- •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
4.2.2. Parallel connection of elements
Let us consider an electrical circuit with a parallel connection of a resistances r , r , ..., r , inductances L , L , ..., L ,capacitances C , C , ... , C and current sources j , j , ... , j (Fig. 4.3).
Fig. 4.3
According to the of Kirchhoff law for the currents we get to the instantaneous values
(4.26)
or
(4.27)
and finally
(4.28)
where:
(4.29)
(4.30)
(4.31)
(4.32)
I.e. by parallel connection оf resistances or inductances the value of inverse equivalent resistance or inductance equals to the sum of the inverse values of each parallel connected resistances or inductances.
By parallel connection of capacitances the equivalent capacitance equals to the sum of parallel-connected capacitances.
By parallel connection of the current sources value equivalent current source equals to the algebraic sum of the values of each parallel-connected current sources.
Parallel connection ideal voltage sources is impossible.
Similar relations can get ratios for complex conductances and current sources.
(4.33)
On the rules of parallel connection elements is based device current divider (Fig. 4.4).
Fig. 4.4
Here
(4.34)
or through the resistance (Fig.4.5)
Fig. 4.5
Here
(4.35)
The ratio of (4.35) expresses a rule of "the alien resistance": current in one of two parallel-connected resistances equals to the total current, divided by the sum of these resistances and multiplied by the other ("alien") resistance.