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Why
single-ended tube amplifiers?
Eduardo
de Lima
Single-ended triode tube
amplifiers are not that hard to find anymore. You can buy them from many
companies, or you can build one with parts that are offered by dozens of
suppliers. The high-end audio press has been talking about them for the past
several years and it is very strange that something that goes against all
the traditional ways of designing, building and evaluating amplifiers has
stayed with us for such a long time. We should be shocked by how good a
single ended tube amplifier can sound when paired with a compatible
loudspeaker. It can sound like music and at the same time it can exhibit
very high distortion and very low output power on the test bench.
To say that people enjoy
distorted or heavily colored sound does not seem to be a reasonable answer.
If we take a look at who has been using these amplifiers, we will have to
agree that among them we can find many highly respected listeners who have a
keen ear and years of live music exposure. There must be some very good
reasons for this apparent paradox. What I intend to do in this article is to
suggest and explain one important reason which I believe has not been given
enough attention up till now. After about five years of designing and
building amplifiers with single-ended circuits and the loudspeakers to work
with them as a system, I would like to describe some of the things I have
observed. They may be useful in solving the giant puzzle that tubed
single-ended amplifiers have presented to us.
2. Some Good Reasons
....
Looking back at what has been
said about this subject -- and leaving the "people like second harmonic
distortion and colored sound" type of argument aside -- we are left
with many important articles, which touch on the subject of distortion in
amplifiers in general, and some about the specific single-ended case. The
articles by Norman Crowhurst reprinted in Glass Audio (The Amplifier
Distortion Story part I & II, GA 6/95 & 1/96) and the article by
Lynn Olson, in Valve (12/97) have several reasons for the good sound
of amplifiers without feedback. They are very important for understanding
the true meaning of the distortion specifications in power amplifiers. The
articles by Scott Frankland in Stereophile ( SE vs. PP - starting
12/96 ) are very good references about all the problems and benefits of
single-ended topology. This may have its roots in the old question of tube
versus transistor sound that is very well covered by the 1972 AES paper
written by Russell Hamm. It points to many important effects and explains
the difference between tube and transistor amplifiers. The paper looks at
the overload characteristics of tubes, transistors and integrated circuits
and describes several issues related to this behavior, including some
psychoacoustic ones. It helps to explain why tubes may sound better than
transistors. This same paper explains why tube amplifiers usually can play
louder than a transistor amplifier of the same specified output power before
we reach for the volume control to turn it down.
Although the arguments used
by Mr. Hamm can also be used to partially clarify the non-feedback
single-ended tube amplifier case, a lot is left to be explained. Mr. Hamm
was looking at the reasons why tube amplifiers sounded different (and
usually better) than transistor amplifiers. But these tube amplifiers were,
as a group, very well engineered and had very decent specifications. They
had been designed following the rules of the low distortion/low output
impedance paradigm and the main difference in sound between them and
transistor amps could be related to the way they behave when overloaded.
Looking backwards, we may think that part of the difference in sound could
also be explained by the fact that tube circuits change their operating
point with age - specially with the passive components of the 50's and 60's
- and the chances that a tube amplifier would run at some point in its life
with some imbalance were certainly high. But in any controlled test, the
first thing you should do is to make sure that your equipment is within the
specifications! Therefore if the imbalance was one of the reasons why a tube
amplifier sounded better, people would never find it in controlled tests. On
the other side, most early single-ended amplifiers have been built entirely
by ear, in complete disregard to the low distortion/low output impedance
paradigm. The better sounding ones always seemed to have high distortion
when finished. More than once have we heard single-emded amplifier designers
comment that "whenever some of the specifications got better, the sound
got worse" .
Almost all of the dozens of
articles that I have read in favor of single-ended amplifiers try to show
that even with the high distortion and high output impedance, there are a
lot of good reasons for them to sound good. Yes, they should sound good despite
these awful specifications. And there are some very good points in
these comments. I have tried to list all apparently correct technical
reasons for the good sound of single-ended amplifiers given in several
articles and to broadly classify them. After doing it, I finally arrived at
five great groups:
- SE amps usually have
extremely simple circuits with a very low parts count. With all the
current vision that passive components do affect the sound, it is easy
to conclude that the less the better.
- Most single ended amps do
not use global negative feedback. Negative feedback can improve the
measured distortion levels but usually change considerably the
distortion spectrum and may produce serious problems with complex loads
and during recovery from overload.
