For the given series RLC circuit, damping ratio (ξ) and undamped natural frequency (ωn) are respectively given by
The characteristic equation of series RLC circuit is given by:
Comparing with s2 + 2ξωns + ωn2 = 0, we have:
For the unity feedback control system shown below, by what factor the amplifier gain K should be multiplied so that the damping ratio is increased from 0.3 to 0.9?
Characteristic equation of given system is
1 + G(s) H(s) = 0
For ξ1 = 0.3, gain = K1
For ξ2 = 0.9, gain = K2
A system is shown in figure below
What are the values of K and α respectively so that the system has a damping ratio of 0.7 and an undamped natural frequency ωn of 4 rad/sec ?
From given block diagram, the closed loop transfer function is
Characteristic equation is,
s2 + (2 + Kα) s + K = 0
Here, ωn = -√K
and 2ξωn = (2 + kα)
Given, ωn = 4 rad/sec
∴ K = 16
Also, ξ = 0.7
∴ 2 x 0.7 x 4 = (2 + 16α)
or, (2 + 16α) = 5.6
α = 3.6/16 = 0.225
For type-2 system, the steady state error due to parabolic input is
For parabolic input,
= Finite for type-2 system
So, ess = Some finite value
The open loop transfer function of a unity-feedback control system with unit step input is given by:
The settling time of the system for 5% tolerance band is,
Characteristic equation is: 1 + G(s) H(s) = 0 or, s2 + s + 1 = 0
For 5% tolerance band, settling time is
The integral of a ramp function becomes
Consider a unity feedback control system as shown in figure below:
The error signal e(t) for a unit ramp input for T ≥ 0 is
Here, A = -T, 6 = 1, C = T2
Taking inverse Laplace transform on both sides, we get:
C(t) = (-T + t + Te-t/T)
∴ Error signal,
e(t) = c(t) - r(t)
= c(t) - t = - T + Te-t/T
= T(1 - e-t/T)
A second order system has a closed loop transfer function
If the system is initially at rest and subjected to a unit step input at t = 0, the third peak in the response will occur at
For 3rd peak, n = 5
(3rd overshoot is the third peak)
A closed servo system is represented by the differential equation
where c is the displacement of the output shaft r is the displacement of the input shaft anc e = r - c.
The damping ratio and natural frequency for this system are respectively
or, s2C(s) + 8s C(s) = 64 [R(s) - C(s)]
Hence, ωn = 8 rad/sec
and 2ξωn = 8 or ξ = 0.5
The poles of a closed loop control system art located at -1 ± j.
For the given system match List-I (Time domain specifications) with List - II (Values) and select the correct answer using the codes given below the lists
List - I
A. Damping ratio
B. Undamped natural frequency (rad/s)
C. Damping factor
D. Settling time (in secs)
List - II
Given, poles are at s = -1 ± j ∴ Characteristic equation is
(s+1)2 + 1 =0
or, s2 + 2s + 2 = 0
Comparing above equation with
we have, ωn = √2 rad/sec
and 2ξωn = 2
Undamped natural frequency,
Damping factor = ξωn = 1
Settling time (For 2% tolerance band) is
Damped natural frequency is
= 1 rad/sec
Consider the block diagram of a control system shown below.
What is the value of gain k such that the system is non oscillatory yet has the lowest possible settling time to step input?
Here, ωn = 10 rad/s and 2ξωn > = 100 K
For no-oscillation and to have lowest possible settling time ξ = 1.
∴ K = 1/5 = 0.2
Assertion (A): The "dominant pole" concept is used to control the dynamic performance of the system, whereas the “insignificant poles” are used for the purpose of. ensuring that the controller transfer function can be realized by physical components.
Reason (R): if the magnitude of the real part of a pole is at least 5 to 10 times that of a dominant pole or a pair of complex dominant poles, the pole may be regarded as insignificant in so far as the transient response is concerned.
Both assertion and reason are true. However, the correct explanation of assertion is that the poles having real part magnitude 5 to 10 times of the magnitude of real part of dominant poles doesn't affect the transient response (dynamic performance) of the system to a great extent due to which they are termed as insignificant poies and hence are neglected for transient stability study purpose.
Assertion (A): The step-function input represents an instantaneous change in the reference input.
Reason (R): If the input is an angular position of a mechanical shaft, a step input represents the sudden rotation of the shaft.
(a) Both A and R are true and R is a correct explanation of A.
(b) Both A and R are true but R is not a correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Consider the following statements related to second order control system with unit step input:
1. The right-half s-plane corresponds to negative damping.
2. Negative damping gives a response that doesn’t grows in magnitude without bound with time.
3. The imaginary axis corresponds to zero damping.
4. The maximum overshoot is often used to measure the relative stability of a control system.
5. Higher the damping in the system larger is the peak overshoot in the transient response of the system.
Which of the above statements is/are correct?
Negative damping gives a response that grows in magnitude without bound in time. Hence, statement-2 is false.
Since therefore higher is the damping in the system, less is the peak overshoot of the system. Hence, statement-5 is not correct
Which of the following is not true regarding addition of a pole to the forward-path transfer function?
With addition of a pole to the forward-path transfer function, bandwidth will reduce.
Since therefore rise time will increase.
Since rise time therefore stability will decrease.
For a second-order prototype system, when the undamped natural frequency is increased, the maximum overshoot of the output will
We know that, maximum overshot is
Mp is independent of wn. So Mp ∝ ξ only.
Consider a unity feedback system shown in figure below where, K > 0 and a > 0.
What are the values of k and a respectively so that the system oscillates at a frequency of 2 rad/sec?
Now, 1 + G(s)H(s) = 0
or, s3 + as2 + (2 + K) s + (1 + K) = 0
Routh’s array is
For oscillation, we have:
Also, oscillating frequency will be given by roots of A2(s) = 0.
We see that only option (d) is matching, i.e. for K = 2 and a = 0.75,
The transient response of a system is mainly due to
A second order system has a damping ratio ξ and undamped natural frequency of oscillation ωn respectively. The settling time at 5% of tolerance band of the system is
For 2% tolerance band,
For 5% tolerance band,
The unit impulse response of a linear time invariant second order system is
g(t) = 10e-8t sin 6t (t ≥ C)
The natural frequency and the damping ratio of the system are respectively
Given, c(t) = g(t) = 10e-8t sin6t
and ξωn = 8
= 10 rad/sec