Ch 3: Methods of

AnalysisEC Fundamentals 85-234

Introduction• We have studied the basic concepts and the basic laws that

govern the electrical world we live in. • We have also learned to apply Ohm’s Law, and Kirchhoff’s

Voltage and Current Laws for circuit analysis• Nodal analysis is the systematic approach using Kirchhoff's

Current Law (KCL)• Mesh analysis is the systematic approach using Kirchhoff's

Voltage Law (KVL)

Nodal Analysis• Nodal Analysis provides a general procedure for analyzing

circuits using node voltages as the circuit variables. • This is very convenient and reduces the number of equations

one must solve.• Steps to Determine Node Voltages:

1) Select a node as the reference node. Assign voltages v1 , v2 , … , vn-1 to the remaining n-1 nodes. The voltages are referenced with respect to the reference node.2) Apply KCL to each of the non-reference nodes. Use Ohm’s Law to express the branch currents in terms of node voltages.3) Solve the resulting equations and obtain unknown node voltages

• The reference node is commonly called the ground because it has zero potential.

• The number of non-reference nodes is equal to the number of independent equations that we will derive.

• Current flows from a higher potential to a lower potential in a resistor

Nodal Analysis with Voltage Sources

CASE 1:• If a voltage source is connected between the reference node and

non-reference node, set the voltage at the non-reference node equal to the voltage source

CASE 2:• If the voltage source is connected between 2 non-reference

nodes, these two nodes form a supernode and apply both KCL and KVL to find the node voltages

Nodal Analysis with Voltage Sources

• A supernode is formed by enclosing a (dependent or independent) voltage source connected between two non-reference nodes and any elements connected in parallel with it

• A supernode may be regarded as a closed surface

• Properties of a supernode: The voltage source inside the supernode provides a

constraint equation needed to solve for the node voltages A supernode has no voltage of its own A supernode requires the application of both KCL & KVL

Figure 3.13Applying (a) KCL to the two supernodes, (b) KVL to the loops

Mesh Analysis• Mesh Analysis provides another general procedure for analyzing

circuits using mesh currents as the circuit variables.• It reduces the number of equations that must be solved

simultaneously• Recall: a loop is a closed path with no node passed more than

once. Similarily a mesh is a loop that does not contain any other loop within it

• We use KVL (Kirchhoff’s Voltage Law) to find unknown currents• We can only use mesh analysis on planar circuits – a circuit that

can be drawn in 2D with no branches crossing each other

Mesh Analysis• A nonplanar circuit

Mesh Analysis

• In Figure 3.17 paths abefa and bcdeb are meshes but path abcdefa is not a mesh.

• In mesh analysis, we are interested in applying KVL to find the mesh currents in a given circuit

• The direction of the mesh current and loop is arbitrary• Steps to Determine Mesh Currents:

1) Assign mesh currents i1 , i2 , … , in to the n meshes2) Apply KVL to each of the n meshes. Use Ohm’s law to express the

voltages in terms of the mesh currents3) Solve the resulting n simultaneous equations to get the mesh

currents

Mesh Analysis with Current Sources

• Applying mesh analysis with current sources may appear complex but is easier than before.

• A supermesh results when two meshes have a (dependent or independent) current source in common

• As shown in Figure 3.23b we create a supermesh as the periphery of the two meshes and treat it differently.

• This is because mesh analysis applies KVL which requires we know the voltage across each branch (and we don’t know the voltage across a current source in advance).

• A supermesh must satisfy KVL like any other mesh. So applying KVL to Figure 3.23b gives us

−20+6 𝑖1+10 𝑖2+4 𝑖3=0

We apply KCL to a node in the branch where the two meshes intersect and that gives us

Solving both equations we get

𝑖2=𝑖1+6

and

Properties of a Supermesh:1) The current source in the supermesh provides the constraint

equation necessary to solve for the mesh currents2) A supermesh has no current of its own3) A supermesh requires the application of both KVL and KCL

So which method is better?• Both nodal and mesh analysis provide a systematic way of

solving a complex network. • The choice of the better method is dictated by 2 factors:

1) The nature of the particular networkNetworks that contain many series-connected elements,

voltage sources, or supermesh are better suited for mesh analysis. Networks with parallel-connected elements, current sources,

or supernode are better suited for nodal analysis2) The information that is required

If node voltages are required, then apply nodal analysis. If branch or mesh current is required, then use mesh analysis