11/21/2004
Example The B-Field of a Coaxial Transmission Line
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Example: The B-Field of
Coaxial Transmission Line
Consider now a coaxial cable, with inner radius a :
I0
I0
aˆz
The outer surface of the
inner conductor has radius a,
the inner surface of the
outer conductor has radius
b, and the outer radius of
the outer conductor has
radius c.
Jim Stiles
The Univ. of Kansas
b
a
c
Dept. of EECS
11/21/2004
Example The B-Field of a Coaxial Transmission Line
2/6
Typically, the current flowing on the inner conductor is equal
but opposite that flowing in the outer conductor. Thus, if
current I0 is flowing in the inner conductor in the direction aˆz ,
then current I0 will be flowing in the outer conductor in the
opposite (i.e., −aˆz ) direction.
Q: Hey! If there is current, a magnetic flux density must
be created. What is the vector field B ( r ) ?
A: We’ve already determined this (sort of) !
Recall we found the magnetic flux density produced by a hollow
cylinder—we can use this to determine the magnetic flux
density in a coaxial transmission line.
A coaxial cable can be viewed as two hollow cylinders!
Q: I find it necessary to
point out that you are
indeed wrong—the inner
conductor is not hollow!
A: Mathematically, we can view
the inner conductor as a hollow
cylinder with an outer radius a
and an inner radius of zero!
Jim Stiles
The Univ. of Kansas
Dept. of EECS
11/21/2004
Example The B-Field of a Coaxial Transmission Line
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Thus, we can use the results of the previous handout to
conclude that the magnetic flux density produced by the
current flowing in the inner conductor is:
Binner
⎧ I 0 µ0
⎪
⎪ 2πρ
⎪
(r ) = ⎨
⎪I µ
⎪ 0 0
⎪⎩ 2πρ
I 0 µ0
⎛ ρ 2 − 02 ⎞
ˆ
a
ρ aˆφ
=
⎜ 2
2 ⎟ φ
2
a
π
a
0
2
−
⎝
⎠
aˆφ
ρ <a
ρ >a
⎡Webers ⎤
⎢⎣ m 2 ⎥⎦
Likewise, we can use the same result to determine the magnetic
flux density of the current flowing in the outer conductor:
Bouter
⎧
⎪
0
⎪
⎪
⎪
2
2
⎪ −I 0 µ0 ⎛ ρ − b ⎞ ˆ
a
(r ) = ⎨
⎜ 2
2 ⎟ φ
c
b
2
πρ
−
⎝
⎠
⎪
⎪
⎪
−I 0 µ0
⎪
aˆφ
⎪
2πρ
⎩
ρ <b
b < ρ <c
⎡Webers ⎤
⎢⎣ m 2 ⎥⎦
ρ >c
Note the minus sign is due to direction of the current ( −aˆz ) in
the outer conductor.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
11/21/2004
Example The B-Field of a Coaxial Transmission Line
4/6
We can now apply superposition to determine the total magnetic
flux density in a coaxial transmission line! Specifically:
J ( r ) = Jinner ( r ) + Jouter ( r )
if
then
B ( r ) = Binner ( r ) + Bouter ( r )
Note due to the piecewise nature of these solutions, we must
evaluate this sum for 4 distinct regions:
1) ρ < a (in the inner conductor)
2) a < ρ < b (in the region between the conductors)
3) b < ρ < c (in the outer conductor)
4) ρ > c (outside the coaxial cable)
ρ <a
B ( r ) = Binner ( r ) + Bouter ( r )
I 0 µ0
ρ aˆφ + 0
2π a 2
I µ
= 0 20 ρ aˆφ
2π a
=
Jim Stiles
The Univ. of Kansas
Dept. of EECS
11/21/2004
Example The B-Field of a Coaxial Transmission Line
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a < ρ <b
B ( r ) = Binner ( r ) + Bouter ( r )
I 0 µ0
2π ρ
I µ
= 0 0
2π ρ
=
aˆφ + 0
aˆφ
b < ρ <c
B ( r ) = Binner ( r ) + Bouter ( r )
I 0 µ0
−I 0 µ0 ⎛ ρ 2 − b 2 ⎞
aˆφ +
aˆ
=
⎜ 2
2 ⎟ φ
2π ρ
2πρ ⎝ c − b ⎠
I µ
= 0 0
2π ρ
⎛ c 2 − ρ2 ⎞
aˆ
⎜ 2
2 ⎟ φ
⎝c −b ⎠
ρ >c
B ( r ) = Binner ( r ) + Bouter ( r )
=
I 0 µ0
−I µ
aˆφ + 0 0 aˆφ
2π ρ
2πρ
=0
Jim Stiles
The Univ. of Kansas
Dept. of EECS
11/21/2004
Example The B-Field of a Coaxial Transmission Line
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Summarizing, we find the total magnetic flux density to be:
⎧ I 0 µ0
⎪ 2π a 2
⎪
⎪
⎪I µ
⎪ 0 0
⎪ 2π ρ
⎪
B (r ) = ⎨
⎪
⎪ I 0 µ0
⎪ 2π ρ
⎪
⎪
⎪0
⎪⎩
ρ aˆφ
ρ <a
aˆφ
a < ρ <b
⎛ c 2 − ρ2 ⎞
aˆ
⎜ 2
2 ⎟ φ
⎝c −b ⎠
⎡Webers ⎤
⎢⎣ m 2 ⎥⎦
b < ρ <c
ρ >c
B( ρ )
a
b
c
ρ
The magnetic flux density and the electric field outside of a
coaxial transmission line are zero!
Jim Stiles
The Univ. of Kansas
Dept. of EECS