gujarat technological university · gujarat technological university b.e. sem-iii (all branches)...

1

Seat No.: _____ Enrolment No.______

GUJARAT TECHNOLOGICAL UNIVERSITY B.E. Sem-III (All Branches) Examination December 2009

Subject code: 130001 Subject Name: Mathematics III Date: 15 /12 /2009 Time: 11.00 am – 2.00 pm

Tota l Marks: 70

Instructions: 1. Attempt all questions. 2. Make suitable assumptions wherever necessary. 3. Figures to the right indicate full marks.

Q.1 Do as directed 14

(a) Solve: xy

dx

dy =+ .

(b) Evaluate the integral: ∫

∞

−0

2 )exp( dxx .

(c) Find 2cos2sin ttL . (d) State the generating function and integral representation for the Bessel

function )(xJ n .

(e) Prove that:

−= xx

x

xxJ cos

sin2)(

2

3 π .

(f) Show that: )(

5

3)(

5

213

3 xPxPx += .

(g) Find the Fourier transform of the function

<<

=otherwise

axkxf

,0

0,)(

Q.2 (a) By using the method of Laplace transform solve the initial value

problem: teyyy −=+′+′′ 2 , 1)0( −=y and 1)0( =′y .

07

(b) Solve the following differential equations (i) 02 2 =+ dyxdxxy

(ii) xey

dx

dy 2=−

(iii) y

xy

dx

dy −=+

02

02

03

OR (b) (i) Using the relationship between the beta and gamma functions,

simplify the expression ),(),(),( qpnmBpnmBnmB +++ .

(ii) Express ∫ −1

0

)1( dxxx pnm in terms of Gamma function.

(iii) State Legendre duplication formula. Hence prove that

.22

1,

2

1),( 411 mmmmBmmB −−=

++ π

02

02

03

2

Q.3 (a) Solve the initial value problem :

02 =−′+′′ yyy , 4)0( =y and 5)0( −=′y 05

(b) Given the functions

xe and xe−

on any interval [a, b]. Are these functions linearly independent or dependent?

04

(c) Using the method of variation of parameter solve the differential equation: xyy sec=+′′ .

05

OR Q.3 (a)

Prove that: )()]([ 11

1 xJxxJxdx

dn

nn

n ++

+ = . 05

(b) Attempt (any three). 09 (i) Express the polynomial 32 23 −−+ xxx in terms of Legendre

polynomials.

(ii) Show that ∫−

=1

1

0)()( dxxPxP nm , if nm ≠ .

(iii) By using generating relation of Legendre polynomials, evaluate )1(−nP .

(iv) Obtain the value of ∫−

=1

1

2 0)( dxxPn .

Q.4 (a) Find the Fourier series of the function ππ <<−= xxxf ,)( 2

. 05

(b) Obtain the Fourier series of periodic function 22,21,2)( ==<<−= Lpxxxf .

05

(c) Obtain the Fourier transform of the function )exp( 2ax− . 04

OR Q.4 (a) Using the method of undetermined coefficients, solve the differential

equation: 284 xyy =+′′ .

05

(b) Using the method of series solution, solve the differential equation: 0=+′′ yy .

04

(c) Find the steady state oscillation of the mass-spring system governed by the equation: tyyy 2cos2023 =+′+′′ .

05

Q.5 (a) Attempt (any two) 04

(i) Evaluate:

−+−

)3()2(

11

ssL .

(ii) Evaluate:

++−

186

32

1

ssL .

(iii) By using first shifting theorem, obtain the value of )1( 2 tetL + .

(b) Find the value of

(i) sin ttL ω (ii) 11∗ where ∗ denote convolution product.

04

3

(c) (i) Evaluate:

+

−−

22

21

πs

esL

s

.

(ii) Using convolution theorem, obtain the value of

+−

)4(

12

1

ssL .

06

OR

Q.5 (a) Find the solution ),( yxu of the partial differential equation

0=+ yyxx vu by method of separation of variables.

07

(b) Attempt (any one). (i) Prove that Laplacian u in polar coordinate is

2

2

22

22 11

θ∂∂+

∂∂+

∂∂=∇ u

rx

u

rx

uu .

(ii) Find the potential inside a spherical capacitor consisting two metallic hemispheres of radius 1 ft separated by a small slit for reasons of insulation , if the upper hemisphere is kept at 110 volts and lower hemisphere is grounded.

07

***********

1


GUJARAT TECHNOLOGICAL UNIVERSITY B.E. Sem-III Remedial Examination March 2010

Subject code: 130001 Subject Name: Mathematics -3

Date: 09 / 03 / 2010 Time: 11.00 am – 02.00 pm Total Marks: 70

Instructions: 1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

Q.1 (a) (1) Find the solution of differential equation ( ) 02 =++ dyeydxeyxx ,

where ( ) 10 −=y .

02

(2) Find the solution of differential equation xsinyy 324 =+′′ by method of

undetermined coefficient.

02

(3) Find

−−

+−−

2

102

1

ss

sL .

03

(b) (1) If possible, find the series solution of yy ′=′′ . 03

(2) Find the Fourier series of ( ) xxxf += , ππ <<− x 04

Q.2 (a) (1) Find the particular solution of xxxyyy 6655223 +−=+′−′′ . 02

(2) Evaluate dxex

nxam

∫∞

−

0

. 02

(3) Solve the partial differential equation xyx uu −= . 03

(b) (1) Evaluate ( ) ( )∫

−

−+1

1

11 dxxxnm , where 0>m , 0>n are integers.

03

(2) Find the solution of Wave equation

2

22

2

2

x

uc

t

u

∂

∂=

∂

∂under the conditions

(i) ( ) 00 =t,u , for all t, (ii) ( ) 01 =t,u for all t,

(iii) ( ) ( )( )

<<−

<<==

12112

21020

xifxk

xifxkxf,xu (iv) ( ) 0

0

==

∂

∂

=

xgt

u

t

.

04

OR

(b) (1) Find general solution of xsecyy 39 =+′′ by method of variation of

parameter.

03

(2) Get the Laplacian operator in cylindrical coordinates. 04

Q.3 (a) (1) Find

( )

+

++−

1

2223

231

ss

ssL .

03

(2) State Convolution theorem and use it to evaluate Laplace inverse

of( )222

ass

a

+.

04

(b) (1) Find the Laplace transform of half-wave rectification of tsinω defined

by ( )

<<

<<=

ω

π

ω

πω

πω

20

0

tif

tiftsin

tf where ( )tfn

tf =

+

ω

π2for all integer n .

03

2

(2) Find a series solution of differential equation 02 =+′+′′ yxyyx . 04

OR

Q.3 (a) (1) Find

−

−

814

31

s

sL .

03

(2) By Laplace transform solve, atsinKyay =+′′ 2 . 04

(b) (1) Find the inverse transform of the function

+

2

2

1s

wln .

03

(2) Find a series solution of differential equation ( ) 02 =+′−′′− yyxyxx . 04

Q.4 (a) (1) Solve the differential equation xcosexsinyy =+′ . 03

(2) Solve the Legendre’s equation ( ) ( ) 01212 =++′−′′− ynnyxyx for 0=n . 04

(b) (1) Write the Bessel’s function of the first kind. Also derive ( )xJ 0 and ( )xJ1

from it.

03

(2) Prove that ( ) ( )xJxJ 10 −=′

. 04

OR

Q.4 (a) (1) Solve the differential equation

2

22

3

6x

eyxy

x−

=+′ , where ( ) 01 =y . 03

(2) Obtain the Legendre’s function as a solution of ( ) 02212 =+′−′′− yyxyx . 04

(b) (1) Discuss the linear independency/dependency of Bessel’s functions ( )xJ n and ( )xJ n− .

03

(2) Show that ( ) ( ) ( )xJxxJxJ 1

1

01

−−=′

. 04

Q.5 (a) (1) Solve ( ) 433322 −=+− xlnyDxDx . 03

(2) Find Fourier series expansion of ( ) 22

xxf = , ( )ππ <<− x 04

(b) (1) Prove that

>

<<=

−∫∞

π

πππ

xif

xif/dwwxsin

w

wcos

0

021

0

. 03

(2) Find Fourier sine series of ( ) xxf −= π , ( )π<< x0 . 04

OR

Q.5 (a) (1) Solve ( )

2

22

44x

eyDD

x−

=++ . 03

(2) Sketch the function ( ) π+= xxf , ( )ππ <<− x where ( ) ( )xfxf =+ π2 and

find its Fourier series.

04

(b) (1) Find the Fourier cosine integral of ( ) xkexf

−= , where 0>x , 0>k . 03

(2) Find Fourier cosine series of ( ) xexf = , ( )Lx <<0 . 04

*************

1


GUJARAT TECHNOLOGICAL UNIVERSITY B.E. Sem-III Regular / Remedial Examination December 2010

Subject code: 130001 Subject Name: Mathematics – 3 Date: 11 /12 /2010 Time: 10.30 am – 01.00 pm

Total Marks: 70




Q.1 Do as directed.

a) Solve : yyyx +=′ 2

b) Find a second order homogeneous linear differential equation for which the

functions2x , 2x xlog are solutions.

c) Find the convolution of t and te .

d) Evaluate : ∫

1

0

3

4 1log dx

xx

e) Solve : 022 =+′+′′ yyy , ( ) 10 =y , 02=

πy .

f) Find ( )( )

−+−

32

11

ssL .

g) Compute :

2

7,

2

9β

14

Q.2 (a) Using the method of variation of parameters find the general solution of the

differential equation

( ) xexyDD 2

3

2 312 =+− .

05

(b)

Attempt all.

1) Solve the initial value problem ( ) 231 1 +=+−′ − xyxy , ( ) 11 −= ey .

2) Find the orthogonal trajectories of the curve cxy += 2.

3) Find a basis of solution for the differential equation ,02 =+′−′′ yyxyx if

one of its solutions is xy =1 .

09

OR

(b) Attempt all.

1) Solve : ( ) 4213

1

3

1xxyy −=+′ .

2) Solve the initial value problem ,0=+ RIdt

dIL 0)0( II = , where R ,L and I0

being constants.

3) Prove that

=−

∫ 2

1,

5

2

5

1

1

1

05

βx

xdx.

09

Q.3 (a) Using Laplace transforms solve the initial value problem ,2sin tyy =+′′

( ) ( ) 10,20 =′= yy .

05

2

(b)

Find the Fourier cosine series of the periodic function

( ) ( ) LpLxxxf 2,0; =<<= . Also sketch ( )xf and its periodic extension.

05

(c) Using the method of undetermined coefficient, find the general solution of the

differential equation .325102 2 +=+′+′′ xyyy

04

OR

Q.3 (a) Find the Fourier series of the periodic function ( ) xxf ππ sin= , ( )10 << x ,

12 == Lp .

05

(b) Solve the initial value problem 2484 22 ++=+′′ − xeyy x, ( ) 20 =y , ( ) 20 =′y . 05

(c) Find the complex Fourier series of the function ( ) xxf = , ( )π20 << x ,

π22 == Lp .

04

Q.4 (a) Find a series solution of the differential equation ( ) 02232 =−+′+′′ yxyxyx by

Frobenious method.

06

(b) Find the Laplace Transforms of

1) πsin2t t 2) ( )2−tue t

04

(c) Find the inverse Laplace Transformation of

1) 22

2

π+

−

s

se s

2) bs

as

++

log

04

OR

Q.4 (a) Attempt all.

1) Express ( ) 13 ++= xxxf in terms of Legendre’s polynomials.

2) Show that ( ) ( )∫−

=1

1

,0dxxPxP nm .nifm ≠

06

(b) Find the general solution of the equation ( ) xxyxDDx cos22 322 =+− . 04

(c) State Convolution theorem and use to evaluate

( )

+

−222

1 1

ωsL .

04

Q.5 (a) Using the method of separation of variables, solve the partial differential equation

yxx uu 16= .

06

(b)

Show that ∫∞

−

>

=

<

=+

+

0

2

0;

0;2

0;0

1

sincos

ifxe

ifx

ifx

dxx

xπ

πω

ωωωω

05

(c) Prove that ( ) ( ) ( )xJ

xxJxJ 101

1−=′ .

03

OR

Q.5 (a) Using Laplace transform, find the solution of the initial value problem

xtt

u

x

ux =

∂∂

+∂∂

, ( ) ,0;00, ≥= ifxxu ( ) .0,0,0 ≥= ifttu

06

(b) Find the Fourier Transforms of the Function ( )

<

>=

−

0;0

0;

ifx

ifxxexf

x

. 05

(c) Show that ( ) ( ) ( )xPxP n

n

n 1−=− .Hence find ( ).1−nP 03

*************

1


GUJARAT TECHNOLOGICAL UNIVERSITY B.E. Sem-III Remedial Examination May 2011

Subject code: 130001 Subject Name: Mathematics-III Date: 31-05-2011 Time: 10.30 am – 01.30 pm

Total Marks: 70




Q.1 Do as Directed

i) Solve the Euler-Cauchy equation 2 '' '2.5 2.0 0x y xy y− − = 2

ii) Find the Laplace transforms of

sinwt

t

2

iii) Solve ' 2y xy= by power series method. 2

iv) Find the Fourier cosine and sine transforms of the function

( )

0

k if o x af x

if x a

< <=

>

2

v) Write duplication formula. Use it to find the value of

1 3

4 4

2

vi) Write Abel-Liouville formula. Use it to check that the set

2, , logx x x x is a basis for some third order linear ordinary

differential equation.

2

vii) Obtain 1 1

logLs

−

2

Q.2 (a) i) Find the order and degree of the differential equation 1

2

sindy

y xdx

+ =

1

ii) To solve heat equation

22

2

u uc

t x

∂ ∂=

∂ ∂. How many initial and boundary

conditions requires.

2

iii) Evaluate

1

3 2

1

( ) ( )p x p x dx−∫

2

iv) Prove that

( 1, )

( , )

m n m

m n m n

ββ

+=

+

2

2

(b) i) Solve 2 2( ) cosD a y ecax+ = 4

ii) Solve

2

1 ydy e

dx x x+ =

3

OR

(b) i) Solve ( )4 2 2 42 cosD a D a y ax+ + = 4

ii) Obtain the second linearly independent solution of

'' '2 0xy y xy+ + = given that 1

sin( )

xy x

x= is one solution.

3

Q.3 (a) Solve 3 ''' 2 '' 1

2 2 10x y x y y xx

+ + = +

4

(b) Solve the initial value problem by method of undetermined

coefficients ''' '' ' ' ''3 3 30 , (0) 3, (0) 3, (0) 47,xy y y y e y y y−+ + + = = = − = −

4

(c) Solve the simultaneous equations: Using Laplace

transform , sin given (0) 1, (0) 0tdx dyy e x t x y

dt dt− = + = = =

6

OR

Q.3 (a) Solve 2 '' ' 44 6 21x y xy y x−− + = 4

(b) Solve the nonhomogeneous Euler-Cauchy equation 3 ''' 2 '' ' 43 6 6 logx y x y xy y x x− + − = by Variation of parameters

method.

