Electrical – what causes voltage harmonics

What are the sources of harmonics? Why are they generated?

Assume that we have a voltage waveform of the form:

\$v_s(t) = \cos(\omega t) \$

In words, the source voltage waveform is composed of a single sinusoid of (angular) frequency \$\omega\$.

If this waveform is the input to a linear circuit, the output will also be composed of a single sinusoid of the same frequency as the input.

For example, a linear voltage amplifier scales the input signal by some constant \$A_v \$:

\$v_o(t) = A_v v_s(t) = A_v \cos(\omega t)\$

Now, consider what happens when the amplifier is non-linear. For example:

\$v_o(t) = A_v v_s(t) + 2\alpha v^2_s(t)\$

This amplifier has a 2nd order non-linearity. By a simple trigonometry identity, we have:

\$v_o(t) = A_v \cos(\omega t) + \alpha[1 + \cos(2\omega t)]\$

See what happened? The output is no longer composed of a single frequency but, due to the non-linear term, now has a DC component as well as 2nd harmonic component.

If instead of a 2nd order non-linearity, the amplifier had a 3rd order non-linearity:

\$v_o(t) = A_v v_s(t) + 4\beta v^3_s(t) \$

you might guess that a 3rd harmonic will be generated. Let's see:

\$v_o(t) = (A_v + 3\beta)\cos(\omega t) + \beta \cos(3\omega t)\$

Note that the 3rd order non-linearity creates a 3rd harmonic as well as an additional 1st order term.

Essentially, even-order nonlinearities generate even harmonics while odd-order nonlinearities generate odd harmonics.

Now, a symmetric circuit, such as a complementary push-pull circuit, generates odd-order harmonics for the reason that the even-order nonlinearities cancel.

An example of a circuit that creates 2nd order harmonics is a single-ended FET (a square-law device) amplifier.

Note: I am not an expert in harmonics.

It's valid to ask why the harmonics levels are set to the exact level they are.

I am currently reading AS 61000.3.6 (internationally, IEC 61000.3.6) which is about the nitty-gritty details of how the utilities calculate the harmonics emission limits for MV/HV/EHV customers.

Basically, it goes like this:

• Compatibility levels: Equipment is designed to tolerate / be immune to harmonic distortion up to some level, the compatibility level. If we can keep the power system harmonic levels below the compatibility levels, the equipment will work. (The compatibility levels are defined in IEC 61000.2.2 and IEC 61000.2.12.)

  The harmonic effects being controlled are long-term effects (overheating of cables, motors, transfomers, capacitors, etc.) over a period of minutes, and short-term effects on electronic devices over a period of seconds.

• Planning levels: If we keep the amount of harmonic emissions at 'Timbuktu substation' to '10 units' or less, we will meet the compatibility levels. (IEC 61000.3.6 goes into detail about how to calculate '10 units'.)

• Apportionment of planning level to individual customers: Bob's Factory is using '5 units' of power, out of '10 units' available at 'Timbuktu substation'. We will allow Bob's Factory to emit a maximum of '5 units' of harmonics.

The key point is that the objective of the harmonics emission limits is to ensure that the compatibility limits aren't violated.

If each customer respects the harmonics emission limit assigned to them, then the harmonics emission level of the system as a whole will be less than the compatibility level, and everyone's equipment will work. If you exceed your emissions limit, you'll probably bugger it up for everyone.

EDIT:

You ask why there are limits on harmonics, even within your own islanded installation, where you won't be affecting other customers.

From Schneider's Cahier Technique 199 Power Quality we have the following list of negative effects from harmonics:

From Schneider's Cahier Technique No. 152, Harmonic disturbances in networks, and their treatment, we have a few more effects not mentioned above:

2.1 Instantaneous effects

Harmonic voltages can disturb controllers used in electronic systems. They can, for example, affect thyristor switching conditions by displacing the zero-crossing of the voltage wave (see IEC 146-2 and Schneider Electric "Cahier Technique" n°141).

...

Interference on communication and control circuits (telephone, control and monitoring)

Disturbances are observed when communication or control circuits are run along side power distribution circuits carrying distorted currents...

Some of these effects might be severe. For example, if harmonics were so bad as to cause interference on a critical control circuit, causing maloperation of that circuit, that would be bad regardless of its effect (or not) on other customers.

So, from this perspective, the limits are there to force you to consider electromagnetic compatibility with vulnerable equipment in your own installation.

Anecdotally, dirty power supplies cause all kinds of intermittent and hard-to-find issues.

My boss told me a story about how they installed a plant full of VSD's (VVVF drives) with no harmonic filtering. The problem was that substations would trip on earth fault (?) seemingly at random when drives were started up - including substations on the far side of the plant to the VSD being started!

Eventually it was determined that the relays were mal-operating due to harmonic currents. From memory, the harmonic current were flowing through the entire plant via earth paths, including structural metalwork and metal cable trays, which is why a VSD on one side of the plant was able to affect a substation on the other side. The electrical paths had to be broken up by replacing some sections of metal tray with plastic trays, and so on.

In the end the problem was only solved after an expensive course of rectification works was performed. Again, the limits are there to ensure that you have to consider harmonics from the start, so this doesn't happen to you!