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**T****cal Calibration**

*Calibration/Validation of Data/Transient Loggers, Recorders and Acquisition Systems*

*This is a Tcal calibration article. Follow this link to return to the Tcal homepage.*

If you are a non-technical reader you may want to jump straight to the examples and skip the introductory parts of this article. The examples are all from the sample data-set. It is possible to recreate most of this using the sample data and the spreadsheets (particularly using the "valPlot" tab in the range spreadsheet). Some analysis was done by hand for this article so sometimes spreadsheet may round to a slightly different value.

**Percentage Error**

*The errors are presented as percentages.*

A data logger produces a voltage that may be combined with other numbers to give a final result. In many cases it is therefore easier to use a percentage. Tcal first calculates errors as a voltage, then divides it by the total span (for that range) which makes it a fraction and then multiplies it by 100 to turn it into a percentage.

The use of the span to describe the percentage error is typical for data loggers and so for consistency is used by Tcal.

**Least Squares Fit**

*A standard way of measuring a relationship, in this case between the input and the measured voltage.*

This produces a "line of best fit" for the data collected when the test voltages were input to the instrument.

A least squares fit is done to the five data points. It is not necessary to understand this to understand the process and interpret the results. The x-axis is the input voltage and the y-axis is the measured voltage. The standard Excel functions have not been used due to problems in the past with Excel having bugs with the in-built least squares functions.

**Linearity**

*A standard name for how well a set of points lie on a line. There are a number of ways to quantify this. A maximum-error approach is taken.*

Tcal finds the maximum difference between a measured voltage value and the line of best fit. This maximum value is used for the "Linearity" error. This is seen as a good approach as it is conservative and gives the maximum likely error regardless of the voltage being measured.

**DC Offset**

*A standard name for the error for an input of zero volts.*

This is where the line of best fit passes through the y-axis. It is the value that the trend in the instrument response gives to a zero voltage. It is possible to use a zero-correction to correct this error.

For example, a pre-trigger section of data might be used to provide data for a zero-correction.

DC Offset is a component of the Inaccuracy error.

**Inaccuracy**

*This name is used to represent the error caused by the line of best fit being wrong.*

The Inaccuracy as reported by Tcal is the maximum difference between the best fit line and the input signal.

Inaccuracy includes the DC Offset error.

**How to Combine the Errors (Ignoring RMS Noise)**

*Using these errors to create realistic numbers for an application.*

The maximum likely error (assuming five data points is sufficient which is generally the case) is the maximum difference between the input signal and the best fit line plus the maximum difference between the best fit line and the voltage measured. This then gives the maximum difference between the input voltage and the measured voltage which is what we are generally interested in. So, add the Linearity and the Inaccuracy to get a value for the maximum likely error (ignoring noise).

**Making a Zero-Correction**

*Checking the response of a measurement system to a zero input is often done to correct the origin. If this is done the errors due to the instrument can be treated differently.*

If you do a zero-correction then the component of the Inaccuracy that is due to the origin being in the wrong place is corrected. In this case the Inaccuracy can be adjusted by subtracting the DC Offset.

**RMS Noise**

*This is treated last as it is different type of error than the others.*

The RMS Noise is discussed in more detail in its own article. In many calibration systems, including Tcal, the ground (or zero voltage) value is used to find the RMS Noise. This means any noise from the wires and the rest of the calibration setup do not add to the final result.

**Example Calibration Results**

*Examples from the sample Tcal dataset.*

Examples from different ranges and channels from the Tcal sample dataset are now presented. The Inaccuracy, Linearity and DC Offset are given as voltages and not as percentages so you can practise finding the values on the graphs.

The Inaccuracy is the maximum distance between the best-fit line and the input voltage. On the unadjusted graph the Inaccuracy is generally too small to see. On the difference graph the Inaccuracy can be seen as the maximum distance between the straight line and the x-axis.

The Linearity is the maximum distance between the best-fit line and an average measured voltage. On the unadjusted graph the Linearity is generally too small to see. On the difference graph the Linearity can be seen as the maximum distance between a cross (representing a measured voltage) and the best-fit line.

The DC Offset is the error of the best-fit line for a zero input voltage. On the unadjusted graph the DC Offset is generally too small to see. On the difference graph the DC Offset can be seen as the y-intercept of the best-fit line.

**Example from Sample Dataset, +/-10V, Channel 3**

*Results from processing the sample dataset. Graphs are from the valPlot tab in the range spreadsheet.*

Average voltage readings with calibration results from sample dataset, +/-10V range, channel 3.

Average voltage readings with calibration results from sample dataset, +/-10V range, channel 3. The y-axis shows difference between measured voltage and input voltage.

The calibration results were 0.0022V Inaccuracy, 0.0018V Linearity and -0.0018V DC Offset. The error bars are a combination of 0.0002V from RMS Noise and 0.004V from other errors.

