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Electromagnetic Induction & Inductors

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Revision of Electromagnetic Induction and Inductors

(Much of this material has come from Electrical & Electronic Principles & Technology by John Bird)

Magnetic Field

A permanent magnet is a piece of ferromagnetic material and if freely suspended will align in

a north and south direction. The north seeking end being called the north pole and the south the

south pole. The area around the magnet is called the magnetic field and its existence can easily

be demonstrated using iron filings and paper. See below.

Magnetic Flux and Flux Density

Magnetic flux is the amount of magnetic field (or number of lines of force) produced by a

magnet. The magnetic flux is given the symbol ϕ and its unit is the weber (Wb).

If the magnet has a cross-sectional area A, then it is possible to define a flux density (B), where

B =ϕ

A. The unit of magnetic flux density is the tesla (T) and 1T = 1 Wb/m2.

Electromagnets

These usually consist of an iron core with a current carrying coil wound around it and have a

similar flux as a permanent magnet. It was found by Faraday that the flux in an electromagnet

is directly proportional to the current (I), the number of turns (N) and the length of the magnetic

path (l), This means that for a given electromagnet the value of IN

l is constant and is called its

magnetising force and denoted H. Therefore, we can write

Magnetising force = H =IN

l (Ampere turns/metre)

The length of the magnetic circuit is difficult to establish because the flux will flow through

the surrounding air as well as the core. However, if the coil is wound on a toroid as shown

below, the flux will be entirely within the core and the length of the circumference.

Now if the coil is wound on a core which has no magnetic properties we find that the ratio B

H is

12.5 x 10-7. It is given the symbol µo and called the absolute permeability of free space.

If we use a magnetic material for the core, often iron, the ratio is dramatically changed and a

relative permeability µr is used to correct the values.

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B

H= μoμr

Where μoμr = μ and is called absolute permeability

Introduction to Electromagnetic Induction

Electromagnetic induction is the production of a potential difference across a conductor when

it is exposed to a varying magnetic field. It is generally considered a discovery attributed to

Faraday.

When a conductor is moved across a magnetic field such that it cuts the lines of force (flux),

an e.m.f. is produced in the conductor. If the conductor forms part of a closed circuit, an electric

current will flow. This is known as electromotive induction.

This can be demonstrated with a magnet and a coil connected to a galvanometer (sensitive

ammeter) as shown below.

An increase in the speed of movement, a stronger magnet, or an increase in the number of turns,

all increase the amount of current flow.

Faraday’s Laws

1) An induced e.m.f. is set up whenever a magnetic field linking the circuit changes.

2) The magnitude of the induced e.m.f. is proportional to the rate of change of magnetic

flux linking the circuit.

Lenz’s Law

The direction of an induced e.m.f. is always such that it tends to set up a current opposing the

motion or the change of flux responsible for inducing that e.m.f.

An alternative method to Lenz’s law for determining relative directions is given using

Fleming’s Right Hand Rule. See below.

Magnet moved at constant speed will

produce a reading as shown.

When magnet moved in other direction,

direction of current is reversed

Alternatively, could hold magnet stationary

and move the coli.

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In a generator, conductors forming an electric circuit are made to move through a magnetic

field. By Faraday’s law an e.m.f. is induced in the conductor and thus a source of e.m.f. is

produced.

In the diagram below and e.m.f. will be induced by the movement of the conductor between

the magnets.

Induced e.m.f is given by E = Blν volts

Where B = flux density in teslas

l = length of conductor in the magnet in metres

ν = velocity of conductor in m/s

If the conductor moves at an angle θ to magmetic field instead of 90o as shown above, then

E = Blν sin θ volts.

Consider now a rotating loop in a magnetic field as shown below.

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In the diagram, the direction of current flow has come from Fleming RH rule and the total e.m.f

is now E = 2NBlν sin θ volts.

Inductance

This is the name given to the property of a circuit whereby there is an e.m.f induced into the

circuit by a change in flux linking produced by a current change.

When the e.m.f. is induced in the same circuit as that in which the current is changing, the

property is called self-inductance and is denoted L. When the e.m.f. is induced in a circuit by

a change of flux due to current changing in an adjacent circuit, the property is called mutual

inductance and denoted M. The unit of inductance is the henry (H)

A circuit has an inductance of one henry when an e.m.f. of one volt is induced in it by a current

changing at a rate of one ampere per second.

