Specifying A/D and D/A Converters Specifying A/D

Specifying A/D and D/A Converters Specifying A/D
National Semiconductor
Application Note 156
February 1976
The specification or selection of analog-to-digital (A/D) or
digital-to-analog (D/A) converters can be a chancey thing
unless the specifications are understood by the person
making the selection. Of course, you know you want an accurate converter of specific resolution; but how do you ensure that you get what you want? For example, 12 switches,
12 arbitrarily valued resistors, and a reference will produce a
12-bit DAC exhibiting 12 quantum steps of output voltage. In
all probability, the user wants something better than the expected performance of such a DAC. Specifying a 12-bit
DAC or an ADC must be made with a full understanding of
accuracy, linearity, differential linearity, monotonicity, scale,
gain, offset, and hysteresis errors.
This note explains the meanings of and the relationships
between the various specifications encountered in A/D and
D/A converter descriptions. It is intended that the meanings
be presented in the simplest and clearest practical terms.
Included are transfer curves showing the several types of
errors discussed. Timing and control signals and several binary codes are described as they relate to A/D and D/A
converters.
Accuracy is sometimes considered to be a non-specific
term when applied to D/A or A/D converters. A linearity
spec is generally considered as more descriptive. An accuracy specification describes the worst case deviation of the
DAC output voltage from a straight line drawn between zero
and full scale; it includes all errors. A 12-bit DAC could not
have a conversion accuracy better than g (/2 LSB or g 1
part in 212 a 1 ( g 0.0122% of full scale) due to finite resolution. This would be the case in Figure 1 if there were no
errors. Actually, g 0.0122% FS represents a deviation from
100% accuracy; therefore accuracy should be specified as
99.9878%. However, convention would dictate 0.0122% as
being an accuracy spec rather than an inaccuracy (tolerance or error) spec.
Accuracy as applied to an ADC would describe the difference between the actual input voltage and the full-scale
weighted equivalent of the binary output code; included are
quantizing and all other errors. If a 12-bit ADC is stated to be
g 1 LSB accurate, this is equivalent to g 0.0245% or twice
the minimum possible quantizing error of 0.0122%. An accuracy spec describes the maximum sum of all errors including quantizing error, but is rarely provided on data
sheets as the several errors are listed separately.
MEANING OF PERFORMANCE SPECS
Resolution describes the smallest standard incremental
change in output voltage of a DAC or the amount of input
voltage change required to increment the output of an ADC
between one code change and the next adjacent code
change. A converter with n switches can resolve 1 part in
2n. The least significant increment is then 2 bn, or one least
significant bit (LSB). In contrast, the most significant bit
(MSB) carries a weight of 2b1. Resolution applies to DACs
and ADCs, and may be expressed in percent of full scale or
in binary bits. For example, an ADC with 12-bit resolution
could resolve 1 part in 212 (1 part in 4096) or 0.0244% of
full scale. A converter with 10V full scale could resolve a
2.44mV input change. Likewise, a 12-bit DAC would exhibit
an output voltage change of 0.0244% of full scale when the
binary input code is incremented one binary bit (1 LSB).
Resolution is a design parameter rather than a performance
specification; it says nothing about accuracy or linearity.
Specifying A/D and D/A Converters
Specifying A/D and D/A
Converters
TL/H/5612 – 1
FIGURE 1. Linear DAC Transfer Curve Showing
Minimum Resolution Error and Best Possible Accuracy
AN-156
C1995 National Semiconductor Corporation
TL/H/5612
RRD-B30M115/Printed in U. S. A.
Quantizing Error is the maximum deviation from a straight
line transfer function of a perfect ADC. As, by its very nature, an ADC quantizes the analog input into a finite number
of output codes, only an infinite resolution ADC would exhibit zero quantizing error. A perfect ADC, suitably offset (/2
LSB at zero scale as shown in Figure 2 , exhibits only g (/2
LSB maximum output error. If not offset, the error will be
b1
a0
LSB as shown in Figure 3 . For example, a perfect 12-bit
ADC will show a g (/2 LSB error of g 0.0122% while the
quantizing error of an 8-bit ADC is g (/2 part in 28 or
g 0.195% of full scale. Quantizing error is not strictly applicable to a DAC; the equivalent effect is more properly a
resolution error.
