## Archive for May, 2011

### Euphonic Distortion?

Tuesday, May 17th, 2011

Why is it that every time you mention that a piece of audio equipment sounds better than another, the spectre of “euphonic distortion” is invoked?  Is it possible that it is really that simple?  You just add some 2nd harmonic distortion and poof!  Now it sounds better?

Enough hand waving already.  They say low order harmonic distortion makes it sound better, I say it doesn’t.  Who’s right?  Well everybody prepare yourselves because…

That’s right, science!  And there’s nothing you can do to stop me – mwaah ha ha ha!

First off let’s start simple.  Some folks say it sounds better when you add low order harmonic distortion.  One of these words in particular stands out to me: “add”.  Why does this word in particular jump out?  It’s due to the importance of level matching.  Most audiophiles know how critical it is to accurately match audio equipment if there’s any hope to perform a fair comparison, but it seems many people have forgotten in this one area.

If you add distortion, then you must adjust the level so that the RMS level matches that of the original signal!

As a concrete example, let’s consider a very simple audio amplifier: A single bipolar junction transistor.  Here’s an outline of the process to follow:

• Determine the appropriate function of the distortion mechanism
• Determine the average value in order to remove any dc component
• Determine the RMS value and apply a multiplier to compensate

Here is the equation for the single bipolar transistor stage when passing a sine wave of angular frequency $\omega$:

$f(t)=\frac{1}{\alpha }\exp(\alpha \sin(\omega t))$

Where $\alpha$ is the degree of nonlinearity – i.e. lower distortion for smaller $\alpha$ and higher distortion for larger $\alpha$.

Integrating over a full cycle to get the average (i.e. dc) value:

$f_{AVG}=\frac{1}{2\pi }\int_{0}^{2\pi }\frac{1}{\alpha }\exp(\alpha \sin(x))dx=\frac{1}{\alpha}I_{0}(\alpha)$

Where $I_{0}(\alpha)$ is a modified Bessel function of the first kind.  Now subtract this dc component from the original equation for the transistor stage to get the ac value:

$f_{AC}(t)=\frac{1}{\alpha }(\exp(\alpha \sin(\omega t))-I_{0}(\alpha))$

From this the RMS value can now be calculated.  Here is a general expression for calculating the RMS value of a function:

$f_{RMS}=\left [ \frac{1}{t_{2}-t_{1}}\int_{t_{1}}^{t_{2}} f^{2}(t)dt \right ]^{1/2}$

Now using this to calculate the RMS value of the single stage bipolar transistor when passing a sine wave:

$f_{RMS}=\left [ \frac{1}{2\pi }\int_{0}^{2\pi}f_{AC}^{2}(t)dt \right ]^{1/2}=\frac{1}{\alpha}\left [I_{0}(2\alpha)-I_{0}^{2}(\alpha) \right ]^{1/2}$

Here’s a plot of this $f_{RMS}$ function versus the distortion parameter $\alpha$:

So now to make the RMS level of the distorted sine wave equal to that of the undistorted sine wave, you must multiply the amplitude of the pure sine wave by the above $f_{RMS}$ expression.  You could instead divide the distorted sine wave by the same value – the point is to make the two RMS levels equal.

For clarity here are the two expressions with differing distortion, but equal RMS level – again $\alpha$ is the distortion parameter (distortion → 0 as $\alpha$ → 0):

$f_{UNDISTORTED}(t)=\frac{1}{\alpha}\left [I_{0}(2\alpha)-I_{0}^{2}(\alpha) \right ]^{1/2} \sin(\omega t)$

$f_{DISTORTED}(t)=\frac{1}{\alpha }(\exp(\alpha \sin(\omega t))-I_{0}(\alpha))$

As a further point of interest, the nature of the distortion mechanism considered above is that of a nice descending spectrum of harmonics, as shown in the following FFT plot for $\alpha$=0.1:

I’m still doing a bit more work with this, so it is by no means anywhere close to conclusive.  However, it is at least an important, and objective, difference that must be accounted for in order to compare “apples to apples” when evaluating the subjective effect of different distortion mechanisms.

### The Scent of an Amplifier

Monday, May 16th, 2011

Recently I tried my hand at perfumery.  As with most disciplines there is so, so much more to it than you would ever think.  Interestingly enough, it reminds me very much of audio amplifier design (funny how many things tend to do that for me…).  Here is a brief summary of my first attempt:

• Took a look at the original Eu de Cologne recipe as a starting point (or at least the closest thing I could find to it on the internet).
• Ordered up the necessary essential oils: lemon, orange, tangerine, bergamot, lime, grapefruit, neroli, lavender, rosemary, thyme, petitgrain, jasmine.
• Added amounts that seemed “about right” to a base of Grey Goose vodka.  Let it rest for a while to let all the scents come together nicely.

