A second order system has natural frequency 3rad/sec and unity damping ratio. Identify its transfer function.
System Transfer Function Derivation
To identify the transfer function of a second-order system, we need to recall its standard form. The standard transfer function for a second-order system is given by:
$$ G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} $$
Where:
- \( \omega_n \) is the natural frequency of the system.
- \( \zeta \) is the damping ratio of the system.
Given System Parameters
The question provides us with the following parameters for the second-order system:
- Natural frequency (\( \omega_n \)): 3 rad/sec
- Damping ratio (\( \zeta \)): Unity (which means \( \zeta = 1 \))
Calculating the Transfer Function
Now, we will substitute these given values of \( \omega_n \) and \( \zeta \) into the standard second-order system transfer function formula:
$$ G(s) = \frac{(3)^2}{s^2 + 2(1)(3)s + (3)^2} $$
Let's perform the calculations for each term:
- Numerator: \( \omega_n^2 = (3)^2 = 9 \)
- Coefficient of \( s \) in the denominator: \( 2\zeta\omega_n = 2(1)(3) = 6 \)
- Constant term in the denominator: \( \omega_n^2 = (3)^2 = 9 \)
Substituting these calculated values back into the formula, we get the transfer function:
$$ G(s) = \frac{9}{s^2 + 6s + 9} $$
Comparing with Options
Let's compare our derived transfer function with the given options to identify the correct one:
| Option | Transfer Function | Match |
|---|---|---|
| 1 | \( \frac{1}{{{s^2} + 3s + 9}} \) | No |
| 2 | \( \frac{9}{{{s^2} + 2s + 9}} \) | No |
| 3 | \( \frac{9}{{{s^2} + 6s + 9}} \) | Yes |
| 4 | \( \frac{3}{{{s^2} + 3s + 3}} \) | No |
Based on the comparison, the calculated transfer function \( \frac{9}{s^2 + 6s + 9} \) perfectly matches Option 3.
Important Questions from Time Response Analysis
Which of the following is correct for over-damped and under-damped system, respectively?
What will be the time response expression for a standard first order system having unit step function \(\frac{1}{s}\) as the input
Statement (I): All the systems which exhibit overshoot in transient response will also exhibit resonance peak in frequency response.
Statement (II): A large resonance peak in frequency response corresponds to a large overshoot in transient response.
The steady state error of a control system can be minimized by:
Usually the control system used should have
