Class Co-ordinator: Ms. Chaya.N.S

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SL.

NO.


SUBJECTS

CODE

STAFF

PAGE NO

1.


INDEX

--

--

0
1

2.


METHODS OF APPLIED
MATHEMATICS
-
II

MA
-
251

Ms.BSA

2


8

3.


NETWORK SYNTHESIS

E
E
-
252

Ms. KP/CAS

9


11

4.


SIGNALS AND SYSTEMS

EC
-
253

Dr. SHL

12


20

5.


CONTROL SYSTEMS

EE
-
254

Ms. KRS

21


30

6.


LINEAR INTEGRATED
CIRCUITS

EC
-
255

Mr. RC

31


33

7.


MICROPROCESSORS


EC
-
256

Ms. SR

34



Class Co
-
ordinator: Ms. Chay
a.N.S




4th sem

Question Bank

INDEX

P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


2



METHODS OF APPLIED MATHEMATICS
-
II


QUESTION BANK


Subject Code : MA
-
251

Faculty: Ms.
BSA


Complex Analysis


1. Sho
w that the function f(z)=
xy

is everywhere continuous but is not analytic

2, Show that each of the following functions is not analytic at any point


(i)
z

(ii)
2
z

3. Determine the

analytic function whose real part is


(i)
1
3
3
3
2
2
2
3




y
x
xy
x


(ii)
2
2
log
y
x



(iii)
y
x
y
y
x
x
sinh
cos
cosh
sin


4. Find the regular function whose imaginary part is


(i)
2
2
y
x
y
x




(ii)
y
e
x
sin


(iii)
)
2
sin
2
cos
(
y
y
y
x
e
x



5. Find the analytic function z=u +iv if


(i) 2u+v=
)
sin
(cos
y
y
e
x



(ii)u
-
v=
2
2
4
y
xy
x
y
x




6. Find the invariant points of the transformation
1
1



z
z
w

7. Find the transformation

which maps the points
-
1 , i , 1 of the z
-
plane onto 1,i
-
1 of the w
-
plane

8. Find the bilinear transformation which maps 1.i.
-
1 to 2, i,
-
2 respectively. Find the fixed and critical points of the
transformation

9. Show that bilinear transformation w=
4
3
2


z
z

maps the circle x
2
+y
2
-
4x=0 into the line 4u+3=0

10.Under the transformation w=1/z


(a) find the image of
2
2


i
z


(b) show that the image of the hyperbola
1
2
2


y
x

is lemniscates


2
cos
2











P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


3


Unit
-
3



NUMERICAL METHODS


1

Use Bisection method to find a real root of the following equations:

1. x
3
-

2x

=
R===M=in=R=stages==,=up=to=P=decimal=places
=
2.==x=log
10

x = 1.2 between 2 & 3 up to 3 decimal places

3. x
3
-

5x
-

4 = 0 in the range (2,3) up to 3 decimal places

4. x
3
-

3x

=
R===M=in=the=range=E2,2.RF=correct=to=2=decimal=places=
=
R.==x
3
-

3x+ 1 = 0 in the range (1, 2) correct to 3 decimal places

2

Use Regula

=
䙡lsi=method=to=find=a=real=root=of=the=following=
equationsW=
=
E=correct=to=P=decimal=places=F
=
N.==sinx=

=
cosh=x=+=N==M===
=
2.==x
2
-

log
e
x

=
N2===M===
=
P.==cosx=
-
=
Px=+=N===M===
=
4.==e
x
-

3x = 0

5. x
4
+ 2x
2
-

16x + 5 = 0 in (0,1)

6. 2x + log
10
x = 7 in (3.5,4)

3

Use Newton Raphson method to find a real
root of the following equations:

(correct to 4 decimal places ).

1

x =
12


2

Cos x =
x


3

x
3
+ 5x + 3 = 0 in (1,2)

4

x
2
+ 4 sin x = 0

5

x
5
-

3.7 x
4

+ 7.4 x
3


=
NM.U=x
2

+ 10.8 x

=
S.U===M
=
4
=
䙩nd=the=negative=root=of==x
3
-

2x + 5 = 0 correct to 3 decimal places using Newton Raphson method.

5

Find an iterative formula for finding
N
3

where N is a positive real number using Newton Raphson
method. Hence evaluate
10
3

correct to 4 dec
imal places.

6

Write the difference table for the following set of values

x:

1

2

3

4

5

y:

2

2.301

2.477

2.778

2.699


7


I f f (x ) i s a c u b i c p o l y n o mi a l f i n d t h e mi s s i n g n u mb e r.

x:

0

1

2

3

4

y:

5

1

--

35

109



P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


4


8

Find the values of f (22) and

f (42) from the following table:

x:

20

25

30

35

40

45

f(x):

354

332

291

260

231

204


9

Find f(x) when x = 0.1604 from the table given below :

x:

0.1 6 0

0.1 6 1

0.1 6 2

f ( x ):

0.1 5 9 3 1 8 2 1

0.1 6 0 3 0 5 3 5

0.1 6 1 2 9 1 3 4


10

Apply Newton

=
dregory=forward=diff
erence=formula=to=find=f=ERF=given=that=
=
f=EMF===2,====f=E2F===T,===f=E4F===NM,===f=ESF===N4,===f=EUF===N9,==f=ENMF===24.
=

=
ff=f=ETRF===24S,====f=EUMF===2M2,===f=EURF==NNU,===f=E9MF===4M,=find===f=ET9F=using=
=
kewton=

=
dregory=forward=difference=formula
=

=
䙩nd=the=polynomial=approximating=f=ExF==using=kewton
-
=
dregory=forward=difference
=
formula,=given=f=E4F===N,=f=ESF===P,=f=EUF===U,=f=ENMF===2M.
=

=
Apply=kewton
-
=
dregory=backward=difference=formula=to=find=the=polynomial======approximating=f=ExF,=
given=f
=
EMF===2,=f=ENF===P,=f=E2F===N2,=f=EPF===NS.
=

=
䙩nd=f=ETF=rsing==kewton=

=
dregory=backward=difference=formula=given==that=
=
f=E2F===T,==f=E4F===NS,==f=ESF===2N,==f=EUF===24,=f=ENMF===PM,=f=EN2F===PR.
=

=
Apply Stirling’s formula to find f (35) given
that=
=
f=E2MF===RN2,==f=EPMF===4P9,==f=E4MF===P4S,==f=ERMF===24P
=

=
Apply Bessel’s formula to find f (6) given that
=
f=E=N=F===T,==f=EPF===NS,==f=E=R=F===2N,=
=
f=E=T=F===24,=
=
f=E=9=F===PM,=
=
fENNF=PR.
=

=
Employing Stirling’s formula, find the polynomial th
at=approximates==f=ExF=given=that
=
=
f=E=N=F===N,==f=E=2=F===
-
N,==f=E=PF===N,==f=E4F====
-
N,==f=E=R=F===N.
=

=
qhe=values=of=e
x

for different x are tabulated below. Find the approximate value

of e
1.4

using (i) Stirling’s formula and (ii) Bessel’s formul
a
=
=

=
N.N
=
N.2
=
N.P
=
N.4
=
N.R
=
N.S
=
N.T
=
e
x
:

