engr367 (induc
TRANSCRIPT
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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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Example of CalculatingSelf-Inductance
Exercise 1 (continued)
Find: the self-inductance of
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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Example of CalculatingSelf-Inductance
Exercise 1 (continued)
Find: the self-inductance of
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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Example of CalculatingMutual Inductance
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 mHL
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Example of CalculatingMutual Inductance
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 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.