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QUESTION: 1

Match List - I (Roots in s-plane) with List - II (Corresponding impulse response) and select the correct answer using the codes given below the lists:

List - I

List - II

Codes:

Solution:

For A: Two conjugate poles on imaginary axis i.e. it’s impulse response = sin at (marginally stable).

For B: Two poles at origin i.e. impulse response = t u(t) (unstable system).

For C: Two complex conjugate poles with negative real parts. So, its impulse response = er^{-at} sin bt (stable system).

For D: Two imaginary complex double roots. So, impulse response = t sin bt. (unstable system).

QUESTION: 2

The transfer function of a system is the ratio of

Solution:

QUESTION: 3

The transfer function is applicable to

Solution:

Transfer function is applicable to only linear and time-variant systems.

QUESTION: 4

The DC gain of the system represented by the following transfer function is

Solution:

Given transfer function is

Converting above transfer function in time-constant form, we get:

Hence, dc gain is

QUESTION: 5

The unit impulse response of a control system is given by c(t) = -te^{-t} + e^{-t}. Its transfer function is

Solution:

Given, c(t) = -te^{-t} + e^{-t} (impulse response)

We know that,

QUESTION: 6

The impulse response of an initially relaxed linear system is e^{-3t} u(t). To produce a response of te^{-3t} u(t), the input must be equal to

Solution:

We know that, impulse response

For given system,

Now, c(t) = t e^{-3t}

So, R{s) = C(s) x (s + 3)

or, r(t) = Required input = e^{-3t} u(t)

QUESTION: 7

A linear time invariant system, initially at rest when subjected to a unit step input gave response c(t) = te^{-t}(t ≥ 0). The transfer function of the system is

Solution:

Given, c(t) = te^{-t}

Now, r(t) = u(t) (unit step input)

∴ R(s) = 1/s

So,

QUESTION: 8

A linear time invariant system having input r(t) and output y(t) is represented by the differential equation

The transfer function of the given system is represented as

Solution:

Taking Laplace transform of the given differential equation on both sides, we get

2s^{2} Y(s) + sY(s) + 5 Y(s) = R(s) + 2e^{-s} R(s)

or,

QUESTION: 9

The open loop transfer function of a system is

A closed loop pole will be located at s = -12

when the value of K is

Solution:

QUESTION: 10

The singularities of a function are the points in the s-plane at which the function or its derivates

Solution:

QUESTION: 11

Assertion (A): The final value theorem cannot be applied to a function given by

Reason (R): The function s F(s) has two poles on the imaginary axis of s-plane.

Solution:

The final value theorem is valid only if sF(s) does not has any poles on the jω axis and in the right half of the s-plane. Hence, both A and R are true and R is a correct explanation of A.

QUESTION: 12

When two time constant elements are cascaded non-interactively then, the overall transfer function of such an arrangement

Solution:

QUESTION: 13

The unit impulse response of a unity feedback control svstem whose Open Iood transfer function

Solution:

Given, O.L.T.F.

For unit impulse response, R(s) = 1

or, c(t) = 2(e^{-t }- te^{-t}) + te^{-t}

= 2e^{-t} - te^{-t} = (2- t) e^{-t}

Thus, impulse response, c(t) = (2 - t) e-^{t} u(t)

QUESTION: 14

The impulse response of a linear time invariant system is a unit step function. The transfer function of this system would be

Solution:

Impulse response,

c(t) = u(t)

QUESTION: 15

Consider the function where F(s) is the Laplace transform of f(t) is equal to

Solution:

Using final value theorem

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