Mesh Analysis


Nodal analysis applies KCL to find unknown voltages in a given circuit, while mesh analysis applies KVL to find unknown currents. Mesh analysis is not quite as general as nodal analysis because it is only applicable to a circuit that is planar. A planar circuit is one that can be drawn in a plane with no branches crossing one another; otherwise it is nonplanar. A circuit may have crossing branches and still be planar if it can be redrawn such that it has no crossing branches. F or example, the circuit in Fig. 3.15(a) has two crossing branches, but it can be redrawn as in Fig. 3.15(b). Hence, the circuit in Fig. 3.15(a) is planar. However, the circuit in Fig. 3.16 is nonplanar , because there is no w ay to redraw it and a void the branches crossing. Nonplanar circuits can be handled using nodal analysis, but they will not be considered in this text.

Figure 3.15 : (a) A planar circuit with crossing branches,
(b) the same circuit redrawn with no crossing branches


Figure 3.16 A nonplanar circuit

To understand mesh analysis, we should first explain more about what we mean by a mesh.

A mesh is a loop that does not contain any other loops within it.
Figure 3.17 A circuit with two meshes

In Fig. 3.17, for example, paths abefa and bcdeb are meshes, but path abcdefa is not a mesh. The current through a mesh is known as mesh current. In mesh analysis, we are interested in applying KVL to find the mesh currents in a given circuit.
In this section, we will apply mesh analysis to planar circuits that do not contain current sources. In the next section, we will consider circuits with current sources. In the mesh analysis of a circuit with n meshes, we take the following three steps.


Steps to Determine Mesh Currents:

  1. Assign mesh currents i1, i2, . . . , in to the n meshes.
  2. Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of the mesh currents.
  3. Solve the resulting n simultaneous equations to get the mesh currents.

To illustrate the steps, consider the circuit in Fig. 3.17. The first step requires that mesh currents i1 and i2 are assigned to meshes 1 and 2. Although a mesh current may be assigned to each mesh in an arbitrary direction, it is conventional to assume that each mesh current flows clockwise.
As the second step, we apply KVL to each mesh. Applying KVL to mesh 1, we obtain

For mesh 2, applying KVL gives 

Note in Eq. (3.13) that the coefficient of i1 is the sum of the resistances in the first mesh, while the coefficient of i2 is the negative of the resistance common to meshes 1 and 2. Now observe that the same is true in Eq. (3.14). This can serve as a shortcut way of writing the mesh equations.
Notice that the branch currents are different from the mesh currents unless the mesh is isolated. To distinguish between the two types of currents, we use i for a mesh current and I for a branch current. The current elements I1, I2, and I3 are algebraic sums of the mesh currents. It is evident from Fig. 3.17 that 

Example:

find the branch currents I1, I2, and I3 using mesh analysis.

Solution:

We first obtain the mesh currents using KVL. For mesh 1
for mesh 2, 
Using the substitution method, we substitute Eq. (3.5.2) into Eq. (3.5.1), and write :