The equivalent complex circuit

⇐ ПредыдущаяСтр 6 из 8Следующая ⇒

In section 3.4 presents the general procedure for calculation by the complex amplitudes method. In this case we have to build the equivalent complex circuit (ECC). Let us consider the procedure of obtaining the ECC.

In Fig. 4.29 circuit shows of the power of three-phase asynchronous motor (AM) is shown. Here three motor windings are presented by resistances r₁ , r₂ , r₃ , inductances L₁ , L₂, L₃, and capacitance C₁ , C₂, C₃ . A food circuit is represented by three-phase voltage source e₁, e₂ , e₃

Fig. 4.29

Circuit load is presented resistances r , r , r . Sources of voltages e , e , e form a three phase system. Write down their images through the complex amplitudes

(4.199)

The complex impedance of the separate phase of the circuit

(4.200)

From (4.199), (4.200) get ECC (Fig. 4.30)

Fig. 4.30

Here currents

(4.201)

Here the procedure of obtaining the ECC is the following:

1) select the conventional-positive direction of the currents in the branches of the original circuit;

2) instantaneous values of harmonic currents, voltages and EMF submit in a united trigonometric form of a record: all through the sine or all through the cosine functions;

3) the instantaneous values of voltages and currents, voltages and EMF replace their images in the form of complex amplitudes, the direction of the voltages and currents in the ECC coincide with the direction of these values in the original circuit;

Thus, the order of calculation of the circuits by the method of complex amplitudes can be defined as follows:

1) make up the equivalent complex circuit (ECC);

2) work out a system of algebraic equations in the complex form and solve it;

3) pass from images of unknown quantities in the complex form to the originals in a real form (to obtain the instantaneous values of these quantities);

4) verify the correctness of the calculation by conditions of active and reactive powers balance.

Let us consider the main methods of making and solve complex equations of the circuit.

Method of Kirchhoff's equations

The method is based on direct application of Kirchhoff laws for currents and voltages. By this the maximum number of equations of Kirchhoff's law for the current is composed. The remaining equations are based on Kirchhoff's law for the voltage. Independent values are currents in the branches. The number of equations in the system equals the number of branches of the network.
You can determine the following procedure for the calculation by the method of Kirchhoff's equations:

1) select the conditional positive direction of the currents in the branches;

2) select the independent nodes of the network, select the basic node, as the basic node it is advisable to take node, which converges the largest number of branches;

3) to construct q - 1 equations of Kirchhoff's law for the current for independent nodes (q -the total number of the network nodes);

4) select the independent loops of the network, it must be remembered that a branch with an ideal current source does not form a separate loop;

5) select the direction of path-tracing of the loops;

6) to construct p - q + 1 equations of Kirchhoff’s law for the voltages for independent loops (p - the number of the network branches);

7) the obtained system of equations of Kirchhoff's laws for the currents and the voltages solve together, find the currents of branches; if signs of calculated currents in some of the branches were negative, the actual direction of the currents in these branches are opposite of selected;

8) using the found values of currents and resistances branches determine the voltage of branches.

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)

(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.

Method of the nodal voltages

Method of the nodal voltages also referred to as the method of determining the values. Defining values through which is then calculated currents in the branches of the network, are nodal voltages, between of independent and base nodes.

The essence of method of the nodal voltages consists in working out and solving an equations system of Kirchhoff's law for the currents. Such equations are working out for independent nodes.

Let us consider the previously analyzed the network of electric circuit on fig. 4.3. Convert the voltage sources with the EMF E , E into current sources

(4.216)

Convert as well the impedances in resistance in admittances

(4.217)

Denote units I - I I I, 0. Get the converted scheme (fig. 4.34). Construct a graph of the scheme (4.35). At the same time as the branches of the count take voltage between nodes. Voltage U , U , U) between the nodes I, I I, I I I and node 0 (fig. 4.34) are denoted through U , U , U respectively (fig. 4.35). Let us tree of the graph on the branches U , U , U (solid lines). Point out the chords of the graph U , U , U (dotted lines).

Fig. 4.34 Fig. 4.35

From the graph of Fig. 4.35 it is shown the voltages U - U of the network branches can be expressed through the voltages U , U , U :

(4.218)

Nodes I - I I I of the networks art called independent nodes, node 0 - basic. Voltages U , U , U passing between the independent nodes and basic, are called nodal voltages. From (4.218) it is shown the voltage of branches can be expressed through the nodal voltages.

The number of nodal voltages, as well as the number of independent nodes, less than the number of branches of the network. Hence the order of equation system is reduced and the calculation is simplified.

Let us determine nodal voltages on the network of fig. 4.34. Write down the equations of Kirchhoff's law for the current of nodes I - I I I.

(4.219)

(4.220)

System (4.220) can be rewritten in the form

(4.221)

Here the values

(4.222)

Are called self- admittances of the I - th, I I- th and I I I - th nodes respectively. Thus, self-admittance of the nodes represents the sum of the branches admittances, converging to this node. Values

(4.223)

are called mutual admittances between I-st and I I-nd, I-st and I I I-th and I I-nd and I I I-th nodes respectively. Thus, the mutual admittance between two nodes is the sum of the admittances of the branches of connecting these nodes are taken with the sign " minus".

Currents

(4.224)

are called nodal currents of the I-st ,I I-nd and I I i -th nodes respectively. They represent the sum of the currents of the current sources, converging to a given node and taken with the sign "plus", if the currents are directed to a node, and with the sign "minus", if the currents are directed from the node.

In the general case of N independent nodes, you can write the equations system

(4.225)

or in the matrix form

(4.226)

That is

(4.227)

where

(4.228)

- matrix of the nodal admittances (MNA),

(4.229)

- matrix - column of the nodal voltages (MNV),

(4.230)

- matrix - column of the nodal currents (MNC).

Solve of the system (4.226) we can with help determinants

(4.231)

where: - the determinant of MNA;

, ,..., N – determinants, obtained from by replacing the 1 - st, 2 - nd, ... , N - th column of a matrix - column MNC.

Having determined by (4.231) nodal voltages we can be calculate the branch voltages on (4.218), and then the currents of the branches.

Thus, it is possible to determine the following procedure for the calculation method of the nodal voltages:

1) select the conventional- positive direction of the currents in branches,

2) convert all voltage sources in equivalent current sources, and resistances - into conductances,

3) select the conventional- positive direction of the branch voltages,

4) select the independent nodes of the circuit, indicate the basic node,

5) work out the system of nodal equations in matrix form for independent nodes,

6) to solve the system of nodal equations with respect to the nodal voltages,

7) express branch voltages through the nodal voltages,

8) find branch currents through a branch voltages,

9) using Kirchhoff's law for the currents, find the currents in the branches of the original circuit, which were converted from voltage sources in the current sources.

The method of loop currents and the method of nodal voltages are dual.

Дата добавления: 2021-03-18; просмотров: 65; Мы поможем в написании вашей работы!

Поделиться с друзьями:

⇐ Предыдущая 1 2 3 4 567 8 Следующая ⇒

Мы поможем в написании ваших работ!