Advanced chapters of theoretical electro-engineering.
SPbTU, IE, Prof. A.G. Kalimov 2022
1
Lecture 7
Time dependent electromagnetic fields
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Diffusion of electromagnetic fields in conducting media.
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Diffusion of electromagnetic fields.
Consider electromagnetic field in a conductor.
The displacement currents are neglected. No free charges.
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B |
divB 0 |
B H |
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curlE |
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Basic equations: |
t |
divJ 0 |
J E |
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curlH J |
E |
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divE 0 |
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Differential equations for the electric field intensity:
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E |
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H |
J |
E |
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t |
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t |
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Mathematical |
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E |
transformation: |
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E E E |
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4
Diffusion of electromagnetic fields.
Diffusion equations for the electromagnetic field characteristics:
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E |
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J |
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B |
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H |
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A |
E |
t |
J |
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t |
B |
t |
H |
t |
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A |
t |
Such equations (applied to scalar variables) describe processes of the particle diffusion, thermal processes.
One-dimensional equations: (here we assume that only x- component of the E and y- component of H exist)
2 E |
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E |
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z2 |
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2 H y |
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H y |
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t |
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5
Periodic electromagnetic field in the conductors.
Quasi-stationary approach:
Ex Em sin( t E ); H y Hm sin( t H )
Applying the complex method:
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j E |
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j H |
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Em Eme |
Hm Hme |
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Differential equations in complex form:
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d |
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E |
m |
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d Hm |
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dHm |
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j E |
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dz2 |
j Hm |
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m |
Em |
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6
Periodic electromagnetic field in the conductors.
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d |
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Equation: |
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Hm |
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dz2 |
j Hm 0 |
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Solution for the complex field intensity: |
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z |
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Hm A1e |
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A2e |
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or |
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Hm A1e |
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1 j |
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j |
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j |
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j |
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Parameter α: |
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1 j |
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j |
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e |
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e |
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Using designation: |
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k |
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kz |
e |
jkz |
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2 |
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Hm |
Hm0e |
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Solution for the field intensity: |
H Hm0e kz sin( t H0 |
kz) |
7
Penetration of the electromagnetic field into a conductor
1 |
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- the penetration length |
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k |
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Amplitude of the electromagnetic wave dumps according to exponential dependence:
H Hm0e kz sin( t H0 kz 4 )
8
The skin effect.
Penetration of the electromagnetic wave into the conducting plate
Material - copper
Width = 5 cm
Frequency= 200 Hz
Penetration depth = width of the skin layer
2
9
Poynting 's Theorem.
10