An Algorithm for 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:
For a circuit with n nodes and m independent voltage sources:
- The A matrix:
- is (n+m)x(n+m) in
size, and consists only of known quantities.
- the nxn part of
the matrix in the upper left:
- has only passive elements
- elements connected to ground appear only on the diagonal
- elements not connected to ground are both on the diagonal and
off-diagonal terms.
- the rest of the A matrix (not included in the nxn
upper left part) contains only 1, -1 and 0 (other
values are possible if there are dependent current and voltage
sources; I have not considered these cases. Consult Litovski
if interested.)
- The x matrix:
- is an (n+m)x1 vector that
holds the unknown quantities (node voltages and the currents through
the independent voltage sources).
- the top n elements are the n node voltages.
- the bottom m elements represent the currents through the m
independent voltage sources in the circuit.
- The z matrix:
- is an (n+m)x1 vector that
holds only known quantities
- the top n elements are either zero or the sum and difference
of independent current sources in the circuit.
- the bottom m elements represent the m independent
voltage sources in the circuit.
The circuit is solved by a simple matrix manipulation:
Though this may be difficult by hand, it is straightforward and so is
easily done by computer.
Obviously, the notation used does not change the solution. However
the convention described below will make it quite easy to develop the matrices
necessary for solution of the circuit.
- 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.
These rules are somewhat restrictive (more so than they need to be) but
they make development of the algorithm easier while still allowing quite a
bit of freedom.
These rules are easily explained with an example (Example
3 from previous page):
This circuit is labeled according to the guidelines above. Ground
is node 0 and the other two nodes are labeled 1 and 2. The voltage and
current sources have labels with no underscores and the current through the
voltage source Vs1 is labeled I_Vs1. We will discuss how to use this
diagram (with these labels) to generate the MNA equations below.
There are three matrices we need to generate, the A matrix, the x
matrix and the z matrix. Each of these will be created by combining
several individual sub-matrices. To motivate the rules for generating the
matrices we will consider the two sample circuits below (example
2 and example
3 from previous
web page, with Notational Convention as above). I
simply state the resulting MNA equations, but in the following text we will show
how to generate each one.
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Case 1
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| Circuit Diagram |
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Case 2
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| Circuit Diagram |
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The A matrix will be developed as the combination of 4 smaller
matrices, G, B, C, and D.
The A matrix is (m+n)x(m+n)
(n is the number of nodes, and m is the number of independent
voltage sources) and:
- the G matrix is nxn and
is determined by the interconnections between the passive circuit elements
(resistors)
- the B matrix is nxm and
is determined by the connection of the voltage sources.
- the C matrix is mxn and
is determined by the connection of the voltage sources. (B and
C are closely related, particularly when only independent sources
are considered).
- the D matrix is mxm and
is zero if only independent sources are considered.
Rules for making the G matrix
The G matrix is an nxn matrix formed in two steps
- Each element in the diagonal matrix is equal to the sum of the
conductance (one over the resistance) of each element connected to the
corresponding node. So the first diagonal element is the sum of
conductances connected to node 1, the second diagonal element is the sum
of conductances connected to node 2, and so on.
- The off diagonal elements are the negative conductance of the element
connected to the pair of corresponding node. Therefore a resistor
between nodes 1 and 2 goes into the G matrix at location (1,2)
and locations (2,1).
Demonstrations
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Case 1
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| Circuit Diagram |
MNA Equations |
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Case 2
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| Circuit Diagram |
MNA Equations |
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If an element is grounded, it will only have contribute to one entry in
the G matrix -- at the appropriate location on the diagonal. If
it is ungrounded it will contribute to four entries in the matrix -- two
diagonal entries (corresponding to the two nodes) and two off-diagonal
entries.
Rules for making the B matrix
The B matrix is an nxm matrix
with only 0, 1 and -1 elements. Each location in the matrix
corresponds to a particular voltage source (first dimension) or a node
(second dimension). If the positive terminal of the ith voltage
source is connected to node k, then the element (i,k) in the B
matrix is a 1. If the negative terminal of the ith voltage
source is connected to node k, then the element (i,k) in the B
matrix is a -1. Otherwise, elements of the B matrix are zero.
Demonstrations
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Case 2
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| Circuit Diagram |
MNA Equations |
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If a voltage source is ungrounded, it will have two elements in the B
matrix (a 1 and a -1 in the same column). If it is grounded it will
only have one element in the matrix.
Rules for making the C matrix
The C matrix is an mxn matrix
with only 0, 1 and -1 elements. Each location in the matrix
corresponds to a particular node (first dimension) or voltage source (second dimension). If the positive terminal of the ith voltage
source is connected to node k, then the element (k,i) in the C
matrix is a 1. If the negative terminal of the ith voltage
source is connected to node k, then the element (k,i) in the C
matrix is a -1. Otherwise, elements of the C matrix are zero.
In other words, the C matrix is the transpose of the B
matrix. (This is not the case when dependent sources are present.)
Case 1:
Case 2:
Rules for making the D matrix
The D matrix is an mxm
matrix that is composed entirely of zeros. (It can be non-zero if
dependent sources are considered.)
Case 1:
Case 2:
The x matrix holds our unknown quantities and will be developed as the combination of 2 smaller
matrices v and j.
It is considerably easier to define than the A matrix.
The x matrix is (m+n)x1
(n is the number of nodes, and m is the number of independent
voltage sources) and:
- the v matrix is nx1 and hold
the unknown voltages
- the j matrix is mx1 and holds
the unknown currents through the voltage sources
Rules for making the v matrix
The v matrix is an nx1 matrix formed
of the node voltages. Each element in v corresponds to the
voltage at the equivalent node in the circuit (there is no entry for ground
-- node 0).
For example if a circuit has three nodes, the v matrix is
For a circuit with n nodes we get
and so on.
Rules for making the j matrix
The j matrix is an mx1 matrix,
with one entry for the current through each voltage source. So if
there are two voltage sources V1 and V2, the j matrix will be:
Demonstrations for the x matrix.
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Case 1
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| Circuit Diagram |
MNA Equations |
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Case 2
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| Circuit Diagram |
MNA Equations |
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The z matrix holds our independent voltage and current sources and will be developed as the combination of 2 smaller
matrices i and e.
It is quite easy to formulate.
The z matrix is (m+n)x1
(n is the number of nodes, and m is the number of independent
voltage sources) and:
- the i matrix is nx1 and
contains the sum of the currents through the passive elements into the
corresponding node (either zero, or the sum of independent current
sources).
- the e matrix is mx1 and holds
the values of the independent voltage sources.
Rules for making the i matrix
The i matrix is an nx1 matrix
with each element of the matrix corresponding to a particular node.
The value of each element of i is determined by the sum of current
sources into the corresponding node. If there are no current sources
connected to the node, the value is zero.
Rules for making the e matrix
The e matrix is an mx1 matrix
with each element of the matrix equal in value to the corresponding
independent voltage source.
Demonstrations for the z matrix.
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Case 1
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| Circuit Diagram |
MNA Equations |
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Case 2
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| Circuit Diagram |
MNA Equations |
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We can now write out the full matrix solutions for both cases that we have
been developing.
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Case 1
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| Circuit Diagram |
MNA Equations |
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Case 2
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| Circuit Diagram |
MNA Equations |
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If you are still unsure about the application of the algorithm, some more
examples are available.
So far we have only dealt with resistors and independent voltage and current
sources. The next document will show that adding capacitors and inductors
is trivial, and adding op-amps only adds minimally to the difficulty.
Next