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TECHNICAL INFORMATION
Effective Use of the Math Function Module (1)
Math function modules (computation units) are widely used in various process instrumentation systems. M-System offers various kinds of modules. The following article is to explain M-System’s procedure of gain and bias calculation methods.

Models available are M2ADS (adder), M2SBS (subtractor), JF and JFK.
Adder-subtractor
Figure1Addition-subtraction is necessary in cases where a total value indication for liquid flow through multiple pipelines or that of liquid level for several reservoir tanks is required. An adder-subtractor (M-System’s Math Function Module, Model JF or JFK) is used for this purpose.

Practical application examples are shown in Figures 1, 2 and 3. Figure 1 shows an example that indicates the total liquid flow, Flow 1 + Flow 2. Figure 2 shows an example that indicates the total amount of liquid of two cylindrical reservoir tanks, Level 1 + Level 2. Figure 3 shows another example that indicates the total quantity of liquid of two spherical reservoir tanks by the help of the linearizer that is able to convert liquid level of a spherical tank to the corresponding liquid amount. The Model JFX1 is a highly accurate Linearizer that provides a 100 segment poly-gonal line approximation representing this level.

Gain Calculations — Addition-subtraction
Figure2The Math Function Module does not have an indicator, therefore an engineering unit scale is n ot used. Its input and output signals are limited to 4 – 20 mA DC or 1 – 5 V DC. Computations within the module are carried out using numerical values of 0 – 1 or 0 – 100%. When input and/or output conditions change, the preset constants in the module must also be changed which can involve somewhat complicated procedures. That is to say, gains and biases for the computation equation s hould be re-calculated. In these calculations, the specific engineering unit scalings should be used for bo th input and output signal ranges.

Figure3As a typical example, Figure 4 shows the calculation procedure for the application shown in Figure 1. The input and output signal ranges for this example are as follows:

Flow 1: 0 – 100 m3/H
Flow 2: 0 – 200 m3/H
Output: 0 – 300 m3/H

In this case, all engineering unit scales start at zero which means that bias need not be considered. Therefore, the following equation is used for the Math Function Module:

X0 = K1 x X1 + K2 x X2
Where, X0 = 0 – 1, Output Signal
X1 = 0 – 1, Input Signal (Flow 1)
X2 = 0 – 1, Input Signal (Flow 2)

Figure4In most basic applications, the equation: Output = Flow 1 + Flow 2 is used. However, in this example the output range becomes 0 – 2 using this formula, instead of the 0 – 1 intended range. Because of this, values for gains K1 and K2 must be determined. The following equations are used to determine their values:


The actual calculation for the example shown in Figure 4 is:

JF

Gain Calculations — Biased Signal Range
Figure5An example of gain calculations for biased signal ranges is shown in Figure 5. The input and output signal ranges in engineering units are as follows:
Flow 1: 100 – 200 m3/H
Flow 2: 100 – 300 m3/H
Output: 200– 500 m3/H

When both Flow 1 and Flow 2 are 0%, the following is true:

Output = Flow 1 + Flow 2
200 m3/H = 100 m3/H + 100 m3/H
X0 = K1 x X1 + K2 x X2
0 = 0 + 0

In this case, only the gains K1 and K2 need to be considered in this application. Each gain is calculated by using the following:



In this example, these are calculated as follows:



These results are the same as those shown in Figure 4. This means that the same computation equation can be used for both examples because the Math Function Module does not concern itself with the actual engineering units.

Output Bias Calculation
Figure6When the input signal range has a bias, and at the same time the output signal range starts form zero without a bias, the math equation becomes slightly more complex. A typical example is shown in Figure 6. The input and output signal ranges in engineering units are as follows:
Flow 1: 100 – 200 m3/H
Flow 2: 100 – 300 m3/H
Output: 0– 500 m3/H

In this example, the output bias B in the following equation should be calculated.

X0 = K1 x X1 + K2 x X2 + B

First, the two inputs must be added in their respective engineering units at their 0% values.

Output = 100 m3/H + 100 m3/H = 200 m3/H

The resulting bias value (B) is calculated as follows:

The gains K1 and K2 are calculated in the same manner as shown in the preceding example:



In this example, the output signal is never less than 40% (200 m3/H) because the minimum value of each input signal is 0% (100 m3/H as shown above).

Input Bias Calculation
Figure7The example used for the output bias calculation can be also used as an example for an input bias calculation. This is shown in Figure 7. The input and output signal ranges in engineering units are as follows:
Flow 1: 100 – 200 m3/H
Flow 2: 100 – 300 m3/H
Output: 0– 500 m3/H

The following equation is used for this example:

X0 = K1 x (X1 ± B1) + K2 x (X2 ± B2)

Where, X0: Output
K1: Gain of Input 1
X1: Input 1
B1: Bias of Input 1
K2: Gain of Input 2
X2: Input 2
B2: Bias of Input 2

The input bias values are calculated as follows:

Input Bias
Then,


The gains K1 and K2 are calculated easily in two steps. For Flow 1, the span (including the bias) is 200 m3/H (Range: 0 – 200 m3/H) versus the span of the actual signal, 100 m3/H (Range: 100 – 200 m3/H). The first step is to determine the value for the gain k1:


Next, the output gain k2 is calculated:

The resultant gain K1 can be obtained by multiplying k1 and k2 as follows:

K1 = k1 x k2 = (1/2) x (2/5) = 2/10 = 1/5

The gain K2 for Flow 2 is determined by the same procedure:

K2 = k3 x k4 = (2/3) x (3/5) = 6/15 = 2/5

The next explanation of the Math Function Module will be in the multiplier-divider role as it relates to temperature and pressure compensation of flow measurements.


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