**What Coefficients Do**

ASTM E74, ISO 376, and other standards may use calibration
coefficients to characterize the performance characteristics of continuous
reading force-measuring equipment better. These standards may use
higher-order fits such as a second or third-order fit (ISO 376) or ASTM E74
allows higher-order fits up to as high as a 5^{th} order. Both Standards use observed data, and they both fit
the data to a curve. In simple terms, these higher-order fits give instructions
on how best to predict an
output given a measured input. This assumes multiple measured values of
the force-measuring device are grouped closely together. When calibrating
to a standard such as ASTM E74, or ISO 376, we often refer to this grouping as
between run variance. The higher the variance, the larger the reproducibility
error in the standard uncertainty becomes, which raises the Lower Limit Factor
(ASTM E74) or Reproducibility b (ISO 376). That output can either be the Force
at a specific response or the Response when the Force is known.

We can use these equations to predict the appropriate coordinate
more accurately for any point in the measurement range, typically above the
first non-zero point on the curve. The ASTM E74 standard has additional
requirements for higher-order fits. To use a fit above the 2^{nd}-degree (or quadratic) requires the force-measuring device to have
a resolution that exceeds 50,000 counts. Almost any force-measuring
device can be characterized using standard equations; many will remember from
algebra classes they may have had in high school.

Figure 1 Load Cell Curve Using the Equation for
a Straight Line Where Y - Values are Force and X -Values are Response (What the
Instrument Reads).

**Straight Line Fits **

Straight-line equations such as y = mx + b is common for
force-measuring devices shown in figure 1. One would typically find the
slope of the line, which could predict other points along the line. The
standard formula of y = mx + b, where m designates the slope of the line, and
where b is the y-intercept that is b is the second coordinate of a point where
the line crosses the y-axis. We can modify this formula to use coefficients,
and it would become Force = B_{1 }* (Response) + B_{0}. If we switched the x and y-axis, we would develop a
formula for solving for Response = A_{1 }* (Force) + A_{0} Figure 1 shows us a force-measuring device with a consistent
linear behavior. Some devices may deviate from the line that we cannot
precisely, or maybe we cannot predict precisely enough for the measurements we
need to make. Thus, a straight line may introduce additional
errors. It is often dependent on how linear the force-measuring device is
to determine if a straight line gives us enough precision. Typically,
this is characterized as nonlinearity; the error on most good force-measuring
equipment is often less than 0.05 %.

Figure 2 Data To Plot the Line in Figure 1 Which
Would Produce B_{0 }and B_{1 }Coefficients

When we use this equation, y is the Force applied, and x is the
output of the force-measuring device. If one wanted to solve for the output of
the readings when they know the Force applied, they could decide to plot the
actual readings against the Force applied, which is just changing what x is and
what is y, or they can use the same formula and solve for x. To do this,
we take the formula y = Mx + B and solve for x. The formula then becomes
Mx = (y-B), which we then divide (y-B) by m. Thus, x = (y-B)/M.
Pretty simple, right? When we get to higher-order equations, the formulas
get more complicated.

Figure 3 Using Least Squares Method or
Higher-Order Equations

**Using Least Squares and Higher-Order Equations**

Like a simple straight line, this regression
analysis method begins with a set of data points that are plotted on an x- and
y-axis graph. ASTM E74, ISO 376, and other standards use the method of least
squares because it is the smallest sum of squares of errors. It is the
best approximate solution to an inconsistent matrix, often involving multiple
x- values. The method used in ASTM E74 will contain a formula that is a bit
more complex than a straight line. Section 7.1.2 of the ASTM E74 standard
states, "A polynomial equation is fitted to the calibration data by the
least-squares method to predict deflection values throughout the verified range
of Force. Such an equation compensates effectively for the nonlinearity of the
calibration curve. The standard deviation determined from the difference of
each measured deflection value from the value derived from the polynomial curve
at that Force provides a measure of the error of the data to the curve fit
equation. A statistical estimate, called the lower limit factor, LLF, is
derived from the calculated standard deviation and represents the width of the
band of these deviations about the basic curve with a probability of
approximately 99 %."[1] The polynomial equation often uses
higher-order equations to minimize the error and best replicate the line.
Figure 3 above shows a plot from the actual readings in mV/V and fit to a 3rd
order equation. Instead of using the equation for a straight line (y=mx+b),
we have a formula that uses x values that are raised to higher powers, such as
Response(mV/V) = A_{0} +A_{1} * F + A_{2} * F^{2} where: A_{0} = 0.0614, A_{1} = 2415, A_{2} = -1.4436, and A_{3} = 0.17379. These are
often called coefficients. On a calibration report, they are often
referred to as A_{0}, A_{1}, A_{2}, A_{3}. Let us look at these
numbers and dissect them from a calibration report.

