Question

A stable real linear time-invariant system with single pole at p, has a transfer function $H\left(s\right)=\frac{s^2+100}{s-p}$
with a dc gain of 5. The smallest positive frequency, in rad/s at unity gain is closed to:

• A. 11.08
• B. 78.13
• C. 8.84
• D. 122.87

Answer 1

The correct option is C 8.84

$H\left(s\right)=T.F=\frac{s^2+100}{s-p}$
D.C. gain =5
$\Rightarrow \frac{100}{-p}=5=p=-20$
$H\left(j\omega \right)=\frac{-{\omega }^2+100}{j\omega +20}$
$|H\left(j\omega \right)|=\frac{-{\omega }^2+100}{\sqrt{{\omega }^2+400}}$
$\frac{-{\omega }^2+100}{\sqrt{{\omega }^2+400}}=1$
$\Rightarrow \omega =8.84rad/sec$.
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