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ANALYTIC FUNCTIONS
Necessary and sufficient condition: Cauchy-Riemann
equations: A necessary and sufficient condition that a
function w=f(z)=u(x,y)+iv(x,y), where
z=x+iy, be analytic in a region R is that the Cauchy-Riemann
equations
∂u ∂v ∂u ∂v= & = −
∂x ∂y ∂y ∂x
are satisfied in R.
If a function, w=f(z)=u(x,y)+iv(x,y) where z=x+iy, is analytic
in a region R, then
(i) (∂u/∂x)=(∂v/∂y) & (∂u/∂y)= −(∂v/∂x).
(ii) (dw/dz)=f1(z)=(∂u/∂x) − i(∂u/∂y) or (dw/dz)=f
1(z)=(∂v/∂y) + i(∂v/∂x) or
(dw/dz)=f1(z)=(∂u/∂x) + i(∂v/∂x) or (dw/dz)=f 1(z)=(∂v/∂y)
− i(∂u/∂y).
(iii) (∂w/∂x)=(∂u/∂x) + i(∂v/∂x)⇒ i(∂w/∂x)=i(∂u/∂x)
− (∂v/∂x)⇒ i(∂w/∂x)=i(∂v/∂y) +
(∂u/∂y)⇒ i(∂w/∂x)=(∂w/∂y).
(iv) u(x,y) and v(x,y) are harmonic.
(v)
u(x,y)=α and v(x,y)=β where α and β are any
constants, are orthogonalfamilies of curves.
Harmonic functions: A function f(x,y) is harmonic if it
satisfies the Laplace’s equation i.e.,
∂2f ∂2f∇2f = + = 0
∂x2 ∂y2
Orthogonal functions:
The families of curves u(x,y)=α & v(x,y)=β are
orthogonal if
∂u ∂v∂x ∂x
∂u ∂v∂y ∂y
= −1
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Complex integration: Cauchy’s integration formula: If
f(z) is analytic inside and on a closed curve C and
a is any point inside C, then
f(z)
(z−a)n
f n-
(a)
dz = 2πi(n−1)!
∫ C
where f n-1
(a) is the value of the (n-1)th derivative of f(z) at z=a and
the integral is taken inthe counterclockwise sense around C.
Important notes: (i) |z−a|=r means a circle of radius r with
center at z=a. (ii) If z=a is notinside C, then
f(z)
(z−a)n dz = 0∫ C Residue theorem: If
f(z) is analytic on and inside a closed curve C and is of the
form
m j
q(z)
f(z)=
k
∏[z-p j]
j=1
then
res[f(z)]
z=p j
k
∫ C f(z)dz=2π iΣ j=1
where
m j-1
m j-1
m j 1 d
res[f(z)]= lim (z− p j) f(z)z=p j
(m j−1)! z→ p j dz
Jacobian of a transformation: If we define
w=f(z)=u(x,y)+iv(x,y)
then the determinant
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∂(u,v) (∂u/∂x) (∂u/∂y)=
∂(x,y) (∂v/∂x) (∂v/∂y)
is called the Jacobian of the transformation f(z).
Bilinear transformation: The bilinear transformation that
maps the points z1, z2 and z3
on the z-plane into the points w1, w2 and w3 on the
w-plane is
[(w-w1)(w2-w3)/(w-w3)(w2-w1)]=[(z-z1)(z2-z3)/(z-z3)(z2-z1)]
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ORDINARY DIFFERENTIAL EQUATIONS
Type I: Differential equations with variables
separable: For the differential equation of the form
(dy/dx)=f(x)g(y)
the solution would be obtained by (i) separating the variables
and (ii) integrating i.e.,
∫ [dy/g(y)]=∫ f(x)dx+C.
Type II: Differential equations reducible to Type I: For
the differential equation of the form
(dy/dx)=f(ax+by+c)
the solution would be obtained by (i) making the
substitution
z=ax+by+c
and hence (dz/dx)=a+b(dy/dx)
(ii) separating the variables and (iii) integrating i.e.,
∫ {dz/[bf(z)+a]}=x+C.
Type III: Homogeneous differential equations: For the
differential equation of the form
(dy/dx)=f(y/x)
which is called the homogeneous differential equation, the
solution would be obtained by(i) making the substitution
y=vx
and hence (dy/dx)=v+x(dv/dx)
(ii) separating the variables and (iii) integrating i.e.,
∫ {dv/[f(v)−v]}=lnx+C.
Type IV: Differential equations reducible to Type III: For
the differential equation of the form
(dy/dx)=(a1x+b1y+c1)/(a2x+b2y+c2)
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the solution would be obtained by (i) making the
substitutions
x=X+h & y=Y+k
(ii) choosing the constants h & k such that
a1h+b1k+c1=0 & a2h+b2k+c2=0
(iii) thereby reducing the given differential equation to
(dY/dX)=( a1X+b1Y)/(a2X+b2Y)
which is homogeneous, (iv) making the substitution
Y=VX
and hence (dY/dX)=V+X(dV/dX)
(v) separating the variables and (vi) integrating i.e.,
∫ {(a2+b2V)/[(a1+b1V)−V(a2+b2V)]}dV=lnX+C.