3.004

3.320

3.669

4.055

4.482

4.953

5.474



P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


5


19

The table below gives the distance in nautical miles of the visible horizon for the given

heights, in feet, above the earth’s surface. Find d when x = 410 feet.
=
eeightExFW
=
㄰N
=
ㄵN
=
㈰2
=
㈵2
=
㌰P
=
㌵P
=
㐰4
=
aistanceEdFW
=
NM.SP
=
NP.MP
=
NR.M4
=
NS.UN
=
NU.42
=
N9.9M
=
2N.2T
=

20

Estimate the values of (i) sin 38
0

and (ii) sin 22
0

from the table given below:

x(degrees):

0

10

20

30

40

sin x:

0

0.17365

0.34202

0.50000

0
.64279


21

Given the following table, estimate the values of y at x = 2.2 using


(i) Stirling’s formula and (ii) Bessel’s formula
=

=
M
=
N
=
2
=
P
=
4
=

=
N
=
N.NRUR
=
N.2UNT
=
N.PSSM
=
N.4MUU
=

22

Use Lagrange’s interpolation formula to find the value
=
of=y=at=x===NM=from=the
=
following=tableW
=

=
R
=
S
=
9
=

=

=

=

=

=

=

23

Solve f (x) = 0 using Lagrange’s interpolation formula given that
=
f=EPMF===
-
PM,==f=E=P4=F===
-
NP,==f=E=PU=F===P,==f=E=42=F===NU.
=

=
Use Lagrange’s interpolation formula to fit a
=
polynomial=to=the=dataW
=

=
M
=
N
=
P
=
4
=

=
-

=
M
=
S
=

=

25

Given log 654 = 2.8156, log 658 = 2.8182, log 659 = 2.8189, log 661 = 2.8202,

find log 656 using Lagrange’s interpolation formula.
=

=
Given the following table, find f(2.2) using Newton’s d
ivided=difference=formulaW
=
=

=
N.M
=
2.M
=
2.R
=
P.R
=
4.M
=
fExFW
=
U4.2U9
=
UT.NPU
=
UT.TM9
=
UU.PSP
=
UU.RSU
=

27

If
2
1
)
(
x
x
f


show that the first divided differences of a, b, c is
2
2
2
c
b
a
ca
bc
ab



28

Apply Newton’s divided difference formula to fin
d=f=EUF==given=that=
=
f=E4=F====4U,==f=E=R=F===NMM,==f=E=T=F====294,==f=E=NM=F===9MM,==f=E=NN=F===N2NM.
=
P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


6

29

Use Trapezoidal rule to evaluate:




a)

2
0

e
x
dx ; 6 i nterval s b)

2
/
0

si n
2
x dx ; 6 intervals c)

2
1


x dx ; 8 intervals .


30

Evaluate
2
1
0
1
x
dx



usin
g Trapezoidal rule with h = 0.2. Hence determine the value of

.

31

Evaluate
dx
e
x
2
1
0



by dividing the range into 4 equal parts using Trapezoidal rule

32


Use Simpson’s one
-
third rule to evaluate:

a)


1
0
3
1
x
dx

; 6 intervals b)
dx
x
sin
2
/
0



; 6 intervals c)

2
1


x dx ; 8 intervals


33

Find the value of
3
/
1
2
log
from
dx
x
x
3
2
1
0
1


using Simpson’s one
-
third rule with

h = 0.25

34

A curve passes through the points as given in the table .Find the area bounded by the

curve, x axis, x = 1 and x= 9 using (i) Trapezoidal rule (ii) Simpson’s 3/8
th

rule

x

1

2

3

4

5

6

7

8

9

y

0.2

0.7

1

1.3

1.5

1.7

1.9

2.1

2.3


35

Use Simps
on’s three


eighth rule to evaluate :

a)

5
.
1
1
1
3

x
e
x

dx ; 6 intervals b)

2
.
5
4

log x dx ; 6 intervals c)
2
1
0
1
x
dx


; 6 intervals



36

Use Weddle’s rule to evaluate





a)
2
6
0
1
x
dx



b)
dx
x
log
4
.
1
2
.
0


c)
5
4
3
0


x
dx

( 6 intervals in each case )

37

Evaluate
x
dx


1
1
0

using (
1)
Trapezoidal rule (2) Simpson’s one
-
third rule

(3) Simpson’s three


eighth rule and (4) Weddle’s rule. Find the error in each method by

comparing with actual integration up to 4 places of decimals. Take h = 1/6 for all

cases.


P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


7






38


Use Taylor’s
series=method=to=solve=the=following=W
=
=======
(a) y’ = x y
1/3

; y(1) = 1 Find y(1.1) and y(1.2).


(b) y’ = xy + y
2

; y(0) = 1 Find y at x = 0.1 and 0.2.


(c) y’ = x
2

+ y
2


=
2===;======

yEMF=====N===䙩nd====y=at=x===M.N==and===M.2=
=

=
Use Modified Euler’s method to solve the following.
=
=======
EaF==dy===E=xy=

=
N=Fdx==;======yENF=====2.==䙩nd==yEN.2F===taking===h=====M.N.
=
=======
(b) y’ = 3x + y/2 ; y(0) = 1. Find y(0.1)
===
and==yEM.2F.
=
=======
(c) y’ = 1

=
2xy========;=======yEMF=====N===䙩nd==yEM.2F====taking==h===M.N.
=

=
Use Modified Euler’s method to solve the following:
=
=======
(a) y’ = y

=

====
;======yEMF=====N.==
=
䙩nd===yEM.NF====and===yEM.2F.
=

=
====
y
=
=======

) y’ =
2y==
+=x
3

; y(1) = 0.5

Find y(1.2) and y(1.4).


x


(c) y’ = y + x
2

; y(0) = 1.


Find y(0.2) and y(0.4).

41

Use Runge

=
hutta==E=fourth=order=F=method==to=solve=the=following.=W
=
=======
(a) y’ = x + y
2

; y(0) = 1 Find y (0.2) in steps of 0.1.


(b) y’ =


2
+ y
2
) ; y (1) = 1.5 Find y (1.2) in steps of 0.1.


(c) y’ = 3x +

======
;==y=EMF=====N.===䙩nd==y=EM.2F=taking===h===M.N
=
=================
==============
2
=

=
rse=ounge=

=
hutta==E=fourth=order=F=method==to=solve=the=following.=W
=
EaF==
y
x
dx
dy


1
=
,=y=E=M=F===N=,=find=y=E=M.N=F
=
=
EbF==
2
1
x
x
y
dx
dy


,=y=E=N=F===N=,=find=y=E=N.2=F=with=h===M.N
=










P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


8
















































PARTIAL DIFFERENTIAL
EQUATIONS

1


Form the P.D.E by eliminating the arbitrary constants a, b, c appearing in the following relations:


_________


(a) z = xy + y


x
2

-

a
2

+ b (b) z = axe
y

+ (1/2)a
2

e
2y

+ b





(c)
x
2

+

y
2

+

z
2

= 1 (d) z= (x
-
a)
2
+ (y
-
b)
2


a
2

b
2

c
2


2

Form the P.D.E by eliminating the arbitrary functions in the following relations:


(a) xy + yz + zx = f (z/(x + y))



(b) x + y + z = f (x
2

+ y
2

+ z
2

)


(c) z = f (x) + e
y

g (x)


(d) x = f (z) + g (y).


