A phase lead compensating network ha its transfer function \(G_c (s)=\frac{10(1+0.04s)}{(1+0.01s)}\).The maximum phase lead occurs at a frequency of

Phase-lead compensator:​

The transfer function of the phase-lead compensator is

\(G\left( s \right) = \frac{{α \left( {1 + sT} \right)}}{{\left( {1 + α sT} \right)}}\)

\(G\left( {jω } \right) = \frac{{α \left( {1 + jω T} \right)}}{{\left( {1 + jα ω T} \right)}}\)

∠ G(jω ) = ∅ = Tan-1 ωT - Tan-1 ωαT  

\(∅ = {\tan ^{ - 1}}\frac{{ω T - ω α T}}{{1 + {{\left( {ω α } \right)}^2}T^2}}\)

Corner frequencies are

ωc1 = 1/T rad/sec

ωc2 = (1/ αT)  rad/sec

The maximum phase lead occurs at the geometric mean of its corner frequencies.

ωm = √(ωc1 x ωc2)

ωm = 1/ T√ α 

At ω = ωm ; ∅ = ∅m

 \({∅ _m} = \;{\tan ^{ - 1}}\frac{{1 - α }}{{2\surd α }}\)

Calculation:

Given that

\(G_c (s)=\frac{10(1+0.04s)}{(1+0.01s)}\)

Compare with the transfer function of phase-lead compensator,

T = 0-04

αT = 0.01

ωm = 1/ T√ α 

\(ω_m={\sqrt{\frac{1}{\alpha T}}{\frac{1}{{T}}}}\)

ω_m = 50 rad/sec