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# Load Cell Terminology and Formulas to Calculate Them

Non-linearity, Hysteresis, Non-Repeatability, Static Error Band, and Creep are common load cell terminology typically found on a load cell specification sheet. Although several more terms are used to describe load cells' characteristics and performance, these four are the most common specifications found on calibration certificates.

When broken out individually, these terms can help you select the suitable load cell for an application. Some of these terms may not be as important today as they were years ago because better meters that overcome inadequate specifications are available. One example is Non-Linearity. An indicator capable of multiple span points can significantly reduce the impact of a load cell's non-linear behavior.

Non-linearity: The quality of a function that expresses a relationship that is not one of direct proportion. For force measurements, Non-Linearity is the algebraic difference between the output at a specific load and the corresponding point on the straight line drawn between the outputs at minimum and maximum load. It is usually expressed in units of % of full scale. It is usually calculated between 40 - 60 % of the full scale.

Non-linearity is one of the specifications that would be particularly important if the indicating device or meter used with the load cell only has a two-point span, such as capturing values at zero and capacity or close to capacity. The specification gives the end-user an idea of the anticipated error or deviation from the best fit straight line. However, suppose the end-user has an indicator capable of multiple span points and uses coefficients from an ISO 376 or ASTM E74 calibration. In that case, the non-linear behavior can be corrected, and the error significantly reduced.

One way to calculate Non-Linearity is to use the slope formula or manually perform the calibration by using the load cell output at full scale minus zero and dividing it by force applied at full scale and 0. For example, a load cell reads 0 at 0 and 2.00010 mV/V at 1000 lbf. The formula would be (2.00010-0)/ (1000-0) = 0.002. This formula gives you the slope of the line, assuming a straight-line relationship. Some manufacturers take a less conservative approach and use higher-order quadratic equations.

Plot the Non-Linearity baseline as shown below using the force applied slope + Intercept or y = mx +b formula. If we look at the 50 lbf point, this becomes 50 0.0020001 +0 = 0.100005. Thus, at 50 lbf, the Non-Linearity baseline is 0.100005.

To find the Non-Linearity percentage, take the mV/V value at 50 lbf minus the calculated value and divide by the full-scale output multiplied by 100 to convert to a percentage. Thus, the numbers become ((0.10008-0.100005)/2.00010) *100) = 0.004 %.

Non-Linearity Shortcomings

Non-linearity is a great way to visualize how much a measuring device deviates from an "ideal" device. However, all points may be perfectly linear, but if the full-scale point is non-linear itself, the rest of the points will appear to be non-linear.

Some manufacturers use higher-order equations to improve their non-linearity specification. Therefore, it is important to ask them how they calculate Non-Linearity.

Morehouse uses the more conservative straight-line approach method.

Non-Linearity Calculations

Calculate Slope

Slope = (0start(force) – FullScale(force)) / (0start(response) – FullScale(response))

Calculate Intercept

Intercept = FullScale(force) – Slope x FullScale(response)

Calculate Non-Linearity per Response

Non-Linearity = (Point(force) – (Slope x Point(response) + Intercept)) / FullScale(force)

Hysteresis: The phenomenon in which the value of a physical property lags changes in the effect causing it. An example is when magnetic induction lags the magnetizing force. For force measurements, Hysteresis is often defined as the algebraic difference between output at a given load descending from the maximum load and output at the same load ascending from the minimum load.

Hysteresis is normally expressed in units of % full scale. It is normally calculated between 40 - 60 % of full scale. The graph above shows a typical Hysteresis curve where the descending measurements have a slightly higher output than the ascending curve.

If the end-user uses the load cell to make descending measurements, they may want to consider the effect of Hysteresis.

Hysteresis Shortcomings

Errors from Hysteresis can be high enough that if a load cell is used to make descending measurements, it must be calibrated with a descending range. The difference in output on an ascending curve versus a descending curve can be significant.

Morehouse's calibration lab sampled several load cells from five manufacturers, and the results were recorded. The differences between the ascending and descending points varied from 0.007 % (shear web type cell) to 0.120 % on a column type cell.

