4.5. The of calculation methods of harmonic current circuits

4.5.1. Features of harmonic current circuits calculation

As indicated above, the method of direct solution of the differential equations can be used to calculate the simplest electrical circuits, for example, the circuits of the first order. However even in the case of such circuits can be seen the application the method of complex amplitudes significantly simplifies the calculations. By calculation of complicated branched circuits of the second and higher orders, for example, a resonant transistor amplifier, in which the transistor in the linear mode is replaced by the equivalent circuit of substitution, or electric power supply circuit of three-phase asynchronous motor, in which each motor phase is the circuit of the second order, the application of the direct method of differential equations solution is practical impossible. The only method of calculation of such circuits is a method of complex amplitudes.

Side by side with the harmonic current circuits, in practice the direct current circuits are widely used. The calculation of such circuits requires decision only of algebraic equations systems. By comparison with direct current circuits calculation of harmonic current circuits has the following features:

1) the dependence of the currents and voltages on the frequency;

2) the dependence of impedances and conductances on frequency;

3) the need to take into account the phase relations between the currents and voltages;

4) the circulation of reactive power between the active and passive elements of the circuit;

5) the resonant phenomena.

The dependence of resistances on the frequency can be represented by Table 4.2.

Table 4.2

From the Table 4.2 it is visible that inductance L in the harmonic current circuit is resistance jωL, and in direct current circuit is equivalent to short-circuit of the circuit part, so as in direct current circuit it is excepted frequency ω = 0. The capacity C of the harmonic current circuit is resistance 1/ jωC and in the direct current circuit is equivalent to break the circuit (no load). Active resistance r is one and the same resistance r in circuits of the direct and harmonic current.

4.5.2. The equivalent complex circuit

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.