4

(c) i) Find the Laplace transform of the function ( ) sin , 0f t wt t= ≥ 3

ii) Using Convolution theorem, find the inverse Laplace transform of

( )22 2

1

s a+

3

Q.4 (a) Express 4 3 22 3 4 5x x x x− + − + in terms of Legendre’s

polynomials, by using Rodrigue’s Formula.

4

(b) i) Find the generalized Fourier series expansion of ( ), 0 3f x x< <

arising from the eigenfunction of '' ' '0, 0 ; (0) 0, ( ) 0y y x l y y lλ+ = < < = =

4

ii) Obtain the value of 32

( )J x 3

(c) Show that

1

21

( ) 2

2 11 2

nnp x

dx tnxt t−

=+− +

∫ 3

OR

Q.4 (a) Find Power series solution of the equation 2 '' '(1 ) 0x y xy py− − + = ,

p is an arbitrary constant.

4

(b) Find the series solution of '' ' 0xy y xy+ + = 7

(c) Find the power series solution of the equation 2 '' '( 1) 0x y xy xy+ + − = about an ordinary point.

3

3

Q.5

(a) Find the Fourier transform of

21 1( )

0 1

x if xf x

if x

− <=

> and use it to

evaluate 3

0

cos sincos

2

x x x xdx

x

∞ − ∫

4

(b) Find the Fourier series for ( ) sin in f x x xπ π= − < < 4

(c) If the string of length L is initially at rest in equilibrium position and

each of its points is given the velocity. 0

3 2sin cos

x xu

L L

π π

Where0 at 0x L t≤ ≤ = , determine the displacement ( , )u x t .

6

OR

Q.5 (a)

Find half –rang cosine series for

, 02

( )

, 2

x x

f x

x x

π

ππ π

< <= − < <

4

(b) Find Fourier series for the function ( )f x given by

21 ; 0

( )2

1 ; 0

xx

f xx

x

ππ

ππ

+ − ≤ ≤= − ≤ ≤

Hence deduce that 2

2 2 2

1 1 1...

1 3 5 8

π+ + + =

4

(c) A rod 30 cm long has its end A and B kept 020 C and 080 C

respectively until steady state conditions prevail. The temperature at

each end is suddenly reduced to 00 C and kept so. Find the

resulting temperature function ( , )u x t from the end A.

6

*********

1

Seat No.: _________ Enrolment No._______________

GUJARAT TECHNOLOGICAL UNIVERSITY BE SEM-III Examination-Dec.-2011

Subject code: 130001 Date: 22/12/2011

Subject Name: Mathematics-III

Time: 2.30 pm -5.30 pm Total marks: 70

Instructions:

1. Attempt all questions.



4. Follow usual conventions.

Q. 1 (a) Do as directed :

1 Verify that y=ex (a cos x + b sin x) is a solution of y" + 2y' + 2y =0, (2) where a and b are constatnts.

2. Solve : 9yy' + 4x =0 (2) 3. Verify that the functions x -1/2 and x3/2 form a basis of solutions of (3) 4x2 y" – 3y=0 ; and solve it when y(1)=3, y' (1) = 2.5 (b) Do as directed :

1. Find the Laplace Transform of (2)

f(t) = 0 , 0≤ t <2 = 3 , when t ≥2

2. Find the general solution of ( D2 +1) y = 0. (2) 3. Show that cos mx and sin nx are orthogonal on -л ≤ x ≤ л , where m and n (3)

are integers. Q. 2 (a) Do as directed :

1. Show that u=sin 9t sin (1/4) x is a solution of a one dimensional wave (2) equation.

2. Determine if x=1 is a regular singular point of (1-x2) y" – 2 xy' + n(n+1) y=0, (2) where n is a constant. 3. Show that Γ(m+1) =m!, where Γ is the Gamma function and m is positive (3) integer.

(b) Do as directed : 1. Solve the IVP : xy'+y =0, y(2)= -2 . (2) 2. Solve the Bernoulli equation y'+y sinx=ecosx (3) 3. Solve : y"+4y'+4y=0, y(0)=1, y'(0)=1 (2) OR (b) Do as directed :

1. Test for exactness and solve : [ (x+1)ex –ey ] dx – x ey dy=0 , y(1)=0 (3)

2

2. Find the general solution : 16y"– 8y' +5y=0 (2) 3. Solve : (x2D2 -3xD +4 ) y =0 , y(1)=0, y'(1)=3 (2) Q. 3 (a) Solve the non-homogeneous equations (7)

1. y"-3y'+2y=ex 2. y"+y= sec x. (b) Do as directed : 1. Define the terms : Laplace Transform of f(t) , and its Inverse Transform. (2) 2. Using the Beta and Gamma functions evaluate the integral (3) 1

∫ (1-x2)n dx, where n is a positive integer. -1 3. Find the Laplace Transform of cos2 (at), where a is a constant. (2)

OR Q. 3 (a) Do as directed : 1. Find the general solution : y"'-3y" + 3y' – y = 4et (4) 2. Solve : y"'-y" +100y'-100y=0, y(0)=4, y'(0)=11, y"(0)= - 299 (3) (b) Do as directed : 1. Find the Laplace Transform of f(t)=sinh(ωt) , t≥0 (3) 2. Find the Laplace Transform of (4) (5 s2 +3s – 16)/(s-1) (s-2) (s+3) Q. 4 (a) Do as directed : 1. Solve the IVP using the Laplace Transform : y"+4y=0, y(0)=1, y'(0)=6. (3) 2. Find the Inverse Laplace Transform of (6+s)/(s2+6s+13) , use Shifting (4) theorem. (b) Do as directed 1. Find a power series solution in powers of x of y'+2xy=0 (4) 2. Derive the Legendra Polynomials P0(x)=1 and P2(x)=(1/2) (3x2-1) from the (3) Rodrigue’s formula. OR Q. 4 (a) Do as directed : 1. Applying the Binomial theorem to (x2-1)n and differentiating n times – or (3) by any other method - derive the Rodrigue’s formula Pn(x)= 1/(2n n! ) dn / dxn [ (x2-1)n ] 2. Find a series solution of y"+y=0 near x=0 (4) (b) Do as directed : 1. Solve by Frobenious method at x=0 : x (x-1) y" +(3x-1) y'+y=0 (4) 2. What is the Bessel’s function Jυ(x) of the first kind? Write the formula. (3) And show that J0' (x) = -J1(x)

3

Q. 5 (a) Find the Fourier series expansions of 1 f(x)=x, -л ≤ x ≤ л , f (x+2л ) = f(x) (3) 2 f(x)=x2 , -2 ≤ x ≤ 2 (4) (b) Derive the one dimensional wave equation that governs small vibrations (7) of an elastic string. Also state physical assumptions that you make for the system. OR Q. 5 (a) Do as directed 1 Define the terms : Fourier Transform and its Invertse. Give details. (3) 2 Find the Fourier Transform of e raised to –ax2 , where a>0 (4)

(b) Derive the expression for the Laplacian operator in cylindrical (7) coordinates from its expression in rectangular coordinates.

**********

1


GUJARAT TECHNOLOGICAL UNIVERSITY BE SEM-III Examination May 2012

Subject code: 130001 Subject Name: Mathematics - III

Date: 14/05/2012 Time: 02.30 pm – 05.30 pm Total Marks: 70 Instructions:

1. Attempt all questions. 2. Make suitable assumptions wherever necessary. 3. Figures to the right indicate full marks.

Q.1 (a)

(1)

(2)

(3)

(4)

Attempt all quations:

Solve the differential equation xyyxdxdyxy +++=1

Find the general solution of 08118 2

2

4

4

=+− ydx

yddx

yd

Find particular solution of ( )

,cosh1

12 x

Dy

+= where

dxdD =

Find the value of 43

41ΓΓ

04

(b) (1)

(2)

(3)

(4)

(5)

Attempt the following equations: Determine the singular points of differential equation

( ) ( ) 02322 2 =−+′+′′− yxyxyxx and classify them as regular or irregular. Find half range cosine series for ( ) xexf = in ( )1,0 . Find the fourier sine transform of ( ) 0,32 >+= −− xeexf xx .

Solve : ( ) ⎥⎦⎤

⎢⎣⎡ +=⎥⎦

⎤⎢⎣⎡ ++

dxdyxyy

dxdyxyx 12 Evaluate :

xdxx 11

0

4 cos−∫

10

Q.2 (a) (1)

(2)

(3)

Attempt the following quations: Find the Laplace transform of ( ) atttf sinh2=

Find the Laplace transform of ( )⎩⎨⎧

Π>Π<<

=ttt

tf,sin

0,0

Find the inverse Laplace transform of ( )( )15235

2 −+++

ssss

02

02

03

(b) (1)

(2)

Attempt the following quations : Solve the differential equation : ( ) ( ) 022 2222 =−++ xdyyxydxyx . Find the solution of differential equation 065 =+′−′′ yyy with initial condition ( ) 21 ey = and ( ) 231 ey =′ .

03

02

2

(3) Find the Laplace transform of ( )t

tcos1 − 02

OR (b)

(1)

(2)

Attempt the following quations: Using Laplace transform solve the differential equation

texdtdx

dtxd t sin522

2−=++ where ( ) 00 =x and ( ) 10 =′x .

Find the series solution of ( ) 091 2 =−′+′′+ yyxyx .

03

04

Q.3 (a)

(1)

(2)

(3)

Attempt the following quations

Solve: tt eetydt

yddt

yd++=+− 2

2

2

4

4

cos2

Solve: 5

2

2

2

44xey

dxdy

dxyd x

=+−

The Bessel equation of of order zero is 02'"2 =++ yxxyyx then (i) find the roots of the indicial equation (ii) show that one solution for 0>x is ( )xJcy 00=

where, ( ) ( )( )∑ −

+= 22

2

0 !211

nxxJ

n

nn

03

03

04

(b) Find fourier series for ( )

⎩⎨⎧

Π≤≤≤≤Π−Π−

=xx

xxf

0,0,

and show that ...........51

31

11

8 222

2

+++=Π

04

OR Q.3 (a)

(1)

(2)

(3)

Attempt the following quations

Solve: xeydxdy

dxyd

dxyd x 3cos53

2

2

3

2

=++−

Solve: ( ) ( ) ( ) )( xydxdyx

dxydx +=++++ 1logcos411 2

22

Find the series solution using by Fobenius method 0=−′+′′ yyyx

03

03

04

(b) Find fourier series for ( ) 22 xxxf −= in the interval ( )3,0 . 04

Q.4 (a)

(1)

(2)

(3)

Attempt the following quations :

Solve the differential equation ( )2cos2232

2xey

dxdy

dxyd x=+− Solve the

differential equation ( ) 222 43 xyXDDX =+− given that ( ) 11 =y and ( ) .01 =′y

Evaluate : ( ) ( ) dxxx 41

41

7

3

73 −−∫

03

03

02

(b) (1)

(2)

Attempt the following quations: Prove that in usual notation ( ) ( ) ( ) ( )xJxJxJxJ nnnn 22 24 +− +−=′′

Find Laplace transform of (i) ( )23 −− tue t , (ii) udue ut

cos0

−∫

03

03

3

OR

Q.4 (a)

(1)

(2)

(3)

Attempt the following quations:

Solve the differential equation ecxdxdy

dxyd cos3

3

=+ by method of variation of

parameters.

Solve : ( ) 2

22

144

xeyDD

x

+=+− where

dxdD =

Evaluate : ( ) dxxx∫1

0

3log

03

03

02

(b) (1)

(2)

Attempt the following equation:

Solve the differential equation ( ) ( ) ( ) 10,00,42

2

=′==+ yytfydt

yd by laplace

transform

where (i) ( )⎩⎨⎧

><<

=1,0

10,1t

ttf (ii) ( ) ( )2−= tHtf

Find the fourier transform of 2

2x

e− is 2

2λ−

e

03

03

Q.5 (a)

(1)

(2)

Attempt the following equation: Find half Range cosine series for sinx in (0,П ) and show that

4

.........71

51

311 Π

=+−+− And using parseval’s Identity prove that

168.........

7.51

5.31

3.11 2

222222

−Π=+++

Solve 022

2

=∂∂

+∂∂

−∂∂

yz

xz

xz

by the method of separation of variables

05

04

(b) A tightly stretched string with fixed end points x = 0 and x = L is initially

Given the displacement ⎟⎠⎞

⎜⎝⎛ Π=

Lxyy 3

0 sin If it is released from rest from this

position then find the displacement y

use the equation 2

22

2

2

xya

ty

∂∂

=∂∂

05

OR Q.5 (a)

(1)

(2)

Attempt the following equation:

If ( )( )⎪

⎪⎩

⎪⎪⎨

⎧

Π≤≤Π

−Π

Π≤≤

=xxm

xmxxf

2,

20,

then show that

( ) ...........5

5sin3

3sin1

sin4222 −+

⎩⎨⎧ −

Π=

xxxmxf

Determine the solution of one dimensional heat equation

2

22

xuc

tu

∂∂

=∂∂

where the boundary condition

are ( ) ( ) 0,0,,0 >== ttLutu and the initial condition is

05

04

4

( ) ,0, xxu = , L being the length. ( )Lx <<0 (b)

Solve the equation 2

2

xuk

tu

∂∂

=∂∂

for the condition of heat along

a rod without

radiation subject to the condition (i) 0=∂∂

xu

for 0=x and tx = (ii) 2xlxu −= at 0=t and for all x

05

*************

Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY B. E. - SEMESTER – III • EXAMINATION – WINTER 2012

Subject code: 130001 Date: 09-01-2013 Subject Name: Mathematics - III Time: 10.30 am – 01.30 pm Total Marks: 70 Instructions:


Q.1 (a) Find a Fourier Series for f(x) = x2 , where π20 ≤≤ x 05

(b)

ππππ

<<−=<<−+=

xxxxxfIf

0,0,)(

and xallforxfxf ,)2()( π+= then expand f(x) in a Fourier Series.

05

(c) Find a half range sine series for ππ <<−= xxxf 0,)( 04

OR (c) Find a Fourier Series for a periodic function f(x) with a period 2,where

10,101,1)(<<=<<−−=

xxxf

04

Q.2 (a) Find the inverse Laplace transforms of

81)(

)2)(3)(1(1635)( 4

32

−−+−−+

ssii

sssssi

05

(b) State the Convolution theorem on Laplace transforms and using it find

( )⎥⎦⎤

⎢⎣

⎡+

−

41

21

ssL

05

(c) Find the Laplace transforms of (i) cos22t and (ii) t3 cosh2t 04 OR (c) Find the Laplace transforms of the half wave rectifier

ωπ

ωπ

ωπω

2,0

0,sin)(

<<=

<<=

t

tttf

and ⎟⎠⎞

⎜⎝⎛ +=

ωπ2)( tftf

04

Q.3 (a) Define Beta function. Prove that (i) π=

21 (ii) B( m , n ) = B( m , n + 1 ) + B( m + 1 , n )

05

(b) State the Duplication formula.

