Linear and discrete systems analysis Jntu Model Paper{Www.Studentyogi.Com}

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Code No: R05310205
Set No. 1
III B.Tech I Semester Regular Examinations, November 2007
LINEAR AND DISCRETE SYSTEMS ANALYSIS
(Electrical & Electronic Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks

1. (a) An engineer has designed the electrical circuit as shown in gure 1a to supply
power from the voltage source to the three resistive loads.

Figure 1a
De ne state variables and nd the corresponding state and output equations
in matrix form for this electric circuit.
(b) Determine transfer function for the system described by the following di er-
ential equation ¨ + 4 + 3 = + [8+8]

2. A 2 ohm resistive load is supplied from a full wave recti er connected to 230V,
50Hz single phase supply. Determine the average and rms values of load current.
Also nd out the proportion of DC power and AC power to the total power in the
load. Investigate the e ect of adding an inductance in series with the load.[8+4+4]

3. (a) Explain the concept of continuous spectrum.
(b) Find the Fourier Transform for the following functions as shown in Figure 3(i),
3(ii). [6+10]


Code No: R05310205
Set No. 1

(i) & (ii) Figure 3
4. (a) Determine the current in a series RL circuit shown in below driven by a Square
wave voltage source of amplitude 1 and half period T/2 = 1.

Figure 4a

(b) Find v(t) for a system whose ( ) = s+4 s(s+1)2 (s+3) . using of Heaviside theorem.
[8+8]

5. (a) State and explain the properties of positive real function.
(b) Check whether given polynomial ( ) = 2 4 + 5 3 + 6 2 + 2 + 1 is Hurwitz
or not. [8+8]

6. Find the networks for the following functions in one Foster and one Cauer form

(a) ( ) = (s+1)(s+3)
(s+4)(s+2)
[2 × 8]
(b) ( ) = 2(s+0.5)(s+4) s(s+2) .


Code No: R05310205
Set No. 1
7. The sampling theorem with Ts= M that is m(t)= 8 ( s) n( ) where
n=-8
wM (t-nTs ) , n=0 ±1 ±2 ±8. Show that n( ) is orthogonal
n( ) = sin wM (t-nTs )

over the interval -8 8 and 8 n( ) k( ) = s nk. [16]
-8

8. Show that if x(n) is a right-sided sequence and X(z) convergence for some value of
z, then the ROC of X(z) is of the form | | max or 8 | | max where max is
the maximum magnitude of any of the poles of X(z). [16]


Code No: R05310205
Set No. 2

1. (a) The natural response of a certain system is described by the homogeneous
state equation.
1+71-2=0

2 + 12 1 = 0

Show that the state transition matrix can be written as

( ) = 4 -4t -3 -3t -3t - -4t
12 -4t -12 -3t 4 -3t -3 -4t
(b) Show that the state variable formulation for the Circuit as shown in gure 1b
can be written as
d c = -8 3 + -8 8 [8+8]
dt -5 0 c 05x
L L y

Figure 1b
2. (a) A series RLC circuit with R=25W, L = H and C = 10mF is energized with
a source ( ) = 15 sin 100 + 20 sin 200 + 5 sin 200 Determine the e ective
value of current and average power consumed by the circuit.
(b) What is meant by Fourier series of a non-sinusoidal periodic waveform? Ex-
plain the signi cance of the term “Half wave symmetry” used in determining
the Fourier series of a given waveform. [8+8]

3. (a) Does the FT of external (- + ) signal cos ot exist? If no, give reasons.
Derive FT of cos ot. Draw the spectral density function.
(b) Find the fourier transform for the doublet pulse and normalized ganssion pulse
as shown in gure 3a, gure 3b. [8+8]


Code No: R05310205
Set No. 2
(a) Figure 3a

Figure 3a
(b) Figure 3b

Figure 3b

4. (a) Find the Inverse LT of
( ) = 2s3 -9s2 +4s+10
s2 -3s+4
(b) Use the LT, nd the voltage across the capacitors, y(t) for the RC circuit
-2t ( ) and
shown in gure 4b, in response to the applied voltage ( ) = 3
5
initial condition (0-) = -2

Figure 4b
5. (a) Check if the polynomial ( ) = 2 4 + 5 3 + 6 2 + 2 + 1 is Hurwitz or not.

(b) Check whether the function ( ) = 2s2 +2s+1 s3 +2s2 +s+2 is a positive real function.
[8+8]

6. Indicate which of the following functions are either RC, RL, or LC impedance
functions. Give reasons.

(a) ( ) = (s+1) (s+3)
s(s+4)

(b) ( ) = (s+3) (s+7)
(s+2) (s+5)

(c) ( ) = s2 +4s+3
s2 +6s+8
[4 × 4]
(d) ( ) = s2 +5s+6 s2 +s .
7. Show that if the sampling rate is equal to or greater than twice the highest message
frequency, the message m (t) can be recovered from the natural sampled signal xns(t)


Code No: R05310205
Set No. 2
8. (a) How is the region of convergence de ned for a nite duration signal?
(b) Derive the di erentiation property in Z-domain.
(c) Explain the relationship between S-plane & Z-plane. [4+4+8]


Code No: R05310205
Set No. 3

1. (a) Write the state equations for the circuit as shown in gure 1a.

