electrotechnics - utcluj.rousers.utcluj.ro/~claudiah/electrotechnics/cursuri/engl...21 /20 curs 5...

1 /20 Curs 5 Electrotehnică

Electrotechnics

Lect. dr. eng. ec. Claudia CONSTANTINESCU

Electrical Engineering Faculty/ Departament of Electrotechnics and Measurements

Email: [email protected]

http://users.utcluj.ro/~claudiah/

mailto:[email protected]

http://users.utcluj.ro/~adina/


1. Inductivities, metods of calculation

2. Magnetic circuit law


1. Inductivities,

methods of calculation


1.1. Inductors. Self inductivity and

mutual inductivity


Inductor

Inductor = a reactive electronic circuit component which has as an essential

electric parameter it’s self inductance, L. The most important property of a coil

consists in the fact that it can accumulate energy.

Wireless alimentation system for a mobile phone

Receptor Emitter

The most important property of an inductor is the fact that it can accumulate

magnetic energy.

❖ Images with different inductors (coils) :


Obs.

o measurement unit: , Henry; −SI

L H

o symbol:

o If in the proximity of the circuits there are only linear mediums (µ =

const.), the inductance depends only on the dimensions and shape of

the circuit and on the magnetic permeability of the medium;

o In the case of the ferromagnetic materials (µ dependent on H),

inductivity is not constant, but it depends on the curent (thus on H).

The inductance of an electrical circuit is defined as a ratio between the

total magnetic flux, Ψ, which crosses an area limited by the contour of a

circuit and the current i which produces this flux (any other currents are

considered null): = 0L

i

def

Inductivity = represents the extent to which a coil accumulates

magnetic energy for a certain value of the current in the circuit.

Inductivity


Self and mutual inductances

❖ Self inductance of circuit 1

(with and ):01 i 02 =i

Two circuits wih N1 and N2 turns are considered and it is assumed that

only the first circuit is crossed by the current i1. The fascicular flux

produced by current 1 passing through a turn of circuit 1 is named Φf11

and the fascicular flux produced by current 1 passing through a turn of

circuit 2 is called Φf21.

11 1 11fN =

11

11

1

0Li

=

▪total flux through circuit 1, Φ11 :

where:

11

11 1

1

0fL Ni

=

It is called self inductivity L11 of the circuit the positive ratio between the total flux

through circuit 1 Φ11, produced by the current of that circuit (in the direction

associated after the right-hand thumb rule to the current) and current i1 producing

it.


❖ Self inductivity of circuit 2 (with and ) is defined as :

22 2 22fN =

22

22

2

0Li

=

▪Total flux through circuit 2, Φ22 :

where:

22

22 2

2

0fL Ni

=

10i =

20i

❖ Mutual inductivity L21, between circuit 2 and 1:

21

21

1

0Li

=

Is called mutual inductivity L21 , between circuits 1 and 2, the ratio between

the total flux Φ21 produced by current 1 passing through 2, and the current i1producing it.


❖ Mutual inductivity L12, between circuits 1 and 2 is defined as:

12

12

2

0Li

=

12 1 12fN = 12

12 1

2

fL Ni

=

❖ The reciprocity relation:

12 21L L=

21 2 21fN = 21

21 2

1

fL Ni

=

➢ Between the two mutual inductivities L21 and L12 the reciprocity

relation is followed, namely:


❖ Magnetic coupling coefficient k of two circuits (inductors):

1221 12

11 22 11 22 1 2

0 1LL L M

kL L L L L L

= = =

❖ Disspersion coefficient σ:

2

2 11 22 21 12 1 2

11 22 1 2

1L L L L L L M

kL L L L

− −

= − = =

Obs.

o the possible notations for the self inductivity are: L11 , L22 ,

Lkk , L1, L2, Lk ;

o the possible notations for the mutual inductivity are: L12 ,

L21 , Lkj , Ljk , M ;


1.2. Inductivity

calculation methods


1) It is supposed that there is a circuit (inductor) passed by the current i ;

2) The magnetic induction, B, is determined in different points:

first the magnetic field intensity is determined, mostly by using

Ampere’s theorem:

where the current linkage (solenatia) θ:

where N= inductor’s number of turns

then from the relation , B is calculated;

3) we determine the fascicular magnetic flux:

and then the total magnetic flux:

4) The inductivity is determined with the relation:

=H ds

I. Direct method

=B H

A

B dA =

Li

=

❑Calculation steps:

