- •Advanced chapters of theoretical electro-engineering.
- •Lecture 7
- •Diffusion of electromagnetic fields in conducting media.
- •Diffusion of electromagnetic fields.
- •Diffusion of electromagnetic fields.
- •Periodic electromagnetic field in the conductors.
- •Periodic electromagnetic field in the conductors.
- •Penetration of the electromagnetic field into a conductor
- •The skin effect.
- •Poynting 's Theorem.
- •Electromagnetic Field Energy.
- •The rate of decrease of the electromagnetic field energy in a closed volume.
- •The rate of decrease of the electromagnetic field energy in a closed volume.
- •Poynting 's Theorem.
- •The Poynting vector.
- •Energy flows in the electromagnetic field
- •Transmission of energy along the wires with current.
- •Transmission of energy in a DC line.
- •Transmission of energy in a DC line.
- •Transmission of energy in a DC line.
- •The field picture near the wires with current.
- •Energy flows in static fields.
- •The momentum of the electromagnetic field.
- •The momentum of the electromagnetic field.
- •The momentum of the electromagnetic field.
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: |
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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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Mathematical |
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transformation: |
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E E E |
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Diffusion of electromagnetic fields.
Diffusion equations for the electromagnetic field characteristics:
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E |
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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)
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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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E |
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d Hm |
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j E |
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Periodic electromagnetic field in the conductors.
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Equation: |
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dz2 |
j Hm 0 |
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Solution for the complex field intensity: |
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Hm A1e |
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or |
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1 j |
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Parameter α: |
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Using designation: |
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k |
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kz |
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jkz |
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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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2 |
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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