### AVR120: Characterization and Calibration of the

```AVR120: Characterization and Calibration of the
ADC on an AVR
Features
•
•
•
•
Understanding Analog to Digital Converter (ADC) characteristics
Measuring parameters describing ADC characteristics
Temperature, frequency and supply voltage dependencies
Compensating for offset and gain error
8-bit
Microcontrollers
Application Note
1 Introduction
This application note explains various ADC (Analog to Digital Converter)
characterization parameters given in the datasheets and how they effect ADC
measurements. It also describes how to measure these parameters during
application testing in production and how to perform run-time compensation for
some of the measured deviations.
A great advantage with the Flash memory of the AVR is that calibration code can
be replaced with application code after characterization. Therefore, no code space
is consumed by calibration code in the final product.
Rev. 2559D-AVR-02/06
2 Theory
Before getting into the details, some central concepts need to be introduced. The
following section (General ADC concepts) can be skipped if quantization, resolution,
and ADC transfer functions are familiar to the reader.
2.1 General ADC concepts
The ADC translates an analog input signal to a digital output value representing the
size of the input relative to a reference. To get a better basis for describing a general
ADC, this document distinguishes between ideal, perfect and actual ADCs.
An ideal ADC is just a theoretical concept, and cannot be implemented in real life. It
has infinite resolution, where every possible input value gives a unique output from
the ADC within the specified conversion range. An ideal ADC can be described
mathematically by a straight-line transfer function, as shown in Figure 1.
Figure 1. Transfer function of an ideal ADC
Output value
To define a perfect ADC, the concept of quantization must be introduced. Due to the
digital nature of an ADC, continuous output values are not possible. The output range
must be divided into a number of steps, one for each possible digital output value.
This means that one output value does not correspond to a unique input value, but a
small range of input values. This results in a staircase transfer function. The
resolution of the ADC equals the number of unique output values. For instance, an
ADC with 8 output steps has a resolution of 8 levels or, in other words, 3 bits. The
transfer function of an example 3-bit perfect ADC is shown in Figure 2 together with
the transfer function of an ideal ADC. As seen on the figure, the perfect ADC equals
the ideal ADC on the exact midpoint of every step. This means that the perfect ADC
essentially rounds input values to the nearest output step value.
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Figure 2. Transfer function of a 3-bit perfect ADC
Output value
The maximum error for a perfect ADC is ±½ step. In other words, the maximum
quantization error is always ±½ LSB, where LSB is the input voltage difference
corresponding to the Least Significant Bit of the output value. Real ADCs have other
sources of errors, described later in this document.
2.2 Conversion ranges
The ADC in Atmel’s AVR devices can be configured for single-ended or differential
conversion. Single-ended mode is used to measure input voltages on a single input
channel, while differential mode is used to measure the difference between two
channels. Regardless of conversion mode, input voltages on any channel must stay
between GND and AVCC.
When using single-ended mode, the voltage relative to GND is converted to a digital
value. Using differential channels, the output from a differential amplifier (with an
optional gain stage) is converted to a digital value (possibly negative). A simplified
illustration of the input circuitry is shown in Figure 3.
Figure 3. Simplified ADC input circuitry
Single-ended conversion mode
Single input
(A)
Conversion
circuitry
Differential conversion mode
Positive input
Conversion
circuitry
Negative input
(B)
To decide the conversion range, the conversion circuitry needs a voltage reference
(VREF) to indicate the voltage level corresponding to the maximum output value.
According to the datasheets, VREF should be at least 2.0 V for standard devices, while
devices operating down to 1.8V can use a reference voltage down to 1.0V. This
applies both to single-ended and differential mode. Consult the datasheet for details.
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2.2.1 Single-ended conversion range
Single-ended conversion feeds the input channel directly to the conversion circuitry,
as shown in Figure 3A. The 10-bit ADC of the AVR therefore converts continuous
input voltages from GND to VREF to discrete output values from 0 to 1023.
