Ch 3: Methods
of Analysis
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 nonreference 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 3.13
Applying (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 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
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 + 6𝑖1 + 10𝑖2 + 4𝑖3 = 0
We apply KCL to a node in the branch where the two meshes
𝑖2 = 𝑖1 + 6
intersect and that gives us
Solving both equations we get
𝑖1 = −3.2𝐴 and 𝑖2 = 2.8𝐴
Properties of a Supermesh:
1) The current source in the supermesh provides the constraint
equation necessary to solve for the mesh currents
2) A supermesh has no current of its own
3) 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 network
Networks 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 analysis
2) 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