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Home

Technical Articles

The
Sweet Spot

Nelson Pass 10/10/09

Introduction

Every audiophile understands the concept of the “sweet spot”, that happy
balance of conditions that elicits the best possible sound. It can be a
particular listening spot, the best positioning of loudspeakers, or the
fortuitous combination of components which complement each other perfectly. It
should be no surprise that the desire for the best performance takes the search
for the sweet spot into the interiors of the components themselves.

This article will concern itself with finding the sweet spot for each gain
device in audio amplifiers. It is a commonly held belief in audio that the best
amplifiers are composed of one or more active gain stages, each made as
intrinsically linear as possible before negative feedback is applied to further
improve performance.

Taken to the extreme, linear stage = good / negative feedback = bad. The
latter half of that statement is the subject of controversy, but there is no
argument about linear stages being good.  Achieving intrinsically low
distortion in active devices (tubes or transistors) is usually not easy. We can
use good parts, see to it that they have adequate voltage and current available,
and use them in topologies that play to their strengths. These are the basics,
and this is what you generally see when you examine schematics of popular
amplifiers.

However each individual device has a sweet spot where its performance is the
best, often dramatically better, and finding that sweet spot is a powerful
technique for maximizing the performance.  I am writing about this because
over the years of communicating with DIYers and many professionals I have
discovered that this is a poorly understood concept, and I have seen many
circuits which have failed to take advantage of it.

For every gain device, there is a number which characterizes the amount of
gain. When we talk about distortion we are talking about the alteration of the
amount of gain in the device as the conditions change. The gain of a tube or
transistor is altered primarily by changes in voltage across the device, current
through the device and the temperature of the device. Changes in any of these
conditions changes the gain, and all gain changes make the distortion. More
precisely, they are the distortion.

Of course the concept of a sweet spot depends on having an idea what of what
constitutes the best performance. It could be that you want the lowest measured
distortion, a particular mix or phase of harmonics in the distortion waveform,
the greatest efficiency, greatest power or simply the best subjective experience
when you listen to it. The sweet spot is whatever you want – after all, you are
the designer.

I Brake For Tutorials
If you don’t mind a little basics, let’s look at
examples of gain devices in generic circuits, starting with a triode:

This illustration shows simplified examples of the two most common of gain
stages using any kind of gain device, and the device pins are the Grid, Cathode,
and Plate. In both cases the input signal appears at the Grid and variation of
voltage between the Grid and Cathode causes a change in the current flow from
the Plate to the Cathode. (All the examples you will see are simplified, and
omit DC bias that may be present on the pins except for the V+ supply).

In Common-Cathode mode, the tube develops both current and voltage gain, and
the output is taken off the plate, inverted in phase and usually increased in
amplitude.

In Common-Plate mode, the tube seeks merely to have the Cathode voltage
follow the Grid voltage, so the output voltage is ideally the same as the input,
but with some current gain. That is why it’s called a follower.

For both modes we would normally look for the sweet spot, the conditions
giving the best performance, by varying the values of the supply V+, the value
of the load resistance, and the bias current through the device. Inevitably
there will be some combination of these which is the best. Sometimes one of the
circuit conditions is fixed – often it is the load value, in which case we vary
the other conditions – the supply voltage and the bias.

In all of the examples we will look at later, I have chosen to simply look
at Common-Plate (or Common-Drain / Common-Collector) performance, but the same
ideas apply to Common-Cathode (or Source or Emitter) operation.

Actually, Common-Drain circuits would seem to offer the least opportunity
for improvement - as followers enjoying 100% degenerative feedback, they already
have a much lower distortion character than Common-Drain circuits. But we will
see that even they can be improved, often dramatically, if you find the sweet
spot.
Here’s the example of the JFET counterparts to the tube circuits seen
above:

Here the pins are Gate, Source, and Drain. In the Common-Drain follower, the
input voltage pretty much follows the output voltage, and in the Common-Source
amplifier, the output is inverted and has both voltage and current gain
available.

Here’s the MOSFET example, and it has the same pin designations as the
JFET.
And last,

And last, here is the example of a bipolar transistor, where the pins are
Base, Emitter and Collector:

The choice of a particular type of gain device - tube or transistor - is
often arbitrary, but ideally it plays to the particular strength of the device
for a given application. In principle you can substitute one type for another,
but as a practical matter, there is usually a good reason for the selection of
one over the others.

It’s doesn’t matter in terms of the idea of finding the sweet spot – the
terrain changes at bit, but the game remains the same.

{pagebreak}

Let’s Get to the Sweet Spot

For any of the above circuit types and devices, if you start with reasonable
textbook values for the circuit, you will get reasonable performance. If you
start to play with the values a little bit, you will find that the performance
changes for better or worse. Often you will find a combination of voltage and
resistor values which give a lot better than the generic performance. That’s the
sweet spot.

