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

310 Chapter 8 Matrix algebra
The branch currents arethen
I a
=I 1
=278mA
I b
= I 2
−I 1
= −281 −278 = −559 mA
I c
=I 3
−I 1
=619−278=341mA
I d
= −I 2
= 281 mA
I e
= I 2
−I 3
= −281 −619 = −900 mA
I f
=I 3
=619mA
An alternative approach to analysing an electrical network is to use the node voltage
method which is often simply called nodal analysis. This technique is fundamental to
manycomputerprogramswhichareusedtosimulateelectricalcircuits,suchasSPICE.
We introduce thistechnique by means of an example.
Engineeringapplication8.4
Analysinganelectricalnetworkusingthenodevoltagemethod
The node voltage method utilises the notion that ‘islands’ of equal potential lie between
electrical components and sources.The procedure isas follows:
(1) Pick a reference node. In order to simplify the equations this is usually chosen
tobethenodewhichiscommontothelargestnumberofvoltagesourcesand/or
the largestnumber of branches.
(2) Assignanodevoltagevariabletoalloftheothernodes.Iftwonodesareseparated
solelybyavoltagesourcethenonlyoneofthenodesneedbeassignedavoltage
variable. The node voltages areall measured with respect tothe reference node.
(3) At each node, write Kirchhoff’s current law in terms of the node voltages. Note
that once the node voltages have been calculated it is easy to obtain the branch
currents.
We will again examine the network of Figure 8.2, but this time use the node voltage
method.ThenetworkisshowninFigure8.4withnodevoltagesassignedandbranch
currentslabelled.Thereferencenodeisindicatedbyusingtheearthsymbol.Writing
Kirchhoff’s current lawfor each node, weobtain:
node a
node b
I a
=I a
V b
+E 1
−V a
R 1
= V a −V d
R 2
V b
R 2
+E 1
R 2
−V a
R 2
=V a
R 1
−V d
R 1
V a
(R 1
+R 2
)−V b
R 2
−V d
R 1
=E 1
R 2
I a
+I b
+I d
=0
V b
+E 1
−V a
+ V b −V c
R 1
R 3
+ V b +E 2 −V e
R 5
= 0