Type V: Linear differential equations of first order: For
the differential equation of the form
(dy/dx)+Py=Q
which is a linear differential equation of first order, the
solution would be obtained by (i)identifying P & Q in the given
differential equation (ii) finding the integrating factor (IF)
IF=e∫ Pdx
(iii) multiplying the given differential equation by IF and (iv)
integrating i.e.,
ye∫ Pdx
=∫ Qe∫ Pdx+C.
Type VI: Bernoulli’s differential equations: For the
differential equation of the form
(dy/dx)+Py=Qyn
which is called the Bernoulli’s equation, the solution would be
obtained by (i) dividingthe given differential equation by y
n (ii) making the substitution
z=y1-n
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and hence (dz/dx)=(1-n)y1-n
(dy/dx)
(iii) reducing the given differential equation to
(dz/dx)+Pz(1-n)=Q(1-n)
and (iv) following Type V.
Type VII: Exact differential equations: For the
differential equation of the form
Mdx+Ndy=0
which is the exact differential equation as it satisfies the
condition that (∂M/∂y)=(∂ N/∂x),the solution would be obtained
by (i) integrating M wrt x as if y were constant (ii)integrating
those terms in N which do not occur in
{∂[∫ Mdx]/∂y}
wrt y and (iii) equating their sum to a constant i.e.,
∫ Mdx+∫ { N−[∂(∫ Mdx)/∂y]}dy=C.
Type VIII: Linear differential equations with constants
coefficients:
For the differential equation of the form
a0(dny/dx
n)+a1(d
n-1y/dx
n-1)+…+any=Q
the solution would be obtained by (i) replacing
(dk y/dx
k ) by m
k for k=0,1,2,…,n to obtain
a characteristic equation (or as sometimes called, an auxiliary
equation) of the form
a0mn+a1m
n-1+…+an=0
(ii) finding the n roots, m1, m2, …, mn of the above
characteristic equation (iii) finding ageneral solution (or as
sometimes called a complimentary function) of the form
(iiia) CE=C1exp(m1x)+C2exp(m2x)+…+Cnexp(mnx) if the roots are
real anddistinct
OR(iiib) CE=[C1+C2x+C3x
2+…+Ck x
k-1]exp(m1x)+Ck+1exp(mk+1x)+…+Cnexp(mnx) if
the first k roots are equal to m1 and the other (n-k) roots
are real and distinctOR
(iiic) CE=[C1cosβx+C2sinβx]eαx+C3exp(m3x)+…+Cnexp(mnx) if the
first tworoots are complex conjugate pairs α±βi and the other (n-2)
roots are real anddistinct.
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(iii) finding the particular solution as follows:(iiia) If
Q=eax, then the particular solution would be of the form
PI=[1/f(D)]eax=[eax/f(a)] if f(a)≠0OR
PI=[1/f(D)]eax=[xk eax/f k (a)] if
f(a)=f 1(a)=…=f k-1(a)=0.
(iiib) If Q=sin(ax) or cos(ax), then the particular solution
would be of the form
PI=[1/f(D2)]Q=[1/f(−a2)]Q if f(−a2)≠0
OR
PI=[1/f(D2)]Q=[x
k /f
k (−a2)]Q if f(−a2)=f 1(−a2)=…=f k-1(−a2)=0.
(iiic) If Q=xm, then the particular solution would be of the
form
PI=[1/f(D)]Q=f –1
(D)Q {Hint:(1+x)−1
=1−x+x2−x3+… & (1−x)−1=1+x+x2+x3+...}
(iiid) If Q=eax
R, where R is any function of x, then the particular solution
would be of the
form
PI=[1/f(D)]Q=eax
[1/f(D+a)]R.
(iiie) If Q=Q1+Q2+…+Qn, then the particular solution would be of
the form
PI=(PI for Q1)+(PI for Q2)+…+(PI for Qn).
and (iv) finding the total solution by adding them i.e.,
y=CE+PI.
Type IX: Differential equations expressed as a polynomial in
(dy/dx):
(a) Differential equations solvable for p=(dy/dx): For the
differential equation ofthe form
f(p)=pn+Q1 p
n-1+Q2 p
n-2+…+Qn=0
where p=(dy/dx) and Q1, Q2,…,Qn are functions of x & y,
the solution would be obtained by (i) factorizing the
polynomial f(p) to obtain
f(p)=(p-R 1)(p-R 2)…(p-R n)=0
where R 1, R 2,…, R n are functions of x
& y, (ii) finding the primitive solutions
f 1(x,y,c1),f 2(x,y,c2),…, f n(x,y,cn) by equating
each factor to zero and (iii) finding the product of the
primitive solutions to obtain a solution of the form
f 1(x,y,c1)f 2(x,y,c2)…f n(x,y,cn)=0 or
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f 1(x,y,c)f 2(x,y,c)…f n(x,y,c)=0
by replacing c1, c2,…, cn by c without any loss of
generality.
(b) Differential equations solvable for y: For the
differential equation of the form
y=f(x,p)
where p=(dy/dx), the solution would be obtained by (i)
differentiating the given equationwrt x (ii) separating the
variables (iii) integrating to obtain a value for p and (iv)
substituting the value of p in the given differential
equation.
(c) Differential equations solvable for x: For the
differential equation of the form
x=f(y,p)
where p=(dy/dx), the solution would be obtained by (i)
differentiating the given equationwrt y (ii) separating the
variables (iii) integrating to obtain a value for p and
(iv)substituting the value of p in the given differential
equation.