- There is no need for a
phase splitter stage as in push-pull tube designs. The signal does not
need to be inverted and recombined at the output stage. It is not hard
to agree that it should be better to avoid processing the audio signal
this way.
- A single ended output
stage has to work in pure class A always. There is no way to make a
single ended output stage work in any other class (AB or B). These other
classes of operation require, by definition, that we switch the
conduction from one device to another and in a single ended output stage
there isn't another device.
- The output transformer
will be very different. It will not need to rely on the balance between
each half of the primary and will have an air gap that changes favorably
its distortion spectrum.
3. ..... and One More
Reason
Here I will take the opposite
approach and show that not despite but due to the high
distortion (and also due to the high output impedance), single-ended
amplifiers can be a sensible way of putting together a system with a
performance that in some areas, with present day technology, may be
extremely expensive or out of reach using the traditional methods of
designing amplifiers and loudspeakers.
What follows from here needs
one comment. I am assuming that people always prefer the sound from a system
that has lower distortion. Lower distortion of any kind! I can see no reason
for this not be true. I agree that psychoacoustic tests may find that some
kind of distortion is good or may improve a particular aspect of the
reproduction, but I believe that in the long run, we all look for less
distortion. But, in any case, even if we are looking for some distortion
effect, everything that I will say will still hold true and may even help us
to achieve this target.
4. Total System Distortion
What is the distortion that
we actually hear? I think everyone will agree that we hear the distortion
that reaches our ears. There are several types of distortion. Harmonic
distortion, intermodulation distortion, frequency distortion, phase
distortion, time distortion, dispersion distortion and so on. And they are
usually interrelated. It is easy to see that we can quickly get lost with
all this. But we have agreed on something very important. It is the
distortion that gets to our ears that matters. We will see why I am
repeating this in a while. This distortion is the addition of all the
distortions in the sound system. We should add the distortion from the
microphone, mic preamp, mixer, recording gear, mastering gear, the medium
(tape, CD, LP or whatever), the source (tape, CD, turntable/arm/cartridge or
whatever), preamp, amplifier and finally the loudspeaker and the room.
When dealing with amplifiers,
the total harmonic distortion (THD) has always been the preferred
specification for them. And the high measured THD is the main criticism that
single-ended amplifiers have to face. Therefore, lets examine its case.
What is the best way to
obtain the lowest distortion sound at our ears? Make everything perfect! No
distortion at all. This is much more easily said than done! Although most
parts of a state of the art digital based system can be made to have very
low measured harmonic distortion, this cannot be said of the speakers. At
the present technology, moving coil drivers will have much higher THD than
any other electronic device in a system. And if we do not count the room,
the loudspeaker is the last element in the system. The total harmonic
distortion that reaches our ears is the sum of the whole system distortion
up to the amplifier plus the loudspeaker distortion. And in our state of the
art digital based system with a very low distortion amplifier most of the total
system THD we will measure will be the loudspeaker distortion. But still
we can hear the sound of different low distortion amplifiers, DACs or any
other low distortion piece of equipment.
Why can this happen ? Moving
coil speakers drivers are essentially a very simple electro-mechanical
system and they normally have a very high total harmonic distortion
specification compared to any other device in this digital based chain. But if
we examine the distortion spectrum of any reasonable speaker we will see
that all the distortion is low order. In most cases it is mainly second
harmonic with some third harmonic also appearing with lower levels. The
higher order terms will only become noticeable near the limit of power.
On the other side the so called low distortion electronics may have high
order harmonic distortion (and sometimes transient and/or non harmonic
related distortions ) that even in relatively small amounts may be noticed
by the ear. This will not be shown as a high THD reading. And they will came
through the speaker unmasked because of the low order distortion spectrum
that most loudspeakers present.
What will happen when we put
a high distortion tube single ended amplifier in the same system in place of
the low distortion amplifier ? Single ended tube amplifiers are very simple
electronic devices. They normally have a high harmonic distortion
specification. But if we examine the distortion spectrum of most single
ended amplifiers we will see that all the distortion is low order. In most
cases it is mainly second harmonic with some third harmonic also appearing
with lower levels. The higher order terms will only become noticeable near
the limit of power. I have repeated the same words as above when
describing the speaker driver distortion because this is the key. Single
ended amplifiers usually do not have problems like high order or transient
distortions and have low order distortions just like loudspeakers. If we
understand that second harmonic distortion is a mathematical construction
that happens to describe an aspect of nature and has a precise definition
that does not take in account if it was generated electrically or
mechanically or by whatever means, we can take a look at how do the
distortion from the single ended amplifiers adds to the loudspeaker
distortion and I believe we will have discovered where many pieces of our
puzzle should be placed.
