Obtain the fourier transform for the data using the fourier

Question

Question 1:

a) In relation to Fourier analysis state the meaning and significance of

i) odd and even functions

ii) half-wave symmetry {i.e. f(t+Π)= -f(t)}.

Illustrate each answer with a suitable waveform sketch.

b) State by inspection (i.e. without performing any formal analysis) all you can about each of the periodic waveforms shown in FIGURE 1 in terms of their Fourier series when analysed about t = 0.

 

Question 2:

a) If the firing angle is set for α = Π / 3 estimate the power dissipated in the bulb if it is rated at 100 W and the voltage source is 230 V @ 50 Hz.

b) An anomaly that can occur in controlled rectification is drift of the firing angle on one half cycle, so causing an asymmetrical waveform, as illustrated opposite. State the effect, if any, this would have on the harmonic content of the waveform.

 

c) Sketch the waveform defined below and explain how you would obtain its Fourier series:

f (ωt) = 0 for 0 ≤ ωt ≤ Π/2
f (ωt) = Vsin(ωt) for Π/2 ≤ ωt ≤ Π
f (ωt) = 0 for Π ≤ ωt ≤ 3Π/2
f (ωt) = Vsin(ωt) for 3Π/2 ≤ ωt ≤ 2Π

Develop the analysis as far as you are able.

Question 3:

i) Obtain the Fourier Transform for the data using the Fourier Analysis tool of Excel. The transformed data should commence in cell D2.

ii) Identify the principal frequencies in the current waveform.

iii) Estimate the total harmonic distortion [THD] present in the current waveform using the formula:

THD(I) = 1/I₁√(_n=2∑^max (I_n)² x 100%)

where I₁ is the r.m.s. value of the fundamental current, In the r.m.s value of the nth harmonic and n(max) is the number of the highest measurable or significant harmonic.

[Note the vertical axis of the spectrum graph is scaled in (current) 2.]

iv) Attempt to synthesise the shape of the original waveform from its principal harmonics [e.g. sketch the waveforms of the harmonics on the same time axis and add them together].

Question 4: Sketch, on a set of common axes, waveforms to represent the transient response of circuits having transfer functions with the following parameters:

a) ζ = 0.5, ω_o = 1 x 10³ rad s^-1

b) ζ = 0.2, ω_o = 2 x 10³ rad s^-1

c) ζ = 2, ω_o = 1 x 10³ rad s^-1

Question 5:

a) With the aid of a table of standard transforms, determine the Laplace Transform of:

a) 10 + 3t² + sin4t

b) 4e^-3t sin2t

b) With the aid of a table of standard transforms, determine the Inverse Laplace Transform of:

a) 5/s³ + 12/s-4

b) 3s + 9/(s+3)²+7

Question 6:

a) Draw the Laplace form of the input portion of the circuit, as represented in FIGURE 4(c).

b) Derive an expression for the Laplace transfer function,

T(s) = Δv_L(s)/Δi_p(s), of the circuit of FIGURE 4(c).

c ) Express ΔvL as a function of time (i.e. the transient response of the voltage ΔvL when ip is subject to a step change.

d) Using the values given in TABLE A, estimate the time taken for the voltage vL to reach its steady state value if the current ip is subject to a step change of 2 nA.

Cp 1400 pF

Cc 250pF

RL  5MΩ