
Keywords: voltage references, voltage reference, DAC reference, three terminal reference, reference accuracy, DAC, digital to analog converter, DAC design examples, error budget
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Calculating the Error Budget in Precision DigitaltoAnalog Converter (DAC) Applications
By: 
David Fry, Strategic Applications Engineering Manager 
© Sep 25, 2008, Maxim Integrated Products, Inc.


Abstract: This application note analyses the parameters that affect the errors in precision digitaltoanalog 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.
Download associated spreadsheet.
Overview
When designing a
digitaltoanalog 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 3terminal 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.
OutputVoltage 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
_{MAX}  T
_{MIN}) × TC = (85 + 40) × ±5 = ±625ppm
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 highaccuracy applications. Note that this is more important with some DAC topologies such as
R2R 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
_{MIN} to T
_{MAX}. It is specified as a ratio of the two voltages and expressed in ppm:
TEMPHYST = 10
^{6} × (ΔV
_{REF}/V
_{REF})
Where ΔV
_{REF} is the change in reference voltage caused by the temperature cycle.
LongTerm Stability
This is the change in reference output voltage vs. time, specified in ppm/1000 hours. Cumulative drift beyond a 1000hour interval is not generally specified, but is usually much lower than the initial drift. An application's longterm stability can be improved by PCBlevel burnin. A typical outputvoltage longterm stability characteristic is shown in
Figure 2.
Figure 2. Typical outputvoltage longterm stability.
Output Noise Voltage
This defines the voltage noise at the reference output. The 1/f component is specified in µV
_{PP} over a 0.1Hz to 10Hz
bandwidth, and the
wideband noise is usually specified in µV
_{RMS} over a 10Hz to 10kHz bandwidth.
DAC Specifications
Only bufferedvoltageoutput DACs are discussed here, as the key points about error calculations are easier to illustrate with this architecture. Currentoutput 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 R2R 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 voltagereference selection: referenceinputvoltage 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 outputvoltage range. For example, a 12bit 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 fullscale 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 DACoutput and ADCinput voltages are nearly equivalent for a given digital code.
Output error is often specified as a onesided value (in LSBs at the DAC resolution), but it actually implies a doublesided error (
Figure 3). For example, a 12bit 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 12bit 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_{2}(Error)
So in this example:
Accuracy = 12  log_{2}(4) = 10 bits
Therefore, one can only get to within 1 LSB at 10bit resolution (±4mV = ±4/4096 = ±1/1024) of any ideal DAC output value.
Sources of system
gain error include:
 Reference initial error
 Referenceoutput temperature coefficient
 Reference temperature hysteresis
 Reference longterm stability
 Reference load regulation
 Reference line regulation
 Reference output noise
 DAC gain error
 DAC offset error
 DAC gainerror temperature coefficient
Other sources of system error include:
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 gainerror 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 endpoint 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 transferfunction linearity. Endpoint 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 12bit 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 noiseinduced 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 
Main Design Objectives 
Low cost, loose accuracy 
High absolute accuracy and precision 
Onetime calibrated, low drift 
Low voltage, battery powered, moderate accuracy 
Example Application 
Consumer audio device 
Lab instrument 
Digital offset and gain adjustment 
Portable instrument 
DAC 
MAX5304, 10bit single 
MAX5170, 14bit single 
MAX5154, 12bit dual 
MAX5176, 12bit single 
Minimum ReferenceInput Resistance 
18kΩ 
18kΩ 
7kΩ (two shared 18kW inputs) 
18kΩ 
Output Voltage 
0 to 2.5V 
0 to 4.096V 
0 to 4.000V 
0 to 2.048V 
DAC Output 
Force/sense 
Fixed gain = 1.638 
Fixed gain = 2 
Fixed gain = 1.638 
Power Supply 
5V (varying), 4.5V (min), 5.5V (max) 
5V (constant), 4.95V (min), 12V available 
5V (constant), 4.75V (min), 5.25V (max) 
3V (varying V_{BATT}), 2.7V (min), 3.6V (max) 
Temperature Range 
0°C to +70°C (commercial) 
0°C to +70°C (commercial) 
40°C to +85°C (extended) 
+15°C to +45°C (< commercial) 
Signal Bandwidth 
10Hz to 10kHz 
DC to 1kHz 
DC to 10Hz 
10Hz to 10kHz 
DAC Calibration 
None 
Burnin, plus annual (gain and offset) 
Onetime factory (gain and offset) 
None 
Maximum Target Error 
16 LSB at 10 bits (6bit accuracy) 
2 LSB at 14 bits (13bit accuracy) 
4 LSB at 12 bits (10bit accuracy) 
8 LSB at 112 bits (9bit 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 supplyvoltage and the DAC's outputvoltage 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 tradeoff 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 powersupply range. Some DACs do not allow the reference input to go all the way to the powersupply 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 steadystate output current. In addition, one can use the maximum powersupply voltage, as this will determine the maximum powersupply 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 DesignA example, V
_{DD} is 5V and the output range is 0 to 2.5V. Thus, a 2.5V 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 lowcost design.
