chapter 32 inductance
TRANSCRIPT
What is inductance? Inductance is the property in an electrical circuit
where a change in the electric current through thatcircuit induces an electromotive force (EMF) thatopposes the change in current.
What is inductor?
An inductor or a reactor is a device that canstore energy in a magnetic field created by theelectric current passing through it.
An inductor's ability to store magnetic energy ismeasured by its inductance, in units of henries.
Typically an inductor is a conducting wireshaped as a coil, the loops helping to create astrong magnetic field inside the coil due toAmpere's Law
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Aluminum and most metals do not conduct electricity
as good as copper.
Insulators are materials that have just the opposite
effect on the flow of electrons. They do not let
electrons flow very easily from one atom to another.
Insulators are materials whose atoms have tightly
bound electrons. These electrons are not free to roam
around and be shared by neighboring atoms.
Some common insulator materials are glass, plastic,
rubber, air, and wood.
Inductors are one of the basic electronic components
used in electronics where current and voltage change
with time, due to the ability of inductors to delay and
reshape alternating currents.2
In everyday speak inductors are sometimes called
chokes, but this refers to only a particular type and
purpose of inductor.
Source = source emf and source current
Induced
Induced current:
Induced current is the current generated in a loop due
to changing magnetic flux.
Induced emf:
Induced emf is the work done per unit charge inproducing an induced current.
Electronic symbol for inductor
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1st example
Consider a circuit consisting of a switch, a resistor,
and a source of emf (Fig. above)
Switch closed – the source current does not
immediately jump from zero to its maximum value
/R.
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S
B
R
I
I
L
• As the source current increases with time, the
magnetic flux through the circuit loop due to this
current also increases with time.
• This increasing flux creates an induced emf in the
circuit.
• The direction of the induced emf would cause an
induced current in the loop.
• This would establish a magnetic field that would
oppose to the change in the source magnetic field.
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• Thus, the direction of the induced emf is opposite the
direction of the source emf; this results in a gradual
rather than instantaneous increase in the source current
to its final equilibrium value.
• This effect is called self-induction because the
changing flux through the circuit and the resultant
induced emf arise from the the circuit itself.
• The emf L set up in this case is called a self-induced
emf or a back emf.
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An inductor is a circuit element that stores magnetic
field.
If the magnetic field is changing, i.e. the current is
changing.
It will have an induced EMF across it with a
magnitude proportional to the rate of change of
current:
ε di/dt
ε = -L (di/dt)
The proportionality constant L is called the
inductance of the device.
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It is a property of the device depends on geometry or
windings and does not depend on the current.
Inductance is measured in units of “henrys”, where
1 henry = 1 volt-second/ampere.
As per Lenz’s Law, the sign of the EMF is
determined such that it opposes the change in the
magnetic flux through the device.
When going from point “a” to point “b” on each end
of the device, the EMF is given by
ε = -N (dΦ/dt) --------------------- (1)
ε = -L (di/dt) -----------------------(2)
(1) = (2) N Φ = L i
L = N Φ / I9
For a solenoid, B = μ0ni where n is the number of
turns per unit length n = N/ l
L = NΦ / i
L = (nl) (BA) / i
= (nl) (µo ni)(A) / i
= n2 µo Al
L = n2 µo V
Where V is the volume of the solenoid
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For a toroid, B= (µoNi) / 2 r
Φ = B ∫ dA
= B ∫ hdr
= Bh ∫ dr
= [(µoNi) h / 2 ] ∫r2r1 r
= [(µoNi) h / 2 ] ln (r2/r1)
L= NΦ / i
= [(µoN2) h / 2 ] ln (r2/r1)
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r1
r2
A series connection of resistor and inductor
Switch S is thrown closed at t=0 the current in the
circuit begins to increase – and a back emf that
opposes the increasing current is induced in the
inductor.
