Sahithyan's S2 — Theory of Electricity

Coupled Circuits

For all the definitions below, means the number of turns.

Self inductance

When the magnetic field produced by a coil causes an emf on itself. Denoted by . When the length and area are constants, .

Mutual coupling

Mutual coupling between coils exist when one (secondary coil) is in the magnetic field created by the other coil (primary coil).

When a time-varying current flows in the primary coil, a time-varying flux is produced, which produces a back emf .

Magnetic field

Magnetic flux density

Measure of strength and direction of the magnetic field. Denoted by . Measured in tesla () or or .

Magnetic flux

Denoted by . Measured in weber ().

For a coil having self inductance, turns, carrying current , the generated is given by:

Magnetic field strength

Aka. magnetic field intensity. Denoted by (henry). Measured in ampere per meter ().

Magnetic permeability

Measure of magnetization on a material when a magnetic field is applied. Depends on the material. Denoted by . Meausred in or .

Flux linkage

Denoted by .

Magnetomotive force

A force acted on a coil carrying current. Denoted by .

Here:

- number of turns
- current in the coil

It’s similar to electromotive force in electrical circuits.

Reluctance

Reluctance of a path for magnetic flux:

Here:

- Length of the path
- Permeability
- Cross-sectional area

The above equation can be thought of the equation of resistance in electrical context. is used instead of .

is similar to in electrical context.

Fringing

Flux lines in the air gap tend to bow out. Thus the effective area of air gap is larger than the cross sectional area of the core.

Laws

Faraday’s Law

The magnetic flux passing through a surface A is given by the surface integral:

Ampere’s Law

Line integral of magnetic field intensity around a closed path is equal to the sum of the currents owing through the surface bounded by the path.

When is constant (magnitude and direction) along the path, the above equation reduces to .

When is constant and the path has turns, .

Mutual inductance

When 2 coils are coupled, part of the magnetic flux produced in the primary coil links with secondary coil.

Coefficient of coupling

Ratio between the produced magnetic flux and linked magnetic flux. Denoted by .

Induced emf

Since the produced flux is time-varying, an emf is induced in the second coil.

In the linear region of magnetization characteristic:

Here is the mutual inductance.

Practically, coupling between the primary and secondary coils is identical to the coupling between secondary and primary coils.

Energy stored

The last component is the effective energy stored in the mutual inductance.

The mutual inductance energy is:

Added; if produced fluxes aid each other
Subtracted; if produced fluxes oppose each other

Equivalent inductance

In series

The mutual inductance is:

Added; if produced fluxes aid each other
Subtracted; if produced fluxes oppose each other

In parallel

The mutual inductance in the denominator is:

Subtracted; if produced fluxes aid each other
Added; if produced fluxes oppose each other

T-junction

If 2 coils are given in a T-junction, the circuit has to be changed to include a 3rd inductor. If the coils were aiding, the 3rd inductor will be , otherwise . The 3rd inductor value will be subtracted from the other 2 inductors.

Dot notation

One terminal of the coils is marked with a dot. If both currents enter or exit from the dotted terminals, the fields aid; mutual inductance is positive. Otherwise the fields oppose; mutual inductance is negative.