Even if you do not like
mathematics you should look at the Appendix where I show how second harmonic
distortion will sum. This is not at all intuitive at first sight and I will
resume in plain words some of what all the math tells:
If two devices have only
second harmonic distortion in the same quantity (lets say 1%) what happens
if we connect them one after the other ?
The result will depend on the
phase of this second harmonic in relation to the fundamental in each of the
devices. (this phase can be anything from zero to 360° or if you prefer
from -180° to +180° , always spanning a 360° range). At this point I
should say that the multiplication will generate 3rd and 4th harmonics but
at much lower levels. Lets look at the 2nd harmonic:
- If the two devices are
just equal, with the same phase relation between the distortion and the
fundamental, the result will be a total of 2.0% of 2nd harmonic
distortion.
- A 30° difference in the
relation of the 2nd harmonic and the fundamental between the two devices
will produce a total of 1.93% of 2nd harmonic distortion.
- A 60° difference will
produce a total of 1.73% of 2nd harmonic distortion.
- A 90° difference will
produce a total of 1.41% of 2nd harmonic distortion.
- A 120° difference will
produce a total of 1.0% (exactly the same distortion as any of the
devices alone)
- A 150° difference will
produce a total around 0.52% (about half of the distortion of any of the
devices alone)
- If there is 180°
difference in the relation of the 2nd harmonic and the fundamental
between the two devices the sum will be zero 2nd harmonic (yes that is
right, in this ideal situation there would be NO 2nd HARMONIC DISTORTION).
(*)
(*) The total distortion
cancelation of the second harmonic may not occur if the complete output of
the amplifier with 2nd harmonic allows DC to pass and the distortion
producing mechanisms are DC coupled to the output. Then some 2nd harmonic
will result that will not be cancealled. But it should only be noticeable at
very high levels since it has a coeficient with E4.
But what are the chances that
we can get this complete distortion cancellation, as it is normally called ?
Very few, just like there are very few chances that the distortions will
fully add so we would have 2.0%. If the difference between the phase of the
2nd harmonics is between 0° and ± 120° there will be an increase in the
total system distortion compared to the distortion of one device alone up to
a maximum of 2.0%. If the difference is between ± 120° and ± 180° there
will be a reduction in the total system distortion down to a minimum of
zero! Of course the audibility of different levels of distortion may not
follow the same relation as these numbers suggest, but there is a great
possibility that the reduction may be more noticeable than the increase.
And what are the chances that
we have one condition or the other ? Lets start by imagining that for any
frequency the phase of the second harmonic could be any, with no preferred
behavior. We could say that we would have a 66% (2x120/360 = 2/3)chance of
increasing the distortion over one device distortion alone and a 33%
(2x(180-120)/360 = 1/3) chance of decreasing. Real world single ended
amplifiers and real world speakers will definitely show one preference for
one case or the other at several frequency ranges. And now we have the
really great thing about all this: If there is more distortion in the
system than in the speaker alone in most of the frequency range, just
reverse the polarity of the connection between the amplifier and the speaker
(This is the ONLY place to invert it) and you will have shifted 180° the
amplifier. Now you have a good chance of getting less distortion than you
started with and probably less distortion than the speaker alone! This is
usually clearly audible. (You can flick the polarity switch in your DAC
to restore absolute polarity. This has nothing to do with it. It also has
nothing to do with reversing the primary windings of the output transformers.
It may change the relation of several important parameters to ground
altering the frequency response). We can say that including the possibility
of reversing the polarity, at any given frequency, we would have a 66%
chance of decreasing the distortion compared to the distortion of one device
alone against a 33% chance of increasing it.
Obviously things could not be
that simple. I have described an ideal case. There are three things to
remember. First I am talking about one frequency. When you change the
polarity you changed it for all frequencies. Second, what I have just said
assumes that the two devices (amplifier and loudspeaker) have the same
amount of 2nd harmonic distortion. Third, I have assumed that the first
device (amplifier) distortion is independent of the reversal of the load
presented by the second device (loudspeaker). In real world conditions these
two last facts are not completely true, but nevertheless all I have
described occurs partially at all frequencies all the time you hook a single
ended amplifier with a loudspeaker and sometimes it produces music!
