What is the transfer function of a phase lag compensator? The values of ∝ and τ are given as ∝ > 1 and τ > 0:

Concept:

The transfer function of the phase lag compensator is given by

\(G\left( s \right) = \frac{{1 + \tau s}}{{1 + a\tau s}}\)

∝ > 1 and τ > 0

\(G\left( s \right) = \frac{\tau ({s + \frac{1}{\tau}})}{\alpha \tau ({s + \frac{1}{\alpha \tau}})}\)

\(G\left( s \right) = \frac{ ({s + \frac{1}{\tau}})}{\alpha ({s + \frac{1}{\alpha \tau}})}\)

For a lag compensator poles are more closer to the origin than zeros. 

Important Points

The transfer function of phase lag compensator is given by

\(G\left( s \right) = \frac{{1 + Ts}}{{1 + aTs}}\)

\(G\left( {j\omega } \right) = \frac{{1 + j\omega T}}{{1 + j\omega aT}}\)

Phase angle, \(\angle G\;\left( {j\omega } \right) = {\tan ^{ - 1}}\omega T - {\tan ^{ - 1}}a\omega T\)

ϕ = tan-1 ωT – tan-1 aωT

The maximum phase lag occurs at ωm­ such that \(\frac{{d\phi }}{{d\omega }} = 0\)

\(\Rightarrow {\omega _m} = \frac{1}{{T\sqrt a }}\)

It is a Geometric mean of its two corner frequencies

\({\omega _m} = \sqrt {\frac{1}{T} \times \frac{1}{{aT}}} = \frac{1}{{T\sqrt a }}\)

The maximum phase lag,

\({\phi _m} = {\tan ^{ - 1}}\omega T - {\tan ^{ - 1}}a\omega T\)

\(= {\tan ^{ - 1}}\left( {\frac{1}{{T\sqrt \alpha }}} \right)T - {\tan ^{ - 1}}\left( {\frac{1}{{T\sqrt \alpha }}T} \right)\)   

\(= {\tan ^{ - 1}}\frac{1}{{\sqrt a }} - {\tan ^{ - 1}}\sqrt a\)

\({\phi _m} = {\tan ^{ - 1}}\left( {\frac{{\frac{1}{{\sqrt a }} - \sqrt a }}{{1 + \sqrt a \cdot \frac{1}{{\sqrt a }}}}} \right)\)

\({\phi _m} = {\tan ^{ - 1}}\left( {\frac{{1 - a}}{{2\sqrt a }}} \right)\)