Time Domain Specifications MCQ Quiz in తెలుగు - Objective Question with Answer for Time Domain Specifications - ముఫ్త్ [PDF] డౌన్‌లోడ్ కరెన్

Concept:

The damping ratio (ζ = c/cc) is a system parameter, that can vary from un-damped (ζ = 0), under-damped (ζ < 1) through critically damped (ζ = 1) to over-damped (ζ > 1)

Overdamped System: ζ > 1

This is the equation of aperiodic motion i.e. the system cannot vibrate due to over-damping. The magnitude of the resultant displacement approaches zero with time.

Underdamped: ζ < 1

This resultant motion is oscillatory with decreasing amplitudes having a frequency of ωd. Ultimately, the motion dies down with time.

Critical Damping: ζ = 1

The displacement will be approaching zero in the shortest possible time.

Concept:

The characteristic equation of a general second-order differential equation is given by:

s² + 2ζω_ns + ω_n² = 0

ζ = Damping ratio

This expression can also be represented as:

s2 + 2αs + ωn2 = 0

where α = damping factor given by:

α = ζ ωn

Application:

$\begin{array}{l}{\omega }_n=20\sqrt{10},\alpha =\zeta {\omega }_n=70\\ \Rightarrow \zeta =\frac{70}{20\sqrt{10}}=1.10\\ T\left(s\right)=\frac{{\omega }_n^2}{s^2+2{\omega }_{n}s+{\omega }_n^2}\\ =\frac{4000}{s^2+140s+4000}\\ \frac{C\left(s\right)}{R(s)}=\frac{4000}{\left(s+100\right)\left(s+40\right)}\\ C\left(s\right)=\frac{1}{s}.\frac{4000}{\left(s+100\right)\left(s+40\right)}\\ =\frac{1}{s}+\frac{\frac{2}{3}}{\left(s+100\right)}+\frac{\frac{-5}{3}}{\left(s+40\right)}\\ c\left(t\right)=1-\frac{5}{3}{e}^{-40t}+\frac{2}{3}{e}^{-100t}\end{array}$

Concept:
The damping ratio symbol is given by ζ and this specifies the frequency response of the 2nd order general differential equation.

Using the damping ratio, one can know the damping level of a system.

We have,

$G(s)=\frac{4}{(s+19)(s+4)}$

⇒ $G(s)=\frac{4}{s^2+23s+76}$

Characteristics Equation is given by:

1 + G(s) = 0

$1+\frac{4}{s^2+23s+76}=0$

$s^2+23s+80=0$

Comparing with C.E. of the second-order system: 

$s^2+2\zeta \omega s+{\omega }^2=0$

$\omega =\sqrt{80}$

$2\zeta \omega =23$

$\zeta =\frac{23}{2\sqrt{80}}=1.285$

Time-domain specification (or) transient response parameters:

Delay time (td): It is the time taken by the response to change from 0 to 50% of its final or steady-state value.

$c\left(t\right){|}_{t=t_d}=0.5$

$t_d\simeq \frac{1+0.7\xi }{{\omega }_n}$

Rise time (t­r): It is the time taken by the response to reach from 0% to 100% Generally 10% to 90% for overdamped and 5% to 95% for the critically damped system is defined.

$c\left(t\right){|}_{t=t_r}=1=1-\frac{e^{-\xi {\omega }_n{t}_r}}{\sqrt{1-{\xi }^2}}\sin \left({\omega }_n{t}_r+\phi \right)$

$t_r=\frac{\pi -\phi }{{\omega }_d}$

Peak Time (tp): It is the time taken by the response to reach the maximum value.

${\frac{dc\left(t\right)}{dt}|}_{t=t_p}=0,t_p=\frac{\pi }{{\omega }_d}$

$=\frac{\pi }{{\omega }_n\sqrt{1-{\zeta }^2}}$

Maximum (or) Peak overshoot (Mp): It is the maximum error at the O/P

$M_p=c\left(t_p\right)-1,M_p=e^{\left(\frac{-\xi \pi }{\sqrt{1-{\xi }^2}}\right)}$

$\mathrm{\%}{M}_p=\frac{c\left(t_p\right)-c\left(\mathrm{\infty }\right)}{c\left(\mathrm{\infty }\right)}\times 100\mathrm{\%}$

If the magnitude of the input is doubled, then the steady-state value doubles, therefore Mp doubles, but % Mp, tr, tp remains constant.

Settling time (Ts): It is the time taken by the response to reach ± 2% tolerance band as shown in the fig above.

$e^{-\xi {\omega }_n{t}_s}=\pm 5\mathrm{\%}\left(or\right)\pm 2\mathrm{\%}$

 $t_s\simeq \frac{3}{\xi {\omega }_n}$ for a 5% tolerance band.