(e) ax + by + cz = f (x
2

+ y
2

+z
2
)


(f) z = f (x+ay) +


(x
-
ay)


(g) z = y
2
+ 2f (1/x + log y)
.


3

(a) Find the P.D. E of all spheres of radii ‘r’ and centers on the plane z = 0.



(b) Find the P.D. E of all planes which are at a constant distance from the origin.

4

Solve the following equations:


(a)
a
y
x
y
x
z





2


(b)
xy
x
z



2
2

(c)
)
3
2
cos(
2
3
y
x
y
x
z







(d)
z
y
z



2
2
, gives that when y=0, z = e
x

and
x
e
y
z






(e)
x
e
t
x
u
t
cos
2






5

Solve the following linear P.D.E s:


(a) p

-

2q = 2x
-

e
y

+ 1


(b) x( y
-

z) p + y( z
-

x) q = z(x
-

y)


(c) x
2
p + y
2
q = (x + y) z


(d) p + q = x + y + z.


(e) z(p
-
q) = x + y + z


(f) z(px
-

qy) = y
2

-

x
2


(g) (x + 2z)p + (4zx
-

y) q = 2x
2

+ y


(h) ( x
2

-
y z) p + ( y
2

-

zx) q = z
2

-
x y


(i) pzx + qxy = xy.


6

Solve the following by Charpit’s method:

(a)

(p
2

+ q
2
)y = qz

(b)

2z + p
2

+ qy + 2y
2

= 0

(c)

z = p
2
x + q
2
y

(d)

1 + p
2

= qz

(e)

P(p
2

+ 1) + (b
-

z)q = 0

(f)

z
2

= pqxy

(g)

px
y + pq + qy = yz

(h)

q + xp = p
2

P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


9

NETWORK SYNTHESIS


QUESTION BANK



Subject Code : EE 252 Faculty: Ms. KP/CAS


1.

Determine the z,y,h
,T parameters for the following ckt.




2.

Find z parameters for the following



3.

Find the z and h parameters of the CE transistor represented by its T ckt model.



4.

Find the T parameter of the following configuration.



5.

Test
wether the polynomial is Hutwirtz.


a) s
3
+s
2
+2s+2


b) s
7
+2s
6
+2s
5
+s
4
+4s
3
+8s
2
+8s+4


6. Detemine wether the folloeing is p.r.


a) F(s)=2s
2
+2s+4 /(s+1)(s
2
+2)



P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


10



7. Indicate whether the following are L
-
C,R
-
L or L
-
C impedence functions


a) Z(s)= s
3
+2s/s
4
+4s
2
+3 b)s
2
+6s+8/s
2
+4s+3


8 . Synthesize the following voltage ratio


a)V
2
/V
1
=(s+2)(s+4)/(s+3)(3s+4)

b)V
2
/V
1
=s2
-
s+1/s2+s+1


9. Synthesize the foll. Functions


Z
21
= 1/s
3
+3s
2
+3s+2


10. . Find the z and h parameters of the CE transistor represented by its T ckt model.



11. Given Z(s)=(s
2
+Xs)/(s
2
+5s+4)


a) What are the restrictions on X for Z(s) to be p.r functions.


b) Choose the numerical value of X and synthesize them.

12. Perform continued fraction expansion on the ratio


Y(s)=s
3
+2s
2
+3s+1/s
3
+s
2
+2s+1

13.
Find the networks for the following functions


Z(s)=(s+1)(s+4)/s(s+2)

14. Synthesize Z(s)=(s+2)(s+4)/(s+1)(s+5)

15. Synthesize L
-
C driving point impedence


Z(s) = 6s
4
+42s
2
+48/s
5
+18s
3
+48s and determine the el
ement values of the network




16. An impedence function has a pole zero pattern shown below. If Z(
-
2)=3, synthesize the imedence in a
Foster form.






17. Synthesize a constant resistance lattice terminated in a 1 ohm resistor.


V2/V1=s
2
-
s+1/s
2
+s+1


18. Synthesize the following functions into the form shown in the figure


Z
21
= 1/s
3
+3s
2
+3s+2




P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


11

19. Find the poles of the system functions with n=3 , n=4 and n=5 Butterworth characteristics.

20. Synthesize

the n=3 linear phase filter as a transfer impedence terminated in a 1
-
ohm load.

21. Determine the asymptotic rate of fall off in the stopband of optimum filters.

22. What do you mean by rise time ,Ringing, Settling time , Delay time and overshoot

23. Comp
are Butterworth with Chebyshev filters.

24. How do you reduce the overshoots in the filters .

25. What do you mean by frequency transformation.

26. find the networks for the following functions. Both Foster and Ladder form


a)Z(s)= (s+1)(s+4)
/s(s+2)


b)Z(s)=3(s+1)(s+4)/(s+3)


27. tanhs=3s/(s
2
+3).Synthesize networks for the functions above whose input impedence s approximate to
tanhs.

28. When to you call a polynomial as Hurwitz.

29. Properties of Hurwitz polynomial.

30. What are

the properties of positive real functions

31.
Why Z
-
parameters are called as open circuit impedence (Z) parameter.

32. Define driving point impedence at port 1 with port 2 open.

33. Define open circuit forward transfer impedence.

34. Give the condition fo
r reciprocly for Z parameters.

35. Why Y parameters are called as short circuit admittance parameters.

36. What are called as sending end and receiving end in case of ABCD parameters.

37. What are the applications of cascaded ABCD parameters.

38.Give the e
xpression of h
-
parameters in terms of Z
-
parameters

39. Give the expression of ABCD parameters in terms of Y
-
parameters.

40.Derive the expression for Z
-
parameters interms of Y
-
parameters.

41. Dervie the expression for transmission parameters in terms of Z
-
p
arameters

42. The hybrid parameters of a 2 port network are h11 = 1k; h12 = 0.003; h21=100; h22 = 50. find


r2 and Z parameters of network.

43.What is a Hurwitz polynomial.





* * * * *


















P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


12

SIGNALS & S
YSTEMS


QUESTION BANK



Subject Code : EE 252 Faculty: Dr. SHL


1.

what are even and Odd signals

2.

Find the even and odd components of the following signals

a.

t
cos
t
sin
t
sin
t
cos
)
t
(
x




b.

4
3
2
g
9
t
5
t
3
1
)
t
(
x





c.

t
10
t
cos
)
t
1
(
)
t
(
x
3
3



3.

What are periodic and A periodic signals. Explain for both continuous and discrete cases.

4.

Determine whether the following signals are periodic. If they are periodic find the fundamental period.

a.

2
))
t
2
(
(cos
)
t
(
x



b.






5
5
k
)
k
2
t
(
w
)
t
(
x

where
)
t
(
w

is shown below

c.








k
)
k
3
t
(
w
)
t
(
x

for
)
t
(
w

shown in figure.1

d.

)
n
2
cos(
)
n
(
x


e.

n
2
cos
)
n
(
x



5.

Define energy and power of a signal for both contin
uous and discrete case.

6.

Which of the following are energy signals and power signals and find the power or energy of the signal
identified.

a.












otherwise
0
2
t
1
,
t
2
1
t
0
,
t
)
t
(
x

b.












otherwise
0
10
n
5
,
n
10
5
n
0
,
n
)
n
(
x

c.









