# Rules for Forming MNA Matrices without OpAmps

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.

### The A matrix

The A matrix is (m+n)x(m+n) and will be developed as the combination of 4 smaller matrices, G, B, C, and D. • 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

1. 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.
2. 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).

#### 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 (k,i) in the B 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 B matrix is a -1.  Otherwise, elements of the B matrix are zero.

#### 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).  For each indendent voltage source, if the positive terminal of the ith voltage source is connected to node k, then the element (i,k) in the C 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 C matrix is a -1.  For each op-amp let the positive input terminal be at node k and negative terminal at node j;  the corresponding (ith) row of the C matrix has a 1 at location corresponding to the positive terminal (i,k), and a -1 at the location corresponding to the negative terminal (i,j).  Otherwise, elements of the C matrix are zero.

#### 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.)

### The x matrix

The x matrix is (m+n)x1 and holds our unknown quantities.  It will be developed as the combination of 2 smaller matrices v and j. • the v matrix is 1xn and hold the unknown voltages
• the j matrix is 1xm and holds the unknown currents through the voltage sources

#### Rules for making the v matrix

The v matrix is an 1xn 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).

#### Rules for making the j matrix

The j matrix is an 1xm 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:

### The z matrix

The z matrix is (m+n)x1 z matrix and holds our independent voltage and current sources.  It will be developed as the combination of 2 smaller matrices i and e.  It is quite easy to formulate. • the i matrix is 1xn 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 1xm and holds the values of the independent voltage sources.

#### Rules for making the i matrix

The i matrix is an 1xn 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 1xm matrix with each element of the matrix corresponding to a voltage source.  If the element in the e matrix corresponds to an independent source it is set equal to the value of that voltage source.  If the element corresponds to an op-amp, then its value is set to zero.

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