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> Amplifier
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> Digital
Potentiometers > APP 864

Nov 28, 2001

Keywords: EPOT Applications: Gain Adjustment in
Op-Amp Circuits
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APPLICATION NOTE 864

EPOT Applications: Gain Adjustment in Op-Amp
Circuits

Variable
gain amplifiers often use a mechanical potentiometer to set
the gain. An example is a volume control dial. However, when
the analog
signal path is digitally controlled, a digital
potentiometer might be used to control gain. This article
discusses the use of the digital potentiometer to form
digitally controlled gain or filter blocks.

General
Throughout this discussion of digital
potentiometers, the term EPOT is used rather than digital pot,
EEPOT or E²POT. The term EPOT describes the Maxim series of
volatile digital potentiometers, while the other terms
describe a variety of non-volatile
and volatile devices manufactured on one of several different
processes. The specifics of how these different processes
affect digital potentiometer performance are outside the scope
of this article. While the equations given below can be
applied to all digital potentiometer circuits, the special
considerations relating to voltage
ratings, current
ratings are intended specifically for Maxim's EPOT
devices.

Even though the designer needs to adhere to
the standard constraints associated with semiconductor
devices, (i.e., absolute maximum ratings) there are a couple
of characteristics specific to EPOTs that the engineer should
keep in mind when using them in op-amp circuits.

One
of the first things to remember is that the tolerance of the
end-to-end or absolute resistance
of digital potentiometers is usually specified for a maximum
of ±25% to ±30%. This is mainly due to process variations and
a technique for handling this is discussed throughout this
article.

Additional things to consider are the
characteristics of the EPOT's "wiper". The "wiper" in an EPOT
is effectively a string of CMOS
switches and will exhibit the typical characteristics.
Important specifications for the wiper include its resistance,
the variation of its resistance, its maximum current rating
and its temperature
coefficient. For example, the wiper resistance of the MAX5400
is specified for a typical value of 250 Ω but with a maximum
value of 800 Ω. The absolute maximum current rating of the
wiper (as well as the rest of resistor in the EPOT) is 1mA.
Care should be taken not to exceed this current rating as
excessive current can damage the device. Lastly, the
temperature coefficient for the wiper is typically about
±300ppm/°C.

One final EPOT consideration is how it will
behave near either extreme of the adjustment range. For
example, when an EPOT is set to one of its higher or lower
settings the ESD
structures, such as wiper characteristics, can begin to
drastically affect performance. A general rule of thumb is to
try and keep the EPOT set to between 10% and 90% of full
scale.

In general, the principles described here apply
to all of the EPOTs in the Maxim line. The 256 tap EPOT has
been chosen for illustration purposes. Internally, an EPOT is
viewed as a string of resistors with CMOS switches "tapping"
off that string. In linear EPOTs the spacing along the
resistor string is even, in logarithmic
taper EPOTs, the spacing would be logarithmic. Therefore,
an EPOT with a given number (m) of taps has one fewer
resistors (m-1). For the purposes of discussion in this
article, the variable "n" refers to the absolute setting of
the EPOT (i.e., 0 to 31 or 0 to 255 and so on) and "N" refers
to the ratio of the setting of the EPOT to its maximum value
(i.e., 0 to 1). For example, in a linear
taper EPOT with 256 taps a setting of n=89 would give
N=0.3490, which is 89/255. "Re" refers to the total end-to-end
resistance of the EPOT, thus an EPOT can be thought of as 2
resistors with values N × Re and (1-N) × Re. Furthermore, gain
equations are shown in this article as a function of the
variable "N" as G(N). The total available range of adjustment
is shown as G(0)<=G(N)<=G(1) or as
G(1)<=G(N)<=G(0), as applies.

Non-Inverting Configurations
Circuit 1 is a
non-inverting gain configuration using the EPOT as a variable
resistor. In this circuit, the wiper is connected to one side
of the EPOT effectively shorting out the (1-N) × Re side of
the device. As a result, the feedback current flows through
wiper and the portion of the EPOT correlating to N × Re. This
circuit has a nice linear gain adjustment curve with respect
to N, but some of the EPOT's characteristics give this circuit
a few disadvantages.

Figure
1a. Circuit 1.

One disadvantage is that because the
feedback current flows through the wiper, its resistance (and
that resistance's variance with temperature) as well as its
maximum current rating (typically about 1mA) may limit the
applications for this circuit. Another disadvantage is that
the end-to-end resistance, Re, is a part of the gain equation
causing the ±25% tolerance of Re to contribute up to a ±25%
variance in the gain. Mathematically, the maximum possible
contribution of a 25% error in Re to the overall gain is 25%.
Of course, as the Re/Ri term in the gain equation becomes
smaller and the "1" term begins to dominate the gain, the
error as a function of the tolerance of Re becomes smaller.

Figure
1b. Gain of circuit 1 with Re=50k and
Ri=10k.

