ELEC2134 Circuits and Signals S1 2018 Mid-semester Exam Total marks: 80 Time: 100 minutes This paper contains 3 pages and may not be retained This is a closed-book exam, only UNSW-approved calculators permitted Fourier Transform pair (for continuous-time signal) deXtxdtetxX tjtj )( 2 1 )()()( Fourier Series: () = 0 + ∑ cos(0) + sin(0) ∞ =1 The fundamental frequency of x(t) is 0 and T is the period of the signal x(t). …3,2,1) 2/ 2/ sin()( 2 ) 2/ 2/ cos()( 2 00 ndtt T T ntx T bdtt T T ntx T a nn Or () = ∑ 0+∞ = ∞ where = 1 ∫ () 0 + for any QUESTION 1 [22 Marks] For the circuit in Figure 1, if the source frequency is 15.9 Hz, determine i) The admittance seen by the source (in phasor form). ii) The phase angle between the input voltage Vin and current Iin (using the voltage as reference). iii) The total power factor. iv) Sketch the power triangle, clearly labelling the real (average), reactive and apparent power, and any relevant angles. v) The type and value of the circuit component which, when connected in parallel with the components shown in Figure 1, will raise the total power factor to unity. 2.5 50 mH Vin = 2 0 V Iin Figure 1. Name:___________________ Student ID:_______________ Signature:________________ Page 2 of 3 QUESTION 2 [18 Marks] (a) Consider the function f(t) shown in Figure 2. Given that ∫ () = 1 2 cos() + sin() and ∫ () = 1 2 sin() cos(), determine the trigonometric Fourier series of f(t), up to the second harmonic only (i.e. up to n = 2), showing all working. (10 marks) Figure 2 (b) If () = ||, derive () using the definition of the Fourier transform. (8 marks) ========== Answer Questions 3 and 4 in a separate answer booklet ========== QUESTION 3 [26 Marks] (a) For the series
RLC circuit shown in Figure 3, assuming that the source is operating at frequency = 0, where 0 is the resonant frequency: (6 marks) i) Sketch an example phasor diagram, clearly labelling the phasor voltages across each element (including the source) and giving the values of all relevant angles. Note that you are not required to provide numerical values for the phasor amplitudes. ii) Explain in words the relationships between the voltage phasors at = 0 in your phasor diagram. a b R LC Vi Vo Figure 3 (b) For the circuit in Figure 4, i) Sketch the magnitude response of the voltage gain, i.e. | () () | versus for ≥ 0, labelling all important features, and evaluating at least two values of . (4 marks) 1 10 H vin + – + – vout Figure 4 t 1 f(t) A/2 -1 -A/2 2 Page 3 of 3 ii) If a voltage waveform with amplitude spectrum |Vin(ω)| shown in Figure 5 is applied as the input to the circuit in Figure 4, sketch the amplitude spectrum of the output voltage |Vout(ω)|. (4 marks) Figure 5 (c) If the voltage shown in Figure 5 is applied to a 1 resistor only, determine the energy dissipated by that resistor. (4 marks) (d) Consider the circuit shown in Figure 6. Determine: (8 marks) i) The load current IL ii) The average power absorbed by the 20 load. j12 8 j10 100 0 V j15 + – 20 -j12.4 IL I1 I2 Figure 6. QUESTION 4 [14 Marks] (a) If () = ( 1) + ( + 1), derive () using the definition of the Fourier transform. (4 marks) (b) Determine the signal f(t) whose Fourier transform F( ) is shown in Figure 7. (8 marks) F(ω) ω 1 20 -1 10 2-2 Figure 7 (c) Given the definition of the Fourier transform, prove from first principles that ∫ 2() = ∞ ∞ 1 2 ∫ |()|2 ∞ ∞ , i.e. Parseval’s Theorem. Show all working. (2 marks) END OF PAPER 1 2 4 |Vin(ω)| 2 5