Any applied input voltage greater than the reference voltage VREF will return the
maximum value (1023 using 10-bit ADC), and any negative input voltage will return 0.
2.2.2 Differential conversion range
Differential conversion feeds the two input channels to a differential amplifier with an
optional gain stage. The output from the amplifier is then fed to the conversion logic,
as shown in Figure 3B. Voltage differences from -VREF to +VREF therefore results in
discrete output values from –512 to +511. The digital output is represented in 2’s
complement form. Even when measuring negative voltage differences, the voltages
applied to the input channels themselves must stay between GND and AVCC.
Voltage differences smaller than -VREF will return the minimum value (-512 using 10bit ADC), and voltage differences greater than +VREF will return the maximum value
(512 using 10-bit ADC).
Note that some devices cannot measure negative differences, e.g. ATtiny26.
2.3 The need for calibration
The total error of the actual ADC comes from more than just quantization error. This
document describes offset and gain errors and how to compensate for them. It also
describes two measures for non-linearity, namely differential and integral nonlinearity.
For most applications, the ADC needs no calibration when using single ended
conversion. The typical accuracy is 1-2 LSB, and it is often neither necessary nor
practical to calibrate for better accuracies.
However, when using differential conversion the situation changes, especially with
high gain settings. Minor process variations are scaled with the gain stage and give
large parameter differences from part to part. The uncompensated error is typically
above 20 LSB. These variations must be characterized for every device and
compensated for in software.
At first sight 20 LSB seems to be a large value, but does not mean that differential
measurements are impractical to use. Using simple calibration algorithms, accuracies
of typically 1-2 LSB can be achieved.
2.4 Absolute error
The absolute error is the maximum deviation between the ideal straight line and the
actual transfer function, including the quantization steps. The minimum absolute error
is therefore ½ LSB, due to quantization.
Absolute error or absolute accuracy is the total uncompensated error and includes
quantization error, offset error, gain error and non-linearity. Offset, gain and nonlinearity are described later in this document.
Absolute error can be measured using a ramp input voltage. In this case all output
values are compared against the input voltage. The maximum deviation gives the
absolute error.
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Note that absolute error cannot be compensated for directly, without using e.g.
memory-expensive lookup tables or polynomial approximations. However, the most
significant contributions to the absolute error - the offset and gain error - can be
compensated for.
Be aware that the absolute error represent a reduction in the ADC range, and one
should therefore consider the margins to the minimum and maximum input values to
avoiding clipping keeping the absolute error in mind.
2.5 Offset error
The offset error is defined as the deviation of the actual ADC’s transfer function from
the ideal straight line at zero input voltage.
When the transition from output value 0 to 1 does not occur at an input value of ½
LSB, then we say that there is an offset error. With positive offset errors, the output
value is larger than 0 when the input voltage approaches ½ LSB from below. With
negative offset errors, the input value is larger than ½ LSB when the first output value
transition occurs. In other words, if the actual transfer function lies below the ideal
line, there is a negative offset and vice versa. Negative and positive offsets are shown
in Figure 4.
Figure 4. Examples of positive (A) and negative (B) offset errors
Output value
Output value
+1½ LSB offset
-2 LSB offset
(A)
(B)
Since single-ended conversion gives positive results only, the offset measurement
procedures are different when using single-ended and differential channels.
2.5.1 Offset error – single-ended channels
To measure the offset error, increase the input voltage from GND until the first
transition in the output value occurs. Calculate the difference between the input
voltage for which the perfect ADC would have shown the same transition and the
input voltage corresponding to the actual transition. This difference, converted to LSB,
equals the offset error.
In Figure 5A, the first transition occurs at 1 LSB. The transition is from 2 to 3, which
equals an input voltage of 2½ LSB for the perfect ADC. The difference is +1½ LSB,
which equals the offset error. The double-headed arrows show the differences. The
same procedure applies to Figure 5B. The first transition occurs at 2 LSB. The
transition is from 0 to 1, which equals an input voltage of ½ LSB for the perfect ADC.
The difference is -1½ LSB, which equals the offset error.
