Transcript Inductance - UTK Department of Physics and Astronomy
Chapter 30 Inductance
Physics 231 Lecture 10-1 Fall 2008
Magnetic Effects
As we have seen previously, changes in the magnetic flux due to one circuit can effect what goes on in other circuits The changing magnetic flux induces an emf in the second circuit Physics 231 Lecture 10-2 Fall 2008
Mutual Inductance
Suppose that we have two coils, Coil 1 with N
1
Coil 2 with N
2
turns and turns Coil 1 has a current i
1
a magnetic flux,
B2
which produces , going through one turn of Coil 2 If i
1
changes, then the flux changes and an emf is induced in Coil 2 which is given by
2
N
2
d
B
2
dt
Physics 231 Lecture 10-3 Fall 2008
Mutual Inductance
The flux through the second coil is proportional to the current in the first coil
N
2
B
2
M
21
i
1
where M
21
is called the
mutual inductance
Taking the time derivative of this we get
N
2
d
B
2
dt
M
21
di
1
dt
or
2
M
21
di
1
dt
Physics 231 Lecture 10-4 Fall 2008
Mutual Inductance
If we were to start with the second coil having a varying current, we would end up with a similar equation with an
M 12
We would find that
M
21
M
12
M
The two mutual inductances are the same because the mutual inductance is a geometrical property of the arrangement of the two coils To measure the value of the mutual inductance you can use either
2
M dI
1
dt
or
1
M dI
2
dt
Physics 231 Lecture 10-5 Fall 2008
Units of Inductance
1 Henry 1 V sec Amp 1 J Amp 2
Physics 231 Lecture 10-6 Fall 2008
Self Inductance
Suppose that we have a coil having N turns carrying a current I That means that there is a magnetic flux through the coil This flux can also be written as being proportional to the current
N
B
L I
with L being the
self inductance
having the same units as the mutual inductance Physics 231 Lecture 10-7 Fall 2008
Self Inductance
If the current changes, then the magnetic flux through the coil will also change, giving rise to an induced emf in the coil This induced emf will be such as to
oppose
the change in the current with its value given by
L dI dt
If the current I is increasing, then If the current I is decreasing, then Physics 231 Lecture 10-8 Fall 2008
Self Inductance
There are circuit elements that behave in this manner and they are called
inductors
and they are used to oppose any change in the current in the circuit As to how they actually affect a circuit’s behavior will be discussed shortly Physics 231 Lecture 10-9 Fall 2008
What Haven’t We Talked About
There is one topic that we have not mentioned with respect to magnetic fields Just as with the electric field, the magnetic field has energy stored in it We will derive the general relation from a special case Physics 231 Lecture 10-10 Fall 2008
Magnetic Field Energy
When a current is being established in a circuit, work has to be done If the current is i at a given instant and its rate of change is given by di/dt then the power being supplied by the
external
source is given by
P
V L i
The energy supplied is given by
di L i dt dU
Pdt
The total energy stored in the inductor is then
U
L I
0
i di
1 2
L I
2
Physics 231 Lecture 10-11 Fall 2008
Magnetic Field Energy
This energy that is stored in the magnetic field is available to act as source of emf in case the current starts to decrease We will just present the result for the energy density of the magnetic field
u B
1 2
B
0 2
This can then be compared to the energy density of an electric field
u E
1 2 0
E
2
Physics 231 Lecture 10-12 Fall 2008
R-L Circuit
We are given the following circuit and we then close S 1 leave S 2 open and It will take some finite amount of time for the circuit to reach its maximum current which is given by
I
R
Kirchoff’s Law for potential drops still holds
Physics 231 Lecture 10-13 Fall 2008
R-L Circuit
Suppose that at some time t the current is i The voltage drop across the resistor is given by
V ab
i R
The magnitude of the voltage drop across the inductor is given by
V bc
L di dt
The sense of this voltage drop is that point b is at a higher potential than point c so that it adds in as a negative quantity
i R
L di dt
0
Physics 231 Lecture 10-14 Fall 2008
R-L Circuit
We take this last equation and solve for di/dt
di dt
L
R i L
Notice that at t = 0 when I = 0 we have that
di dt initial
L
Also that when the current is no longer changing, di/dt = 0, that the current is given by
I
R
as expected But what about the behavior between t = 0 and t =
Physics 231 Lecture 10-15 Fall 2008
R-L Circuit
We rearrange the original equation and then integrate
i
di
/
R
R L dt
The solution for this is
i
0
i i
'
di
/ '
R
