- •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
4.4.2. Series connection of the magnetic - coupled coils
Two coils with inductances L , L , included series among themselves, are on Fig. 4.22. Here R , R - active resistances of the coil. Here also M = M = M.
Fig. 4.22
In according to Kirchhoff’s low for the voltages to figure 4.22 we can write down
(4.165)
(4.166)
Where R = (R + R ), L = (L + L + 2M) - equivalent resistance and inductance of the two series connected magnetic – coupled coils.
In the complex form we can obtain from (4.165), (4.166)
(4.167)
Here
(4.168)
In the recorded expressions, as above, the sign "plus" corresponds to aiding and "minus " – to opposite connection of coils. It is seen by aiding connection equivalent inductance of the two magnetic coupled coils
(4.169)
is more on the value of 2M
(4.170)
and by opposite connection is lesser on the value of 2M
(4.171)
equivalent than inductance of the two magnetic not coupled coils. This property is used in variometer - device for smooth regulation of the inductance. It consists of two series-connected coils, one of which is put in another and has the ability to rotate on the other. Changing location of the coils axes from zero to 180 degrees, we can smoothly change the inductance in the range from aiding L + L + 2M to opposite L + L - 2M connection of coils, i.e. in the amount 4M.
Fig. 4.23
In according to (4.167) vector diagrams can be construct for aiding and opposite connection of coils (Fig. 4.23). Here Fig. 4.23.a corresponds to the aiding connection. Here the voltage of self-induction j L I and mutual induction j M I of the first coil, as well as the voltage of self-induction j L I and mutual induction j M I of the second coil are summed up. As a result phase of angle for two coils , as well as the resulting phase angles of the both magnetic coupled coils are positive ( > 0, > 0, > 0). Each of the coils and the two coils are generally have the inductive nature. Fig. 4.23,b corresponds to the opposed connection. Here the voltage of mutual induction j M I of the first coil is deducted from the voltage of self-induction j L I of this coil, and the voltage mutual induction j M I of the second coil is deducted from the voltage of self-induction j L I of this coil. However, as for both coils we have inequalities (4.172) it is still the phase angles > 0, > 0, > 0.
(4.172)
That is, each coil separately and both coils are generally inductive nature. Fig. 4.23, also applies to an opposite connection, however voltage mutual induction of the first coil j M I is more than a voltage of self-induction of this coil. As a result of the combined reactance of this coil is negative, the angle < 0 and the first coil have capacitive nature, that it is behaved as capacity. Reactance the second coil remains positive, angle > 0, the second coil has inductive nature. The resulting inductance of both coils nevertheless has inductive nature. Such effect, when one of the coils has capacitive nature, is then, when the inductance L of the second coil is significantly more than inductance L of the first coil, which is possible at W > W .