The theory of the Otala amplifiers, and thus the basis for the Electrocompaniet
amplifiers, has been detailed in many papers, mostly by Matti Otala, but also
by several others. At Electrocompaniet these theories were extended, and a
theoretical framework for our amplifiers was established. The following pages
give you an insight into both the original theories by Dr. Matti Otala, and
also the theories built up in the first years of Electrocompaniet. Some of the
thoughts are solely mine, however — don’t assume that Electrocompaniet of
today will vouch for all I say here :-)
This is the original outline for the Theory section. Most items below were
only ever sketched as topic headings on the old site and never actually
written up; they’re kept here as a record of the plan, marked (not yet
written).
On TIM, DIM, nonlinearity and distortion
TIM — this was the starting point of Otala’s theories, and his own idea
from the beginning. In the early ’70s many people had already noticed that
the sound of the new transistor amplifiers was inferior to the sound of
tube amplifiers. How could that be, when the transistor amps had less
distortion than the tubes?
DIM — what it is, and how it relates to TIM. This was when everybody
tried to find measurement methods. (not yet written)
The debates and the quarrels — but Otala was right! (not yet written)
“All distortions are equal, but some are more equal than others.” That was
the feeling at Electrocompaniet at that time, and the beginning of an
understanding of how these distortion mechanisms interacted, and how they
affected the sound — and that the basis of all distortion is nonlinearity.
The nonlinearity is the source, the distortions the symptoms. (not yet
written)
A theoretical framework for building good-sounding amplifiers
A theory of single stages — a single stage is the basis
for all amplification, and at Electrocompaniet a model for the single stage,
and “our way”, evolved.
A theory of multiple stages — multiple stages put together do not behave as
N times a single stage; they interact, and in many cases even counteract.
(not yet written)
A theory of feedback — many people are either pro-feedback or fully against
feedback. At Electrocompaniet we learned to live with feedback in all forms,
local and loop, and learned how to get it as a friend and not an enemy.
(not yet written)
Output stages — that current, what does it do? (not yet written)
How do we calculate the nonlinearity
Calculations of input-stage nonlinearity — this is
simple, once you know how; a table summarises everything.
Calculations of the nonlinearity of multiple stages. This is more
complicated, but by following a set of rules, and abiding by the general
model we use, it is not that difficult. (not yet written)
Calculations of distortions — one thing is the nonlinearity, another is how
much distortion of the different types your amp will measure. (not yet
written)
Designing for lowest distortion. Note that there very seldom exists one
optimum point — there are just too many variables, so you have to use some
creativity and a feel for what’s going on. (not yet written)
How the frequency plane and time plane interconnect (not yet written)
Time-response behaviour (not yet written)
Poles and zeroes (not yet written)
Compensation of an amplifier (not yet written)
Other factors
The damping factor — or output impedance, what is it? (not yet written)
Why there are no simple solutions (not yet written)
Power-supply interaction (not yet written)
RFI — what can it do (not yet written)
The importance of the components (not yet written)
Subsections of Theory
A Theory of Single Stages
The general idea
When you design something, you’re moving your ideas and abstract thoughts
into the real world. Using electronic components, they will never act
exactly equal to your ideas. So the best approach is to approximate their
behavior, and make them work as close to your ideas as possible. Further, if
you choose your models skillfully, you will get a behavior from the device
which very closely mimics what you’re after.
Therefore, the sequence is: Model ⇒ Design ⇒ Measurement ⇒ Listening — and you
circle this sequence until you’re happy :-)
The walkthrough below is rather detailed, to show the general outline of the
procedure. The other derivations are less rigorous.
A model for a single stage
We model a single stage as a voltage-controlled current source. In our ideal
world this means we want an active device working as a transconductance
device. A transistor (and also a tube) is very close to this ideal. But one
has to further improve the circuitry around the active device, in order to
make it behave as closely as possible to this ideal.
What this means is that a perfect transistor, and thus a perfect stage, will
have infinite input impedance, no reverse coupling from output to input,
infinite output impedance, and a finite and constant (with respect to both
the signal and the environment) transfer conductance.
