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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

nto be the number of nodes (not including ground) andmto 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.

The A matrixThe

Amatrix is (m+n)x(m+n) and will be developed as the combination of 4 smaller matrices,G,B,C, andD.

- the
Gmatrix isnxnand is determined by the interconnections between the passive circuit elements (resistors)- the
Bmatrix isnxmand is determined by the connection of the voltage sources.- the
Cmatrix ismxnand is determined by the connection of the voltage sources. (BandCare closely related, particularly when only independent sources are considered).- the
Dmatrix ismxmand is zero if only independent sources are considered.## Rules for making the G matrix

The

Gmatrix is annxnmatrix 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
Gmatrix at location (1,2) and locations (2,1).## Rules for making the B matrix

The

Bmatrix is anmxnmatrix 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 theith voltage source is connected to nodek, then the element (i,k) in theBmatrix is a 1. If the negative terminal of theith voltage source is connected to nodek, then the element (i,k) in theBmatrix is a -1. Otherwise, elements of theBmatrix are zero.## Rules for making the C matrix

The

Cmatrix is annxmmatrix 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 theith voltage source is connected to nodek, then the element (k,i) in theCmatrix is a 1. If the negative terminal of theith voltage source is connected to nodek, then the element (k,i) in theCmatrix is a -1. Otherwise, elements of theCmatrix are zero.In other words, the

Cmatrix is the transpose of theBmatrix. (This is not the case when dependent sources are present.)## Rules for making the D matrix

The

Dmatrix is anmxmmatrix that is composed entirely of zeros. (It can be non-zero if dependent sources are considered.)

The x matrixThe

xmatrix is (m+n)x1 and holds our unknown quantities. It will be developed as the combination of 2 smaller matricesvandj.

- the
vmatrix isnx1 and hold the unknown voltages- the
jmatrix ismx1 and holds the unknown currents through the voltage sources## Rules for making the v matrix

The

vmatrix is annx1 matrix formed of the node voltages. Each element invcorresponds to the voltage at the equivalent node in the circuit (there is no entry for ground -- node 0).## Rules for making the j matrix

The

jmatrix is anmx1 matrix, with one entry for the current through each voltage source. So if there are two voltage sources V1 and V2, thejmatrix will be:

The z matrixThe

zmatrix is (m+n)x1zmatrix and holds our independent voltage and current sources. It will be developed as the combination of 2 smaller matricesiande. It is quite easy to formulate.

- the
imatrix isnx1 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
ematrix ismx1 and holds the values of the independent voltage sources.## Rules for making the i matrix

The

imatrix is annx1 matrix with each element of the matrix corresponding to a particular node. The value of each element ofiis 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

ematrix is anmx1 matrix with each element of the matrix equal in value to the corresponding independent voltage source.