Circuit 1 is suited for cost sensitive
systems that can live with the somewhat loose tolerance in
overall gain. As with many of the circuits seen here, this
includes circuits like volume or brightness
control where the user is going to iteratively adjust the
circuit according to personal preference. Another useful
application would be in an automatic
gain control circuit where either digital or analog
automatic gain control is used to make a gain compensation for
a system output. For example, an APD
circuit may use either digital or analog automatic gain
control to control the biasing of the photodiode with high
accuracy. In such a circuit, the automatic gain control
technique takes the tolerance of Re (in the EPOT) out of the
equation.

Another possible configuration is shown in
circuit 2. Here the wiper is connected to the high impedance
input of the op-amp using the EPOT as a sort of variable
voltage divider. By connecting the wiper to this high-Z input,
the only current that flows through the wiper will be the
input bias current of the op-amp. This current is typically
very small (usually between pA and uA) so that any variance in
the several hundred Ω resistance of the wiper will be
negligible.

Figure
2a. Circuit 2.

In addition, the Re terms on the top
and bottom of the gain equation cancel out leaving just a
function of N. This removes the impact of end-to-end
resistance from the gain equation
altogether.

Unfortunately, this circuit has one big
disadvantage. The gain adjustment is non-linear with respect
to N. A plot of G(N) for circuit 2 is shown in Figure 2. By
deriving the slope of G(N) as G'(N) = 1/(1-N)², it can be seen
that the slope is also a function of N and therefore isn't
even piecewise linear over a select range of N. In the section
on Inverting Configurations, several ways to use this circuit
and counteract its inherent non-linearity are
discussed.

Figure
2b. G(N) for circuit 2.

Circuit 2 is useful in
many of the same systems mentioned for circuit 1 (i.e., audio
control, brightness control, AGC loops etc.). In addition to
the applications seen in circuit 1, the lack of a dependence
on Re and the resistance of the wiper makes this an excellent
circuit for calibration adjustments. Also notice that in
circuit 2, the gain steps are heavily concentrated from 1 to
2, but that much higher gains are still possible. While rare,
some calibration applications may call for a period of high
gain followed by a period of very finely tuned low gain. An
example might be in the measurement of air pressure in a blood
pressure cuff. In such a system, when inflation is initiated,
the pressure is very low and the measurement is primarily used
to detect faults (high gain). Once the cuff is fully inflated
the measurements are used to detect minute changes in cuff
pressure vs. time. The AGC loop using circuit 2 can be used to
optimize ADC
dynamic
range during blood pressure measurement. The calibration
is done at production to compensate for mechanical and sensor
deviations.

Inverting Configurations
The principles used in the
non-inverting configurations can also be applied to inverting
op-amp circuits. In circuit 3 the gain is simply the apparent
resistance in the feedback path over the input resistance.

Figure
3a. Circuit 3.

This circuit has the benefit of
having a linear gain adjustment as a function of N, as shown
in Figure 3b.

Figure
3b. G(N) for circuit 3 with Re=50k and Ri=10k.

This
circuit has the same disadvantages as those seen in circuit 1.
It will have a slightly different dependence on the tolerance
of Re because the gain equation is slightly different; the
gain will vary as much as ±25% as the tolerance on Re varies.
This circuit is going to be useful in inverting applications
similar to those already mentioned for circuits 1 and
2.

Just like circuit 2, circuit 4 is not susceptible to
the disadvantages mentioned for circuits 1 and 3. Because the
Re terms on the top and bottom cancel each other out of the
gain equation, the gain depends only on N. As a result, errors
in the absolute resistance are effectively cancelled out in
circuit 4, making it attractive for use with EPOTs that have a
wide tolerance for the end-to-end resistance. In addition, an
op-amp's input bias current is usually in the range of pA to
uA. With so little current flowing through it, the wiper's
resistance, drift with temperature and maximum current spec
will not be a factor in this circuit.

Figure
4a. Circuit 4.

The gain for this circuit is shown
in Figure 4b. This circuit has the same drawback as circuit 2,
its gain adjustment is going to be non-linear over the entire
range of N. This inherently non-linear gain adjustment does
not prevent this from being useful. This circuit will fit very
well where an inverting amp is needed in the same applications
as for circuit 2. For example, this might fit well in
applications with user feedback, such as volume control or
display brightness control circuits and also in calibration
circuits or AGC circuits.

Figure
4b. G(N) for circuit 2 with a 256 tap
EPOT.

However, some systems require even gain steps
as a function of N. Below, two approaches are given for
linearizing the gain of circuit 4.

Linearizing Circuit 4's Gain with Resistors
In circuit
5, resistors have been added to the input path and the
feedback path to "flatten out" the plot of the gain. In the
plot of G(N) for this circuit, both its gain and the gain for
circuit 4 are shown. The last 50 points have been omitted for
clarity. This plot shows how the gain curve can be "flattened"
at the expense of adjustment range.

Figure
5a. Circuit 5.

By adding resistors to both the
input path and the feedback path, the minimum gain (where N=0)
is raised and the maximum gain (where N=1) is lowered
effectively compacting the available gain range. For example,
in the plot shown the gain can be adjusted from 1 to 3 rather
than from the 0 to 255 corresponding circuit 4.

Figure
5b. G(N) for circuit 5 with Ri=50k and Rf=100k (yellow). G(N)
for circuit 4 is also shown (blue) for
illustration.