0
5
.
0
t
5
.
0
t
cos
5
)
t
(
x

d.









otherwise
0
4
n
4
,
n
sin
)
n
(
x

7.

The raised cosine pul
se is shown in fig.1 and is defined by






















0
1
w
1
)
t
(
cos
2
1
)
t
(
x

Determine the total energy of the signal x(t)

P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


13



8.

Explain with examples various operations that can be performed on dependent and independent variables of
a continuous time function and discrete

time function.


9.

A sinusoidal signal x(t)=3 cos
)
6
/
t
200
(



is passed through a square law device defined by the input
output relation y(t)=x
2
(t).


10.

A signal x(t) as shown in fig. 1 is applied to a differentiator. Determine

1. Output of the dif
ferentiator

2. Total energy of the output






11.

A rectangular pulse x(t) is defined by






otherwise
0
T
t
0
A
)
t
(
x

This pulse is applied to an integrator whose output is defined by





1
0
d
)
(
x
)
t
(
y

find the total energy of the output.


12.


A Trapezoidal si
gnal as shown in fig. 2 is time scaled producing output y(t)=x(t). Sketch y(t) for a=5, a=0.2.



P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


14

13.

Let x9t) and y(t) be given in Figs. P1.14(a) and (b), respectively. Carefully sketch the following signals:

(a)

x(t)y(t


1)

(b)

x(t)y(2


t)

(c)

x(2t)y(
2
1
t + 1)

(d)

x(4


t)y(t)


Time
-
domain representations for LTI Systems


1.

Show that if x(n) is input of a linear time invariant system having impulse response h(n), then the output of the
system due to x(n) is







k
)
k
n
(
h
)
k
(
x
)
n
(
y


2.

Use the definitio
n of convolution sum to prove the following properties

1.

x(n) * [h(n)+g(n)]=x(n)*h(n)+x(n)*g(n) (Distributive Property)

2.

x(n) * [h(n)*g(n)]=x(n)*h(n) *g(n) (Associative Property)

3.

x(n) * h(n) =h(n) * x(n) (Commutative Property)


3.

Prove that absolute summabilit
y of the impulse response is a necessary condition for stability of a discrete
time system.


4.

Write a differential equation description relating the output to the input following electrical circuit as shown in
fig 3 and fig.4





Fourier representation for

signals, The continuous and discrete time Fourier Transforms


1.

A continuous time periodic signal x(t) is real valued and has a fundamental period T=8, the non
-
zero Fourier
series coefficients for x(t) are
j
4
a
a
,
2
a
a
3
3
1
1







Express x(t) in form







0
k
k
k
k
)
1
w
(
cos
A
)
t
(
x

P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


15

2.

For a continuous time signal (periodic)

)
t
3
/
5
(
sin
4
)
t
3
/
2
cos(
2
)
t
(
X






determine the fundamental frequency
0
w

and Fourier series
coefficients
k
a

such that






k
jkw
k
0
e
a
)
t
(
x

3.

Consider the following
three continuous time periodic signals whose Fourier series representation are as
follows:










100
0
1
50
2
.
2
1
)
1
(
k
jk
k
t
e
x






100
100
50
2
2
.
)
(
cos
)
(
k
t
jk
e
k
t
x

















100
100
k
t
50
2
jk
3
e
.
2
k
sin
j
)
t
(
x


Using Fourier Series properties

14.

Which of the three signals is / are real valued

i.

Which of t
he three signals is / are even

4.

State and prove the properties of continuous time Fourier Series

5.

Obtain an expression to express a continuous time periodic signal in Fourier series representation in (i)
Trigonometric form (ii) Exponential form

6.

Find the com
plex exponential Fourier series representation for the rectangular waveform as shown in fig.5



7.

State and prove parsevels relation for continuous time periodic signals

8.

Obtain an expression for Fourier series representation of a discrete time periodic sign
als in complex
exponential form.

9.

What is Gibbs phenomenon? State and prove the properties of discrete time Fourier series.

10.

Obtain Fourier transform of

i.

0
a
for
)
t
(
u
.
e
)
t
(
x
)
t
(
a




ii.

0
a
for
,
e
)
t
(
x
t
a




iii.

|||
)
t
(
)
t
(
x






P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


16

11.

Consider a rectangular pulse







1
1
T
|
t
|
,
0
T
|
t
|
,
1
)
t
(
x

Obtain Fourier transform of x(t)

12.

Obtain the analysis and synthesis equation of Fourier Transform of n periodic signal.

13.

Obtain x(t) if x(jw)






w
|
w
|
for
0
w
|
w
|
for
1

14.

Obtain an expression for Fourier transform of a periodic signal

15.

State

and prove properties of continuos time Fourier transform.

16.

(a) Use partial fraction expansions to determine the inverse FT for the following signals:

(a)

X(j

) =
6
5
)
(
12
5
2






j
j
j

(b)

3
4
4
)
(
2






j
j
X


(c)

2
3
)
(
)
(
2







j
j
j
j
X

(d)

)
4
](
2
3
)
[(
6
4
)
(
)
(
2
2












j
j
j
j
j
j
X

(e)

5
6
)
(
14
12
)
(
2
)
(
2
2









j
j
j
j
j
X

(f)

2
)
2
(
1
2
)
(





j
j
j
X




(b) Use the tables of transforms and properties to find the FTs of the following signals:


(a)
)
(
)
sin(
)
(
2
t
u
e
t
t
x
t





(b)
]
2
[
3
)
(



t
e
t
x


(c)













t
t
t
t
t
x




)
2
sin(
)
sin(
2
)
(


(d)
))
(
)
sin(
(
)
(
2
t
u
t
te
dt
d
t
x
t





(e)



d
t
x
t
)
sin(
)
(






(f)










2
4
)
(
1
2
t
u
e
t
x
t


(g)



















t
t
t
t
dt
d
t
x


)
2
sin(
*
)
sin(
)
(

17.

(a) Evaluate the following quantities:

(a)






d
e
j
2
|
4
1
1
|
1



P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


17

(b)
2
2
)
4
/
(
sin
k
k
k







(c)


d
2
2
)
1
(
4






(d)














19
0
2
2
20
sin
20
11
sin
k
k
k



(e)
dt
t
t
2
2
)
(
sin







(b) For the signal x(t) shown in Fig.P3.23, evaluate the following quantities without explicitly computing X(j

):

(a)


d
j
X
)
(





(b)


d
j
X
2
|
)
(
|





(c)



d
e
j
X
j
2
)
(





(d) arg{X(j

)}

(e) X(j0)


(c) Let x[n]




DTFT
x(e
j

), where x(e
j

) is depicted in Fig. P3.24. Evaluate the following without
ex
plicitly computing x[n]:

(a)

x[0]

(b)

arg{x[n]}

(c)





n
n
x
]
[

(d)

2
|
]
[
|




n
n
x

(e)





n
n
i
e
n
x
)
4
/
(
]
[


(d) Prove the following properties:

(a)

The FS symmetry properties for:

(i) Real
-
valued time signals.

(ii) Real and even time signals.

(b)

The DTFT time
-
shift property.

(c)

The DTFS frequency
-
shift property.