Therefore, the induced e.m.f in a coil of N turns is E = −Ndϕ

dt volts, where dϕ is the change in

flux in Webers and dt is the time taken for the flux to change is seconds (dϕ

dt is rate of change

of flux).

Alternatively, the induced e.m.f in a coil of inductance L henry’s is E = −LdI

dt, where dI is the

change in current in amperes and dt is the time taken for the current to change in seconds (dI

dt is

rate of change of current).

The minus sign in each equation reminds us of its direction (given by Lenz’s Law)

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Inductors

A component called an inductor is used in an electrical circuit when the property of inductance

is required. Any plain coil of wire is an inductor, although more often than not the coil will be

wound on a magnetic core.

Factors affecting the inductance of an inductor include:

The number of turns of wire – the more, the higher the inductance

The coils cross-sectional area – the greater, the higher the inductance

The presence of a magnetic core – this increases the concentration of the magnetic field

and hence the inductance is increases

The way the turns are arranged – a short, thick wire will have higher inductance that a

long, thin one.

Energy Stored in and Inductor

Inductors have the ability to store energy (W) in the magnetic field and this is given by

W =1

2LI2 joules

Where L is inductance measures in henry’s (H) and I is the current in amperes

Inductance of a Coil

The proofs are not given here, but the inductance of a coil may be calculated a number of ways,

and the relevant equations are given below.

L =Nϕ

I where N is the number of turns, ϕ is the flux linkage and I is the current.

We also have

L =N2

S where N is the number of turns and S is a property of a coil called reluctance and is

given by the equation:

S =l

µoµrA where µo is absolute permeability of free space and has a value of 12.5 x 10-7 H/m,

µr is the relative permeability, and l is the magnets length in m.

Worked Example 1

A flux of 25mWb links with a 1500 turn coil when a current of 3A passes through the coil.

Calculate (a) the inductance of the coil, (b) the energy stored in the magnetic field and (c) the

average e.m.f. induced if the current falls to zero in 150ms.

(a) Inductance L =Nϕ

I=

1500×25×10−3

3= 12.5H

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(b) Energy stored W =1

2LI2 =

12.5×32

2= 56.25J

(c) Induced e.m.f. E = −LdI

dt= −12.5 (

3−0

150×10−3) = −250v

Or we could use E = −Ndϕ

dt= −1500 (

25×10−3

150×10−3) = 250v

Worked Example 2

A 750 turn coil of inductance 3H carries a current of 2A. Determine the flux linking the coil

and the e.m.f. induced in the coil then the current collapses to zero in 20ms.

We have that L =Nϕ

I, rearranging we obtain ϕ =

LI

N=

3×2

750= 8 × 10−3Wb = 8mWb

Also E = −LdI

dt= −3 (

2−0

20×10−3) = −300v

We could also have used E = −Ndϕ

dt

Worked Example 3

A silicone ring is wound with 800 turns, the ring having a mean diameter of 120mm and a

cross-sectional area of 400mm2. If when carrying a current of 0.5A the relative permeability is

found to be 3000, determine (a) the self-inductance of the coil, (b) the induced e.m.f. if the

current is reduced to zero in 80ms.

(a) Inductance L =N2

S where S =

l

µoµrA=

π×120×10−3

12.5×10−7×3000×400×10−6 = 2.51 × 105A/Wb

Therefore, the self-inductance is L =N2

S=

8002

2.51×105 = 2.55H

(b) Induced e.m.f. E = −LdI

dt= −2.55 (

0.5−0

80×10−3 = 15.93v)

Quick Recap on Relevant Equations

Equation Units

E = −Ndϕ

dt

Volts (v)

E = −LdI

dt

Volts (v)

W =1

2LI2

Joules (J)

L =Nϕ

I

Henry (H)

L =N2

S

Henry (H)

S =l

μoμrA

1

Henry (H−1or A/Wb)

𝜇𝑜 = 12.5 × 10−7 Henry/metre (H/m)

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Tutorials Problems

1) A flux of 30mWb links with a 1200 turn coil when a current of 5A is passed through

the coil. Determine (a) the inductance of the coil, (b) the energy stored in the magnetic

field, and (c) the average e.m.f. induced is the current is reduced to zero in 0.20s.

(7.2H, 90J, 180v)

2) A coil of 2500 turns has a flux of 10mWb linking with it when carrying a current of

2A. Calculate the coil inductance and e.m.f. induced in the coil when the current

collapses to zero in 20ms.