Design B. High Accuracy and Precision
A 2.5V reference is chosen for the DesignB 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 referencevoltage 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
_{DD} level is 4.95V. Thus the highest reference voltage that could be used is 4.95V  1.4V = 3.55V, as the DAC reference input is limited to (V
_{DD}  1.4V).
Design C. OneTime Calibrated, Low Drift
In the DesignC 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 referencevoltage options if the
MAX5156 force/sense DAC is used. Note that the referenceinput upperlimit voltage is 4.75V  1.4V = 3.35V.
Design D. Low Voltage, Battery Powered, Moderate Accuracy
The minimum V
_{DD} is 2.7V in the DesignD example, so the largest reference voltage 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 worstcase reference voltage, including all error terms, remains below 1.3V, or the specification for the DAC referenceinput 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 referenceinput voltage of most Maxim DACs is limited to V
_{DD}  1.4V, the dropout voltages can 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 VoltageReference DeviceSelection 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 lowcost 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 buriedzener references are the initial candidates. These references have low temperature coefficients, excellent longterm 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 MAX6325 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 = 2/2
^{14}  1 = 2/16383 = 1.22 × 10
^{4} = 122ppm) while leaving margin for the other error sources. If the DesignB accuracy requirements are relaxed slightly, the MAX6225 Agrade device (2ppm/°C, max, tempco) would reduce the reference cost by more than half.
A 12V supply is conveniently present in the DesignB 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) bandgapbased references could be considered, but a slight relaxation of the system specifications would be required.
Design C. OneTime Calibrated, Low Drift
The
MAX6162 and
MAX6191 Agrade devices are considered for Design C because of their low tempco (5ppm/°C max):
Total error budget is 4 LSB at 12 bits = 4/4096 × 106 = 977ppm
Required tempco ≤ 977ppm/(85  (40)) ≤ 7.8ppm/°C
Available error beyond tempco = 977ppm  5ppm/°C × 125°C = 352ppm
Note that the MAX6162 and the MAX6191 both have 2mV (977ppm) initial accuracy. This specification is not a concern with the 2.048V reference, because the outputvoltage range is only 0 to 4.000V and a gain calibration is planned for this design. The reference initial accuracy is set to zero in the spreadsheet. The MAX6162 (5mA of output current drive) and the MAX6191 (500µA of output current drive) can both drive the 293µA referenceinput current that results when the two MAX5154 DAC reference pins are tied together (2.048V/[14kΩ14kΩ]). However, the MAX6162 has more margin if additional loads are connected to the reference output. The MAX6162 does have higher quiescent current than the MAX6191 (120µA versus 35µA, max), but this is not a deciding factor as Design C is not power constrained. After reviewing the initial specifications, it is clear that either device is probably acceptable. However, the MAX6162 is the first choice due to its higher output current. If further analysis shows the MAX6162 to be marginally unacceptable, the MAX6191 could be considered because it has slightly better loadregulation, temperaturehysteresis, and longtermstability specifications.
Design D. Low Voltage, Battery Powered, Moderate Accuracy
Following the approaches used in the other examples, the total error for Design D is found to be 3907ppm (10
^{6} × 16/4095). Over the narrow +15°C to +45°C temperature range, a tempco of at most 130.2ppm/°C (3906ppm/30°C) can be tolerated. However, one must be careful not to make a common mistake here. Remember that these example exercises are dealing with the box method. So start again and ignore the 130ppm/°C that was just erroneously calculated. Using the guideline from Design A to allocate roughly half of the error budget to the tempco, then reasonable, conservative reference choices are the
MAX6012 (A and B grades are 15ppm/°C and 25ppm/°C, respectively, over 0°C to +70°C) and the
MAX6190 (A, B, and C grades are 5ppm/°C, 10ppm/°C, and 25ppm/°C, respectively, over 40°C to +85°C). These parts are considered because they have a maximum quiescent current of 35µA, which is appropriate for the lowpower needs of Design D.
