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

8.12 Analysis of electrical networks 307
Solutions
( ) )
1
1
1 (a)t ,t(
1
−1
3
( )
1
1
(c) t ,t(
1
1)
2
( ) (
1
(d) t
(
− 1 + √ )
,t
13 /3
( 1
(b) t
1)
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 1 1
2 (a)t⎝5⎠,t
⎝12⎠,t
⎝0⎠
1 5 1
⎛ ⎞ ⎛ ⎞
1 1
(b)t ⎝−5.0981⎠,t
⎝ 0.0981⎠,t
2.7321 −0.7321
)
(
1 √13−1 )
/3
⎛ ⎞
1
⎝−3⎠
−3
⎛ ⎞
1
− 1 ⎛ ⎞ ⎛ ⎞
1 1
(c)t ⎜ 3⎟
⎝
− 7 ⎠ ,t ⎝−2⎠,t
⎝−0.8⎠
1 1.4
3
⎛ ⎞
1 ⎛ ⎞ ⎛ ⎞
4
1 1
(d)t ⎜ 9⎟
⎝
− 1 ⎠ ,t ⎝ 0.6⎠,t
⎝ 1⎠
−0.3 0.5
3
⎛ ⎞
1 ⎛ ⎞ ⎛ ⎞
19
1 1
(e)t ⎜ 6 ⎟
⎝
− 2 ⎠ ,t ⎝4⎠,t
⎝2⎠
6 4
3
8.12 ANALYSISOFELECTRICALNETWORKS
Matrix algebra is very useful for the analysis of certain types of electrical network. For
suchnetworksitispossibletoproduceamathematicalmodelconsistingofsimultaneous
equations which can be solved using Gaussian elimination. We will consider the case
whenthenetworkconsistsofresistorsandvoltagesources.Thetechniqueissimilarfor
othertypesofnetwork.
In order to develop this approach, it is necessary to develop a systematic method
for writing the circuit equations. The method adopted depends on what the unknown
variables are. A common problem is that the voltage sources and the resistor values are
knownanditisdesiredtoknowthecurrentvaluesineachpartofthenetwork.Thiscan
beformulatedasamatrix equation. Given
where
V=RI ′
V = voltage vectorforthe network
I ′ = currentvectorforthe network
R = matrix ofresistor values
theproblemistocalculateI ′ whenV andRareknown.I ′ isusedtoavoidconfusionwith
the identitymatrix.
Anysizeofelectricalnetworkcanbeanalysedusingthisapproach.Wewilllimitthe
discussion to the case whereI ′ has three components, for simplicity. The extension to
larger networks is straightforward. Consider the electrical network of Figure 8.2. Mesh
currentshavebeendrawnforeachoftheloopsinthecircuit.Ameshisdefinedasaloop
that cannot contain a smaller closed current path. For convenience, each mesh current
is drawn in a clockwise direction even though it may turn out to be in the opposite
directionwhenthecalculationshavebeenperformed.Thenetcurrentineachbranchof
thecircuitcanbeobtainedbycombiningthemeshcurrents.Thesearetermedthebranch