The result?  Nice, but not wonderful.  Why?  What did I do that was so different than anybody else?  What are the variables I have control over?  Thinking this over for a bit I came to the following realizations:

• Simplicity is best!  Whether creating a new culinary experience, a cologne, or an amplifier you should only add the essential.
• More is not always better, even if it’s more good stuff.  You must have a specific purpose in mind to add even one more thing.
• Quality is crucial!  All essential oils are not created equal.  I later purchased some very good (and expensive) petitgrain and jasmine.
• There is no substitute for quality when trying to create the best.  It is a necessary, but not sufficient, requirement.
• Iteration is necessary!  You can keep it simple and use good quality ingredients, but you still need to try and try again.
• This is how we learn and how true genius comes about (1% inspiration and 99% perspiration…)

Already you probably see how this applies to the design of audio electronics – specifically audio amplifiers in my case.  Lessons to take from this for amplifier design:

• “Perfection is achieved, not when there is nothing more to add, but when there is nothing left to take away.” – Antoine de Saint Exupery.
• Do not add more and more stuff, even if high quality, instead take away everything you can while maintaining the fundamental functionality.
• Good components are crucial.  Do not waste your efforts by pinching pennies in this area – you will ultimately pay even more in service costs and reputation.
• For example, a “thick film” resistor in your signal chain can introduce a surprising amount of distortion, yet only saves you less than a penny!
• Question: “How do I get to Carnegie Hall?”  Answer: “Practice, practice, practice.”
• Prototype, prototype, prototype.  That amazing zone where theory meets reality.  Do lots of it – it’s fun and results in beautiful audio amplifiers!

As a parting thought, here is another area of perfumery with surprising parallels to audio electronics.  There are three distinct categories of perfume scents, based on the time frame in which they assert themselves: The top notes, the middle or “heart” notes and finally the base notes.

Top Notes

These are the scents that form the initial impression of the perfume.  They dissipate quickly, but are important for the same reason all initial impressions are important.  Audio equipment also has “top notes”.  This is what you first noticed when you first heard the equipment, but which faded from your conscious notice after the break in period.  Now only your friends notice these notes when hearing your system for the first time.  Although if you haven’t listened to the equipment for a while you may be reminded.  These are the traits that it is easy for your brain to adapt to.

Heart Notes

These are the primary scents showcased by the perfume.  Once the ephemeral top notes have dissipated this is mostly what you smell.  With an audio system this is what you are left with after the break-in period.  This is the true nature of the gear – it is for these notes that you love it or hate it.  These are the traits of your audio equipment that are not as readily “filtered out” by your brain.  Maybe that’s a good thing and maybe it isn’t – it all depends on how much you enjoy that particular trait!  It is for these notes that a brand of audio equipment becomes famous or infamous.

Base Notes

The base notes are there to anchor the scent, and they do so in a well designed perfume, however they can also be the unpleasant scents left behind once the top and heart notes have dissipated.  Your audio equipment, which one is it?  Does your listening session start off great, only to lead to disinterest after a while?  “Listener fatigue” perhaps?  Or is your audio equipment the kind that invites you in and doesn’t let go until you really must be moving on to other things…  After everything else has dissipated, this is why you either keep the equipment, or put it up for sale.

### The Olympus

Friday, May 6th, 2011

The newest member of the Hephaestus Monaural Amplifier series is now available:  The “Olympus“!  The name is a reflection of the position the Olympus takes at the top of the HMA series.  It is also at the top of the audiophile amplifier world I believe, but my opinion toward both my children and my amplifiers is somewhat biased.  🙂

The Olympus boasts an impressive 2000 Wrms into 4O while still maintaining the miniscule 3.5 pound package of the HMA series.  It also makes use of the premium chassis and connectors of the Keledones.  Best and most important of all though, it maintains the amazing sound of the HMA series, so there’s no worry of sacrificing warmth and beauty for power and headroom.

The Forge page has been updated to better call out the differences between the members of the HMA series.  The naming convention has been simplified too – moving away from engineering-inspired names, such as the “HMA-1000”, toward, well, more name-like names!  The series now consists of the “Hephaestus”, “Aphrodite”, “Keledones” and “Olympus”.