LINEAR SYSTEMS ANALYSIS.pdf

Code No:- V3107/R07

III B.Tech I Semester Supplementary Examinations, April/May - 2012

LINEAR SYSTEMS ANALYSIS

(Electrical and Electronics Engineering)

Time: 3 Hours Max Marks: 80

Answer any FIVE Questions

All Questions carry equal marks

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1. a) Write state equations for the network shown using graph theory.

b) What is state transition matrix? Discuss about its properties.

2. a) Find the Fourier transform of 0tsin x(t))( ω=i tjw

eii 0 x(t))( =

b) Determine the Fourier series of the function

f (t) = t 0 < t < π

= 0 π < t < 2 π

3. a) The voltage applied to a load is

tttv 1131sin70754cos140377sin215 ++=

)601131sin(3)45754(cos4)30377(sin15 000−+−+−= ttti Determine the average

power in the load and effective value of current and voltage.

b) A voltage ∑∞

− Π

−+=

1

sin)2

(2

)(n

in tnn

kktV ω is applied to the circuit shown below.

Determine i (t)

1 of 2

R1 R2

C2

C1

L

is

2F i(t)

25 Ω Vin (t) 3H

Set No: 1

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Code No:- V3107/R07

4. Determine the current i(t) in a series RLC circuit consisting of R=5Ω, L=1 H and

C=(1/4)F When each of the following voltages are applied:

(i) ramp voltage 12 r (t-2) (ii) step voltage 3 u(t-3). Assume that the circuit is

initially relaxed.

5. a) Discuss the properties of positive real function.

b) Test if the polynomial 812623

+++ sss is Hurwitz.

6. a) The driving point impendence of a network is given by

6

149

26)(

24

3

++

+=

ss

sssZ .Find

the first form of Cauer network.

b) The driving point impedance of a one port LC network is given by

)4(

)16()9(2)(

2

22

+

++=

ss

sssZ obtain the Foster form of LC network realization.

7. a) Define Auto-correlation and explain properties of it.

b) Explain the principle of natural sampling with neat schematics.

8. a) Determine the inverse of the Z-transform )5.0()1(

32)(

2

−−

+=

zz

zzzX

b) Determine the Z-transform of the ramp sequence Annx =)( and the sequencen

rAnnx =)( .

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Set No: 1



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1. a) Obtain the state equation for the circuit shown below.

b) Determine the state transition matrix of the system described by

)(1

0

43

21

2

1

2

1tu

x

x

x

x

+

−−=

•

•

2. a) Find the Fourier transform of the functions ta

etxi−

=)()( ttxii =)()(

b) Obtain the Fourier series for the square waveform shown

3. A series RLC circuit has an applied voltage of

.942sin75628sin100314sin150)( ttttV ++= If R =5Ω, L=5 mH and C=50µf.

Determine the effective current and average power.

1 of 2

3F

Vc

1Ω

iL

4Ω 6Ω

2H

Set No: 2

V

π 2π

-V

t

f(t)



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Code No:- V3107/R07

4. a) A 20 V step voltage is applied across a RC Series circuit at t=0, Find i(t) at t=0+ and

obtain the value of dt

di at t = 0+. Assume R=100 Ω, C=100 µF.

b) If )(2)(1 tutf = and ),()(3

2 tuetft−

= determine the convolution between f1(t) and

f2(t).

5. a) Find the limits of K so that the polynomial Ksss +++ 5614 23 may be Hurwitz.

b) Test whether the function 52

42

++

+

ss

s is positive real or not.

6. The driving point impedance of a one port LC network is given by

)16(

)25()4(8)(

2

22

+

++=

ss

sssZ . Obtain the first and second foster form of equivalent

networks.

7. a) Consider a filter with ω

ωj

H+

=1

1)( and input ).()( 2

tuetxt−

= Find the energy

spectral density of the output.

b) Define cross-correlation function and explain the properties of it.