(d)

Linearity for the FT.

(e)

The DTFT convolution property.

(f)

The DTFT modulation property.

(g)

The DTFS convolution property.

(h)

The FS modulation property.

(i)

The Parseval relationship for the FS.


18.


The form of the FS rep
resentation presented in this chapter,






k
ot
jk
e
k
X
t
x

]
[
)
(

is termed the exponential FS. In this problem we explore several alternative, yet equivalent, ways of expressing
the FS representation for real
-
valued periodic signals.



P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


18

(a)

Trigonometric form.

(
i) Show that the FS for a real
-
valued signal x(t) can be written as


x(t) = a
o

+





1
cos(
k
k
k
a

o
t) + b
k

sin (k

o
t)

where a
k

and b
k

are real
-
valued coefficients.

(ii) Express a
k

and b
k

in terms of X[k]. Also express X[k] in
terms of a
k

and b
k
.

(iii) Show that



dt
t
x
T
a
T
o
)
(
1
)
(



tdt
k
t
x
T
a
o
T
k

cos
)
(
2
)
(




tdt
k
t
x
T
b
o
T
k

sin
)
(
2
)
(



(iv) Show that b
k

= 0 if x(t) is even and a
k

= 0 if x(t) is odd.


(b)

Polar form.

(i) Show that the FS for a real
-
valued signal x9t0 can be written a
s


)
cos(
)
(
1
k
o
k
k
o
t
k
c
c
t
x










where c
k

is the magnitude (positive) and

k

is the phase of the kth harmonic.

(ii) Express c
k

and

k
as functions of X[k].

(iii) Express c
k

and

k
as function of

a
k

and b
k
from (a).


1.

The output of a system in response to an input
)
t
(
u
.
e
)
t
(
x
t
2



is
)
t
(
u
.
e
)
t
(
y
t


. Find the frequency
response and impulse response of this system.

2.

Find the frequency response and impulse response of a system described
by the differential equation

)
t
(
x
)
t
(
x
dt
d
2
)
t
(
y
2
)
t
(
y
dt
d
3
)
t
(
y
dt
d
2
2





3.

Find the frequency response and impulse response of a discrete time system described by the difference
equation y(n
-
2)+5y(n
-
1)+6y(n)=8x(n
-
1)+18x(n)

4.

Determine the Fourier transform of periodic signal as s
hown in fig.6. Sketch the magnitude and phase
spectra.



P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


19


5.

Find the Fourier transform of impulse train given by








n
)
nT
t
(
)
t
(
p

T is fundamental period


6.

Obtain an expression that relates discrete time Fourier transform and Discrete time Fourier se
ries.

7.

Find both DTFS & DTFT for the signal.
)
2
/
(
sin
4
)
3
/
8
/
3
(
cos
2
)
(
n
n
n
x







8.

Show that x(n) * h(n)


)
e
(
H
)
e
(
x
jn
jn

where x(n) are the input and impulse response of a LTI systems
and
)
e
(
x
jn

and
)
e
(
H
jn

are their respective DFTF

9.

If
)
n
(
u
)
2
/
j
(
)
n
(
h
n


is impulse response of a LTI system, then obtain the response of this system to the
input
)
3
/
n
(
cos
3
)
n
(
x






10.

Consider x(n)=cos
)
n
16
/
7
(

)
n
16
/
9
(
cos



using modulation prope
rty. Evaluate the effect of
computing the DFTF using only 2M+1 values of x(n) where |n|

M

11.

Consider x(t) =cos
t

. Assume that this signal is sampled at internals T=1/4, T=1 & T=1/2. Find the Fourier
Transform of t
he sampled data for all the three cases.


Z


TRANSFORMS


1.

State and prove the properties of unilateral Z transform

2.

Discuss the role of ROC in Z transforms

3.

Find Z transform of

1.
)
n
(
u
)
2
/
1
(
)
1
n
(
u
)
n
(
X
n






2. Determine the Z
-
Transform of the sequence,

{u
n
} =
{a
n
},


3. Determine the Z
-
Transform of the sequence,

{u
n
} = {n}.

where a is a non
-
zero constant.

4.

Determine Z transform, ROC pole and zero location of x(t) for

i.

)
n
(
u
)
3
/
1
(
)
n
(
u
)
2
/
1
(
)
n
(
x
n
n




ii.

)
n
(
u
e
)
n
(
x
n
0
j



5.

Find Z transform of

X(n)={n(
-
1/2)
n

x(n)} * (1/4)
-
n

u(n)

X(n)=a
n

cos (

0
n)

6.

Find inverse Z transform of

1
z
2
1
2
1
z
2
1
1
1
)
z
(
x








Assume a. signal is casual

b. Signal has DTFT


4
Z
2
Z
2
4
Z
4
Z
10
Z
)
z
(
x
2
2
3







with ROC

|
z
|

7.


Determine the inverse Z
-
Transform of the function,


P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


20

)
3
)(
2
)(
1
(
)
5
(
10
)
(





z
z
z
z
z
z
F

8.


Determine the inverse Z
-
Transform of the function,


)
(
1
)
(
a
z
z
F



9.

Determine the inverse Z
-
Transform of the function,

)
3
)(
1
(
)
1
2
(
4
)
(




z
z
z
z
F

10.

Find the inverse Z transform of

)
z
e
9
.
0
1
(
)
z
e
9
.
0
1
(
Z
1
)
z
(
H
1
4
/
j
1
4
/
j
1











11.

A system is described by the difference

equation

)
2
n
(
x
8
/
1
)
1
n
(
x
4
/
1
)
n
(
x
)
2
n
(
y
4
1
)
1
yn
)
n
(
Y










Find the Transfer function of the Inverse system

Does a stable and causal Inverse system exists

12.

Sketch the magnitude response for the system having transfer functions.

13.

Certain simple difference equations may be solved by v
ery elementary methods.

For example, to solve

u
n+1

− (n + 1)u
n

= 0, subject to the boundary condition that u
0

= 1,

we may rewrite the difference equation as

u
n+1

= (n + 1) u
n
.

By using this formula repeatedly, we obtain

u
1

= u
0

= 1, u
2

= 2 u
1

= 2,

u
3

= 3 u
2

= 3 × 2, u
4

= 4u
3

= 4 × 3 × 2, . . . .

Hence,

u
n

= n!








* * *

*

P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


21

CONTROL SYSTEMS


QUESTION BANK


Subject Code : EE 254

Faculty:

Ms. KRS


Unit 1: Chapter: 2
-

Mathematical Models of Systems (T: 2.1 to 2.7) &

Chapter:3
-

State Variable Models T: 3.1 to 3.7


1.

Explain System, Control System, Open loop and closed loop systems. Discuss the merits and demerits of
Open loop and closed loop
systems.

2.

Differentiate between


(i) Linear and Non
-
linear systems (ii) Continuous and Discrete data systems.

3.

What are the effects of feedback on Control System?

4.

Define Transfer function of a control system.

5.

What are analogous systems? What are the advan
tages of studying non
-
electrical systems in terms of
their electric analogs?

6.

For what purpose feedback is used in control systems? Mention the effect of feedback on:
-


(i) Stability (ii) Overall Gain (iii) Disturbance and (iv) Sensitivity of control s
ystem.

7.

