Home Random Page


CATEGORIES:

BiologyChemistryConstructionCultureEcologyEconomyElectronicsFinanceGeographyHistoryInformaticsLawMathematicsMechanicsMedicineOtherPedagogyPhilosophyPhysicsPolicyPsychologySociologySportTourism






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)

 

or

 

(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.

 


Date: 2015-12-18; view: 656


<== previous page | next page ==>
The method of loop currents | The main theorem of the circuit theory
doclecture.net - lectures - 2014-2024 year. Copyright infringement or personal data (0.014 sec.)