Real transistors are not quite as good as this, but we can improve on the
transistor in order to make it behave more like this ideal. Doing this will
normally make the stage perform better in all respects, but keep in mind that
all rules will turn back on you at a certain stage. There is no such thing as
a free lunch.
If such a stage is voltage-driven, we will reduce the nonlinearity from all
“leakages” back to the input, be it input impedance or reverse coupling. The
dominating nonlinearity will then be the transconductance, which is easy to
control.
A transistor in a common-emitter coupling is the starting point. It has, in
principle, the behavior described above. The following rules exist:
Linear behavior:
Transfer conductance, given by:
$$g_m = \frac{I_e}{V_T}$$
$I_e$ is the emitter DC current and $V_T$ is the voltage equivalent of
temperature, normally equal to 25 mV — the exact formula is $kT/q$, where $k$
is Boltzmann’s constant, $q$ is the charge of an electron, and $T$ is the
absolute temperature in Kelvin.
The inverse of the transconductance is called the dynamic resistance, called
$r_e$.
Current amplification: $H_{FE} = I_c/I_b$, derived for a particular current
$h_{fe} = i_c/i_b$, which applies for small-signal currents around a
quiescent point $I_c$.
Input impedance: $r_{in} = h_{fe} \cdot r_e$
The dominating nonlinear mechanism lies in the transconductance. Since this
is a single stage (not a differential stage) it will generate a smooth series
of harmonics (if stimulated with a pure sinusoid).
If the input signal is given as $x$ (where $x$ can be e.g. $\sin(\omega t)$),
then the output $y$ will be:
$$y = a_1 x + a_2 x^2 + a_3 x^3 + \cdots$$
And $x$ is a relative parameter which must always obey $|x| < 1$ in order for
the series to converge. If $|x| < 1$ then it follows that $|y| < 1$. The
output current is $i_e$ and the output parameter is then $i_e/I_e$. The input
signal generating a current of $i_e$ is $u_{in}$, from the formula
$i_e = u_{in} \cdot g_m$. It then follows from the transconductance formula
that the input parameter we seek is $u_{in}/V_T$.
So, given the input $x$, defined as $u_{in}/V_T$, and the output $y$, defined
as $i_e/I_e$ — what do these things mean?
If $u_{in}/V_T$ exceeds 1, then the varying part of the current exceeds the
quiescent current $I_e$, and the stage is clipping. When the stage is
clipping, our formula breaks down. If we want to find the distortion when the
stage is clipping, we’ll have to resort to Fourier analysis.
To make this into a practical case: assume a transistor running at a current
of 1mA. The $g_m$ is then 1/25 siemens (the inverse of ohm, the unit for
transconductance, although “mhos” — ohm reversed — is also used). An input
signal of 1 mV will then generate an output current of 1/25 mA = 40 µA. But
now note: this is the first-order approximation. As can be seen from the
series expansion above, we also have second- and third-order components. The
first-order coefficient should be pretty close to $g_m$, but what are the
other two coefficients?
The real equation relating input voltage to output current is
$I_e = I_s \cdot \exp\left(\frac{U_{be}}{V_T} - 1\right)$, called the
Ebers-Moll equation. $I_s$ is the “leakage” current, but don’t bother about
it — we’ll soon enough get rid of it. We are interested in finding the
equation for the behavior around the quiescent point. We do this by adding
small deviations $i_e$ and $u_{be}$ to the equation above. Resolving this, we
get the much simpler equation: $i_e/I_e = \exp(u_{be}/V_T) - 1$, and its
inverse: $u_{be}/V_T = \ln(1 + i_e/I_e)$. These equations are called the
signal equations, and will be used to get the coefficients for the series
expansion above. We’ll first make a series expansion of the first equation,
and we get:
We now introduce the efficiency parameter $n_i$. The point of introducing
this parameter is to generate simpler formulas for calculating the
distortion, and to gain a better understanding of how the distortion and
other transistor parameters are coupled.
