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Calculating the Error Budget in Precision Digital-to-Analog Converter
(DAC) Applications
David Fry, 应用工程师经理
Abstract: This application note analyses the parameters that affect the errors in precision digital-to-analog converter (DAC) applications. The analysis focuses on
the factors introduced by both the data converter and the voltage reference. It describes the calculations required to select the data converter and the reference to meet the system’s target specifications. The calculations are available in a
spreadsheet.
Overview
When designing a digital-to-analog converter (DAC) system, the DAC specifications and its voltage reference work in tandem to produce the overall system performance. Consequently, selection of both DAC and reference should be made
together. The components' specifications can be traded off against each other to ensure that system specifications are met at the lowest cost.
This application note focuses on Maxim's 3-terminal voltage references and
precision DACs. To design a system, one must first understand how the parts are specified and then how their performance characteristics interact. Voltage references and DACs have many specifications. Only those factors relevant to the error budget will be discussed here.
Voltage Reference Specifications
Initial Accuracy
This is the output voltage tolerance, ignoring any effects of temperature, input voltage, and load. Temperature is normally +25?C.
Output-Voltage Temperature Coefficient
This is the change in reference output voltage, measured for a given change in temperature and specified in ppm/?C. Maxim uses the box method. The shape of the change vs. temperature characteristic is not specified; only the limits of this function are specified. The limits of the output voltage do not necessarily coincide
with the limits of temperature. So, to calculate the maximum change, multiply the
temperature coefficient by the temperature range for the part. Thus to illustrate, if a part has a temperature coefficient of 5ppm/?C, specified from -40?C to +85?C, the
maximum deviation over temperature would be:
ΔV = (T - T) × TC = (85 + 40) × ?5 = ?625ppm MAXMIN
It is generally best to select a device that is specified over the required temperature range, rather than a broader range. For instance, the MAX6025A is specified as a
15ppm/?C reference over 0?C to +70?C. This reference value works out to 1050ppm over the range. If, however, one chose a reference specified over the -40?C to +85?C range, a reference that is 1050/125 = 8.4ppm/?C or better would be required. Note that some devices are specified over several temperature ranges.
A graphical example of the box method is shown in Figure 1. Two different example
curves are shown, both of which satisfy the 5ppm/?C specification over -40?C to
+85?C.
Figure 1. Example temperature characteristics.
With series references, therefore, it is generally not possible to relate voltage drift and temperature so that one can calculate the drift over a specific range other than that over which the part is specified.
Line Regulation
This term defines the incremental change in output voltage for a change in input
voltage. It is normally defined in terms of µV/V.
Load Regulation
This term defines the incremental change in output voltage for a change in load
current. Some DACs may not buffer the reference input. Therefore, as the code changes, the reference input impedance will also change, causing a change in reference voltage. This change is generally small, but should be considered in high-accuracy applications. Note that this is more important with some DAC topologies such as R-2R ladders, while resistive string topologies are less
susceptible.
Temperature Hysteresis
This is the change in reference voltage at +25?C after the temperature is cycled from T to T. It is specified as a ratio of the two voltages and expressed in ppm: MINMAX
6TEMPHYST = 10 × (ΔV/V) REFREF
Where ΔV is the change in reference voltage caused by the temperature cycle. REF
Long-Term Stability
This is the change in reference output voltage vs. time, specified in ppm/1000 hours. Cumulative drift beyond a 1000-hour interval is not generally specified, but
is usually much lower than the initial drift. An application's long-term stability can be
improved by PCB-level burn-in. A typical output-voltage long-term stability
characteristic is shown in Figure 2.
Figure 2.Typical output-voltage long-term stability.
Output Noise Voltage
This defines the voltage noise at the reference output. The 1/f component is specified in µV over a 0.1Hz to 10Hz bandwidth, and the wideband noise is usually P-P
specified in µV over a 10Hz to 10kHz bandwidth. RMS
DAC Specifications
Only buffered-voltage-output DACs are discussed here, as the key points about error calculations are easier to illustrate with this architecture. Current-output DACs
are typically used in a multiplying configuration (MDAC) to provide variable gain; they usually require external op amps to buffer the voltage generated across a fixed resistor.
