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

4.3.2. Harmonic current circuit with series connection of R, L – elements

The ratio for this circuit can be obtained from the expressions of section 4.3.1. in with C . Indeed we get

(4.100)

that is, the capacitance C in Fig. 1.9.a you can replace by the short-circuited jumper. Then, from (4.80) we get

(4.101)

or

(4.102)

From (4.83) we can write down

(4.103)

where

(4.104)

Reactance

(4.105)

Impedance

(4.106)

The phase angle

(4.107)

The current in the circuit and voltages across resistance r and inductance L are defined by the expressions (4.88) - (4.91). The expression for the instantaneous values of the current and voltages are identical with expressions (4.93) - (4.95). In Fig. 4.15 vector diagrams for the voltages and current are shown for the circuit (a) and for the resistances (b).

Fig. 4.15

4.3.3. Harmonic current circuit with series connection of R, C elements

The ratio for this circuit can be obtained from the expressions of section 4.3.1 when L = 0. Indeed we get

(4.108)

that is the inductance L in Fig. 1.9.a you can replace by the short-circuit jumper. Then, from (4.80) we get

(4.109)

or

(4.110)

From (4.83) we can write down

(4.111)

where

(4.112)

Reactance

(4.113)

Impedance

(4.114)

The phase angle

(4.115)

The current in the circuit and voltages across resistance R and capacitance C are defined by the expressions (4.88) - (4.90), (4.92). The expressions for the instantaneous values of the current and voltages are identical with expressions (4.93) – (4.94), (4.96). In Fig. 4.16 presents vector diagrams for the voltage and current are shown for the circuit (a) and resistances (b).

Fig. 4.16

4.3.4. Harmonic current circuit with a parallel connection of R, L, C elements

Let us consider the networks of Fig. 1.9.b, which is dual circuit of Fig. 1.9.a. Obviously, all of the ratio for this network may be obtained from the expressions of sections 4.3.1- 4.3.3 by the dual replacement. Let the energy source creates a current

(4.115,a)

According to the Kirchhoff law for the currents we get

(4.116)

or

(4.117)

Write the image to (4.117) in the complex form

(4.118)

where

(4.119)

- complex admittance of the circuit;

(4.120)

- susceptance of the circuit;

(4.121)

- admittance of the circuit;

(4.122)

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

Now from (4.118) we get

(4.123)

That is

(4.124)

That is the phase angle φ corresponds to the same expression (4.89). The current in conductance g

(4.125)

That is current in the active conductance g in according to (4.123), (4.125) is in phase with the voltage and lags behind on the angle φ an current source.

The current in the capacitance C

(4.126)

That is current in capacitance C in according to the (4.123), (4.126) ahead of the phase voltage on the angle φ.

The current in the inductance L

(4.127)

That is the current in the inductance L in according to (4.123), (4.127), lags behind of the voltage on the angle φ .

Passing on from the complex image to the original, we will obtain from (4.123), (4.125) - (4.127)

(4.128)

(4.129)

(4.130)

(4.131)

In Fig. 4.17 shows vector diagrams for the r, L, C - circuit in Fig. 1.9,b. Here in Fig. 4.17,a the current I is ahead of the voltage U_m. Angle φ from the current to voltage, is negative. In Fig. 4.17.b current lags behind voltage U_m. The angle φ is positive. The circuit as a whole has inductive nature.

Fig. 4.17

Dividing all values of vector diagrams in Fig. 4.17 on the voltage U_m , we get the corresponding vector diagrams for conductances (Fig. 4.18). Here the angle is measured from the admittance of Y. Vector diagrams Fig. 4.18.a and b for capacitance (φ < 0) nature of the load are equivalent. Also equivalent to vector diagrams in Fig. 4.18.c and d for inductive (φ > 0) nature of the load are equivalent. Triangles OAB in Fig. 4.18 - triangles of conductances.

4.3.5. Harmonic current circuit with a parallel connection of R, C elements.

The ratio for this circuit can be obtained from the expressions of section 4.3.4.при L → ∞. Indeed we get

(4.132)

Fig. 4.18

that is the inductance L in Fig. 1.9.b, you can replace break in a branch with inductance. Then from (4.11.b) we get

(4.133)

or

(4.134)

From (4.118) we obtain

(4.135)

where

(4.136)

Susceptance

(4.137)

Admittance

(4.138)

The phase angle

(4.139)

The voltage across the terminals of the circuit and the currents in the conductance g and capacitance C are defined by the expressions (4.123) - (4.126). The expressions for the instantaneous values of voltages and currents are identical with (4.128) - (4.130). Fig. 4.19 shows vector diagrams for voltages and currents in the circuit (a) and conductances (b).

Fig. 4.19

4.3.6.Harmonic current circuit with a parallel connection of R, L elements

The ratio for this circuit can be obtained from the expressions of section 4.3.4.при C = 0. Indeed we get

(4.140)

that is, the capacitance C in Fig. 1.9.b you can replace by the gap of branches with a capacity. Then from (4.116) we obtain

(4.141)

or

(4.142)

From (4.118) we receive

(4.143)

where

(4.144)

Susceptance

(4.145)

Admittance

(4.146)

The phase angle

(4.147)

The voltage on the terminals of the circuit and the currents in the conductance g and inductance L are defined by the expressions (4.123) - (4.125), (4.127). The expression for the instantaneous values of voltages and currents are identical with expressions (4.128) - (4.129), (4.131). In fig. 4.20 vector diagrams for voltage and currents in the circuit (a) and conductances (b) are given.

Fig.4.20

Date: 2015-12-18; view: 703