The MAX6190C price is in the same range as the MAX6012B. Either part will probably work in the application. However, the MAX6012A is especially attractive, because it is available in a 3pin SOT23 package, which is ideal for a small, batterypowered, portable instrument.
A quick check of the MAX6012A reveals the tempcorelated error to be 1050ppm (70°C × 15ppm/°C). The initial error of 3200ppm (0.32%) also needs to be considered, as there is no trimming planned for this design. The sum of these two errors is 4250ppm, which is already outside the 3906ppm design limit. Because the MAX6012A is insufficient, the 3pin SOT23 package is abandoned. The MAX6190A is chosen as a starting point, because its initial error of 1600ppm and 5ppm/°C leave enough room for the other error terms. Reference output current is not a concern for this design, because the MAX6190 can supply 500µA (>> 69µA design requirement).
Step 3. Final Specification Review and ErrorBudget Analysis
With the preliminary selection of references complete, it is now time to verify the remaining specifications, which include referenceload regulation; inputline regulation; outputvoltage temperature hysteresis; outputvoltage longterm stability; and output noise voltage. The analysis is shown in the spreadsheet segment below (
Figure 7).
Figure 7. This portion of the spreadsheet helps calculate the remaining specifications and, ultimately, the error budget.
Each example is analyzed, focusing on the specifications that apply to that particular design. It is most convenient to do the errorbudget accounting in parts per million (ppm), although this could be done equivalently in other units such as %, mV, or LSBs. It is also important to apply the proper scaling and to use the proper normalization factor to obtain the correct error values. Referenceerror terms can be calculated relative to the reference voltage or the DAC output voltage. For example, assuming a reference error of 2.5mV (e.g., noise, drift, etc.) and a reference voltage of 2.5V, then:
Reference output error = 10^{6} × 2.5mV/2.5V = 1000ppm
Assuming that the DAC output
amplifier has a gain of 2.0, then both the error and the reference voltage are scaled. This produces the same result at the DAC output (5V fullscale range):
DAC output error = 10^{6} × (2.5mV × 2)/(2.5V × 2) = 1000ppm
Into this section of the spreadsheet, enter the reference specifications for: temperature hysteresis, longterm stability, load regulation, line regulation, and output noise. Also enter DAC specifications for: INL, gain error, gain tempco, and noise.
The spreadsheet will calculate values for: worstcase error, rootsumsquare (RSS) error, worstcase error margin, and RSS error margin. It is important to think of how the errors can stack up. Some very accurate applications may be very difficult to meet if a worstcase analysis is used. If one can assume that errors may not be correlated, then the RSS method can often be used. Yet, theoretically, some boards that result may not be as accurate as they should be.
Design A. Low Cost, Loose Accuracy
No calibration or trimming is planned for Design A, so the MAX6102 initial error of 4000ppm (or 0.4%) directly becomes part of the budget, as does the 4450ppm due to the voltagereference tempco (70°C × 65ppm/°C). The typical MAX6102 outputvoltage temperaturehysteresis specification is also used directly in the error budget. (Remember that this is a typical value, if the design ultimately has marginal accuracy.) For outputvoltage longterm stability, assume twice the MAX6102 1000hour specification (2 × 50ppm = 100ppm). This is fairly conservative, as it is usually much better after the first 1000 hours. A conservative estimate here at least partially offsets the typical specification used for temperature hysteresis.
To calculate the variation in reference voltage caused by load regulation, one must know the worstcase range of currents that the voltage reference supplies to the DAC's reference input. Recall that in Step 2 above, the maximum DAC reference current that the MAX6102 would have to drive is 140µA. The minimum current is close to 0 since the MAX5304 in an R2R ladder. The MAX5304 reference input is effectively an open circuit (several GΩ input impedance) when the DAC code value is 0. This means that the total outputcurrent variation that the MAX6102 sees is 140µA. This value should be used for the loadregulation calculation:
Loadregulation error 
= 140µA × 0.9mV/mA 
= 126µV (max) 
= 10^{6} × 126µV/2.5V 
= 50ppm (max) 
In general, it is best to be conservative and use the maximum output current directly for the loadregulation calculation. There is a possible exception: if you are trying to extract the last bit of accuracy from a design and both the maximum and minimum DAC referenceinput resistance values are well specified. This approach results in a smaller loadregulation error because of the smaller ΔI
_{REF}.
