Transfer Function MCQ Quiz - Objective Question with Answer for Transfer Function - Download Free PDF

System 1:

$y\left(t\right)=\frac{d}{dt}x\left(t\right)+2x\left(t\right)$ 

By applying the Laplace transform,

Y(s) = s x(s) + 2 x(s)

$\Rightarrow \frac{Y\left(s\right)}{X\left(s\right)}=\left(s+2\right)$ 

⇒ H₁(s) = s + 2

System 2:

The transfer function of high pass filter is,

$H_2\left(s\right)=\frac{sRC}{1+sRC}$ 

τ = RC = 1 sec

$H_2\left(s\right)=\frac{s}{1+s}$ 

Both the systems are connected in cascade connection.

H(s) = H₁(s) H₂(s)

$=\left(s+2\right)\times \frac{s}{\left(s+1\right)}$ 

$\Rightarrow \frac{Y\left(s\right)}{X\left(s\right)}=\frac{s\left(s+2\right)}{\left(s+1\right)}$ 

x(t) = e^-2t

$\Rightarrow X\left(s\right)=\frac{1}{\left(s+2\right)}$ 

$Y\left(s\right)=\frac{s\left(s+2\right)}{\left(s+1\right)}\times \frac{1}{\left(s+2\right)}=\frac{s}{\left(s+1\right)}$ 

$\Rightarrow Y\left(s\right)=1-\frac{1}{\left(s+1\right)}$ 

Calculation

The transfer function is defined as the ratio of the Laplace transform of output to the Laplace transform of the input keeping initial conditions zero.

Given, $\frac{\mathrm{d}\mathrm{y}(\mathrm{t})}{\mathrm{d}\mathrm{t}}+\mathrm{y}(\mathrm{t})=3\mathrm{x}(\mathrm{t}-3)\mathrm{u}(\mathrm{t}-3)$

where, y(t) = Output

x(t) = Input

Taking Laplace on both sides:

$sY(s)+Y(s)=3X(s)e^{-3s}$

$(s+1)Y(s)=3e^{-3s}X(s)$

$\frac{Y(s)}{X(s)}=\frac{3{\mathrm{e}}^{-3\mathrm{s}}}{\mathrm{s}+1}$

Concept:-

The transistor function is given by = $\frac{K\left(S+Z_1\right)\left(S+Z_2\right)---}{\left(S+P_1\right)\left(S+P_2\right)---}$

Where K = dc gain Z₁, Z₂,--- = zeros

P₁, P₂, --- = Pole location

Calculation:

The roots of the transfer function is -3, -4. 

Transfer Function:

• The transfer function of a system is defined as the ratio of the Laplace transform of output to the Laplace transform of input where all the initial conditions are zero.
• The transfer function for the system is unique & it is applicable only for linear time-invariant (LTI) systems.

Signal flow graphs:

• Signal flow graphs are used to find the transfer function of the control system by converting the block diagrams into signal flow graphs
• It cannot be used for nonlinear systems.

Block diagrams:

• A block diagram is used to represent a control system in diagram form.
• Each element of the control system is represented with a block and the block is the symbolic representation of the transfer function of that element.

State variables:

• A state variable is one of the sets of variables that are used to describe the mathematical "state" of a dynamical system.
• The state variable method is used for analysis of time-variant systems.

Concept:

• The impulse response of any system is known as the transfer function.
• The transfer function is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input by assuming initial conditions to be zero.

Explanation:

T.F. = $\frac{Y(s)}{X(s)}=H(s)$

T.F. = $\frac{Y(s)}{X(s)}=1$ 

Taking Inverse Laplace on both sides:

Concept:

• The impulse response of any system is known as the transfer function.
• The transfer function is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input by assuming initial conditions to be zero.

Explanation:

T.F. = $\frac{Y(s)}{X(s)}=H(s)$

T.F. = $\frac{Y(s)}{X(s)}=1$ 

Taking Inverse Laplace on both sides:

Concept:

Poles of closed-loop system = zeroes of the characteristic equation.

Characteristic equation is given by:

1 + G(s) H(s) = 0

Order: Highest power of characteristic equation.

Type: It is obtained by observing the number of open loop poles occuring at origin.

Analysis:

$G(s)=\frac{K}{s^2\left(sT+1\right)}$

H(s) = 1

The characteristic equation will be:

$1+\frac{k}{s^2\left(sT+1\right)}=0$

s² (sT + 1) + k = 0

s³ T + s² + k = 0

The highest power of the above characteristic equation is 3

So, order = 3

Type = 2

Concept:

Consider a second-order control system with unity feedback: 

Closed-loop transfer function is given by

$\frac{C\left(s\right)}{R\left(s\right)}=\frac{{\omega }_n^2}{s^2+2\xi {\omega }_n+{\omega }_n^2}$

 where,

C(s) = Output response of CLTF

R(s) = Input to the system

ωn = Undamped natural frequency in rad/sec

ζ = Damping ratio

Application:

Given:

Input R(t) = δ(t) implies R(s) = 1,

Closed Loop Transfer function $\frac{C\left(s\right)}{R\left(s\right)}=\frac{20(s+4)}{s^2+8s+15}$ 

Substitute R(s) = 1 in above CLTF, We get

$C(s)=\frac{20(s+4)}{s^2+8s+15}$

$C(s)=\frac{20(s+4)}{(s+3)(s+5)}$

$C(s)=\frac{10}{s+3}+\frac{10}{s+5}$

Taking Inverse Laplace Transform on both sides,

C(t) = 10 e^-3t u(t) + 10 e^-5t u(t)

C(t) = 10 (e-3t + e-5t) u(t)

Concept:

A transfer function is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input by assuming initial conditions are zero.