TL/H/5612 – 3
FIGURE 4. Linear, 1 LSB Scale Error
Gain Error is essentially the same as scale error for an
ADC. In the case of a DAC with current and voltage mode
outputs, the current output could be to scale while the voltage output could exhibit a gain error. The amplifier feedback
resistors would be trimmed to correct the gain error.
Offset Error (zero error) is the output voltage of a DAC with
zero code input, or it is the required mean value of input
voltage of an ADC to set zero code out. (See Figure 5 .)
Offset error is usually caused by amplifier or comparator
input offset voltage or current; it can usually be trimmed to
zero with an offset zero adjust potentiometer external to the
DAC or ADC. Offset error may be expressed in % FS or in
fractional LSB.
FIGURE 2. ADC Transfer Curve, (/2 LSB Offset at Zero
TL/H/5612 – 4
FIGURE 5. DAC Transfer Curve, (/2 LSB Offset at Zero
TL/H/5612–2
Hysteresis Error in an ADC causes the voltage at which a
code transition occurs to be dependent upon the direction
from which the transition is approached. This is usually
caused by hysteresis in the comparator inside an ADC. Excessive hysteresis may be reduced by design; however,
some slight hysteresis is inevitable and may be objectionable in converters if hysteresis approaches (/2 LSB.
Linearity, or, more accurately, non-linearity specifications
describe the departure from a linear transfer curve for either
an ADC or a DAC. Linearity error does not include quantizing, zero, or scale errors. Thus, a specification of g (/2 LSB
FIGURE 3. ADC Transfer Curve, No Offset
Scale Error (full scale error) is the departure from design
output voltage of a DAC for a given input code, usually fullscale code. (See Figure 4 .) In an ADC it is the departure of
actual input voltage from design input voltage for a full-scale
output code. Scale errors can be caused by errors in reference voltage, ladder resistor values, or amplifier gain, et. al.
(See Temperature Coefficient.) Scale errors may be corrected by adjusting output amplifier gain or reference voltage. If the transfer curve resembles that of Figure 7 , a scale
adjustment at */4 scale could improve the overall g accuracy compared to an adjustment at full scale.
2
linearity implies error in addition to the inherent g (/2 LSB
quantizing or resolution error. In reference to Figure 2 ,
showing no errors other than quantizing error, a linearity
error allows for one or more of the steps being greater or
less than the ideal shown.
Differential Non-Linearity indicates the difference between actual analog voltage change and the ideal (1 LSB)
voltage change at any code change of a DAC. For example,
a DAC with a 1.5 LSB step at a code change would be said
to exhibit (/2 LSB differential non-linearity (see Figures 6 and
7 ). Differential non-linearity may be expressed in fractional
bits or in % FS.
Differential linearity specs are just as important as linearity
specs because the apparent quality of a converter curve
can be significantly affected by differential non-linearity
even though the linearity spec is good. Figure 6 shows a
curve with a g (/2 LSB linearity and g 1 LSB differential nonlinearity while Figure 7 shows a curve with a 1(/4 LSB linearity and g (/2 LSB differential non-linearity. In many user applications, the curve of Figure 7 would be preferred over that
of Figure 6 because the curve is smoother. The differential
non-linearity spec describes the smoothness of a curve;
therefore it is of great importance to the user. A gross example of differential non-linearity is shown in Figure 8 where
the linearity spec is g 1 LSB and the differential linearity
spec is g 2 LSB. The effect is to allow a transfer curve with
grossly degraded resolution; the normal 8-step curve is reduced to 3 steps in Figure 8. Similarly, a 16-step curve (4-bit
converter) with only 2 LSB differential non-linearity could be
reduced to 6 steps (a 2.6-bit converter?). The real message
is, ‘‘Beware of the specs.’’ Do not ignore or omit differential
linearity characteristics on a converter unless the linearity
spec is tight enough to guarantee the desired differential
linearity. As this characteristic is impractical to measure on
a production basis, it is rarely, if ever, specified, and linearity
is the primary specified parameter. Differential non-linearity
can always be as much as twice the non-linearity, but no
more.
