4.4.3. 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

(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)

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).