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