0195136047.pdf

72 CIRCUIT ANALYSIS TECHNIQUES

R 3

R 1 R 2

Node O

Node A Node B

V 1

+

−

V 2

+

−

Figure 2.2.1Circuit for illustration of nodal-voltage method.

I 1 = G 1 =

O Reference

AB

V (^1) V
AO^ =^ VA VBO^ =^ VB
VAB = VA − VB
R 1

• +−
  −


• 
  

  −
  I 2 =VR^2
  2
  1
  R 1 G^2 =
  1
  R 2
  G 3 =R^1
  3
  Figure 2.2.2Redrawn Figure 2.2.1 for node-voltage method of analysis.
  An examination of these equations reveals a pattern that will allow nodal equations to be
  written directly by inspection by following the rules given here for a network containing no
  dependent sources.

  
  

1. For the equation of nodeA, the coefficient ofVAis the positive sum of the conductances
   connected to nodeA; the coefficient ofVBis the negative sum of the conductances
   connected between nodesAandB. The right-hand side of the equation is the sum of
   the current sources feeding into nodeA.


2. For the equation of nodeB, a similar situation exists. Notice the coefficient ofVBto be
   the positive sum of the conductances connected to nodeB; the coefficient ofVAis the
   negative sum of the conductances connected betweenBandA. The right-hand side of the
   equation is the sum of the current sources feeding into nodeB.
   Such a formal systematic procedure will result in a set ofNindependent equations of the
   following form for a network with (N+1) nodes containing no dependent sources:
   G 11 V 1 − G 12 V 2 − ··· − G 1 NVN = I 1
   −G 21 V 1 + G 22 V 2 − ··· − G 2 NVN = I 2
   ..
   .

..

.

−GN 1 V 1 − GN 2 V 2 − ··· + GNNVN = IN (2.2.4)

whereGNNis the sum of all conductances connected to nodeN,GJK=GKJis the sum of all

conductances connected between nodesJandK, andINis the sum of all current sources entering

nodeN. By solving the equations for the unknown node voltages, other voltages and currents in

the circuit can easily be determined.