Show that ∫∫∞ −∞

− =×00 22

2

2 πdxx

edxexx

x

05

(c) Show that (i) ),()2()()( 111 nmBadxxaxa nma

a

nm −+

−

−− =−+∫

(ii) ∫ ⎟⎠⎞

⎜⎝⎛=

−

1

05 2

1,52

51

1Bdx

xx

04

OR Q.3 (a) State the necessary and sufficient conditions to be exact differential

equation. Using it, solve 0)( 332 =+− dyyxdxyx 05

(b) Using the method of variation of parameters , solve xecdxdy

dxyd cos3

3

=+ 05

(c) Solve (i) yxyxdxdy sinsincoscos −=

(ii) xxydxdy 2sintan =+

04

Q.4 (a) Using the method of undetermined coefficients , solve 32

2

2

6366 xxxydxdy

dxyd

−+=−+

05

(b) Solve ⎟⎠⎞

⎜⎝⎛ +=++

xxy

dxydx

dxydx 11022 2

22

3

33 05

(c) Solve 22

2

2xey

dxdy

dxyd x

=+− 04

OR Q.4 (a) (i) If one of the solutions of 0,064 2

12 >==+′−′′ xxyisyyxyx then

determine its second solution. (ii) Solve : 0100100 =−′+′′−′′′ yyyy

05

(b) Prove that, (i) xx

xJ sin2)(21 π

= (ii) ⎟⎠⎞

⎜⎝⎛ −= x

xx

xxJ cossin2)(

23 π

05

(c) Show that (i) )()()(1 xJxnxJxJ nnn =′−−

(ii) 1)0(0 =J 04

Q.5 (a) Express 322)( 234 −−++= xxxxxf in terms of Legendre’s polynomials. 05

(b) Find a power series solution of 02

2

=+ ydx

yd 05

(c) Classify the singularities for following differential equations

(i) 0)3(62 2

22 =+++ yx

dxdyx

dxydx

(ii) 0)12()1( 22 =+′−+′′+ yxyxyxx

04

OR

Q.5 (a) Solve two dimensional Laplace’s equation 02

2

2

2

=∂∂

+∂∂

yu

xu ,using the

method separation of variables.

05

(b) A rod of length L with insulated side is initially at uniform temperature 1000C. Its ends are suddenly cooled at 00C and kept at that temperature. Find the temperature u(x, t).

05

(c) Find the Fourier transform of 0,00,)(

<=>= −

xxxexf x

04

************

1/2


GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTER–III • EXAMINATION – SUMMER 2013

Subject Code: 130001 Date: 01-06-2013 Subject Name: Mathematics-III Time: 02.30 pm - 05.30 pm Total Marks: 70 Instructions:


Q.1 Do as directed 14

(a) Find the Laplace transform of tt 2sin2 (b)

Evaluate

25,

27β

(c) Solve ( ) 0sec1tan 2 =−+ ydyeydxe xx (d) Find general solution of 0''' =− yy (e) Define Convolution and unit step function. (f) Discuss singularities of 0')1(3'')1(3 =+−−− xyyxyxx (g) Express 133)( 2 ++= xxxf in terms of Legendre polynomial

Q.2 (a) By power series method solve 022)1( 2

22 =+−− y

dxdyx

dxydx

07

(b) (i) Prove that [ ] )()( 1 xJxxJxdxd

nn

nn

−= 04

(ii) Define Gamma function and Evaluate ∫

∞−

0

26 xex 03

OR (b) (i) Solve 3)1( xyx

dxdyx =++ 03

(ii) Prove that )()()12()()1( 11 xnPxxPnxPn nnn −+ −+=+ 04 Q.3 (a) Define Laplace transform and find Laplace transform of

(i) 2133 tet t ++ − (ii) te t 2sin 22−

07

(b) Find inverse laplace transform of

(i) )3)(2)(1(

23 2

++++

ssss (ii)

)1(22

23

23

+++

ssss

07

OR Q.3 (a) State and prove convolution theorem 07

(b) Find inverse laplace transform of

(i)

−+

11log

ss (ii)

54)2(

2

4

+++−

ssse s

07

Q.4 (a) (i) Using method of variation of parameter solve 2

3

'2'' xeyyy x=+− 04

(ii) Solve ( ) xeyDD x 3sin6 22 =−+ 03 (b) (i) Find the Fourier series of xxxf +=)( , where ),( ππ−∈x 04

(ii) find Fourier series of 3)( xxf = where ),( ππ−∈x 03 OR

2/2

Q.4 (a) (i) Solve )sin(ln4'4''2 xyxyyx =−+ 04 (ii)Using method of undetermined coefficients solve xeyyy x +=+− '2'' 03 (b) Solve using Frobenius method 03')1('')1(2 =+−+− yyxyxx 07

Q.5 (a) Using method of separation of variables solve u

yu

xu 22

2

+∂∂

=∂∂ ;

yexuu 31,0 −+=

∂∂

= when 0=x

07

(b) A rod 30cm long has its end A and B kept C°20 and C°80 respectively until steady state condition prevail. The temperature at each end is suddenly reduce to C°0 and kept so. Find the resulting temperature function ),( txu from end A.

07

OR Q.5 (a) Find the Fourier transform of

2axe− where .0>a 07 (b)

Find Fourier cosine integral of

><<

=axif

axifxxf

00

)( 07

*************

1


GUJARAT TECHNOLOGICAL UNIVERSITY B.E. Sem-III Examination December 2009

Subject code: 130901 Subject ame: Circuits and etworks Date: 19 / 12 /2009 Time: 11.00 am – 1.30 pm

Total Marks: 70




Q.1 (a) State and explain (i) Thevenin’s theorem and (ii) Norton’s theorem in

brief giving suitable examples. 06

(b) What are Y-parameters and Z-parameters? Derive the expression for Z

parameters in terms of Y parameters and vice versa. 06

(c) How inductor and capacitor will behave at t = 0 and at t = ∞. Draw

equivalent networks. 02

Q.2 (a) What is duality? Prepare a list of dual quantities encountered in electrical

engineering. Describe the procedure to draw dual of a network. 07

(b) Determine the current through 4Ω resistor branch of the network given in

Fig 1. using mesh analysis 07

OR

(b) In the network of Fig.2 using node analysis find V2 which results in zero

current through 4Ω resistor. 07

Q.3 (a) A network with magnetic coupling is shown in Fig.3. For the network

M12=0 Formulate loop equations for this network using KVL. 04

(b) Determine the equivalent inductance at terminals A-B for circuit in Fig.4 02

(c) Explain the rules for source transformation technique. For the network

shown in Fig.5 determine the numerical value of current i2 using source

transformation technique.

08

OR

Q.3 (a) State and explain the maximum power transfer theorem. Derive the

condition for maximum power transfer to the load for d.c. circuits. 06

(b) For the network shown in Fig.6 determine the value of RL for maximum

power transfer. What will be the value of power transfer under this

condition?

08

Q.4 (a) For the network shown in Fig.7 switch K is closed at time t = 0 with zero

inductor current and zero capacitor voltage. Solve for 10

(i) V1 and V2 at t = 0+

(ii) V1 and V2 at t = ∞

(iii) dV1/dt and dV2/dt at t = 0+

(iv) d2V2/dt

2 at t = 0

+

(b) In the network of Fig. 8 steady state is reached with switch K open. At t =

0 switch K is closed. Find i(t) for the numerical values given. 04

OR

Q.4 (a) State the procedure to obtain solution of a network using Laplace

transform technique. State its advantages over classical method. 06

(b) For the circuit shown in Fig. 9 obtain the transform of the generator

current I(s). 03

2

(c) A series R-L-C circuit having initially zero inductor current and zero

capacitor voltage is excited by a 20V d.c. source. Find i(t). Assume R =

9Ω, L = 1H and C = 0.05F.

05

Q.5 (a) What is meant by poles and zeros of network function? State its important

features and explain its physical significance. 07

(b) Obtain ABCD parameters for the network shown in Fig. 10 07

OR

Q.5 (a) Give the definition of the following: 04

(i) Graph (ii) Branch (iii) Node (iv) Tree

(b) Draw the graph for the circuit shown in Fig.11. Prepare the incidence

matrix A and partition it into a matrix containing all passive branches Ap

and a matrix containing independent current sources branches Ag.

Formulate the branch admittance matrix Yp and hence find node

admittance matrix Yn

10

***********

1



Subject code: 130901 Subject ame: Circuit & etworks Date:10 / 03 /2010 Time: 11.00 am – 01.30 pm

Total Marks: 70




Q.1 (a) Define Charge, Current, Potential difference, Voltage, Node, Loop and

Independent source.

07

(b) Using source shifting and source transformation find out the voltage Vx in

the figure.

.

Figure for 1(b)

07

Q.2 (a) Explain Substitution theorem. 07

(b) Draw the Thevenin’s equivalent of the circuit shown in figure and find

current through load resistance(between terminal bb).

07

Figure for 2(b)

OR

(b) Find the current in the 5 ohm resistor for the circuit shown in figure using

Norton’s theorem.

07

Figure for 2(b)

Q.3 (a) Explain KCL and KVL using suitable example. 07

(b) Using mesh analysis obtain the current through the 10 V battery for the

circuit shown in figure.

07

2

Figure for 3(b)

OR

Q.3 (a) Explain Millman’s theorem. 07

(b) Find current and voltage drop through 5 ohm resistor in network shown in

figure.

07

Figure for 3(b)

Q.4 (a) Derive expression for rise of current and decay of current in RL series

circuit excited by DC voltage source. Discuss the role of time constant in

each.

07

(b) In figure steady state condition is reached with 100 V DC source . At t=0,

switch K is suddenly opened. Find the expression of current through the

inductor. Also find current through the inductor at t=0.5 second.

Figure for 4(b)

07

OR

Q.4 (a) Draw and explain equivalent circuit of two port network using h-

parameters.

07

(b) Find the Y-parameter for the circuit shown in figure.

Figure for 4(b)

07

Q.5 (a) Derive inter relationship between incidence matrix (A), fundamental tie set

matrix (Bf) and fundamental cut set matrix (Qf).

07

3

(b) For a resistive network shown in figure, draw graph and tree of the

network. Also develop the fundamental cut-set matrix.

07

Figure for 5(b)

OR

Q.5 (a) State the procedure to obtain solution of a network using laplace transform

method.State advantage of laplace method over classical method.

07

(b) What is meant by poles and zeros of a network function? What is the

significance of poles and zeros? Discuss the restrictions on locations of

poles and zeros of transfer functions.

07

*************

1



Subject code: 130901

Subject Name: Circuits and Networks Date: 14 /12 /2010 Time: 10.30 am – 01.00 pm

Total Marks: 70




Q.1 (a) Explain the terms ( i ) Linear (ii) Bilateral ( iii) Passive (iv) Reciprocal

(v) Time invariant (vi) Lumped parameter and (vii) Dual with reference to Network.

07

(b) Write down voltage and current relationships in resistor, inductor and capacitor.

Obtain these relationships in “s” domain also. State assumptions if any in obtaining

the relationship.

07

Q.2 (a) (i) Explain about voltage sources and current sources. Include ideal, practical,

independent and dependent sources in your explanation.

04

(ii) Using Nodal analysis find voltage V1 and V2 for the circuit shown in Figure 1. 03

(b) Explain in brief about source transformation and Find Norton’s equivalent circuit

for the network shown in Figure 2 and obtain current in 10Ω resistor.

07

OR

(b) Obtain Thevenin’s equivalent circuit for the network shown Figure 3 and find the

power dissipated in RL= 5Ω resistor. Find RL for maximum power transfer from the

source and compute maximum power that can be transferred i.e. Pmax.

07

Q.3 (a) Find the voltage across 1KΩ resistor in the circuit shown in Figure 4, using

superposition theorem.

07

(b) Obtain the response vC(t) and iL(t) for the source free RC and RL circuits

respectively. Assume initial voltage V0 and initial current I0 respectively.

07

OR

Q.3 (a) For the circuit shown in Figure 5, the switch “S” is at position “1” and the steady

state condition is reached. The switch is moved to a position “2” at t = 0. Find the

current i(t) in both the cases, i.e. with switch at position 1 and switch at position 2.

07

(b) How do one classify that the given circuit is of first order or second order? Obtain

second order circuit models for series RLC and parallel RLC circuits in time

domain and in “s” domain.

07

Q.4 (a) Obtain the Laplace Transform for f1(t)=t and f2(t)= te-at

07

(b) State the final value theorem of Laplace Transform and find the final value of the

function f(t) =5u(t) + 10e-t using final value theorem. Under what conditions the

final value theorem cannot be used ? Give one example.

07

OR

Q.4 (a) What is an impulse function ? For the network function H(s) given below, Find the

impulse response h(t). 1

H(s) =

s2+ 4s +1

07

(b) Obtain currents I1(s), I2(s) and V0(s) for circuit shown in Figure 6. 07

2

Q.5 (a) Find Z- parameters for the network shown in Figure 7. 07

(b) Explain about hybrid parameters for two port network and state where do one make

use of these parameters.

07

OR

Q.5 (a) ABCD parameters are also known as transmission parameters and they are derived

from the basic two port network parameters. Show that, for reciprocal linear time

invariant two port network, AD-BC =1.

07

(b) Explain about linear oriented graph, Incidence Matrix and Circuit Matrix. Show

Kirchoff’s Laws in Incidence Matrix formulation and Circuit Matrix formulation.

07

1



Subject code: 130901 Subject Name: Circuits and Networks Date: 25-05-2011 Time: 10.30 am – 01.00 pm

Total Marks: 70



3. Symbols and notations have conventional meaning unless stated.


Q.1 (a) Give the relation between energy (E) and power (P). Derive the

equations for the energy stored in a capacitor (C) and an inductor (L)

using P=VI.

07

(b) Prove the maximum power transfer theorem for a practical voltage

source (Vs, Rs). What is the maximum power that can be delivered if

Vs=20 V and Rs=1 Ohm?

07

Q.2 (a) Derive a tree of the graph of the network in Fig.1. Determine the node

voltages V1 and V2, using the mesh analysis.

Figure 1 Network of Q. 2. Resistance values are in Ohms.

07

(b) Determine the node voltages V1 and V2 in the network shown in Fig. 1,

by applying the superposition theorem.

07

OR

(b) In Fig. 1, if 1 Ohm resistance is changed to 1.2 Ohm then determine the

source-voltage for compensating for the change.

07

Q.3 (a) Solve for the nodal voltages V1, V2, V3 and V4 as shown in the

network in Fig. 2, using the nodal analysis.

Figure 2 Network of Q. 3. Resistance values are in Ohms.

07

(b) In Fig. 2, if 2V source is replaced by an open circuit then find

Thevenin’s and Norton’s equivalent circuits across V2 and V3.

07

2

OR

Q.3 (a) Find the equivalent inductance for the series and the parallel connections

of L1 and L2 if their mutual inductance is M.

07

(b) State Millman’s theorem. Obtain the equivalent of a parallel connection

of three branches each with a voltage source and a series resistance, (2V,

1 Ohm), (3V, 2 Ohm) and (5V, 2 Ohm).

07

Q.4 (a) Define the time-constant of RL and RC networks and explain the

significance of the time-constant.

07

(b) Explain how to determine the initial conditions in an RL network and

the current i(t) based on these conditions.

07

OR

Q.4 (a) Obtain the loop-current i(t) in the RC network in Fig. 3, by solving the

differential equation of the loop.

07

(b) Obtain the voltage across the capacitor Vc(t), in the LC circuit in Fig. 4

using Laplace transform technique if Vc(0)=2V.