Figure 6 shows a 3-bit DAC transfer curve with no more
than g (/2 LSB non-linearity, yet one step shown is of zero
amplitude. This is within the specification, as the maximum
deviation from the ideal straight line is g 1 LSB ((/2 LSB
resolution error plus (/2 LSB non-linearity). With any linearity
error, there is a differential non-linearity (see below). A g (/2
LSB linearity spec guarantees monotonicity (see below) and
s g 1 LSB differential non-linearity (see below). In the example of Figure 6 , the code transition from 100 to 101 is the
worst possible non-linearity, being the transition from 1 LSB
high at code 100 to 1 LSB low at 110. Any fractional non-linearity beyond g (/2 LSB will allow for a non-monotonic transfer curve. Figure 7 shows a typical non-linear curve; non-linearity is 1(/4 LSB yet the curve is smooth and monotonic.
FIGURE 6. g (/2 LSB Non-Linearity (Implies 1 LSB
Possible Error), 1 LSB Differential Non-Linearity
(Implies Monotonicity)
TL/H/5612 – 6
FIGURE 8. g 1 LSB Linear,
g 2 LSB Differential Non-Linear
TL/H/5612 – 5
Monotonicity. A monotonic curve has no change in sign of
the slope; thus all incremental elements of a monotonically
increasing curve will have positive or zero, but never negative slope. The converse is true for decreasing curves. The
transfer curve of a monotonic DAC will contain steps of only
positive or zero height, and no negative steps. Thus a
smooth line connecting all output voltage points will contain
no peaks or dips. The transfer function of a monotonic ADC
will provide no decreasing output code for increasing input
voltage.
FIGURE 7. 1(/4 LSB Non-Linear,
(/2 LSB Differential Non-Linearity
Linearity specs refer to either ADCs or to DACs, and do not
include quantizing, gain, offset, or scale errors. Linearity errors are of prime importance along with differential linearity
in either ADC or DAC specs, as all other errors (except
quantizing, and temperature and long-term drifts) may be
adjusted to zero. Linearity errors may be expressed in % FS
or fractional LSB.
3
Figure 9 shows a non-monotonic DAC transfer curve. For
the curve to be non-monotonic, the linearity error must exceed g (/2 LSB no matter by how little. The greater the linearity error, the more significant the negative step might be.
A non-monotonic curve may not be a special disadvantage
is some systems; however, it is a disaster in closed-loop
servo systems of any type (including a DAC-controlled
ADC). A g (/2 LSB maximum linearity spec on an n-bit converter guarantees monotonicity to n bits. A converter exhibiting more than g (/2 LSB non-linearity may be monotonic,
but is not necessarily monotonic. For example, a 12-bit DAC
with g (/2 bit linearity to 10 bits (not g (/2 LSB) will be monotonic at 10 bits but may or may not be monotonic at 12 bits
unless tested and guaranteed to be 12-bit monotonic.
(a) Full-Scale Step
TL/H/5612 – 7
TL/H/5612 – 8
FIGURE 9. Non-Monotonic
(Must be l g (/2 LSB Non-Linear)
(b) 1 LSB Step
Settling Time is the elapsed time after a code transition for
DAC output to reach final value within specified limits, usually g (/2 LSB. (See also Conversion Rate below.) Settling
time is often listed along with a slew rate specification; if so,
it may not include slew time. If no slew rate spec is included,
the settling time spec must be expected to include slew
time. Settling time is usually summed with slew time to obtain total elapsed time for the output to settle to final value.
Figure 10 delineates that part of the total elapsed time
which is considered to be slew and that part which is settling
time. It is apparent from this figure that the total time is
greater for a major than for a minor code change due to
amplifier slew limitations, but settling time may also be different depending upon amplifier overload recovery characteristics.
