Inductancia: Propiedades y Cálculo de Inductores, Monografías, Ensayos de Materiales

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INDUCTANCIA

Inductancia

El

inductor

es un elemento de un circuito

que guarda energía en el campo magnéticoque rodea a sus alambres portadores decorriente.

Del mismo modo que un capacitor guardadicha energía en el campo eléctrico formadoentre sus placas cargadas.

El inductor se caracteriza por su inductancia

, la cual depende de la forma de

dicho inductor.

Inductancia

We used a coil and the solenoid assumption tointroduce the inductance. But the definitionholds for all types of inductance, including a straightwire.

Any conductor has capacitance and inductance.

L

L

dI

dt

≡ −

E

An inductor is usually made of a coil to make a largeinductance (more loops = more flux). The circuitsymbol isThe self-induced emf through this inductor under achanging current

I

is given by:

L

dI

L

dt

= −

E

Unidades de la inductancia

The SI unit for inductance is the

henry

(H)

Named for Joseph Henry:

1797 – 1878

American physicist

First director of the Smithsonian

Improved design of electromagnet

Constructed one of the first motors

Discovered self-inductance

A

s

V

H

Cálculo de la inductancia

i

N

L

B

Φ

=

Por ser

Φ

B

proporcional a la corriente i, la razón de dicha

ecuación

no depende

de i y, por consiguiente, la

inductancia (como la capacitancia) depende sólo de laforma del dispositivo.

N

Φ

B

conexiones de flujo

C

á

lculo

de la

inductancia

de un

solenoide

I

When a current flows through a coil,there is magnetic field established.If we take the solenoid assumptionfor the coil:

E

E

L

0

B

nI

μ

=

When this magnetic field fluxchanges, it induces an emf, E

L

,

called self-induction:

(

)

(

)

0

2

0

B

L

d

NAB

d

NA

nI

d

dI

dI

n V

L

dt

dt

dt

dt

dt

μ

Φ

μ

= −

= −

= −

= −

≡ −

E

or:

L

dI

L

dt

≡ −

E

This defines the inductance

L

, which is a constant related only to the coil.

The self-induced emf

ε

L

is generated by (changing) current in the coil.

According to Lenz’s Law, the emf generated inside this coil is always opposingthe change of the current which is delivered by the original emf

ε

.

For a solenoid:

2

0

L

n V

μ

=

Where n

: # of turns per unit length.

N

: # of turns in length l.

A

: cross section area

V

: Volume for length l.

Magnetic Field of a Toroid

The toroid has

N

turns of

wire

Find the field at a point atdistance

r

from the center

of the toroid (loop 1)

There is no field outsidethe coil (see loop 2)

2

2

o

o

d

B

π

r

μ

N

μ

N

B

π

r

⋅

=

=

=

∫

B

s

r

r



(

)

I

I

La inductancia de un toroide

Magnetización

B

B

r

r

m

La permeabilidad de lamayor parte de losmateriales comunes(excepto losferromagnéticos) tienevalores cercanos a 1.

Con respecto a otrosmateriales que no sonferromagnéticos, lapermeabilidad puededepender de propiedadescomo la temperatura y ladensidad del material, perono del campo

B

0

.

Para los ferromagnéticos

κ

m

depende del campoaplicado

B

0

.

Put inductor

L

to use:

the RL

Circuit

An

RL

circuit contains a

resistor

R

and an inductor

L

.

There are two cases as in a RC

circuit (charging and

discharging) but in an

RL

circuit one changes current, notelectric charge.

Current increases:

When S

2

is connected to

position

a

and when switch S

1

is closed (at time

t

= 0), the

current through

R

and

L

begins

to increase

Current decreases:

When S

2

is connected to

position

b

.

RL

Circuit

When switch S

2

is moved to

position

b

, the original current

disappears. The self-induced emfwill try to prevent that change, andthis determines the emf direction(Lenz Law).

τ

−

−

=

≡

Rt L

t

ε

ε

I

e

e

R

R

(

)

=

=

E

0

R

I t

Solve for the current

I

, with initial

condition that

we find

0

=

dt

dI

L

IR

Energy stored in an inductor

The increasing current

I

from the

battery supplies power not only to theresistor, but also to the inductor. FromKirchhoff’s loop rule, we have

=

d I

ε

I R

L

dt

Multiply both sides with

I

:

=

2

d I

ε

I

I

R

LI

dt

This equation reads: power

battery

=power

R

+power

L

So we have the rate of energy increase in the inductor as:

=

L

dU

d I

LI

dt

dt

Solve for

U

L

:

=

=

∫

2

0

1 2

I

L

U

LId I

LI

Stored energy type andthe Energy Density of a Magnetic Field

Given

U

L

=

½

L I

2

and assume (for simplicity) a solenoid with

L =

μ

o

n

2

V

Since V is the volume of the solenoid, the magnetic energydensity, u

B

is

This applies to any region in which a magnetic field exists (notjust the solenoid)





=

=









2

2

2

1 2

2

L

o

o

o

B

B

U

μ

n V

V

μ

n

μ

2

L

B

o

U

B

u

V

So the energy stored in thesolenoid volume

V

is

magnetic (

B

) energy.

And the energy density isproportional to

B

2

.