Relationship between incidence matrix, loop matrix and cut set matrix

For the same circuit, if the numbers and directions of the branches and nodes are the same, there is a certain connection between the correlation matrix, loop matrix and cut set matrix written out.

For the directed graph shown in Figure 7-5-1, select branches 1, 2, and 3 as branches, and make single branch cut sets as shown in the figure. Then, its basic circuit matrix and basic cut set matrix can be written as follows:

Figure 7-5-1

Multiplying by on the left gives:

That is:

（7-5-1）

According to the matrix properties, another form can be obtained:

（7-5-2）

These two equations reflect the relationship between the basic cut set matrix and the basic loop matrix in the network with the same number.

The general proof of formula 7-5-1 can be briefly described as follows: Let , then any element in D is , the subscript j represents the jth single-branch loop, k represents the kth cut set, and represents the multiplication of the i- branch element in the j-th loop and the i- branch element in the k-th cut set . Obviously, if the i- branch is not included in both the j -loop and the k -cut set, the product must be zero. The number of branches included in both the j -loop and the k- cut set must be an even number. Because if all the branches in the k -cut set are removed, the circuit is divided into two independent parts. If a closed loop spans two parts of the circuit, it is obvious that the number of branches connecting the two parts (included in the k- cut set) must be an even number. For example, for the network shown in Figure 7-5-1, the number of branches included in both cut set 1 and loop 1 (a single-branch loop composed of branch 4) is 4 and 1.

For branches that appear in pairs in the loop and the cut set, if the directions of the two branches are consistent with the loop (in this case, the two elements in the corresponding row have the same sign), then the two branches must have one positive and one negative direction with the cut set (in this case, the two elements in the corresponding row have different signs), and the value of must be zero. Conversely, if the directions of the two branches are one positive and one negative direction with the loop, then they must have the same sign relative to the cut set direction, and their product is also zero. It can be seen that the elements in the matrix D are all zero, so formula (7-5-1) can be derived.

If the network branch numbering is strictly arranged in the order of tree branch first and then link branch, then formula (7-5-1) can be written as:

That is:

（7-5-3）

Where, represents the loop matrix submatrix composed of tree branches; represents the cut set matrix submatrix composed of connected branches.

For the circuit in Figure 7-5-1, if node 4 is set as the reference node, its association matrix is ​​written as:

Multiplying A on the left gives:

That is:

(7-5-4) or (7-5-5)

In fact, if the cut set is selected to surround only one node, and the cut set direction leaves the node, then the cut set formed in this way is the association matrix A , which means that the association matrix is ​​nothing more than a form of the cut set matrix. From formula (7-5-1), it can be seen that formula (7-5-4) is established.

If the branch numbering is done by tree branch first and then connected branch, the association matrix can be expressed as , where represents the submatrix composed of all tree branch elements, and represents the submatrix composed of connected branch elements. Formula (7-5-4) can be described as:

Multiply the above formula on the left to get:

That is:

（7-5-6）

Based on this, the basic loop matrix can be written as:

（7-5-7）

From this expression, it can be seen that for a circuit whose branch numbering adopts the method of first tree branch and then connecting branches, its basic loop matrix can be obtained through the correlation matrix.

Similarly, from equations (7-5-3) and (7-5-6), we can get, , so the basic cut set matrix can be expressed as: (7-5-8)

It can be seen from the formula that the basic cut set matrix can be obtained from the incidence matrix.

When using computer-aided calculation to establish state equations, it is often difficult to directly write the loop matrix or cut set matrix, but it is very convenient to derive the association matrix. Therefore, in practical applications, the loop matrix and cut set matrix are often obtained from the association matrix through equations (7-5-7) and (7-5-8).