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Output value
Output value
Figure 5. Positive (A) and negative (B) offset errors in single-ended mode
+1½ LSB offset
-1½ LSB offset
(A)
(B)
The measurement procedure can be formalized as described in Figure 6.
Figure 6. Flowchart for measuring single ended offset errors
Offset measurement
Set input voltage to 0
Store current output
value as A
Store input voltage as
Actual
Calculate input voltage
required for perfect
ADC to change output
from A to B
Increase until output
value changes
Store calculation result
as Perfect
Store current output
value as B
Offset error equals
(Perfect - Actual)
converted to LSB
Finished
To compensate for offset errors when using single ended channels, subtract the
offset error from every measured value. Be aware that offset errors limit the available
range for the ADC. A large positive offset error causes the output value to saturate at
maximum before the input voltage reaches maximum. A large negative offset error
gives output value 0 for the smallest input voltages.
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2.5.2 Offset error - differential channels
With differential channels, the offset measurement can be performed much easier
since no external input voltage is required. The two differential inputs can be
connected to the same voltage internally and the resulting output value is then the
offset error. Since this method gives no exact information on where the first transition
occurs, it gives an error of ½ to 1 LSB (worst case).
To compensate for offset errors when using differential channels, subtract the offset
error from every measured value.
2.6 Gain error
The gain error is defined as the deviation of the last output step’s midpoint from the
ideal straight line, after compensating for offset error.
After compensating for offset errors, applying an input voltage of 0 always give an
output value of 0. However, gain errors cause the actual transfer function slope to
deviate from the ideal slope. This gain error can be measured and compensated for
by scaling the output values.
Run-time compensation often uses integer arithmetic, since floating point calculation
takes too long to perform. Therefore, to get the best possible precision, the slope
deviation should be measured as far from 0 as possible. The larger the values, the
better precision you get. This is described in detail later in this document.
The example of a 3-bit ADC transfer functions with gain errors is shown in Figure 7.
The following description holds for both single-ended and differential modes.
Figure 7. Examples of positive (A) and negative (B) gain errors
111
111
110
+1½ LSB error
101
100
011
Output code
Output code
110
100
011
010
010
001
001
000
-1½ LSB error
101
000
0/8 1/8 2/8 3/8 4/8 5/8 6/8 7/8 AREF
(A)
0/8 1/8 2/8 3/8 4/8 5/8 6/8 7/8 AREF
(B)
To measure the gain error, the input value is increased from 0 until the last output
step is reached. The scaling factor for gain compensation equals the ideal output
value for the midpoint of the last step divided by the actual value of the step.
In Figure 7A, the output value saturates before the input voltage reaches its
maximum. The vertical arrow shows the midpoint of the last output step. The ideal
output value at this input voltage should be 5.5, and the scaling factor equals 5.5
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divided by 7. In Figure 7B, the output value has only reached 6 when the input
voltage is at its maximum. This results in a negative deviation for the actual transfer
function. The ideal output value for the midpoint of the last step is 7.5 in this case.
The scaling factor now equals 7.5 divided by 6.
The measurement procedure is illustrated in Figure 8.
Figure 8. Flowchart for measuring gain errors
Gain measurement
Set input voltage to 0
Increase until last
output step is reached
Find midpoint of last
step based on previous
step lengths
Use actual output value
as denominator
Use ideal output value
as nominator
Compensation factor =
nominator /
denominator
Finished
2.7 Non-Linearity
When offset and gain errors are compensated for, the actual transfer function should
be equal to the transfer function of perfect ADC. However, non-linearity in the ADC
may cause the actual curve to deviate slightly from the perfect curve, even if the two
curves are equal around 0 and at the point where the gain error was measured. There
are two methods for measuring non-linearity, both described below. Figure 9 shows
examples of both measures.
