### ETC AB-065

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SUPERPOSITION: THE HIDDEN
DAC LINEARITY ERROR
As More DACs Become Available With Resolutions of 12 Bits and Greater, Users Should Know the
Causes and Effects of Superposition Error on Relative and Absolute Accuracy and What to Do to
Minimize It.
A digital-to-analog converter (DAC) translates digital signals to analog signals. For example, a 12-bit DAC takes a
12-bit binary number, called an input code, and converts it
into one of 4,096 analog output voltages or currents. When
the contribution to the output voltage or current of each
individual bit is independent of any other, it means that the
device exhibits no superposition error or that “superposition
holds.” For a DAC with little or no superposition error, the
linearity error for any given code will relate to the linearity
error at some different code. This allows you to determine
the worst case linearity error, and the digital code where that
error occurs, with a very simple test.(1)
ment of that code. The linearity error (sometimes called
relative accuracy, integral linearity, nonlinearity or endpoint linearity) is defined as the maximum error magnitude
that occurs.
Now consider the relationship between the individual bit
errors (εi) and the linearity error. There exists some digital
input code (b1, b2 . . . bn) that yields the maximum linearity
error (EMAX) and the one’s complement of this code (b1,
b2...bn), that must yield an error of the same magnitude but
in the opposite direction (–EMAX). The relative magnitude
and polarities of the errors determine which actual input
code has the most linearity error. For the error to be maximum, all of the error terms must be additive and the
following proves true:
However, if the DAC under test has excessive superposition
error, this simple test will give erroneous results; therefore,
you must test all digital codes to determine the worst case
error and code. Superposition error, or bit interaction, often
is significant in converters with a resolution of 12 to 16 bits.
If the error becomes large enough, a DAC may fail to meet
a 1/2LSB linearity error or relative accuracy specification
even with each individual bit adjusted perfectly. This specification becomes important in many applications such as
automatic test equipment or precision voltage standards
where the absolute value of the output voltage must remain
within specified limits after calibration of offset and gain
errors.
For a DAC with low superposition, the following equation
determines the output voltage, if we assume that the offset
and gain errors have been removed:
 b1 (1/2 + ε1 ) + b 2 ( 1/4 + ε 2 ) +...
V O = V FS 
,
 + b n (1/2n + ε n )

(1)
where ει x VFS equals the linearity error associated with the
ith bit and bi, equals the value (0 or 1) of the ith bit of the DAC
input code. Since the analog output error with all input code
bits off (000...000) and all input bits on (111...111) has been
adjusted to 0 the summation of all the bit errors,
( ε1 + ε 2 + ε 3 ... ε n ) or  ∑ ε i  ,
E MAX + −E MAX = b1ε1 + b 2 ε 2 +...
+ b n ε n + b1ε1 + b 2 ε 2 +... +b n ε n
(
1983 Burr-Brown Corporation
(
)
)
(3)
but bi + bi = 1, making the maximum linearity error:
[
]
E MAX = 1/2 ε1 + ε 2 +... + ε n .
(4)
This result proves interesting because it relates the maximum linearity error to the individual bit errors; therefore,
you can evaluate a DAC by simply measuring the output
error associated with n digital input codes instead of all of
the 2n possible combinations.(2,3)
Stated another way, the sum of the positive bit errors should
equal in magnitude the sum of the negative bit errors when
the gain and offset errors have been removed. Any difference in these magnitudes indicates the presence of a superposition error. If this difference proves greater than approximately 1/10 of an LSB (JDEC standard for superposition
error), further testing may become necessary to determine
the accuracy of the DAC. However, a superposition error of
more than l/10LSB does not by itself imply that a DAC
cannot meet a linearity specification of, say, ±l/2LSB; it
simply means that you must conduct a more elaborate test to
determine the worst case linearity error and digital input
code where that error occurs.
(2)
becomes zero. This means that the errors are symmetrical or,
in other words, for every possible input code there exists an
equal and opposite error associated with the one’s comple-
)
+ bn + bn εn ;
n
i=i
(
2 E MAX = b1 + b1 ε1 + b 2 + b 2 ε 2 +...
AB-065
1
Printed in U.S.A. February, 1987
A 3-BIT DAC
The data in Table II came from a monolithic bipolar 12-bit
DAC. Note that the difference here between the positive bit
errors (+550µV) and the negative bit error (–1,650µV)
equals –1.lmV or almost 1/2LSB. In this situation, superposition does not hold and you cannot say anything definite
about linearity with the data available.
