Question

Statement (I): In a prototype, second-order system the rise time t_r and bandwidth are inversely proportional.

Statement (II): Increasing ω_n increases bandwidth while t_r reduces

1.

Both Statement (I) and Statement (II) are individually true and Statement (II) is the correct explanation of Statement (I)

2. Both Statement (I) and Statement (II) are individually true but Statement (II) is not the correct explanation of Statement (I)
3. Statement (I) is true but Statement (II) is false
4. Statement (I) is false but Statement (II) is true 

Answer 1

Correct Answer - Option 1 :

Both Statement (I) and Statement (II) are individually true and Statement (II) is the correct explanation of Statement (I)

Statement (I):

Rise time (tr): It is defined as the time required for a pulse or signal to increase from one specified value (as 10 %) of its amplitude to another (as 90%).

3-dB Bandwidth: The frequency at which the power level of the signal decreases by 3 dB from its maximum value is called the 3 dB Bandwidth, The 3 dB Bandwidth is the frequency at which the signal amplitude reduces by 3 dB i.e. becomes half its value.

The relationship between rise time (tr) and bandwidth is given by:

\(B.W. = \frac{{0.35}}{{{t_r}}}\)

Thus in a second-order system the rise time tr and bandwidth are inversely proportional. (Statement I is correct)

Statement (II):

We know that,

\(ω_d =ω_n\sqrt{1-\xi^2}\)

Increasing ω_n causes an increase in ω_d and thus bandwidth increases and rise time reduces. 

Hence, Both Statement (I) and Statement (II) are individually true and Statement (II) is the correct explanation of Statement (I)