Question

For a LTI system consisting of two integrators, the output of the first integrator is inverted and fed as input to the sigma at beginning and output of the second integrator is also inverted and fed as input to the sigma at the beginning. The input is x (t) and output y (t), the transfer function is ______________

(a) \(\frac{1}{s+4}\)

(b) \(\frac{1}{s^2+4s+3}\)

(c) \(\frac{1}{s^2-4s-3}\)

(d) \(\frac{1}{s^2-4s+3}\)

The question was posed to me in a national level competition.

This intriguing question comes from Average Power and Energy of a Signal topic in section Fourier Series of Signals and Systems

Answer 1

Right option is (b) \(\frac{1}{s^2+4s+3}\)

For explanation I would say: Transfer function of linear loop = \(\displaystyle\frac{\frac{1}{s}}{1 + \frac{4}{s}}\)

= \(\frac{1}{s+4}\)

∴ Transfer function of the overall system = \(\displaystyle\frac{\frac{1}{s(s+4)}}{1 + \frac{3}{s(s+4)}}\)

= \(\frac{1}{s^2+4s+3}\).