ENGR367 (Inductance)
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Transcript of ENGR367 (Inductance)
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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
m N
L I
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
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
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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
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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
m N N S L I
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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
W W 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
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
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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 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
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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 l L
I I L
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
r d d
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 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
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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
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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
N M
I
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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
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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
r N D N 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 mH L
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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 M I l
M
M
L L
individual solenoid)
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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
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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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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
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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.