LINEAR SYSTEMS ANALYSIS

Code No: R7310206 1
III B.Tech. I Semester(R07) Regular Examinations, December 2009
LINEAR SYSTEMS ANALYSIS
(Electrical & Electronics Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE questions
All questions carry equal marks
? ? ? ? ?
1. (a) Explain what is meant by state variable and mention the advantages of state space ap-
proach.
(b) The transfer function of a system is G(s) = 2
(s+1)(s+3) obtain the state variable represen-
tation of the system.
(c) Write the state equation for the circuit shown in ¯gure 1.
Figure 1:
2. (a) The input voltage in volts to a series RL circuit is
e(t) = 180 sin(314t + 100) + 56 sin(942t + 350) + 18
The value of R and L are 18­ and 0.0413H. Determine:
i. The expression for the current.
ii. The rms value of voltage and current.
iii. The power factor of the circuit.
(b) Find the trigonometric series for the wave form shown in ¯gure 2.
Figure 2:
3. (a) Obtain the Fourier transform of unit step function.
(b) State and explain the properties of Fourier transform.
(c) Show that the Fourier transform of ±(t), the impulse function has constant magnitude.
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4. (a) Find the Laplace transform of the following waveform shown in ¯gure 3.
Figure 3:
(b) For the network shown in ¯gure 4 ¯nd the voltage across the inductor 4H.
Figure 4:
5. (a) State the properties of Hurwitz polynomial.
(b) Check whether the following functions are positive real function or not.
i. Z(s) = s+3
s+2.
ii. Z(s) = s2+7s+70
s(s+10) .
6. (a) Obtain the second Foster form for a network has impedance.
Z(s) = s(s2+10)
(s2+4)(s2+16)
(b) Explain how the removal of pole at in¯nity of an impedance Z(S) can realize an element
in the network.
7. (a) State and explain Sampling Theorem.
(b) What are the e®ects of Under Sampling?
(c) Write short notes on the reconstruction of signal from its sample.
8. (a) State the properties of the Region Of Convergence(ROC).
(b) Find the Z-transform of the following:
i. Unit step function.
ii. Unit impulse function.
(c) Find the inverse Z-transform of the following:
X(Z) = 1+2Z+3Z2+4Z3+5Z4
Z4 .
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Code No: R7310206 1
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Code No: R7310206 2
III B.Tech. I Semester(R07) Regular Examinations, December 2009
LINEAR SYSTEMS ANALYSIS
(Electrical & Electronics Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE questions
All questions carry equal marks
? ? ? ? ?
1. (a) What are the state variables chosen in analysis of electrical circuits?
(b) Write state variable equation for the following di®erential equations:
d2y
dt2 + 5dy
dt + 6y = sin t + 5e¡t.
(c) Obtain the state variable equation for the network shown in ¯gure 1.
Figure 1:
2. (a) Find the RMS and average values of the periodic function shown in ¯gure 2.
Figure 2:
(b) The voltage V (t) = 4
¼ £sin 2¼t
1 + sin 6¼t
3 + sin 10¼t
5 + :::::::::::::::::1¤is applied to circuit of R =
4­ in series with an inductance of L = ¡1
¼ ¢H.
Calculate the average power and power factor.
3. (a) State and explain any four properties of Fourier transform.
(b) Obtain the Fourier transform of the following:
i. Signum function.
ii. Rectangular pulse.
iii. Ramp signal.
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4. (a) Obtain the L.T. of the following waveform shown in ¯gure 3.
Figure 3:
(b) Find the current through the resistance R2 if the applied voltage is Vi = V ¡at
e to the circuit
shown in ¯gure 4.
Figure 4:
5. (a) Test whether the following polynomial is Hurwitz or not.
i. H(S) = S4 + S3 + 5S2 + 3S + 4.
ii. H(S) = S3 + 2S2 + 3S + 6.
(b) Test whether the following function is a positive real function or not.
N(s) = s3+s2+3s+5
s2+6s+8 :
6. (a) Synthesis given function in II Cauer form
F(S) = 2(s+1)(s+3)
S(S+2) .
(b) State the properties of RL impedance of RC admittance functions.
7. Write short notes on the following:
(a) Sampling theorem.
(b) Aliasing.
(c) Properties of correlation function.
8. (a) What are the di®erences between continuous and discrete time signals?
(b) Find the sequence corresponding to the following Z-transformed function given by
X(Z) = Z3¡4:8
Z(Z¡0:2)(Z¡0:4) .
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Code No: R7310206 2
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Code No: R7310206 3
III B.Tech. I Semester(R07) Regular Examinations, December 2009
LINEAR SYSTEMS ANALYSIS
(Electrical & Electronics Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE questions
All questions carry equal marks
? ? ? ? ?