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Figure 9. Example of a non-linear ADC conversion curve
111
110
Output code
Output code
110
111
½ LSB wide,
DNL = -½ LSB
101
100
011
010
101
100
011
Max INL = +¾ LSB
010
1½ LSB wide,
DNL = +½ LSB
001
000
001
000
0/8 1/8 2/8 3/8 4/8 5/8 6/8 7/8 AREF
(A)
0/8 1/8 2/8 3/8 4/8 5/8 6/8 7/8 AREF
(B)
2.7.1 Differential non-linearity
Differential Non-Linearity (DNL) is defined as the maximum and minimum difference
between the step width and the perfect width (1 LSB) of any output step.
Non-linearity produces quantization steps with varying widths. All steps should be 1
LSB wide, but some are narrower or wider.
To measure DNL, a ramp input voltage is applied and all output value transitions are
recorded. The step lengths are found from the distance between the transitions, and
the most positive and negative deviations from 1 LSB are used to report the
maximum and minimum DNL.
2.7.2 Integral non-linearity
Integral Non-Linearity (INL) is defined as the maximum vertical difference between
the actual and the perfect curve.
INL can be interpreted as a sum of DNLs. E.g. several consecutive negative DNLs
raise the actual curve above the perfect curve as shown in Figure 9A. Negative INLs
indicate that the actual curve is below the perfect curve.
The maximum and minimum INL are measured using the same ramp input voltage as
in DNL measurement. Record the deviation at each conversion step midpoint and
report the most positive and negative deviations as maximum and minimum INL.
2.7.3 Measurements and compensation
It is important that DNL and INL values are measured after offset and gain error
compensation. If not, the results will be infected by the offset and gain error and thus
not reveal the true DNL and INL.
Non-linearity cannot be compensated for with simple calculations. Polynomial
approximations or table lookups can be used for that purpose. However, the typical
DNL and INL values are ½ LSB for the 10-bit ADC of the AVR, and are rarely of any
concern in real life applications.
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2.8 Temperature, frequency and supply voltage dependencies
When using the internal voltage reference with the ADC, the accuracy of this
reference must be considered. The internal voltage reference is proportional to the
bandgap voltage, which is characterized in the devices’ datasheets. The
characteristics show that the bandgap voltage is slightly dependent on supply voltage
and operating temperature.
The ADC accuracy also depends on the ADC clock. The recommended maximum
ADC clock frequency is limited by the internal DAC in the conversion circuitry. For
optimum performance, the ADC clock should not exceed 200 kHz. However,
frequencies up to 1 MHz do not reduce the ADC resolution significantly.
Operating the ADC with frequencies greater than 1 MHz is not characterized.
2.9 Bandwidth and input impedance
When using single-ended mode, the ADC bandwidth is limited by the ADC clock
speed. Since one conversion takes 13 ADC clock cycles, a maximum ADC clock of 1
MHz means approximately 77k samples per second. This limits the bandwidth in
single-ended mode to 38.5 kHz, according to the Nyquist sampling theorem.
When using differential mode, the bandwidth is limited to 4 kHz by the differential
amplifier. Input frequency components above 4 kHz should be removed by an
external analog filter, to avoid non-linearity’s.
The input impedance to VCC and GND is typically 100 MΩ. Together with the output
impedance of the signal source, this creates a voltage divider. The signal source
should therefore have sufficiently low output impedance to get correct conversion
results.
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3 Implementation
An example setup for calibrating the ADC is shown in Figure 10.
Figure 10. Production calibration setup
Calibration
Test fixture
High accuracy
DAC
Calibration
algorithm
EEPROM
Production test
control
Test firmware
AVR
Calibration parameters
End application
Offset and gain
compensation
EEPROM
AVR
Application
firmware
During the production test phase, each device’s ADC must be characterized using a
test setup similar to this. When the test fixture is ready for calibrating the AVR, the
tester signals the AVR to start calibrating itself. The AVR uses the test fixture’s high
accuracy DAC (e.g. 16-bit resolution) to generate input voltages to the calibration
algorithm. When calibration is finished, the parameters for offset and gain error
compensation are programmed into the EEPROM for later use, and the AVR signals
that it is ready for other test phases.
