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Nodal Analysis. Discussion D2.3 Chapter 2 Section 2-7. Nodal Analysis. Interested in finding the NODE VOLTAGES, which are taken as the variables to be determined For simplicity we start with circuits containing only current sources. Nodal Analysis Steps.

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Nodal analysis

Nodal Analysis

Discussion D2.3

Chapter 2

Section 2-7


Nodal analysis1
Nodal Analysis

  • Interested in finding the NODE VOLTAGES, which are taken as the variables to be determined

  • For simplicity we start with circuits containing only current sources


Nodal analysis steps
Nodal Analysis Steps

  • Select one of the n nodes as a reference node (that we define to be zero voltage, or ground). Assign voltages v1, v2, … vn-1 to the remaining n-1 nodes. These voltages are referenced with respect to the reference node.

  • Apply KCL to each of the n-1 non-reference nodes. Use Ohm’s law to express the branch currents in terms of the node voltages.

  • Solve the resulting simultaneous equations to obtain the node voltages v1, v2, … vn-1.


Example
Example

Select a reference node as ground. Assign voltages v1, v2, and v3 to the remaining 3 nodes.


Example1
Example

Apply KCL to each of the 3 non-reference nodes (sum of currents leaving node is zero).

Node 1:

Node 2:

Node 3:


Example2
Example

Now express i1, i2, …i5 in terms of v1, v2, v3 (the node voltages). Note that current flows from a higher to a lower potential.


Nodal analysis

Node 1:

Node 2:

Node 3:


Nodal analysis

Node 1:

Node 2:

Node 3:



Nodal analysis

is an (n –1) x (n –1) symmetric conductance matrix

is a 1 x (n-1) vector of node voltages

is a vector of currents representing “known” currents


Writing the nodal equations by inspection
Writing the Nodal Equations by Inspection

  • The matrix G is symmetric, gkj = gjk and all of the off-diagonal terms are negative or zero.

The gkk terms are the sum of all conductances connected to node k.

The gkj terms are the negative sum of the conductances connected to BOTH node k and node j.

The ik (the kth component of the vector i) = the algebraic sum of the independent currents connected to node k, with currents entering the node taken as positive.




Nodal analysis

MATLAB Run

v1

v2

v3



What happens if we have dependent current sources in the circuit
What happens if we have dependent current sources in the circuit?

  • Write the nodal equations in the same way we did for circuits with only independent sources. Temporarily, consider the dependent sources as being independent.

  • Express the current of each dependent source in terms of the node voltages.

  • Rewrite the equations with all node voltages on the left hand side of the equality.


Example3
Example circuit?

Write nodal equations by inspection.


Example4
Example circuit?


Example5
Example circuit?


Example6
Example circuit?


Nodal analysis

Note: the circuit?G matrix will no longer necessarily be symmetric


Nodal analysis for circuits containing voltage sources that can t be transformed to current sources
Nodal Analysis for Circuits Containing Voltage Sources That Can’t be Transformed to Current Sources

  • Case 1. If a voltage source is connected between the reference node and a nonreference node, set the voltage at the nonreference node equal to the voltage of the source.

  • Case 2. If a voltage source is connected between two nonreference nodes, assume temporarily that through the voltage source is known and write the equations by inspection.


Example7
Example Can’t be Transformed to Current Sources

Assume temporarily that i2 is known and write the equations by inspection.


Nodal analysis

There appears to be 4 unknowns ( Can’t be Transformed to Current Sourcesv1, v2, v3, and i2) and only 3 equations. However, from the circuit

or

so we can replace v1 (we could also replace v2) and write



Nodal analysis

Test with numbers the LHS yields

v1

v2

v3

Noting that


Nodal analysis

Test with numbers the LHS yields

v1

v2

v3

Unknowns:


Nodal analysis

MATLAB Run the LHS yields

v1

v2

v3

v2

V

v3

V

i2

A


Nodal analysis

PSpice Simulation the LHS yields

v2

MATLAB:

v3

i2