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Chapter 4. THE CALCULATION OF THE HARMONIC CURRENT CIRCUIT

As a rule, the calculation of electrical circuits includes the following stages:

1) representation of the real electric circuit with the help of theoretical model - equivalent electrical circuit;

2) obtaining mathematical model - drawing up of a linearly independent equations system, unknown variables in which are the voltages and currents - the equations of the electrical equilibrium;

3) the solution of equations system of the electrical equilibrium and bringing the odtained results in compliance with the calculated model,

At the first stage, each element of electric the circuit (generator - a source of energy, the electric motor, resistor, capacitor, etc.) is replaced by the simplified model, consisting only of idealized passive and active elements. At the second stage an equations system of the electrical equilibrium is drawn up on the basis of the adopted system of independent variables that describe the electromagnetic processes in the studied circuit. Finally in the third stage this system is solved and the response of a circuit to the external influence is determined, or the ratio between these values record.

# 4.1. The basic laws of the electrical engineering

Ohm's Law (1.8) for the part of the circuit can be written as follows in the general form

(4.1)

where i, u, Z are respectively instantaneous values of current and voltage and impedance of the part of the circuit.

There are also two Kirchoff's Laws.

Kirchhoff's Law for currents: the sum of the currents in a node is zero.

(4.2)

where: ik - the instantaneous value of the k-th branch current, converging to a given node; n - total number of branches, converging to a given node.

So, for node 2 in the network of Fig. 2.1 can be written

i2 i4 i5 j1 = 0 (4.3)

Here directed current to the node (i2) is recorded with the sign "plus" and currents directed from the node (i4, i5, j1) are recorded with the sign "minus".

Kirchhoff's Law for currents (4.2) can be formulated and for the cross-section : the sum of the currents in the cross-section is equal to zero. Then in (4.2) i – current of k- th branch and n - the number of this cross-section branches. So for the cross-section q q of Fig. 2.3 we can write down

(4.4)

Here currents, the direction of which coincides with the direction of cross-section (i3), recorded with the sign "plus", the currents, the direction which is opposite to the direction of cross-section (i1, i4, i5, j1) - with the sign "minus".

Kirchhoff's Law for currents can be recorded for the given network with the help of topological matrices. So, for the nenwork of Fig. 2.1 using topological matrix (2.2), we can write down

(4.5)

where I- matrix-column of branches currents

(4.6)

or

(4.7)

Using the matrix cross-sections matrix Da(2.12), we can write down for network of Fig. 2.1

(4.8)

or

(4.9)

Here Kirchhoff's law for currents recorded for the section: k - k, l - l, m - m, n - n, p - p, g – g, r – r of network graf of Fig. 2.3.

Kirchhoff's Law for voltages: the sum of the voltages in the loop is equal to zero

(4.10)

where: u - instantaneous value of the voltage of the k - th branch, belonging to the given loop; n - total number of branches of the loop.

So, for the loop r - e - r - r of network i in Fig. 2.1, you can write down, bypassing the loop in the direction of clockwise

(4.11)

Here voltage, coinciding in the direction to the direction of loop path-tracing (u , u ), recorded with the sign "plus", voltage opposite to the direction of looh path-tracing (u , e ) - with the sign "minus".

Kirchhoff's Law for voltages can be recorded for the given network with the help of topological matrices. As for the network of Fig. 2.1, using the loop matrix (2.5), we can write down

(4.12)

where u - matrix - column of branch voltages

(4.13)

Here the path to the voltage source isolated in a separate branch. Then we get by (4.12)

(4.14)

Here Kirchhoff's law for voltages recorded for the loops: r - e - r - r , r - e - r - r - r , r - r - r - r - r -r -r , r - e - r - r , r - e - r - r - r , r - r - r - r of the network of Fig. 2.1.

In the calculations of circuit harmonic current laws Ohm's laws and Kirchhoff's laws are written down in the complex form.

For the expression of (4.1) you can record the image through the complex amplitudes

(4.15)

So Ohm's law is written down in the complex form.

For the expression (4.2) you can record the image through the complex amplitudes

(4.16)

So Kirchhoff's law for currents is written down in the complex form.

For the expression (4.10) you can record the image through the complex amplitudes

(4.17)

So, Kirchhoff's law for the voltage is written downin the complex form.

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