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Definite integrals: b a
1. ∫ f(x) dx = −∫ f(x) dxa ba a
2. ∫ f(x) dx = ∫ f(a-x) dx0 0
2a a a
3.
∫ f(x) dx =
∫ f(x) dx +
∫ f(2a-x) dx
0 0 0
2a a
4. ∫ f(x) dx = 2∫ f(x) dx [if f(2a-x) =
f(x)]0 0
2a
4A.
∫ f(x) dx = 0 [if f(2a-x) = -f(x)]
0
na a
5. ∫ f(x) dx = n∫ f(x) dx [if f(a+x) =
f(x)]0 0
b a
1. ∫ f(x) dx = −∫ f(x) dxa b
a a
6. ∫ f(x) dx = 2∫ f(x) dx [if f(x) = f(-x)
i.e., f(x) is even]-a 0
a a
6. ∫ f(x) dx = 2∫ f(x) dx [if f(x) = f(-x)
i.e., f(x) is even]-a 0
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a
6A.
∫ f(x) dx = 0 [if f(x) = - f(-x) i.e., f(x) is
odd]
-a
b a
7. ∫ f(x) dx = ∫ f(a+b-x) dxa 0
Bernoulli’s formula:
∫ uv = uv-u1
v1+u
2
v2-u
3
v3+...+(-1)
k
u
k
vk +...
where uk =k
th derivative of u & vk =k
th integral of v.
Increasing and decreasing functions:
First derivative test:
(i) Find f l(x).
(ii) Find the critical points of f(x) by setting
f l(x)=0.
(iii) Arrange the critical points in the ascending
order.
(iv)
Partition the domain of f(x) into open intervals using the
critical points.(v) Find the sign of f l(x) in each open
interval. (a) If the sign of f l(x) is +ve in anopen interval,
then the function f(x) is increasing in that interval. (b) If
the
sign of f l(x) is –ve in an open interval, then the
function f(x) is decreasing in
that interval.
Local extrema: (relative extrema):
First derivative test:
(i) Find f l(x).
(ii)
Find the critical points of f(x) by setting f
l
(x)=0.(iii)
Find the sign of f l(x) in the immediate, preceding and
succeeding
neighbourhoods of each of the critical points. (a) If the sign
of f l(x) changes
from +ve to –ve around a particular critical point, then that
point is the pointof local or relative maxima. (b) If the sign of
f
l(x) changes from –ve to +ve
around a particular critical point, then that point is the point
of local or relativeminima.
Second derivative test:
(i) Find f l(x).
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(ii) Find the critical points of f(x) by setting
f l(x)=0.
(iii) Find the sign of f ll(x) at each of the
critical points. (a) If the sign of f ll(x) is –ve
at a particular critical point, then that point is the point of
local or relativemaxima. (b) If the sign of f ll(x) is +ve at
a particular critical point, then that
point is the point of local or relative minima.
Rolle’s theorem and The mean value theorem:
(i) Let f be continuous on the closed interval [a, b] and
differentiable on the openinterval (a, b). If f(a)=f(b), then there
is at least one point c in (a, b) such that
f l(c)=0.
(ii) If f is continuous on the closed interval [a, b] and
differentiable on the openinterval (a, b), then there exists at
least a point c in (a, b) such that
f l(c)=[f(b) – f(a)]/(b-a)
Tangents & normals to a curve:
Tangent: Tangent to a curve y=f(x) at a point (x1, y1) on
the curve is given by
(y−y1)=m(x−x1)
where m=(dy/dx)|x1, y1
Normal: Normal to a curve y=f(x) at a point (x1, y1) on the
curve is given by
(y−y1)=(−1/m)(x−x1)
where m=(dy/dx)|x1, y1
Angle between two curves:
(i) Find the points of intersection of the curves given by
substituting one into theother.
(ii)
Find the slope of tangent to each curve at any point of
intersection. Let them be m1 & m2.
(iii) The angle between the curves is given by
θ=tan-1(m1-m2)/(1+m1m2)
Taylor series:
f(a+h)=f(a)+[h/1!]f l(a)+[h2/2!]f ll(a)+...to
∞
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Maclaurin series:
f(x)=f(0)+[x/1!]f l(0)+[x2/2!]f ll(0)+... to
∞
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MATRICES & DETERMINANTS
1. Row matrix: A matrix with only one row is called a
row matrix. A=[aij]1×n.
2. Column matrix: A matrix with only one column is
called a column matrix.
A=[aij]m×1.
3. Null matrix: A matrix with only zero elements is
called a null matrix.
4. Transpose of a matrix: Transpose of a matrix
A is the matrix A itself with rows
and columns interchanged. If A=[aij]m×n, then
AT=[a ji]n×m. Properties: (i)
(A+B)T=A
T+B
T (ii) (AB)
T=B
TA
T
5. Symmetric matrix: A matrix A is said to be
symmetric if AT=A.
6. Skew-symmetric matrix: A matrix A is said to
be skew-symmetric if AT=−A.
7. Square matrix: A matrix with equal number of rows
and columns is called a square
matrix. A=[aij]n×n.
8. Singular matrix: An n×n square matrix A is called
a singular matrix if the rank ofA i.e., ρ(A)
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17. Unitary matrix: A complex square matrix A is
called a unitary matrix ifA(A*)T=(A*)TA=I.
18.