Several techniques for
increasing the distortion reduction, extending it to most frequency ranges,
and improving the power tracking, can be used, including some very
sophisticated ones. I believe that most of the tube swapping and several
transformer coupling techniques own a good part of their success to this
decrease in the total system distortion. But the easiest thing you can do is
to invert your connection between the output stage and the loudspeaker (This
is the only place to invert the connection if you are looking for this
effect). If you try to invert the connection of your single ended amplifier
with your speaker you certainly will hear a difference. Sometimes it could
be hard to tell which is best because you may lower the distortion at one
frequency range and increase at another. Usually there is definitely a
preferred way of hooking the amplifier. But you can be sure of one thing,
this effect I described shows that by looking at what is often referred as
its weakness (the high harmonic distortion) a single ended amplifier can be
better in a system than the "straight (perfect) wire with gain"
utopia or than any perfect amplifier! The fact that a single ended amplifier
produces 2nd harmonic distortion in quantities close to a loudspeaker is the
fundamental reason why the system can have less total 2nd harmonic
distortion if the relative phases happen to be on the "good" side.
And you can always try to find the preferred polarity of the connection. It
is not uncommon that, although differently, both connections may sound
better than a normal low distortion amplifier because the reductions will
occur at different frequency ranges but may be more noticeable to the ear
than the increase in distortion. Also do not forget that these amplifiers
are usually free of other defects like high order harmonics, transient
distortions and instabilities .....
The reduction of the other
harmonics (from 3rd up) may also happen, but the conditions and requirements
certainly are not as simple as the ones for 2nd harmonic. It seems to be
much more difficult to happen by chance and to be level dependent. This
deserves further investigation.
All the math in the appendix
is for 2nd harmonic distortion and is valid for one frequency. The way the
distortion varies with the frequency and the way it "breaths" with
the power levels will be very important to how effective the effect can be.
But all this certainly explains why single ended amplifiers are one of the
better ways to have a low distortion system.
5. Some Consequences
Lets see some of the strange
facts that we can explain based on what we have seen:
- The preference
for full range drivers to use with single ended amplifiers
is very easily explained. In a two way loudspeaker the crossover will
direct the fundamental to one driver and the 2nd harmonic to the other
over a range of frequencies. The drivers will not benefit from the
effect I described over a fairly large range of frequencies. The
crossover will also disrupt the original phase relation between
harmonics and fundamental in the amplifier signal reaching the driver.
This may be good or bad but it seems that it can do more harm than good.
The dispersion characteristics and spatial positioning make very hard
for the same effect to happen effectively in the acoustic side. Also the
different amount of distortion of two different driver units increases
the chance that at least one of them will not benefit from the effect.
Loudspeakers with three or more crossover points can reduce this "single
ended" effect considerably. This is also one of the advantages of
using active crossovers.
- Each loudspeaker,
depending on its efficiency and distortion levels will mate better with
single ended amplifiers of corresponding different power ratings and
distortion characteristics.
The famous case of 2A3 amps sounding better than 300Bs with Lowthers
could be such case.
- Even technically
correct global negative feedback when applied in small quantities can
make the sound worse, if the good sound of the system relies on this
effect. The best amplifier for each loudspeaker is not a "perfect
one" but one which has distortions that complements the speaker
distortion. When negative feedback is introduced the single ended
amplifier is no longer a simple electronic device and the increase in
2nd harmonic distortion with power will not follow so closely the
increase in the loudspeaker distortion.
- SE Amplifiers
with less stages usually sound better.
The distortion interaction between stages can cancel the 2nd harmonic
inside the amplifier and also make the distortion increase with power
follow a more complex rule.
6. Some Problems
The 2nd Harmonic distortion
is less offensive to the ears because of its musical relation to the
fundamental but also because of the way it adds as it goes along a system.
Single ended amplifiers can be made such that they will form a system with
the loudspeaker that will explore this synergy fully. Obviously the
injection of 2nd harmonic in any place of the chain can produce the same
results if it can follow the behavior of the speaker distortion with
frequency and power. This becomes progressively harder to achieve as we go
backwards in a system. Single ended amplifiers are almost a natural way of
doing it. Tube preamps or tube gadgets to "smooth" the sound of
CDs can try, just as, may be, LPs do, but going further back in the chain is
certainly wrong. Injecting second harmonic distortion in the recording is
something that may help the sound of the recording in the particular system
used for monitoring but it introduces forever something strange that
certainly will not work with other systems.
