DOT CONVENTION

Three coils A, B and C wound on a common magnetic circuit are shown in Figure 1. Coil A has N_{A} turns with terminals A_{1} and A_{2}. The sense of winding is shown in the figure. Similarly, the coil B has N_{B} turns with terminals B_{1} and B_{2} and coil C has N_{C} turns with terminals C_{1} and C_{1}. The terminals are marked, arbitrarily on the coils. Let us consider that a current I_{A} enters terminal A_{1}. Looking from top, the current encircles the core counter-clockwise. According to corkscrew rule the flux is directed upwards through the coil. The flux through the core is directed clockwise through the core. A dot is placed quite arbitrarily at terminal A_{1}. This can be done for any coil. Once the dot is placed on one coil, dots on the terminals of other remaining coils cannot be placed arbitrarily now.

Dots on other coils are now automatically decided according to the sense of the winding. Now we are to determine the position of dots on the terminals of the remaining windings corresponding to the dot placed at one of the terminals of winding A, which is at A_{1}. Dot on the other winding is placed on such a terminal that a current entering through the dotted terminal magnetizes the core in the same direction as the flux created by current entering the dot on the first coil A. Thus, if the dot were placed at B_{2}, a current entering this terminal would produce a flux in opposition to the flux produced by current I_{A}. Thus, B_{2} is not the dotted terminal corresponding to the dot at terminal A_{1}. While, if the current enters terminal B_{1}, the magnetization of the core is also clockwise. Thus, terminal B_{1} is the dotted terminal on coil B. Similarly, C_{1} is the dotted terminal on coil C. Thus, A_{1}, B_{1} and C_{1} are dotted terminals on the three coils which are mutually coupled. Now we can make a statement that “currents entering through the dotted terminals of all the coupled coils develop the flux (apply mmf) so that the fluxes due to each coil are additive.”

This also means, that due to rate of change of the common flux emfs are induced in all the coils which have similar polarity at all the dotted terminals.

MUTUAL COUPLING IN A SIMPLE MAGNETIC CIRCUIT

The self-inductance of a circuit is intimately associated with the magnetic field linking the circuit. The self-inductance emf may be thought of as the emf induced in the circuit by a magnetic field produced by the circuit current.

Since a magnetic field exists in the region around the current that develops it, there is also may a possibility that an emf be induced in the other circuits linked by the field. Two circuits linked by the same magnetic field are said to be *coupled* to each other. The circuit element used to represent *magnetic coupling* is shown in Figure 2 and is called *mutual inductance*. It is represented by symbol M and is measured in henrys. The volt-ampere relationship is one which gives the induced emf in one circuit by a current in another and is given as

A similar equation can, of course, be written giving an emf e_{1} induced by a current i_{2}. The two dots, called polarity markings, in Figure 2 are used for indicating the direction of the magnetic coupling between the two coils: note from the figure and from equation given above that by matching the dots, the directions of currents and voltage drops are made to correspond with those of Figure 3 for self-inductance.

If currents are present in both coupled circuits, emfs of self-inductance and mutual inductance are induced in each circuit. The self-induced emf, have the directions shown in Figure 4 the mutually induced emfs, follow the pattern of Figure 2. Figure 3 shows two such elements. Each coil is characterized by its own self-inductance; the combination has a mutual inductance indicating the coupling between the coils. The Volt-ampere relationships for this case are given below:

……(1)

And ……(2)

Coupling between two closed circuits permits the transfer of energy between the circuits through the medium of the mutual magnetic field. This phenomenon is the basis on which transformers operate.

COEFFICIENT OF COUPLING

When two coils are placed near each other, all the flux produced by one coil does not link with the other coil, only a certain portion (say, K) of flux produced by one coil links with other coil, K being less than unity. K is called the *coefficient of coupling*.

Flux created in coil. A due to current of i_{1} amperes,

Flux linking with coil

Coefficient of mutual inductance,

……(3)

Coefficient of self-inductance of coil A,

……(4)

Coefficient of self-inductance of coil B,

……(5)

Multiplying Equations. (4) and (5) and taking square root of both sides we get

…..(6)

Comparing Equations (3) and (6) we get

Or …….(7)

When the coils are tightly coupled i.e. when the flux due to one coil links with the other coil completely, the coefficient of coupling, K is unity and the coefficient of mutual inductance M is given as

……(8)

When the flux due to one coil does not link with the other coil at all, the value of coefficient of coupling, K is zero.

INDUCTORS INSERIES

Figure 5 represents two inductors (inductance coils) in series with the axis of one coil perpendicular to the axis of the other. In this case there is no mutual inductance i.e. M = 0. If the coils are connected is series, represented as L_{1} and L_{2} in Figure 6, so that when the current enters the dot end of coil L_{1} and leaves, it must enter L_{2} at its dotted end, the fluxes of two coils will add. Such a series connection is known as series aiding. If the connections to inductor L_{2} are reversed so that the current must enter its undotted end, their fluxes will oppose each other. This series connection *is known as series opposing. *It has been assumed that the axes of the inductance coils are on the same straight line.

Series-aiding. For the series aiding connections, the total emf induced in each of the coil L_{1} and L_{2} is due to coil’s self-inductance and the emf induced by the other coil. So

Similarly

So, the total induced emf in the circuit is given as

Or = Total inductance of the circuit, L

Or …..(9)

Series Opposing. For the series opposing connections the mutually induced emf opposes the self-induced emf. So

And

Or

Or ……(10)

Inductor in parallel

Let us consider two coils of inductance L1 and L_{2} connected in parallel as shown in figure 8

The supply circuit divides into two components i_{1} and i_{2 }following through the coils.

i.e.

or

Self-induced emf in coil A,

Mutually induced emf in coil A due to change of current in coil B,

Where M is the mutual coefficient of inductance

Resultant emf induced in coil A,

Similarly, resultant emf induced in coil B

As both coils are connected in parallel, therefore, resultant emf induced in both of the coils must be equal

Or

Or ……(11)

Or

……(12)

If L is the equivalent inductance of the combination then induced emf

Since induced emf in parallel combination = Induced emf in either of the coils.

Or

Substituting from equation (11) and from equation (12) we get

Or

Or

When mutual flux helps the individual flux

Or

When mutual flux opposes the individual flux.