Adjoint of a square matrix: Adjoint of a square matrix
A is the transpose of A
with each element replaced by its cofactor. If A=[aij]n×n, then
adjA=[Aij]n×nT, where
Aij is the cofactor of the element
aij. Properties:
(i) A(adjA)=(adjA)A= detA 0 -------------0
=detAIn×n 0 detA -------------0
0 detA
(ii)
det(adjA)=(detA)n-1, where n is the order of A
(iii) adj(AB)=adjB adjA (iv)
adj(adjA)=(detA)n-2A
(v) det(adj(adjA))=[(detA)n-1
]n-1
19. Inverse of a square matrix: Inverse of a square
matrix A is defined asA
-1=[adjA]/detA. Properties: (i) AA
-1=A
-1A=I (ii) (AB)
-1=B
-1A
-1
20. Equivalent matrices: Two matrices A and
B are said to be equivalent (written asA~B) if one can be
obtained from the other through a number of elementary row orcolumn
transformations.
Rank of a matrix: Rank of a matrix A is the order of
highest-ordered non-zero minor of
A and denoted as ρ(A).
Eigen values (or characteristic roots) of a square
matrix: Eigen values of an n×nsquare matrix A are the
roots of the characteristic equation det(A−λI)=0.
Properties:
i. An n×n square matrix has n non-zero Eigen
values.ii. The sum of Eigen values of a square matrix is
equal to the sum of the principal
diagonal elements.
iii.
The product of Eigen values of a square matrix is equal to its
determinant value.iv.
Eigen values of AT are the Eigen values of A.
v. Only a singular matrix has zero as an Eigen
value.vi. The Eigen values of a triangular matrix are the
diagonal elements themselves.
vii. If λ1, λ2,…, λn are the Eigen values of a matrix
A, then those of A2 are λ12,λ22,…, λn2.
viii. If λ1, λ2,…, λn are the Eigen values of a
matrix A, then those of k A, where k is ascalar, are
k λ1, k λ2,…, k λn.
ix. If λ is an Eigen value of a non-singular matrix
A, then (a) that of A-1 is λ-1 & (b)that of
adjA is (λdetA).
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x. Cayley-Hamilton theorem: A square matrix
satisfies its own characteristicequation.
Eigen vectors of a square matrix: The Eigen vectors of an
n×n square matrix A are theset of values satisfying the vector
equation (A−λ)x=0.
Properties:
i. For each Eigen value λi, i=1,2,…,n, of an n×n square
matrix A, there correspondsan Eigen vector xi, i=1,2,…,n, that
satisfies the vector equation (A−λi)xi=0.
ii. If xi is an Eigen vector of A corresponding to
the Eigen value λi, then k xi, wherek is any constant, is also
an Eigen vector of A corresponding to the same Eigen
value λi.
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NUMERICAL METHODS
(I) Curve fitting:
(i) Fitting a straight line: To fit a straight line of the
form
y=ax+b
evaluate the constants a & b using the given data on x &
y and the normal equations
Σy=aΣx+nbΣxy=aΣx2+bΣx.
(ii) Fitting a parabola: To fit a parabola of the form
y=ax2+bx+c
evaluate the constants a, b & c using the given data on x
& y and the normal equations
Σy=aΣx2+bΣx+ncΣxy=aΣx3+bΣx2+cΣxΣx2y=aΣx4+bΣx3+cΣx2.
(iii) Fitting an exponential curve: (a) To fit an
exponential curve of the form
y=ae bx
(1) take logarithm on both sides of the given equation to
obtain
logy=logae bx
⇒logy=loga+bxloge⇒Y=A+Bx
where Y=logy, A=loga & B=bloge and (2) evaluate the
constants A & B by method (I).
(b) To fit an exponential curve of the form
y=ax b
(1) take logarithm on both sides of the given equation to
obtain
logy=logax b
⇒logy=loga+blogx⇒Y=A+bX
where Y=logy, A=loga & X=logx and (2) evaluate the constants
A & b by method (I).
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(iv)Curve fitting-Method of moments: Given that
x
y
to fit a curve of the form y=f(x), equate the nth moment of the
observed y to the nthmoment of the expected y i.e.,
n xn+½∆x
∆xΣxn-1y=∫ xn-1f(x)dxxn-½∆x
(II) Theory of Equations: Relations between the roots
and the coefficients of the equation f(x)=0: Let α1,
α2,..., αn be the roots of the equation
f(x)=a0xn+a1x
n-1+...+an=0
Then
n
Σαi= −(a1/a0)i=1
n
Σαiα j= (a2/a0)i=1,j=1
i≠ jn
Σαiα jαk = −(a3/a0)i=1,j=1,k=1
i≠ j≠kand so on, where
n
Σαi= sum of the rootsi=1
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n
Σαiα j= sum of the roots taken 2 at a timei=1,j=1
i≠ jn
Σαiα jαk = sum of the roots taken 3 at a
timei=1,j=1,k=1
i≠ j≠k
and so on. Generally
Σα1α2...αk = (−1)k (ak /a0), k=1,2,3,...,n
where
Σα1α2...αk =sum of the roots taken k at a timeTo form the
equation whose roots are the roots of the given equation f(x)=0,
each
multiplied by the same constant: Let α1, α2,..., αn be
the roots of the equation
f(x)=a0xn
+a1xn-1
+...+an=0
Then, the equation whose roots are mα1, mα2,..., mαn is
found to be
f 1(x)=a0xn+ma1x
n-1+...+m
nan=0
To form the equation whose roots are the roots of the given
equation f(x)=0, each
diminished by the same constant: Let α1, α2,..., αn be
the roots of the equation
f(x)=a0xn+a1x
n-1+...+an=0
Then, the equation whose roots are α1±h, α2±h,..., αn±h is found
to be
f 1(x)=A0xn+A1x
n-1+...+An=0
where An, An-1,..., A0 are the successive remainders when
f(x) is divided by x±h.