If you have a distortion analyzer, you could simply run through the range of
combination of values and select the result you like best. If you simply want
good measurements, you might be able to stop there. If you are looking for
better subjective performance, you might find this a good place to start your
listening.

What creates the existence of the sweet spot in gain devices? To see where
this comes from, we will look at the characteristics of some tubes and
transistors.

Let’s start with a simple big triode, the 300B. This curve shows the Plate
to Cathode current as a function of Grid voltage, with each curve representing a
fixed value of plate voltage.

Looking at this curve, we see that the lines are curved – the more current,
the faster they rise. Below is a different view of the same character, but with
the current plotted as a function of the Plate voltage with each line
representing a fixed value for Grid voltage.

These lines are also curved with a shape very similar to the previous
graph.
A perfectly linear tube will have these lines in both graphs
perfectly straight and equally spaced. Any deviation from that would be
distortion. Looking at the curved lines, we can see that there is quite a bit of
distortion.

For a given Plate voltage, the current increases exponentially with the Grid
voltage, in what is known as a square law characteristic, resulting in second
harmonic distortion. For a given Grid voltage, the current increases
exponentially with the Plate voltage, also a square law and also creating second
harmonic distortion.

In this curve the Y axis is still the plate current, but the X axis is now
the Plate voltage, and each line represents the Grid voltage. It looks familiar
– just as the plate current is an exponential function of Plate voltage, it is
also an exponential function of the Grid voltage. This is a particularly
important observation with respect to finding the sweet spot.

{pagebreak}

So Here’s the Deal:

Ordinarily in a tube amplifying an AC signal, a positive change in Grid
voltage causes an increase in Plate current and is accompanied by a decrease in
Plate voltage. The gain due to Plate current increases, and the gain due to
Plate voltage decreases. The opposite happens when the Grid voltage goes
negative.

These two gain variations tend to cancel each other, resulting in more
constant gain and lower distortion. If you choose your conditions so that the
simultaneous gain increases and decreases are equal, you can lower the
distortion a lot.

This technique is often referred to as load-line cancellation. It is called
that because the range of the device’s operation sits on a line in the transfer
curve, and the particular shape and position of that line results in minimum
distortion.

By some criteria, this would be the sweet spot. At this point you will
probably find that your second harmonic has largely disappeared and you are left
with some third harmonic. This is because you can’t completely cancel two square
law distortions without leaving a cubic trace – the third harmonic.

So how much distortion reduction are we talking about? On a clear day, we
can think in terms of a 90% reduction, as seen in the following curve, where a
triode follower is operated at a high plate voltage giving the upper blue
distortion curve. Without changing the load or the current, I simply reduced the
Plate voltage until
the gain dependence on Plate voltage matched the gain
dependence on Plate current (as determined by lowest distortion at 1 volt) and
ran the red curve.

I think we can agree that this is a significant reduction, roughly
equivalent to what we would see with 20 dB of negative feedback, except that we
have not attached a negative feedback loop and the output gain and impedance of
the circuit remain about the same.

Because of the square law characteristics of the devices, the primary
distortion is second harmonic in content, and when we cancel two second
harmonics we find that the remainder is a third harmonic. The original second
harmonic looks like this on an oscilloscope viewing the output of a distortion
analyzer:

After cancellation, the harmonic content looks like this:

If your criteria instead is a particular harmonic content and/or harmonic
phase, a deviation from this point in either direction will give you second
harmonic back with its phase depending on which way you went, and you can tweak
this in relation to the amplitude of the third harmonic if you like.

There is a reason why triodes are popular in single-ended applications –
their gain dependence on Plate voltage is quite strong and makes for easy
load-line cancellation.

Load line cancellation is available in the other types of gain devices,
including pentodes, JFETs, MOSFETs and bipolar transistors, but they take more
subtle advantage of load-line cancellation. In many cases it is easier to find
the sweet spot with push-pull operation. We will see real world examples of both
single-ended and push-pull later, but first we will consider the characteristics
of the other devices.

This technique also applies to pentode tubes, although it will not be quite
the same. The transfer curve of the pentode is much the same with regard to grid
voltage, but the current variation due to Plate voltage is different. Here is
the transfer curve of a pentode:

Here we see a family of curves which indicate that the gain
(transconductance) increases exponentially with Plate current, but has less
variation with respect to Plate voltage as compared to the Triode. You can still
vary the voltage, current and load parameters to find the sweet spot, but it
will be in a different place.

