082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

318 Chapter 8 Matrix algebra
example, large networks for the distribution of electricity from power stations to
points of consumption can be very complicated to analyse due to the large number
of loops involved. This example illustrates the method by making use of the circuit
given in Engineering application 8.3. However, it should be borne in mind that this
approach isof mostvalue when analysing much larger networks.
Use Jacobi’s method and the Gauss--Seidel method to obtain approximate solutions
for the node voltages of the electrical network examined in Engineering
application 8.3.
Solution
The node voltage equations are
6V a
−4V b
−2V d
= 12 3V a
+12V c
−19V d
= 0
15V a
−31V b
+10V c
+6V e
= 57
2V b
−11V c
+6V d
= −12
These can be rearranged togive
4V b
−9V e
= −8
V a
= 2V b +V d +6
3
V b
= 15V a +10V c +6V e −57
31
V c
= 2V b +6V d +12
11
V d
= 3V a +12V c
19
V e
= 4V b +8
9
The results of applying Jacobi’s method with aninitial guess of
V (0)
a
=V (0)
b
=V (0)
c
=V (0)
d
=V (0)
e
= 0
are shown in Table 8.4. Convergence was achieved to within 0.001 after 44 iterations.TheresultsofapplyingtheGauss--SeidelmethodareshowninTable8.5.Convergence
was achieved to within 0.001 after 21 iterations. Clearly the Gauss--Seidel
method converges more rapidly than Jacobi’s method.
Table8.4
Node voltages derived from Jacobi’smethod.
Iterationno.(n)
V (n)
a
V (n)
b
V (n)
c
V (n)
d
V (n)
e
0 0.0000 0.0000 0.0000 0.0000 0.0000
1 2.0000 −1.8387 1.0909 0.0000 0.8889
.
20 2.8710 0.4802 2.1379 1.8265 1.0738
.
44 2.9679 0.5243 2.1990 1.8578 1.1215