The parameter is defined as:
$$n_i = \frac{i_e}{I_e}$$
For a bipolar stage without local feedback, as the stage discussed above,
$n_i$ is equivalent to $u_{be}/V_T$, where we only consider the linear part
of the series expansion. The equation above can then be written as:
If the input signal is a sinusoid, then the following equations hold (ask via
the about page if you’d like the proof), where $\text{2ndh}$ is
the second-order harmonic distortion, and $\text{3rdh}$ is the third-order
harmonic distortion:
It is interesting to do the same exercise for FET transistors. The result is
a simpler series, with only first- and second-order components. By inserting
the efficiency parameter, one gets the following equation for the FET’s
second-harmonic distortion:
$$\text{2ndh} = \frac{n_i}{8}$$
Half the amount of the bipolar transistor. Some people have argued that the
FET is a much more linear device than the bipolar. This equation shows that
to be only a partial truth — there is only a 6dB improvement.
Local feedback
It is well known that local series current feedback (read: inserting an
emitter resistor) reduces the distortion. The feedback factor can be written
as:
$$D = 1 + g_m R_e = 1 + \frac{I_e R_e}{V_T}$$
and the resulting efficiency/distortion equations are then:
If the current source feeding the emitters has infinite output impedance, $c$
approaches 0. The formula for $c$ is $c = i_k/(2i_e)$. More information on
this is in my AES paper (not yet ported to this site). There used to be an
ActiveX Single Stage Calculator here as well — it’s retired along with the
rest of the site’s old ActiveX controls.
Summary
Stage type
2nd harmonic
3rd harmonic
Bipolar single stage
$n_i/(4D)$
$n_i^2/(12D)$
FET single stage
$n_i/(8D)$
Ideally zero
Bipolar differential stage
$(m+c) \, n_i/(4D)$
$n_i^2/(12D)$
Open and Closed Loop Frequency Response
The open-loop frequency response is the frequency response of the amplifier
with no feedback — before feedback, or with the feedback network deliberately
broken.
The closed-loop frequency response is the frequency response of the amplifier
with feedback.
These two are closely related. The theoretical closed-loop frequency response
is equal to the open-loop frequency response times the amount of feedback. If
you have 40dB (100 times) of feedback, and an open-loop response of 1kHz, the
closed-loop frequency response is 100kHz.
The formula relating these two is:
$$f_{cl} = \frac{f_{ol}}{1 + A_{ol} D}$$
where $D$ is the feedback factor and $A_{ol}$ is the open-loop gain. The
total denominator expression is what we call feedback.
The open-loop frequency response is determined by the internal compensation
(intended or not) of the amplifier. Many amplifiers are designed with one
stage having a very high output impedance, so the stray capacitance of that
stage’s output determines the open-loop frequency response. For integrated
circuits the open-loop frequency response is either specified, or you can see
it graphically as a function of gain — in the latter case, look at the
maximum gain, which means zero feedback.
Just to remind you: the closed-loop gain $A_{cl}$ is related to the open-loop
gain $A_{ol}$ in exactly the same way as the frequency response, although
inversely.
$$A_{cl} = \frac{A_{ol}}{1 + A_{ol} D}$$
At Electrocompaniet the thinking favored a large open-loop bandwidth. This is
also my opinion, but I feel it shouldn’t be larger than necessary. There is
always a tradeoff, and if you go for too high an open-loop bandwidth, you
reduce the possible amount of feedback you can have. My thinking is that as
long as the open-loop bandwidth is high enough, you should use the rest of
your gain for feedback. This will give you a more optimal design, because the
overall distortion will be reduced.
What determines the open-loop bandwidth
Mostly it is determined by the last voltage-amplification stage. The
collectors of this stage (assuming transistor amplifiers, which these
articles are all about :-)) are connected to the bases of the drivers of the
output stage. The input impedance of these drivers is normally very
nonlinear, and strongly frequency-dependent. This means you can very well get
a major pole here which varies strongly with signal level and the load
(loudspeaker and cables) of the output stage. The solution to this is to
voltage-drive the output stage, thus loading down the amplification stage.
This will also have the effect of pushing up the cutoff frequency at this
point. The EC amplifiers have this pole around 500kHz. The benefits of a
voltage-driven output stage are described elsewhere on this site, including
in an AES paper (not yet ported to this site).