Advanced chapters of theoretical electro-engineering. Lecture 3
SPbTU, IE, Prof. A.G. Kalimov 2022 |
1 |
Static magnetic field.
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Variables and units
Variable |
symbol |
Units |
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Flux density |
B |
Tesla |
[T] |
Field intensity |
H |
Ampere/ meter |
[A/m] |
Magnetic permeability |
μ |
Henry/meter |
[H/m] |
Inductance |
L |
Henry |
[H] |
Flux |
Φ |
Weber |
[Wb] |
Flux linkage |
ψ |
Weber |
[Wb] |
Scalar magnetic potential |
Um |
Ampere |
[A] |
Vector magnetic potential |
A |
T· m (Wb/m) |
[T·m] |
Magnetization |
M |
Ampere/ meter |
[A/m] |
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Main Relations
B H
B 0 H 0 M
M H ( r 1)
r 0
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Magnetic flux density
Definition of the flux density vector
F Q B v
Flux |
B dS |
B d |
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S |
ds |
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There are no magnetic charges in the nature |
B ds 0 |
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S |
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Biot–Savart’s Law
Due to Biot-Savart’s law the flux density induced by the current sources may be expressed as:
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0dV |
J r |
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dB |
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J |
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4 r3 |
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r
For the line current: |
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dS |
dl |
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0i |
dl r |
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dB |
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4 r3 |
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Ampere’s Law
(закон полного тока)
The magnetic field intensity
- magnetic permeability
Integral |
form of |
I H dl |
the |
Ampere’s |
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Law |
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l |
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I is the current crossing the surface limited by the contour
B H
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J curl H |
Differential form |
J H |
or |
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Scalar magnetic potential
Main relations: |
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curl H J |
divB 0 |
B H |
In general case the H – field is not potential:
H dl i
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l |
Nevertheless outside the space with currents: |
curl H 0 |
It is possible to introduce the scalar potential: |
H Um |
Um |
- the unit is – А (Ampere) |
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The cut in the space
I |
H dl 0 |
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l |
H Um
Um
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Laplace equation for the scalar magnetic potential
Basic equations: |
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divB 0 |
B H |
H grad(Um ) |
In general case:
Um 0
For the medium with the constant magnetic permeability:
Um 0 - Laplace equation
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