Focusing discussion on the reference voltage, the main characteristic of this DAC
architecture is the varying DAC reference input resistance vs. DAC code. Many DACs are implemented using an R-2R ladder. The resistance of the ladder will change with DAC code. If the reference drives the ladder directly, the reference must have
sufficient load regulation to avoid introducing errors. Care must be taken to ensure that the voltage reference can source enough current at the DAC's minimum reference input resistance. Note that some DAC configurations will draw virtually
zero current from the reference at DAC code 0. Hence, switching from code 0 to code 1 can create a large current transient in the reference.
Two other DAC specifications are important to voltage-reference selection:
reference-input-voltage range and DAC output gain. These specifications will define the reference voltage for the particular application.
Output Error and Accuracy Specifications
Output error is defined as the deviation from an ideal output voltage that would be provided by the perfect match of voltage reference and DAC. It is important to note that this article addresses absolute accuracy, meaning that everything is referenced to an ideal DAC output-voltage range. For example, a 12-bit DAC code 4095 should
produce an output of 4.096V with a reference voltage of 4.096V; any deviation from this is an error. This performance contrasts to relative accuracy, where the full-scale
output is defined more by the application than by an absolute voltage. Consider another example: a ratiometric system where an ADC and a DAC with equal
resolution share a reference. It may not matter (within reason) what the actual reference voltage is, as long as the DAC-output and ADC-input voltages are nearly
equivalent for a given digital code.
Output error is often specified as a one-sided value (in LSBs at the DAC resolution),
but it actually implies a double-sided error (Figure 3). For example, a 12-bit DAC
with a 4.096V output range has an ideal LSB step size of 4.096V/4095 ~ 1mV. If the specified output error in this case is 4 LSBs at 12-bit resolution, this means that the
DAC output at any code could be ?4 LSBs (or ?4mV) from the ideal value.
Consequently, accuracy is defined by how many actual bits are available to reach a
desired output voltage with at most 1 LSB of error: Accuracy = DAC Resolution - log(Error) 2
So in this example:
Accuracy = 12 - log(4) = 10 bits 2
Therefore, one can only get to within 1 LSB at 10-bit resolution (?4mV = ?4/4096
= ?1/1024) of any ideal DAC output value.
Sources of system gain error include:
; Reference initial error
; Reference-output temperature coefficient
; Reference temperature hysteresis
; Reference long-term stability
; Reference load regulation
; Reference line regulation
; Reference output noise
; DAC gain error
; DAC offset error
; DAC gain-error temperature coefficient Other sources of system error include:
; DAC integral nonlinearity (INL)
; DAC output noise
Figure 3. Data show how errors compound to define the system DAC transfer
function.
Although the target error applies over the entire DAC code range, most of the error sources mentioned above cause an effective gain-error variation that is largest near
the full scale (highest DAC codes) of the transfer function (Figure 3). Gain errors reduce with decreasing DAC code value; these errors are halved at midscale and virtually disappear near code zero, where offset error dominates. Error sources that do not exclusively affect the gain error and apply equally over most of the DAC code range include DAC integral nonlinearity (INL) and output noise.
INL is typically defined using one of two methods: absolute linearity or end-point
linearity (Figure 4). The offset error is removed and the gain error is normalized
before the INL is measured. Absolute linearity compares the DAC linearity to the ideal transfer-function linearity. End-point linearity uses the two measured end
points to define the linearity (a straight line is drawn between these points); all
other points are compared to this line. In either case, INL should be included in the error analysis. In the latter case, the DAC INL error is zero at the end points, but can be present at DAC code words just inside these values. As an example, for a 12-bit
DAC with INL defined between the end points of 0V and 4.095V (full scale), the INL specification applies to DAC codes near 0 and 4095. For maximum error calculations, it is reasonable to add the DAC's INL and noise-induced output errors
to the previously mentioned gain errors that are most severe near code 4095. Some DACs are specified with differing INL values over the range of codes. DACs are often used in applications where the whole code range is not used, and devices specified
in this way can provide better performance over a smaller code range.
