App-20 IM 701450-01E
Differentiation and Integration (DIF, DDIF, INTG, and IINTG)
Differentiation (DIF, DDIF)
The computation of the first order and second order differentiation uses the 5
th
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 (DIF)
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 (sec) (example h = 200 × 10
–6
at 5 kHz)
1
12h
1
12h
1
12h
1
12h
1
12h
1
12h
1
12h
• Equation for Second Order Derivative (DDIF)
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
Appendix 4 User-Defined Computation
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