07

Q.5 (a) Determine the voltage across the capacitor in the RLC circuit as shown

in Fig. 5, if R=400 Ohm using Laplace transform.

07

(b) Determine the poles of a series RLC circuit, if R=120 Ohm, L=10 mH

and C=1 micro-F. Sketch the pole-plot and comment on the nature of the

response.

07

OR

Q.5 (a) Explain the short-circuit admittance and the open-circuit impedance

parameters for a two port network.

07

(b) Draw a tree of the network in Fig. 6 taking the branches denoted by (b2),

(b4), and (b5) as tree branches. Give the fundamental loop matrix.

Determine the matrix loop equation from the fundamental loop matrix.

07

Figure 3 RC

network of Q.4

(a).

Figure 4 LC

network of Q. 4

(b).

Figure 6 RLC circuit of Q. 5 (a).

Figure 5 Network of Q. 5 (b).

Branch-impedances are in Ohms.

*************

1




Subject Name: Circuits and Networks

Time: 2.30 pm -5.00 pm Total marks: 70 Instructions:




Q.1 (a) (i) Find Vx using node analysis for the network in Fig.1

(ii) A series RLC circuit with zero inductor current and zero capacitor

voltage is excited by 50V dc source, find i(0+) and di(0+)/dt. Take R=20Ω,

C=10µF, L=1H.

03

04

(b) State KVL and find loop currents i1, i2 and i3 using loop analysis for the

network in Fig.2.

07

Q.2 (a) In a series RLC circuit of Fig.3 v(t) = 6 e -2t

volts.switch K is closed at t = 0.

find current i(t) using laplace transformation method. Assume zero initial

conditions.

07

(b) (i) Derive Laplace transform of derivatives and integrals.

(ii) Find Laplace transform of cosωt.

05

02

OR

(b) Describe Laplace transformation method for solving differential equations,

state its advantage over the classical method.

07

Q.3 (a) Using super position theorm find voltae Vx for the network shown in Fig.4. 07

(b) Find Z parameter for the the two port network shown in Fig.5 07

OR

Q.3 (a) State thevenins theorm, find Rth and Vth. for the network shown in Fig.6 07

(b) Find ABCD parameter for the two port network shown in Fig.7 07

Q.4 (a) Explain duality and find equivalent dual network of the circuit given in

Fig.8

07

(b) In the network of Fig.9 a steady state is reached with switch K open. At t=

0 switch is closed , for the element values given determine va (0-) , va(0+).

07

OR

Q.4 (a) State and explain (i) Reciprocity theorm (ii)Nortons Theorm. 07

(b) (i) Draw transform representation in terms of impedance for inductor

with initial current.

02

(ii) For the circuit in Fig.10 switch K is moved from position a to b at

t = 0, having been in position a for long time before t=0. capacitor C2

is uncharged initially, Find particular solution for i(t) and v2(t) for t > 0.

05

Q.5 (a) Explain incident matrix of a linear oriented graph with example. 07

(b) For the network shown in Fig.11 all sources are time invariant, find the

branch current I using source transformation method.

07

OR

Q.5 (a) Explain circuit matrix of a linear oriented graph and kirchhoff’s laws in

fundamental circuit matrix formulation.

07

(b) Discuss dot convention of coupled coils and write kirchhoff’s voltage law

equations for the network in Fig.12

07

**********

1



Subject code: 130901 Subject Name: Circuits and Networks



Q.1 (a) State and Explain the “The Maximum Power Transfer Theorem” .Also

derive the condition for maximum power transfer to the load for D.C. and A.C. Circuit.

07

(b) Explain the various Two port parameters in brief. Hence derive the expression of ABCD parameters in terms of Z parameters.

07

Q.2 (a) Explain the “Dot Convention Rule” for the magnetically coupled

Network. Explain the method to put the Dots on different linked Coils using suitable example.

07

(b) (1)Formulate the Loop equations for Network shown in fig-1 (2) Find Voltage drop across x-y for the fig-2

07

OR (b) Explain following in Brief: Ideal and Practical Energy source

Using the Node Voltage analysis, Find the current in all resistors in fig-3.

07

Q.3 (a) Explain following in Brief: Tree, Graph and Link, Active and Passive

elements ,Lumped and Distributed parameters, 07

(b) For the Network shown in fig-4, Draw the oriented Graph and all possible trees. Also prepare (1)The Incidence Matrix.(2)Tie-set Matrix.(3)F-cut set Matrix.

07

OR Q.3 (a) Explain various source transformation techniques. Using Source

transformation techniques find current “i1” in the network shown in fig-5..

07

(b) (1) State and Explain in brief: Thevenin’s Theorem 07 (2) Determine the Inductance between terminal for the 3 coils system

shown in fig-6.

Q.4 (a) (1)Explain: The concept of Duality.

List all analogous quantities used in Duality.. (2) State the Initial and final condition of R,L and C at t=0+ and t=∞. (Initially all are uncharged and put across the source).

07

(b) In the Network shown in fig-7, a steady state is reached with switch k open with V=100v, R1=10 ohm, R2=R3=20 ohm, L=1 h, C=1μF. At time t=0 switch k closed. Determine (1) voltage across C before switch is closed and its polarity (2) i1 and i2 at t=0+. (3) d i1/dt and d i2/dt at t=0+. (4) d i1/dt at t=∞.

07

2

OR Q.4 (a) State the procedure to obtain the solution of Laplace Transform

Technique. State its advantages over classical method. State only Initial and Final value theorem.

07

(b) In fig-8, i1 is flowing as shown and switch k is closed at time t=0, placing 10 ohm resistor in parallel with series combination of R=10 ohm and L=2 h. Find the resulting currents. Use Laplace Transform Technique.

07

Q.5 (a) Explain the Poles and Zeros of the Network function. State its

important features and explain its physical significance. 07

(b) Find the Z parameters of the Network shown in fig-9.Hence derived ABCD parameters from Z parameters.

07

OR Q.5 (a) Explain the various types of Interconnections of the Two port

networks in brief. 07

(b) Find the current in the 4 ohm resistor in fig-10 using Thevenin’s Theorem and Super position theorem.

07

*************

1



Subject code: 130901 Date: 10-01-2013 Subject Name: Circuits and Networks Time: 10.30 am – 01.00 pm Total Marks: 70 Instructions:


Q.1 (a)

(b)

State and prove Thevenins Theorem, find Rth and Vth. for the network shown in Fig.1 Find ABCD parameters for the two port network shown in Fig.2. Also derive Y parameters from ABCD parameters

07 07

Q.2 (a)

(b)

Explain the terms ( i ) NonLinear (ii) Uniilateral ( iii) Passive (iv) Reciprocal (v) Time variant (vi) Lumped parameter and (vii) principal of Duality. Write down voltage and current relationships in resistor, inductor and capacitor .Also mention the initial and final condition for R,L and C components in the different cases.

07 07

OR

(b) Explain the “Dot Convention Rule” for the magnetically coupled Network using network shown in Fig-3.Also formulate KVL equations.

07

Q.3 (a) Explain the formulation of graph , tree and Incidence Matrix using

suitable example. Hence discuss the procedure of forming reduced Incidence Matrix and its advantages.

07

(b) For the Network shown in fig-4, Draw the oriented Graph and all possible trees. Also prepare (1)The Incidence Matrix. (2) Fundamental Tie set Matrix. (3)Fundamental cut set Matrix.

07

OR

Q.3

(a)

Explain various source transformation techniques. Using Source transformation techniques find current “i” in the network shown in fig-5

10

2

(b) Explain following in Brief: Ideal and Practical Energy source 04

Q.4 (a) Explain The Laplace Transformation method. Find Laplace Transform of Unit Step, and exponential function.

07

(b) In the Network shown in fig-6 ,the switch k is closed at t=0 ,connecting voltage Vo sinwt to the parallel RL-RC circuit. Find

(1) di1/dt and (2) d i2/dt at t=0+

07

OR

Q.4 (a) State and explain various Network Functions. 07 For the resistive two port network of fig-7,determine the numerical

values for G12(s),Z12(s) and α12(s).

Q.5 (a) State and explain the Initial and final value theorem. 07

(b) Find the particular solution for the current using laplace transformation in the n/w shown in fig-8.The switch k is closed at t=0.Assume zero initial conditions in the elements.

07

OR

Q.5 (a) State and explain Superposition Theorem. Hence using this find Vab in fig-9.

07

(b) Find the current through the 2V source in fig-10 using Node voltage analysis.

07

*************



Subject Code: 130901 Date: 04-06-2013 Subject Name: Circuits and Networks Time: 02.30 pm - 05.00 pm Total Marks: 70 Instructions:


Q.1 (a) Define following terms: 07

(1) Linear and Nonlinear networks (2) Lumped and Distributed networks

(3) Passive and Active networks (4) Dependent source

(b) For magnetically coupled network shown in Fig.-1, find dot-convention and

Write the KVL equations. 07

Fig.-1

Q.2 (a) Find currents through the resistors in the network of Fig.-2 using mesh analysis. 07

Fig.-2

(b) Determine the current through 2Ω resistor of Fig.3 using source transformation. 07

Fig.3

Fig.-4

OR (b) In the network of Fig.-4 , determine the node voltages V1, V2 and V3 using node

analysis. 07

Q.3 (a) In the network of Fig.-5 the switch K is opened at t=0. Find the values of V, dV/dt

and d2V/dt2 at t=0+ if I=10A, R=10Ω and L=1H. 07

Fig.-5

Fig.-6

(b) Derive necessary derivations for source free series R-L-C circuit. 07 OR

Q.3 (a) In the network of Fig.-6, the switch K is moved from 1 to 2 position at t=0, steady

state having previously been attained. Determine the current i(t). 07

(b) In the network of Fig.-7, if the switch has remained in position A for a long time and

then moves to position B at t=0. Find and plot Vc(t) for t ≥ 0 for R2= 405Ω. 07

Fig.- 7 Fig.-8

Q.4 (a) State and explain Norton’s theorem. 07

(b) Discuss Duality in detail. 07 OR

Q.4 (a) Determine the current in 1 Ω resistor of the network of Fig.-8 using Thevenin’s theorem. 07 (b) Derive relationship between Z-parameters and Y-parameters. Discuss Reciprocity and symmetry of network in brief. 07

Q.5 (a) In the network of Fig.-9 , the switch K is moved from position a to b at t=0 (Steady state existing). Solve for the current i(t) using Laplace transformation method. 07

Fig.-9

Fig.-10

(b) Find Z-parameters for the network of Fig.- 10 . 07

OR

Q.5 (a) Explain the concept of poles and zeros and their significance. 07

(b) Define Sub-graph. For the circuit shown in Fig.-11, draw the graph and write the

(i) incidence matrix and (ii) cutset matrix. 07

Fig.-11



Subject code: 130701 Subject ame: Digital Logic Design Date: 21 /12 /2009 Time: 11.00 am – 1.30 pm

Total Marks: 70




Q.1 (a) Convert the following numbers to decimal 07

(i) (10001.101)2 (ii) (101011.11101)2 (iii) (0.365)8

(iv) A3E5 (v) CDA4 (vi) (11101.001)2 (vii) B2D4

(b) Perform the operation of subtractions with the following binary

numbers using 2′ complement

07

(i) 10010 - 10011 (ii) 100 -110000 (iii) 11010 -10000

Q.2 (a) Obtain the simplified expressions in sum of products for the

following Boolean functions:

07

(i) F(A,B,C,D,E) =∑(0,1,4,5,16,17,21,25,29)

(ii) A′B′CE′ + A′B′C′D′ +B′D′E′ + B′C D′

(b) Demonstrate by means of truth tables the validity of the following

Theorems of Boolean algebra

07

(i) De Morgan’s theorems for three variables

(ii) The Distributive law of + over-

OR

(b) Implement the following Boolean functions 07

(i) F= A (B +CD) +BC′ with NOR gates

(ii) F= (A + B′) (CD + E) with NAND gates

Q.3 (a) Design a combinational circuit that accepts a three bit binary

number and generates an output binary number equal to the square

of the input number.

07

(b) Discuss 4-bit magnitude comparator in detail 07

OR

Q.3 (a) With necessary sketch explain full adder in detail 07

(b) Design a combinational circuit that generates the 9′ complement of a

BCD digit,

07

Q.4 (a) Discuss D-type edge- triggered flip-flop in detail 07

(b) Design a counter with the following binary sequence:0,4,2,1,6and

repeat (Use JK flip-flop)

07

OR

Q.4 (a) Design a counter with the following binary sequence:0,1,3,7,6,4,and

repeat.(Use T flip-flop)

07

(b) (i)With neat sketch explain the operation of clocked RS flip 05

(ii)Show the logic diagram of clocked D 02

Q.5 (a) With necessary sketch explain Bidirectional Shift Register with

parallel load.

07

(b) Draw the state diagram of BCD ripple counter, develop it’s logic

diagram, and explain it’s operation.

07

OR

Q.5 (a) Construct a Johnson counter with Ten timing signals. 07

(b) Discuss Interregister Transfer in detail 07 ***********

1



Subject code: 130701 Subject ame: Digital Logic Design Date:10 / 03 /2010 Time: 03.00 pm – 0.5.30 pm

Total Marks: 70




Q.1 (a) Define: Digital System.

Convert following Hexadecimal Number to Decimal :

B28, FFF, F28

Convert following Octal Number to Hexadecimal and Binary:

414, 574, 725.25

07

(b) Define : Integrated Circuit and briefly explain SSI, MSI, LSI and VLSI 07

Q.2 (a) Draw the logic symbol and construct the truth table for each of the

following gates.

[1] Two input NAND gate [2] Three input OR gate

[3] Three input EX-NOR gate [4] NOT gate

07

(b) Give classification of Logic Families and compare CMOS and TTL

families

07

OR

(b) Explain SOP and POS expression using suitable examples 07

Q.3 (a) Design a 4 bit binary to BCD code converter 07

(b) Design a full adder circuit using decoder and multiplexer 07

OR

Q.3 (a) Write short note on EEPROM, EPROM and PROM 07

(b) Define: [1] Comparator [2] Encoder [3] Decoder

[4] Multiplexer [5] De-multiplexer [6] Flip Flop [7] PLA

07

Q.4 (a) Draw and explain the working of following flip-flops

[1] Clocked RS [2] JK

07

(b) Convert SR flip-flop into JK flip-flop 07 OR

Q.4 (a) Design sequential counter as shown in the state diagram using JK flip-flops

Clockwise direction to follow

07

000

010

100

011 111

110

001

2

(b) State and explain the features of register transfer logic 07

Q.5 (a) Explain the working of 4 bit asynchronous counter 07

(b) Explain memory unit 07

OR

Q.5 (a) Explain the design of Arithmetic Logic Unit 07

(b) Explain Control Logic Design 07

*************



Subject code: 130701 Subject Name: Digital Logic Design Date: 15 /12 /2010 Time: 10.30 am – 01.00 pm

Total Marks: 70




Q.1 (a) Convert the following Numbers as directed:

(1) (52)10 = ( )2

(2) (101001011)2 = ( )10

(3) (11101110) 2 = ( )8

(4) (68)10 = ( )16

07

(b) Reduce the expression:

(1) A+B(AC+(B+C’)D) (2) (A+(BC)’)’(AB’+ABC) 07

Q.2 (a) Simplify the Boolean function:

(1)F(w,x,y,z) = ∑ (0,1,2,4,5,6,8,9,12,13,14)

(2)F(w,x,y) = ∑ (0,1,3,4,5,7)

07

(b) Explain with figures how NAND gate and NOR gate can be used as Universal gate. 07

OR

(b) Simplify the Boolean function:

(1) F = A’B’C’+B’CD’+A’BCD’+AB’C’

(2) F =A’B’D’+A’CD+A’BC

d=A’BC’D+ACD+AB’D’ Where “d ” indicates Don’t care conditions.