Because the power supply is specified as varying for this example, one must consider the effects of input line regulation on the MAX6102 reference. The powersupply range is specified as 4.5V to 5.5V. From this, a conservative referencevoltage lineregulation calculation is possible:
Loadregulation error 
= (5.5V  4.5V) × 300µV/V 
= 300µV (max) 
= 10^{6} × 300µV/2.5V 
= 120ppm (max) 
The final voltagereferencerelated error term to consider is the effect of reference outputnoise voltage. Conveniently, Design A has a signal bandwidth (10Hz to 10kHz) that corresponds to the exact MAX6102 noisevoltage bandwidth. Consequently, the wideband noisevoltage specification of 30µV
_{RMS} is used directly (that is, bandwidth scaling is not required). Comparing the load and lineregulation values (126µV and 300µV, respectively), it is apparent that noise is not a major contributor in this design. Using crude approximations to get numbers for the error analysis, one can assume an effective peak noise value of ~42µV (30µV × √2), which corresponds to 17ppm (10
^{6} × 42µV/2.5V) with the DAC gain of 1. This analysis is trying purposefully to keep the noise calculations simple; a more detailed analysis can be performed if the relative error of the noise is larger or if the design is marginal. Remember that noise is specified as a typical value when judging design margin.
Consider now the relevant MAX5304 DAC specifications that impact accuracy at, or near, the upper end of the code range. The DAC INL value of ±4 LSB (at 10 bits) is used directly. Treating it as a singlesided quantity, as with the other error terms in our analysis, one arrives at a value of 3910ppm (10
^{6} × 4/1023). Similarly, the DAC gain error is specified as ±2 LSB and results in an error of 1955ppm (10
^{6} × 2/1023). The final MAX5304 DAC specification to be considered is gainerror tempco, which gives a typical error of 70ppm (70°C × 1ppm/°C). The DAC output noise is not specified for the MAX5304, so it is ignored, probably without adverse consequences in this 6bitaccurate system.
When all of the error sources are summed, the result is a worstcase error of 14902ppm which, although fairly close, meets the targeterror specification of 15640ppm. When confronted with this marginal situation, one can rationalize that the design will probably never have an error of this magnitude, because the error specification assumes worstcase conditions for most parameters. The RSS approach gives an error of 7474ppm, which is valid if the errors are uncorrelated. Some error sources may be correlated, so the truth probably lies somewhere between these two numbers. But regardless of the approach, the DesignA requirements have been met.
Design B. High Accuracy and Precision
The initial error of the Agrade MAX6225 is 0.04% or 400ppm, which exceeds Design B's entire 122ppm error budget. Because this application has gain calibration, virtually all of this reference initial error can be removed. This latter decision assumes that the calibration equipment has sufficient (~1µV) accuracy and the trim circuit has enough precision. The tempco contribution is calculated as 70ppm (70°C × 1ppm/°C), and the typical temperature hysteresis value of 20ppm is used directly. The longterm stability specification of 30ppm is also used rather than a more conservative number, because the instrument in this application has an initial burnin as well as an annual calibration.
Applying the same assumptions used in Design A, then Design B's reference output current variation is 140µA (coincidentally, the same number as in Design A). In this case, the MAX6225 data sheet specifies the load regulation in ppm/mA. To use the spreadsheet, convert this value to mV/mA, which leads to the following loadregulation error calculation:
Loadregulation error 
= 6ppm × 2.5/1000 
= 0.015mV/mA 
= 140µA × 0.015mV/mA/2.5 
= 0.8ppm (max) 
The power supply is specified as constant in this application, so the line regulation is assumed to be 0ppm. The precise bounds are not defined, but this does not matter since calibration will remove any errors. Note that it would be < 1ppm even if the power supply were not constant, as long as the regulation remains within the specified 4.95V to 5.05V range, because the MAX6225 lineregulation specification is 7ppm/V max. Hence, zero is entered into the spreadsheet.
