A6Recognise the properties of complex waves in terms of odd & even symmetry and cycle repetition. Question 1 Statewhich symmetry and harmonics are likely to be present (odd, even or both)in Figures 1 to 5 below. State also whether there are any phase shifts (+/- /2 or +/- ) and their approximate periodicity. `[1 mark each] Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 B4Analyse complex waves using Fourier series. C2Synthesise a complex wave from its Fourier equation. Question 2 a) Obtain the Fourier series up to the 11th harmonic for the function defined by: Assume periodicity of the function outside of the given range. [3] b) Comment on the characteristics of the Fourier series coefficients [1] Question 3 a)Obtain the Fourier series up to the9th harmonic for the following function of period 2π. [3] b)Using the series obtained in a) plot the function up to the 9th harmonic over a period of 2π (using the tabular method). [10] A7Use Laplace transforms to represent transient circuits in the s-plane. B5Apply transient analysis to first and second order series and parallel circuits. C3Evaluate the circuit response to given inputs for first and second order systems. Question 4 A coil of 4Ω and inductance 100mH is connected in parallel with a 10Ω resistor. The whole combination is then connected in series with a 15Ω resistance and a 10V step input source. Assuming zero initial conditions, provide: a)An s-plane representation of the circuit diagram [1] b)A Laplace representation of the circuit current [2] c)A time-domain equation of the circuit current [3] d)A time-domain expression for the inductor voltage [3] e)An evaluation of main current behaviour [2] Question 5 A series circuit consists of a 10Ω resistance, 500mH inductance and a 10µF capacitor that are connected via single pole switch to a 1V step input. Assuming zero initial conditions when the switch is closed, provide: a)ans-plane representation of the circuit diagram [1] b)a Laplace representation of the circuit impedance [2] c)a time-domain equation of the current flowing in the circuit [3] d)a time-domain expression for the capacitor voltage [3] e)an evaluation of the damping state of the circuit. Suggest modifications that could change the damping behaviour. [2]
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