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

312 Chapter 8 Matrix algebra
These equations can be writteninthe matrix formAV =B, whereAisthe matrix
⎛
⎞
R 1 +R 2 −R 2 0 −R 1 0
R 3 R 5 −R 1 R 3 −R 1 R 5 −R 3 R 5 R 1 R 5 0 R 1 R 3
⎜ 0 R
⎝ 4 R 6 −R 4 R 6 −R 3 R 6 −R 3 R 4 R 3 R 6 0 ⎟
R 4 R 7 0 R 2 R 7 −R 4 R 7 −R 2 R 7 −R 2 R 4 0 ⎠
0 R 8 0 0 −R 5 −R 8
and
⎛ ⎞
⎛
⎞
V a
E 1
R 2
V b
E 1
R 3
R 5
+E 2
R 1
R 3
V =
⎜V c ⎟ and B=
⎜ −E 3
R 3
R 4 ⎟
⎝V d
⎠
⎝ 0 ⎠
V e
−E 2
R 8
TheequationswouldgenerallybesolvedbyGaussianeliminationtoobtainthenode
voltages and hence the branch currents.
Using the component values from Engineering application 8.3, these equations
become
⎛
⎞ ⎛ ⎞ ⎛ ⎞
6 −4 0 −2 0 V a
12
15−31 10 0 6
V b
⎜ 0 2 −11 6 0
⎟ ⎜V c ⎟
⎝ 3 0 12 −19 0⎠
⎝V d
⎠ = 57
⎜−12
⎟
⎝ 0⎠
0 4 0 0 −9 V e
−8
Use of a computer package avoids the tedious arithmetic associated with Gaussian
elimination and yields
V a
= 2.969 V b
= 0.5250 V c
= 2.200 V d
= 1.858 V e
= 1.122
Itisthen straightforward tocalculate the branch currents:
I a
= V a −V d
=278mA I
R b
= V b −V c
= −558 mA
2
R 3
I c
= V c −V d
R 4
=342mA I d
= V e
R 8
= 281 mA
I e
= V c −E 3
R 6
= −900 mA I f
= V d
R 7
= 619 mA
Compare these answers withthose of Engineering application 8.3.
It is possible to analyse electrical networks containing more complex elements
suchascapacitors,inductors,activedevices,etc.,usingthesameapproach.Theequationsaremorecomplicatedbutthetechniqueisthesame.Oftenitisnecessarytouse
iterative techniques in view of the size and complexity of the problem. These are
examined inthe following section.
8.13 ITERATIVETECHNIQUESFORTHESOLUTION
OFSIMULTANEOUSEQUATIONS
The techniques met so far for the solution of simultaneous equations are known as direct
methods, which generally lead to the solution after a finite number of stages in
the calculation process have been carried out. An alternative collection of techniques is