Inductance (Индуктивность)

Inductance is a coefficient between the current and the flux linkage (потокосцепление).

Units: Henry [Hn]

The magnetic energy stored in a coil is derived as:

Mutual inductance (Взаимная индуктивность)

The flux which is induced in second coil is proportional to the current which induces magnetic field.

A coefficient between the current in one coil and a flux (flux coupling) in another coil is called a ‘mutual inductance’

Reciprocity principle (принцип взаимности):

It means that the value of the mutual inductance does not depend on the order of induces.

Inductance of thin contours (Индуктивность тонких контуров)

We shall consider a round coil, consisting of one wire. The radius of the wire itself is much smaller than the radius of the contour.

In such case total magnetic field may be split in two parts. The first part – field which is circulate outside the wire (external magnetic field). And the part of the magnetic field which circulate inside wire.

Grey surface crossed by the external flux.

11. Magnetic field intensity inside a cylindrical conductor (Напряженность магнитного поля внутри цилиндрического проводника)

Field intensity inside a cylindrical conductor (Напряжённость поля внутри цилиндрического проводника)

I nfinitely long cylindrical wire with the radius of R, and the current of i.

The field intensity inside the conductor at the point with the radius of r satisfies the Ampere Law.

- current which crosses chosen contour.

Directions of the intensity vector and vector are the same.

This integral may be replaced by:

Current density:

Field intensity:

12. Internal inductance of a thin conductor (Внутренняя индуктивность тонкого проводника) Flux linkage of a thin current layer (Потокосцепление тонкого слоя с током)

Consider a thin cylinder with the radius of r and the thickness of dr

T he current inside this layer is:

The flux per unit length:

In principle we assume that somewhere there should be wire with the opposite current.

After integration:

The flux linkage of a thin layer:

Total internal flux linkage:

or:

Internal inductance of a thin conductor (Внутренняя индуктивность тонкого проводника)

Transformation of the last integral:

Taking into account the limits:

Final expression:

Internal inductance:

13. Inductance of a two wire transmission line (Индуктивность двухпроводной линии).

F lux density induced by a wire:

External flux:

R – radius of the wire; R<<d

Similar flux induced be the second wire.

The total external flux:

Flux linkage:

Inductance:

Internal inductance:

External inductance:

14. Variable separation method in a cylindrical coordinate system (Метод разделения переменных в цилиндрической системе координат). Application of Laplace’s equation (Применение уравнения Лапласа).

The variable separation method starts with assumption (предположение): we can present the potential or unknown function as a product of two special functions – function that depends on radius and function that depends on angle.

Cylindrical coordinate system; variables:
Laplace’s equation in cylindrical system:
Presentation of the potential:
Laplace’s equation: (просто подставляем)
or: (разделили обе части на RS/r2)

- function that depends on radius

- function that depends on angle

There is only solution: both sides are the constants, that’s only function which satisfy this strange condition.