TF = L[output]/L[input]

$TF=\frac{C\left(s\right)}{R\left(s\right)}$

For unit impulse input i.e. r(t) = δ(t)

⇒ R(s) = δ(s) = 1

Now transfer function = C(s)

Therefore, the transfer function is also known as the impulse response of the system.

Transfer function = L[IR]

IR = L-1 [TF]

Calculation:

$\frac{d^{2}y}{d{x}^2}+y\left(t\right)=x\left(t\right)$

By aplying the Laplace transform,

s² Y(s) + Y(s) = X(s)

$\Rightarrow \frac{Y\left(s\right)}{X\left(s\right)}=\frac{1}{s^2+1}$

Impulse response will be the inverse Laplace transform of the above transfer function.

By applying the inverse Laplace transform,

⇒ y(t) = sin t

Therefore, the impusle response of the gien system is a sinusoid.

Concept:

A transfer function is defined as the ratio of Laplace transform of the output to the Laplace transform of the input by assuming initial conditions are zero.

TF = L[output] / L[input]

$TF=\frac{C\left(s\right)}{R\left(s\right)}$

So that transfer function of the system is used to calculate the output for a given input.

For unit impulse input i.e. r(t) = δ(t)

⇒ R(s) = δ(s) = 1

Now transfer function = C(s)

Therefore, transfer function is also known as impulse response of the system.

Transfer function = L[IR]

IR = L-1 [TF]

Application:

Given-

TF = H(s) = 1/s,

Input is unit step, R(s) = 1/s

Now output for t > 0 can be calculated as

C(s) = H(s) x R(s)

C(s) = (1/s) x (1/s)

C(s) = 1 / s²

L^-1[C(s)] = t

C(t) = t = ramp function

Concept:

A transfer function is defined as the ratio of Laplace transform of the output to the Laplace transform of the input by assuming initial conditions are zero.

TF = L[output]/L[input]

$TF=\frac{C\left(s\right)}{R\left(s\right)}$

For unit impulse input i.e. r(t) = δ(t)

⇒ R(s) = δ(s) = 1

Now transfer function = C(s)

Therefore, the transfer function is also known as the impulse response of the system.

Transfer function = L[IR]

IR = L-1 [TF]

Calculation:

Given differential equation is,

$\frac{dy\left(t\right)}{dt}+y\left(t\right)=\delta \left(t\right)$

⇒ sY(s) + Y(s) = 1

$\Rightarrow Y\left(s\right)=\frac{1}{1+s}$

⇒ y(t) = e^-t u(t)

Given:

Forward path gain = G

Feedback path gain = H

Input signal = R(s)

Output signal = C(s)

Error signal = E(s)

C(s) = G × E(s)

Error signal = Input signal - Feedback signal

E(s) = R(s) - H × C(s)

E(s) = R(s) - H × G × E(s)

E(s) + H × G × E(s) = R(s) 

E(s) [1 + GH] = R(s)

$\frac{E(s)}{R(s)}={\displaystyle \frac{1}{(1+GH)}}$

By using the block-reduction technique:

Step 1:

Blocks having value 2 and $\frac{4}{s}$ are in parallel.

Step 2:

Blocks having value $\left(2+\frac{4}{s}\right)$ and $\frac{1}{s}$ are cascaded.

Step 3:

Noe its a negative UFB system:

$\frac{Y(s)}{R(s)}=\frac{2s+4}{s^2+2s+4}$

Transfer function,

$\frac{V_o\left(s\right)}{V_i\left(s\right)}=\frac{1-s}{1+s}$

V_i (t) = V_m sin(ωt)

V_o (t) = V_m sin(ωt + ϕ)

Here, ϕ = tan^-1 (-ω) - tan^-1(ω)

= - tan¹(ω) - tan^-1 (ω)

= - 2 tan^-1(ω)

At ω = 0, 

⇒ ϕ = - 2 tan-1(0) = 0 (since tan^-1(0) = 0 )

At ω = ∞,

⇒ - 2 tan-1(∞) = - 2 × π/2 = - π (since tan-1(∞) = π/2)

Transfer Function:

• The transfer function of a system is defined as the ratio of the Laplace transform of output to the Laplace transform of input where all the initial conditions are zero.
• The transfer function for the system is unique & it is applicable only for linear time-invariant (LTI) systems.

Signal flow graphs:

• Signal flow graphs are used to find the transfer function of the control system by converting the block diagrams into signal flow graphs
• It cannot be used for nonlinear systems.

Block diagrams:

• A block diagram is used to represent a control system in diagram form.
• Each element of the control system is represented with a block and the block is the symbolic representation of the transfer function of that element.

State variables:

• A state variable is one of the sets of variables that are used to describe the mathematical "state" of a dynamical system.
• The state variable method is used for analysis of time-variant systems.