4.3. The simplest harmonic current circuit

4.3.1. Harmonic current circuit with series connection of r , l , c elements

Let us consider the networks of Fig 1.9.a. Let voltage ”u” of source is changed according to the harmonic law. Let write down his in the complex form.

(4.79)

According to the Kirchhoffs low for the voltage we get

(4.80)

or

(4.81)

Integral-differential equation (4.81) is the equation of electric balance for the circuit Fig. 1.9.a. Let the current

(4.82)

Write down the equation image (4.81) in the complex form

(4.83)

where

(4.84)

- the complex impedance of a circuit;

(4.85)

- reactance impedance of a circuit;

(4.86)

- impedance circuit

(4.87)

- phase angle of the circuit - the angle of phase shift between the current and the voltage in the circuit.

Now from (4.83) we get

(4.88)

That is

(4.89)

The voltage on resistance R

(4.90)

That is, the voltage across the active resistance r, in according to (4.88), (4.90), is in phase with the current and lags behind on the angle φ an applied voltage to the circuit .

The voltage across the inductance L

(4.91)

That is, the voltage across the inductor L, in according to (4.88), (4.91), ahead of the current phase on the angle .

The voltage across the capacitance C

(4.92)

I.e. voltage across the capacitance C. in according to (4.88), (4.92), lags behind of the current phase on the angle π/2.

Passing on from the complex image to the original, we will obtain from the expressions (4.88), (4.90) - (4.92)

(4.93)

(4.94)

(4.95)

(4.96)

In Fig. 4.11 vector diagrams for the r, L, C - circuit in Fig. 1.9.a is shown. Here in Fig. 4.11.a the voltage U is ahead of the current I on the angle . The angle from the current to voltageis positive. The circuit as a whole has inductive nature. On Fig. 4.11.b voltage U is lagging from the current I . The angle is negative. The circuit as a whole has capacitive nature.

Fig. 4.11

Dividing all values of vector diagrams in Fig. 4.11 on the current I , we obtain the corresponding vector diagrams for resistance (Fig. 4.12). Here the angle is measured from the active resistance of the r to the complex impedance Z. Vector diagrams in Fig .4.12.a and b for inductive ( > 0) nature of the load are equivalent. Also vector diagrams in fig. 4.12.c and d for capacitive ( < 0) the nature of the load are equivalent. Triangles OAB in Fig. 4.12- triangles of resistance.

Fig. 4.12

Inductive reactance x_L = ω L and capacitive reactance x_C = 1/ ω C depend on the frequency ω. Diagrams of dependences for the resistance r, inductive reactance x_L, capacitive reactance x_C, reactive reactance x = ω L - 1/ ω C and impedance Z are depicted in Fig. 4.13

Fig. 4.13

It is visible, we get at the frequency

(4.97)

(4.98)

(4.99)

This mode is called voltage resonance and will be discussed in detail below.

Vector diagrams at the frequency are shown in Fig.4.14.a (foe currents and voltages) and Fig. 4.14.b (for resistances).

Reactive power at voltage resonance is equal to zero. Power in the circuit is of pure active.

Fig. 4.14