(III) Numerical solution of equations: Iteration:(i) Fixed-point
iteration method: (ii) Newton or Newton-Raphson iteration
method: Given a function
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f(x)=0
which is continuous and has a continuous derivative, and a
starting value x0, the solutionis found by (i) computing a
sequence, x0, x1, x2,…, recursively from the relation of the
form
xn+1=xn−[f(xn)/f 1(xn)]
if f 1(xn)≠0 and (ii) terminating the process on any of the
following termination criteria:i. after N steps (N given
& fixed) or
ii. if |xn+1−xn| ≤ ε (ε > 0 given)
oriii. if |f(xn)| ≤ α (α > 0
given).
The last value in the sequence is the solution.
(IV) Interpolation: Prediction of value of a function
from a given set of its values:
Interpolation for a set of equally spaced values of independent
variable:
(i) Newton’s forward interpolation formula: Given the
data
x x0 x1 x2 … xny=f(x) y0 y1 y2 … yn
(1) evaluating the forward difference table
x y ∆y ∆2y … ∆n-1yx0x1 x2 ...
xn-2xn-1 xn
y0 y1 y2 ...
yn-2yn-1 yn
∆y0 ∆y1
.
.
.
.
∆yn-2 ∆yn-1
∆2y0 .....
∆2yn-2
… ∆n-1y0
where xi=x0+ih, i=0,1,2,…,n, the approximate value of y=f(x) for
any value of x can beestimated using the Newton’s forward
interpolation formula
y=y0+[(x-x0)/1!h]∆y0+[(x-x0)(x-x1)/2!h2]∆2y0+…+[(x-x0)(x-x1)…(x-xn-1)/n!hn]∆ny0 y=y0+[p/1!]∆y0+[p(p-1)/2!]∆2y0+[p(p-1)(p-2)/3!]∆3y0+…+[p(p-1)…(p-n+1)/n!]∆ny0
where p=[(x-x0)/h] & h=|xn+1-xn| i.e., the increment of
x.
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Error in polynomial interpolation: The error
in the polynomial interpolation isgiven by
y(x)−ϕ(x)=[(x-x0)(x-x1)...(x-xn)/(n+1)!]∆n+1y(c)
Error in Newton’s forward interpolation formula: The
error ε in the above Newton’s interpolation formula is
given by
ε=[p(p-1)…(p-n)/(n+1)!]h n+1∆n+1y0.
(ii) Newton’s backward interpolation formula: Given the
data
x x0 x1 x2 … xn
y=f(x) y0 y1 y2 … yn
(1) evaluating the forward difference table
x y ∆y ∆2y … ∆n-1yx0x1 x2 ...
xn-2xn-1 xn
y0 y1 y2 ...
yn-2yn-1 yn
∆y0 ∆y1
.
.
.
.∆yn-2 ∆yn-1
∆2y0 .....
∆2yn-2
… ∆n-1y0
where xi=x0+ih, i=0,1,2,…,n, the approximate value of y=f(x) for
any value of x can beestimated using the Newton’s forward
interpolation formula
y=yn+[(x-xn)/1!h]∆yn+[(x-xn)(x-xn-1)/2!h2]∆2yn+…+[(x-xn)(x-xn-1)…(x-x1)/n!hn]∆nyn y=yn+[p/1!]∆yn+[p(p+1)/2!]∆2yn+[p(p+1)(p+2)/3!]∆3yn+…+[p(p+1)…(p+n-1)/n!]∆nyn
where p=[(x-xn)/h] & h=|xn-xn-1| i.e., the increment of
x.
Error in Newton’s backward interpolation formula: The
error ε in the above Newton’s interpolation formula is
given by
ε=[p(p+1)…(p+n)/(n+1)!]hn+1∆n+1yn.
Interpolation for a set of unequally spaced values of
independent variable:
Interpolation-for a set of either equally or unequally spaced
values of
independent variable: (iii) Lagrange’s interpolation
formula: Given the data
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x x0 x1 x2 … xny=f(x) y0 y1 y2 … yn
evaluate
n n n
y=Σ[ Π(x-x j) / Π(xi-x j)]yi i=0 j=0,j≠i
j=0,j≠in
y=Σ[ϕ(x)f(xi)/(x-xi)ϕ/(xi)] i=0
where
n
ϕ(x)=Π(x-x j) j=0
ϕ/(xi)=(d/dx)ϕ(x) at x=xi
(V) Numerical integration:(i) Trapezoidal rule of
integration: By the Trapezoidal rule of integration
b
∫ f(x)dx≈(1/2)h[y0+yn+2(y1+y2+…+yn-1)]a
Procedure: i. divide the interval (a,b) into n
equal subintervals,
(x0,x1),(x1,x2),(x2,x3),…,(xn-1,xn), each of length
h=[(b−a)/n]ii. putting a=x0 and b=xn, compute
y0=f(x0),y1=f(x1),…,yn=f(xn) and
iii. compute (1/2)h[y0+yn+2(y1+y2+…+yn-1)].