7. The High Output
Impedance
Another criticism which is
often raised against single ended amplifiers is that they have a high output
impedance. This is also a very important point because if an amplifier has a
high output impedance the frequency response of the amplifier and
loudspeaker together will be affected by the loudspeaker input impedance.
This input impedance is normally anything but constant. I have described in
great detail all that happens in two articles in Glass Audio (SE Amplifier
Output Impedance I & II - GA 3/97 & GA 6/97). In these articles I
have also considered the effects of the finite primary inductance and I have
shown that in certain conditions this might help reduce the problems of the
high output impedance changing the bass alignment of the loudspeaker. I
believe that the two most important conclusions are:
- If the speaker has its low
frequency design done with the high output impedance of a SE amplifier
taken in account you can not only correct the small signal low end
frequency response but actually get an added bonus with an improved
relation between size, low frequency extension and efficiency.
- The values of the primary
inductance of the output transformers usually can be smaller than
calculated for a fixed resistor values if you are driving real world
loudspeakers.
- The high output impedance
can have several ill effects on a system, but if the system is designed
taking it in account it will actually benefit from it. And certainly
there are the cases where it just happen to "fix" a bad
design.
8. Conclusion
The really good sonic
performance of a well put together system using single ended amplifiers can
be understood if we consider the whole system performance. A system made
using single ended amplifiers is a perfectly valid way of achieving a very
good sound also from the measurements point of view. We just have to measure
the correct parameter. The output of the whole system. The other way
of achieving a low distortion system is to use very low distortion
loudspeakers with almost perfect amplifiers, everything used below their
full power range, but this tends to be also costly and may be at least as
difficult to achieve as a correctly assembled SE system. If we take in
account the distortion reducing mechanism and the low frequency advantage
that the high output impedance may give, it is possible to design SE systems
which can have a measured acoustic performance equivalent or better than the
best examples of the traditional high end systems.
Appendix - How Do
Distortions Add and Multiply
When we talk about harmonic
distortion we usually say a number, like 1%. We need to remember that this
number is an extreme simplification of what is going on and as we will see
it may not represent anything at all without further information about it.
First we need to know what harmonics we are talking about. 2nd, 3rd, 4th,
.... what is the proportion of each of these. We need to know the spectrum
of the distortion. We should have a number associated with each of the
harmonics and if we look at the definition of total harmonic distortion we
can see how we will add all the harmonics to arrive at one number that
represents the THD.
THD = (H22 + H32 + H42 + ............ ) 1/2 (1)
where H2 = % of 2nd harmonic, H3 = % of 3rd harmonic ...
If you do some calculations
you will see that, just as an example, if we add 1% of 2nd Harmonic plus 1%
of 3rd harmonic we will have a THD of 1.4%. But we may arrive at this 1.4%
of THD in many ways. It may represent the summation of 1% of 2nd with 1% of
3rd as we said above or the same addition of any two harmonics. It also may
simply be 1.4% of 2nd harmonic or 1.4% of 3rd harmonic or 1.4% of 10th
harmonic. It may also represent that we have around 0.44% of each of ten
harmonics (something like 2nd + 3rd + 4th +.......+ 11th)!! You can be sure
that the same 1.4% would sound completely different in each case.
As you could see to add the
harmonics to get the final THD figure we needed only the number which
represents the quantity of each of the harmonics. But it is very easy to see
why this THD number can not have any correlation with what we hear, except,
perhaps, when comparing different THD numbers from devices with the same
relation between harmonics.
But what happens if we need
to add (actually multiply) the distortion from two sources. How can we
calculate the final distortion of a system composed of an amplifier and a
loudspeaker as in our case. Now things are a bit more strange. It is not
enough to know the spectrum of the distortion of each of the two devices. We
can not just add the 2nd harmonic of the first device with the 2nd harmonic
of the second and so on, and arrive at a new spectrum. Although it is not
necessary to calculate the total THD number from the spectrum we need one
more information about each harmonic in the spectrum to actually multiply
the distortion of two devices. We need to know the phase of the distortion
of each harmonic relative to the fundamental in that particular device.
Because we need to multiply each of the harmonics of the first device with
all of the harmonics of the second devices taking the phase of each one in
account. It requires only basic trigonometry and looking at the simplest
case of all, multiplying the distortion of two devices which have only
second harmonic distortion, is very useful. After all, loudspeakers and
single ended amplifiers, as we already said, are devices which generate
mostly 2nd harmonic distortion, quite a bit less of third and nothing else
when used at power levels within their ratings.
