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:

\(\begin{array}{l} {ω _n} = 20\sqrt {10} ,α = {ζ ω _n} = 70\\ \Rightarrowζ= \frac{{70}}{{20\sqrt {10} }} = 1.10\\ T\left( s \right) = \frac{{{ω_n ^2}}}{{{s^2} + 2{ω _n}s + ω _n^2}}\\ = \frac{{4000}}{{{s^2} + 140s + 4000}}\\ \frac {C\left( s \right)}{R(s)} = \frac{{4000}}{{\left( {s + 100} \right)\left( {s + 40} \right)}}\\ C\left( s \right) = \frac{1}{s}.\frac{{4000}}{{\left( {s + 100} \right)\left( {s + 40} \right)}}\\ = \frac{1}{s} + \frac{{\frac{2}{3}}}{{\left( {s + 100} \right)}} + \frac{{\frac{{ - 5}}{3}}}{{\left( {s + 40} \right)}}\\ c\left( t \right) = 1 - \frac{5}{3}{e^{ - 40t}} + \frac{2}{3}{e^{ - 100t}} \end{array}\)

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: 

\(s^2+2\zeta ω s+ω^2=0\)

\(\omega=\sqrt{80}\)

\(2\zeta \omega=23\)

\(\zeta=\frac{23}{2\sqrt{80}}=1.285\)

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.

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

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

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.

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

\({t_r} = \frac{{\pi - \varphi }}{{{\omega _d}}}\)

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

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

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

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

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

\(\% {M_p} = \frac{{c\left( {{t_p}} \right) - c\left( \infty \right)}}{{c\left( \infty \right)}} \times 100\% \)

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.

\({e^{ - \xi {\omega _n}{t_s}}} = \pm 5\% \;\left( {or} \right) \pm 2\% \)

 \({t_s} \simeq \frac{3}{{\xi {\omega _n}}}\) 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

\(T(s) = \frac {C(s)}{R(s)}=\frac{ω _n ^2 }{s^2 +2ζω_n+ω_n^2}\)

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 

\(T_s = 4 T = \frac{4}{\zetaω_n}\)

Important formulae:

• Peak overshoot = M_p = \(e^{-\frac{\zeta π }{\sqrt{1-\zeta^2}}} \)
• Damped frequency = \(ω_d=ω_n\sqrt{1-\zeta^2}\) 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.

\({e^{ - \xi {\omega _n}{t_s}}} = \pm 5\% \;\left( {or} \right) \pm 2\% \)

 \({t_s} \simeq \frac{3}{{\xi {\omega _n}}}\) 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 \({\omega _n} = \sqrt {10}\) ; \(2\xi {\omega _n} = 5\)

\({t_s} = \frac{{4 \times 2}}{5} = \frac{8}{5} < 2\;sec.\)

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:

\(t_s = {4\over ζω_n}\)

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

\(t_s = {3\over ζω_n}\)

Calculation:

Given, Tolerance = 2%

ζ = 0.1

ω_n = 10

\(t_s = {4\over 0.1\times10}\)

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