Because the bandwidth for Design B is specified as DC to 1kHz, one must consider both the 1.5µV
_{PP} low frequency (1/f) noise and the 2.8µV
_{RMS} broadband noise specified from 0.1Hz to 10Hz and 10Hz to 1kHz, respectively. Using the same crude RMStopeak approximation as Design A, and adding the two peaknoise terms together, the total noise estimate is 2ppm at the reference output ([[0.75µV + 2.8µV
_{RMS} × √2]/2.5V] × 10
^{6}). Again, to put the values into the spreadsheet, convert to ppm. Notice that this is the same value that one would obtain if it were calculated at the DAC output. This is because the equation would be multiplied by 1.638/1.638 to rescale everything to 4.096V. It is worth mentioning that the peaknoisesum method used here is fairly conservative, yet the total error contribution is still relatively small. An RSS approach is probably more accurate, because the two noise sources are most likely uncorrelated. Still, this smaller value would be even more "in the noise" compared to the peakvalue approach.
All that remains for the DesignB analysis is to include the DAC error terms. The INL for the Agrade MAX5170 DAC is specified as ±1 LSB, which is 61ppm and exactly half of the 122ppm error budget of ±2 LSB at 14 bits. The DAC gain error is specified as ±8 LSB worst case, but this error is removed completely by the gain calibration mentioned earlier. As with the reference, one can set the gain error to zero in the spreadsheet. The calibration works as follows:
 The DAC is set to a digital code where the ideal output voltage is known (for example, decimal DAC code 16380 should produce precisely 4.095V at the output).
 The reference voltage is then trimmed until the DAC output voltage is at this exact value, even if the reference voltage itself is not 2.500V.
The MAX5170 DAC does not list a gain tempco, although the gain error is specified over the operatingtemperature range. Because the gain error is calibrated out at only one temperature, Design B should be tested to ensure that the gain does not drift excessively over temperature. The final consideration is the MAX5170 DAC output noise, whose typical peak noise is roughly estimated as 1ppm ([10
^{6} × √(1000Hz ×
π/2) × 80nV
_{RMS}/√Hz × √2]/4.096V).
Ultimately, the final worstcase accuracy is 184ppm (≈3 LSB at 14 bits), which does not quite meet the accuracy target of 122ppm. In contrast, the RSS accuracy is acceptable at 100ppm. Based on these numbers, the design can be considered a success. It has illustrated the important points and is close to the target accuracy with several conservative assumptions. In a realworld application, this design could be accepted as is, or the accuracy requirements could be relaxed slightly. Alternatively, a more expensive reference could be used if this design were not acceptable.
Design C. OneTime Calibrated, Low Drift
The initial error of the Agrade MAX6162 is 0.1%, which consumes the entire DesignC error budget of 977ppm. However, like Design B this is at least partially calibrated out. Note that the uncalibrated +4.096V MAX5154 DAC fullscale output voltage exceeds the required +4.000V output range, and that the DAC has 1mV resolution even though only ±4mV of accuracy is required. Therefore, it is possible to do a digital calibration on the DAC input digital codes to remove some of the reference's initial error and the DAC's gain error.
The digital gain calibration is best demonstrated with an example. Assume that the DAC output voltage needs to be at the fullscale value of 4.000V, but the ideal decimal DAC code of 4000 results in a measured output of only 3.997V due to various errors in the system. Using digital calibration, a correction value is added to the DAC code to produce the desired result. In this example, when the DAC output voltage of 4.000V is required, a corrected DAC code of 4003 is used instead of 4000. This gain calibration is scaled linearly across the DAC codes, so it has little effect at the lower codes and more impact on the upper codes.
The digital gain calibration accuracy is limited by the 12bit resolution of the DAC, so the best one can expect is ~±1mV or 244ppm (10
^{6} × 1mV/4.096V) of error after the calibration has been applied. Note that the accuracy is calculated on a 4.096V scale in this example to maintain consistency. It could be calculated relative to the +4.000V output range if required by the application; the error would be slightly higher.
If the required output range in this example were 4.096V, there are other options that could be used to always bias the uncalibrated DAC output voltage above 4.096V. In this manner, the digital gain calibration scheme described in this example could be employed. Such options include the following:
 Use an adjustable reference whose output is always above 4.096V when all circuit tolerances are considered.
 Use a force/sense DAC with the gain set slightly higher than necessary.
 Add an output buffer with gain.
The MAX6162 reference tempco error is calculated as 625ppm (125°C × 5ppm/°C), and the typical temperaturehysteresis value of 125ppm is used directly. The longtermstability specification is doubled to a more conservative 160ppm, because no burnin is specified for the application and the reference is never calibrated once it leaves the factory.
