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What is linearisation – what is it and why do I need it?

Flow rate versus frequency

Linearisation is a process that enhances the accuracy of any repeatable flowmeter.
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This article will show the amount of accuracy increase that can be expected from optimising linearisation points in terms of number and position.

A standard calibration of a VFF flowmeter will involve 10 calibration points. These are always spread out along the customer’s operating flow range. They will be distributed in favour of lower flow rates – where changes in raw meter accuracy are higher.

Below is an example of a flowmeter calibration curve. The results of a calibration are a table of flow rates and the corresponding pulses per litre. In an ideal world, there would be a hundred or a thousand such points so that the complete curve could be plotted and for any given flow rate the pulses per litre would be known. In practice, there are usually 10 to 20, distributed as shown.

A plot of flow rate against frequency would be an apparently straight line relationship. The angle of the straight line would be the ‘meter factor’ or the average number of pulses per litre.

To make this more visual, it’s common to plot the flow rate against the ‘pulses per litre’. This emphasises the changes as the flow rate increases.

Flowmeter Linearisation Fig-2

Flowmeter meter factor or ‘k’ factor plotted against flow rate.

Better still is a plot of flow rate against accuracy using the ‘meter factor’ as a zero. The biggest positive excursion defines the positive error and the biggest negative excursion is the negative error.

Flowmeter Linearisation Fig-3

Flowmeter linearity expressed as percentage error versus flow rate.

If the flowmeter is repeatable then we can use a technique called linearization. In other words, if the curve (i.e. the relationship between flow rate and pulses per litre) is the same the next time around, then the flow rate can be calculated from the frequency output using the specific pulses per litre value rather than an average pulses per litre value. If the flow rate is identical to one of the original calibration points there is no adjustment. At other flow rates a calculation is required. In between the points most linearization systems use simple linear interpolation. For example, if the flow rate is halfway between two original calibration points then an average of the pulses per litre for those two points is used.

Flowmeter Linearisation Fig-4

Linearisation is normally linear interpolation between two points. The difference between the (brown) linear interpolation and the (blue) real value is the subsequent error.

The error is represented here by the length of the red arrow.

Now the measurement error is no longer compared to a single meter factor from min to max but a linearity curve built into the meter.

By carefully selecting the calibration flow rates along the range the remaining error can be minimised. In the example below, the actual curve (in blue) is shown plus the linear approximation (in brown).

Flowmeter Linearisation Fig-5

The meter character and the linear interpolation between linearisation points for the whole range of the meter.

Magnified in scale:Here is how this linearization improves the accuracy, on the same scale:

Flowmeter Linearisation Fig-6

Results of the linearisation.

Magnified in scale:

Flowmeter Linearisation Fig-7

The result of linearisation. Now between +0.19% and -0.28%.


And here is the result, if there are 4 calibration points at the lowest flow rates rather than 2.

Flowmeter Linearisation Fig-8

Concentrating the calibration and linearisation points at low flows, in this example, has a further impressive improvement to overall linearity.

As a result of linearization, the overall error has been reduced from max error: 1.19%, min error: -3.98% to max error: +0.19%, min error: -0.28% representing an 11x improvement. Selecting the calibration points carefully improves this further. Every calibration point has a cost so there are diminishing returns. If the repeatability is +/- 0.25% then 10 points overall are normally enough.

If enough calibration time is available and the repeatability of the system is considerable then more points can be selected for the linearisation table. In this extension of the example above 21 points are used rather than just 10. Arranged carefully throughout the flow range, the net effect is to reduce the max/min to 0.08%, a 64x improvement over the non-linearised version. If the repeatability of the system was ±0.1% then no more points are required.The flowmeter response (blue) is almost perfectly matched by 21 carefully selected calibration points (brown).

Flowmeter Linearisation Fig-9

The flowmeter response (blue) is almost perfectly matched by 21 carefully selected calibration points (brown).

Flowmeter Linearisation Fig-10

The final analysis. Total non-linearity reduced from 5.17% to 0.08%.

This article shows that linearisation with a flowmeter instrument can significantly improve the linearity of the flowmeter system. In this example, if the meter is repeatable, then 6 to 60 times improvements are possible.

Litre Meter Ltd

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