Bronkhorst E-7000 - Manual (Page 36)

BRONKHORST HIGH-TECH B.V.
page 36 9.17.004
3.11 Calibration with polynomial functions
3.11.1 General information
A normally calibrated device will have linearized transfer functions. This means that real flow/pressure and
setpoint are proportional to the output signal (0-5 V, 0-10 V, 0-20 mA or
4-20 mA).
A polynomial function is a method of approximation which mathematically describes a continues transfer
function.
By means of a few samples, a polynomial function can be obtained.
After determining the polynomial function, the original calibration points and an infinite amount of values in
between, can be calculated with high accuracy.
In a system where pressure- and/or flow meters and -controllers should be readout and set with high
accuracy, these polynomial functions often are used for approximation of their transfer function. For instance
the function which describes the relation between output voltage and measured flow.
3.11.2 General form
The general form of a polynomial function of the n-nd degree is as follows:
Y = a0 + a1 · X + a2 · X
2
+ a3 · X
3
+ ……+ an · X
n
Where 'a0' to 'an' are polynomial parameters, which can be calculated.
When you have 'n + 1' measure-points, they can be approximated by means of a 'n-nd' degree polynomial
function.
3.11.3 Polynomial functions of sensor signal and setpoint
By means of a calibration at Bronkhorst High-Tech B.V. several measure points will be used to obtain a
polynomial function.
The form of this function is:
Y = a + b · X + c · X
2
+ d · X
3
In which 'Y' is the measured value in correct unit (e.g. flow in [ln/min]) and 'X' is the value of output signal in
correct unit (e.g. Uout in [V]).
Characters 'a - d' are polynomial parameters, which can be obtained by a mathematical program. These
parameters can be filled in and the polynomial function is completed.
Now a flow can be calculated out of the sensor output signal of the device.
When a controller is calibrated, also a polynomial function for setpoint is calculated.
This will be the inverse function of the polynomial function of the sensor signal.
The form of this function is:
Z = e + f · Y + g · Y
2
+ h · Y
3
In which 'Y' is the measured value in correct unit again (e.g. flow in [I / min
n
] and Z is the value of the setpoint
signal in the correct unit (e.g. [V]).
Characters 'e - h' are parameters, which can be filled in.
Now the desired setpoint signal can be calculated out of the amount of flow at which the controller has to
operate.