The first graph shows that visually it looks like an accurate instrument. The graph of difference between measured and input voltage gives a closeup view of the accuracy of the instrument. The instrument can be seen to slightly underestimate all voltages. It passes through the y-axis at -0.002V which is consistent with the DC Offset being -0.0018V. The Inaccuracy is at a maximum on the right-hand-side of the graph due to the negative slope of the line. Coincidentally the Linearity is at a maximum here as well because this is where the measured voltage is furthest from the best-fit line.

**Example from Sample Dataset, +/-10V, Channel 5**

*Results from processing the sample dataset. Graphs are from the valPlot tab in the range spreadsheet.*

Average voltage readings with calibration results from sample dataset, +/-10V range, channel 5.

Average voltage readings with calibration results from sample dataset, +/-10V range, channel 5. The y-axis shows difference between measured voltage and input voltage.

The calibration results were 0.0017V Inaccuracy, 0.0017V Linearity and -0.0011V DC Offset. The error bars are a combination of 0.0002V from RMS Noise and 0.004V from other errors.

The point where the line-of-best-fit is furthest from the x-axis is for the largest negative input voltage. The largest Linearity error is for zero volts where it can be seen by eye to be approximately 0.0015V which is consistent with the actual value of 0.0017V.

**Example from Sample Dataset, +/-10V, Channel 6**

*Results from processing the sample dataset. Graphs are from the valPlot tab in the range spreadsheet.*

Average voltage readings with calibration results from sample dataset, +/-10V range, channel 6.

Average voltage readings with calibration results from sample dataset, +/-10V range, channel 6. The y-axis shows difference between measured voltage and input voltage.

The calibration results were 0.0021V Inaccuracy, 0.0018V Linearity and 0.0021V DC Offset. The error bars are a combination of 0.0001V from RMS Noise and 0.004V from other errors.

For this channel the measured voltages are always slightly too high compared to the input voltage. This cannot be seen on the overview graph but the graphs of differences shows it. If you are doing analysis like this it is important to remember that this is exaggerating a very small error.

**Example from Sample Dataset, +/-1V, Channel 1**

Average voltage readings with calibration results from sample dataset, +/-1V range, channel 1.

Average voltage readings with calibration results from sample dataset, +/-1V range, channel 1. The y-axis shows difference between measured voltage and input voltage.

The calibration results were 0.0015V Inaccuracy, 0.0002V Linearity and 0.0002V DC Offset. The error bars are a combination of 0.00001V from RMS Noise and 0.0004V from other errors.

This range and channel shows a slight systematic error in the measurement that varies with the input voltage. The Inaccuracy is a measure of how far the input voltage is from the best-fit-line which is why the -80% input voltage is used to give the Inaccuracy. The Inaccuracy looks by eye to be approximately 0.0015V which is consistent with the calculated value of 0.0015V.

**Example from Sample Dataset, +/-1V, Channel 2**

Average voltage readings with calibration results from sample dataset, +/-1V range, channel 2.

Average voltage readings with calibration results from sample dataset, +/-1V range, channel 2. The y-axis shows difference between measured voltage and input voltage.

The calibration results were 0.0006V Inaccuracy, 0.0007V Linearity and -0.0001V DC Offset. The error bars are a combination of 0.00001V from RMS Noise and 0.0004V from other errors.

Similar features can be seen in this example to the others. The -80% input voltage must be furthest from the best-fit line as that has been selected by the analysis to be the Linearity error.

**Example from Sample Dataset, +/-0.1V, Channel 7**

Average voltage readings with calibration results from sample dataset, +/-0.1V range, channel 7.

Average voltage readings with calibration results from sample dataset, +/-0.1V range, channel 7. The y-axis shows difference between measured voltage and input voltage.

The calibration results were 0.0006V Inaccuracy, 0.0007V Linearity and -0.0001V DC Offset. The error bars are a combination of 0.000006V from RMS Noise and 0.00004V from other errors.

These graphs are of voltage and not percentage on the y-axis. This means that at first glance the errors can look smaller for this range but this is only because the voltages being measured are themselves smaller. The error as a percentage of the input voltage is actually higher for this range.

**Example from Sample Dataset, +/-0.01V, Channel 4**

Average voltage readings with calibration results from sample dataset, +/-0.01V range, channel 4.

Average voltage readings with calibration results from sample dataset, +/-0.01V range, channel 4. The y-axis shows difference between measured voltage and input voltage.

The calibration results were 0.0015V (0.90%) Inaccuracy, 0.0002V (0.34%) Linearity and 0.0002V (-0.46%) DC Offset. The error bars are a combination of 0.0000001V (0.00%) from RMS Noise and 0.000004V (0.02%) from other errors.

This range shows the data points being a long way from the line of best fit compared to the size of the error bars. This instrument must therefore have a significant error that varies from point-to-point that can not be averaged away: this will be making the Linearity error high. Additionally, a high DC Offset error can be seen. The high Inaccuracy error cannot be seen by eye on these particular graphs as the units for the x- and y-axes are different.