Figure 1a
(b) Solve the state equations.
-t
d
dt = -2 0 0 -4 +
2 -t
with initial conditions
0 =1 [8+8]
1.
0

2. A 2 ohm resistive load is supplied from a full wave recti er connected to 230V,
50Hz single phase supply. Determine the average and rms values of load current.
Also nd out the proportion of DC power and AC power to the total power in the
load. Investigate the e ect of adding an inductance in series with the load.[8+4+4]

3. (a) Evaluate the following integrals and functions.

i. a [ 2 + 1] ( )
-a

ii. 5 [ 3 + 4 + 2] ( - 1)
3
(b) Find the FT for the following functions.
i. Unit step function
ii. Dirac Delta function. [8+8]

4. (a) In the RC circuit shown in gure 4a, switch is closed at time t = 0. Determine
the current i(t) after the switch is closed. Assume that there is no change in
the capacitor before switching.


Code No: R05310205
Set No. 3

Figure 4a
(b) Find the Inverse LT of the following function.
( ) = 5S-12S2 +4S+13 .
[8+8]

5. (a) Find the range of value of ‘a’ so that ( ) = 4 + 3 + 2 + + 3 is Hurwitz.

(b) Find the even and o dd parts of impedance function ( ) = 1+s+s2 (4+s+s2 ) and hence
nd out Re Z(s) and Im Z(s) is positive real as well as realizable. [8+8]

6. Synthesize the following functions in Cauer form.

(a) ( ) = s3 +2s2 +s+1
s3 +s2 +s
[2 × 8]
(b) ( ) = s3 +2s2 +2s+1 s4 +s3 +3s2 +s+1 .
7. A periodic signal f(t) shown in the gure 7 is transmitted through a system with
transfer function H(w). For these di erent values of T ( = 2 3 3 6),
nd the power density spectrum and the power of the output signal. Calculate the
power of the input signal f(t). [16]

Figure 7

8. The output y(n) of a discrete-time LTI system is found to be 2(1/3)n ( ) when the
input x(n) is u(n)


Code No: R05310205
Set No. 3
(a) Find the impulse response h(n) of the system.
(b) Find the output y(n) when the input x(n) is 1 n ( ). [8+8]
2


Code No: R05310205
Set No. 4

1. (a) Write the matrix state equation for the network shown in gure 1a.

Figure 1a
(b) Find the response of the circuit as shown in gure 1b. [8+8]

Figure 1b
2. (a) Determine the Fourier series expansion for triangular waveform shown in gure
2a.

Figure 2a


Code No: R05310205
Set No. 4
(b) Determine the exponential form of Fourier series for the following function
shown in gure 2b. [8+8]

Figure 2b

3. (a) A signal -3t ( ) is passed through an ideal low pals lter with cut o frequency
of 1 rad per second.
i. Test whether the input is an energy signal.
ii. Find the output energy.
(b) Write short notes on “Power density spectrum”. [8+8]

4. (a) A step DC current of 5 ampers is applied at time t = 0 to a parallel RLC
circuit shown in gure 4a, consisting of resistor = 1 7 , induction L =
0.1 Henry and Capacitor C =1 Farad. Determine the voltage V(t) across the
circuit. Assume zero change across the capacitor C.

Figure 4a

(b) Current I(s) in a network is given by ( ) = 2S+3 S2 +3S . Find i(t), the current at
any time. [8+8]

5. (a) Show that the function ( ) = (s+2)(s+4) (s+1)(s+3) is positive real.
(b) Check if the polynomial ( ) = 4 + 3 + 2 2 + 2 + 24 is Hurwitz or not.
[8+8]


Code No: R05310205
Set No. 4
6. An impedance function is given by ( ) = s(s+2)(s+5) (s+1)(s+4) Find the R-L representation
of

(a) Foster-I and II forms.
(b) Cauer-I and II forms. [2 × 8]

7. (a) Explain how sampling is done in the case of band pass signals and how the
message is reconstructed from its samples.
(b) A band pass signal has a center frequency fo and extends from fo-5 KHz to fo+5
KHz. The signal is sampled at a rate fs = 25 KHz. As the center frequency
fo varies from fo = 5KHz to fo = 50KHz. Find the ranges of fo for which the
sampling rate is adequate. [8+8]

8. (a) Find the fundamental perio d of following signals if they are periodic?

i. ( ) = ( pn2 4)

ii. 8 [ ( - 4 ) - ( - 2 - 4 )]
m=-8
(b) Determine the energy, E for the following sequences:
n-6 ( - 6)
i. 1( ) = 1
2

ii. 2( ) = 4 1/3 n-1 - 3 1/4 n-3 ( - 5)

iii. 3( ) = -n-1 ( )
iv. 4( ) = ( - 1) ( ). [4 + 4 + 4 × 2]

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