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Basic Laws of Electric Circuits

Nodal Analysis

Lesson 6

Basic Circuits

Nodal Analysis The Concept.

- Every circuit has n nodes with one of the nodes

being - designated as a reference node.

- We designate the remaining n 1 nodes as

voltage nodes - and give each node a unique name, vi.

- At each node we write Kirchhoffs current law

in terms - of the node voltages.

1

Basic Circuits

Nodal Analysis The Concept.

- We form n-1 linear equations at the n-1 nodes
- in terms of the node voltages.

- We solve the n-1 equations for the n-1 node

voltages.

- From the node voltages we can calculate any

branch - current or any voltage across any element.

2

Basic Circuits

Nodal Analysis Concept Illustration

Figure 6.1 Partial circuit used to illustrate

nodal analysis.

Eq 6.1

3

Basic Circuits

Nodal Analysis Concept Illustration

Clearing the previous equation gives,

Eq 6.2

We would need two additional equations, from

the remaining circuit, in order to solve for V1,

V2, and V3

4

Basic Circuits

Nodal Analysis Example 6.1

Given the following circuit. Set-up the

equations to solve for V1 and V2. Also solve

for the voltage V6.

Figure 6.2 Circuit for Example 6.1.

5

Basic Circuits

Nodal Analysis Example 6.1, the nodal equations.

Eq 6.3

Eq 6.4

6

Basic Circuits

Nodal Analysis Example 6.1 Set up for solution.

Eq 6.3

Eq 6.4

Eq 6.5

Eq 6.6

7

Basic Circuits

Nodal Analysis Example 6.2, using circuit

values.

Figure 6.3 Circuit for Example 6.2.

Find V1 and V2.

At v1

Eq 6.7

At v2

Eq 6.8

8

Basic Circuits

Nodal Analysis Example 6.2 Clearing Equations

From Eq 6.7

V1 2V1 2V2 20

or

Eq 6.9

3V1 2V2 20

From Eq 6.8

4V2 4V1 V2 -120

or

-4V1 5V2 -120

Eq 6.10

Solution V1 -20 V, V2 -40 V

9

Basic Circuits

Nodal Analysis Example 6.3 With voltage source.

Figure 6.4 Circuit for Example 6.3.

At V1

Eq 6.11

At V2

Eq 6.12

10

Basic Circuits

Nodal Analysis Example 6.3 Continued.

Collecting terms in Equations (6.11) and (6.12)

gives

Eq 6.13

Eq 6.14

11

Basic Circuits

Nodal Analysis Example 6.4 Numerical example

with voltage source.

Figure 6.5 Circuit for Example 6.4.

What do we do first?

12

Basic Circuits

Nodal Analysis Example 6.4 Continued

At v1

Eq 6.15

At v2

Eq 6.16

13

Basic Circuits

Nodal Analysis Example 6.4 Continued

Clearing Eq 6.15

4V1 10V1 100 10V2 -200

or

14V1 10V2 -300

Eq 6.17

Clearing Eq 6.16

4V2 6V2 60 6V1 0

or

-6V1 10V2 60

Eq 6.18

V1 -30 V, V2 -12 V, I1 -2 A

14

Basic Circuits

Nodal Analysis Example 6.5 Voltage super node.

Given the following circuit. Solve for the

indicated nodal voltages.

super node

Figure 6.6 Circuit for Example 6.5.

When a voltage source appears between two nodes,

an easy way to handle this is to form a super

node. The super node encircles the voltage source

and the tips of the branches connected to the

nodes.

15

Basic Circuits

Nodal Analysis Example 6.5 Continued.

Constraint Equation

V2 V3 -10

Eq 6.19

At V1

Eq 6.20

At super node

Eq 6.21

16

Basic Circuits

Nodal Analysis Example 6.5 Continued.

Clearing Eq 6.19, 6.20, and 6.21

Eq 6.22

7V1 2V2 5V3 60

-14V1 9V2 12V3 0

Eq 6.23

Eq 6.24

V2 V3 -10

Solving gives

V1 30 V, V2 14.29 V, V3 24.29 V

17

Basic Circuits

Nodal Analysis Example 6.6 With Dependent

Sources.

Consider the circuit below. We desire to solve

for the node voltages V1 and V2.

Figure 6.7 Circuit for Example 6.6.

In this case we have a dependent source, 5Vx,

that must be reckoned with. Actually, there is a

constraint equation of

Eq 6.25

18

Basic Circuits

Nodal Analysis Example 6.6 With Dependent

Sources.

At node V1

At node V2

The constraint equation

19

Basic Circuits

Nodal Analysis Example 6.6 With Dependent

Sources.

Clearing the previous equations and substituting

the constraint VX V1 - V2 gives,

Eq 6.26

Eq 6.27

which yields,

20

circuits

End of Lesson 6

Nodal Analysis

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