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# Nodal Analysis - PowerPoint PPT Presentation

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

Discussion D2.3

Chapter 2

Section 2-7

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

• 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.

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

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

Node 1:

Node 2:

Node 3:

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.

Node 2:

Node 3:

Node 2:

Node 3:

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

• 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.

v1

v2

v3

v1

v2

v3

MATLAB:

• 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.

Example circuit?

Write nodal equations by inspection.

Example circuit?

Example circuit?

Example circuit?

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

• 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.

Example Can’t be Transformed to Current Sources

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

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

Test with numbers the LHS yields

v1

v2

v3

Noting that

Test with numbers the LHS yields

v1

v2

v3

Unknowns:

MATLAB Run the LHS yields

v1

v2

v3

v2

V

v3

V

i2

A

PSpice Simulation the LHS yields

v2

MATLAB:

v3

i2