4.10 Complex Networks
153
and are shown in the table:
Function
Q 14
Q 12
Q 24
Q 34
Q 23
F 1
dF 1 /dQ 14
dF 1 /dQ 12
dF 1 /dQ 24
dF 1 /dQ 34
dF 1 /dQ 23
F 2
dF 2 /dQ 14
dF 12 /dQ 12
dF 2 /dQ 24
dF 2 /dQ 34
dF 2 /dQ 23
F 3
dF 3 /dQ 14
dF 3 /dQ 12
dF 3 /dQ 24
dF 3 /dQ 34
dF 3 /dQ 23
F 4
dF 4 /dQ 14
dF 4 /dQ 12
dF 4 /dQ 24
dF 4 /dQ 34
dF 4 /dQ 23
F 5
dF 5 /dQ 14
dF 5 /dQ 12
dF 5 /dQ 24
dF 5 /dQ 34
dF 5 /dQ 23
So, using the values in the example:
Function
Q 14
Q 12
Q 24
Q 34
Q 23
F 1
1
−1
0
0
0
F 2
0
1
−1
0
−1
F 3
0
0
0
−1
1
F 4
2 R 14 · |Q 14 |
2 R 12 |Q 12 |
2 R 24 |Q 24 |
0
0
F 5
0
0
2 R 24 |Q 24 |
2 R 34 |Q 34 |
−2 R 23 |Q 23 |
The initial numerical values Q ij(o) are chosen arbitrarily but in compliance with
the laws of mass and energy conservation at the nodes:
Table of initial values
Branch
14
12
24
34
23
R ij
0.4
0.3
0.8
0.2
0.1
Q ij(o)
−60.0
90.0
30.0
−90.0
60.0
These values are then substituted into the matrix function for the system [F i ] and
the following results are obtained:
Function matrix for initial values
Function
Q 14
Q 12
Q 24
Q 34
Q 23
Ind. term. = Q a
F i
F 1
−60.0
−90.0
0
0
0
150.0 =
0.0
F 2
0
90.0
−30.0
0
−60.0
0 =
0.0
F 3
0
0
0
90.0
60.0
−150.0 =
0.0
F 4
−1440
2430
720
0
0
0 =
1710.0
F 5
0
0
−720
−1620
360
0 =
−1980.0
The Jacobian matrix is then calculated: