76 CIRCUIT ANALYSIS TECHNIQUES

`is traditionally chosen. Branch currents can be found in terms of mesh currents, whose solution`

is obtained from the independent simultaneous equations. The number of necessary equations in

the mesh-analysis method is equal to the number of independent loops or meshes.

All current sources with shunt conductances will be replaced by their corresponding Thévenin

equivalents consisting of voltage sources with series resistances. Let us illustrate the method by

considering a simple, but typical, example, as shown in Figure 2.2.3.

Replacing the current source with shunt resistance by the Thévenin equivalent, Figure 2.2.3 is

redrawn as Figure 2.2.4, in which one can identify two elementary loops, or independent meshes.

By assigning loop or mesh-current variablesI 1 andI 2 , as shown in Figure 2.2.4, both in the

clockwise direction, one can write the KVL equations for the two closed paths (loops)ABDAand

BCDB,

LoopABDA:I 1 R 1 +(I 1 −I 2 )R 2 =V 1 −V 2 or (R 1 +R 2 )I 1 −R 2 I 2 =V 1 −V 2 (2.2.5)

LoopBCDB:I 2 R 3 +(I 2 −I 1 )R 2 =V 2 −V 3 or −R 2 I 1 +(R 2 +R 3 )I 2 =V 2 −V 3 (2.2.6)

Notice that currentI 1 exists inR 1 andR 2 in the direction indicated;I 2 exists inR 2 andR 3 in the

direction indicated; hence, the net current inR 2 isI 1 −I 2 directed fromBtoD. An examination of

Equations (2.2.5) and (2.2.6) reveals a pattern that will allow loop equations to be written directly

by inspection by following these rules:

- In the first loop equation with mesh currentI 1 , the coefficient ofI 1 is the sum of the

resistances in that mesh; the coefficient ofI 2 is the negative sum of the resistances common

to both meshes. The right-hand side of the equation is the algebraic sum of the source

voltage rises taken in the direction ofI 1. - Similar statements can be made for the second loop with mesh currentI 2. (See also the

similarity in setting up the equations for the mesh-current and nodal-voltage methods of

analysis.)

`Such a formal systematic procedure will yield a set ofNindependent equations of the`

following form for a network withNindependent meshes containing no dependent sources:

`R 3 I 3`

`R 1 R 2`

`+`

`−`

`+`

`−`

`V 1 V 2`

`Figure 2.2.3Circuit for illustration of mesh-`

current method.

`+`

`−`

`+`

`−`

`I 1 I 2`

`V 1 V 3 V 3 = I 3 R 3`

`R 1 R 3`

`AB`

`D`

`C`

`+`

`−`

`V 2`

`R 3`

`Figure 2.2.4Redrawn Figure 2.2.3 for mesh-`

current method of analysis.