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)
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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.
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(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 .
Parallel connection of magnetic coupled coils
There are two coils with inductances L , L connected in parallel among themselves (Fig 4.24). Here r1 , r2 - resistance of these coils. Here also decided to M = M = M.
Fig. 4.24
In according to Kirchhoff’s law for the voltages to Fig. 4.24 we can be written in the complex form
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(4.172,a)
or at M =M = M we get
(4.173)
Let us write down the system (4.173) in the matrix form
(4.174)
where
(4.175)
Hence
(4.176)
where
( (4.177)
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Now from (4.176), (4.177)
(4.178)
(4.179)
(4.180)
When r = r = 0 we obtain
(4.181)
Here
(4.182)
- equivalent inductance of the two in parallel coils.
In (4.182) the upper sign in the denominator (minus) corresponds to the aiding connection and the lower (plus) – opposite connections of the coils. Hence it is seen by aiding connection equivalent inductance
(4.183)
more, and by opposite connection equivalent inductance
(4.184)
less than the equivalent inductance of two parallel not connected coils (M = 0)
(4.185)
Referred to in section 4.4.2 variometer can be built and by parallel connection of magnetic coupled coils. By angle change between of the coils axes from zero to 180 degrees inductance changes in the range from L to L , that is by value
(4.186)
With L = L we obtain
(4.187)
that is, by value M. Hence it follows, construction of variometer is more expedient to use the series connection of the magnetic - coupled coils.
Let us (4.173) construct vector diagrams using (4.173) to aiding and opposed connection of the magnetic соupled coils (Fig. 4.25).
Fig. 4.25
Here Fig. 4.25.a corresponds to the aiding connectio. When the voltages 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 geometrically summed up, giving as a result the voltage U applied to the coils. The phase angle of the coils , as well as the resulting phase angle are positive ( >0, > 0, > 0). Each of the coils and the two coils are generally inductive nature. Fig. 4.25.b corresponds to the opposed connection. When the voltages of mutual induction j M I of the first coil and j M I of the second coil have the opposite direction compared to the Fig. 4.25.a and as well as geometrically summed up accordingly with the voltages of self-induction j L I of the first coil and j L I of the second coil. However, in contrast to the series connection of magnetic-coupled coils, when they are connected in parallel a capacitive effect in one of the coil does not arise. Always each coil separately and both coils are generally inductive nature ( > 0, > 0, > 0).
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