Explain Transfer function. Derive the transfer function of an armature controlled D.C. motor.

8.

For the following mechanical systems, write the differential equations governing the dynamic behavior of
the system and obtain the (
f
-
v

and
f
-
I, T
-
v and T
-
i)
analogous electrical networks.

(i) (ii)




(iii) (iv)




P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


22

(v)

(vi)







9.

For the mechanical system shown in fig., draw the force voltage analogous electrical system & determine
the displacements as the function of time at A & B; also draw approximatel
y these displacements. Given,


F(t)=δ(t) N, M
2
=
24 kgms , l
1
=100cm, M
1
=4kg,D
2
=60N/m/s,


l
2
=25cm, D
1
=5N/m/s, K
2
=1.25cm/N (spring compliance)

10.

Draw a block diagram representation for the electric circuit shown in fig and evaluate the transfer function
E
o
(s)/E
i
(s).



11.

A switching circuit is used to convert one level of dc voltage to an output dc voltage. The filter circuit to
filter out the high frequencies is shown in fig. Determine the transfer function
.


12.

Determine the transfer function

if R
1

= R
2

= 100k
Ω; C
1
=10μF and C
2
=5μF.


P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


23


13.

For the mechanical system shown below

i.

Draw the mechanical network

ii.

Write the differential equation of the system (12)



14.

Define the term “Transfer Function”. The unit step response of single
loop unity feedback control system
is given by, C(t) = 1
-
1.25 e
-
2t
+0.5e
-
10t.

Determine its closed loop and open loop transfer functions.

15.

Determine the TF of the electrical network shown below. Assume zero initial conditions




16.

What are the analogous systems? What are the advantages of studying non
-
electrical systems in terms
of their electric analogs? (05)

17.

Draw the F
-
V analogous mechanical system for the electric network shown below.



18.

F
or the system shown in fig,

i.

Identify the Type of C(s)/E(s)

ii.

Find the values of K
p
, K
v

and K
a

iii.

If r(t)=10u(t), find the steady state values of the output.



P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


24

19.

Draw the signal flow graph for the block diagram shown in fig and evaluate
C(s)/R(s), using Mason’s
formula.



20.

The suspension system for 1 wheel of an old fashioned pickup truck is illustrated in fig. The mass of the
vehicle is m1 & the mass of the wheel is m2. The suspension spring has a spring constan
t k1, & the tyre
has a spring constant k2. The damping constant of the shock absorber is b. Obtain the transfer function
Y1(s)/X(s), which represents the vehicle response to bumps in the road.


21.

Obtain C(s)/R(s) of the system shown below by using block di
agram reduction method.



22.

For the signal flow graph shown in fig find the transfer function using Mason’s
formula.


23.

Find
.



P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


25

24.

The block diagram of a feedback control system is shown in fig. Find:
-



(i)The transfer function Y(s)/E(s) with N=0 and (ii) The transfer function Y(s)/N(s) with R=0


25.

Determine the transfer function of a system whose block diagram is given in fig. using the block diagram
reduction technique.


26.

Draw the signal flow graph

for the system of equation given below and obtain the over all transfer function
X
6
/X
1

using MGF

X
2
=G
1
X
1


H
1
X
2


H
2
X
3


H
6
X
6

X
3
=G
1
X
1
+G
2
X
2


H
3
X
3

X
4
=G
2
X
2
+G
3
X
3
-
H
4
X
5


X
5
=G
4
X
4
-
H
5
X
6

X
6
=G
5
X
5



27.

Construct the block d
iagram for the circuit shown in fig below, where V
s

and i
L

are the input and output
variables respectively. Determine the Transfer function.




Unit
-
2 Chapter:4
-

Feedback Control System Characteristics (T: 4.1 to 4.6) &



Chapter:5
-

The Performance of Feedback Control Systems (T: 5.1 to 5.8)


28.

For the block diagram shown in fig.
-



(i) What type of system does C(s)/E(s) represent? (ii) Find C(s)/R(s)

(iii) Find the position, velocity &

acceleration error constants (iv) If r(t)=10u(t), evaluate C(s)

P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


26


29.

The open loop transfer function of a unity feedback single loop control system is given by,



i.

Find the value of K so that the steady state
error for a unit parabolic input is <=0.1

ii.

For the value of K found in part (i), verify whether the closed loop system is stable or
otherwise.

30.

For the UFB systems:(a)

(b)

(c)

determine:
-

(i) Type (ii) Order (iii
) Static error constants (iv) Steady state error for the following inputs
-


(a)

(b)

(c)


31.

A unity FBCS has the forward path TF
. The input r(t)=1+5t is applied to the system.

It is desired that the steady state value of the error should be

0.1 for the given input function.
Determine the minimum value of K to satisfy the requirement.


32.

A unity FB system has

i.

For a unit ramp input, it is desired e
ss
<=
0.2, find K

ii.

Determine e
ss

if input r(t)=2+4t+t
2
/2

33.

For the unity feedback Control System shown below, starting from the fundamentals derive an expression
for the error signal e(t). Assume that the system is under damped (0<
<1) and the in
put is a unit step
signal.



34.

Find the open loop transfer function of an equivalent prototype, single loop unity feedback system second
order, whose step response is as shown in fig.



35.

With diagra
m, explain the time domain specifications.

36.

A mercury in a glass thermo meter has an overall transfer function of
. If the thermo meter
requires 1 min. to indicate 95% of its final value for a unit step input, determine the value of
.

37.

Obtain an expression for:
-

(i) Rise time (ii) Delay time (iii) Peak time (iv) Settling time (v) peak
overshoot

38.

Define a second order system and hence determine the Transfer function of the system.

39.

Define damping factor and explain its effect
on the performance of a second order system.

40.

Derive an expression for response of a second order system for a step input.

P.E.S.I.T

DEPT. OF TE IIISEMESTER


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27

41.

The closed loop poles of a system are
. Calculate the damping ratio and damped frequency of
oscillations of the system. A
lso find the % overshoot with a unit step input.

42.

The OLTF of a UFB system is
. Determine for a unit step input, (i) damping ratio
(ii) damping factor (iii) peak time (iv) Max. overshoot (v) peak value of response.

43.

For a

spring mass damper system shown in fig (a) an experiment was conducted by applying a force of
2Newtons to the mass. The response x(t) was recorded using an xy plotter and the experimental result is
as shown in fig(b). Find the values of M, K and B. (08)




44.

Consider unity FBCS, whose OLTF is given by
. Obtain the response to step input. For
the same, calculate rise time, maximum peak overshoot, peak time and settling time.

45.

A feedback control system is described
by the forward path and feedback path transfer functions,
G(s)=16/(s
2
+4s+16) and H(s)=Ks respectively. It has a damping ratio of 0.8. Determine the overshoot of
the system and the value of K for a unit step input.

46.

Explain why many control systems are desi
gned with higher gain (and hence lower damping).

47.

Consider a UFBCS, whose OLTF is given by
.Obtain the response to step input. For the
same, calculate the rise time, maximum peak overshoot, peak time and settling time.


Unit
-
3: Chapter: 6
-

The

Stability of Linear Feedback Systems (T: 6.1 to 6.4) &


Chapter: 7
-

The Root Locus Method (T: 7.1 to 7.4)

48.