Figure 4. DAC INL measurement.
DAC and Reference Design Examples
To illustrate the steps involved with voltage reference selection for DACs, a few
design examples cover a range of applications (Table 1). The design steps are broken into individual sections by design examples (i.e., Design A through Design
D). A spreadsheet was developed to calculate the various steps and produce the
results. In the spreadsheet, cells with blue text should be entered by the designer.
Cells with red text show calculated results.
Table 1. Requirements for DAC Design Examples
Parameter Design A Design B Design C Design D
Low voltage, High absolute One-time Main Design Low cost, battery powered, accuracy and calibrated, Objectives loose accuracy moderate precision low drift accuracy
Digital offset Example Consumer Portable Lab instrument and gain Application audio device instrument adjustment
MAX5304, MAX5170, MAX5154, MAX5176, 12-bit DAC 10-bit single 14-bit single 12-bit dual single
Minimum 7kΩ (two
Reference Input 18kΩ 18kΩ shared 18kW 18kΩ
Resistance inputs)
Output Voltage 0 to 2.5V 0 to 4.096V 0 to 4.000V 0 to 2.048V
Fixed gain = Fixed gain = DAC Output Force/sense Fixed gain = 2 1.638 1.638
5V (varying) 5V (constant) 5V (constant) 3V (varying V) BattPower Supply 4.5V (min), 4.95V (min), 4.75V (min), 2.7V (min),
5.5V (max) 12V available 5.25V (max) 3.6V (max)
-40?C to Temperature 0?C to +70?C 0?C to +70?C +15?C to +45?C +85?C Range (commercial) (commercial) (< commercial) (extended)
Signal 10Hz to 10kHz DC to 1kHz DC to 10Hz 10Hz to 10kHz Bandwidth
One-time Burn-in, plus factory DAC Calibration None annual None (gain and (gain and offset) offset)
16 LSB at 10 4 LSB at 12 2 LSB at 14 bits Maximum bits bits 8 LSB at 112 bits (13-bit Target Error (6-bit (10-bit (9-bit accuracy) accuracy) accuracy) accuracy)
Step 1. Voltage Ranges and Reference Voltage Determination
When selecting a voltage reference for a DAC application, the first task is to evaluate the supply-voltage and the DAC's output-voltage ranges. A section of the
spreadsheet is shown below (Figure 5). To simplify the design examples described
above, DACs have already been chosen, so their output gain is not a variable that one would trade-off in a real design.
Figure 5. The error calculation spreadsheet assists in balancing the tradeoffs between a DAC and voltage reference.
First, enter the values for the maximum output voltage and power-supply range.
Some DACs do not allow the reference input to go all the way to the power-supply
rails, so the reference overhead can be entered. In addition, enter the minimum DAC input resistance. Thus there are four calculated parameters that can be used for reference selection: maximum reference voltage, minimum dropout, and
maximum steady-state output current. In addition, one can use the maximum power-supply voltage, as this will determine the maximum power-supply voltage
which the reference can accept. The calculated output gain is often provided by an external op amp, but may be internal as in Design B.
Design A. Low Cost, Loose Accuracy
For the Design A example, V is 5V and the output range is 0 to 2.5V. Thus, a 2.5V DD
reference is used and the MAX5304 force/sense output is set to unity gain (OUT and FB pins shorted). A lower voltage reference could be used with a higher, externally set gain, but the approach here saves the two resistors for a low-cost design.
Design B. High Accuracy and Precision
A 2.5V reference is chosen for the Design B example. The MAX5170 gain is fixed at
1.638 and a final output voltage range of 0 to 4.096V is required. If a lower reference voltage is desired for Design B, a MAX5171 DAC could be used and its
output force/sense gain could be set higher than 1.638 with external resistors. Note that the minimum V level is 4.95V. Thus the highest reference voltage that could DD
be used is 4.95V - 1.4V = 3.55V, as the DAC reference input is limited to (V - DD
1.4V).