JFETs have the exponential current vs Gate voltage but enjoy a wider variety
of transfer characters situated between triode and pentode and are generally
well suited to exploit the concept of a sweet spot in single-ended Class A
circuits.

JFETs come in both enhancement mode and depletion mode, the primary
difference being the DC voltage required on the Gate to turn them on. Depending
on the dimensions of the Gate structure on the chip can exhibit pentode or
triode characteristics, or an in-between “mixed mode”.

Following is the curve of an enhancement mode power JFET:

And here is the curve of a depletion mode power JFET:

You will note that these aren’t the tiny little signal JFETs that everyone
is accustomed to. These are examples of the new generation of high power JFETs
coming out of the labs. In the 1970’s when Yamaha and Sony produced power JFET
devices for use in their own amplifiers it created some excitement among
audiophiles. Unfortunately it seems that they were ahead of their time, and now
those parts are quite rare.

Fortunately totally new devices show promise in modern power amplifiers both
as switches and linear amplifying parts – tomorrow’s transistors (at tomorrow’s
prices).

For power JFETs the Ids dependence on Vds is partly a function of the Gate
channel depth on the chip. The pentode-like character is seen in high current
power “Vertical” JFETs and the triode-like character is seen in SITs, (Static
Induction Transistor).

The enhancement mode power JFETs do well in single-ended Class A
applications and can take advantage of load-line cancellation, and my samples of
depletion mode JFETs have a more pronounced voltage dependence, and are thus
even more interesting.

Here is an example of a JFET follower whose characteristic is mostly
“pentode-like”. The upper curve reflects the distortion at V+ voltage near the
device rating, and the lower curve shows the distortion cancellation available
by searching for the “sweet spot” at a lower supply voltage. You can see that at
1 volt, the distortion has been reduced by about 85%.

Alternatively we could have adjusted the bias current or the load
resistance, and we also could have tried for best performance at some higher
voltage that 1 volt.

This example was of one of those “little” JFET devices, but it works with
parts of all sizes. The thing to remember is that this cancellation is there
somewhere, and you might want to go looking for it. Also keep in mind that the
sweet spot is a bit different for every part, even within the same part types,
and you will want to consider expending the effort for each individual part if
you want the absolute in performance.

This is important when you change out tubes in an amplifier, as every
tube-o-phile knows they are all different. Setting them to a generic bias
current does not guarantee the sweet spot for a given tube. You might want to
consider how you can vary the supply voltage and the bias while evaluating the
performance.

Below is an example of a power MOSFET’s transfer curve:

Power MOSFETs have a character somewhat similar to a pentode. The Plate has
been replaced by the Drain, the Grid has become the Gate, and the Cathode is now
the Source. We see from this diagram that the gain increases with Ids but there
is not a lot of dependence on Vds above a few volts. At just a of couple volts
we operate in what is known as the “linear” region of the MOSFET, where there is
strong dependence. You can work the load-line in this region effectively, but it
is not a popular technique for a number of good practical reasons. Fortunately,
for many MOSFETs operating in Class A, the distortion of the current transfer
curve is low enough that you can still find the cancellation for a pentode type
voltage characteristic – just go looking for it.

And of course, someone is going to ask, “What about bipolar transistors?” It
so happens that bipolars have a sweet spot also. Here is an example of the
current gain figure of an ordinary NPN transistor vs Collector current.

There is also a dependence on Vce known as the Early effect, a slightly
straight line dependency increasing with voltage. As with other devices, finding
the sweet spot for a bipolar device means locating the point where these
variations result in distortion cancellation.

Here we see the gain vs current curve of an NPN power transistor showing
some gain dependence vs Collector current. Above a couple of volts, the
variation is much less and forms a fairly linear straight line with voltage.

Of course you can work these two curves. The following distortion curve
shows
a follower circuit like the others, but with a bipolar transistor. The
blue line is with a high supply voltage, and the red is the reduced voltage. The
reduction in distortion is less dramatic, but it’s still about 7 or 8 dB
better:

{pagebreak}

Push-Pull Operation

Of course, single-ended Class A is not the only way to operate gain devices.
Large reductions in distortion can be had by operating gain devices in
push-pull, where two similar devices operate in balanced opposition such that
their second harmonic characteristic is naturally cancelled. Below on the left,
you see a pair of JFET followers operated push-pull:

So where’s the sweet spot for this sort of circuit? It’s there if you go
looking for it. The following sets of curves show three techniques for getting a
pair of these devices into the sweet spot.

