Question

A positive feedback system has \(G\left( s \right) = \frac{{{k_1}}}{{s + p}}\) and H(s) = k₂ /s. The loop transfer function is
1. \(\frac{{{k_1}{k_2}s}}{{s + p}}\)
2. \(\frac{{{k_1}}}{{{k_2}\left( {s + p} \right)}}\)
3. \(\frac{{{k_2}}}{{{k_1}\left( {s + p} \right)}}\)
4. \(\frac{{{k_1~s}}}{{s\left( {s + p} \right) - {k_1}{k_2}}}\)

Answer 1

Correct Answer - Option 4 : \(\frac{{{k_1~s}}}{{s\left( {s + p} \right) - {k_1}{k_2}}}\)

Concept:

The transfer function of the closed-loop transfer function, with the positive value of feedback gain is given by \(\frac{{G\left( s \right)}}{{1 - G\left( s \right)H\left( s \right)}}\)

The overall gain of the transfer function increases with the positive value of feedback gain.

The transfer function of the closed-loop transfer function, with a negative value of feedback gain, is given by \(\frac{{G\left( s \right)}}{{1 + G\left( s \right)H\left( s \right)}}\)

The overall gain of the transfer function decreases with the negative value of feedback gain.

Calculation:

Given that, \(G\left( s \right) = \frac{{{k_1}}}{{s + p}}\) and H(s) = k₂ /s

For the positive feedback system, the closed-loop transfer function is

\(TF = \frac{{\frac{{{k_1}}}{{s + p}}}}{{1 - \frac{{{k_1}}}{{s + p}} \times \frac{{{k_2}}}{s}}}\)

\(=\frac{{{k_1~s}}}{{s\left( {s + p} \right) - {k_1}{k_2}}}\)