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**Static characterization**- 1. ADC transition points
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As discussed in the previous chapter, the linearity parameter calculations of an A/D converter are based on the transition points (or trip-points) of the device.

The following parameters will be discussed:

To explain the linearity parameters of an A/D converter, the plot below can overlay some example ADCs with an ideal 4 bits ADC. The 1/2 LSB option shows a ADC where the first transition point starts at a halve LSB in stead of 1 LSB. The plot can show five different kinds of ADC data:

- 1) ADC 1: an ADC with only an offset error
- 2) ADC 2: an ADC with only a gain error
- 3) ADC 3: an ADC with an offset, gain and linearity error, no missing codes
- 4) ADC 4: an ADC with an offset, gain and linearity error and 1 missing code (code 8)
- 5) Random ADC data: random errors. Use the button "New ADC data" for a new ADC

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1/2 LSB Click on a point

The following ADC data presentations are available:

- Trip-points: The transfer plot of the ideal ADC and example ADC (blue line) are overlayed
- End point overlay: Overlay end point line and ADC error related to the end point line (blue line)
- Best fit overlay: Overlay best fitting line and ADC error related to the best fitting line (blue line)
- End point error: The error of the example ADC compared to the end point reference line (error in LSBs)
- Best fit error: The error of the example ADC compared to the best fitting reference line (in LSBs)
- Differential error: The error of each step of the example ADC (LSBs)
- Total unadjusted error: The error of the example ADC compared to the ideal line(in LSBs)

For the trip-point (1), end point overlay (2) and best fit overlay (3), the x-axis can diplay voltages or LSBs. The Differential error (6) shows the error in of each step. The first and last step are not included.

For the first three presentations, the y-axis shows (ADC output) codes and the x-axis (ADC input) voltages or LSBs. For the other 4 presentations the x-axis shows trip-points (point 1 is the first trip-point or the transition from 0 to 1), except the Differential error plot (6) as mentioned above. The y-axis shows the error in LSBs.

To determine the error parameters of an ADC, it is necessary to have a reference line. There are two common used reference lines:

- End point line
- Best fitting line

**Figure 1**

The end point line is a straight line between the first transition point and the last transition point. So only the first and last points are used for the calculation of the reference line. The first and last point of the End point error plot (4) will always be zero. In End point overlay mode (2) the first and last transitions are equal to the device first and last transition points.

The best fitting line calculation uses all transition points. The least-squares linear regression algorithm is used. The equation of the best fitting line (y = ax + b) is:

The best fitting line will be exactly in the center of all errors. In the best fitting error plot representation (5), the sum of all errors above the zero line is equal to the sum of all errors below the zero line (the zero line is the best fitting reference line). The best fitting line will always have a better INLE result, but it is more common to use the end point line.

The end point overlay presentation (2) corresponds with the left plot of Figure 1. The best fit presentation (3) corresponds with the right plot.

The offset error is the error of the first transition point (or trip-point) from the ideal transition point (end point calculation).

For the best fitting line calculation the offset error is the offset of the best fitting reference line (related to the ideal transfer line).

**Examples:**

ADC 1: The end point reference line is y = 1.000x + 0.250. The offset of the end point reference line from the ideal first trip-point is -0.25 LSB. See also the end point overlay presentation (2) and select x-axis = LSB. The first trip-point of the end point reference line is 0.25 LSB less than the trip-point of the ideal line.

ADC 4: Select the best fit overlay presentation (3) and x-axis = LSB. The offset of the best fit reference line (orange line) is 0.90 LSB from the ideal first trip-point.

A better term for offset error would be the zero scale error. The term offset implies that all conversions are off by an equal amount. In the case of a strong non-linearity near the zero scale value, this definition may be misleading, and the less ambiguous zero scale error would be a better term.

The full scale error is the error of the last transition point (or trip-point) from the ideal transition point (end point full scale error). It is equal to the sum of the gain error and offset error.

**Examples:**

ADC 1: The full scale error is equal to the offset error: -0.25 + 0.00 = -0.25 LSB.

ADC 2: The full scale error is equal to the gain error: 0.00 + -0.70 LSB = -0.70 LSB.

