ENGR367 (Inductance)

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Electromagnetics

ENGR 367Inductance



Introduction

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

circuit?

► Answer: It all depends on current and fluxlinkages!



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



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



Self-Inductance

►Formula by Definition

Applies to linear magnetic materials only

Units:

flux linkage

current through each turn

m N

L I

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



Inductance of Coaxial Cable

►Magnetic Flux

► Inductance

(as commonly used in transmission line theory)

0

ˆ ˆ( ) ( )2

ln( / )2 2

m

S S

b d

a

I B dS d dz

I Id d dz b a

ln( / ) [H]

2

or ln( / ) [H/m]2

m d

L b a

I L

b ad



Inductance of Toroid

►Magnetic FluxDensity

►Magnetic Flux

► If core small

vs. toroid

2 [T] [Wb/m ]2

NI B

m

S

B dS

2

0

0

(if )2

where S cross section area of the toroid core

m B S

NIS S



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

m N N S L I



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

W W LI L

I



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

NI H d

N I N I W dv dS dz

d d

N I S W S ad

2

2

2 H

W N S L

I d



Internal Inductanceof a Long Straight Wire

►Significance: an especially important issuefor HF circuits since

►Energy approach (for wire of radius a)

L L L Z 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

I W B Hdv d d dz

a

I d dz

a

I I l a l

a



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 l L

I I L

l



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

r d d

L b a b a

L




► 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




► Exercise 1 (continued)


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 N N S N L S S S S

d d d d d

S

S

L

7 2)(500)

3.2 mH

0.50



Example of Estimating Inductance:Structure with Irregular Geometry

► Exercise 2: Approximate the inductance per unit length of

the 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

N L

l N

L

l



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

N M

I



Mutual Inductance

►Expressed 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



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

r N D N S

L l l

L




►Exercise 3 (continued)

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 mH L




►Exercise 3 (continued)

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 M I l

M

M

L L

individual solenoid)



Summary

►Inductance 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



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



Summary

►Numerical inductance may be evaluated by

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

dimensions and permeability of material Approximation based on a curvilinear square

method if axial symmetry exists (uniform cross

section) and a magnetic field map is drawn



References

►Hayt & Buck, Engineering Electromagnetics,7/e, McGraw Hill: Bangkok, 2006.

►Kraus & Fleisch, Electromagnetics with Applications, 5/e, McGraw Hill: Bangkok,1999.

►Wentworth, Fundamentals ofElectromagnetics with Engineering

Applications, John Wiley & Sons, 2005.

ENGR367 (Inductance)

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