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

308 Chapter 8 Matrix algebra
R 1 R 2
+
E 1
I 1
–
R 4
R 5
R 3
R 6 R 7
I 2 I 3
–
E 2 E 3
+ R 8
+
–
Figure8.2
An electrical network with meshcurrents
shown.
currents.Theconceptofameshcurrentmayappearslightlyabstractbutitdoesprovide
aconvenientmechanismforanalysingelectricalnetworks.Wewillexamineanapproach
that avoids the use of mesh currents later inthis section.
ThenextstageistomakeuseofKirchhoff’svoltagelawforeachofthemeshesinthe
network.Thisstatesthatthealgebraicsumofthevoltagesaroundanyclosedloopinan
electrical network iszero. Therefore the sum of the voltage risesmustequal the sum of
voltage drops. When applying Kirchhoff’s voltage law it is important to use the correct
sign for a voltage source depending on whether or notitis‘aiding’ a mesh current.
For mesh 1
E 1
= I 1
R 1
+I 1
R 2
+ (I 1
−I 3
)R 4
+ (I 1
−I 2
)R 3
E 1
= I 1
(R 1
+R 2
+R 4
+R 3
)+I 2
(−R 3
)+I 3
(−R 4
)
For mesh 2
−E 2
−E 3
= I 2
R 5
+ (I 2
−I 1
)R 3
+ (I 2
−I 3
)R 6
+I 2
R 8
−E 2
−E 3
= I 1
(−R 3
)+I 2
(R 5
+R 3
+R 6
+R 8
)+I 3
(−R 6
)
For mesh 3
E 3
= (I 3
−I 2
)R 6
+ (I 3
−I 1
)R 4
+I 3
R 7
E 3
= I 1
(−R 4
)+I 2
(−R 6
)+I 3
(R 6
+R 4
+R 7
)
These equations can be written inmatrix form as
⎛ ⎞ ⎛
⎞ ⎛ ⎞
E 1 R 1 +R 2 +R 4 +R 3 −R 3 −R 4 I 1
⎝−E 2 −E 3
⎠ = ⎝ −R 3 R 5 +R 3 +R 6 +R 8 −R 6
⎠ ⎝I 2
⎠
E 3
−R 4 −R 6 R 6 +R 4 +R 7 I 3
Engineeringapplication8.3
Calculationofmeshandbranchcurrentsinanetwork
Consider the electrical network of Figure 8.3. It has the same structure as that of
Figure8.2butwithactualvaluesforthevoltagesourcesandresistors.Branchcurrents
aswellasmeshcurrentshavebeenshown.Calculatethemeshcurrentsandhencethe
branch currents forthe network.