Download Lecture Note (ppt) - the GMU ECE Department

Fall 2016
ECE 305 Electromagnetic Theory
Chapter 8 Magnetic Forces, Materials, and Devices
Qiliang Li
Dept. of Electrical and Computer Engineering,
George Mason University, Fairfax, VA
1
• 8.1 Introduction
2
• 8.2 Forces due to Magnetic Fields
3
4
• Read example 8.1, 8.2
• Example 8.3
5
• Example 8.4
6
§8.3 Magnetic Torque and Moment
The torque T on the loop is the vector product
of the moment arm r and the force F .
7
8
9
§8.4 A magnetic Dipole
• A bar magnet or a small filamentary current
loop is referred as a magnetic dipole
– Magnetic vector potential
10
§8.4 A magnetic Dipole
Use:
11
FIGURE 8.7 The B lines due to magnetic dipoles: (a) a
small current loop with m = IS, (b) a bar magnet with
m = Qmℓ.
12
§8.5 Magnetization in Materials
• The magnetization M, in amperes per meter, is
the magnetic dipole moment per unit volume
13
• In the materials
FIGURE 8.12 Magnetic dipole moment in a volume
∆ν: (a) before B is applied, (b) after B is applied.
14
• M is defined as
15
§8.6 Classification of Materials
16
§8.7 Magnetic Boundary Conditions
17
§8.8 Inductors and Inductances
• Magnetic flux: Ψ =
𝑩 ∙ 𝑑𝑺
• Flux linkage 𝜆 = 𝑁Ψ
• If the medium is linear, 𝜆 ∝ 𝐼 or 𝜆 = 𝐿𝐼
𝜆
𝐼
• L is a constant – inductance 𝐿 = =
• Magnetic energy 𝑊𝑚 =
• Or 𝐿 =
𝑁Ψ
𝐼
1
𝐿 𝐼2
2
2𝑊𝑚
𝐼2
18
continue
• Consider magnetic interaction between two
circuits: mutual inductance M12 is defined:
𝜆12 𝑁Ψ12
𝑀12 =
=
𝐼2
𝐼2
• The energy is
1
1
2
2
𝑊𝑚 = 𝐿1 𝐼1 + 𝐿2 𝐼2 ± 𝑀12 𝐼1 𝐼2
2
2
• Inductance L=Lin + Lext
• It can be shown that: 𝐿𝑒𝑥𝑡 𝐶 = 𝜇𝜀
* + or – sign depends on I1 and I2 flow such that the B fields of the
two circuits strengthen each other or not.
19
Procedure to find the self-inductance L:
• Choose a suitable coordinate system
• Let the inductor carry current I
• Determine B from Biot-Savart’s law (or from
Ampere’s law if symmetry exits), then
calculate Ψ = 𝑩 ∙ 𝑑𝑺
𝜆
𝐼
• Finally, find L from 𝐿 = =
𝑁Ψ
𝐼
20
§8.9 Magnetic Energy
• In electric field 𝑊𝐸 =
1
2
𝑫 ∙ 𝑬𝑑𝑉 =
1
2
𝜖𝐸 2 𝑑𝑉
• Similarly, magnetic energy in the field of an
1
inductor is 𝑊𝑚 = 𝐿 𝐼2
2
• Or 𝑊𝑚 =
𝑤𝑚 𝑑𝑉,
Where 𝑤𝑚 =
1
𝑩
2
∙𝑯=
1
𝜇𝐻2
2
21
22
• What is the magnetic energy in a circuit?
or
or
magnetic energy
23
Example 8.10 calculate the self-inductance per
unit length of an infinitely long solenoid
Solve:
𝑁𝐼
𝐵 = 𝜇𝐻 = 𝜇𝐼𝑛 = 𝜇
X X X X X X X X
𝑙
(remember K=current/length)
Ψ = 𝐵𝑆 = 𝜇𝐼𝑛S
′
𝜆
𝑙
Linkage per unit length 𝜆 = = 𝑛Ψ = 𝜇𝑛2 𝐼𝑆
So 𝐿′ =
𝜆′
𝐼
= 𝜇𝑛2 𝑆 (H/m)
24
Example 8.11 Determine the self-inductance of a
coaxial cable of inner r=a and outer r=b.
Solve:
From previous example
X
Eq. 7.29
𝜇𝐼𝜌
2
𝑩𝟏 =
𝒂𝝓
1
2
2𝜋𝑎
𝜇𝐼
𝑩𝟐 =
𝒂𝝓
2𝜋𝜌
25
Method 1
26
27
Method 2
This method is more straight forward.
28
Example 8.12: determine the inductance per
unit length of a two-wire transmission line with
separation distance d. Each wire with r=a.
Solve:
Method 1
d
29
Method 2
L = 2(𝐿𝑖𝑛 + 𝐿𝑒𝑥𝑡 )
30
Example 8.13 Two coaxial circular wires of radius
a and b (b>a) are separated by distance h (h>>a,
b)as shown in Fig. 8.13. Find the mutual
inductance between the wires
Solve:
b
P
h
a
31
32
• Section 8.10, 8.11 and 8.12: self-reading
• Example of application is levitating trains
33
§8.12 Application Note – Hall Effect
What is Hall Effect?
Which way should
the charges go?
34