For the characteristic equations given below determine the number of roots with positive real part

iii.

s
6
+s
5
+3s
4
+2s
3
+5s
2
+3s+1=0

iv.

s
8
+
s
7
+4s
6
+3s
5
+14s
4
+11s
3
+20s
2
+9s+9=0

49.

Explain: (i) Order and type of system (ii)Absolute stability and relative stability

50.

The open loop transfer function of a single loop, unity feedback control system is given by


i.

Find the value of K f
or which the closed loop system is stable.

ii.

Find the value of K for which the closed loop poles are more negative than
-
1.

51.

For a single loop, unity feedback system, the open loop transfer function is given by


G(
s)=K(s+2)(s+3)/s(s+1). Show that the complex part of the root locus is a circle. Identify its centre and
radius.

52.

Derive the condition on the impulse response so that the system is bounded input bound output (BIBO)
stable.

53.

A unity FB system has

using RH criteria, find the range of K for stability. Also find
. and
.

54.

Determine the range of values of K (K>0) such that the characteristic equation is:
s
3
+3(K+1) s
2
+(7K+5)s+(4K+7)
=0, has roots more negative than s=
-
1.

55.

State the different rules for the construction of root locus.

56.

Sketch the root locus diagram of a control system having


P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


28


57.

For a UFB system

. Sketch the root locus showing all

the details. Comment on the


Stability. Determine the value of K for a damping ratio of 0.5.

58.

Sketch the root locus for a UFB system
. Comment n the stability. Determine the
range of stability and frequency of sustained oscillations for
critical value of K.

59.

Determine the range of values of K if
. Sketch the root locus.

60.

Draw the root locus for
. Comment on the stability of the system. Also, explain the stability
of the system when a zero is added at s=
-
1.

61.

Sketch t
he root locus, given
.Comment on the stability.

62.

The open loop TF of a unity FBCS is given by
.Determine the value of K that will
cause sustained oscillations in the closed loop system. Also find the frequency of sustained oscillat
ions.

63.

Obtain the equation of root locus for a unity FBCS whose forward path TF
. Show that the
root locus is a circle centered at (
-
b,0) and of radius √b
2
-
ab.

64.

For a negative feedback control system
.and
. Obtain the root locus
for


the root of the characteristic equation and plot the same using scale of 1 unit of real s=2cm & 1 unit of



imaginary s=2cm.

65.

A unity FBCS has G(s)=K/[s(s+2)(s+5). Sketch the root locus and show on it

(i) Break away point (ii) Line for ξ=0.5 and value of k for this damping ratio


(iii) The frequency at which the root locus crosses the imaginary
axis and the corresponding value of K





Unit
-
4
-

Chapter: 8
-

Frequency Response methods(T: 8.1 to 8.6) &


Chapter: 9
-

Stability in the Frequency Domain (T: 9.1 to 9.5)

66.

The open loop TF of a unity FBCS is given by,

Determine the values
of K so that the system will have a phase margin of 40 degree. What will be the gain margin? Use Bode
plot.

67.

With diagram, explain the frequency domain specifications.

68.

Given

Draw the polar plot and hence dete
rmine if system is stable and its gain
margin and phase margin.

69.

The OLTF of an unity FBCS is

i.

Find the values of K and so that M
r
=resonant peak=1.04 and
r
=resonant frequency=11.55rad/sec

ii.

For the values of K and a found in part (i
), calculate the settling time and bandwidth of the system.

70.

Define asymptotic stability. Prove that for BIBO stability, the roots of the characteristic equation must all
lie in the left
-
half of s
-
plane.

71.

A FBCS is described by
.

Construct an asymptotic log magnitude plot
and an exact phase plot. There from determine gain margin and phase margin and comment on the
stability of the closed loop system.

72.

Define the terms ‘Gain Margin’ and ‘Phase Margin’. Explain how these can be d
etermined from polar
plots. (06)

73.

Compute analytically the gain margin and phase margin if
.

P.E.S.I.T

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29


74.

The open loop TF of a unity FBCS is
. Draw the bode asymptotic magnitude plot


and phase plot. From the graph
-

i.

Find gain lim
it

ii.

Determine the value of K for a gain margin of 10dB. What is the corresponding phase margin

iii.

Determine the value of K for a phase margin of 40
0
. What is the corresponding gain margin



75.

Find the TF which has the asymptotic bode magnitude plot shown in
fig below:



76.

Plot the bode magnitude and phase diagrams for the open loop transfer function
.

Discuss the stability range of the closed loop system.

77.

Obtain the open loop transfer function of the control system whose

Bode magnitude plot is shown in fig.


78.

State and explain Nyquist stability criterion .

79.

Sketch the Nyquist plot of a unity feedback control system having the open loop TF
..
Determine the stability of the system using Nyquist stability criter
ion.

80.

For a prototype second
-
order system, whose closed loop TF is given by M(s)=ωs2/s2+2 ξωs+ω2, derive
an expression for the resonant peak (M
r
) and the frequency ω
r

at which it occurs. (08)

81.

The open loop TF of a FBCS is
. Draw the polar plot and

determine the restriction
on K for stability.




Unit
-
5
-
Chapter: 10
-

The Design of Feedback Control Systems ( T: 10.1 to 10.4,10.6,10.8) & Chapter: 11
-

The Design of State Variable Feed Back Systems (T: 11.1 to 11.3, 7.7)


82.

Obtain the transfer f
unction of
-

(i) Phase Lead network (ii) Phase Lag network (iii) Lead
-
Lag network.

83.

Discuss the basic characteristics of Lead, Lag and Lag
-
Lead compensation.

84.

Given

and
. Design a compensator for the system so that the static veloc
ity error
constant
, the Phase Margin is at least

and the Gain Margin is at least 10dB.

85.

Given

and
. Design a compensator for the system so that the static
velocity error constant
, the Phas
e Margin is at least

and the Gain Margin is at least
10dB.

86.

Distinguish between Lead, Lag and Lag
-
Lead compensation. Draw the response curves of an
uncompensated system and compensated system (Lead, Lag,Lag
-
Lead) for a unit step and unit ramp

inputs.

P.E.S.I.T

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30

87.

The circuit shown in fig called a lead
-
lag filter. Find the transfer function V2(s)/V1(s). when R1=100ohm,
R2=200kohm and C2= 0.1uF.


88.

Determine the controllability and observability of the systemdescribed the state equation. Find out the
transf
er function and draw the block diagram.







89.

Define Controllability and Observability of a system.

90.

Given

and
, check whether the system is Controllable?


91.

A dc motor has a transfer function of
. Determine whether the system is
Controllable and Observable?

92.

Consider the transfer function defined by

. Derive the controllable canonical form of the state space
representation.

93.

Discuss the Controllability of the following system



94.

A state space representation of a system in the controllable canonical form is given by,



and

----
(1)


The same system may be represented ny the following stste space equation which is in the observable



canonical form,

and

----
(2)


Show that the state space equation (1) gives a system that is state controllable and the state space
equation (2)


gives a system that is not completely state controllable.

95.

Sho
w that the following system is not completely observable


and


Where

,
,
,


*

* * **


P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


31

LINEAR INTEGRATED CIRCUITS THEORY


QUESTION BANK



Subject Code: EC
-
255

Faculty: Mr.

