Gerard Borg
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(1) Gauss’ Law Given that capacitance is defined as the ratio of charge Q to voltage V
from two points in a circuit where +Q is located at one point and and −Q is located
at another, use Gauss’ law to show that the capacitance of a parallel plate capacitor
with plates of area A and separation d is given by,
C =
ǫo A
where ǫo is the permittivity of free space.
(2) A solenoid is a coil of wire in which the turns are wound on a former such that the
turns touch each other to form a perfect coil. Assuming that the number of turns
per unit length is n′ show that the magnetic field within the solenoid is given by the
following equation,
B = µo n′ I
where I is the current flowing in the solenoid and µo is the permeability of free space.
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(3) Consider a metallic hoop of radius a, that is connected to an oscilloscope across a
break in the hoop as shown in the following figure. Assuming that a uniform magnetic
field of magnitude Bo Tesla and rotating at frequency fo Hz links the hoop, derive an
expression for the voltage observed on the oscilloscope.
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(1) Impedance and Matching Networks
Answer the following questions and take note of the HINT provided at the end.
(a) Compute the internal impedance of copper wire of .3 mm radius, and length 30
mm at f = 330 MHz.
(b) Compute the external inductance of a strand of copper wire of 300 µ radius and
length 1 cm.
(c) The length of wire in (b) is used to connect an input BNC connector to the top
of a double sided print circuit board. The under side of the board is connected
to ground.
Under excitation at 10 MHz it is discovered that there is a short circuit from
the BNC to ground.
• Explain in simple terms why there could be a short circuit.
• Assuming that the dielectric constant of the print circuit board ǫr = 6 and its
thickness is 0.6 mm compute the area of the print circuit board that would
lead to the short circuit.
(d) Suppose that we are to match a source impedance of Zs = 100 + j126 Ohms to a
load ZL = 1000 Ohms in parallel with 2 pF. Design a matching network for 100
HINTS: You may assume the following:
• For any metal, the impedance per square is given by Zs = Rs (1 + j) where
Rs = σδ
• The conductivity of copper, σ = 5.80 × 107 Mho/m.
• The skin depth δ =
√ 1
πµ0 σf
where f is the frequency (Hz).
• The permeability of free space µ0 = 4π × 10−7 W ebers/m.
• The external inductance of a cylindrical wire is given by Le = 0.002l(loge (2l/a) −
0.75) (µH) where l is the length (cm) and a is the radius (cm).
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3 Transmission lines and transmission line transformers
(1) Transmission Lines
(a) An ideal coaxial transmission line has a characteristic impedance of 50Ω. If the
line is excited at one end by a signal source producing a sine wave at frequency
1 GHz and the cable dielectric material has a dielectric constant of 2.25 and zero
loss tangent, answer the following,
(1) Compute the wavelength of the current and voltage waves on the line. You
may assume that the velocity of light in free space is 3 × 108 m/s.
(2) Compute the inductance and capacitance per unit length on the lines.
(3) What is the insertion loss of the line?
(4) If the line is terminated in an impedance Z (where Z 6= Z0 ), describe what
happens to the incident current and voltage waves as they arrive at the
impedance Z at the end of the line.
(5) Derive an expression for the reflection coefficient in terms of Z and Z0 .
(6) Define the Voltage Standing Wave Ratio (VSWR) in terms of the maximum
and minimum voltage amplitudes on the line.
(7) Express the VSWR in terms of the reflection coefficient.
(b) You need to measure the impedance ZA , of a VHF log-periodic aerial at 45 MHz.
In order to improve the radiating efficiency, the aerial is mounted on the roof of
a tall building and is fed by an unknown length of Zo = 50Ω coaxial cable. The
antenna impedance must be measured at the feed point on the roof but there is
no access for your Vector Network Analyser.
Assuming that you can at least disconnect the antenna from the cable on top of
the roof, derive a mathematical formula and a measurement procedure to measure the antenna impedance in the absense of knowledge about the length of the
coaxial cable. Show all working.
HINT: You may use the following formula that gives the impedance Zin measured
at the input to a transmission line in terms of the impedance ZL terminating the line
and the phase shift θ along the line.
Zin = Zo
ZL + jZo tan θ
Zo + j ZL tan θ
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(2) Transmission Line Transformers
Answer the following questions taking into account the HINT below.