An example illustrating the relationship between linearity
error and the individual bit errors for a 3-bit DAC appears in
Figure 1a. Any deviation in the DAC output from the
straight line drawn between all bits off and all bits on
indicates a linearity error. With the superposition error less
than 1/10LSB, the error pattern will appear as symmetrical
around midscale as indicated.
Figure 1b shows a transfer characteristic for a 3-bit DAC
which exhibits superposition error. Note that, in this example, the symmetrical error pattern around midscale no
longer exists. You must consider the difference between the
electrical sum and the algebraic sum of the bit errors when
determining whether to use a more comprehensive test.
TESTING ALL INPUT CODES
When the short-cut method of measuring linearity error
does not prove sufficient, you can develop a high speed
measurement circuit capable of testing all 2n code combinations. A simple schematic of this type of tester appears in
Figure 2. The binary counter has n + 1 stages to provide a
binary count from 0 to 2n – 1 and to reset the counters at the
end of the count. The reference DAC and the x 10 error
amplifier must have combined settling times to ±1/10LSB
of less than 10µs since the system clock must operate at
20kHz to have a flicker-free display. For a 12-bit converter,
a complete cycle takes 50µs x 4,096 counts or approximately l00ms. The output of the nth counter stage also
displays on the scope to indicate the midscale transition
point, and offset and gain adjustment potentiometers are
provided to zero the end points of the error display.
The data in Table I came from a 12-bit hybrid DAC. Note
that, for this test, the full scale voltage was increased to
10.2375V, making the ideal bit weights, starting at the LSB,
equal to 2.5mV, 5.0mV, l0.0mV... 2.560V, and finally
5.12V for the MSB. You can memorize these numbers easily
and calculate the error voltages quickly by inspection. The
difference between the algebraic sum of the positive bit
errors (320µV) and negative bit errors (–310µV) equals only
10µV, which indicates a low superposition error. Thus, the
maximum linearity error becomes 1/2 x (320 + 310) =
315µV.
DAC Output (V)
DAC Output (V)
8.75
8.75
7.50
7.50
6.25
6.25
Midscale
5.00
5.00
3.75
3.75
2.50
2.50
1.25
1.25
0
Algebraic Sum
of Code
Digital
Input
0
001
010
011
100
101
110
111
001
0.50
0.25
0
–0.25
–0.50
Error (LSB)
Error (LSB)
Electrical Value
of Code
010
011
100
101
110
111
0.50
0.25
0
–0.25
–0.50
FIGURE 1. Both of These 3-Bit DAC Transfer Functions Exhibity Errors. Linearity Error (a) exists for input codes 001, 010,
101 and 110; note the symmetry of the errors about midscale. Superposition errors (b) lack symmetry about
midscale.
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INPUT CODE IDEAL OUTPUT (V) ACTUAL OUTPUT(V) ERROR(µV)
INPUT CODE IDEAL OUTPUT (V) ACTUAL OUTPUT(V) ERROR(µV)
All Bits “On”
All Bits “Off”
Bit 1 (MSB)
Bit 2
Bit 3
Bit 4
Bit 5
Bit 6
Bit 7
Bit 8
Bit 9
Bit 10
Bit 11
Bit 12 (LSB)
0
0
–50
–180
–60
–20
+40
+40
+40
+50
+100
+20
+20
+10
All Bits “On”
All Bits “Off”
Bit 1 (MSB)
Bit 2
Bit 3
Bit 4
Bit 5
Bit 6
Bit 7
Bit 8
Bit 9
Bit 10
Bit 11
Bit 12 (LSB)
+320
–310
+10
Positive Sum
Negative Sum
Difference
Positive Sum
Negative Sum
Difference
+10.23750
0
5.12000
2.56000
1.28000
0.64000
0.32000
0.16000
0.08000
0.04000
0.02000
0.01000
0.00500
0.00250
+10.23750
0
5.11995
2.55982
1.27994
0.63998
0.32004
0.16004
0.08004
0.04005
0.02010
0.01002
0.00502
0.00251
+10.23750
0
5.12000
2.56000
1.28000
0.64000
0.32000
0.16000
0.08000
0.04000
0.02000
0.01000
0.00500
0.00250
+10.23750
0
5.11927
2.55928
1.27996
0.64013
0.32013
0.16003
0.07987
0.03997
0.02000
0.01008
0.00512
0.00256
0
0
–730
–720
–40
+130
+130
+30
–130
–30
0
+80
+120
+60
+550
–1650
–1100
TABLE I. In This Data from a 12-Bit Hybrid DAC, the FullScale Voltage Was Increased to 10.2375V, Making the Ideal Bit Weights, Starting at the LSB,
Equal to 2.5mV, 5.0mV, 10.0mV...2.560V and
Finally 5.12V for the MSB. You can easily
memorize these numbers and quickly calculate
the error voltages by inspection.