RC


Unit 1

1.

Explain the block diagram of the op
-
amp

2.

List the applications of the op
-
amp


3.

Derive the voltage gain expression of an inverting amplifier.

4.

Show how an op
-
amp can be used as an inverting / noninverting Adder.

5.

Show how an op
-
amp can be used a
n inverting averaging amplifier.

6.

Derive a subtractor expression using an op
-
amp.


7.

With a neat circuit diagram, input & output waveforms and the transfer characteristics, explain the working of
inverting / noninverting comparator.


8.

With a neat circuit di
agram, input & output waveforms and the hysteresis curve, explain the working the
inverting Schmitt trigger (positive feedback).


9.

Design a Schmitt trigger for the UTP = +4V and LTP =
-
4V

10.

Design a Schmitt trigger for the UTP = +2V and LTP =
-
4V

11.

Explain Ze
ro
-
crossing detector with Hysteresis.

12.

Explain inverting / noninverting voltage level detector with hysteresis.


13.

Explain the working operation of the instrumentation amplifier.


14.

A three input non inverting summer has V1 = 1V DC, V2 =
-
0.2V
DC, V3 is a 2V peak 100Hz
sine wave.


Resistors R1, R2, R3 at the input of the non inverting terminal and the feedback resistor Rf are 20K


each.
The resistor connected from the inverting terminal to ground, R is 5 K

. Determine the output voltage
. Draw
the circuit diagram with all the component values.


15.

Draw the circuit diagram of an inverting voltage level detector with hysteresis and explain its operation with
necessary equations and waveforms. Design a similar circuit with Vut = 12V and Vlt =
8v. Assume ±Vsat =
±15V.


16.

What is a voltage follower? What is the purpose of having an amplifier with a gain of one? Name any one
application of the voltage follower.


Unit 2


1.

With a neat circuit diagram explain the working of the voltage to current conv
erter with floating load using an
op
-
amp.


2.

With a neat circuit diagram explain the working of the constant current to a grounded load using an op
-
amp.

3.

Show how an op
-
amp can be used as integrator and differentiator.

4.

With a neat circuit diagram explain t
he free
-
running oscillator using op
-
amp.


5.

Derive the expression for the frequency of the free
-
running oscillation


6.

With a neat circuit diagram explain the working operation of the one shot multivibrator and also derive the
expression of the output pulse
width.


7.

With a neat circuit diagram explain the working operation the triangular wave generator using op
-
amp.


8.

With a neat circuit diagram explain the working operation saw
-
tooth wave generator using op
-
amp.


9.

A relay coil which requires 1.0mA is to cont
rolled by a microprocessor whose output current is 10µA. Design
an op
-
amp based circuit to drive the relay coil from the microprocessor. Draw the circuit with all the
component values.



P.E.S.I.T

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32


Unit 3

1. Design a suitable circuit to obtain the following tran
sfer characteristics


2.

With a neat circuit diagram, input & output waveforms and transfer characteristics explain the working
operation of the precision rectifier.


3.

Explain the working operation the absolute value circuit.

4.

Explain the working operation t
he peak detector using op
-
amp.


5.

Design a suitable circuit to obtain the following transfer characteristics












6. Design a suitable circuit to obtain the following transfer characteristics












7.

With a suitable circuit explain the workin
g operation of the precision clipper with negative reference.


8.

What is the difference between a comparator and a dead zone circuit? Explain the working of a dead zone
circuit with negative output.


Unit 4

1.

With a design procedure, design a 1
st

order LPF f
or a cutoff frequency of 2.5KHz.


2.

With a neat circuit diagram, explain

40dB/Decade low pass Butterworth filter and also give the design
procedure for the same.


3.

With a neat circuit diagram, explain

60dB/Decade HPBF and also give the design procedure fo
r the same.


4.

Define Quality factor.

5.

Design narrowband band pass filter.


6.

With a frequency response explain Notch filter.

7.

Write the ideal and practical frequency response of LPF, HPF, BPF, BRF

8.

Given the design procedure of a notch filter.


9.

An audio pr
eamplifier needs to reproduce signals with frequencies as high as 20KHz. The maximum output
swing is 10V peak. State with reasons if 741 can be used for this application.


10.

What is the difference between the open loop and closed loop gain of an op
-
amp? Com
pare with necessary
equations.


P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


33


Unit 5

1.

Explain the internal functional block diagram of the 555 timer.

2.

Explain all the pins of the 555 IC timer.


3.

With a neat circuit diagram and output & capacitor voltage, explain the working operation of the free run
ning
oscillator.


4.

Design and explain a free running oscillator for the duty cycle less than 50%.

5.

Derive a frequency expression for a free running oscillator.


6.

With a neat circuit diagram, explain one
-
shot operation and drive an expression for the output

pulse width.

7.

Design a monostable multivibrator for the pulse width of 2ms.


8.

Define Resolution, offset error and gain error of DAC


9.

With a neat circuit diagram explain the working of the R
-
2R ladder DAC and also derive the output voltage
expression.


10.

D
efine Resolution, Quantization error, offset error, gain error and linearity error


11.

Explain the working operation of integrating ADC.

12.

Explain the working operation of the Successive Approximation ADC.

13.

With a neat circuit diagram expla
in the 3
-
bit flash type ADC.

14.

Define aperture time and aperture error.


15.

With a neat circuit diagram, explain the working operation of the sample and hold circuit.


16.

Design a timer based left/right turn indicator (automobile flasher) for which the blinking

cycle is 1 second. The
supply voltage is 10 V. draw the circuit diagram and insert the component values.


17.

Define resolution of digital to analog converter. An 8
-
bit DAC wired for unipolar operation has a full scale
voltage from 0 to 5.12 V. what is


a.

Res
olution of the DAC

b.

The output voltage change per bit

c.

The ideal full scale output voltage when all logic 1s are applied at the input


d.

What is the analog output for the digital input code 00010110?




*

* * * *



P.E.S.I.T

DEPT. OF TE IIISEMESTER


AUTONOMOUS


34



MICROPROCESSORS


QUESTION BANK



Sub
ject Code: EC256

Faculty: Ms.SR




1.

Explain the programming model of intel 8086 processor.

2.

Explain the flag register of 8086 processor.

3.

Explain real mode memory ad
dressing.

4.

Explain the different data addressing modes of 8086 processor.

5.

Explain program addressing mode.

6.

Explain stack memory addressing modes with examples.

7.

Explain the different string data transfer instructions with examples.

8.

Explain the different arit
hmetic and logical instructions with examples.

9.

Explain the shift and rotate instructions with examples.

10.

Explain the DOS function calls used in programming.

11.

Explain conditional and unconditional jump instructions with examples.

12.

Explain loop instructions wi
th examples.

13.

With neat diagram explain the pin outs of 8086 IC and explain the functions of each pin.

14.

Explain how the address and data bus is multiplexed in 8086.

15.

Explain the different interrupts that occur in 8086.

16.

Explain the action taken when an interru
pt occurs.

17.

Explain 8255 PPI with a neat diagram.

18.

Explain 8254 software programmable timer/counter with a neat diagram.

19.

Explain 8259 Programmable interrupt controller with a neat diagram.

20.

Explain 8237 DMA controller with a neat diagram.





*

* * * *