Design C's worstcase referenceoutput current variation is found to be 293µA (2.5V/[14kΩ14kΩ]) remember that there are two DACs driven by the reference), which is used directly in the loadregulation calculation:
Loadregulation error 
= 293µA × 0.9mV/mA 
= 264µV (max) 
= 10^{6} × 264µV/2.048V 
= 129ppm (max) 
Because referenceload regulation is proportional to the reference output voltage, it can be calculated at either the voltage reference (264µV/2.048V) or the DAC output ((2 × 264µV)/(2 × 2.048V)).
The power supply is constant in this application, so the line regulation is assumed to be 0ppm. With the bandwidth for Design C specified as 0.1Hz to 10Hz, half of the 22µV
_{PP} lowfrequency (1/f) noise specification (peak value) is used to arrive at a noise contribution of 5ppm at the reference output (10
^{6} × (22µV/2)/(2.048V)). As mentioned previously, the same 5ppm answer is obtained if the calculation is referred to the DAC output, because the equation is just multiplied by 2.0/2.0.
Moving to the MAX5154 DAC error terms, the Agrade INL is ±0.5 LSB, which is 122ppm on the 12bit scale. The DAC gain error is ±3 LSB (244ppm), but is ignored because it was already accommodated in the digital reference/DAC gain calibration mentioned earlier in this step. It should not be counted twice. The MAX5154 gainerror tempco has a typical value of 4ppm/°C, which gives a total of 500ppm (125°C × 4ppm/°C). The DAC output noise is not specified for the MAX5154, so it is ignored. Recognize, now, that this could present a problem, but experience with Design B indicates that DAC noise is usually a relatively small contributor to the total error. Measurements can be performed to confirm this assumption.
The worstcase error for Design C is calculated as 1980ppm, and the RSS error is 861ppm. With a targeterror specification of 977ppm, the current design is marginally acceptable at best, especially given that some typical values were used and the DAC output noise was not considered. Some options for improvement follow:
 Use the MAX6191 instead of the MAX6162. The MAX6191 has better load regulation (0.55µV/µA vs. 0.9mV/mA), temperature hysteresis (75ppm vs. 125ppm), and longterm stability (50ppm vs. 115ppm). The end result would be a 1750ppm worstcase error and an 823ppm RSS error, which is a net change of 230ppm and 38ppm, respectively. This is a slight improvement, but may not be enough.
 Reexamine the overall systemaccuracy specifications to determine if any parameters can be relaxed. The existing design could be the best choice in terms of accuracy vs. cost.
 Reduce the temperature range if the entire extended range is not needed. For example, if the range can be reduced from 40°C to +85°C down to 10°C to +75°C, the worstcase error drops to 1505ppm and the RSS error becomes 648ppm. This happens because much of the error budget is consumed by the reference tempco (625ppm) and the DAC's gainerror tempco (500ppm). Although only one of these error terms is below the 977ppm target, the comfort level increases considerably compared to the original MAX5154/MAX6162 design.
 If an 8V or greater supply is available, consider the MAX6241 4.096V reference and the MAX5156 DAC (the force/sense version of the MAX5154) set to unity gain. This combination is slightly more expensive, but produces an approximate worstcase error of 956ppm and an RSS error of 576ppm, both of which are under the 977ppm totalerror target.
 Consider other DACs that have typical gain tempcos as low as 1ppm/°C.
Design D. Low Voltage, Battery Powered, Moderate Accuracy
No calibration or trimming is planned for Design D, so the Agrade MAX6190 initial error of 1600ppm (10
^{6} × 2mV/1.25V) is used directly in the error budget, along with 625ppm (125°C × 5ppm/°C) for the tempco error. The 75ppm temperature hysteresis is also used directly; the risk of using this typical specification is at least partially offset by the reduced operatingtemperature range (+15°C to +45°C). Once again, the 1000hour longterm stability is doubled to 100ppm as a conservative estimate of the drift, as there is no burnin in this application.
The loadregulation error is again calculated from the assumed worstcase MAX5176 DAC referenceinput current of 69µA:
Loadregulation error 
= 69µA × 0.5µV/µA 
= 34.5µV (max) 
= 10^{6} × 34.5µV/1.25V 
= 28ppm (max) 
The power supply varies between 2.7V and 3.6V in this design, so the MAX6190 lineregulation specification of 80µV/V (max) must be included in the analysis:
Lineregulation error 
= (3.6V  2.7V) × 80µV/V 
= 72µV (max) 
= 10^{6} × 72µV/1.25V 
= 58ppm (max) 
As with Design C, the bandwidth for Design D is specified as 0.1Hz to 10Hz, so half of the 25µV
_{PP} lowfrequency (1/f) noise specification is used to arrive at a peak noise contribution of 10ppm at the reference output (10
^{6} × [12.5µV/1.25V]). The same 10ppm referenceinduced noise term is expected at the DAC output, because the reference voltage and noise see the same DAC gain.