(12.5H, 1.25kv)

3) An iron ring has a cross-sectional area of 500mm2 and a mean length of 300mm. It is

wound with 100 turns and its relative permeability is 1600. Calculate (a) the current

required to set up a flux of 500µWb in the coil, (b) the inductance of the system and (c)

the induced e.m.f. if the field collapses in 1ms.

(1.492A, 33.5mH, -50v)

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Current Growth and Decay in an L-R Circuit

Current Growth

A typical L-R circuit is shown below. When an inductor and resistor are connected in this way

we get a similar charging and discharging effect as with capacitors. The mathematics of this is

not considered here, however below the circuit are shown the plots of VL, VR, and I against

time during charging, together with the relevant equations.

Worked Example

The winding of an electromagnet has an inductance of 3H and a resistance of 15Ω. When it is

connected to a 120v d.c. supply, calculate (a) the steady-state value of current flowing in the

winding, (b) the time constant of the circuit, (c) the value of the induced e.m.f. after 0.1s, (d)

the time taken for the current to rise to 85% of its final value and (e) the value of the current

after 0.3s.

(a) Steady-state current I =V

R=

120

15= 8A

(b) Time constant τ =L

R=

3

15= 0.2s

(c) Induced e.m.f after 0.1s VL = Ve−t τ⁄ = 120 × e−0.1 0.2⁄ = 72.78v

(d) Time for current to rise to 85% of final value i = 0.85I and i = I(1 − e−t τ⁄ )

Decay of induced voltage

VL = Ve−t τ⁄

Growth of resistor voltage

VR = V(1 − e−t τ⁄ )

Growth of current flow

i = I(1 − e−t τ⁄ )

Where τ is the time constant and

equal to L

R

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Therefore 0.85I = I(1 − e−t/0.2)

Hence 0.85 = (1 − e−t/0.2)

Therefore e−t/0.2 = 1 − 0.85 = 0.15

or et/0.2 =1

0.15

Hence t

0.2= ln

1

0.15 → t = 0.379s

(e) Current after 0.3s i = I(1 − e−t τ⁄ ) = 8(1 − e−0.3/0.2) = 6.215A

Current Decay

When a series connected L-R circuit is connected to a d.c. supply as shown with S in position

A, a current I =V

R flows after a short time, creating a magnetic field (ϕ α I) associated with the

inductor. When S moves to position B, the current value decreases, causing a decrease in the

strength of the magnetic field. Flux linkages occur, generating voltage VL, equal to Ldi

dt. By

Lenz’s law, this voltage keeps current i flowing in the circuit, its value being limited by R.

Since V = VL + VR, 0 = VL + VR and VL = −VR. In other words, VLandVR are equal in

magnitude but opposite in direction. The current decays exponentially to zero and since VR is

proportional to the current flowing, VR decays exponentially to zero. Since VL = VR, VL also

decays exponentially to zero. The decay curves are similar to those we saw for capacitors and

the relevant equations are given below.

The equations representing the decay transients are:

Voltage decay

VL = VR = Ve−t τ⁄

Current decay

i = Ie−t τ⁄

Tutorial Problems

1) A coil having an inductance of 6H and a resistance of RΩ is connected in series with a

resistor of 10Ω to a 120v d.c. supply. The time constant of the circuit is 300ms. When

steady-state conditions have been reached, the supply is replaced instantaneously by a

short-circuit. Determine (a) the resistance of the coil, (b) the current flowing in the

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circuit one second after the shorting link has been replaced in the circuit, and (c) the

time taken for the current to fall to 10% of its initial value.

(10Ω, 0.214A, 0.691s)

2) The field windings of a 200v d.c. machine has a resistance of 20Ω and an inductance

of 500mH. Calculate (a) the time constant of the field winding, (b) the value of current

flow one time constant after being connected to the supply, and (c) the current flowing

50ms after the supply has been switched on.

(25ms, 6.32A, 8.65A)

3) An inductor has negligible resistance and an inductance of 200mH and is connected in

series with a 1kΩ resistor to a 24v d.c. supply. Determine (a) the time constant of the

circuit and the steady-state value of current flowing in the circuit, (b) the current

flowing in the circuit at a time equal to one time constant, (c) the voltage drop across

the inductor at a time equal to two time constants, and (d) the voltage drop across the

resistor after a time equal to three time constants.

(0.2s, 24mA, 15.17mA, 3.248v, 22.81v)

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