07

Q.3 (a) With logic diagram and truth table explain the working of 3 to 8 line decoder. 07

(b) With logic diagram and truth table explain the working JK Flipflop.Also obtain its

characteristic equation. How JK flip-flop is the refinement of RS flip-flop? 07

OR

Q.3 (a) Design a counter with the following binary sequence:

0, 4,2,1,6 and repeat. Use JK flip-flops 07

(b) With logic diagram and function table explain the operation of 4 to 1 line

multiplexer. 07

Q.4 (a) What is the function of shift register? With the help of simple diagram explain its

working. With block diagram and timing diagram explain the serial transfer of

information from register A to register B.

07

(b) With respect to Register Transfer logic, explain Interregister Transfer with

necessary diagrams. 07

OR

Q.4 (a) With logic diagram explain the operation of 4 bit binary ripple counter. Explain the

count sequence. How up counter can be converted into down counter? 07

(b) Prepare a detailed note on: Instruction Codes. 07

Q.5 (a) What is scratchpad memory? With diagram explain the working of a processor unit

employing a scratchpad memory. 07

(b) Briefly explain control organization. With diagram explain control logic with one

flip-flop per state. 07

OR

Q.5 (a) Draw the block diagram of a processor unit with control variables and explain its

operation briefly. 07

(b) With simple diagram explain the working of control logic with sequence register

and decoder. 07

*************

1



Subject code: 130701 Subject Name: Digital Logic Design Date: 27-05-2011 Time: 10.30 am – 01.00 pm

Total Marks: 70


Q.1 Answer the following 14

(i) Draw symbol and construct the truth table for three input Ex-OR gate.

(ii) What is the principle of Duality Theorem?

(iii) Explain briefly: standard SOP and POS forms.

(iv) What are Minterms and Maxterms?

(v) Define: Noise margin , Propagation delay

(vi) Give comparison between combinational and Sequential logic circuits

(vii) What is race-around condition in JK flip-flop?

Q.2 (a) (i) Explain NAND and NOR as an universal gates (04)

(ii) Convert decimal 225.225 to binary ,octal and hexadecimal (03) 07

(b) (i) Implement Boolean expression for Ex-OR gate using NAND gates only

(04)

(ii) convert decimal 8620 into BCD , excess-3 code and Gray code.

(03)

07

OR

(b) (i) Simplify the following Boolean function using K-map

F( w,x,y,z) = ∑( 1 , 3 , 7 , 11 , 15 ) (04)

with don’t care conditions d( w,x,y,z ) = ∑( 0, 2 ,5 )

(ii) Draw logic diagram , graphical symbol , and

Characteristic table for clocked D flip-flop (03)

07

Q.3 (a) Design a combinational circuit whose input is four bit binary number and

output is the 2’s complement of the input binary number.

07

(b) Design a full-adder with two half-adders and an OR gate 07

OR

Q.3 (a) Design a BCD to decimal decoder 07

(b) What is multiplexer? Implement the following function with a multiplexer:

F(A,B,C,D) = ∑(0 , 1 , 3 , 4 , 8 , 9 ,15 ) 07

Q.4 (a) Write short note on : Read Only Memory (ROM) 07

(b) A combinational circuit is defined by functions:

F1(A,B,C) = ∑( 3 , 5 , 6, 7 )

F2(A,B,C) = ∑( 0 , 2 , 4, 7 )

Implement the circuit with PLA having three inputs ,four product term and

two outputs

07

OR=

Q.4 (a) Give classification of counters and explain asynchronous 07

2

4-bit binary ripple counter

(b) Explain briefly:

(i) logic and shift micro-operations

(ii) fixed-point binary data and floating-point data

07

Q.5 (a) Draw block diagram of a 4-bit arithmetic logic unit. Design an

adder/subtractor circuit with one selection variable S and two inputs A and

B .when S = 0 circuit performs A+B, when S = 1 circuit performs A – B by

taking the 2’s complement of B

07

(b) Draw and explain block diagram of microprograme control. 07

OR

Q.5 (a) Simplify the following Boolean function using tabulation Method and

draw logic diagram using NOR gates only

F(w,x,y,z ) = ∑( 0 ,1 , 2 , 8 ,10 ,11,14,15 )

07

(b) Explain working of master-slave JK flip-flop with necessary logic diagram

, state equation and state diagram 07

*************

1




Subject Name: Digital Logic Design





Que. 1 ( a) Convert the decimal number 225.225 to binary , octal and hexadecimal. (06)

( b) Explain briefly : SOP & POS , minterm & maxterm , canonical form , (05)

propagation delay, fan out

( c ) Represent the decimal number 8620 in BCD , Excess-3 , and Gray code (03)

Que.2 ( a) Design a combinational circuit whose input is a four bit number and whose (08)

Output is the 2’s complement of the input number

OR

(a) Simplify the following Boolean function by using Tabulation method. (08)

F = Σ ( 0,1,2,8,10,11,14,15 )

( b ) Draw symbol and truth table for four input EX-OR gate. Explain NAND (06)

and NOR as an universal gate

Que.3 ( a ) Design BCD to Excess-3 code converter using minimum number (08)

of NAND gates

OR

( a ) Simplify Boolean function F ( w,x,y,z ) = Σ ( 0,1,2,4,5,6,8,9,12,13,14 ) using

K-map and Implement it using (i) NAND gates only (ii) NOR gates only (08)

( b ) Explain the working of the Master Slave J K flip-flop (06 )

OR

( b ) Explain Arithmetic micro operations

Que.4 ( a ) Explain working of 4-bit binary ripple counter (07)

( b ) Draw and explain block diagram of 4-bit bidirectional shift register with ( 07 )

Parallel load

OR

Que. 4 ( a ) What is meant by multiplexer ? Explain with diagram and truth table

the Operation of 4-to-1 line multiplexer (07)

( b ) What is meant by decoder ? Explain 3-to-8 line decoder with diagram

and truth table (07)

Que.5 ( a ) Explain the procedure followed to analyze a clocked sequential circuit (10)

With suitable example

2

( b ) Define : state table , state equation , state diagram , input & output equations (04)

OR

Que. 5 ( a ) Draw and explain logic diagram of arithmetic logic unit ( ALU ) (08)

( b ) What is the difference between hardwired control and micro program control ?

write advantages and disadvantages of each method (06)

**********

P.T.O.



Subject code: 130701 Subject Name: Digital Logic Design

Date: 09/05/2012 Time: 02.30 pm – 05.00 pm Total Marks: 70


Q.1 (a) Convert the Decimal Number 250.5 to base 3, base 4, base 7 & base 16. 04 (b) Given Boolean function

F= x y + x′ y′ + y′ z 1. Implement it with only OR & NOT gates 2. Implement it with only AND & NOT gates

05

(c) Design the Combinational Circuits for Binary to Gray Code Conversion. 05

Q.2 (a) Determine the Prime Implicants of following Boolean Function using Tabulation

Method. F(A,B,C,D,E,F,G)=∑(20,28,38,39,52,60,102,103,127)

07

(b) Explain Design Procedure for Combinational Circuit & Difference between Combinational Circuit & Sequential Circuit.

04

(c) Express following Function in Product of Maxterms F(x,y,z)= ( xy + z ) ( y + xz )

03

Q.3 (a) Construct 4*16 Decoder with help of 2*4 Decoder. 05 (b) Discuss 4 bit BCD Adder in Detain. 05 (c) Explain Master Slave Flip Flop through J.K Flip Flop 04

OR Q.3 (a) Design Sequential Circuit with J.K. Flip Flops to satisfy the following state

equation. A( t + 1 ) =A′ B′ CD + A′ B′ C + ACD +AC′ D′ B(t+1)= A′ C + CD′ + A′ BC′ C(t + 1) = B D(t +1)=D′

07

(b) Explain 4 bit Magnitude Comparator. 07

Q.4 (a) Explain 4bit binary ripple counter. 07 (b) Explain Arithmetic addition and arithmetic subtraction. 04 (c) Brifley explain processor unit with a 2-port memory 03

P.T.O.

============================= Best of Luck ==============================

OR Q.4 (a) Define the different mode of operation of registers & explain any two in details. 07 (b) How many flip flops are required to build a shift register to store following

numbers. i) Decimal 28 ii) Binary 6 bits iii) Octal 17 iv)Hexadecimals A

04

(c) Explain Macro operations Versus micro operations 03

Q.5 (a) Explain 4-bit up-down binary synchronous counter. 07 (b) Explain comman cathode types seven segments displays. 03 (c) Simplify the following Boolean function using K-Map.

F=A′B′C′+B′CD′+A′BCD′+AB′C′ 04

OR Q.5 (a) Explain Johnson Counters. 07 (b) Write the Comparisons between Hard wired control and micro programmed

Controls. 03

(c) Design a combination circuits for a full adder. 04

1



Subject code: 130701 Date: 04-01-2013 Subject Name: Digital Logic Design Time: 10.30 am – 01.00 pm Total Marks: 70 Instructions:


Q.1 Convert the decimal number 250.5 to base 3, base 4, base 7, base 8 and

base 16 14

Q.2 (a) Show how a full-adder can be converted to a full-subtractor with the

addition of one inverter circuit. 07

(b) Design a combinational circuit with four input lines that represent a decimal digit in BCD and four output lines that generate the 9’s complement of the input digit.

07

OR (b) Construct 4x16 decoder with two 3x8 decoders. 07

Q.3 (a) Find the complement of the following Boolean function and reduce to a minimum number of literals. B’D + A’BC’ + ACD + A’BC

07

(b) Obtain the simplified expressions in sum of products using K-map: x’z + w’xy’ + w(x’y + xy’)

07

OR Q.3 (a) Simplify the following Boolean function by means of the tabulation

method: F(A,B,C,D,E,F,G) = (20,28,38,39,52,60,102,103,127)

07

(b) Explain JK Flipflop. What is the disadvantage of it and how it can be eliminated?

07

Q.4 (a) Design a counter with the following binary sequence:

0, 1, 3, 7, 6, 4 and repeat. Use T flipflop. 07

(b) Explain BCD Ripple counter and draw its logic diagram and timing diagram.

07

OR Q.4 (a) Explain in detail bidirectional shift register with parallel load. 07 Q.4 (b) Explain PLA with necessary diagrams. 07

Q.5 Explain arithmetic, logic and shift microoperations in detail. 14

OR Q.5 (a) Explain bus organization for four processor register and ALU connected

through common buses. 07

(b) Distinguish between microprogram control and hard-wired control. 07

*************

1/1



Subject Code: 130701 Date: 27-05-2013 Subject Name: Digital Logic Design Time: 02.30 pm - 05.00 pm Total Marks: 70 Instructions:


Q.1 (a) Convert the decimal number 250.5 to base 3, base 4, base 7and base 16. 07

(b) Perform the subtraction with the following decimal numbers using 1’s compliment and 2’s compliments. (a) 11010-1101 , (b) 10010-10011

07

Q.2 (a) Simplify the following Boolean functions to a minimum numbers of literals.

(a) xyz+x’y+xyz’ and (b)(A+B)’(A’+B’)’ 07

(b) Obtain the truth table of the function F= xy+xy’+y’z 07 OR (b) Implement the Boolean functions. 07

Q.3 (a) Implement the Boolean functions.(a) xyz+x’y+xyz’ (b) (A+B)’(A’+B’)’ and (c) F= xy+xy’+y’z with logic gates.

07

(b) Show that the dual of the exclusive-OR is equal to its compliment. 07 OR

Q.3 (a) Obtain the simplified expression in sum of product for the following Boolean functions. (a) F= ∑(0,1,4,5,10,11,12,14) and (b) F=∑(11,12,13,14,15).

07

(b) Implement the functions F=∑(1,3,7,11,15) with don’t care conditions d=∑(0,2,5) Discuss the effect of don’t care conditions.

07

Q.4 (a) Explain half and full adders in detail. 07

(b) Design and implement BCD to excess 3 code converter. 07 OR

Q.4 (a) What is the difference between serial and parallel transfer? What type of registers are used in each case?

07

(b) Design a synchronous BCD counter with JK flip flops. 07

Q.5 (a) Explain a 4 to 1 line multiplexer in detail. 07 (b) Explain PLA in detail. 07 OR

Q.5 (a) Explain scratchpad memory in detail. 07 (b) Explain D type positive edge triggered flip flop. 07

*************

1



Subject code: 131101 Subject ame: Basic Electronics Date: 17 /12 / 2009 Time: 11.00 am – 1.30 pm

Total Marks: 70




Q.1 (a) Define following terms:

(i) Electron Volt (eV).

04

(ii) Mobility of charge carries.

(iii) Barrier potential.

(iv) Voltage equivalent of temperature.

(b) Explain energy band diagram of insulator, semiconductor and

conductor.

05

(c) Explain following for npn transistor. 05

(i) Current components.

(ii) Regions of operation according to biasing condition

Q.2 (a) Draw the circuit diagram of full wave bridge rectifier and give its input

and output waveforms. Also derive the expression for the d.c. current.

07

(b) Explain Hall effect. Derive expression of Hall voltage and state its

applications.

07

OR

(b) A bar of n type silicon has length of 5 cm and circular cross sectional

area of 10 mm2. When it is subjected to a voltage of 1 V along its

length, the current flowing through it is 5 mA. Calculate the

concentration of free electrons and drift velocity of electrons. Assume

mobility of free electrons to be 1300 cm2/V-s.

07

Q.3 (a) Compare zener and avalanche break down. 04

(b) What is transition capacitance in p-n junction diode? Give its physical

significance.

05

(c) State the use of clipping circuits. Discuss with neat sketch working of a

biased parallel clipper.

05

OR

Q.3 (a) Explain principle of operation of a Photodiode. 04

(b) Discuss piece-wise linear model of a diode. 05

(c) Compare V-I characteristics of Silicon and Germanium p-n junction

diode.

05

Q.4 (a) Draw CE transistor configuration and give its input and output

characteristics. Also derive the relation between current gain of CE,

CB and CC configurations.

07

(b) Give constructional details of JFET and give its characteristics. Why

FET is called voltage controlled device?

07

OR

2

Q.4 (a) Explain the operation of Emitter follower amplifier. Why is it named as

emitter follower?

07

(b) Give points of difference between BJT and FET. Also explain FET as

voltage variable resistor.

07

Q.5 (a) State the need of biasing. Discuss voltage divider bias circuit and

mention its advantages.

07

(b) What is the difference between voltage amplifier and power amplifier?