Focusing now on the MAX5176 DAC error terms, the Agrade INL is ±2 LSB, which is 488ppm on the 12bit scale.
The DAC worstcase gain error of ±8 LSB with a 5kΩ load translates to 1953ppm at 12 bits. Like the MAX5170 in Design B, the MAX5176 does not specify a gainerror tempco. This is not a concern in Design D for two important reasons: it is not a lowdrift design calibrated at one temperature, and the maximum DAC gain error is specified over the entire operatingtemperature range. The final consideration is the MAX5176's DAC output noise. The estimated typical peak value can be shown to be negligible ([10
^{6} × (√10Hz ×
π/2) × 80nV
_{RMS}/√Hz × √2]/2.048V) ≈ 0.22ppm.
As with Designs B and C, the worstcase error of 4462ppm exceeds the 3906ppm target error, while the 2580ppm RSS error is well below the target. Based on these numbers, Design D is considered successful, because it comfortably meets the requirements from an RSS standpoint and has demonstrated the important design concepts. If further improvement is desired, alternative DACs should be considered first, because the MAX6190 is the best lowpower voltage reference available with an output below 1.3V (caused by the V
_{DD}  1.4V limitation on DAC reference inputs) and such lowquiescent current (35µA).
DAC VoltageReference Design Summary
This article has demonstrated a design procedure for DAC voltagereference selection involving the three steps:
Step 1. Voltage ranges and referencevoltage determination. The powersupply voltage and the DAC outputvoltage range were used to determine viable referencevoltage and DAC gain options.
Step 2. Initial voltagereference deviceselection criteria. Candidate voltage references were considered. Design focus was on reference voltage (determined in Step 1), initial accuracy, tempco, and reference output current. From these candidates, an initial device was selected.
Step 3. Final specification review and errorbudget analysis. The selected voltagereference and DAC requirements were established. To meet the design goals, iteration between Steps 2 and 3 may be required. When following the design procedure described above, it is convenient to do the error analysis in ppm and to understand how it relates to other systemaccuracy and error measures (
Table 2).
Table 2. Error Analysis in PPM Relative to Other Standard DAC System Specifications
±LSB Accuracy (Bits) 
±1 LSB Error (ppm) 
±1 LSB Error (%) 
±16Bit Error LSB 
±14Bit Error LSB 
±12Bit Error LSB 
±10Bit Error LSB 
±8Bit Error LSB 
±6Bit Error LSB 
16 
15.25878906 
0.001525879 
1 
0.25 
< 0.25 
< 0.25 
< 0.25 
< 0.25 
15 
30.51757813 
0.003051758 
2 
0.5 
< 0.25 
< 0.25 
< 0.25 
< 0.25 
14 
61.03515625 
0.006103516 
4 
1 
0.25 
< 0.25 
< 0.25 
< 0.25 
13 
122.0703125 
0.012207031 
8 
2 
0.5 
< 0.25 
< 0.25 
< 0.25 
12 
244.140625 
0.024414063 
16 
4 
1 
0.25 
< 0.25 
< 0.25 
11 
488.28125 
0.048828125 
32 
8 
2 
0.5 
< 0.25 
< 0.25 
10 
976.5625 
0.09765625 
64 
16 
4 
1 
0.25 
< 0.25 
9 
1953.125 
0.1953125 
128 
32 
8 
2 
0.5 
< 0.25 
8 
3906.25 
0.390625 
256 
64 
16 
4 
1 
0.25 
7 
7812.5 
0.78125 
512 
128 
32 
8 
2 
0.5 
6 
15625 
1.5625 
1024 
256 
64 
16 
4 
1 
5 
31250 
3.125 
2048 
512 
128 
32 
8 
2 
4 
62500 
6.25 
4096 
1024 
256 
64 
16 
4 
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APP 4300: Sep 25, 2008
TUTORIAL 4300,
AN4300,
AN 4300,
APP4300,
Appnote4300,
Appnote 4300