TABLE II. Note That the Difference Between the Positivie
Bit Errors (+550µV) and the negative bit errors
(–1,650µV) in This Data from a Monolithic Bipolar DAC, Equals –1.1mV or Almost 1/2LSB.
In this situation, superposition does not hold and
you cannot say anything definite about linearity
with the amount of data available.
This tester works well for 8-, 9-, and 10-bit converters. For
a 12-bit DAC, the 4,096 segments displayed on the CRT are
spaced so close together that the switching transients create
a wide band of noise making it difficult to tell if the
converter meets its specification, especially with a linearity
error near the ±l/2LSB limit. One way around this problem,
if you assume that the errors contributed by the last four bits
of the DAC are small, entails inhibiting these bits with the
AND gates shown in Figure 2; this reduces the binary count
to 256 and also gives each count 16 times longer for the
glitches to settle out. You can make other improvements to
this tester such as automatic offset and gain error nulling, a
sample/hold deglitcher to remove the glitches at the error
output and a go/no go window comparator to test the
linearity error at each binary count.
SOURCES OF SUPERPOSITION ERROR
Generally, superposition error in monolithic and hybrid converters results from the feedback resistor, Rf, changing in
value as the output voltage varies from 0V to +10V. This
apparent nonlinearity comes from the variable power dissipation that occurs in this resistor which can produce a
temperature rise (self-heating) of as much as 1°C to 2°C in
some DACs. This in turn changes the absolute value of the
feedback resistor since it will have a temperature coefficient
(TC) of between 50ppm/°C and 300ppm/°C for a thin-film
material and over 1,000 ppm/°C for a monolithic diffused
resistor. This problem generally does not occur in discrete
data converters because the physical size of the feedback
resistor is so large that the temperature rise, and therefore the
resistance variation, remain extremely small. In a monolithic
converter, however, with real estate at a premium, the mass
of the feedback resistor is often so small the large temperature rise will occur for even small changes in power dissipation due to self-heating
To determine if the feedback resistor is at fault, substitute a
low TC external resistor for the internal feedback resistor of
the DAC and see if the nonlinearity disappears. The oscilloscope photograph in Figure 4 shows the results of using the
test circuit shown in Figure 2, the same DACs whose transfer
functions appear in Figures 3a and 3b respectively, with the
internal feedback resistors being replaced by low TC external
resistors. Note that the DAC errors in both cases are now
almost evenly distributed about the center of the oscilloscope
which indicates that the major cause of the superposition
error has been removed. You cannot use an external feedback
resistor, of course, in most practical applications because it
will cause excessive gain drift since it will not track the
internal diffused or thin-film reference resistor with variations in time and temperature.
Using the test circuit shown in Figure 2, and three different
12-bit DACs produced the oscilloscopic photographs in
Figure 3. The offset and gain errors have been nulled at the
left and right portions of the photographs, respectively; and
the linearity error appears as the deviation from the horizontal center line of the scope, with the vertical sensitivity
l/2LSB per division. The digital input to the MSB indicates
the mid-range and full-scale binary counts
The DAC errors displayed in Figure 3a appear symmetrically about the center of the scope, indicating very little
superposition error; while those in Figure 3b are almost all
positive which indicates a moderate amount of superposition error. Figure 3b shows why some manufacturers specify
linearity error as the maximum deviation from a best fit
straight line rather than straight line through the end points
You can see, in this example, that a linearity error specification of ±l/2LSB proves easier to meet when using the
best fit straight-line method. In a DAC with symmetrical
error patterns, as shown in Figure 3a, a straight line through
the end points becomes the same as a best fit straight line.
3
D
QD
C
QC
B
QB
A
En
Zero
+VCC
One
Shot
QA
Gain
–VCC
74121
D
C
Min/Max
3 Each 7408
B
Bit 1
QD
MSB
QC
QB
A
En
QA
D
C
Min/Max
+5V
B
16-Bit
Current
Output
Reference
DAC
QD
QC
QB
A
En
5kΩ
IOUT
Error
Out
QA
OPA111
D
C
Min/Max
+5V
B
QD
QC
QB
A
En
Bit 12
QA
1
+5V
4 Each
74191
Scope
Synchronization
LSB
12
5kΩ
DAC
Under Test
Clock
FIGURE 2. This Tester Works Well For 8-, 9-, and 10-Bit Converts. For a 12-bit DAC, the 4096, segments displayed on the
CRT are spaced so close together that the switching transients create a wide band of noise making it difficult to
tell if the converter meets its specification, especially with linearity error near the ±1/2LSB limit. One way around
this problem, if you assume that the errors contributed by the last four bits of the DAC are small, entails inhibiting
these bits with the AND gates shown.
