University of California, Berkeley
EECS 117
Spring 2004
Prof. A. Niknejad
Final Exam(closed book)
Friday, May 21, 2004
Guidelines: Closed book and notes. You may use a calculator. Do not unstaple the exam.
Warning: Illustrations not to scale.
1. (18 points) The initially charged transmission line shown below is suddenly
disconnected from the source at t = 0. The line has a characteristic impedance of
Z1 = 50Ω and Rs = 25Ω. The line has a propagation time delay of td .
t = 0+
+
V0
−
Rs
Z 1 td
`
(a) Draw the bounce diagram for the circuit for 0 < t < 4tp .
(b) Sketch the voltage waveform at the output.
(c) Sketch the voltage waveform at the input across Rs .
2
2. (15 points) A top secret government laboratory wishes to isolate its mainframe
computer from potential outside eavesdropping. Specify the thickness of walls shown
below to prevent outside detection if the minimum detectable signal power is
10 pW/m2 . The walls are made of a conductive material with resistivity
ρ = 5 × 10−2 Ω-m. The measured electric field strength on the inner surface of the
wall is 10 mV/m at a frequency of f = 1 GHz. You may assume that the computer
radiates its power isotropically (equally in all directions) and the incoming radiation
onto the surface of the walls is a TEM mode plane wave.
t
(a) Calculate the amount of reflected and transmitted field into the conductive walls
as a function of t.
3
(b) Estimate the thickness t to prevent detection of the leakage radiation.
4
3. (20 points) Two long parallel wires carry an AC current as shown below. A
rectangular loop is placed symmetrically in between the wires in the plane of the
wires. The inner loop is at a distance d = 3 mm from the wires. The loop has a
length ` = 10 mm and a width of w = 15 mm.
d
I1
`
w
d
I1
(a) Calculate the magnetic field in the space between the wires as a function of the
current in the parallel lines. The inner loop is terminated in an open circuit as
shown.
5
(b) Calculate the magnetic flux crossing the inner loop and the mutual inductance.
(c) Calculate the voltage pickup in the inner loop if the wires carry the AC current
waveform shown below. The maximum excursion of the current I1 = 1 A and
the period of oscillation is 1 µs.
I1
t
6
(d) Calculate the peak force on the inner loop if a DC current of 1 mA is forced into
the inner loop while the current in the long wires is of the triangular form
described above but mismatched by 5%.
(e) If the loop is terminated in a resistive load, indicate the direction of current flow
in the loop relative to the phase of the current in the long wires. Explain.
7
4. (15 points) Consider the following capacitor sandwich structure with ² 1 6= ²2 . The
interface between the dielectrics is charged with a fixed surface charge
ρs = 10 mC/m2 The thickness of the structure is negligible in comparison with the
width and so you may neglect fringing fields.
ρs
t1
²1
++++++++++++++++++++++++++
t2
²2
(a) Calculate the electric fields in region 1 and region 2 when an external voltage
Vs = 10 V is applied to the structure.
8
(b) What’s the charge density induced on the top and bottom plates?
(c) Show that the charge density q on the top plate can be expressed with an
equation similar to a capacitor q = Cv + q0 , where q0 is a fixed charge density.
Calculate C and q0 .
9
(d) Alternatively, we can interpret the fixed charge as producing a threshold voltage
for the capacitor, such that q = C(v − vt ). Calculate vt .
(e) Calculate the electrostatic energy in the volume of the capacitor structure for
Vs = 0. How does this compare with the value calculated from circuit theory?
Explain.
10
5. (20 points) Calculate each question succinctly.
(a) Calculate the SWR on each transmission line (Z0 = 48Ω, Z1 = 24Ω, Z2 = 6Ω,
Rs = 48Ω, RL = 3Ω, β0 = 2π, β1 = π, β2 = 2π, `0 = 100m, `1 = 1.5m,
`2 = .75m).
Rs
Z 0 β0
Z 1 β1
Z 2 β2
`0
`1
`2
RL
(b) Write down five inductive termination impedances ZL = R + jX (X > 0) which
result in an SWR of 3 on a 50 Ω transmission line.
11
(c) The current density in a circular conductor of radius r0 is given by
r
J(r) = J0
r0
Calculate the conductive loss per unit length per unit Ampère of current in the
wire.
(d) Demonstrate that in the low frequency limit, a short section of a shorted
transmission line shown below can be modeled as an equivalent resistor Req .
Compute the value of Req in terms of the distributed line parameters R0 , G0 , L0 ,
C 0.
Z
β
`
12
α
(e) Draw an equivalent circuit for small frequency offsets from the frequency
f = 100 MHz. Assume the wave velocity on the line is c = 1 × 108 m/s and
` = 1m. Do not calculate the equivalent lumped parameter values but simply
draw an equivalent circuit.
Z
β
`
13
α
6. (12 points) For the cross-section shown below, we wish to find the propagation
characteristics of a TEM wave. Assume that the conductors have zero resistance and
the dielectric and magnetic losses are negligible at the frequency of operation.
t
w
g
² µ
h
ŷ
x̂
(a) Setup the equations for the electrostatic potential in the cross-section of the
plane by writing φ(x, y) = f (x)g(y). What boundary conditions must be
satisfied?
(b) What’s the propagation constant and wave impedance?
14
(c) Assume that we have determined the electrostatic potential from part (a) and in
rectangular coordinates, denote this potential as φ(x, y). Write the magnetic
and electric fields in terms of the potential function φ(x, y).
15
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