YOKOGAWA DL750 - Page 689

App-16
IM 701210-06E
Appendix 5 User-Defined Computation
Differentiation and Integration (DIF, DDIF, INTG, and IINTG)
Differentiation (DIF, DDIF)
The computation of the first order and second order differentiation uses the 5th order
Lagrange interpolation formula to derive a point of data from the 5 points around the
point.
The figure below shows data f
0
to f
n
with respect to sampling time x
0
to x
n
. The
derivative and integrated value corresponding to these data points are computed as
follows:
x 0 x 1 x 2 x 3 x 4 x k x n-3 x n-2 x n-1 x n
f
f
0
f 1
f 2
f
3
f 4
f k
f n-4
f n-3
f n-2
f n-1
f n
n-4
x
• Equation for First Order Derivative
Point x 0 f 0 ' = [–25f 0 + 48f 1 – 36f 2 + 16f 3 – 3f 4 ]
Point x
1 f 1 ' = [–3f 0 – 10f 1 + 18f 2 – 6f 3 + f 4 ]
Point x
2 f 2 ' = [f 0 – 8f 1 + 8f 3 – f 4 ]
Point x
k f k ' = [f k-2 – 8f k-1 + 8f k+1 – f k+2 ]
Point x
n-2 f n-2 ' = [f n-4 – 8f n-3 + 8f n-1 – f n ]
Point x
n-1 f n-1 ' = [–f n-4 + 6f n-3 – 18f n-2 + 10f n-1 + 3f n ]
Point x
n f n ' = [3f n-4 – 16f n-3 + 36f n-2 – 48f n-1 + 25f n ]
h = ∆x is the sampling interval (s) (example h = 200 ×10
–6
for 5 kHz)
1
12h
1
12h
1
12h
1
12h
1
12h
1
12h
1
12h
• Equation for Second Order Derivative
Point x0 f0" = [35f0 – 104f1 + 114f2 – 56f3 + 11f4]
Point x
1 f1" = [11f0 – 20f1 + 6f2 + 4f3 – f4]
Point x
2 f2" = [–f0 + 16f1 – 30f2 + 16f3 – f4]
Point x
k fk" = [–fk-2 + 16fk-1 – 30fk + 16fk+2 – fk+2]
Point x
n-2 fn-2" = [–fn-4 + 16fn-3 – 30fn-2 + 16fn-1 – fn]
Point x
n-1 fn-1" = [–fn-4 + 4fn-3 + 6fn-2 – 20fn-1 + 11fn]
Point x
n fn" = [11fn-4 – 56fn-3 + 114fn-2 – 104fn-1 + 35fn]
1
12h
2
1
12h
2
1
12h
2
1
12h
2
1
12h
2
1
12h
2
1
12h
2