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Electromagnetics

ENGR 367Inductance

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Introduction

Question: What physical parametersdetermine how much inductance aconductor or component will have in a

circuit?

Answer: It all depends on current and fluxlinkages!

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Flux Linkage

Definition:

the magnetic flux generated by a current thatpasses through one or more conducting loops

of its own or another separate circuit

Mathematical Expression:

If the total flux generated by N turns

and # of turns through which passes

then flux linkage (assuming none escapes)

m

m

m

N

N

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Types of Inductance

Self-Inductance (L):

whenever the flux linkage of aconductor or circuit couples with itself

Mutual Inductance (M):

if the flux linkage of a conductor orcircuit couples with another separate one

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Self-Inductance

Formula by Definition

Applies to linear magnetic materials only

Units:

flux linkage

current through each turn

mN

LI

2[Henry] [H] [Wb/A] [T m /A]L

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Inductance of Coaxial Cable

Magnetic Flux

Inductance

(as commonly used in transmission line theory)

0

( ) ( )2

ln( / )2 2

m

S S

b d

a

IB dS d dz

I Idd dz b a

ln( / ) [H]

2

or ln( / ) [H/m]2

md

L b a

IL

b ad

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Inductance of Toroid

Magnetic FluxDensity

Magnetic Flux

If core small

vs. toroid

2[T] [Wb/m ]2

NIB

m

S

B dS

2

0

0

(if )2

where S cross section area of the toroid core

mB S

NISS

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Inductance of Toroid

Inductance

Result assumes that no flux escapes throughgaps in the windings (actual L may be less)

In practice, empirical formulas are often used to

adjust the basic formula for factors such aswinding (density) and pitch (angle) of thewiring around the core

2

0

[H]2

mN N SLI

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Alternative Approaches

Self-inductance in terms of

Energy

Vector magnetic potential (A)

Estimate by Curvilinear Square Field Map method

2

2

21

2

H

H

WW LI L

I

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Inductance of aLong Straight Solenoid

Energy Approach

Inductance

2

. .

2 2 2 2

2 2 0. ( )

2 2

2

1

2 2

where for this solenoid

2 2

where for a circular core2

H

vol vol

d

H

vol S core

H

W B Hdv H dv

NIHd

N I N IW dv dS dz

d d

N I SW S ad

2

2

2H

W N SL

I d

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Internal Inductanceof a Long Straight Wire

Significance: an especially important issuefor HF circuits since

Energy approach (for wire of radius a)

L L LZ X L Z

2

2

. .

22

3

2 4 0 0 0

2 24

2 4

1( )

2 2 2

8

( / 4)(2 )( )

8 16

H

vol vol

a l

IW B Hdv d d dz

a

Id dz

a

I I la l

a

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Internal Inductanceof a Long Straight Wire

Expressing Inductance in terms of energy

Note: this result for a straight piece of wire

implies an important rule of thumb forHF discrete component circuit design:

keep all lead lengths as short as possible

2

2 2

2( )2 16

8

or8

H

I l

W lL

I IL

l

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Example of CalculatingSelf-Inductance

Exercise 1 (D9.12, Hayt & Buck, 7th edition, p. 298)

Find: the self-inductance of

a) a 3.5 m length of coax cable with a = 0.8 mm

and b = 4 mm, filled with a material for which

r = 50.

0

7

ln( / ) ln( / )

2 2(50)(4 10 H/m)(3.5m) 4

ln( )2 0.8

56.3 H

rdd

L b a b a

L

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Exercise 1 (continued)


b) a toroidal coil of 500 turns, wound on a

fiberglass form having a 2.5 x 2.5 cm squarecross section and an inner radius of 2.0 cm

2 7 2 2

0

(1)(4 10 H/m)(500) (0.025m)

2 2 (0.020 m 0.0125 m)

0.96 mH

N S

L

L

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c) a solenoid having a length of 50 cm and 500 turns

about a cylindrical core of 2.0 cm radius in which r =50 for 0 < < 0.5 cm and r = 1 for 0.5 < < 2.0 cm

2 2 22 2

0

2 6 2

2 2 3 2

6 3

( ) (50 )

where (.005 m) 78.5 10 m

and [(.020 m) (.005 m) ] 1.18 10 m

(4 10[(50)(78.5 10 ) 1.18 10 ]

i i o o

i i o o i o

i

o

N S N S NN S NL S S S S

d d d d d

S

S

L

7 2)(500)3.2 mH

0.50

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Example of Estimating Inductance:Structure with Irregular Geometry

Exercise 2:Approximate the inductance per unit length ofthe irregular coax by the curvilinear square method

0

5.5

4(6)

(0.23)(400 nH/m)

(if filled with air or

non-magnetic material)

290 nH/m

S

P

NL

l N

L

l

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Mutual Inductance

Significant when current in one conductorproduces a flux that links through the pathof a 2nd separate one and vice versa

Defined in terms of magnetic flux (m)

2 12

121

12 1 2

2

mutual inductance between circuits 1 and 2

where the flux produced by I that links the path of I

and N the # of turns in circuit 2

NM

I

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Mutual InductanceExpressed in terms of energy

Thus, mutual inductances betweenconductors are reciprocal

12 1 2 0 1 2

1 2 1 2. .

12 21

1 1

and [H/m]

vol vol

M B H dv H H dv

I I I I

M M

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Example of CalculatingMutual Inductance

Exercise 3 (D9.12, Hayt & Buck, 7/e, p. 298)

Given: 2 coaxial solenoids, each l= 50 cm long

1st: dia. D1= 2 cm, N1=1500 turns, core r=75

2nd: dia. D2=3 cm, N2=1200 turns, outside 1st

Find: a) L1=? for the inner solenoid2 22

0 1 11 1 1

1

7 2 2

1

4

(75)(4 10 H/m)(1500) (.02m)

4(.50m)

.133 H = 133 mH

rN DN S

L l l

L

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Find: b) L2 = ? for the outer solenoid

Note: this solenoid has inner core and outer air

filled regions as in Exercise 1 part c), soit may be treated the same way!

2 0.087 H 87 mHL

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Find: M = ? between the two solenoids

1

2 12 2 1 112 12

1

7 2

1 2

using since core 1 is smaller of the two

(75)(4 10 )(1200)(1500) (.02)

4(.50)107 mH

( geometric mean of the self-inductance

of each

S

N N N S

M M MI l

M

M

L L

individual solenoid)

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SummaryInductance results from magnetic flux (m)

generated by electric current in a conductor Self-inductance (L) occurs if it links with itself

Mutual inductance (M) occurs if it links withanother separate conductor

The amount of inductance depends on How much magnetic flux links

How many loops the flux passes through

The amount of current that generated the flux

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Summary Inductance formulas may be derived from

Direct application of the definition

Energy approach

Vector Potential Method

The self-inductance of some common structureswith sufficient symmetry have an analytical result

Coaxial cable Long straight solenoid

Toroid

Internal Inductance of a long straight wire

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SummaryNumerical inductance may be evaluated by

Calculation by an analytical formula if sufficientinformation is known about electric current,

dimensions and permeability of materialApproximation based on a curvilinear square

method if axial symmetry exists (uniform cross

section) and a magnetic field map is drawn

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ReferencesHayt & Buck, Engineering Electromagnetics,

7/e, McGraw Hill: Bangkok, 2006.

Kraus & Fleisch, Electromagnetics withApplications, 5/e, McGraw Hill: Bangkok,1999.

Wentworth, Fundamentals ofElectromagnetics with Engineering

Applications, John Wiley & Sons, 2005.

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