R L
0
t dt
'
R
1
e
R
/
L
t
Which looks like Physics 231 Lecture 10-16 Fall 2008
R-L Circuit
As we had with the R-C Circuit, there is a
time constant
associated with R-L Circuits
L R
Initially the power supplied by the emf goes into dissipative heating in the resistor and energy stored in the magnetic field
i
i
2
R
L i di dt
After a long time has elapsed, the energy supplied by the emf goes strictly into dissipative heating in the resistor Physics 231 Lecture 10-17 Fall 2008
R-L Circuit
We now quickly open S 1 and close S 2 The current does not immediately go to zero The inductor will try to keep the current, in the same direction, at its initial value to maintain the magnetic flux through it Physics 231 Lecture 10-18 Fall 2008
R-L Circuit
Applying Kirchoff’s Law to the bottom loop we get
iR
di L dt
0
Rearranging this we have
di i
R L dt
and then integrating this
0
I
0
di
'
i
'
R L
0
t dt
'
I
I
0
e
R
/
L
t
where I
0
is the current at t = 0 Physics 231 Lecture 10-19 Fall 2008
R-L Circuit
This is a decaying exponential which looks like The energy that was stored in the inductor will be dissipated in the resistor Physics 231 Lecture 10-20 Fall 2008
L-C Circuit
Suppose that we are now given a fully charged capacitor and an inductor that are hooked together in a circuit Since the capacitor is fully charged there is a potential difference across it given by V
c = Q / C
The capacitor will begin to discharge as soon as the switch is closed Physics 231 Lecture 10-21 Fall 2008
L-C Circuit
We apply Kirchoff’s Law to this circuit
L di dt
q C
0
We then have that Remembering that
di dt
d
2
q dt
2
The circuit equation then becomes
i
dq dt d
2
q
dt
2 1
LC q
0
Physics 231 Lecture 10-22 Fall 2008
L-C Circuit
This equation is the same as that for the Simple Harmonic Oscillator and the solution will be similar
q
Q
0 cos(
t
)
The system
oscillates
with angular frequency
1
LC
is a phase angle determined from initial conditions The current is given by
i
dq dt
Q
0 sin(
t
)
Physics 231 Lecture 10-23 Fall 2008
L-C Circuit
Both the charge on the capacitor and the current in the circuit are oscillatory The maximum charge and the maximum current occur
p / 2
seconds apart For an ideal situation, this circuit will oscillate forever Physics 231 Lecture 10-24 Fall 2008
L-C Circuit
Physics 231 Lecture 10-25 Fall 2008
L-C Circuit
Just as both the charge on the capacitor and the current through the inductor oscillate with time, so does the energy that is contained in the electric field of the capacitor and the magnetic field of the inductor Even though the energy content of the electric and magnetic fields are varying with time, the sum of the two at any given time is a constant
U Total
U E
U B
Physics 231 Lecture 10-26 Fall 2008
L-R-C Circuit
Instead of just having an L-C circuit with no resistance, what happens when there is a resistance R in the circuit Again let us start with the capacitor fully charged with a charge Q 0 on it The switch is now closed Physics 231 Lecture 10-27 Fall 2008
L-R-C Circuit
The circuit now looks like The capacitor will start to discharge and a current will start to flow We apply Kirchoff’s Law to this circuit and get
iR
L di dt
q C
0
And remembering that
i
dq dt
we get
d
2
q
dt
2
R L dq
dt
1
LC q
0
Physics 231 Lecture 10-28 Fall 2008
L-R-C Circuit
The solution to this second order differential equation is similar to that of the damped harmonic oscillator The are three different solutions Underdamped Critically Damped Overdamped Which solution we have is dependent upon the relative values of R 2 and 4L/C Physics 231 Lecture 10-29 Fall 2008
Underdamped:
L-R-C Circuit
R
2 4
L C
The solution to the second differential equation is then
q
Q
0
e
R
2
L t
cos 1
LC
R
2 4
L
2
t
This solution looks like The system still oscillates but with decreasing amplitude, which is represented by the decaying exponential This decaying amplitude is often referred to as the
envelope
Lecture 10-30 Fall 2008 Physics 231
L-R-C Circuit
Critically Damped:
R
2 4
L C
Here the solution is given by
q
Q
0 1
R
2
L t
e
R
2
L t
This solution looks like This is the situation when the system most quickly reaches q = 0 Physics 231 Lecture 10-31 Fall 2008
L-R-C Circuit
Overdamped:
R
2 4
L C
Here the solution has the form
q
Q
0
e
R
2
L
2
t
1
R
2 '
L
e
'
t
1
R
2 '
L
e
'
t
with
'
R
2 4
L
2 1
LC
This solution looks like Physics 231 Lecture 10-32 Fall 2008
L-R-C Circuit
The solutions that have been developed for this L-R-C circuit are only good for the initial conditions at t = 0 that q = Q
0
that i = 0 and Physics 231 Lecture 10-33 Fall 2008