Flux linkage of a thin current layer
Consider a thin cylinder with the radius of r and the thickness of dr
The current inside this layer is
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2 r dr |
2irdr |
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R2 |
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dr R |
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The flux linkage per unit length:
R |
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i r |
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int (r) B(r) dr |
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dr |
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2 |
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2 R |
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r |
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(r) i |
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After integration: |
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int |
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4 |
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R |
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Flux linkage of a thin current layer
The flux linkage of a thin layer:
d di |
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di i |
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r |
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1 |
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int |
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2 |
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int |
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4 i |
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R |
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Total internal flux linkage:
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R |
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R |
r |
2 |
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2ir dr |
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d |
1 |
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int |
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int |
4 |
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R |
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R |
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0 |
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dr |
r |
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B |
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i |
R |
r |
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i |
R |
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i |
R |
r3dr |
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1 |
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rdr |
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rdr |
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or: |
int |
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2 R |
2 |
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R |
2 |
2 R |
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2 R |
4 |
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0 |
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0 |
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22
Internal inductance of a thin conductor
Transformation of the last integral:
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i |
R rdr |
i |
R r3dr |
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i |
1 r2 |
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1 |
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r 4 |
R |
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2 |
4 |
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2 |
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2 |
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int |
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2 R 0 |
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2 R 0 |
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2 R 2 |
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4R |
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0 |
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Taking into account the limits: |
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i |
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1 |
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R2 |
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1 |
R4 |
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2 |
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int |
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2 R |
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2 |
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4R |
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Final expression: |
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int |
i |
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8 |
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Internal inductance: |
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int |
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Lint |
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L |
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i |
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int |
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23 |
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External inductance of two-wire transmission line
Flux density induced be a wire:
B(r) i 0
2 r
External flux:
1ext |
i 0 |
d dr |
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i 0 |
ln |
d |
R – radius of the wire; R << d |
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2 |
R r |
2 |
R |
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Similar flux is induced by the second wire.
The total external flux: |
ext i 0 |
ln |
d |
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R |
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24
Inductance of a two-wire transmission line
Flux linkage:
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1 |
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d |
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r |
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ext 2 int i 0 |
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ln |
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R |
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4 |
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1 |
ln |
d |
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r |
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Inductance: |
L 0 |
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R |
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4 |
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Internal inductance: |
Lint |
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4 |
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d |
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External inductance: |
L |
0 |
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ln |
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ext |
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R |
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