This circuit also has the advantage
of connecting the wiper to the high impedance op-amp input.
Additionally, while the gain is not going to be perfectly
linear like circuit 3, any effects the wiper's characteristics
have on performance are taken out of the equation by greatly
reducing the current through the wiper.

The big
drawback to this circuit is that by adding Ri and Rf to the
gain equation, the Re terms that were cancelled out in circuit
4 cannot be cancelled out here. One advantage of circuit 4 was
that the Re terms in the gain equation cancelled each other
out making the large tolerance of the end-to-end resistance
irrelevant. For circuit 5 however, the Re terms cannot be
cancelled out and deviations in the end-to-end resistance of
the EPOT will affect the overall gain. The exact impact of
deviations in the end-to-end resistance will depend on the
ratio of Rf and Ri to Re. Making Rf and Ri large with respect
to Re can minimize this impact, but will hamper the available
adjustment range. In the example used for the plot of Figure
5b, a 25% increase in Re makes the minimum gain 0.88 rather
than 1 and the maximum gain 3.25 rather than 3. A 25% decrease
in Re makes the minimum 1.14 and the maximum 2.75.

A Design Example using Software Compensation to Linearize
Gain
Most systems use a microcontroller to control the
setting of the EPOT and therefore, the gain of the circuits
here are usually controlled by software. In addition, while
this article has used 256 step EPOTs for illustration, many
systems only need 20 to 30 steps in the gain. Using software
and a 256 step EPOT configured as shown in circuit 4, 20 to 30
steps of linear gain adjustment curve can easily be
accomplished by using software compensation. A design example
is shown here for a software compensated, inverting, and
adjustable gain op-amp circuit.

The goal for this
design is to have the following performance characteristics:

No current through the wiper.

A linear gain adjustment curve with better than 3%
error. This is better than 1 LSB
for a 32 tap EPOT.

Eliminate any dependence on absolute resistance Re.

To meet these design requirements, the following
steps are offered. The basic idea is to pick the optimal
number of gain steps between the desired minimum and maximum
gain.

1. Pick a maximum and minimum gain. For
this example a minimum of 2 and a maximum of 10 was chosen.
Empirical testing shows that this method works pretty well for
gains of up to about 20.

2. Find the actual
possible maximum gain. The actual maximum gain is the gain
that will be a solution to the equation

G(n) = n/(256
- n) n=0,1,2...

For this example, solving for a G(n)=10
gives n=232.72. Since only whole values of n are available,
select n=233. The actual maximum gain is then G(233)=233/(256
- 233) = 10.13.

3. Get the slope m. The slope m
could also be called the step size, as it will correlate to
the gain steps between values of n. The recommended method is
to use the average gain change over the last 5 values of n in
the range of interest. In other words,

m=
[G(nmax) - G(nmax -5)]/5

The
decision to use the last 5 values is somewhat arbitrary and
the designer is encouraged to experiment with other values.
For this example, m=[G(233) - G(228)]/5=0.398.

4.
Find the actual possible minimum gain and the number of
steps. In order to be certain that the minimum gain is a
possible solution to the line being created, the equation for
y=mx+b can be solved. Since a discrete system is being used,
simply find the number of steps between the max gain and the
expected min gain (chosen in step 1), where

max steps =
[G(nmax) - Gmin]/m.

For this
example, max steps = [10.13 - 2]/0.398=20.4. Once again, only
a discrete number of steps is available, so round up the
result to increase the maximum number of steps to 21.
Therefore, the actual minimum gain is going to be 21 steps of
0.398 each less than the maximum gain or

Gmin=
G(nmax) - (max steps) × m = 10.13 - (21 × 0.398) =
1.772.

5. Get the intercept b. The value for b
or the y intercept is simply the min gain.

The result
is the equation

f(x)=0.398x + 1.772, where
x=0,1,2...21.

This equation can be used to find the
values of n or the EPOT settings that are used to get this
straight line gain curve. This is done by solving G(N) =
n/(256 - n) = f(x). The actual implementation is up to the
designer, but basically the options are to either use a lookup
table or to solve for the value of n on the fly.

A plot
of the gain for this example is shown below as a function of
x.

Figure
6. Gain curve for software compensated example.

The
deviation from the ideal straight line is plotted below as a
percentage.

Figure
7. Deviation from ideal as a percentage.

Summary
In this article, simple EPOT adjustable gain
block configurations have been discussed, along with the
advantages and disadvantages of each. The EPOT has important
characteristics that must be considered when being used in a
design. One of the cleanest ways of preventing these
characteristics from limiting the performance of the circuit
results in a non-linear gain adjustment curve, which may not
be desirable. Finally, two ways of compensating for that
non-linear curve are presented. This would allow the EPOT to
be used in a way where many of its constraints are not a
factor and linear gain is still achievable.

Related parts

MAX5400

256-Tap SOT-PoT,
Low-Drift Digital Potentiometers in SOT23

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Download,
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APP 864: Nov 28, 2001

AN864, AN 864, APP864,
Appnote864, Appnote 864

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