(ii) Rectangle formulas: By the rectangle formulas
b
∫ f(x)dx≈h[y0+y1+y2+…+yn-1] ora
b
∫ f(x)dx≈h[y0+y1+y2+…+yn] ora
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b
∫ f(x)dx≈h[y1/2+y3/2+…+y(2n-1)/2]a
Procedure: i. divide the interval (a,b) into n
equal subintervals,
(x0,x1),(x1,x2),(x2,x3),…,(xn-1,xn), each of length
h=[(b−a)/n]ii. putting a=x0 and b=xn, compute
y1/2=f(x1/2),y3/2=f(x3/2),…,y(2n-1)/2=f(x(2n-1)/2),
where x1/2, x3/2,…, x(2n-1)/2 are the midpoints of the
subintervals(x0,x1),(x1,x2),(x2,x3),…,(xn-1,xn) respectively
and
iii. compute h[y1/2+y3/2+…+y(2n-1)/2].
(iii) Simpson’s rule of integration: By the Simpson’s rule
of integration
b
∫ f(x)dx≈(1/3)h[y0+yn+2(y1+y2+…+yn-1)+4(y1/2+y3/2+…+y(2n-1)/2)]a
Procedure:
i. divide the interval (a,b) into n equal
subintervals,
(x0,x1),(x1,x2),(x2,x3),…,(xn-1,xn), each of length
h=[(b−a)/n]ii. putting a=x0 and b=xn, compute
y0=f(x0),y1=f(x1),…,yn=f(xn)
iii. compute
y1/2=f(x1/2),y3/2=f(x3/2),…,y(2n-1)/2=f(x(2n-1)/2), where x1/2,
x3/2,…, x(2n-1)/2
are the midpoints of the subintervals
(x0,x1),(x1,x2),(x2,x3),…,(xn-1,xn)respectively andiv.
compute (1/6)h[y0+yn+2(y1+y2+…+yn-1)+4(y1/2+y3/2+…+y(2n-1)/2)].
Limiting errors in the numerical integration: (i) Limiting
error in the Trapezoidal rule: The limiting error in the
Trapezoidal rule
is given by
LE=[(b−a)3/12n2]M2=[nh3/12]M2
where M2 is the greatest value of |f ii(x)| in the
interval (a,b).
(ii) Limiting error in the rectangular rule of
integration: The limiting error in the
rectangular rule is given by
LE=[(b−a)3/24n2]M2=[nh3/24]M2
where M2 is the greatest value of |f ii(x)| in the
interval (a,b).
(iii) Limiting error in the Simpson’s rule of
integration: The limiting error in theSimpson’s rule is given
by
LE=[(b−a)5/180(2n)4]M4=[nh5/(180×24)]M4
where M4 is the greatest value of |f iv
(x)| in the interval (a,b).
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PROBABILITY
Standard distributions: (I) Binomial distribution: (i)
The binomial distribution deals with the Bernoulli’s
experiment. (ii) The Bernoulli’s experiment is any experiment,
which has only two
possible sets of outcomes: favourable and non-favourable.
(iii) The experiment may betried several times. However, the number
of trials must be finite. (iv) Getting a
favourable outcome in a trial is termed as success and
getting a non-favourable outcomein a trial is termed
as failure. Hence, in n trials of an experiment, the
experimenter may
succeed on time or once or twice…or all the n times. (v) Here,
we define p as the probability of succeeding in a trial, q as
that of failing in a trial and X as the random
variable denoting the number of successes in the n trials of the
experiment. (vi) Then, the
probability of succeeding exactly k times in the n trials
of the experiment is found to be
P(X=k)=nCk pk qn-k .
(vii) The mean and variance of a binomial random variable
are
mean=E(X)=npand
variance=V(X)=E(X2)−[E(X)]2=npq.
(Ia) Geometric random variable: Let us suppose that a
Bernoulli’s experiment is
tried until we get a success. Here, we define Y as a random
variable denoting the numberof trials required (to get a success).
Y is called the geometric random variable. If wedefine p as the
probability of succeeding in a trial and q as that of failing in a
trial, then
the probability of succeeding in the kth trial for the first
time (or the probability ofrequiring exactly k trials to get the
first success) is found to be
P(Y=k)=qk-1 p.
The mean and variance of a geometric random variable are
mean=E(Y)=(1/p)
andvariance=V(Y)=E(Y
2)−[E(Y)]2=(q/p2).
(Ib) Hypergeometric random variable: Let us choose n items,
without replacement,randomly from a lot containing T items out of
which F items are favourable. Here wedefine Z as a random variable
denoting the number of favourable items in the n itemschosen. Z is
called the hypergeometric random variable and can take on the
values from 0
to n. We can choose k favourable items from F items in
FCk ways and (n−k) non-favourable items in
(T−F)C(n−k) ways. Thus we can choose k favourable items and
(n−k)non-favourable items from the lot containing F favourable
items in FCk ×(T−F)C(n−k) ways. Since n items can be
chosen from T items in TCn ways, the probability of having
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exactly k favourable items in the n items chosen (i.e.,
probability of choosing exactly kfavourable items) is found to
be
P(Z=k)=[FCk ×(T−F)C(n−k)]/TCn.
