- •4.1. The basic laws of the electrical engineering
- •4.2. Equivalent transformations in electric circuits
- •4.2.1. Series connection of elements
- •4.2.2. Parallel connection of elements
- •4.2.3. Mutual equivalent transformations of the parallel and series connection of elements
- •4.2.4. The transformation of delta – to star – connection and back
- •4.2.5. Conversion circuits with the ideal voltage and current sources
- •4.3. The simplest harmonic current circuit
- •4.3.1. Harmonic current circuit with series connection of r , l , c elements
- •Harmonic current circuit with series connection of r, l – elements
- •4.3.3. Harmonic current circuit with series connection of r, c elements
- •4.3.4. Harmonic current circuit with a parallel connection of r, l, c elements
- •4.3.5. Harmonic current circuit with a parallel connection of r, c elements.
- •4.3.6.Harmonic current circuit with a parallel connection of r, l elements
- •4.4. Inductive - coupled circuit
- •4.4.2. Series connection of the magnetic - coupled coils
- •4.4.3. Parallel connection of magnetic coupled coils
- •4.4.4. Notion of the ideal and the real transformers
- •4.5. The of calculation methods of harmonic current circuits
- •4.5.1. Features of harmonic current circuits calculation
- •4.5.2. The equivalent complex circuit
- •4.5.3. Method of Kirchhoff's equations
- •4.5.4. The method of loop currents
- •4.5.5. Method of the nodal voltages
- •4.6. The main theorem of the circuit theory
- •4.6.1. Superposition theorem
- •4.6.2. Theorem on the equivalent generator
- •4.6.3. Reciprocity theorem
- •4.6.4. Compensation theorem
- •4.6.5. Thellegen theorem
- •4.7. The optimal methods of electrical circuits calculation
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 Um. Angle φ from the current to voltage, is negative. In Fig. 4.17.b current lags behind voltage Um. 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 Um , 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.