Lets see what will happen
when we connect two devices. First we will look at a distortionless
amplifier followed by a loudspeaker which produces 2nd harmonic distortion.
Then we will move on to the case of an amplifier also producing 2nd harmonic
distortion followed by the same loudspeaker.
In the first case device #1 (the
amplifier) is perfect.
For an input x = Ecoswt
(where w = 2pf, f = frequency and E = amplitude) (2)
The output will be:
y = Ax ---> y = AEcoswt, where A is the gain. (3)
This would be the case of the
perfect wire with gain. Now we take the output and use it as input to the
second device (loudspeaker) which will have 2nd harmonic distortion. The
output w of the whole system will be:
w = By + ceiqy2 , where B is the gain (4)
c
is a factor related to the amount of the 2nd harmonic distortion and the e
to the power of iq represents the phase q which is assumed
constant for each w . Substituting (3) in (4) :
w = BAEcoswt + ceiqA2E2cos2wt (5)
Using trigonometry identities
we will get the result:
w = BAEcoswt + A2E2ceiq + A2E2ceiqcos2wt (6) (output of the system)
2 2
The first term represents the
original signal multiplied by B A , the second term (which is
independent of frequency (w )) represents a DC component and the third term
is the second harmonic with an amplitude of
A2 E2 c/2 and a phase q .
It should be remembered that
the phase angle q may be a function of frequency but here we are looking at
just one frequency therefore it is a constant. Also, just one real world
remark, the DC component, in our particular case, could displace the voice
coil of the loudspeaker of its normal position but we should not be
concerned with this right now.
Now we will look at the
output of a system where the amplifier and the speaker both produce 2nd
harmonic distortion. y is the output of the amplifier for an input
like (2) :
y = Ax + deifx2 ----> y = AEcoswt + deifE2cos2wt (7)
where d is the factor
which defines the amount of the 2nd harmonic distortion and f is the phase
of this second harmonic in the amplifier. Again using the same trigonometric
identities we get:
y = AEcoswt + E2deif + E2deifcos2wt (8)
2 2
We will ignore the second
term. It is DC and depending where the distortion is generated inside the
amplifier and the frequency response after it, it will probably not be at
the output. Therefore we consider the output of the amplifier to be:
y = AEcoswt + E2deifcos2wt (9) (output with 2nd harmonic)
2
This is the amplifier output
which we will connect to the loudspeaker:
Substituting (9) in (4) will
give us equation (10) :
w = BAEcoswt + BE2deifcos2wt + ceiq(AEcoswt + E2deifcos2wt)2
2 2
Expanding the terms inside
the parenthesis squared, using a little trigonometry and rearranging the
terms we will get:
w = (BAE + cdei(q+f) AE3)coswt + (fundamental)
2
+ (BE2deif+ A2E2ceiq)cos2wt + (2nd harmonic)
2 2
+ AE3cdei(q+f)cos3wt + (3rd harmonic)
2
+ E4cd2eiqcos4wt + (4th harmonic)
8
+ A2E2ceiq+ E4cd2eiq (DC component)
2 8
As we can see the output even
for this simple 2nd harmonic case is not simple. Lets look at the 2nd term:
- The second term is the 2nd
harmonic distortion. It will be reduced to zero if :
(BE2deif+ A2E2ceiq) = 0
2 2
For this to happen we need
that B d = A2 c and that f - q = 180° .
This actually means that we
need an amplifier that produces the same amount of 2nd harmonic distortion
as the loudspeaker at the same power levels and that the difference between
the relative phases of these distortions is 180° . But, more important, it
also means that if the amplifier and the loudspeaker have comparable 2nd
harmonic distortion specifications and differences in the relative phases
bigger than 120° we will have a distortion reduction that can be very
significant.
Here I need to place some
comments about this derivation and about the values of the other terms:
- The assumption that the
distortion producing mechanism is not DC coupled to the output is an
important and very reasonable one at mid frequencies for the case of SE
amplifiers with output transformers. Anyway if the distortion producing
mechanisms are DC coupled we can not have the ideal case of total 2nd
harmonic distortion cancellation because a term with an E4
in the coeficient will appear. Anyway this should be a very small value
except at very high signal levels.
- The terms which correspond
to the 3rd and 4th harmonics generated by the multiplication also have a
much lower value. Not only they are multiplied by cd or cd2
but they depend on E3 or E4.
Therefore at lower signal levels, were most of the music is, they are
really very small compared to the 2nd harmonic term
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