THE AUSTRALIAN NATIONAL UNIVERSITY Gerard Borg RADIOFREQUENCY ENGINEERING SAMPLE QUESTIONS ENGN4545/6545 Telecommunications Systems Page 1 of 10 1 Electromagnetism (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 d 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. ENGN4545/6545 Telecommunications Systems Page 2 of 10 (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. ENGN4545/6545 Telecommunications Systems Page 3 of 10 2 Impedance (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 MHz. HINTS: You may assume the following: • For any metal, the impedance per square is given by Zs = Rs (1 + j) where 1 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). ENGN4545/6545 Telecommunications Systems Page 4 of 10 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 ENGN4545/6545 Telecommunications Systems Page 5 of 10 (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. ENGN4545/6545 Telecommunications Systems Page 6 of 10 (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. ENGN4545/6545 Telecommunications Systems Page 7 of 10 (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? ENGN4545/6545 Telecommunications Systems Page 8 of 10 4 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, TLN B F = 1 + To 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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