The mean and variance of a hypergeometric random variable
are
mean=E(Z)=n(F/T)
and
variance=V(Z)=E(Z2)−[E(Z)]2=n[F/T][(T−F)/T][(T−n)/(T−1)].
(II) Poisson distribution: If n is very large and p is very
small in the binomialdistribution, then it is called the Poisson
distribution. The probability function is
approximated by the formula
P(X=k)=(λk e−λ)/k!.
The mean and variance of a Poisson random variable are
mean=E(X)=np=λ and
variance=V(X)=E(X2)−[E(X)]2=λ.
(III) Normal distribution: The continuous distribution
described by
1 (x−µ )2
f(x)= exp −(1/2)σ√2π σ2
is called the normal distribution or Gauss distribution. the
normal distribution has adistribution function of the form
1 x (v−µ )2
F(x)= ∫ exp −(1/2) dvσ√2π −∞ σ2
where µ is the mean of the distribution and σ is the
standard deviation of the distribution.The random variable X with
the distribution function, F(x) is called the normal
random variable. Here, we define that
(i) P(X ≤ a)=P(X
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Counting techniques:
(I) Permutation: Permutation is ordering.(i) A set of
n symbols can be arranged in n! ways.
(ii) A set of n symbols can be arranged in
nPr=n(n−1)(n−2)...(n−r+1)=[n!/(n−r)!] ways, considering r
at a time.
(II) Combination: Combination is grouping.
(i) A set of n elements can be split into
nCr=[n(n−1)(n−2)...(n−r+1)/r!]=[n!/(n−r)!r!]
(ii)
The binomial expansion:
n
(x+y)n=Σ nCi xiyn−ii=0
(iii) The number of subsets that can be chosen from a set
of n elements is
2n.
(III) Permutation of a set of n elements with k different
kinds: Let S be a set of
n elements, out of which n1 are alike, n2 are alike,...and
nk are alike so thatk
Σni=ni=1
Then, these n elements can be arranged in
[n!/(n1!n2!...nk !)] different ways.
Random variables: Distribution functions and probability density
functions: Giventhe distribution function F(x) of a random
variable X, the probability density function f(x)of X is found to
be the slope of the distribution function at x
i.e.,f(x)=(d/dx)F(x).
(I) Moments and moment generating functions: (i) The
kth moment of a random variable X is defined as the expected
value of X to the kth power i.e.,
mk =E(Xk )=Σxk p(x) (discrete)
x
∞ mk =E(X
k )=∫ xk f(x)dx (continuous)−∞
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(ii) The moment generating function m(t) for a random
variable X isdefined as
m(t)=E[etX
]=Σetx p(x) (discrete)x
∞ m(t)=E(e
tX)=∫ etxf(x)dx (continuous)−∞
The kth moment of a random variable X in terms of m(t) is found
to be
mk =lim[(dk /dtk )m(t)] t→0
(iii) The factorial moment generating function for a
random variable X isdefined as
ϕ(t)=E[tX]=Σtx p(x) (discrete)x
∞ ϕ(t)=E(tX)=∫ txf(x)dx (continuous)
−∞
Independent events: If A and B are independent events,
then
p(A∩B)=p(A)p(B)
Conditional probability: The probability of the event A,
given that the event B has
occurred, is defined as
p(A/B)=p(A∩B)/p(B)
If A and B are independent, then
p(A/B)=p(A)
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Skewness & Kurtosis:
Measure of skewness=[m32/m2
2] i.e., =[(square of 3
rd moment)/(square of 2
nd moment)]
Measure of kurtosis=[m4/m22] i.e., =[(4th moment)/(square
of 2nd moment)]
(i) B inomial distribution:
Measure of skewness= ±[(q− p)/√(npq)]Measure of
kurtosis=3+[(1−6pq)/npq]
(ii ) Poisson distribution:
Measure of skewness= ±[1/√(np)]Measure of kurtosis=3+[1/np]
(iii) Normal distribution:
Measure of skewness=0Measure of kurtosis=3
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STATISTICS
Coefficient of correlation: Given that
x
y
the coefficient of correlation is defined as
r(x,y)={covar(x,y)/√[var(x)var(y)]}
where
covar(x,y)=mean(xy)−mean(x)mean(y) i.e.,
=[Σ(xy)/n]−[Σ(x)/n][Σ(y)/n]var(x)=mean(x
2)−[mean(x)]2 i.e., =[Σ(x2)/n]−[Σ(x)/n]2
var(y)=mean(y2)−[mean(y)]2 i.e.,
=[Σ(y2)/n]−[Σ(y)/n]2
Properti es of coeff icient of correlation: The
coefficient of correlation
(i) lies between −1 and +1.(ii) is independent of
choice of origin and scale of measurement.(iii) is a measure
of linear relationship.
(iv) is positive if both the variables increase or
decrease; negative if one variableincreases while the other
decreases and vice versa.
Regression coefficient:
Regression coeffi cient of y on x: The regression
coefficient of y on x is defined as
byx=[covar(x,y)/var(x)]
and the regression line of y on x is given by
[y−mean(y)]=byx[x−mean(x)]
Regression coeffi cient of x on y: The regression
coefficient of x on y is defined as
bxy=[covar(x,y)/var(y)]
and the regression line of x on y is given by
[x−mean(x)]=bxy[y−mean(y)]
⇒r(x,y)=√(byx bxy)
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VECTOR CALCULUS
Vector differential operator: The vector differential
operator∇ is defined as
∇=i(∂/∂x)+ j(∂/∂y)+k (∂/∂z).
