Time Response Analysis MCQ Quiz - Objective Question with Answer for Time Response Analysis - Download Free PDF

Explanation:

In system modeling using a transfer function, the roots of the characteristic equation are the Poles of the transfer function.

• The characteristic equation of a system is typically obtained by setting the denominator polynomial of the closed-loop transfer function to zero.
• The poles of a transfer function are the values of 's' that make the denominator of the transfer function equal to zero (assuming no common factors with the numerator).
• The locations of the poles in the s-plane are critical for determining the stability and transient response of a system. If any pole lies in the right half of the s-plane, the system is unstable.

Concept:

The standard second-order system has the characteristic equation:

The roots of this equation determine the nature of the system (overdamped, underdamped, critically damped).

Condition for Complex Conjugate Roots:

Roots are complex conjugates if the discriminant is negative:

So, complex roots exist when:

Such systems are called underdamped systems.

Therefore,  the correct answer is    

Concept:

A standard second-order system is represented as:

Where,

•  = natural frequency
•  = damping coefficient

Given Equation:

Compare with the standard form:

Hence, the correct answer is option 1

Concept:

To compute the steady-state error, use the Final Value Theorem for the error signal:

Given:

Input:

Calculation:

This is a Type-1 system (one pole at the origin), so for a ramp input, the steady-state error is given by:

where

Hence, the correct answer is option 2

Concept:

Time constant 

Calculation:

Taking Laplace transform, we get

40 s X(s) + 2X(s) = 12(s)

Pole will be at -1/20. 

Time constant 

Concept:

KP = position error constant = 

Kv = velocity error constant = 

Ka = acceleration error constant = 

Steady-state error for different inputs is given by

From the above table, it is clear that for type – 1 system, a system shows zero steady-state error for step-input.

Concept:

The output response of a system is equal to the sum of natural response and forced response.

Forced response: The response generated due to the pole of the input function is called the forced response.

Natural response: The response generated due to the pole of system function is called the natural response.

Calculation:

The output y(s) is given as

Converting into partial fractions

Multiply whole equation LHS and RHS by s and put s = 0

A = 1

Multiply whole equation by (s + 1) and put s = -1

B = -2

Multiply whose equation by (s + 3) and put s = -3

C = 1

Taking the ILT, we get:

Concept:

KP = position error constant = 

Kv = velocity error constant = 

Ka = acceleration error constant = 

Steady state error for different inputs is given by

From the above table, it is clear that for type – 1 system, a system shows zero steady-state error for step-input, finite steady-state error for Ramp-input and  steady-state error for parabolic-input.

Calculation:

Velocity error coefficient, 

Standard 2^nd order system is  

Given

Steady state value

Settling time:

It is the time required for the response to reach the steady-state (or) final value and stay within the specific tolerance bands around the final value.

This is explained with the help of the following:

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

Similarly, for 2% tolerance, the settling time is given by:

ζ = Damping ratio

ω_n = Natural frequency

Time-domain specification (or) transient response parameters:

Rise time (t­r): It is the time taken by the response to reach from 0% to 100% Generally 10% to 9% for overdamped and 5% to 95% for the critically damped system is defined.

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

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

Maximum (or) Peak overshoot (Mp): It is the maximum error at the output. 

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

The standard second order system is given by 

Where ξ is damping ratio.

If ξ = 1, then system is critically damped.

If ξ

If ξ > 1, then system is order damped.

By comparing with standard second order transfer function,

ω_n² = 15 ⇒ ω_n = √15

So, it is underdamped system.

ω_n² = 25 ⇒ ω_n = 5

2 ξ ω_n = 10 ⇒ ξ = 1

So, it si critically damped system.

ω_n² = 35 ⇒ ω_n = √35

1\)

So, it is overdamped system.

Concept:

Damped natural frequency of a second order system is given by

Where,

ωn = Natural frequency

ζ = Damping ratio

Calculation:

Given-

ωn = 11 rad/sec

ζ = 0.6

Now, natural damped frequency can be calculated as

ωd = 11 x 0.8

ωd = 8.8 rad/sec  

Steady-state error for the Unit step input is,

= 

Concept:

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

ζ is the damping ratio

ωn is the undamped natural frequency

Characteristic equation: 

Roots of the characteristic equation are: 

α is the damping factor

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