ADC 3: The end point full scale error is -0.25 + -0.20 LSB = -0.45 LSB. Select the trip-point presentation (1) and for the x-axis LSB. The last trip-point is approx. 0.45 LSB before the ideal last trip-point.

ADC 3: The best fit full scale error is approx. -1.5 LSB (-1.18 + -0.28 = -1.48 LSB). See the last trip-point of the reference line (orange line) in the best fit overlay presentation (3). It can be found at approx. 1.5 LSB before the ideal trip-point position.

The gain error is equal to the Full scale error with the offset error subtracted. It is the deviation (of the end point or best fit reference line) from the ideal slope of the transfer characteristic. The slope can be found in the "a" of the reference line y = ax + b. The gain error can be calculated with the equation (N-1)/a - (N-1). Where N is the number of trip-points and N-1 the number of steps between the trip-points.

**Examples:**

The example ADCs are all 4 bits converter with 16 steps and 15 trip-points. E.g. for ADC 2 the (end point) error is (15-1)/1.0526 - (15-1) = -0.70 LSB.

ADC 4: Select the best fit presentation (3) and x-axis = LSB. Reading the full scale error from the plot, an error of approx. 0.3 LSB (exactly 0.34 LSB) can be found (The last trip-point of best fit reference line (orange line) is about 0.3 LSB > than the ideal trip-point). The offset is 0.9 LSB. The gain error is 0.34 - 0.9 = -0.56 LSB. (N-1)/a - (N-1) = (15-1)/1.0417 - (15-1) = -0.56 LSB.

Integral non linearity error describes the departure from a reference line. The offset and gain are not included in the INLE. It is a measure of the straightness of the transfer function. The size and distribution of the DNL errors will determine the integral linearity of the converter. The INLE representation shows the sum of the DNL errors. The INL error is calculatated by:

Where Vtrp(x) is the transistion from code x-1 to x. Vzs is the zero scale voltage (start voltage) of the reference line. ALSB is the actual (or measured) LSB step. The actual LSB step is calculated by ILSB/a , where ILSB is the ideal LSB step and "a" is the angle of the reference line (the "a" of y = ax + b).

**Examples:**

For an INL plot, select the Best fit error (4) or End point error (5) presentation. The maximum deviation from the zero line (the zero line is the reference line) is the INLE.

ADC 3: Select the Best fit error plot (5). The maximum deviation is at trip-point 8 (transition 7 -> 8, see also the Best fit overlay presentation).

ADC 1 does only have an offset error, the INL error is zero.

ADC 2 does only have a gain error, no linearity error.

The maximium deviation of the 1 LSB step. The 1 LSB step for the DNL calculation is based on the measured (or actual) LSB step. The actual 1 LSB step is the ideal LSB divided by "a" (ILSB/a), where "a" is the angle of the reference line (the "a" of y = ax + b). In reality the difference between the actual 1 LSB (1/a) and the ideal 1 LSB step is very small. The DNL is calculated by:

Where ALSB is the actual 1 LSB step. Vtrp(x+1) is the trip-point voltage of code change x to x+1 and Vtrp(x) the trip-point voltage of code change x-1 to x.

A DNLE of -1 or could indicate a missing code. ADC 4 in the plot above is missing code 8. In the differential error presentation (6), an error of -1 LSB can be found.

With the "search trip-point algorithm" option in the ATX7006 calculations, the DNLE can be less than -1 LSB. The trip-point is found before its preceding trip-point. This is mostly due to insufficient measurement resolution, a noisy source or a noisy ADC. With the option "sort codes" a DNLE less than -1 LSB will not occur.

Total Unadjusted Error is a specification that includes linearity errors, gain error, and offset error. It is the worst-case deviation from the ideal device performance. The TUE is calculated by:

Where Vtrp(x) is the transistion from code x-1 to x. Vzs is the zero scale voltage (start voltage) of the (ideal) ADC. ILSB is the ideal LSB step.

Select the Total unadjusted error presentation (7) in the plot for an example. The trip-point presentation (1) will also show the total error of the device related to the ideal converter.

Code Error is the error between the ideal (expected) code and the current code. The code error is the Total Unadjusted in LSB, rounded to the nearest integer.

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