# Calculating tempco and initial accuracy for voltage references

The primary purpose of a voltage reference (V_{REF}) is to establish system accuracy. An analog-to-digital converter (ADC), for example, uses a reference voltage to set its full-scale input level. The following discussion explains how a tradeoff between initial accuracy and temperature coefficient (tempco) can broaden your selection of voltage references while meeting a given requirement for system accuracy. The calculations presented determine precisely the tempco needed for a given value of initial accuracy, and vice versa.

The many other factors having an effect on voltage-reference accuracy include load regulation, line regulation, temperature hysteresis, and long-term drift (also known as stability). These effects are usually secondary, but they should be considered when a reference selected by the basic method (described below) is only marginally acceptable. Several excellent application notes cover these effects, along with the pros and cons of available voltage reference technologies, **Reference 1**.

A typical application specifies a range of analog voltages to be digitized by the ADC. To fit a standard input-voltage range, these analog signals must, in the general case, be anti-alias filtered, buffered, and perhaps scaled in amplitude. Among the typical full-scale values specified for an ADC input, 2.048V and 4.096V are obviously handy in a digital system, because they provide an integral number of millivolts for the bit value. A 12-bit ADC with 4.096V full-scale input, for example, gives bit values of 4.096/(2^{12 }= 4096) = 1mV. An 8-bit ADC in the same system has a “granularity” of 4.096/(2^{8} = 256) = 16mV/bit.

We further assume that the system requires full resolution from the ADC. That is, the output will be reasonably correct and responsive to even a 1-LSB input change. As a consequence, we state that the total allowable conversion error is ^{1}/_{2} LSB. To simplify the discussion we assume a perfect ADC, for which the only error contribution is from the reference. The worst-case error allowed from the V_{REF} is therefore ^{1}/_{2} LSB (8mV for an 8-bit ADC).

*Initial accuracy:* To lock the boundary conditions, we separate variables and temporarily assume the possibility of a V_{REF} with zero tempco, thereby assuming that all of its error comes from its initial accuracy. Note that 8mV of error in a 4.096V output represents an error of 0.195%, and that a zero-tempco reference of that accuracy can have a potential error of 0.195% at any temperature.

*Tempco:* To again consider boundary conditions, assume for the moment a V_{REF} with zero initial error at 25 deg C (where most voltage references are calibrated). All of its error must therefore come from its tempco, which can cause an error of no more than ^{1}/_{2} LSB above or below 4.096V at the operating-temperature extremes. In other words, the tempco must cause no more than 8mV of error at the operating-temperature extreme furthest from 25 deg C, whether hot or cold.

**Going through the math** An actual VREF exhibits both initial-accuracy and tempco errors (obviously), so we take the following approach:

- Determine the V
_{REF}operating temperature range. - Note which temperature extreme is furthest from 25 deg C.
- Base all calculations on that extreme.
- Determine the reference output voltage (V
_{REF}). - Calculate
^{1}/_{2}LSB as a percentage of full scale, which is the total accuracy for a zero-tempco part. An 8mV error in a 4.096V reference, for instance, is 0.195% of full scale. - Calculate the worst-case tempco allowed in ppm/deg C, assuming a perfectly accurate, zero-tolerance part at 25 deg C.
- Find suitable solutions using the technique illustrated below.

As an example, we assume an operating-temperature range of 0 deg C to 70 deg C and a 10 deg C rise inside the enclosure, giving a maximum V_{REF} temperature of 80 deg C. (The minimum is the minimum operating temperature of 0 deg C.) A V_{REF} temperature of 80 deg C is 55 deg C hotter than 25 deg C and 0 deg C is only 25 deg C colder than 25 deg C, so we are concerned in this example with the hot extreme. For the maximum allowed error (0.195%), the tempco (assuming zero inaccuracy at 25 deg C) is 0.195%/55 = .00355% = 35.5 ppm/ deg C. (See discussion below for a reminder on why we convert from percentage to ppm.)

**Figure 1** depicts this situation:

Figure 1: This graph supports the calculation of allowable values for V

_{REF}tolerance, and temperature coefficient, while achieving an 8mV error at 80 deg C.

An actual voltage reference that just meets the requirements mentioned above is represented by any one of the multiple lines converging at the worst-case point in the upper right corner.

The worst-case performance over temperature for the given V_{REF} can be represented by a line that passes through the point 80 deg C, 8mV (with respect to 4.096V). Use the formula for a line (y = mx+b), with variables defined as follows:

y = error (%)

m = temperature coefficient (%/ deg C)

x = temperature difference from 25 deg C

b = initial accuracy at 25 deg C.

Notice that the temperature coefficient in these formulas is in terms of %/deg C. That format provides units that are consistent with error (e), which we express in %. Often, temperature coefficients are so small that they are more easily expressed as parts per million (ppm). The unit “ppm” is 10,000 times smaller than the unit “%”: % means “parts per 100,” and the ratio of 100 to 1,000,000 is 10,000. For convenience, we can re-name the variables as follows:

y becomes e

m becomes TC

x becomes ΔT

b becomes A.

Thus,

e = TC(ΔT) + A.

Solving for the initial accuracy (A) makes this equation more useful,

A = e – TC(ΔT),

as does the solution for temperature coefficient (TC):

TC = (e – A)/ΔT.

For the V_{REF} in our example, all lines defining the various combinations of A and TC must pass through the maximum-error point of (55C, 0.195%):

0.195 = TC(55) + A.

Solving for A:

A = 0.195 – 55TC.

Solving for TC:

TC = (0.195-A)/55.

You can now evaluate a V_{REF} by plugging in its tempco (expressed as %/deg C), and calculating its required accuracy. Alternatively, you can pick a given accuracy and use the second formula to calculate the maximum allowed tempco.

For example, the initial accuracy of a MAX6043BAUT41 reference is 0.1%, which is about half of 0.195%. Its temperature coefficient is 25 ppm/deg C. Is that good enough? Substituting this accuracy in the second formula gives the allowable tempco:

TC = (0.195-A)/55

= (0.195 - 0.1)/55

= .00173 %/deg C

= 17.3 ppm/deg C.

Thus, the 25ppm tempco of this part is not acceptable, because 17.3 ppm or better is needed. Fortunately, the A grade version (MAX6043AAUT41) has a tempco of only 15 ppm/deg C, and its initial accuracy is also better (0.06%). Manufacturers such as Maxim offer a wide range of voltage references, with initial accuracies ranging from 2% down to 0.02%, and with temperature coefficients ranging from 150 ppm down to 1 ppm.

**References**

1. See http://www.maxim-ic.com/References.cfm

**About the author**

*John Mossman* is a Field Applications Engineer at Maxim Integrated Products Inc., Sunnyvale, CA

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