Download Homework #2

EE4609
Spring 2009
Homework #2
1) Consider an infinitely long coaxial cable. The center conductor has an outside
radius of a (m). The outer conductor has an inside radius of b (m). The area in
between the conductors is filled with a dielectric with relative dielectric constant r.
 = Vo
z
=b
=a
r


A battery is connected between the conductors which places a potential across them.
The inner conductor is at the potential, (=a) = Vo while the outer one is at (=b)
= 0.
Use Laplace’s equation to find the electric field (V/m) between the conductors. In
this case, cylindrical coordinates (, , z) are appropriate. Since there is no charge in
the region between the conductors, the electric potential satifies
1     1  2   2 


 

0
      2  2 z 2
2
Because the line is infinitely long and there is symmetry about the z axis, there
should be no variation of  with either z or . Thus the equation reduces to
 2 
1        



0
        
To solve this, integrate (with respect to ) twice. The resulting expression is only a
function of  and contains two integration constants. Use the potentials (boundary
conditions) at  = b and  = a to find those constants.
[The answer is
   
ln  V0 ln b V0

] <---------------------------- corrected
ln a
ln a
b
b
 
 
Now use the relationship
E    
to derive an expression for the electric field in the region between the conductors.
2) Now surround the inner conductor with an imaginary cylindrical surface of
length 1 m and radius  = a+ (i.e., just outside the inner conductor
z
Surface, S,
enclosing inner
conductor
=a
r


Use Gauss’ Law to find the total charge, Q, that exists on a 1 m long section of the
inner conductor
1 2
Q

0 0
D  dS
where dS  ad dzˆ
Then use this result to derive an expression for the capacitance per meter for the
coaxial cable.
3) Now assume a current, I A is flowing down the center conductor. The resulting
magnetic field in the region between the conductors is
I
ˆ A/m
2
Consider a surface of unit length (1 m) in between the conductors as shown below.
H 
1m
b
a
z
Find the total magnetic flux passing through that surface
1 b


0 a
B  dS
where dS  d dzˆ
Then use that flux to derive an expression for the inductance per unit length of the
coax cable (assume r = 1 for the dielectic).
4) Using the expressions for the per unit length coaxial cable capacitance and
inductance, determine;
(a) the cable’s characteristic impedance, Z0 in Ohms
and
(b) the cable’s signal velocity in m/s.
[ Zo 
L
C
v
1
LC
]
5) Let there be a bond wire made of AWG 24 round wire that connects two square
bond pads. The pads and bond wire are located above a large ground plane.
Assume the wire runs parallel to the ground plane. Use for the wire length, the
distance in between the pads. See the figure below.
0.100 in.
AWG 24
round wire
0.005 in.
0.100 in.
square
bond pads
The pads and wire are located 0.005 in. above the large ground plane. A dielectric
slab (not shown) with r = 4.3 separates the pads and wire from the ground plane
(i.e., as on a printed circuit board).
Determine a lumped element equivalent circuit including actual element values (e.g.,
nH, pF, etc.) for this structure. Sketch the equivalent circuit and label the element
values. Ignore any resistance.