$t_s\simeq \frac{4}{\xi {\omega }_n}$ for 2% tolerance band

If ξ increases, rise time, peak time increases and peak overshoot decreases.

Underdamped system:

If a system is said to be underdamped when the damping ratio is less than 1 (ζ < 1).

Characteristics:

The standard second-order transfer function is

$T(s)=\frac{C(s)}{R(s)}=\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_n+{\omega }_n^2}$

For underdamped system,

Damping coefficient α = ζ ω_n 

Time constant T = 1 / α = 1/ (ζ ω_n)

Settling time T_s:

It is defined as the time required by the response to reach and steady within a specified range of 2 to 5 % of its final value.

The underdamped system is shown below.

The system output will close to its input after the duration of 4 times of T.

So the settling time is calculated as 

$T_s=4T=\frac{4}{\zeta {\omega }_n}$

Important formulae:

• Peak overshoot = M_p = $e^{-\frac{\zeta \pi }{\sqrt{1-{\zeta }^2}}}$
• Damped frequency = ${\omega }_d={\omega }_n\sqrt{1-{\zeta }^2}$ rad / sec
• Peak time = t_p = (π / ω_d ) in sec 

Concept:

Time-domain specification (or) transient response parameters:

Settling time (Ts):

It is the time taken by the response to reach ± 2% tolerance band as shown in the fig above.

$e^{-\xi {\omega }_n{t}_s}=\pm 5\mathrm{\%}\left(or\right)\pm 2\mathrm{\%}$

 $t_s\simeq \frac{3}{\xi {\omega }_n}$ for a 5% tolerance band.

$t_s\simeq \frac{4}{\xi {\omega }_n}$ for 2% tolerance band

If ξ increases, rise time, peak time increases and peak overshoot decreases.

Calculation:

We shall consider option (3) since it does not increase the order of the system.

Increasing order of the system makes the system less stable

For 2^nd order system, the characteristic equation is

$1+\frac{\left(2s+8\right)}{\left(s+1\right)\left(s+2\right)}=0$

s² + 3s + 2 + 2s + 8 = 0

s² + 5s + 10 = 0

For 2% settling time $t_s=\frac{4}{\xi {\omega }_n}$

Here ${\omega }_n=\sqrt{10}$ ; $2\xi {\omega }_n=5$

$t_s=\frac{4\times 2}{5}=\frac{8}{5}<2sec.$

Concept:

The transfer function of the standard second-order system is:

$TF=\frac{C\left(s\right)}{R\left(s\right)}=\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$

ζ is the damping ratio

ωn is the undamped natural frequency

Characteristic equation: $s^2+2\zeta {\omega }_n+{\omega }_n^2=0$

The roots of the characteristic equation are:

$-\zeta {\omega }_n\pm j{\omega }_n\sqrt{1-{\zeta }^2}=-\alpha \pm j{\omega }_d$

α is the damping factor

• ζ = 0, the system is undamped
• ζ = 1, the system is critically damped
• 0 < ξ < 1, the system is underdamped
• ζ > 1, the system is overdamped

Additional Information

Important Points:

• If the poles are real and left side of s-plane, the step response approaches a steady state value without any oscillations.
• If the poles are complex and left side of s-plane, the step response approaches a steady state value with the damped oscillations.
• If poles are non-repeated on the jω axis, the step response will have fixed amplitude oscillations.
• If the poles are real and right side of s-plane, the step response reaches infinity without any oscillations.
• If the poles are real and right side of s-plane, the step response reaches infinity without any oscillations.
• If the poles are complex and right side of s-plane, the step response reaches infinity with damped oscillations.

Concept:

Settling time is the time required for the response to reach the steady state and stay within the specified tolerance bands around the final value.

The settling time for 2% tolerance band is given by:

$t_s=\frac{4}{\zeta {\omega }_n}$

The settling time for 5% tolerance band is given by:

$t_s=\frac{3}{\zeta {\omega }_n}$

Calculation:

Given, Tolerance = 2%

ζ = 0.1

ω_n = 10

$t_s=\frac{4}{0.1\times 10}$

t_s = 4 sec

Concept:

Rise time (tr): It is specified as the transition time for an input signal to rise from 0.1 to 0.9 times of signals max output value.

Bandwidth: It is defined as the range of frequencies over which the majority of the signal passes.

In a CRO, for B.W < 1 GHz

tr × BW = 0.35

Calculation:

Given that, Bandwidth (B.W) = 0.9 GHZ

$t_r=\frac{0.35}{B.w}=\frac{0.35}{0.9\times {10}^9}$

⇒ tr = 0.388 × 10-9 sec = 0.38 ns

Rise time (tr): It is specified as the transition time for an input signal to rise from 0.1 to 0.9 times of signal's max output value.

Additional Information

Bandwidth(B.W.): It is defined as the range of frequencies over which the majority of the signal passes.

tr × B.W. = 0.35