1K. A. Saaifan, Jacobs University, Bremen
4. Basic Nodal and Mesh Analysis
4.1 Nodal Analysis
This chapter introduces two basic circuit analysis techniques named nodal analysis
and mesh analysis
For a simple circuit with two nodes, we often have one unknown “voltage between two nodes”
To solve the unknown, applying KCL at this node gives
Adding a node should provide an additional unknown, three-node circuit has 2 unknown
N-node circuit has (N-1) voltages with (N-1) equations.
2K. A. Saaifan, Jacobs University, Bremen
Nodal technique applies the following step
1- Count the number of nodes (N)
2- Designate a reference node
3- Label the nodal voltages (we have N-1 voltages)
3K. A. Saaifan, Jacobs University, Bremen
4- Write KCL equations for the non-reference nodes (currents in = currents out)
5- Organize the equations
6- Solve the system of equations for the nodal voltages
(1)
(2)
4K. A. Saaifan, Jacobs University, Bremen
Using a Cramer's rule and determinants, we have
5K. A. Saaifan, Jacobs University, Bremen
Compute the voltages at each nodeAns:
Write KCL equations for the three nodes Organize the equations
(1)
(2)
(3)
6K. A. Saaifan, Jacobs University, Bremen
Compute the voltage at each nodeAns:
Solve the system of equations for the nodal voltages
Use a Cramer's rule and determinants to solve the system
7K. A. Saaifan, Jacobs University, Bremen
4.2 Nodal Analysis with Supernode
A supernode is formed when a voltage source is the only element connected between two essential nodes1- Define a current through the source and write KCL equations for the two nodes
3- Apply KVL between the two nodes
2- We note that there is no need to determine ivs to solve the circuit
(1)
(2)
Thus, the KCL at the supernode is directly given by
8K. A. Saaifan, Jacobs University, Bremen
Determines the node-to reference voltages.
Node 1 to reference is supernode
Node 2
Node 3 & node 4
Express vx=v2-v1 and vy=v4-v1 in terms of nodal voltages and organize the equations
(1)(2)(3)
Solve to get
9K. A. Saaifan, Jacobs University, Bremen
4.3 Mesh Analysis
In nodal analysis, circuit variables are node voltagesNodal analysis applies KCL to find unknown voltages
In mesh analysis, circuit variables are mesh currentsMesh analysis applies KVL to find unknown currents
Both methods result in a system of linear equations
Mesh analysis is only applicable to a circuit that is planar
Planar vs. Non-planar Circuits
Planar circuit: it can be drawn on a plane surface where no branch cross any other branch (element)
Non-planar circuit there is no way to redraw it and avoid the branches crossing
Planar circuit Non planar circuit
10K. A. Saaifan, Jacobs University, Bremen
Mesh & mesh current A mesh is a property of a planar circuit and it is defined a loop that does not contain any other loops within it
The current through a mesh is known as a mesh current
mesh
mesh
11K. A. Saaifan, Jacobs University, Bremen
4.3 Mesh Analysis
1. Determine if the circuit is a planar circuit. If not, perform nodal analysis instead.
2. Count the number of meshes (M)
3. Label each of the M mesh currents (defining all mesh currents to flow clockwise results in a simpler analysis)
4. Write a KVL equation around each mesh
For mesh 1, we have
or
For mesh 2, we have
(1)
or
The solution is easily obtained
(2)
12K. A. Saaifan, Jacobs University, Bremen
Determine the power supplied by the 2 V source.
We first define two clockwise mesh currents
For mesh 1, we write the following KVL equation
The same for mesh 2, we write
i1 i2
13K. A. Saaifan, Jacobs University, Bremen
Rearranging and grouping terms, we have
and
Solve the both equation yields
i1=1.132 A and i2=-0.1053 A
The 2 V source supplies (2)(i1-i2)=2.4 W
14K. A. Saaifan, Jacobs University, Bremen
4.4 The Supermesh
Similar to the supernode in a node voltage analysis
A supermesh is formed when a current source is the only element connected between two meshes
1- Define a voltage across the source and write KVL equations for the two meshes
and
2- We do not need to evaluate vcs to solve the circuit
3- This leads us to create a supermesh whose interior is that of mesh 1 and mesh 2
4- Finally, the source current is related to the mesh currents,
15K. A. Saaifan, Jacobs University, Bremen
Determine the three mesh currents.
i3
i1-i2
i3-i2
The 7 A independent current source forms a supermesh between mesh 1 and mesh 3
Applying KVL over the supermesh gives
or
KVL for mesh 2
or
i1 i2
i3
i1-i2
i3-i2
i1-i3
i1 i2
16K. A. Saaifan, Jacobs University, Bremen
Homework Assignment 3 P4.8, P4.10, P4.14, P4.22, P4.26, P4.31, P4.36, P4.44