State important features of power amplifier and classify them based on

the position of Q point.

07

OR

Q.5 (a) Discuss h-parameter equivalent circuit for transistor in CE

configuration.

07

(b) State the role of voltage regulators in power supplies? Discuss working

of a series voltage regulator.

07

***********

1




Subject ame: Basic Electronics

Date: 09 / 03 / 2010 Time: 3.00 pm – 05.30 pm

Total Marks: 70




Q.1 (A) What is transition capacitance of a p- n junction diode ? For a step graded

junction prove that the expression for the transition capacitance of a diode is

same as the capacitance of a parallel plate capacitor.

07

(B) How does the designer minimize the percentage variation in IC , due to variation

in ICO and VBE and due to variation in β in transistor amplifier circuit.

07

Q.2 (A) A 230 V , 50 Hz AC voltage is applied to the primary of a 5:1 step down

transformer which is used in a bridge rectifier having a load resistor of a value

470 Ω. Assuming the diodes to be ideal , determine the following

(a) DC output voltage

(b) DC power deliver to the load

(c) Maximum value of output current

(d) Average value of output current

(e) RMS value of output current

(f) Output frequency

(g) PIV of diode

07

(B) Define

(a) Drift velocity of electron

(b) Electric field

(c) Photovoltaic potential

(d) Photo excitation

(e) Photo ionization

(f ) Intensity of electric field

(g) Reverse recovery time of diode

07

OR

(B) Define

(a) Electron volt

(b) Potential

(c) Critical wavelength for semiconductor

(d) Mean life time of carrier

(e) Mobility of electron

(f) Volt equivalent of temperature

(g) Pinch off voltage of FET

07

2

Q.3 (A) A bar of silicon 0.2 cm long has a cross sectional area of 9 × 10-8

m2 , heavily

doped with phosphorus. What will be the majority carrier density resulting from

doping if the bar is to have resistance of 2 kΩ ? Given for silicon at room

temperature :

µn= 0.14 m2/V-sec, µp= 0.05 m

2/V-sec, ni= 1.5 × 10

10 /cm

3, q =1.602 ×10

-19 C

04

(B) Give minimum four comparisons of following semiconductor devices

(1) Tunnel diode with conventional diode.

(2) LED with conventional diode.

(3) LED with photo diode.

06

(C) Prove that current density is proportional to product of charge density , mobility

of charge and electric field intensity.

04

OR

Q.3 (A) A static resistance of 4 Ω is observed in an ideal germanium diode at room

temperature. The current flowing through the diode is 50mA. If the forward

biased voltage is 0.2V , volt equivalent temperature is 26mV , calculate :

(i) Reverse saturation current

(ii) Dynamic resistance of diode

04

(B) Draw output waveform of following circuits. Consider input of 20V (peak to

peak), 10kHz sine wave and assume ideal diode.

(i)

1kΩ

V1

5 V

Vi Vo

+

__

+

(ii)

1kΩ

5 V

Vi Vo

_

++

_

1µF

06

(C) Explain the hall effect in semiconductor. How hall effect is considered in

measurement of mobility and conductivity?

04

Q.4 (A) Derive relationship between αdc and βdc of a transistor. 04

(B) Explain any one circuit which is used to improve the input impedance of the

amplifier.

06

3

(C) Determine whether or not the transistor in below circuit is in saturation. Assume

β =50 and VCE( sat) =0.3V, VBE =0.7V.

10 V 6.8kΩ

1kΩ

2.5 V

04

OR

Q.4 (A) Compare various transistor amplifier configurations. 04

(B) Draw and explain the input and output characteristics of p-n-p silicon transistor

in CB configuration. Indicate cut off , saturation and active regions.

06

(C) Two stage amplifier circuit is mentioned below. Calculate overall voltage gain

Av. Take hie = 2.2K ,hfe = 60 , hre= 2.5 × 10-4

,hoe= 2.5 µA/V, Rc= 3.3 kΩ ,

Re= 4.7kΩ, Rs= 1 kΩ, Vcc=+12V.

1kΩ

3.3kΩ

4.7kΩ

+

_

Vo

+Vcc

+

_Vs

Rs

Rc

Re

Q1

Q2

04

Q.5 (A) For following circuit , calculate the minimum and maximum value of emitter

current when β of transistor varies from 75 to 150 . Also calculate the

corresponding values of collector to emitter voltage. Take VBE = 0.3V, Rb=

10kΩ , Rc= 50 Ω , Re= 100 Ω, Vcc= +6V.

10kΩ

50Ω

100Ω

+6V

Rb

Rc

Re

Q1

06

(B) Explain with neat circuit diagram, the working of a transformer coupled class A

power amplifier.

04

4

(C) Compare FET with BJT in terms of advantages, disadvantages, construction and

operation.

04

OR

Q.5 (A) Describe briefly the construction and working of p channel enhancement

MOSFET. Draw its characteristic and transfer curve.

06

(B) A class B push pull amplifier supplies power to a resistive load of 15Ω. The

output transformer has a turns ratio of 5:1 and efficiency of 78 %.

Assume hfe =25 and Vcc =18V.

Obtain :

(a) Maximum power output

(b) Maximum power dissipation in each transistor

(c) Maximum base current for each transistor.

06

(C) Explain the signification of following parameters in evaluating the regulation

performance of a DC series regulator

(a) Input regulation factor ( Stability factor ) SV

(b) Temperature stability factor ( Temperature co efficient ) ST

02

*************

1




Subject Name: Basic Electronics Date: 13 /12 /2010 Time: 10.30 am – 01.00 pm

Total Marks: 70


Q.1 (a) Answer the following:

(i) What is semiconductor? Define a hole in semiconductor

(ii) State the Pauli exclusion principle

(iii) Sketch the piecewise linear characteristics of p-n diode

(iv ) Define an electron volt (eV)

(v) State the mass-action law as an equation and in word.

(vi) What is cutin voltage? Write approx. value of cutin voltage for silicon

and germanium diode

(vii) Write the equation for the volt-ampere characteristic a photo diode

07

(b) Draw and explain bridge rectifier circuit with capacitorfilter. Draw necessary

waveforms. 07

Q.2 (a) Draw the circuit of CE configuration of transistor. ExplainInput and output

characteristics. Derive α = β / β+1 07

(b) (i) Draw symbol and explain briefly the working principle Breakdown diode and

Tunnel diode 04

(ii)Write principle and applications of light emitting diode 03

OR

(b) (i) Describe the Hall effect. Which properties of a Semiconductor are

determined from Hall effect experiment? 04

(ii) Explain electrical properties of germanium and silicon ( conductivity ,the

mobility and the energy gape) 03

Q.3 (a) Draw following diode circuits with input and output Waveforms:

(i) Voltage doublers circuit 03

(ii) Positive clipping circuit 02

(iii) Negative clamper circuit 02

(b) (i) A 5kΩ load is fed from a bridge rectifier connected with a transformer

secondary whose primary is connected to 460V, 50 Hz supply. The ratio of

number of primary to secondary turns is 2 : 1. Calculate dc load current ,dc

load voltage , ripple voltage and PIV rating of diode,

04

(ii) A 100µF capacitor when used as a filter has 12 V dc Across it with a terminal

load resistor of 2.5kΩ. If the rectifier is full wave and supply frequency is 50

Hz calculate the percentage of ripple in the output

03

OR

Q.3 (a) Explain the h-parameter model of CE amplifier with Bypass resistor RE and

derive the expression for Ai , Av , Ri , Ro 07

(b) Find hre in terms of the CB h-parameters 07

2

Q.4 (a) What is biasing? Why biasing is required for transistor? List biasing methods for

transistor. Draw and explain the circuit of voltage divider biasing 07

(b) Where CC configuration is used? Draw circuit of CC and CB configuration of

transistor. Compare current gain ,voltage gain ,input impedance and output

impedance of both

07

OR

Q.4 (a) A CE amplifier using npn transistor has load resistance RL connected between

collector and Vcc supply of + 16 V For biasing resistor , R1 is connected between

Vcc and base Resistor R2 = 30 kΩ is connected between base and ground. RE =

1kΩ. Draw the circuit diagram and calculate the value of R1 , RC ,stability factor

S if VBE = 0.2V, IEQ = 2 mA , VCEQ = 6 V , α = 0.985

07

(b) Design a fixed bias circuit using silicon npn transistor Which has βdc = 150. The

dc biasing point is VCE = 5V And Ic = 5 mA Supply voltage is 10V.Write

advantages and disadvantages of fixed bias circuit.

07

Q.5 (a) (i) Define the pinch-off voltage Vp .Sketch the depletion region before and after

pinch-off. 03

(ii) Sketch the cross section of a P-channel enhancement MOSFET .Show two

circuit symbol for MOSFET 04

(b) Draw circuit of an idealized class-B push-pull power amplifier and explain its

operation with the help of necessary waveforms. 07

OR

Q.5 (a) (i) Compare different types of power amplifier based on conduction angle ,

position of Q-point , efficiency and distortion 04

(ii) Draw circuit of transistor as a switch 03

(b) A MOSFET has a drain- circuit resistance Rd of 100K and operates at 20 kHz.

The MOSFET parameters are gm = 1.6 mA/V, rd = 44K , Cgs = 3 Pf Cds = 1 pF

,Cgd = 2.8 pF.Calculate the voltage gain of this device .

07

*************

1


GUJARAT TECHNOLOGICAL UNIVERSITY B.E. Sem-III Examination May 2011

Subject code: 131101 Subject Name: Basic Electronics Date:30/05/2011 Time: 10.30am to 1.00pm

Total Marks: 70




Q.1 (a) Prove that the minority carrier concentration, in an n-type semiconductor bar which was momentarily illuminated, decreases exponentially with time.

07

(b) Derive continuity equation and explain its importance. 07

Q.2 (a) When a diode is driven from forward condition to reverse condition, draw and explain waveforms for (1) minority carrier concentration at the junction of the diode (2) current flowing through the diode circuit, and (3) voltage across the diode. Assume resistance (RL) is present in series with diode.

07

(b) A symmetrical 5-kHz square wave whose output varies between +10 V and -10 V is impressed upon the clipping circuit shown in Fig. 1. Assume diode forward resistance (Rf) as zero, diode reverse resistance as (Rr) 2M, diode cut-in voltage (Vγ) as zero. Sketch the steady-state output waveform, indicating numerical values of the maximum, minimum, and constant portions.

07

OR

(b) Design a Zener regulator (Fig. 2) for following specifications: load current IL = 20 mA, output voltage Vo = 5 V, Zener wattage PZ = 500 mW, Input voltage Vi = 12 ± 2 V, and IZ,(min) = 8 mA,

07

Q.3 (a) A silicon transistor with VBE, sat = 0.8 V, β = hFE = 100, VCE, sat = 0.2 V is used in the circuit shown in Fig. 3. Find the minimum value of RC for which the transistor remains in saturation.

07

(b) Derive expressions for AI, Ri, AV, and Yo in terms of CE h-parameters for emitter-follower circuit.

07

OR

Q.3 (a) Represent/derive CC h-parameters (hic and hfc) in terms of CE h-parameters . 07

(b) Explain the base-width modulation and its effect on minority-carrier concentration in the base region of a transistor as well as on the common-base input characteristics of a typical p-n-p transistor.

07

Q.4 (a) Define stabilization factors: S, S’, and S”. Derive expressions for S and S’ for self-bias transistor circuit.

07

(b) Derive an expression for voltage gain (AV) for CS amplifier with an bypassed source resistance RS.

07

OR

Q.4 (a) The fixed-bias circuit is given in Fig. 4 and it is subjected to an increase in

temperature from 25 C to 75 C. If β = 100 at 25 C and β = 125 at 75 C,

determine the percentage change in Q point values (VCE, IC) over the temperature range. Neglect any change in VBE. Take VBE = 0.7 V.

07

2

(b) Draw a structure of p-channel MOSFET. Explain its working for enhancement type. Also draw and explain drain characteristics and transfer curve for the same device.

07

Q.5 (a) Illustrate how the energy levels of isolated atoms in group IV A (e.g., C, Si, Ge, Sn) are split into energy bands when these atoms are brought into close proximity to form a crystal. Draw necessary energy band diagrams.

07

(b) Show that the upper limit of the conversion efficiency (η) for the series-fed class A amplifier is 25 %.

07

OR

Q.5 (a) Draw class B push-pull system and show that the maximum conversion efficiency (η) is 78.5 % for this system.

07

(b) Draw and explain working of the circuit for compensation of VBE using diode. 07

2.5 V

1 MVi Vo

Figure 1 Q:2 (b)

I L

IZR

RLVi

I

Vo

+

−

+

Figure 2 Q:2 (b) OR

10 V

5 V

200 K

Rc

Figure 3 Q:3 (a)

RBI

R

IB

C

C

100 KOhm

600 Ohm

12 V

Figure 4 Q:4(a) OR

*************

1




Subject Name: Basic Electronics





Q.1 (a) Explain the concept of potential energy barrier. 07

(b) State the limitations of Rutherford model and explain Bohr atomic

model.

07

Q.2 (a) Explain the mobility and conductivity using electron-gas theory. Also

derive the expression of current density.

07

(b) Describe the Hall effect and also explain how it is help to determine

the different properties of semiconducting material.

07

OR

(b) Explain the generation of holes and electrons in an intrinsic

semiconductor.

07

Q.3 (a) Explain the formation of barrier potential in open circuited PN

junction diode. Also derive the expression for barrier potential.

07

(b) A diode having internal resistance 20Ω is used for half-wave

rectification. If the applied voltage V=50sin(ωt) and load resistance

RL=800Ω,find:

1) Im, Idc, Irms

2) d.c. output voltage

3) efficiency of rectification.

07

OR

Q.3 (a) Define the rectification and describe the full wave bridge rectifier with

the help of neat circuit diagram and waveforms.

07

(b) The resistivities of two sides of a step graded germanium diode are 2

Ω.cm and 1 Ω.cm for p-side and n-side respectively. Calculate the

height of potential energy barrier Vo. Assume µp=1800 cm2/v.sec,

µn=2100 cm2/v.sec, q=1.6×10

-19 ni= 2.5 ×10

13 per cm

3

07

Q.4 (a) Define following terms:

1) PIV

2) voltage equivalent of temperature

3) electric potential

4) electron volt

5) Ripple factor

6) base spreading resistance

7) pinch off voltage

07

(b) Explain the different types of clipping circuits. 07 OR

2

Q.4 (a) Explain the output characteristic of n-p-n transistor in CE

configuration. Also indicate different regions.

07

(b) Determine h-parameters for the two port network. Also draw the

hybrid model for CE, CB and CC configurations.

07

Q.5 (a) Explain DC load line and Q-point for any transistor configuration.

Also state the necessity of biasing and list biasing methods for

transistor.

07

(b) List the basic configurations of a low frequency FET amplifier.

Explain any one of them with the help of neat circuit diagram and

small signal equivalent circuit.

07

OR

Q.5 (a) Classify the power amplifiers based on the position of Q-point on the

ac load line. Also explain Class-B push-pull amplifier.

07

(b) Explain the principle of operation of JFET. Also compare FET with

BJT.

