4.4. Inductive - coupled circuit

4.4.1. Circuit with magnetic coupling

Electrical circuits, processes which influence each other by means of general electric or magnetic fields, are called coupled. If a magnetic field is general, then these circuits with are magnetic or inductive coupling circuits.

Fig.4.21

Let us consider magnetic flux and magnetic-flux linkage in circuits with magnetic coupling. In Fig. 4.21 two inductive - coupled coils W₁ and W₂ schematically depicted (W₁ , W₂ are the number of turns of the first and second coils). Here current i₁ of coil W₁ creates the self-induction flux of the first coil

(4.148)

where: Фs1- the flux- leakage of the first coil - part of the flux , crossing only turns of the first coil; Ф₂₁- the mutual induction flux of the second coil of part of the flux Ф₁₁ , crossing the turns of the first coil.

Similarly current i₂ of the coil W₂ creates the self-induction flux of the second coil

(4.149)

where: - the flux-leakage of the second coil - part of the flux , crossing only turns of the second coil; - the mutual induction flux of the first coil - part of the flux , crossing the turns of the first coil.

The full magnetic fluxes of coils

(4.150)

(4.150,a)

The product of the flow by the number of turns is called magnetic flux-linkage . Then for the first and second coils we get

(4.151)

(4.152)

where: , flux- linkage of self-induction of the first and the second coils;

, flux-linkage of mutual induction of the first (from second) and the second (from first) coils.

The attitude of the linkages of mutual induction to self-induction of the coil is called the degree of coupling

(4.153)

Here - the degree of coupling of the second coil with the first coil - shows which part of the flux-linkage of the second coil associated with the first coil; - degree of соupling of the first coil with the second coil – shows which part of the flux- linkage of the first coil associated with the second coil.

The ratio of magnetic-flux linkage to the current, its creating, is called inductance L. There are distinguished inductances of self-induction (the first and the second coils)

(4.154)

inductances of the mutual induction (the first and the second coils)

(4.155)

inductances of the leakage (the first and the second coil)

(4.156)

Now degrees of coupling in according of (4.153) - (4.155)

(4.157)

The geometric mean of the degrees of coupling and are called the coefficients of coupling

(4.158)

for the linear circuits M = M = M. Therefore

(4.159)

In according to the electromagnetic induction law voltage across terminals of the inductive coil is the time derivative of its total flux linkage

(4.160)

(4.161)

Here

(4.162)

- voltage of self-induction of the first and second coils;

(4.163)

- voltage mutual induction of the first and second coils.

The expression (4.160), (4.161) can be represented in the complex form

(4.164)

In the expressions (4.150) - (4.152), (4.160), (4.161, (4.164) double sign indicates the flux, flux- linkage or the voltage of the mutual induction ( , , , , , ) of a given coil coincides with the flux, flux-linkage or a voltage of self-induction of the this coil (plus sign) or opposed them (minus sign). Hence aiding and opposed connection of inductive coils.

Aiding connection is called the inclusion of two coils, where their magnetic flux of self-induction and mutual induction coincide in the direction. Opposed - opposite in direction.

Here the same name and different name terminals are distinguished of the coils. The same name terminals are called such terminals of two coils, when the currents of these coils, equally directed to these terminals, create a magnetic flaxes of the one direction, that is the coils are aiding connected. Different name - when the coils are opposed connected.

The same name terminals identifies by points on the circuit diagram. In Fig. 4.21 points on the conclusions of the windings of the W and W point to the same name terminals.