Note that this requires the EESAVE fuse to be programmed, so that the Flash
memory can be reprogrammed without erasing the EEPROM. Otherwise, the ADC
parameters must be temporally stored by the programmer while erasing the device.
3.1 Fixed-point arithmetic for offset and gain error compensation
Floating-point arithmetic is not an efficient way of scaling the ADC values. However,
the scaling factor for gain error compensation will be somewhere close to 1, and
needs a certain precision to yield good compensated ADC values. Therefore, fixedpoint numbers represented by integers can be used.
Since the gain compensation factor certainly never exceeds 2, it can be scaled by a
factor of 214 to fit exactly in a signed 16-bit word. In other words, the scaling factor can
be represented by two bytes as a 1:14 signed fixed-point number.
The equation for both offset and gain error compensation is shown in Equation 1.
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Equation 1.
realvalue = (adcvalue − offset ) ⋅ gainfactor
When the calculation result is truncated to an ordinary integer later, it is always
truncated to the largest integer less than or equal to the result. In order to achieve
correct rounding to the nearest integer, 0.5 must be added before truncating.
Adding 0.5, scaling the equation by 214 and moving the offset correction outside the
parentheses gives Equation 2.
Equation 2.
214 ⋅ realvalue = 214 ⋅ adcvalue ⋅ gainfactor + 214 ⋅ 0.5 - 214 ⋅ offset ⋅ gainfactor
Since the gain factor and offset correction value are constants, further optimization
can be achieved. In addition, if the result is scaled by 22, giving a total scale of 216,
the upper two bytes of the result equals the truncated integer without the need for a
16-step right-shift.
We introduce some constants and summarise it all in Equation 3.
Equation 3.
factor = 214 ⋅ gainfactor ,
correction = 214 ⋅ (0.5 − offset ⋅ gainfactor ) ,
216 ⋅ realvalue = 2 2 ⋅ (adcvalue ⋅ factor + correction)
Using this method, the calibration software calculates the constants factor and
correction and stores them in EEPROM memory. At execution time the
compensation firmware only needs one integer multiplication, one addition and two
left shift operations to correct the ADC values. Using the IAR C compiler with highest
optimization for speed, this takes 42 CPU cycles.
3.1.1 Calibration
The test fixture design is outside the scope of this application note. However, a
flowchart of a calibration firmware in the AVR is given. It uses the external DAC
through the test fixture and runs its own calibration algorithm.
There is no need for using several ADC channels, only switching between single
ended and differential conversion. The ADC parameters are the same regardless of
which channel is used. The multiplexer does not introduce any errors.
The firmware could be implemented as shown in Figure 11.
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Figure 11. Flowchart for calibration firmware
Calibration firmware
Wait for startcalibration-signal from
test fixture
Measure offset error
using external DAC
Measure gain error
using external DAC and
offset compensation
Store parameters in
EEPROM
Send calibratonfinished-signal to test
fixture
Finished
This piece of firmware is programmed into the AVR prior to calibration, and is
replaced by the actual application firmware afterwards. Once again, the EESAVE fuse
must be programmed to preserve the calibration parameters in EEPROM during
Flash reprogramming.
3.1.2 Compensation
The run-time compensation code can be implemented as a small function. Every ADC
measurement is run through this function, which uses the constants factor and
correction .
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Figure 12. Flowchart for offset and gain compensation
Offset and gain
compensation
Get ADC raw value
Multiply by factor
Shift left twice
Extract the two high
bytes
Return
This calculation in Figure 12 can be implemented using the following C function, or
alternatively as a macro:
signed int factor,
signed long correction )
{
}
The parameter constants stored in EEPROM could be copied to SRAM variables
during startup for quicker access later.
4 Literature references
• Robert Gordon – A Calculated Look at Fixed-Point Arithmetic
http://www.embedded.com/98/9804fe2.htm
• Application Note AVR210 – Using the AVR Hardware Multiplier
http://www.atmel.com/dyn/resources/prod_documents/DOC1631.PDF
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