do useful work, but we have to consider the reactive power since this does add to the current that must be supplied and therefore drives the dimensioning of the supply infrastructure. This is why we generally want to keep the power factor of AC loads as close to unity as possible.
But this is not the whole story by any means. Many of the loads we use today are not neat linear loads with sinusoidal waveforms as we see in the textbooks. Instead, they are non-linear loads like the near-ubiquitous switch-mode power supplies which fill our homes and workplaces. These rectify and filter the mains producing weird non-sinusoidal current waveforms with narrow spikes where the rectifier diodes conduct to top up the filter capacitors. The current is clearly in phase with the voltage, but what does this mean for power factor?
The answer, I am sorry to report, is “nothing good”. To work out how bad, let’s look at a simple simulation (Figure 3). Here we have a nominal 240V 50Hz mains supply with a typical mains source impedance. This is rectified and filtered to produce a nominal 330V DC bus. A constant current load of 1A is connected to the DC supply.
This simulation of a simple rectifier and filter as you would see in a typical SMPS allows us to investigate the distortion power factor. The simulator calculates the average DC bus voltage and the input rms current for us.
Running the simulation produced the voltage and current waveforms shown in Figure 4. Here we see the current spikes which are aligned to the peaks of the mains input, along with the capacitor voltage. The simulator tells us that the average DC voltage is 331.5V so the real power into the load must be 331.5W since the load current is 1A.
This is the result of the simulation shown in Figure 3. The input current shows short spikes at the peaks of the input voltage when the rectifier diodes conduct, and the capacitors are charged. The harmonics in the input current contribute to the poor power factor of this type of circuit.
Similarly, the simulator calculates the RMS value of the input current waveform (which is quite difficult do analytically) to be 2.66A. This means the apparent power at the input is 240 x 2.66 = 639VA – quite a lot more than we are seeing in the load!
No prizes for guessing that the missing power is reactive. We know the power factor is the ratio of real to apparent power so we can easily calculate it to be around 0.52 – a pretty terrible number I think you will agree. This circuit draws almost twice the current of an equivalent resistive load. This poor power factor is caused by the “distortion” of the current waveform, or more precisely, by the presence of harmonics in the current waveform.
In fact, the proliferation of this type of power supply over the last few decades is a huge issue for the supply authorities. Regulations are in place in most jurisdictions to address this problem by limiting the harmonic currents that devices can draw. One example is IEC/EN61000-3-2 which applies to many devices rated at 75W or above.
There are a number of methods you can use to correct poor power factor in circuits such as this – either passively, by filtering out higher harmonics with the addition of inductors, or actively, with circuits that control the shape and phase of the current to more closely match a sinusoid in phase with mains voltage.
Sponsor this Article
Andrew Levido (andrew.levido@gmail.com) earned a bachelor’s degree in Electrical Engineering in Sydney, Australia, in 1986. He worked for several years in R&D for power electronics and telecommunication companies before moving into management roles. Andrew has maintained a hands-on interest in electronics, particularly embedded systems, power electronics, and control theory in his free time. Over the years he has written a number of articles for various electronics publications and occasionally provides consulting services as time allows.