Design C. One-Time Calibrated, Low Drift
In the Design C example, the MAX5154 has a fixed gain of 2, so a 2.048V reference provides a nominal 4.096V output at full scale. This voltage must exceed the 4.000V design requirement, so that a gain calibration can be used to scale the voltage down
to the 0 to 4V range. This design also has other reference voltage options if the MAX5156 force/sense DAC is used. Note that the reference-input upper-limit
voltage is 4.75V - 1.4V = 3.35V.
Design D. Low Voltage, Battery Powered, Moderate Accuracy
The minimum V is 2.7V in the Design D example, so the largest reference voltage DD
that could be used is 2.7V - 1.4V = 1.3V. For this example, a 1.25V reference
satisfies the 0 to 2.048V output range, as the MAX5176 gain is 1.638. It is important
that the worst-case reference voltage, including all error terms, remains below 1.3V, or the specification for the DAC reference input voltage will be exceeded. Approximate dropout voltages were calculated for each of the design examples
(Figure 5). All of these voltages are well above the 200mV (or lower) dropout voltages typical of Maxim's voltage references. Because the upper-reference input
voltage of most Maxim DACs is limited to V - 1.4V, the dropout voltages can DD
normally be ignored with these designs if, that is, the DAC and voltage reference use the same positive supply rail. The dropout voltages are approximate, because they were calculated without any error terms such as initial accuracy. Nonetheless, these errors are small compared to typical dropout voltages, and they can be ignored.
Step 2. Initial Voltage Reference Device-Selection Criteria
There are many factors to consider when selecting the optimal reference for each design. To make the procedure manageable, candidate devices will be identified
based on: the reference voltage determined above; an estimate of required initial accuracy; an approximated temperature coefficient; and the reference output current needed for the chosen DAC. These selection criteria are shown in the
spreadsheet segment below (Figure 6). Other factors such as cost, quiescent
current, packaging, and a quick glimpse at the remaining specifications will be used to select a specific initial device for each design. The remaining specifications will be
analyzed in Step 3 to determine if the devices satisfy the overall accuracy requirements.
Figure 6. This portion of the spreadsheet identifies the criteria for selecting the
optimal reference for a design.
Design A. Low Cost, Loose Accuracy
A 2.5V reference was chosen in Step 1 above. The MAX6102 is a very low-cost 2.5V
reference, specified with 0.4% initial accuracy and 65ppm tempco over the
commercial temperature range. It looks as though this could be a good choice for this application. The spreadsheet shows an initial accuracy and tempco error of 8.4 LSB, which is well within the 16 LSB requirement.
Design B. High Accuracy and Precision
Because Design B has such challenging accuracy requirements, the MAX6225 and
MAX6325 buried-zener references are the initial candidates. These references have low temperature coefficients, excellent long-term stability, and low noise. These
devices also have very good initial accuracy, but this specification is unimportant in the case of Design B, because gain errors caused by the DAC and the voltage
reference are calibrated out. Therefore, one can set the reference initial tolerance to zero in the spreadsheet. The MAX6225 and the MAX6335 source 15mA, so driving the MAX5170 DAC reference input (2.5V/18k ~ 140µA, max) is not an issue. The
MAX6325 is chosen because it has the only tempco (70?C × 1ppm/?C = 70ppm, max) that is beneath the overall 122ppm accuracy requirement (2 LSB at 14 bits =
14-42/2 - 1 = 2/16383 = 1.22 × 10 = 122ppm) while leaving margin for the other
error sources. If the Design-B accuracy requirements are relaxed slightly, the
MAX6225 A-grade device (2ppm/?C, max, tempco) would reduce the reference cost by more than half.
A 12V supply is conveniently present in the Design-B example. This requirement
allows use of the MAX6325, which needs an input voltage of at least 8V. If 8V (or higher) is not available in the system, the MAX6166 (A grade) or MAX6192 (A
grade) bandgap-based references could be considered, but a slight relaxation of the system specifications would be required.