\(L{\left( s \right)^.} = \frac{{9{e^{ - 0.1s}}}}{s}\)

\(L\left( {j\omega } \right) = \frac{{9{e^{ - j0.1\omega }}}}{{j\omega }}\)

Phase margin = 180 + ∠ L(jω_{gc})

ω_{gc} is gain cross over frequency.

At ω_{gc}, magnitude of L(jω) is 1

⇒ L(jω_{gc}) = 1

\(\Rightarrow \left| {\frac{{9{e^{ - j0.1\omega }}}}{{j\omega }}} \right| = 1\)

⇒ ω = 9 rad/sec.

Phase margin \(= 180 + \left[ { - 90 - \left( {0.1} \right)\left( 9 \right)\left( {\frac{{180}}{\pi }} \right)} \right]\)

\(180 - 90 - \frac{{\left( {0.1} \right)\left( 9 \right)\left( {180} \right)}}{\pi }\)

= 38.43°