chapter 32 inductance

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.

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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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