8. a) The z-transfer function of a system is given as

18

1

2

1

1)(

2+−

=

zz

zH . Determine its

delta response assuming the system to be initially released.

b) Determine z-transform of)1(

1)(

2+

=ss

sF .

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Set No: 2



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Code No:- V3107/R07







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1. a) Form the state matrix equation for the network shown below.

b) A network is described by the following state equations: ux

xx

+

−=

•

2

0

4.00

10

2

1

If x1 (0) = 1 and x2(0) = 0. Obtian the complete solution of the network.

2. a) Determine the Fourier transform of signum function and unit step function.

b) Obtain Fourier coefficients for the function given by )()( π+= ttf when

f(x))2f(x ; =+<<− πππ t

3. A voltage v(t)=(100 sin 314t +20 sin (942t+300)+5) is applied to a series R-L circuit with

R=10 Ω and L=0.0318H. Find the expression for current, rms voltage, rms current and

power factor of the circuit.

4. a) In a series LC circuit, a 50 V dc is applied at t=0. Find the voltage across the capacitor

at ∞=t . Assume zero initial condition in the circuit elements.

b) In the circuit shown below the switch K is thrown from position A to position B at

time t=0, the current having preciously reached its steady state. Determine )(ti after

switching.

1 of 2

C1

iL ic1

C2 L

R ic2

iR

B

A 5 Ω

10v

K

2H 5 Ω

Set No: 3



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Code No:- V3107/R07

5. a) Test the function 3

2

)1(

7

+

+

s

s for positive realness.

b) Check whether the polynomial 647 234++++ ssss is Hurwitz or not.

6. The driving point impedance of a one port reactive network is given by

)16)(1(

)4(4)(

22

2

++

+=

ss

sssz Obtain the Foster forms of LC network realization.

7. Write shorts notes on

(i) Band pass sampling

(ii)Auto-correlation function

8. a) Discuss briefly about the properties of Z- transforms.

b) Determine the inverse of the following Z-transform 1

2

1

2)(

22−

++

=z

z

z

zzx

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1. a) For the network shown obtain the state equations

b) Using state variable analysis, obtain the complete response of the state vector.

2. a) State and prove the linearity and time scaling properties of Fourier transform.

b) Find the exponential Fourier series of the given waveform.

3. a) A R-L series network with R=10 Ω and L=5H contains a current i(t)=5 sin 1000t+5

sin 3000t+5 sin 5000t. Determine the effective applied voltage, rms current and average

power.

b) Determine the output voltage across the capacitor if the excitation is a current source

i(t)=e-t u(t).

4. For a series RL circuit, a voltage v (t) = (20 cos 5t +30 cos 10t) u (t) is given where u(t)

is the unit-step function. Find the current flowing in the circuit assuming that the circuit

is initially relaxed.

1 of 2

C1

iL

R1

C2

vc1 V(t)

L R

vc2 -

+ + +

- -

4π

A

2π

wt o

f (t)

Set No: 4

2F 1 Ω i(t)=e-t u(t)



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Code No:- V3107/R07

5. a) Test whether the polynomial ssssP ++=35

3)( is Hurwitzian or not.

b) Check the positive realness of the function10

202)(

2

+

++=

s

sssY

.

6. a) Synthesize the function)2(

)3)(1(4)(

+

++=

ss

sssZ , using Cauer form I of realization.

b) Using Foster form II, synthesize the function )9(

)13)(5()(

2

22

+

++=

ss

sssZ

7. a) Find the auto correlation of the periodic function x(t)=X sin wt

b) State and explain the sampling theorem.

8. a) The z-transfer function of a system is given as

18

1

2

1

1)(

2+−

=

zz

zH . Determine its

discrete unit step response assuming the system to be initially relaxed.

b) Determine the inverse z-transform of2)1)(2(

)(−−

=zz

zzx

.

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Set No: 4



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LINEAR SYSTEMS ANALYSIS.pdf

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