(a) The following figure shows a hybrid combiner transmission line transformer terminated at its plus/minus outputs in a pair of impedances of value Z.
(1) Derive expressions for each of the input impedances seen at ports 1 and 2.
(2) Derive expressions for the output voltages V+ and V− in terms of V1 and V2 .
(3) A pair of RF power amplifiers with output signals of equal amplitude and
phase are used to provide V1 = V2 .
(a) What are the output voltages V+ and V− under these conditions in terms
of V1 = V2 .
(b) Suppose that the power amplifier feeding the second input port and providing V2 ceases to operate producing zero output power. Derive expressions for the new V+ and V− .
(c) If the output impedance of the second no longer functional amplifier is
50Ω, what signal is developed across this amplifier’s output terminal via
cross-coupling from V1 on port 1? Show all working.
(b) The following figure shows a broadband 1:4 matching balun that matches a 12.5Ω
source to a 50Ω load.
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(1) Ignoring the capacitances show algebraically that if the output impedance is
Zo then the input impedance is Zo /4. Show all working.
(2) Given tests confirm that the transmission line transformer exhibits broadband operation, describe one defect that it has which is partially eliminated
by the capacitors.
HINT: You may assume for a multifilar transformer wound on an appropriate ferrite
that the sum of the currents entering from the same end of the transformer into all
the windings is equal to zero and that the voltages across each winding are equal.
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(3) T/R Switching PIN diodes are used to control transmit and receive in a half duplex
radio. To do so the transmit and receive paths in the transceiver have to be swapped
by the diodes so that during receive mode, the receiver is connected to the antenna and
the transmitter is disconnected from both the receiver and the antenna. Otherwise
noise from the RF Power amplifier would desensitise the receiver. Similarly in transmit mode, the receiver must be disconnected from the antenna and the transmitter or
otherwise the RF power amplifier would destroy the receiver.
A DC power supply is used to control the conducting state of the PIN diodes. The
control input either reverse biases the diodes to produce an open circuit or forward
biases the diodes to produce a short circuit.
(a) What is the sign of the control voltage in transmit mode?
(b) In transmit mode, explain how RF power is prevented from destroying the receiver. Provide a detailed non-mathematical description of the role played by the
quarter-wave lines.
(c) What is the sign of the control voltage in receive mode?
(d) In receive mode, explain how noise from the RF power amplifier does not reach
the receiver. Provide a detailed non-mathematical description of the role played
by the quarter-wave lines.
(e) Why are parallel tuned circuits used at the diodes?
(f) Why are RC low pass filters connected at the control input?
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Satellites and Link Budgets
(1) Satellite links Determine the required parabolic dish diameter of a 4 GHz (C-band)
earth station antenna if its system temperature is 100K for an SNR ratio of 20 dB
and bandwidth 30 MHz with satellite transponder power = 5 Watt, satellite parabolic
dish diameter = 2m and spacing between satellites = 2o .
(2) Satellite TV Systems A C-band earth station consisting of a 3m satellite dish
connected to a ZINWELL ZCF-D21A LNB (Low Noise Block) is used to receive Indonesian SCTV via the PALAPA C2 geo-satellite. A table of PALAPA C2 carrier
frequencies and expected sky noise temperatures is shown in the following graphics.
Assuming that SCTV has a bandwidth of 6 MHz compute the satellite power for
a satellite dish diameter of 2 m and minimum SNR for analog TV reception of 50 dB.
(3) Now assuming that digital TV transmission is being used with a transmit power of 16
dBm, specify the gain and allowable noise figure of a possible LNA to follow the LNB
that would produce a final SNR of the 20 dB at a power level of 0 dBm.
HINTS for Qs 2 and 3 You may think of the LNB as a combination of LNA
and downconverter. In this case the ZINWELL also includes a small collecting aperture built into the LNB itself. The ZINWELL LNB is located at the focus of the
ground station dish and you may assume it has 100 % aperture efficiency whilse the
dish itself has 50 % aperture efficiency.
According to the ZINWELL datasheet, the LNB has a equivalent noise temperature
TLN B of 30K and a typical conversion gain of 65 dB. You may assume that the noise
factor of the LNB is given by,
F = 1 +
where To is the room temperature.
The figures on the following page show the SCTV carrier frequency and the sky noise
temperature versus observation frequency repectively.
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