N i =

N =


2

1

m

dsR

A=

II. Inductivity calculation with the help of the reluctance

= mm

f

UR

❖ The ratio between the magnetic voltage Um and the fascicular

magnetic flux Фf is called reluctance or magnetic resistance:

m

lR

A=

❖ The magnetic reluctance (resistance) can also be determined with the

relation:

▪μ – the absolute permeability of the

medium;

▪A – the area of the conductor’s section;

where:

▪l – the length of the conductor;

where:


2

m

NL

R=

❖ The self inductivity is equal with the ratio between the square of the

number of turns of the inductor and the total (equivalent) reluctance

of the magnetic circuit:

III. Inductivity calculation with the help of the magnetic

energy

❖ The self inductivity of a circuit (inductor) passed by the current i is:

2

2 mWL

i=

where:

▪Wm – magnetic energy.


IV. Neumann’s formula

= =

1 2

12 1 212

1 124

ds dsL

i R

➢ using this formula we can determine the

mutual inductivity between 2 circuits 1

and 2 with the contours Γ1 and Γ2.

➢ considering 2 filiform circuits with the

contours Γ1 and Γ2 the mutual inductivity

mutuală between the 2 circuits will be

given by the relation:

where:


1.3. Aplications


Solution:

Calculate the self inductivity of a toroidal

inductor, with a square section of area A,

height h (where h = R2 – R1) having the air

gap δ and permeability μ much higher than

the vacuum permeability μ0. The inductor

has N turns passed by the current i.

Problem 1

Direct method

H

r R1

R2

Γμ

δμ0

i i

r

R2

R1

h

A

dA


Problem 2

Direct method

Calculate the self inductivity of a toroidal

inductor, with a square section of area A,

height h (where h = R2 – R1) and

permeability μ much higher than the

vacuum permeability μ0. The inductor has N

turns passed by the current i.

Aplications:

Homework:


2. Magnetic circuit law

(Total current law)


S

mm S

du = +

d t

(1)

At any moment and in any medium, the magnetomotive voltage, ummΓ

along any closed curve Γ is equal with the sum of two terms: the first

is the current linkage θSΓ corresponding to the conduction currents

passing through an open surface SΓ bordered by the curve Γ and the

second term is the electric flux ΨSΓ derived in relation to time through

the same surface SΓ.

❑ General (global) form of the law:


❑ Integral form of the law:

Hds= JdA+ DdAt

S S

d

d

where:

o the first term of the right term of the relation (5) is called the

conduction electrical current through SΓ (current linkage θ):

S

i= J Ad

o the second term of the right term of the relation (5) is called

displacement current ( Hertzian current):

S

di =i = D A

dtH D d

where:o u = H smm d

S

S

= J Ad

o

oS

S

= D Ad

(2)

(3)

(4)

we introduce (2), (3), (4) in (1)

(5)


❖ The derivative of a flux integral of a vector :

( )

= +

D

DdA v divD+rot D v At t

S S

dd

d

divD v=

o The density of the displacement current:

The sum of 3 curents with the following densities:

D

DJ

t

=

o The density of the convection current:vJ v v=

o The density of the theoretical Roentgen current: RJ (D v)rot=

❑ from the local form of the electric flux law:

( )

= +

D

DdA v +rot D v At t

v

S S

dd

d(6)

❑ The most general form of the law:

D v RHds= +i +i +i

D


● is deduced from the global form;

Hds= HdAS

rot

LF

( )D

H J v divD rot D vt

rot

= + + +

❖ in immobile mediums: DH J

trot

= +

The first Maxwell equation

●Stokes theorem is applied:

● (6) is introduced in (5) and the result is:

( )

= + + +

v

DHds dA dA v dA rot D v dA

tS S S S

J

( )

= + + +

v

DHdA dA dA v dA rot D v dA

tS S S S S

rot J

❑ Local form of the law:


ConclusionCML says that the magnetomotive voltage, ummΓ, along any

closed curve Γ is equal with the sum of four types of

independent currents, each of them being obtained in specific

conditions and mediums:

S

i= J Ad

→ the conduction current due to the displacement of the

electrical load under the action of the voltage differences

(in mediums where there is σ);

=

D

Di A

tS

d → the displacement current, appears due to the time

variation of the vector (in mediums where there is ε);

v

S

i = v Av d

→ transport current due to the relative transport speed

of the real load ρv;

R

S

i = rot(D v) Ad

→Roentgen current due to the relative transport speed

of the polarization load ρvp:

mm D v Ru =i+i +i +i

D


The end

(Another step

towards the

Final!!)

☺☺

electrotechnics - utcluj.rousers.utcluj.ro/~claudiah/electrotechnics/cursuri/engl...21 /20 curs 5...

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