Assertion (A) : The phase lag network provides an attenuation of 20 log a (a < 1) at high frequencies having transfer function equal to
Reason (R): The lag network allows to pass high frequencies and low frequencies are attenuated.
Here, reason (R) is false because the lag network allows to pass low frequencies and high frequencies are attenuated.
Assertion (A): Introduction of phase lag network in forward path increases the phase shift.
Reason (R): A phase lag network has pole nearer to the imaginary axis as compared to zero.
The transfer function of a phaselag network is:
The polezero plot is shown below:
Here, pole is nearer to the imaginary axis and origin compared to zero.
Hence, reason (R) is true.
Introduction of phase lag network in forward path reduces the phase shift. Hence, assertion (A) is a false statement.
Match ListI (Type of controllers) with Listll (Alternate names) and select the correct answer using the codes given below the lists:
ListI
A. Twoposition controller
B. Proportional controller
C. Integral controller
D. Derivative controller
ListII
1. Rate controller
2. Reset controller
3. An amplifier with adjustable gain
4. ONOFF controller
Codes:
A B C D
(a) 4 2 3 1
(b) 4 3 2 1
(c) 3 4 2 1
(d) 3 4 1 2
The open loop transfer function of a three action controller (PID controller) is given by
A PID controller is shown below.
Here,
G(s) = O.L.T.F
The action of a PD controller is to
Due to PD controller, a zero is added to the forward path which results in a lead compensator. Therefore, steady state error remais the same whiie rise time of the system is reduced (i.e. stability is increased).
For the derivative feedback control shown in figure below, the damping ratio is equal to 0.7
The value of derivative gain constant is
For the given system, characteristic equation is
or,
s^{2} + 1.4s + 14s T_{d} + 14 = 0
or,
s^{2} + (1.4 + 14T_{d}) s + 14 = 0
Comparing with s^{2} + 2 ξω_{n} + ω_{n}^{2} =0, we have:
ω_{n} = √14 rad/s and 2ξω_{n} = (1.4 + 14T_{d})
or, 2 x 0.7 x √14 = 1.4 + 14 T_{d} (∴ ξ = 0.7, given)
or,
or,
= 0.274
The block diagram of an integral control is shown in figure below.
The steady state error in the above system for a parabolic input is
For the given system, forward path transfer function is
For parabolic input, steady state error is
where,
∴
A system employing proportional plus error rate control is shown in figure below.
The value of error rate control (K_{e}) and 2% settling time for a damping ratio of 0.5 are respectively
The forward path gain of the given system is:
∴ Characteristic equation is
1 + G(s)H(s) = 0
or
or, s^{2} + (2 + 10 K_{e})s +10 = 0
Here,
ω_{n }= √10 rad/s
and 2ξω_{n }= (2 + 10K_{e})
Given, ξ = 0.5
So, 2 x 0.5 x √10 = (2 + 10K_{e})
or,
= 1.16/10 = 0.116
Also, 2% settling time
A feedback control system employing output rate damping is shown in figure below.
What is the steady state error (in radian) for the above system resulting from an unit ramp input if the damping ratio is 0.6?
The characteristic equation of the given system is
1+G(s) H(s) = 0
Here,
∴ Characteristic equation is
10
or, s^{2} + (K_{0} + 2)s +10 = 0
Comparing with s^{2} + 2ξω_{n}s + ω_{n}^{2} = 0 we have
ω_{n} = √10 rad/s
and 2ξω_{n} = K_{0} + 2
Given, ξ = 0.6
∴ 2 x 0.6 x √10 = K_{0} + 2
or, K_{0} = 1.792
For a unit ramp input, e_{ss} = 1/K_{v}
where,
∴
= 0.3792 ≈ 0.38 radian
In the control system shown below, the controller which can give infinite value of steadystae error to a ramp input is
Case1: For integral type
∴
∴ e_{ss} = T_{i}/2
Case2: For derivative type
G(s)H(s) = (10_{s}T_{d})/(s+5)
∴
∴ e_{ss} = 1/K_{v }=_{ }∞
Case3: For proportional type
G(s)H(s) = 10K_{p}/(s+5)
∴
∴
Case4: For proportional plus derivative type
∴
∴ e_{ss} = 1/K_{v} = ∞
The transfer function of the OPAMP circuit shown below is given by:
Redrawing the given circuit in sdomain, we have:
Now,
From virtual ground concept,
Now,
or,
2V_{A}(s) = E_{i}(s) + E_{0}(s)
E_{0}(s) = 2V_{A}(s)  E_{i}(s)
or,
or,
The transfer function of a phase lead compensator is given by where α < 1 and T> 0.
Which of the following is not a correct value of maximum phase shift provided by such a compensator?
Assertion (A): Stepping motors are compatible with modern digital systems.
Reason (R): The nature of command is in the forms of pulses.
Since the nature of command is in the form of pulses, therefore stepping motors are compatible with modern digital systems.
Consider the control system shown in figure below employing a proportional compensator.
If the steadystate error has to be less than two percent for a unit step input, then the value of K_{c} would be
For given system,
With proportional controller,
∴
(for a unit step input)
Given, e_{ss} = less than 2% of input
i.e. e_{ss} < 0.02
Now, characteristic equation is:
1 + G(s)H(s) = 0
or, s^{3} + 4s^{2} + 5s + 2 + 2K_{c} = 0
Routh’s array is:
For stability of given system,
2 + 2K_{c} > 0 or K_{c} > 1
or K_{c} < 9
Hence, 1 < K_{c} < 9 (for stability) ....... (1)
Now,
From equations (1) and (2), we conclude that the minimum value of steady state error = 0.1 when K_{c }= 9.
Also, for all value of K_{c} between 1 and 9, value of e_{ss }will be more than 0.02 i.e. more than 2% of input.
Therefore, there is no such value of K_{c }for given system i.e. can not be determined.
Match ListI (Controllers) with ListII (Transfer functions) and select the correct answer using the codes given below the lists:
Codes:
A B C D
(a) 2 3 4 1
(b) 3 4 1 2
(c) 3 4 2 1
(d) 3 1 4 2
The transfer function of a system is given as:
It represents a
Given,
The polezero plot is shown below.
Since zero is near to origin, therefore it represents a lead network.
Integral feedback control
The industrial controller having the best steady state accuracy is
The transfer function of a phase lead controller is
The maximum value of phase shift provided by this controller is
Given,
Here, T_{1} = 3T and αT_{1} = T
∴ α = T/T_{1 }= T/3T = 1/3
∴
or, ϕ_{m} = 30° = maximum phase shift
With regard to the filtering property, the lead compensator and the lag compensators are respectively
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