# Harmonic Distortion of step-approximation to sine wave

Discussion in 'Electronic Design' started by Norm Dresner, May 16, 2004.

1. ### Norm DresnerGuest

Is there any www-resource which gives values (or simple ways to calculate
the values) for the harmonic distortion of various step-approximations to
sine waves, e.g. 8-bit, 10-bit,...

TIA
Norm

2. ### John JardineGuest

Simply as you find it ...
8 bits can be no cleaner than one part in 256 parts. Therefore THD
=1/256th or 1/2%
12 bits is one part in 4096 parts = 0.025%
ect ect

3. ### Don PearceGuest

Except of course that you naturally generate only a properly dithered
step approximation, in which case there is no harmonic distortion.

d

4. ### John JardineGuest

Yes, a good point. Harmonic distortion can be usefully controlled by adding
natural or man-made dither. But then one is forced to add a rider caveat
and specify a particular range of frequencies that this new THD figure (or
spectrum) is to be measured over. The noise floor grass grows higher and the
S/N must reduce. Kind of spread-spectrummy, or swapping ac volts in exchange
for noise bandwidth. The ultimate reduction to a limit, is dithering a
single bit stream but this now becomes a technically expensive fast DSP
operation rather than simple analogue processing.
I much prefer the direct approach. Knowing beforehand which field my errant
harmonics will choose to graze in and can there pick them off at will. If
the THD is too high then I'll try for more 'real' bits. If my THD is still
too high then tough!, I designed it badly in the first place
regards
john

5. ### Don PearceGuest

What you say is true enough for the harmonics, but unfortunately most
of the discrete product of quantisation is decidedly non-harmonic and
distributed widely up and down the spectrum. From the point of view of
measurement, dither noise is easier to deal with, and from the point
of view of sound quality, dither noise is vastly preferable. Either
way dither seems to win over non-dither.

d

6. ### leggGuest

There are a couple of articles covering n-level power conversion
harmonics, where the number of levels employed is ~ constant and the
phase angle of the steps is considered. I think you are more
interested in other things, if you are only starting to consider 8
weighted levels as a minimum.

I've always thought this 'full-scale' number pretty irrelevant in
audio signals, where the program level is so seldom full scale, and
the distortion is scarcely related to a harmonic, as such. I guess
you've got to start somewhere, and that's how meaningful distortion
measurements used to be made, so......

RL

7. ### Tim ShoppaGuest

If you ignore D/A error... usually integrated D/A converters are specced
to +/-1LSB linearity and +/- 0.9LSB differential linearity
which makes it questionable why you're bothering to dither.

Dithering does make a lot of sense if you have a D/A guaranteed to be
monotonic (most are) and it's part of a feedback loop where you don't care
about specs like THD at all. These are important and
interesting applications but the thought of THD doesn't come up when

Now if you're building a D/A from scratch or can spec the innards
of your D/A to be better than 0.5 or 1LSB then the story can be a little
different.

Tim.

8. ### Tim ShoppaGuest

If you ignore D/A error... usually integrated D/A converters are specced
to +/-1LSB linearity and +/- 0.9LSB differential linearity
which makes it questionable why you're bothering to dither.

Dithering does make a lot of sense if you have a D/A guaranteed to be
monotonic (most are) and it's part of a feedback loop where you don't care
about specs like THD at all. These are important and
interesting applications but the thought of THD doesn't come up when

Now if you're building a D/A from scratch or can spec the innards
of your D/A to be better than 0.5 or 1LSB then the story can be a little
different.

Tim.

9. ### Tim ShoppaGuest

If you ignore D/A error... usually integrated D/A converters are specced
to +/-1LSB linearity and +/- 0.9LSB differential linearity
which makes it questionable why you're bothering to dither.

Dithering does make a lot of sense if you have a D/A guaranteed to be
monotonic (most are) and it's part of a feedback loop where you don't care
about specs like THD at all. These are important and
interesting applications but the thought of THD doesn't come up when

Now if you're building a D/A from scratch or can spec the innards
of your D/A to be better than 0.5 or 1LSB then the story can be a little
different.

Tim.

10. ### Don PearceGuest

I think the OP was talking about distortion inherent in the
quantisation process, rather than simple non-linearities that any
circuit might have. Having said that, it is very many years since I
came across any D/A that exhibited non-monotonicity.

