Note: I have a newer version of these pages (including solutions with dependent sources) at
http://lpsa.swarthmore.edu/Systems/Electrical/mna/MNA1.html.
Some Examples of Modified Nodal Analysis
This document describes an algorithmic method for generating MNA
equations. It consists of several parts:.
Many of the ideas and notation from this page are from Litovski,
though the discussion here is quite simpler because only independent
voltage and current sources are considered.
Recall from the previous document:
MNA applied to a circuit with only passive elements (resistors) and
independent current and voltage sources results in a matrix equation of the
form:
We will take n to be the number of nodes (not including
ground) and m to be the number of independent voltage sources.
Notation
- Ground is labeled as node 0.
- The other nodes are labeled consecutively from 1 to n.
- We will refer to the voltage at node 1 as v_1, at node 2 as v_2 and so
on.
- The naming of the independent voltage sources is quite loose, but the
names must start with the letter "V" and must be unique from any
node names. For our purposes we will require that independent
voltage sources have no underscore ("_") in their names.
So the names Va, Vsource, V1, Vxyz123 are all legitimate names, but V_3,
V_A, Vsource_1 are not.
- The current through a voltage source will be labeled with "I_"
followed by the name of the voltage source. Therefore the current
through Va is I_Va, the current through VSource is I_VSource, etc...
- The naming of the independent current sources is similar; the names must
start with the letter "I" and must no underscore ("_")
in their names. So the names Ia, Isource, I1, Ixyz123 are all legitimate
names, but I_3, I_A, Isource_1 are not.
Review the rules
for forming MNA matrices (if needed).
The example given is from Smith,
Figure 2.8. First the MNA equations will be derived from the
circuit. They will then be derived according to the algorithm in the
previous document -- hopefully the results will agree. The nodes and
sources have been labeled as required, and the current through the voltage
source is defined.
To apply the MNA technique we will need 5 equations (one for each of 4
nodes, and 1 for the independent voltage source). Note that Smith
only required 2 equations -- MNA often requires more equations than other
techniques, but is amenable to computer solution. By
inspection we get:
Using the algorithm, we get:
Putting these together yields:
Careful comparison of this result with the
original result verifies that the two solutions are identical.
The example given is from Smith,
Problem 2.11. First the MNA equations will be derived from the
circuit. They will then be derived according to the algorithm in the
previous document -- hopefully the results will agree. The nodes and
sources have been labeled as required, and the current through the voltage
source is defined.
To apply the MNA technique we will need 3 equations (one for each of 2 nodes, and 1 for the independent voltage source).
By
inspection we get:
Using the algorithm, we get:
If we apply these results to the MNA equation, we get
Careful comparison of this result with the
original result verifies that the two solutions are identical.
The last example is a bit more involved, it has two voltage sources and one
current source. The current source and one of the voltage sources are not
grounded.
First we must label the nodes and define currents through the voltage sources
To apply the MNA technique we will need 5 equations (one for each of 3 nodes, and
2 for the independent voltage sources). By
inspection we get:
Using the algorithm we get:
If we apply these results to the MNA equation, we get
Careful comparison of this result with the
original result verifies that the two solutions are identical.
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