Slew Rate is an inherent limitation of the output amplifier in
a DAC which limits the rate of change of output voltage after
code transitions. Slew rate is usually anywhere from 0.2 to
several hundred volts/ms. Delay in reaching final value of
DAC output voltage is the sum of slew time and settling time
as shown in Figure 10 .
Overshoot and Glitches occur whenever a code transition
occurs in a DAC. There are two causes. The current output
of a DAC contains switching glitches due to possible asynchronous switching of the bit currents (expected to be worst
at half-scale transition when all bits are switched). These
FIGURE 10. DAC Slew and Settling Time
glitches are normally of extremely short duration but could
be of (/2 scale amplitude. The current switching glitches are
generally somewhat attenuated at the voltage output of the
DAC because the output amplifier is unable to slew at a very
high rate; they are, however, partially coupled around the
amplifier via the amplifier feedback network and seen at the
output. The output amplifier introduces overshoot and some
non-critically damped ringing which may be minimized but
not entirely eliminated except at the expense of slew rate
and settling time.
Temperature Coefficient of the various components of a
DAC or ADC can produce or increase any of the several
errors as the operating temperature varies. Zero scale offset error can change due to the TC of the amplifier and
comparator input offset voltages and currents. Scale error
can occur due to shifts in the reference, changes in ladder
resistance or non-compensating RC product shifts in dualslope ADCs, changes in beta or reference current in current
switches, changes in amplifier bias current, or drift in amplifier gain-set resistors. Linearity and monotonicity of the DAC
can be affected by differential temperature drifts of the ladder resistors and switches. Overshoot, settling time, and
slew rate can be affected by temperature due to internal
change in amplifier gain and bandwidth. In short, every
specification except resolution and quantizing error can be
affected by temperature changes.
4
example, a 12-bit BCD system has a resolution of only 1
part in 1000 compared to 1 part in 4096 for a binary system.
This represents a loss in resolution of over 4:1.
Offset Binary is a natural binary code except that it is offset
(usually (/2 scale) in order to represent negative and positive
values. Maximum negative scale is represented to be all
‘‘zeros’’ while maximum positive scale is represented as all
‘‘ones.’’ Zero scale (actually center scale) is then represented as a leading ‘‘one’’ and all remaining ‘’zeros.’’ The comparison with binary is shown in Figure 11 .
Two’s Complement Binary is an alternate and more widely
used code to represent negative values. With this code,
zero and positive values are represented as in natural binary
while all negative values are represented in a twos complement form. That is, the twos complement of a number represents a negative value so that interface to a computer or
microprocessor is simplified. The twos complement is
formed by complementing each bit and then adding a 1; any
overflow is neglected. The decimal number b8 is represented in twos complement as follows: start with binary code of
decimal 8 (off scale for g representation in 4 bits so not a
valid code in the g scale of 4 bits) which is 1000; complement it to 0111; add 0001 to get 1000. The comparison with
offset binary is shown in Figure 11 . Note that the offset
binary representation of the g scale differs from the twos
complement representation only in that the MSB is complemented. The conversion from offset binary to twos complement only requires that the MSB be inverted.
Long-Term Drift, due mainly to resistor and semiconductor
aging can affect all those characteristics which temperature
change can affect. Characteristics most commonly affected
are linearity, monotonicity, scale, and offset. Scale change
due to reference aging is usually the most important
change.
Supply Rejection relates to the ability of a DAC or ADC to
maintain scale, offset, TC, slew rate, and linearity when the
supply voltage is varied. The reference must, of course, remain constant unless considering a multiplying DAC. Most
affected are current sources (affecting linearity and scale)
and amplifiers or comparators (affecting offset and slew
rate). Supply rejection is usually specified only as a % FS
change at or near full scale at 25§ C.
Conversion Rate is the speed at which an ADC or DAC can
make repetitive data conversions. It is affected by propagation delay in counting circuits, ladder switches and comparators; ladder RC and amplifier settling times; amplifier and
comparator slew rates; and integrating time of dual-slope
converters. Conversion rate is specified as a number of conversions per second, or conversion time is specified as a
number of microseconds to complete one conversion (including the effects of settling time). Sometimes, conversion
rate is specified for less than full resolution, thus showing a
misleading (high) rate.
