ECE 144A
Electromagnetic Fields and Waves
Bob York
Why is 50- Coaxial Line so Special Anyway?
Field Analysis of Coax:
The coaxial line segment shown above is ¯lled with a dielectric ², and is assumed to be driven by a potential
di®erence V between the inner and outer conductors, which induces a charge §Q on the surface of each
conductor. The charge will be distributed uniformly along the length ¢z of the coax.
b
a
r
∆z
The electric ¯eld will be radial Zby
From Gauss' law
Z symmetry.ZZZ
° ²E ¢ dS =
½ dV
) Er =
Which gives a voltage
V =¡
Z
a
b
Q
2¼²r¢z
Q
ln(b=a)
2¼²¢z
E ¢ d` =
(1)
(2)
The capacitance per unit length is given by
C=
From Ampµere's law,
I
Q
2¼²
=
V ¢z
ln(b=a)
H ¢ d` = I0
)
[F/m]
HÁ =
I0
2¼r
The inductance per unit length is de¯ned by
ZZ
1
¹
L=
B ¢ dS =
ln(b=a) [H/m]
I¢z
2¼
The characteristic impedance is therefore
r
L
´
Z0 =
=
ln(b=a)
C
2¼
(3)
(4)
(5)
(6)
Power Handling Capacity:
Dielectric breakdown will occur in the region between the two conductors if the electric ¯eld exceeds a certain
critical value. The ¯eld strength is a function of the applied voltage and line geometry. Using (1) and (2)
we can express the electric ¯eld as
V
Er =
(7)
r ln(b=a)
This shows that the ¯eld is largest near the center conductor, so
V
E max =
(8)
a ln(b=a)
The peak power transmitted down the line is then given by
2¼ 2 2
P = V 2 =Z0 =
a E max ln(b=a)
(9)
´
and thus the maximum power °ow is in°uenced by the line geometry. To ¯nd the optimum conductor sizes,
we can look for the value of a which maximizes (9)
¢
@P
@ ¡ 2
/
a ln b ¡ a2 ln a = a [2 ln(b=a) ¡ 1] = 0
(10)
@a
@a
This equation is satis¯ed when b=a = 1:65, which gives an optimum characteristic impedance of Z0 = 30 for maximum power transmission in a coaxial air-line.
Attenuation:
From the distributed circuit model for a transmission-line, we found that the attenuation constant (for
low-loss lines) is
R
GZ0
®¼
+
(11)
2Z 0
2
where R is the series resistance per unit length, and G is the shunt conductance per unit length. Physically,
where does this loss come from? The series resistance R comes from Ohmic losses in the metal conductors.
Using a sheet resistivity of Rs, the then total resistance per unit length is just
µ
¶
Rs 1
1
R=
+
(12)
2¼ b
a
The shunt conductance comes from loss in the dielectric material. If the dielectric has a small conductivity
¾, then a small current can °ow radially through the material according to Jr = ¾E r. The total conduction
current through the dielectric is then
Id = 2¼r¢zJr = 2¼r¢z¾E r
(13)
Using (7), the conductance G is expressed as
I
2¼¾
G=
=
(14)
V ¢z
ln(b=a)
Substituting (12) and (14) into (11), we can ¯nd the optimum line dimensions for lowest attenuation,
@®
@ (1=b ¡ 1=a)
=0 /
@a
@a ln(b=a)
(15)
0 = 1 + a=b ¡ ln(b=a)
This equation is satis¯ed for b=a = 3:6, which gives an optimum characteristic impedance of Z0 = 77 - for
lowest attenuation in a coaxial air-line.
The best compromise:
The expressions for attenuation and power handling are plotted below as a function of characteristic
impedance for a coaxial air-line. An impedance of around 50 - gives the best overall performance.
Normalized Values
2.0
Attenuation
minimum at 77 Ω
1.5
1.0
Power handling
maximum at 30 Ω
0.5
0.0
10
100
Characteristic Impedance