Loop current equation in matrix form

The derivation process of establishing the loop current equations in matrix form is similar to the node voltage method in the previous section. First, define a general branch (typical branch) as shown in Figure 7-6-1, then write out the voltage-current relationship of each branch and express it in matrix form as follows:

(7-7-1)

When the network contains no controlled sources, Z is a diagonal matrix.

Multiply the loop matrix on both sides of equation (7-7-1) and take equation (7-3-5) and equation (7-3-8) into consideration, we can get:

That is:

(7-7-2)

or:

(7-7-3)

In the formula, is called the loop impedance matrix; is the loop current column vector. Formula (7-7-2) and Formula (7-7-3) are the loop current equations in matrix form. The quantities and in the formula can be systematically written out by actual circuits and directed graphs, referring to typical branch directions. The loop current value can be solved by the loop current equations in matrix form , and then the current of each branch can be calculated, and then the voltage of each branch can be calculated by formula (7-7-1) .

Example 7-7-1 The circuit and its directed graph are shown in Figure 7-7-1. Take branches 1, 2, and 3 as tree branches and try to establish the loop current equation in matrix form.

Solution: Choose a single-branch loop as the basic loop, then the basic loop matrix can be written as:

Figure 7-7-1

The branch impedance matrix is:

The column vectors of voltage and current sources are:

The loop impedance matrix is:

Finally, the loop equation in matrix form is:

For circuits with controlled sources, such as component voltage-controlled current sources or component current-controlled voltage sources, the processing method is the same as the node voltage method, and the controlled source situation can be considered in the branch impedance matrix Z or branch admittance matrix Y. Since a group of independent loops must be selected in advance when using the loop current method, it is not as convenient as the node method in practical applications.