PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Direct Current and Transient Analysis 117

R.2.69 The following example illustrates the procedure used to obtain the set of two inde-

pendent loop equations for the circuit shown in Figure 2.8.

V 1 = (R 1 + R 2 ) * I 1 − R 2 * I 2 (loop# 1)

−V 2 = −R 2 * I 1 + (R 2 + R 3 ) * I 2 (loop# 2)

R.2.70 Observe that to obtain the loop equations set, the electric circuit must fi rst be drawn

as a planar network. Also observe that the circuit shown in Figure 2.8 presents two

independent loops (from a total of three loops) in which two equations in terms of

two unknown currents (I 1 and I 2 ) are obtained.

R.2.71 Note that the resulting branch current through R 2 in the circuit of Figure 2.8 is

given by I 1 − I 2.

R.2.72 Note that the structure (and format) of each of the loop equations is given by the

following resulting expression:

for any arbitrary loop x

[][]*

[

voltage sources in loop resistances in

resis

∑∑xx
Ix

∑ tances in branch t xy]*Iy for all possible sy

where the voltage sources (left-hand side of the preceding equation) that generate

power are considered positive (otherwise negative if the source consumes power),

and the branch xy consists of the elements common to loops x and y that carry the

opposing currents Ix and Iy. The branch current in xy is then Ix − Iy.

R.2.73 Node analysis refers to a procedure where a given electrical network that consists

for simplicity of resistors and current sources are expressed in terms of all the

nodal voltages with respect to an arbitrary reference node label ground (with zero

potential). Thus, in a circuit with n nodes, one node is designated as the reference

node (ground), and for the remaining n − 1 nodes (applying KCL), n − 1 equations

can then be expressed in terms of the n − 1 unknown nodal voltages labeled V 1 , V 2 ,

V 3 , V 4 , ..., Vn− 1.

For each one of the nodes, assume that the unknown current directions are

toward the reference node. Then solve the n − 1 nodal equations simultaneously

for all the unknown nodal voltages. Once all the nodal voltages are known, then all

the network voltages as well as all the network currents can be easily evaluated.

FIGURE 2.8

Electrical network with two independent loop currents.

V 1

R 1 R 3

R 2 V^2

I 1 I 2