Inductance

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Transcript Inductance

Inductance
We know already: changing magnetic flux creates an emf
changing current in a coil will induce a current in an adjacent coil
Coupling between coils is described by mutual inductance
2 Φ2 = 21 1
mutual inductance
Current i1 through coil 1 creates B-field and thus flux through coil 2.
If i1=i1(t) then dB/dt  0 and thus d/dt  0 inducing an emf in coil2.
From
2 Φ2 = 21 1
Φ2
1
= 21


For vacuum and material with constant susceptibility
M21 is a constant and given by
Using
ℰ2 = −2
2
Φ2

ℰ2 = −21
1

21
2 Φ2
=
1
Repeating the thought by driving a time dependent current, 2 , through coil 2
2
Induction of emf ℰ1 in coil 1 ℰ1 = −12

1 Φ1
=
2
2
ℰ1 = −12

with
12
1

with
21 =
ℰ2 = −21
2 Φ2
1
For the vacuum case (but also in general) mutual inductance depends only on coil geometry
12 = 21 = 
with
=
1 Φ1 2 Φ2
=
2
1
Mutual inductance is measured in Henry where 1H=1Wb/A=1Vs/A=1Ωs=1J/A2
- Mutual inductance can give rise to cross-talk in electronic circuits 
- Mutual inductance has important applications, e.g., in transformers 
Many applications happen in AC circuits (see textbook).
Here a collection:
Transformers
Metal detectors
Here we have a brief look at the Tesla coil
Current i1 through coil 1 creates B-field
0 1 1
1 =

and flux
Φ1 = 1  = Φ2
solenoid 1 is long compared to solenoid 2
=
2 Φ2 0 1 2 
=
1

M depends only on geometry and in particular
on product N1N2.
To see how a Tesla coil can create a vary large emf let’s have a look to
an example
Drive a current i2(t)through solenoid 2 (blue)
2
1
2
emf induced in solenoid 1 reads: ℰ1 = −

Flux through solenoid 1 is given by Φ1 =
In an operating Tesla coil
high frequency alternating current
creates large amplitudes of di2/dt
and thus large amplitudes of alternating ℰ1
Primary and secondary resonant circuits tuned to
1
same frequency  =
with f= 100 kHz to 1 MHz
2 
see slide 11 for derivation of resonance frequency
in LC-circuit
Hallmark of Tesla coil is the loose/critical coupling (large air gap) between the solenoids 1 and 2 to
prevent damage (insulation between tightly coupled solenoids would experience dielectric
breakdown).
From
http://tesladownunder.com/Tesla_coils_intro.htm
Note, I did not check
the scientific validity
of the information
provided on this
web-site
Self-Inductance and Inductors
The concept that a changing flux induces an emf can also be applied in the
case of a single solenoid
Self-inductance  =
Φ
With ℰ = −

Φ

i
ℰ = −

A device designed to have a particular inductance L is called an inductor
Circuit symbol
Examples from http://upload.wikimedia.org/wikipedia/commons/2/27/Aplikimi_i_feriteve.png
The effect of an inductor in a circuit
Let’s compare resistor and inductor
Current flowing through resistor R
gives rise to potential drop  = 
Emf ℰ
= −
i

opposes current change
Potential drop  = 
Note: Sign opposite to emf


Example: Inductance of an air core toroidal solenoid
1) Determine B from Ampere’s law
r
  = 0  
2) Determine flux through one loop Φ =  = 0
3) Determine emf of solenoid and compare with
 2
Φ
 2  
 = 0
ℰ = −
= −0
2

2 

2

 = 0 
2
ℰ = −
i

Magnetic field Energy
We will see that similar to the electric field there is energy stored in a
magnetic field
Energy stored in an Inductor
Let’s calculate the energy input U needed to establish a current
I in an ideal (zero resistance) inductor with inductance L

With  = 
we obtain for the power, P, delivered to the inductor


 =   = 



For the energy delivered after time, t, we obtain  = 
 ′ ′
0 
Changing integration variable from t to i we obtain

=
  =
0
1 2

2
Energy stored in an inductor when permanent current
I is flowing
We now want to use  =
1 2

2
to see that the energy is stored in the field
(very much in analogy to the transition from energy in a capacitor to energy stored in the electric field)
 2
Let’s recall the inductance L of a toroidal inductor  = 0
2
Volume, V, which is filled with a magnetic field of magnitude  = 0 

2
V= 2   where A is the area of the cross-section

 2

=


for i=I
 = 0
0
2
2
1 2
1  2  2 2
1  2 2
The energy  =  can be expressed as  = 0
(
)
 = 0
2
2
2


2 2
0
1 22
1 2
=
=
2 0
2 0
which yields the energy density u=U/V
1 2
=
2 0
1 2
for vacuum or  =
in a magnetic
2  material
R-L circuits
i
ℰ −  −  = 0

t=0 is time when switch is closed all the
voltage drops across L and thus i(t=0)=0
Kirchhoff’s loop rule
For → ∞ I becomes stationary and is limited only by R
ℰ
=

Solving the differential equation:


0
′
=
ℰ − ′

′
0
Substitution x = ℰ −  yields dx = −
ℰ −
−
ℰ

=


′
0
ℰ
ℰ −Rt
 = (1 − e L )


− ℰ − 
ln
=t

ℰ
ℰ −  =
R
−Lt
ℰe
i
 = ℰ/R
  =  = ℰ/R(1−1/e) =0.63ℰ/R
 = /
t
What happens if we release energy stored in the solenoid
Kirchhoff’s loop rule:
i
 +  = 0

  
− ln
=

0

−

=
ℰ −  − 

′
=
′
0
R
−Lt
0 e
i
0
I0/e
 = /
t
i
=0


′
0
L-C circuits
L-C circuit shows qualitative new behavior
Because there is no power dissipation, energy once
stored in C or L will periodically redistribute between
energy in E-field and B-field
From http://en.wikipedia.org/wiki/LC_circuit
Kirchhoff’s loop rule

i
− − =0



2q
− − 2 =0


2q 1
+
=0
 2 
Compare with harmonic oscillator
=
1