Calculate Hysteresis

Hysteresis = | (Ascending(response) - FullScale(response)) / Descending(response) |

The calculation of non-repeatability is straightforward. First, compare each observed force point's output and run a difference between those points. The formula would look like this: Non-repeatability = ABS(Run1-Run2)/AVERAGE (Run1, Run2, Run3) *100. Do this for each combination or runs, then take the maximum of the three calculations.

Static Error Band: is the band of maximum deviations of the ascending and descending calibration points centered on the best-fit straight line through zero output (0,0). It includes the effects of Non-Linearity, Hysteresis, and non-return to minimum load. SEB is usually expressed in units of % of full scale. Thus, a SEB of 0.02 % of FS would have a maximum error of 0.02 % of its full-scale capacity. SEB is a helpful tool in determining how accurate a load cell is.

SEB is calculated to find a line that results in the slightest maximum error. This line also needs to fit through the origin (0, 0), so only the slope needs to be calculated via (y­1+y2) / (x1+x2). The best approach is to iterate across every pair of percent force applied of full scale (% FS) and the zero adjusted responses.

For each pair, calculate the slope, use the slope to calculate the percent error for all % FS, and take the largest error as that slope's "absolute error" value. Repeat this for all possibilities, taking the slope that has the smallest absolute error value.

SEB Shortcomings

If the load cell is used for ascending measurements and, occasionally, descending measurements are needed. The user may want to evaluate Non-Linearity and Hysteresis separately, as those two definitions may provide a more accurate depiction of the load cell's performance.

What needs to be avoided is a situation where a load cell is calibrated following a standard such as ASTM E74 or ISO 376 and additional uncertainty contributors for Non-Linearity and Hysteresis are added. ASTM E74 has a procedure and calculations that, when followed, use a method of least squares to fit a polynomial function to the data points.

The standard uses a specific term called the Lower Limit Factor (LLF), which is a statistical estimate of the error in forces computed from a force-measuring instrument's calibration equation when the instrument is calibrated following the ASTM E74 practice.

SEB Calculation

Excel Macro Snippet

' Iterate across every permutation of 2 points

For i = 0 To N - 1

' Start at i + 1 to duplicating work, reducing iterations

For j = i + 1 To N - 1

' Prevent checking the same point and dividing by zero

If i <> j And PercentFS(i) + PercentFS(j) <> 0 Then

'tempSlope = (Vj + Vi) / (Rj + Ri)

maxError = 0

tempSlope = (Responses(j + 2, 1) + Responses(i + 2, 1)) / (PercentFS(j) + PercentFS(i))

' Ensure we don't accidentally set the minimum error to 0 or divide by 0

If tempSlope <> 0 Then

For k = 0 To N - 1

tempError = (Responses(k + 2, 1) - tempSlope * PercentFS(k)) / tempSlope

' Take the largest error for this slope

If Abs(tempError) > Abs(maxError) Then

maxError = tempError

slope = tempSlope

End If

Next k

' Find the slope that provides the lowest maximum error

If IsNull(minError) Or Abs(maxError) < Abs(minError) Then

minError = maxError

sebSlope = slope

End If

End If

End If

Next j

Next i

Because of what it captures, Static Error Band might be the most exciting term. If the load cell is always used to make ascending and descending measurements, this term best describes the load cell's actual error from the straight line drawn between the ascending and descending curves.

In the last email, we noted that the end-user might want to consider the effects of Hysteresis unless they use the load cell described above because a Static Error Band would be the better specification. The end-user could likely ignore Non-Linearity and Hysteresis and focus on static error band and non-repeatability.

However, we find that many calibration laboratories primarily operate using ascending measurements and, on occasion, may have a request for descending data. When that is the case, the user may want to evaluate Non-Linearity and Hysteresis separately. When developing an uncertainty budget, use different budgets for each type of measurement, i.e., ascending and descending.

Creep

Creep: The change in load cell signal occurring with time while under load and with all environmental conditions remaining constant.

Load Cell Creep Return: The difference between the load cell signal immediately after removal of a load applied for a specific time interval, environmental conditions, and other variables remaining constant during the loaded interval and the load cell signal before the load application. Load Cell Creep Return is commonly expressed in units of % of applied load over a specified time interval. It is common for characterization to be measured with a constant load at or near capacity.