07

*************

1



Subject code: 131101 Subject Name: Basic Electronics



Q.1 (a) Define the following terms: [7]

(i) Potential (ii) ev unit of energy (iii) Volt-equivalent temperature (iv) Thermal resistance (v) Intermodulation distortion (vi) Mean life time of a carrier (vii) Peak Inverse Voltage

(b) Draw and explain the transistor current components when it is biased in active region. Obtain the generalized transistor equation.

[5]

(c) In n-type semiconductor, concentration of donor atoms is 4.14×1014 atoms/cm3. Calculate the conductivity and resistivity of semiconductor. (Assume: mobility of electron=3800 cm2/volt.sec)

[2]

Q.2 (a) Explain the Hall effect and obtain the expression of Hall coefficient. List the

applications of Hall effect. [5]

(b) Explain the concept of potential barrier and state Bohr’s postulates. [5] (c) Draw the circuit of sampling gate and explain its operation. [4]

OR (c) A germanium diode has a contact potential of 0.2v, while the concentration of

acceptor impurity atoms is 3×1020 per m3, calculate for a reverse bias of 0.1v, the width of the depletion region. If the reverse bias is increased to 10v, calculate the new width of the depletion region. Assuming cross-sectional area of the junction as 1mm2, calculate the transition capacitance values for both cases. Assume Er as 16 for germanium.

[4]

Q.3 (a) Draw circuit of CB transistor and explain its input and output characteristics. [5] (b) Figure No. 1 shows the two way clipper. If the input voltage is sinusoidal source of

16v peak-to-peak, sketch the output waveform. (Assume voltage drop across diodes is 0.7v)

[5]

(c) Sketch the full-wave rectifier circuit and explain its operation. [4] OR

2

Q.3 (a) Verify mathematically that transistor means “transfer resistor”. Explain the working

of phototransistor.

[5]

(b) Figure No. 2 shows an n-p-n transistor. It has the Ico= 2×10-5mA and β=100. Find the transistor currents and value of α of transistor.

[5]

(c) Distinguish: (i) avalanche breakdown and Zener breakdown. [4] (ii) Drift current and diffusion current.

Q.4 (a) Draw the self-bias circuit and explain how it establishes the stable operating point. [5] (b) Draw push-pull arrangement of two transistors and prove that this arrangement can

balance out all even harmonics. [5]

(c) Define thermal runaway. Derive necessary condition to avoid thermal runaway. [4] OR

Q.4 (a) Explain the operation of class A large signal amplifier with circuits and output

waveforms and also derive the expression of output power. [5]

(b) Check the condition to avoid the thermal runaway of a self bias circuit, if Vcc=30v, Rc= 2.0KΩ, Re= 4.7KΩ and collector current Ic=1.5mA and give the comments. (Assume that collector current increases by 0.131mA over temperature range of 25 to 75oC).

[5]

(c) Draw and explain regulated power supply system. [4]

Q.5 (a) Draw following circuits:

(i) Small-signal high frequency equivalent common drain FET amplifier. (ii) Ebers-Moll model of P-N-P transistor.

[6]

(b) Explain the working and characteristics of p-channel enhancement type MOSFET. [5] (c) Datasheet for a JFET indicates that IDSS=10mA and VGS(off) = -4v. Determine the

drain current for VGS= 0v, -1v. [3]

OR Q.5 (a) Draw a transistor amplifier circuit using h parameter and derive expressions for

current gain, voltage gain, input impedance and output impedance. [6]

(b) Consider a single stage CE amplifier (Figure No. 3) with Rs=1KΩ, R1=50KΩ, R2= 2KΩ, Rc= 1KΩ, RL= 1.2KΩ, hfe= 50, hie=1.1KΩ, hoe= 25 µA/V and hre= 2.5×10-4. Find current gain, input resistance, voltage gain and output resistance.

[5]

(c) Draw the circuit of emitter follower. [3]

3

Figure No. - 1

Vo

4

Figure No. - 2

Figure No. - 3

Rc

Rb



Subject code: 131101 Date: 07-01-2013 Subject Name: Basic Electronics Time: 10.30 am – 01.00 pm Total Marks: 70 Instructions:


Q.1 (a) Explain followings:

(i) Electron volt. (ii) Mobility (iii) Barrier potential (iv) Diffusion current (v) Mean life time of a carrier (vi) Graded semiconductor (vii) Intrinsic concentration

07

(b) Explain Hall effect with neat sketch. Discuss how to measure charge density and mobility for a given specimen of semiconductor using Hall Effect?

07

Q.2 (a) Specimen of material is 5 cm long and having radius of 5 mm. Current is

due to electrons whose mobility is 5000 cm2/V.s. Current of 50 mA flows through it when 0.5 Volt is impressed across it. Calculate concentration of free electrons and drift velocity.

07

(b) Explain potential variation in graded semiconductor. 07 OR (b) Derive the flowing equation for current density.

μεnqJ = 07

Q.3 (a) (i) Describe two breakdown mechanisms in a p-n junction diode.

(ii) Why the name varicap is given to varactor diode? Give its two applications.

04 03

(b) A sinusoidal voltage peak value of 10V and frequency 50 Hz is applied at the input of clipping circuit shown in figure below. Draw output voltage waveform and transfer characteristic. Assume both diodes are ideal.

07

OR

1

Q.3 (a) (i) Draw symbol of tunnel diode, Draw VI characteristic of tunnel diode and explain it.

(ii) Explain how Zener diode regulates voltage.

04 03

(b) A sinusoidal voltage peak value of 40V and frequency 50 Hz is applied at the input of a half wave rectifier, No filter is used. The Load resistance is 500 Ω. Neglect cut-in voltage. Diode has Rf = 5 Ω and Rr = . ∞(i) Draw Output voltage waveform and derive expression for DC output

voltage. (ii) Calculate DC value of load current, rms value of load current and

Rectification efficiency.

07

Q.4 (a) (i) Derive relation between α and ß for a transistor.

(ii) Why CE configuration is preferred for amplification? 04 03

(b) Draw a fixed bias circuit. State advantages and disadvantages of fixed bias circuit. Specify components value to have operating point at (9V, 2mA). Take VCC = 12 V and ß = 70.

07

OR Q.4 (a) (i) In npn transistor α = 0.98, IE = 20 mA, ICBO = 3µA. Determine IC, IB, ß

and ICEO (ii) What is early effect in CB configuration? Explain with graph.

04 03

Q.4 (b) Draw collector to base bias circuit and explain its operation. Also state advantages and disadvantages of the circuit.

07

Q.5 (a) Draw Emitter follower circuit. Obtain Hybrid equivalent circuit and derive

expression for current gain. 07

(b) (i) Does thermal runaway take place in FET? Why? (ii) Define parameters of FET and state relationship among them.

03 04

OR Q.5 (a) Determine Av, Ai, Ri and Ro for a common emitter amplifier using a

transistor with hie = 1200 Ω, hfe = 36, hre = 0 and hoe = 2 × 10-6 mho. Use RL = 2.5 K Ω, RS = 500 Ω and neglect the effect of biasing circuit.

07

(b) (i) Explain in what respect a power amplifier differ from a voltage amplifier?

(ii) Prove that the maximum theoretical collector circuit efficiency of a class A amplifier is 50%

03 04

*************

2

1



Subject Code: 131101 Date: 31-05-2013 Subject Name: Basic Electronics Time: 02.30 pm - 05.00 pm Total Marks: 70 Instructions:


Q.1 (a) Define electron volt and draw general energy band diagram for insulator,

semiconductor, and metal. 07

(b) State Bohr’s postulates and derive expression for energy levels of electrons in Joules as a function orbit number surrounding nucleus.

07

Q.2 (a) Explain the term mobility related to charged carriers and derive expression for point form of Ohm’s law.

07

(b) Describe phenomenon of Hall effect with mathematical derivations. What are the different applications of Hall effect?

07

OR (b) Obtain expression for potential difference across a semiconductor with non-

uniform (graded) doping. Using the same, derive an expression for potential difference across open-circuited step-graded p-n junction.

07

Q.3 (a) Define and explain following terms related to diode: 1. Transition capacitance and 2. Diffusion capacitance.

07

(b) Draw double-diode clipper circuit which limits output voltage at two independent levels. Explain its working with necessary waveforms.

07

ORQ.3 (a) Draw diode I-V characteristic and explain diode static and dynamic

resistances. 07

(b) Explain working full-wave rectifier with necessary waveforms. Obtain expression for dc output voltage.

07

Q.4 (a) Indicate and briefly explain various current components flowing in p-n-p

transistor with forward-based emitter junction and reverse-biased collector junction.

07

(b) Define h-parameters, and draw h-parameter equivalent circuit for CE, CB and CC configured transistor.

07

OR Q.4 (a) Draw output and input characteristics for common-base configured transistor.

Explain base-width modulation (Early effect) for the same. 07

(b) Derive expression for small-signal voltage gain of emitter follower circuit in terms of h-parameters.

07

Q.5 (a) What do you understand by bias stability in transistor amplifier circuit?

Explain thermal instability of bias point for the same. 07

(b) Draw structure of n-channel JFET and explain its working. 07 OR

Q.5 (a) Draw and explain working of diode compensation circuit for VBE for self-stabilization in amplifier circuit.

07

(b) Compare FET and BJT devices. Define small-signal parameters of FET and draw low-frequency small-signal model for the same.

07

*************

1

Seat o.: _____ Enrolment o.______

GUJARAT TECHOLOGICAL UIVERSITY B.E. Sem-III Examination December 2009

Subject code: 131701 Subject ame: Electrical Machines

Date: 23 / 12/ 2009 Time: 11.00 am – 1.30 pm

Total Marks: 70

Instructions:




Q.1 (a) Explain the working of a 1-phase transformer. Also derive its e.m.f.

equation. 07

(b) A single phase transformer has 350 primary and 1050 secondary turns. The

primary is connected to 400 V, 50 a.c. supply. If the net cross sectional area

of the core is 50 cm2, calculate the maximum flux density in the core and

induced e.m.f. in the secondary winding.

04

(c) Explain how 3-phase supply can be converted into 2-phase supply using

Scott connection. 03

Q.2 (a) Differentiate between single excited and multiple excited systems. Derive the

expression for magnetic field energy stored in a singly excited system. 07

(b) Define energy and co-energy. Derive the expression for force developed for

current excited and voltage controlled systems 07

OR

(b) Attempt the following:

(i) State: (a) regulation of a transformer (b) types of electromechanical

energy conversion and (c) advantages of open delta (V-V) connections of

transformers.

07

(ii) Develop equivalent circuit of a 1-phase transformer. Draw the phasor

diagrams for no-load and load conditions.

Q.3 (a) Discuss types of 3-phase induction motor based on rotor construction and

explain its working. 07

(b) With reference to 3-phase Induction motor, attempt the following:

(i) Define slip of an induction motor. Explain its slip-torque characteristic.

04

(ii) Briefly explain various methods of speed control of a 3-phase induction

motor. 03

OR

Q.3 (a) The following test results refer to a 14.92 kW, 6 pole, 50 Hz, 400 V,

3-phase induction motor:

No-load test (Line values): 400 V, 11 A, p.f. = 0.2

Blocked rotor test (Line values): 100 V, 25 A, p.f. = 0.4

Draw the circle diagram and determine the full load power factor, slip and

efficiency. Rotor copper loss at standstill is half the total copper loss.

07

2

(b) With reference to induction motor, attempt the following:

(i) Explain “cogging” and “crawling” in a 3-phase induction motor with

their remedies.

03

(ii) Why single-phase induction motor is not self-starting? Explain any one

method to make it self-starting. 04

Q.4 (a) State various advantages of stationary armature in an alternator. Also

differentiate between salient pole and non-salient pole synchronous

machines.

07

(b) Define regulation of an alternator (synchronous generator).

A 3-phase star connected alternator supplies a load of 1000 kW at a power

factor of 0.8 lagging with a terminal voltage of 11 kV. Its armature resistance

is 0.4 ohm per phase while synchronous reactance is 3 ohm per phase.

Calculate the regulation at this load.

07

OR

Q.4 (a) Discuss the conditions to be satisfied before a 3-phase alternator is

synchronized with infinite bus.

Two 3-phase alternators operate in parallel. The rating of one machine is 50

MW and that of the other is 100 MW. Both alternators are fitted with

governors having a droop of 4 %. How will the machines share a common

load of 100 MW?

07

(b) Discuss power angle characteristic of an alternator. Also discuss its operation

at constant load with variable excitation. 07

Q.5 (a) Describe different parts of a d.c. machine; their material and functions with

the help of a neat diagram. 07

(b) Explain the phenomenon of armature reaction in a d.c. machine. Explain

different methods to neutralize the effect of armature reaction. 07

OR

Q.5 (a) Differentiate between self-excited and separately excited d.c. machines.

Draw the load characteristics of shunt, series and compound generators. 07

(b) Discuss load characteristics (current-torque) of d.c. shunt, series, and

compound motors. Also state their applications. 07

***********



Subject code: 131701 Subject ame: ELECTRICAL MACHI ES Date: 11 /03 /2010 Time: 11.00 am – 01.30 pm

Total Marks: 70




Q.1 (a) A d.c. shunt generator delivers 450 Amp at 230V and the resistance of the shunt

field and armature are 50Ω and 0.03Ω respectively. Calculate the generated emf. 07

(b) Explain following characteristic of separately excited d.c. generator. 07

(i) No-load saturation characteristic

(ii) Internal and External characteristic

Q.2 (a) Explain armature reaction of the d.c. machine. Give its remedies also. 07

(b) Explain in brief various methods of speed control of d.c. shunt motors. 07

OR

(b) Explain working principle of d.c. motor. Derive the condition for maximum Power. 07

Q.3 (a) Explain the conditions of parallel operation of three phase transformers. 07

(b) How three phase to two phase transformation of transformer is obtained? 07

OR

Q.3 (a) In no load test of single phase transformer, the following test data were obtained:

Primary voltage: 220V; Secondary voltage: 110V;

Primary current: 0.5A; Power input: 30W.

Find the following:

07

(i) The turns ratio

(ii) The magnetizing component of no-load current

(iii) It’s working (or loss) component

(iv) The iron loss

Resistance of primary winding = 0.6Ω.

(b) Write advantages and applications of auto transformer. 07

Q.4 (a) Drive torque equation for three phase induction motor. Also derive condition for

maximum torque and equation for maximum torque. 07

(b) Draw torque-speed characteristic of induction motor. Also explain change in it with

change in rotor resistance and frequency. 07

OR

Q.4 (a) Explain rotating magnetic field theory. Also explain how three phase induction

motor starts. 07

(b) Explain speed control of three phase induction motor. 07

Q.5 (a) Compare star-delta starter with auto transformer starter for tree phase induction

motor. 07

(b) Explain why single phase induction motor is not self started? Explain the starting

methods for single phase induction motor in brief. 07

OR

Q.5 (a) What is commutation? Give remedies for commutation. 07

(b) Explain various types of losses occurring in a d.c.generator. 07 *************



Subject code: 131701 Subject Name: Electrical Machines Date: 16 /12 /2010 Time: 10.30 am – 01.00 pm

Instructions: Total Marks: 70




Q.1 (a) Derive the e.m.f. equation of a 1-phase transformer.

