INDUCTANCIA

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INDUCTANCIA
Inductancia
El inductor es un elemento de un circuito
que guarda energía en el campo magnético
que rodea a sus alambres portadores de
corriente.
Del mismo modo que un capacitor guarda
dicha energía en el campo eléctrico formado
entre sus placas cargadas.
El inductor se caracteriza por su
inductancia, la cual depende de la forma de
dicho inductor.
Inductancia
Inductancia L:
di
εL = L
dt
Inductancia
We used a coil and the solenoid assumption to
introduce the inductance. But the definition
L≡−
EL
dI
dt
holds for all types of inductance, including a straight
wire. Any conductor has capacitance and inductance.
An inductor is usually made of a coil to make a large
inductance (more loops = more flux). The circuit
symbol is
The self-induced emf through this inductor under a
changing current I is given by:
dI
EL = − L
dt
Unidades de la inductancia
The SI unit for inductance is the henry (H)
V ⋅s
1H = 1
A
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
Inductancia
Vb > Va
Vb < Va
di
Vb − Va = −L
dt
Cálculo de la inductancia
NΦ B
L=
i
NΦB conexiones de flujo
Por ser ΦB 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 la
forma del dispositivo.
Cálculo de la inductancia de un
solenoide
When a current flows through a coil,
there is magnetic field established.
If we take the solenoid assumption E
for the coil: B = µ0 nI
When this magnetic field flux
changes, it induces an emf, EL,
called self-induction:
d ( NAB )
d ( NAµ0 nI )
dΦ B
dI
E =−
=−
=−
= − µ n 2V
L
dt
dt
dI
or: EL ≡ − L
dt
dt
0
dt
I
+
EL
–
≡ −L
For a solenoid: L = µ0 n 2V
dI
dt
Where
n: # of turns per unit length.
N: # of turns in length l.
A: cross section area
V: Volume for length l.
This defines the inductance L, which is a constant related only to the coil.
The self-induced emf εL is generated by (changing) current in the coil.
According to Lenz’s Law, the emf generated inside this coil is always opposing
the change of the current which is delivered by the original emf ε.
La inductancia de un toroide
Recordemos…
Magnetic Field of a Toroid
The toroid has N turns of
wire
Find the field at a point at
distance r from the center
of the toroid (loop 1)
r r
∫ B ⋅ ds = B( 2πr ) = µoN I
µo N I
B=
2πr
There is no field outside
the coil (see loop 2)
La inductancia de un toroide
Inductores con materiales
magnéticos
Recordemos…
Magnetización
La permeabilidad de la
mayor parte de los
materiales comunes
(excepto los
ferromagnéticos) tiene
valores cercanos a 1.
Con respecto a otros
materiales que no son
ferromagnéticos, la
permeabilidad puede
depender de propiedades
como la temperatura y la
densidad del material, pero
no del campo B0.
Para los ferromagnéticos κm
depende del campo
aplicado B0.
r
r
B = κ mB 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, not
electric charge.
Current increases:
When S2 is connected to
position a and when switch S1
is closed (at time t = 0), the
current through R and L begins
to increase
Current decreases:
When S2 is connected to
position b.
RL Circuit
Applying Kirchhoff’s loop rule to the
circuit in the clockwise direction gives
ε −IR −L
dI
=0
dt
Here because the current is increasing,
the induced emf has a direction that
should oppose this increase.
Solve for the current I, with initial
condition that I(t=0) = 0, we find
(
)
(
ε
ε
−Rt L
I=
1− e
≡
1 − e −t τ
R
R
)
Where the time constant is defined as:
L
τ≡
R
Constante de tiempo
inductiva
RL Circuit
When switch S2 is moved to
position b, the original current
disappears. The self-induced emf
will try to prevent that change, and
this determines the emf direction
(Lenz Law).
dI
IR + L = 0
dt
Solve for the current I, with initial
condition that I ( t = 0 ) = E R we find
I=
ε −Rt L ε −t τ
e
≡ e
R
R
Energy stored in an inductor
The increasing current I from the
battery supplies power not only to the
resistor, but also to the inductor. From
Kirchhoff’s loop rule, we have
ε =IR +L
dI
dt
Multiply both sides with I:
εI = I 2 R + LI
dI
dt
This equation reads: powerbattery=powerR+powerL
So we have the rate of energy increase in the inductor as:
dUL
dI
= LI
dt
dt
I
1 2
Solve for UL: UL = ∫ LId I = LI
2
0
Stored energy type and
the Energy Density of a Magnetic Field
Given UL = ½ L I2 and assume (for simplicity) a solenoid with L =
µo n2 V
2
2


1
B
B
UL = µo n 2V 
V
 =
2
2 µo
 µo n 
Since V is the volume of the solenoid, the magnetic energy
density, uB is
UL B 2
uB ≡
=
V
2 µo
So the energy stored in the
solenoid volume V is
magnetic (B) energy.
And the energy density is
proportional to B2.
This applies to any region in which a magnetic field exists (not
just the solenoid)
RL and RC circuits comparison
RL
(
Charging
ε
I=
1 − e −Rt L
R
Discharging
ε
I = e −Rt L
R
Energy
1
UL = LI 2
2
RC
)
−t
ε
I ( t ) = e RC
R
−t
Q RC
I (t ) =
e
RC
Q2 1
UC =
= C (∆V ) 2
2C 2
Magnetic field Electric field
Energy density
B2
uB =
2 µo
1
uE = ε o E 2
2
Energy Storage Summary
Inductor and capacitor store energy through
different mechanisms
Charged capacitor
When current flows through an inductor
Stores energy in the electric field
Stores energy in the magnetic field
A resistor does not store energy
Energy delivered is transformed into thermo energy
Oscilaciones electromagnéticas:
cualitativas
Oscilaciones electromagnéticas:
cualitativas
Oscilaciones electromagnéticas:
cualitativas
Analogía con el MAS
q↔x
i↔v
1/C↔k
L↔m
k
ω = 2πf =
m
1
ω = 2πf =
LC
Oscilaciones electromagnéticas:
cualitativas
Oscilaciones electromagnéticas:
cuantitativas
U = UB + UE
2
1 2 1q
U = Li +
2
2C
U = cte
dU
dt
=0
Oscilaciones electromagnéticas:
cuantitativas
d 2q 1
+
q=0
2
LC
dt
d 2x k
+ x =0
2
m
dt
x = x m cos(ωt + φ )
q = qm cos(ωt + φ )
Oscilaciones electromagnéticas:
cuantitativas
2
2
qm
1q
UE =
=
cos 2 (ωt + φ )
2 C 2C
1 2 1 2 2
2
U B = Li = Lω qm sen (ωt + φ )
2
2
Sustituyendo ω:
qm2
2
UB =
sen (ωt + φ )
2C
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