Figure 4 Calibration Report from
Morehouse Showing a 2^{nd}-degree Equation

The calibration report in Figure 4 shows the
formulas for solving for Response and an additional formula for solving for the
Force. The formula for Force is found by switching the x- and y-axis, as
discussed in the previous section. If one wanted to generate coefficients
to solve for Force or find the B coefficients, one would use the Predicted
Response for the x- values, and Force for the y- values. Morehouse has a
spreadsheet available for download that will use these formulas to help
interpolate values that are not on the calibration certificate.

Figure 5 Morehouse
Spreadsheet

The formula for Response is used when one would
know the target force they wish to achieve and need to know what the device
will need to read at that point. For example, in reading the calibration
certificate in figure 4, if one wanted to apply 20,000 lbf of Force, they would
load the force-measuring device until it would read -3.46174 mV/V.
However, if they would want to apply 21,000 lbf of Force, they would need to
use the equation to solve for Force found above. This equation is **Response**
= -1.122919E^{-03} + -1.728071E^{-11} * (**21,000) **+ -1.117887E^{-11}** * (21,000^2). **Thus, to generate 21,000 lbf,
the device should read -3.63500 mV/V. Morehouse has developed a simple
spreadsheet where anyone can generate load tables and plug these equations in
to solve for either Force or Response. The Morehouse spreadsheet shown in
figure 5 can be found here

**Examining What the Coefficients Mean**

A0 an B0 would be the constant at which the
equation crosses the y-intercept.

Figure 6 Showing the Relationship of Force versus
Response at Very Low Responses

Many customers do not like this as many want to
see 0 displayed on a device when they 0 the instrument. That is not how the
math works. In this example, when the meter reads zero, the force value
will be -6.49365 lbf or sometimes a rounded number, which would be -6.5 lbf as
B_{0} = -6.49356. The reason
for this is the equation is Force (lbf) is equal to B_{0} + B_{1} * (Response)+B_{2} * (Response^2). Simply
put, B_{1}, B_{2}, and higher are multiplied by
the 0 on the meter except for the first one. The meter will read 0.0 when
the Response is equal to A_{0} or
-0.00112 mV/V.

Figure 7 Deviation (Residuals) Between Actual Reading and Fitted Curve
Using a Straight Line Fit

A_{1} * F and B_{1} * R is the linear term; in figure 7 above, we
are showing the deviations from the fitted curve, meaning we draw a straight
line through all of the points and then subtract the predicted Response from
the actual force-measuring device reading. We are using this method because if
we showed three runs of data, with very small changes, the lines would all
blend, as shown in Figure 12. When using a straight line, the deviations
are much larger, and the overall reproducibility is less. The ASTM llf,
which represents a large portion of the reproducibility error, jumps from 0.997
lbf, as shown in Figure 4, to 8.14 lbf or about eight times worse.

*Note: It is important to note that we are using
one example for these graphs. The residual plots may vary depending on the
exact measurements. So, when we say this
plot is better than another, it is specific for this force-measuring device. Generally,higher-order equations will better
replicate how the device performs when compared against a first-order
equation. *

Figure 8 Deviation Between Actual Reading and Fitted Curve Using a Quadratic (2nd-Degree) Fit

A_{2} *_{ }F^{2} and B_{2} *_{ }R^{2} is the quadratic term

A positive quadratic coefficient causes the parabola's
ends to point upward, while a negative cause quadratic coefficient causes the
parabola to point downward. When we characterize the force-measuring
device using a quadratic term, we have an ASMT llf of 0.997 lbf; this will be
the 2^{nd} best fit as only the Quintic
will be a little better.

Figure 9 Deviation Between Actual Reading and Fitted Curve Using a Cubic (3rd-Degree) Fit

A_{3} * F^{3} and B_{3} * R^{3} is the cubic term. This coefficient
functions to make the graph "wider" or "skinnier" or
reflect it, if negative—the greater the coefficient, the skinnier the
graph. The cubic fit is a little worse than the quadratic by about 0.15
lbf.