The back emf is, from Equation
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+
-
I
R
S
a
b
+
-
L
dt
dILL
Because the current is increasing, dI/dt is positive;
thus L is negative
This negative value – reflects the decrease in electric
potential that occurs in going from a to b across the
inductor, as indicated by the +ve and -ve signs (Figure
above).
Apply Kirchhoff’s loop rule (clockwise direction) :
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dt
dILL
0dt
dILIR+
-
I
R
S
a
b
+
-
L
To find the current, I in the circuit as a function of
time.
we change variables for convenience, letting :
so that dx = - dI
With these substitutions, we can write Equation
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IR
x
0dt
dx
R
Lx
dtL
R
x
dx
tL
R
x
xln
o
where xo = the value of x at time t=0
Taking the antilohrarithm :
Because I=0 at t=0, from the definition of x : xo= /R.
Hence, this last expression is equivalent to
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L/Rt
oexx
L/RteR
IR
)e1(R
I L/RtEffect of the inductor
The current does not increase instantly to its final
equilibrium value when the switch is closed but
instead increases according to an exponential
function.
We can also write this expression as
where the constant is the time constant of the RL
circuit
is the time it takes the current in the circuit to reach
(1-e-1) = 0.63 of its final value /R.
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)e1(R
I /t
= L / R
The equilibrium value of the current, which occurs as
t approaches infinity, is /R (by setting dI/dt=0 in Eq.
and solving for the current I).
At equilibrium, the change in the current is zero.
Thus, the current initially increases very rapidly and
then approaches the equilibrium value /R as t
approaches infinity.
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I
R/
R63.0
t
R/L
Taking the first time derivative of this equation
The time rate of change of the current is a maximum
(equal to /L) at t=0 and falls off exponentially to
zero as t approaches infinity
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)e1(R
I /t
/teLdt
dI
t
IL
dt
dI
RL circuit (contains two switches that operate suchthat when one is closed, the other is opened.
S1 closed for a length of time sufficient to allow thecurrent to reach its equilibrium value /R.
The circuit is described by the outer loop
S2 closed, S1 opened – the circuit is described byupper loop
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+-
S1
S2
bLaR
The lower loop no longer influences the behavior of
the circuit – a circuit with no battery ( = 0).
Kirchhoff’s loop rule to the upper loop, we obtain
The solution of the differential equation
where = the emf of the batery and Io= /R is the
current at t=0, the instant at which S2 is closed as S1
is opened
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0dt
dILIR
/t
o
/t eIeR
I
No inductor in the circuit - the current would
immediately decrease to zero
With inductor in the circuit – it acts to oppose the
decrease in the current and to maintain the current.
A graph of the current in the circuit versus time
shows that the current is continuously decreasing
with time
The slop dI/dt is always negative and has its
maximum value at t=0.
The negative slope signifies that L= - L(dI/dt) is now
positive; that is, point a in Figure (pg 19) is at a lower
electric potential than point b.
21
Consider two loops: loop 1 and loop 2 (see Figure
above).
A current I1 flowing through loop 1 will produce a
magnetic field at the position of loop 2 equal
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I1
The magnetic flux through loop 2 is equal to
Here, M21 is called the mutual inductance of the two
loops.
It is a purely geometrical quantity that depends on the
sizes, shapes and relative positions of the two loops.
It does not change if we switch the role of loop 1 and
loop 2:
The flux through loop 2 when we run a current I
around loop 1 exactly the same as the flux through
loop 1 when we same current I around loop 2.
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Besides inducing an emf in a nearby loop, the
changing current in loop 1 also induces an emf in
loop 1. The flux through loop 1 generated by the
current in loop 1 is equal to
The constant of proportionality is called the self
inductance. The unit of inductance is Henrie (H).
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45. Two inductors having self-inductances L1 and L2
are connected in parallel as shown in Figure below.
The mutual inductance between the two inductors is
M. Determine the equivalent self-inductance Leq for
the system (Figure below).
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