Clock Rate is the minimum or maximum pulse rate at which
ADC counters may be driven. There is a fixed relationship
between the minimum conversion rate and the clock rate
depending upon the converter accuracy and type. All factors which affect conversion rate of an ADC limit the clock
rate.
Input Impedance of an ADC describes the load placed on
the analog source.
Output Drive Capability describes the digital load driving
capability of an ADC or the analog load driving capacity of a
DAC; it is usually given as a current level or a voltage output
into a given load.
CODES
Several types of DAC input or ADC output codes are in
common use. Each has its advantages depending upon the
system interfacing the converter. Most codes are binary in
form; each is described and compared below.
Natural Binary (or simply Binary) is the usual 2n code with
2, 4, 8, 16, . . . , 2n progression. An input or output high or
‘‘1’’ is considered a signal, whereas a ‘‘0’’ is considered an
absence of signal. This is a positive true binary signal. Zero
scale is then all ‘‘zeros’’ while full scale is all ‘‘ones.’’
Complementary Binary (or Inverted Binary) is the negative
true binary system. It is identical to the binary code except
that all binary bits are inverted. Thus, zero scale is all
‘‘ones’’ while full scale is all ‘‘zeros.’’
Binary Coded Decimal (BCD) is the representation of decimal numbers in binary form. It is useful in ADC systems
intended to drive decimal displays. Its advantage over decimal is that only 4 lines are needed to represent 10 digits.
The disadvantage of coding DACs or ADCs in BCD is that a
full 4 bits could represent 16 digits while only 10 are represented in BCD. The full-scale resolution of a BCD coded
system is less than that of a binary coded system. For
(a) Zero to a Full-Scale
TL/H/5612 – 9
(b) g Full-Scale
FIGURE 11. ADC Codes
5
Specifying A/D and D/A Converters
Sign Plus Magnitude coding contains polarity information
in the MSB (MSB e 1 indicates a negative sign); all other
bits represent magnitude only. This code is compared to
offset binary and twos complement in Figure 11 . Note that
one code is used up in providing a double code for zero.
Sign plus magnitude code is used in certain instrument and
audio systems; its advantage is that only one bit need be
changed for small scale changes in the vicinity of zero, and
plus and minus scales are symmetrical. A DVM might be an
example of its use.
data is valid. Typically, an EOC output can be connected to
an SC input to cause the ADC to operate in continuous
conversion mode. In non-continuous conversion systems,
the SC signal is a command from the system to the ADC. A
DAC does not supply an EOC signal.
Clock signals are required or must be generated within an
ADC to control counting or successive approximation registers. The clock controls the conversion speed within the
limitations of the ADC. DACs do not require clock signals.
CONTROL
Each ADC must accept and/or provide digital control signals telling it and/or the external system what to do and
when to do it. Control signals should be compatible with one
or more types of logic in common use. Control signal timing
must be such that the converter or connected system will
accept the signals. Common control signals are listed below.
Start Conversion (SC) is a digital signal to an ADC which
initiates a single conversion cycle. Typically, an SC signal
must be present at the fall (or rise) of the clock waveform to
initiate the cycle. A DAC needs no SC signal; however, such
could be provided to gate digital inputs to a DAC.
End of Conversion (EOC) is a digital signal from an ADC
which informs the external system that the digital output
CONCLUSION
Once the user has a working knowledge of DAC or ADC
characteristics and specifications, he should be able to select a converter to suit a specific system need. The likelihood of overspecification, and therefore an unnecessarily
high cost, is likewise reduced. The user will also be aware
that specific parameters, test conditions, test circuits, and
even definitions may vary from manufacturer to manufacturer. For practical production reasons, parameters may not be
tested in the same manner for all converter types, even
those supplied by the same manufacturer. Using information
in this note, the user should, however, be able to sort out
and understand those specifications (from any manufacturer) pertinent to his needs.
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AN-156
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