The method of loop currents

Method of Kirchhoff's equations requires the making and the solution of equations system, the number of which is equal to the number of the network branches. To simplify the process, we can reduce the number of equations system, breaking the calculation of the two stages. At the first stage intermediate, auxiliary variable, called loop currents are calculated. At the second stage through loop currents branches currents are calculated. The loop currents called the defining values, because through these values the currents in the branches are determined. Hence the method of loop currents refer to the method of the determining values.

The essence of the loop currents method consists in working out and solving a system of Kirchhoff's law equations for the voltage. Such equations are worked out for independent loops.

Let us consider the network of the electric circuit of Fig.4.31. Convert the current source J in voltage source E = J Z (Fig. 4.32).

Fig. 4.31

Fig. 4.32

Select the conditionally positive direction in the branches currents of the I - I . Let us denote the nodes of the network 1 -4. We get the graph of the network (Fig. 4.33). Select the tree of the graph on the branches I , I , I (ribs of tree - solid lines ). Point out the chords of the graph (the dotted lines). Let us denote the chords of I - I . Each of chords together with the edges of the graph forms the independent loop. These loops are shown in Fig. 4.32 and identified I - I I I. Currents of chords I - I are called loop currents of the circuits I - I I I in Fig. 4.32. Directions of loop currents are indicated by arrows.

Fig. 4.33

From a graph of Fig. 4.33 it is shown the currents I - I in the branches of the network in Fig. 4.33 can be expressed in terms of contour currents I - I

(4.202)

But the number of loop currents, as well as the number of independent circuits, less than the number of the branches currents. Hence the order of the equations system is reduced and the calculation is simplified.

Define loop currents on the network in Fig. 4.32. We write down the equations in according Kirchhoff’s law for the voltages for the loops I - I I I.

(4.203)

or

(4.204)

System (4.204) can be rewritten in the form

(4.205)

Here the values

(4.206)

are called own impedances of the I – st, II - nd and III -rd loops respectively. Thus, own impedance of the loop is the sum of the impedances of the branches included in this loop. Values

(4.207)

are called mutual impedances between I - th and I-st and II-nd, I-st and III-rd, II-nd and III-rd loops respectively. Thus, mutual impedance between two loops represents the sum of the impedances of the branches included simultaneously in both the circuits, taken with the opposite sign. EMF

(4.208)

are called loop EMF of the I - st and II - nd and III - rd loops respectively. They represent the algebraic sum of EMF, included in the branches of the loop and taken with the sign "plus", if the direction of the EMF coincides with the selected direction of the loop path-tracing, and with the sign "minus", if the direction of the EMF is opposite to the direction of the loop patn-tracing.

In the general case of N independent loops we can write the system of equations

(4.209)

Or in the matrix form

(4.210)

That is

(4.211)

where

(4.212)

- matrix of the loop impedances (MLI)

(4.213)

- matrix - column of the loop currents (MLC)

(4.214)

- matrix – column of the loop EMF (MLE).

Solving the system (4.210) we can use

(4.215)

where: - the determinant of the MLI

- determinants, obtained from by substitution 1-st, 2-nd, ..., N - th column the the matrix - column MLE.

Having determined by (4.215) loop currents we can be calculate the currents in the branches from (4.202).

Thus, it is possible to determine the following procedure for the calculation by the method of loop currents:

1) select the conditional positive direction of the branch currents,

2) convert all current sources into equivalent voltage sources,

3) select the independent loops of the network,

4) select the direction of path-tracing of the loops: all in the direction or all against the direction of motion clockwise;

5) work out a system of loop equations in matrix form for independent loops,

6) solve a system of equations with respect to the loop currents,

7) express currents in the branches through the loop currents;

8) using Kirchhoff's law for the currents, find the currents in the branches of the original circuit, which have been transformed from the current sources into the voltage sources,

Advantage of the method of loop currents: thus the equations are working out only according to the Kirchhoff’s for the voltages, the necessary number of equations is less q - 1 than the method of Kirchhoff's equations.

Date: 2015-12-18; view: 748