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

GATE Electrical Engineering (EE) 2020 Solved Paper

Question : 36 of 65

Marks: +1, -0

A stable real linear time-invariant system with single pole at p, has a transfer function

{H}( {s})=\;{s^{2}+100}/{s-p}

with a dc gain of

5 .

The smallest positive frequency, in rad\/s, at unity gain is closest to:

Solution:

H(s)=\;{s^{2}+100}/{s-p}

D.C. gain

=5

D.C. gain

=H(s) ∣ s=0

=\;{s^{2}+100}/{s-p}|_{s=0}=5

\;{100}/{-p}=5 ⇒ p=-20

H(s)=\;{s^{2}+100}/{(s+20)}

For frequency domain

s=j ω

H(j ω)=\;{100+(j ω)^{2}}/{20+(j ω)}=\;{100-ω^{2}}/{(20+j ω)}

Gain

=|H(j ω)|=|\;{100-ω^{2}}/{20+j ω}|

=\;{100-ω^{2}}/{√{400+ω^{2}}}

Frequency at unity gain

\;{100-ω^{2}}/{√{400+ω^{2}}}=1

(100-ω^{2})^{2}=(400+ω^{2})

Soling this equation

ω=8.84 {rad}{/} {sec}

and

11.08 {rad} {/} {sec}

.
Smallest positive frequency

=8.84 {rad} {/} {sec}

Go to Question:

Prev Question

Next Question