And of course the reasons for dithering are quite unrelated to
failings in linearity of the D/A, but are addressing the fundamental
issue of quantisation distortion, which will otherwise occur in even a
perfect D/A

d

11. ### The PhantomGuest

Can you post the references?

where the number of levels employed is ~ constant and the

12. ### leggGuest

I can post links to the ones I've bothered to save.

RL

13. ### The PhantomGuest

The topic of this thread; n-level power conversion harmonics.

14. ### leggGuest

Don't forget to google on your own time, if you're really interested.

The reasons to avoid or encourage specific kinds of harmonic
distortion, outside of Standards conformance, are probably as
important as the methods outlined to do so.

These are the more obvious ones. Two recent references have lost their
link - if anyone picks up a new link for either of these articles,
posting it (the link) would be a service.

RL

"Opportunities for Harmonic Cancellation with Carrier Based
PWM for Two-Level and Multi-Level Cascaded Inverters"
D.Holmes, B.McGrath
IAS'99

"Minimization of Total Harmonic Distortion
for Square Wave Voltage Source Inverters"
C.Liu1, J.Zhu, V.Ramsden
AUPEC'99
http://www.itee.uq.edu.au/~aupec/aupec99/liu399.pdf

"Optimum Harmonic Reduction with a Wide Range of
Modulation Indexes for Multilevel Converters"
S.Sirisukprasert, J.Lai, T.Liu
IAS'00
http://manuales.elo.utfsm.cl/conferences/seminarios/Eenergy/DATA/48_02.PDF

"Multilevel PWM Methods at Low Modulation Indices"
L.Tolbert, F.Peng, T.Habeytler.
transPE'00
http://www.ece.utk.edu/~tolbert/publications/trans_pe_july_2000.pdf

"Investigation into the Harmonic Behaviour of Multipulse Converter
Systems in an Aluminum Smelter."
S Perera, V J Gosbell, D Mannix, N Gersch,
AUPEC'00.
http://www.itee.uq.edu.au/~aupec/aupec00/perera00.pdf

"Voltage Harmonics Generated by 3-Level Converters
Using PWM Natural Sampling"
O.S‡daba, P.Gœrpide, J.Taberna, I.Morales, L.Palomo
PESC'01

"Harmonic Elimination in Multilevel Converters"
J.Chiasson, L.Tolbert, K.Mckenzie, Z.Du.
IASTED'03.
http://www.ece.utk.edu/~tolbert/publications/iasted_2003_harm_elim.pdf

"A Reduction of the AC Line Current Harmonics in a
Multilevel-Voltage Source Rectifier"
S.Brovanov, S.Kharitonov
Novosibirsk State Technical of University - **no link
KORUS'03

15. ### The PhantomGuest

to the OP, you said, "There are a couple of articles covering n-level
power conversion
harmonics...", and it was because you said "article" rather than
referring to papers you found on the web. I thought that you, like I,
had a collection of xeroxes gotten from the library.

I was already aware of Tolbert and Chiasson's interest in these
topics. Did you notice the paper titled "A Complete Solution to the
Harmonic Elimination Problem" on Tolbert's web site?
http://www.ece.utk.edu/~tolbert/publications/trans_pe_mar_2004_complete.pdf

He seems to have solved the same problem that Don Lancaster
attacks with his Magic Sine Waves method.

17. ### The PhantomGuest

Quite so, which is why I followed what you have quoted above with

" He seems to have solved the same problem that Don Lancaster
attacks with his Magic Sine Waves method."

because that is what Lancaster's Magic Sine Wave method does.

I thought this would explain my digression.

18. ### R.LeggGuest

This does not cover multi-level switching - it covers pulse-width
I am unfamiliar with Lancaster's magic sinewave method. Chiasson, on
the other hand, seems to have 're-discovered' hysteritic control,
though how PWM symmetry is intended to be enforced, on a system that
is variable in any way, is a mystery to me.

RL

19. ### The PhantomGuest

Go have a look at: http://www.tinaja.com/magsn01.asp

There has been some previous discussion on this group, and Google
has some discussion from other forums, but so far no one seems to have
implemented an actual inverter

Chiasson, on

20. ### ddwyerGuest

As well as steps also beware transients between stweps out of D/A caught
me out had to de-glitch with sample and hold.