The core of a 150 kVA, 11000/440 V, 50 Hz, 1-phase core type transformer has a

cross section of 20*20 cm. The maximum core density is not to exceed 1. 3 wb/m2.

Find (i) the number of h.v. and l.v. turns per phase.

07

(b) Discuss the tests to find out the iron loss and copper loss in a transformer with

justifications. 07

Q.2 (a) Discuss open delta connections of transformers with necessary circuit and vector

diagrams. 07

(b) Define regulation of a transformer. Compare conventional and instrument

transformers. 07

OR

(b) Two 1-phase furnaces working at 100 V are connected to 3300 V, 3-phase supply

through Scott connected transformers. Calculate the current in each line of 3-phase

mains when the power taken by each furnace is 450 kW at a power factor of 0.8

lagging. Neglect the losses in transformers.

07

Q.3 (a) Differentiate between singly excited and multi-excited field systems. Derive the

expression for the mechanical force in a current excited system. 07

(b) Explain the phenomena of crawling and cogging in a 3-phase induction motor with

their remedies. 07

OR

Q.3 (a) Differentiate between cage and slip ring induction motor. Explain how the torque is

developed in a 3-phase induction motor. 07

(b) Draw the circle diagram for a 3-phase, 29.84 kW, 415 V, 50 Hz, delta-connected

induction motor from the following data:

No-load test: 415 V; 21 A; 1250 W

Blocked rotor test: 100 V; 45 A; 2730 W

Estimate from the diagram for full load condition, the slip, efficiency and power

factor. The rotor copper loss at standstill is half of the total copper loss.

07

Q.4 (a) What is slip of a 3-phase induction motor? Discuss its slip-torque characteristics. 07

(b) Why a 1-phase induction motor is not self starting? Explain working of split phase

type 1-phase induction motor 07

OR

Q.4 (a) Define salient pole and non-salient pole machines. Why is armature winding of a

synchronous machine stationary? 07

(b) What is synchronisation of alternators? Which conditions must be satisfied for

proper synchronisation of 3-phase alternators? 07

Q.5 (a) Compare field control and armature voltage control methods of speed control of a

DC motor. 07

(b) Explain the phenomena of armature reaction in a DC machine. State its remedies. 07

OR

Q.5 (a) Draw schematic diagram of a dc machine with labels. State the functions of (i) pole

shoe, (ii) commutator and (iii) yoke. 07

(b) Differentiate between self-excited and separately-excited dc machines. Draw the

load characteristic of dc shunt and series generator. 07

*************

1



Subject code: 131701 Subject Name: Electrical Machines Date: 28-05-2011 Time: 10.30 am – 01.00 pm

Total Marks: 70




Q.1 (a) Explain auto transformer and star-delta starter methods for starting of an

Induction Motor.

07

(b) Describe an auto transformer including its points such as definition,

comparision with two winding transformer, saving of copper and its

applications.

07

Q.2 (a) Explain steps for the construction of the circle diagram of an Induction Motor. 07

(b) Describe about crawling and cogging of an Induction Motor. 07

OR

(b) A 10 H.P.(7.46kw) motor when started at normal voltage with a star-delta

switch in the star position is found to take an initial current of 1.7x full load

current and gave an initial starting torque of 35% of full load torque. Explain

what happens when the motor is started under the following conditions (a) an

auto transformer giving 60% if the normal and calculate in each case the value

of starting current and torque in terms of the corresponding quantities at full

load.

07

Q.3 (a) Explain Scott-connection used for the 3-phase to 3-phase transformation in 3-

phase transformer.

07

(b) A single phase 150KVA transformer has efficiency 96% at full load on 0.8

power factor and on half load at 0.8 power factor lagging. Find the following

(1) Iron loss (2) copper loss at full load (3) The load KVA at which maximum

efficiency occurs (4) The maximum efficiency of the transformer at 0.8 power

factor lagging.

07

OR

Q.3 (a) Give Comparison between Synchronous and Induction Motors. 07

(b) What is voltage regulation? How it can be determined by using Zero power

factor method in Synchronous machine?

07

Q.4 (a) What is commutation? State methods of improving commutation and describe

any one in detail.

07

(b) Explain External and Internal characteristics of D.C. shunt generator in brief. 07 OR

Q.4 (a) Explain different methods for speed control of Series Motors. 07

(b) The torque of the load driven by a 400V shunt motor varies as the cube of the

speed. The current taken by the motor is 40A at a certain speed. Calculate the

additional resistance required to be connected in series with the armature

circuit to reduce the speed to 60% of the original speed. The resistance is 0.35

ohm.

07

2

Q.5 (a) Explain the two tests used for determing the losses in single phase transformer. 07

(b) Explain about elementary concepts of Rotating Machines. 07

OR

Q.5 (a) How the Rotating field is produced in an Induction Motor? 07

(b) Which conditions must be satisfied for parallel operation of Alternators?

Explain Synchronizing of single phase Alternators.

07

*************

1




Subject Name: Electrical Machine





Q.1 (a) What do mean of an ideal transformer and derive emf equation of a

single phase transformer. Also define Transformation Ratio.

07

(b) Obtain the equivalent circuit of a 200/400 –V ,50 Hz,1 Phase

Transformer from the following test data

O.C.Test : 200 V,0.7 A,70 W - on L.V. side

S.C. Test : 15 V, 10 A, 85 W - on H.V. side

Calculate the secondary voltage when delivering 5 KW at 0.8 p.f.

lagging , the primary voltage being 200V

07

Q.2 (a) Explain the various losses taking place in a transformer

& Derive the equation for its maximum efficiency. Also define All

Day Efficiency.

07

(b) Derive the condition for Maximum torque for induction motor and

Explain Torque - Slip characteristics.

07

OR

(b) Write & Explain the condition of parallel operation of 3-phase

transformer

07

Q.3 (a) Explain construction and working principle of d.c machine.

07

(b) A d.c. shunt machine while running as generator develops a voltage

of 250 V at 1000 r.p.m. on no-load. It has armature resistance of 0.5

Ω and field resistance of 250 Ω. When the machine runs as motor,

input to it at no-load is 4 A at 250 V. Calculate the speed and

efficiency of the machine when it runs as a motor taking 40 A at 250

V. Armature reaction weakens the field by 4 %.

07

OR

Q.3 (a) Explain the Swinburne’s Test of a d.c. machine for finding losses

with necessary diagram

07

(b) Draw and explain the internal & external characteristics of d.c. shunt

generators.

05

(c) Explain the term ‘Back emf’ in respect to d.c.motor.

02

Q.4 (a) Write different starters used for 3 phase induction motor and explain

any one of them.

07

2

(b) An 18.65 KW, 4 pole , 50 Hz, 3-phase induction motor has friction

and windage losses of 2.5 percent of the output. The full load slip is 4

%. Compute for full load

(1) The rotor cu loss (2) The rotor input (3) The shaft torque (4) The

gross electromagnetic torque

07

OR

Q.4 (a) Explain different speed control methods for 3 phase induction motor. 07

(b) Explain the procedure to construct the circle diagram for induction

motor & how various quantities are measured from circle diagram.

07

Q.5 (a) Define voltage regulation of an alternator & explain any one method

to find the voltage regulation

07

(b) Explain the Various types of cooling method in rotating machine. 07

OR

Q.5 (a) What is Synchronizing of an alternator? Explain any one method

for Synchronizing.

07

(b) What do you mean of hunting in Synchronous Machine 03

(c) Derive equation of emf for an alternator 04

*************

1



Subject code: 131701 Subject Name: Electrical Machines Date: 10/05/2012 Time: 02.30 pm – 05.00 pm

Total Marks: 70 Instructions:


Q.1 (a) Explain working principle of transformer in detail and also derive E.M.F.

equation of transformer. 07

(b) A 50 kVA, 4400/220 V transformer has R1 = 3.45 Ω, R2 = 0.009 Ω. The values of reactances are X1 = 5.2 Ω and X2 = 0.015 Ω. Calculate for the transformer (i) equivalent resistance as referred to primary (ii) equivalent resistance as referred to secondary (iii) equivalent reactance as referred to both primary and secondary (iv) equivalent impedance as referred to both primary and secondary (v) total Cu loss, first using individual resistances of the two windings and secondly, using equivalent resistances as referred to each side. Assume efficiency of the transformer equal to 100%.

07

Q.2 (a) A 120 kVA, 6000/400 V, Y/Y, 3-phase, 50 Hz transformer has an iron loss

of 1600 W. The maximum efficiency occurs at 3/4 full load. Find the efficiencies of the transformer at (i) full-load and 0.8 power factor (ii) half-load and unity power factor (iii) the maximum efficiency.

07

(b) For a singly excited system derive the expression for magnetic field energy stored.

07

OR (b) Explain the various losses taking place in a d.c. machine. 07

Q.3 (a) Explain how rotating magnetic field is produced in 3-phase induction motor.

07

(b) Explain crawling and cogging of an induction motor. 07 OR

Q.3 (a) Explain the general construction and working principle of a single-phase induction motor.

07

(b) Explain the double revolving field theory for a single-phase induction motor.

07

Q.4 (a) Explain the working principle of synchronous machine and derive the

relation between electrical and mechanical angle. 07

(b) Define and state the expressions for (i) Pitch factor (ii) Distribution factor. 07 OR

Q.4 (a) A synchronous generator is connected to an infinite bus. Discuss with the help of phasor diagrams the effect of changing excitation at constant mechanical input.

07

(b) Why is it necessary to run alternators in parallel? Explain clearly the terms synchronizing current, synchronizing power and synchronizing torque of

07

2

synchronous machine.

Q.5 (a) State the different types of d.c. generators and state the applications of each type.

07

(b) With neat diagrams explain the phenomenon of armature reaction in a d.c. machine.

07

OR Q.5 (a) Derive the expression for the electromagnetic torque developed in a d.c.

motor. 07

(b) Explain Swinburne’s test for finding the efficiency of a d.c. machine. 07

*************

Seat No.: ________ Enrolment No.___________ GUJARAT TECHNOLOGICAL UNIVERSITY

B. E. - SEMESTER – III • EXAMINATION – WINTER 2012

Subject code: 131701 Date: 05-01-2013 Subject Name: Electrical Machine Time: 10.30 am – 01.00 pm Total Marks: 70 Instructions:


Q.1 (a) What is commutation? Give remedies for commutation. 07

(b) Write advantages and applications of auto transformer. 07

Q.2 (a) Explain the phenomena of armature reaction of a DC machine. State its remedies.

07

(b) What is slip of a 3-phase induction motor? Discuss its slip- torque characteristics. OR

07

(b) Write and explain the conditions of parallel operation of 3-phase transformer. 07

Q.3 (a) Draw and explain the equivalent circuit of single phase transformer. 07 (b) A 25 KVA transformer has 500 turns on the primary and 50 turns on the

secondary winding. The primary is connected to 3000 V, 50 Hz supply. Find the full load primary and secondary currents, the secondary e.m.f. and the maximum flux in the core. Neglect leakage drops and no load primary current.

07

OR Q.3 (a) Explain how the torque is developed in a 3-phase induction motor. Derive the

equation of torque under running condition. 07

(b) A 30 KVA, 2400/120 V, 50 Hz transformer has a high voltage winding resistance of 0.1 ohm and a leakage reactance of 0.22 ohm. The low voltage winding resistance is 0.035 ohm and leakage reactance is 0.012 ohm . find the equivalent winding resistance, reactance and impedance referred to the (i) high voltage side (ii) low voltage side.

07

Q.4 (a) How three phase to two phase transformation of transformer is obtained? 07

(b) Explain the Swinburne’s test of a d.c. machine for finding losses with necessary diagram.

07

OR Q.4 (a) Explain different methods for speed control of series motors. 07

(b) Write different starters used for 3 phase induction motor and explain any one of them.

07

Q.5 (a) What is synchronizing of an alternator? Explain any one method for

Synchronizing. 07

(b) Differentiate between singly excited and multi-excited field systems. Derive the expression for the mechanical force in a current excited system.

07

OR Q.5 (a) Define and state the expressions for (i) Pitch factor. (ii) Distribution factor for

alternator. 07

(b) What is voltage regulation? How it can be determined by using Zero power factor method in synchronous machine?

07

*************

1/2



Subject Code: 131701 Date: 29-05-2013 Subject Name: Electrical Machines Time: 02.30 pm - 05.00 pm Total Marks: 70 Instructions:


Q.1 (a) Differentiate between self-excited and separately excited d.c. machines.

Draw the load characteristics of shunt, series and compound generators. 07

(b) Explain three point starter for D.C.Shunt motor. 07

Q.2 (a) Derive the E.M.F. equation of single phase transformer and explain effect of turns ratio on output voltage

07

(b) Explain scott connection for transformer with diagram. 07 OR (b) An ideal 25 KVA transformer has 500 turns on the primary winding and

40 turns on the secondary winding. The primary is connected to 3000 V,50 Hz supply. Calculate (1) primary and secondary currents on full load (2) secondary e.m.f. (3) maximum core flux

07

Q.3 (a) Differentiate between single excited and multiple excited systems. Derive

the expression for magnetic field energy stored in a singly excited system. 07

(b) Define energy and co-energy. Derive the expression for force developed for current excited and voltage controlled systems

07

OR Q.3 (a) Discuss power angle characteristic of an alternator. Also discuss its

operation at constant load with variable excitation. 07

(b) Discuss the conditions to be satisfied before a 3-phase alternator is synchronized with infinite bus. Two 3-phase alternators operate in parallel. The rating of one machine is 50 MW and that of the other is 100 MW. Both alternators are fitted with governors having a droop of 4 %. How will the machines share a common load of 100 MW?

07

Q.4 (a) State the type of three phase induction motor. Explain how rotor

rotates when three phase induction motor is connected across three phase supply & Define Slip.

07

(b) A 3-phase ,50 Hz,500V Induction motor with 6 poles gives an output of 20 Kw at 950 rpm with a power factor of 0.8 The mechanical losses are equal to 1 Kw. Calculate for this load(i)slip (ii)rotor copper loss (iii)input if the stator losses are 1500 W(iv)line current

07

OR Q.4 (a) With reference to induction motor, attempt the following

(1) Explain “cogging” and “crawling” in a 3-phase induction motor with their remedies. (2) Why single-phase induction motor is not self-starting? Explain any one method to make it self-starting.

07

(b)

A 4-pole, lap wound D.C shunt generator has a useful flux per pole of 0.07 wb. The armature winding consists of 220 turns each of 0.004 ohm resistance.Calculate the terminal voltage when running at 900 r.p.m if the armature current is 50 amp.

07

2/2

Q.5 (a) Explain the difference between cylindrical and salient pole rotors used

in large alternator . Define (1) pitch factor (2) Distribution factor (3) form factor.

07

(b) Define Voltage regulation of alternator. State various methods to find voltage regulation and Explain any one method in detail.

07

OR Q.5 (a) Discuss power angle characteristic of an alternator. Also discuss its

operation at constant load with variable excitation. 07

(b) Explain synchronization of alternators. Which conditions must be satisfied for proper synchronization of 3-phase alternators?

07

*************

gujarat technological university · gujarat technological university b.e. sem-iii (all branches)...

Documents