- •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
4.6.3. Reciprocity theorem
Reciprocity theorem can be formulated as follows.
If a voltage source with the EMF E or current source J is included in the branch a - b of the linear electric circuits, not containing other energy sources, and creates in the branch c - d current I , than the same voltage source E or current source J ,included in the branch c - d, creates in the branch a - b the same current I.
Proof:
Fig. 4.39
Let us consider Fig. 4.39. Here voltage source E . included in the branch a – b of the passive linear electric circuits, creates in the branch c - d with impedance Z current I (Fig. 4.39,a). Take this source in the branch c - d. Define the current in the branch a - b. Let branch a - b is included in the loop n, and the branch c - d - in the loop k of linear electric circuit. Let's calculate the circuit according to the method of loop currents.
1. Let voltage source E is included to the loop n (4.39.a). Then the current of the k – th loop
(4.259)
where: - the determinant of the system of loop impedance matrix;
- the determinant, resulting from by disclosure of the on the column of loop EMF. As EMF E is included only in the loop n, and the rest of the circuit is passive, then all determinants, except the determinant , equal to zero.
2. Let voltage source E is included in the loop k (Fig. 4.39.b). Then the current of the n th loop
(4.260)
where: - the determinant, resulting in the disclosure of the on the column of loop EMF. As EMF E is included only in the loop k, and the rest of the circuit is passive, then all determinants, except the determinant , equal to zero.
It is known, that the matrix of the loop impedances is symmetric about the main diagonal, that is = . Therefore
(4.261)
The theorem is proved.
As an illustration of reciprocity theorem define the current I in the circuit of fig.4.36, used when considering the principle of the superposition. Current I in Fig.4.36 was been previously defined by the expression (4.242). Take voltage source E into the branch with impedance Z . Choose the direction of E coinciding with the direction of current I . Then the current
(4.262)
that coincides with (4.242).
Reciprocity theorem is valid only for linear passive circuits. For nonlinear and active circuits in the general case, reciprocity theorem is not performed.