First off you can take the “regular” follower shown above and simply vary
the supply voltages a bit. The curve below shows distortion for two examples,
the upper blue curve being with two equal supply voltage values. The lower red
curve shows what can happen if you vary only one of the supply voltages:

Here’s a technique we use at Pass Labs – the use of some single-ended bias
in a push-pull output stage to reduce distortion. The distortion curve below
shows what you can get if you simply put a small value current source in
parallel with one of the JFETs as shown on the middle circuit. The current
source displaces the bias current values on the JFETs until they achieve a more
perfect distortion cancellation:

There is yet another way, shown by the circuit on the right in the diagram,
which involves the use of a trim potentiometer instead of the Source resistors
of the “regular” circuit.

Here we see that a simple adjustment of the pot can drop the distortion by
90% without having significant effect on the other aspects of performance.

{pagebreak}

Temperature

There is one last important variable affecting performance and the location
of the sweet spot – temperature. All of these gain devices have transfer curves
which are some function of the temperature of the device. All those transfer
curves I have shown you will be different if you vary the temperature.

Conceivably you could adjust for the sweet spot by varying the temperature.
More practically, you will want to see to it that our circuits have reached
their ordinary operating temperature before adjusting for the last bit of
performance.

{pagebreak}

Cascoding, Paralleling and Transforming

So what do you do if it’s impractical to find the optimal load-line in a
given circuit? Occasionally the sweet spot occurs at voltage values that are
impractically low, or at currents that are higher than a device can handle for a
given voltage. Here are three things that a designer can do to get into the
zone.

The first is cascoding, where the gain device is coupled with a Common-Gate
/ Common-Grid / Common-Base (depending on the type of device!) tube or
transistor which adds practically no influence of its own but which allows a
more arbitrary DC and AC voltage across the gain device. Here are a couple of
examples, using JFETs in both Common-Drain and Common-Source circuits.

The top JFET is the cascode device, and its Source voltage, which will be
seen by the Drain of the JFET below it, is set by Vref. The idea is that the
cascoding device provides a “voltage umbrella” for the gain JFET, and all manner
of voltages can appear at the output of the circuit while the gain device sees
all, a portion, or none of it.

It’s pretty easy to get into the sweet spot using a cascode, and if you want
to explore this in some more detail, I recommend Zen Variations 9, which you can
download from www.passlabs.com. There we explore getting the lowest possible
distortion from an amplifier using a single stage with a power JFET having a
high current rating, but a low voltage and dissipation rating.

Another thing you can do to get into the zone of the sweet spot is to
parallel devices. There are times when the load is too low (and you can’t change
it) or the voltage is too high, but often you can mitigate this situation by
operating devices in parallel. This way you can limit the dissipation of each
device, and the load appears as a multiple of the number of devices.

A third thing you can do is to use a transformer, which can allow you a
flexible range of voltages and currents through the device while delivering the
appropriate values to the load. Tube amplifiers do it all the time, and it gives
the extra flexibility that then allows you adjust for the sweet spot.

{pagebreak}

There’s a Small Catch

All the loads we have examined are resistors. Many times that is all we need
to work with anyway, but when it comes to loudspeaker drivers and passive
crossovers we have to contend with loads which are also reactive and which vary
in impedance. This can be a problem when the load-line that gives best
performance depends on resistance. A reactive load will take it out of the sweet
spot.

This is particularly true for tube amplifiers operating single-ended Class A
without feedback – much of the performance depends on triode load-line
cancellation. These amplifiers are particularly appreciative of resistive loads,
and if you have this situation, it’s worth considering ways to flatten the load
impedance, giving a more resistive load for the amplifier.

There is another solution to this problem if you employ cascoding. Referring
again to Zen Variations 9, we see there is a technique, which I have dubbed
“Cascode Modulation”, in which the gain device can be made to see controlled
load-line voltage variations which are only a function of the output current,
and not related to the output voltage. In this way, the sweet spot is preserved
into any load reactance or impedance, and it is very easily dialed in by
adjusting a couple of resistor values. In the example of ZV9, we get a 90%
improvement in distortion by simply attaching the cascode’s voltage to a
different spot, without changing any values or any other characteristics.

{pagebreak}

Conclusion

So there we have it. As an audiophile you want the best performance, and you
probably aren’t above spending money and trying tweaks of various sorts to help
you get it. But there are real improvements that can be had without extra
accessories or emptying your wallet.

Amplifying circuitry can be made better without more complexity and without
more feedback. The best part of this for do-it-yourselfers is that these are
things that can be done on your own bench, costing mostly just the time it takes
to tweak the circuit and evaluate the results. Moreover, this approach is not
seen on the factory floor – for most manufacturers it’s simply too time
consuming to do, and the rest probably never heard of it.

So here’s your chance to make simple high quality audio amplifiers better
than ever. Buy yourself a cheap used distortion analyzer and go for it.

Copyright 2009 Nelson Pass

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