1. (a) Why are the currents through the resistances and voltage across a conductance not chosen
as state variables.
(b) Find eAt for A = · ¡1 ¡1
0 1 ¸.
(c) Obtain the state equation for the circuit shown in ¯gure 1.
Figure 1:
2. (a) An alternator gives an output of 100 sin !t + 30 sin !t + 20 sin 5!t
Where ! = 100. If this voltage is applied to a load of 10­ series with 0.01H, ¯nd the current,
average power and power factor of the alternator.
(b) Obtain the RMS and average values of the following functions.
i) f(t) = sin2 t. ii) f(t) = sin t + 20 cos 3t + 3 sin(5t + £¼
4 ¤).
3. (a) State and explain Parseval's theorem.
(b) Obtain the Fourier transform of the decaying exponential function. Also draw the magnitude
and phase spectra of exponential function.
4. (a) State the convolution theorem and determine the Laplace transform of the given function
by using convolution theorem.
G(S) = S
(S+1)(S+3) .
(b) A voltage pulse of 10V magnitude and 5¹ sec duration is applied to the RC network shown
in ¯gure 2. using LT method.
Figure 2:
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5. (a) Test whether the following function is positive real function or not.
N(S) = S3+4s2+7s+3
s3+3s2+5s+6 .
(b) i. Check whether the polynomial s4 + s3 + 7s2 + 6 is Hurwitz or not.
ii. Find the range of values `a' so that
H(S) = S4 + S3 + aS2 + S + 3 is Hurwitz polynomial.
6. (a) Obtain the ¯rst Cauer form of Z(S) = (S+2)(S+4)
(S+1)(S+3) .
(b) Explain the realization of ¯rst foster form of R-c equivalent network deriving necessary
expressions.
7. (a) What is the necessity of sampling process?
(b) What are the di®erent types of sampling methods?
(c) Write short notes on correlation of energy signals.
8. (a) Find the Z-transform of the following:
i) f(t) = t2. ii) F(S) = 1
S(s+a) .
(b) Find the inverse Z-transform of the following:
i) F(Z) = Z
Z+a . ii) F(Z) = Z¡4
(Z¡1)(Z¡2)2 .
? ? ? ? ?
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Code No: R7310206 3
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Code No: R7310206 4
III B.Tech. I Semester(R07) Regular Examinations, December 2009
LINEAR SYSTEMS ANALYSIS
(Electrical & Electronics Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE questions
All questions carry equal marks
? ? ? ? ?
1. (a) What are the properties of state transition matrix?
(b) Evaluate the state transition matrix eit, for the following system equation
X(t): = · 0 1 0
0 0 1
1 0 0 ¸X(t) + · 0
0
1 ¸u(t)
(c) Write state equation in matrix form for the circuit shown in ¯gure 1.
Figure 1:
2. (a) Find the exponential Fourier series for the waveform shown in ¯gure 2.
Figure 2:
(b) The current waveform shown in ¯gure 3 is applied to a circuit containing of 0:01¹F in parallel
with 1K­ through a ¯lter that allows frequencies between 13 and 14 KHz. Find the average
power delivered to 1K­.
Figure 3:
3. (a) State the properties of positive real function.
(b) Check whether the following functions are Hurwitz or not.
i. H(S) = S4 + 6S3 + 2S2 + S + 1.
ii. H(S) = S5 + 4S4 + S3 + 2S2 + S + 1.
4. (a) Synthesize the network in ¯rst foster form
Z(S) = 2(S2+1)(S2+9)
S(S2+4) .
(b) State and explain the properties of L-C immittance functions, deriving necessary expressions.
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5. (a) Find the Fourier transform of the time signal shown in ¯gure 4.
Figure 4:
(b) Obtain the Fourier transform of the double sided exponential signal.
6. (a) Find the voltage across 2­ resistor in the circuit shown in ¯gure 5 using L-T method.
Figure 5:
(b) Find the Laplace transform of the following ¯gure 6.
Figure 6:
7. Explain the following with suitable examples.
(a) Impulse sampling.
(b) Natural and °at top sampling.
(c) Band pass sampling.
(d) Power density spectrum.
8. (a) Find the Z-transform of the following:
i. e¡at = f(x)
ii. F(s) = 1
((s(s+1))
(b) Obtain the inverse Z-transform of the following :
i. X(Z) = 10
(Z¡1)(Z¡2)
ii. X(Z) = 1+Z¡1+Z¡2
1¡Z¡1
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Code No: R7310206 4
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