4
A
B
FIGURE 3. In These Scope Waveform Photographs Showing the Output of the Test Circuit in Figure 2, the Top Traces Indicate
the Linearity Error, and the Bottom Traces Reflect the Status of the MSB of the Input Code. The Hybrid DAC (a)
exhibits little superposition error, while the asymmetry of the linearity error (b) about midscale shows supersposition
eror the monolithic bipolar DAC. The MSB transition marks the horizontal center.
You can minimize the effect of wiring resistance (RW,)
external to the DAC by paying careful attention to the
grounding and connection scheme employed. Figure 5a
shows a correct connection configuration that you can use
with most commercially available DACs to yield maximum
accuracy. You can reduce or eliminate the effects of various
wiring and contact resistances, R1, R2, R3 and R4, as follows:
• R1 appears in series with the feedback resistance and
therefore introduces only a gain error that can be nulled
during calibration.
• R2 appears inside the output amplifier feedback loop and
the loop gain will reduce its effect.
• R3 appears in series with the load resistor and will cause an
error in the voltage across RL. One-half LSB error would
result at full load for R3 = 0.02Ω for a 16-bit DAC. Therefore, if possible, you should sense the output voltage in such
a way as to include R3. Figure 5b illustrates the optimum
connection made possible by the ground sense pin available
on some higher accuracy DACs. In the configuration shown,
R´F = RF and RB = RDAC. This causes rejection of any signal
developed across R3 as a common mode input, and R3 will
not affect the voltage across RL. This configuration will also
reject noise present on the system common.
FIGURE 4. An External Feedback Resistor Can Decrease
Superposition Error for the Monolithic Bipolar
DAC Shown in Figure 3b.
Superposition error or bit interaction can occur in other
ways—by temperature gradients on a monolithic chip which
cause the magnitude of a bit output to be a function of the
state of the other bit switches, or by feedback resistors which
have an appreciable voltage coefficient of resistance (VCR),
such as diffused resistors might. Again, the presence of
superposition error does not mean a DAC will not meet its
linearity specification, but you will need more extensive
testing to verify if it does.
Superposition error, however, is by no means the only
source of linearity error. Pay attention to your wiring whenever you use or test a DAC. When critical portions of a
circuit share the same metallization path (e.g., a metallization path on a monolithic chip or in a wirebond; the contact
resistance of a socket; or the wiring resistance of a test
circuit), varying voltage drops caused by changing current
levels can cause serious errors which could “drown out” any
existing superposition error.
• R4 remains negligible in both circuits with ground connections made as shown.
5
RF
R1
RF
R1
R2
IDAC
R2
RDAC
IDAC
RB
RDAC
RL
R3
±15VDC
Supply
Sense
Output
RB
R3
R4
V+
RL
R'F
R4
VLOGIC
V+
±15VDC
Supply
+5VDC
V–
(A)
VLOGIC
+5VDC
V–
(B)
FIGURE 5. These Connection Diagrams Show How to Reduce the Effects of Wiring and Socket Resistance for a Typical DAC
(a) and a High Accuracy DAC (b). Resistors R1, R2, R3 and R4 represent wiring and contact resistance.
REFERENCES
1. J. Naylor, “Testing Digital/Analog and Analog/Digital
Converters,” IEEE Trans. on Circuits and Systems, Vol.
CAS-25, No. 7, July 1978.
2. T. Cate, “Tom Cate of Burr Brown Speaks Out on D/A
Converter Specs,” EDN/EEE, pp. 34-40, June 1, 1971.
3. P. Prazak, “Progammable Handheld Calculator Computes Digital-to-Analog Converter Errors,” pp.122-127,
Computer Design, June 1978.
The information provided herein is believed to be reliable; however, BURR-BROWN assumes no responsibility for inaccuracies or omissions. BURR-BROWN assumes
no responsibility for the use of this information, and all use of such information shall be entirely at the user’s own risk. Prices and specifications are subject to change
without notice. No patent rights or licenses to any of the circuits described herein are implied or granted to any third party. BURR-BROWN does not authorize or warrant
any BURR-BROWN product for use in life support devices and/or systems.
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