Gradient of a scalar function: The gradient of a scalar
function ϕ(x,y,z) is defined as
gradϕ=∇ϕ=i(∂ϕ/∂x)+ j(∂ϕ/∂y)+k (∂ϕ/∂z).
Properties:
i. Ifϕ(x,y,z)=C represents a level surface S, then
∇ϕ at a point P(x,y,z) on S is avector normal to the surface S
at P.
ii. The magnitude of ∇ϕ is the maximum rate of change
of ϕ and the direction of
∇ϕ is the direction of the maximum rate of change of
ϕ.iii. ∇r n=nr n−2r iv.
∇f(r)=[f /(r)/r]r
v. ∇2r
n=n(n+1)r
n−2
vi. ∇2f(r)=f
//(r)+2[f
/(r)/r]
Divergence of a vector function: The divergence of a vector
function
F=F1i+F2 j+F3k
is defined as
divF=∇•F=(∂F1/∂x)+(∂F2/∂y)+(∂F3/∂z)
Properties:
i. If divF=0, then F is solenoidal.
ii. ∇•(A+B)=∇•A+∇•B iii.
∇•(ϕF)=ϕ(∇•F)+ ∇ϕ • F
Curl of a vector function: The curl of a vector
function
F=F1i+F2 j+F3k
is defined as
curlF=∇×F= i j k(∂/∂x) (∂/∂y) (∂/∂z)
F1 F2 F3
Properties:
i. If∇×F=0, then F is irrotational.ii.
∇×(A+B)=∇×A+∇×B
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iii. ∇×(ϕF)=ϕ(∇×F)+ ∇ϕ×F iv.
∇×(A×B)=[(∇•B)A−(∇•A)B]+[(B•∇)A−(A•∇)B]
v.
∇×(∇×A)=∇(∇•A)−∇2
A Miscellaneous properties:
i. ∇•(A×B)=B•(∇×A)−A•(∇×B)ii. Directional derivative
of a scalar function ϕ(x,y,z) in the direction of a vector
A
is defined as
[(∇ϕ•A)/A]
Vector integration: (I) Line integral: Given that
F=F1(x,y,z)i+F2(x,y,z) j+F3(x,y,z)k
to compute the line integral
∫ C F•dr
along the curve C joining the points P1 and
P2 i. compute the dot product
F•dr=(F1i+F2 j+F3k )•(dx i+dy j+dz
k )=F1dx+F2dy+F3dz=f(x,y,z,dx,dy,dz)ii. convert the
multi-variable function, f obtained in (i) into a function (say,
f 1) of
one variable and its derivative (i.e., either a function of x
& dx alone or of y & dyalone or of z & dz alone or of
some third variable t that relates x, y & z) using the
information given about C andiii. compute the integral
P1
∫ f 1(v)dvP2 where v may be x or y or z or a
third variable t that relates x, y & z.
If the curve C is split into a finite number of curves (say,
C1 joining P1 and P12, C2 joining P21 and
P22,…,Cn joining Pn1 and P2), then the given integral is
also split asfollows:
∫ C
F•dr=∫ F•dr+∫ F•dr+…+∫ F•dr C1
C2 Cn
and the steps (i) through (iii) are followed for each integral
separately.
Important notes:
i. If∇×F=0, then F is conservative. (A necessary
& sufficient condition.)
ii. If∇×F=0, then the line integral ∫ C F•dr is
independent of the path C joining anytwo points P1 and P2.
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iii. If∇×F=0, then F=∇ϕ, where ϕ is any scalar
function.
iv. If∇×F=0, thenx y z
ϕ=∫ F1(x,y1,z1)dx+∫ F2(x,y,z1)dy+∫ F3(x,y,z)dzx1
y1 z1
if we arbitrarily choose a straight line segments from
(x1,y1,z1) to (x,y1,z1) to(x,y,z1) to (x,y,z).
v. If∇×F=0, then the line integral ∫ C F•dr=0 around
any closed path.vi. If∇×F=0, then F•dr=F1dx+F2dy+F3dz is an
exact differential. (A necessary &
sufficient condition.)
(II) Surface integral: Given that
F=F1(x,y,z)i+F2(x,y,z) j+F3(x,y,z)k
to compute the surface integral
∫∫ F•ds=∫∫ F•n dSS S
along the surface S=ϕ(x,y,z)i. compute the unit normal
n to the surface S=ϕ(x,y,z) as n={∇ϕ/|∇ϕ|}
ii.
compute the dot product F•n iii. choose a
differential surface dS on S and find its projection onto any one
of the
planes, z=0, y=0 or x=0 andiv. compute the
integral
∫∫ F•n [projdS/|n•n1|]Rwhere projdS is the projection of dS
onto any one of the planes, z=0, y=0 or x=0and n1 is the unit
normal to the plane containing projdS.
Note: If we project dS onto the plane z=0, then
dS=[projdS/|n•k |] or if we project dSonto the plane y=0, then
dS=[projdS/|n• j|] or if we project dS onto the plane x=0,
thendS=[projdS/|n•i|].
(III) Volume integral: To compute the volume integral
∫∫∫ (VE)dVV
where VE is any vector expression,i. compute the vector
expression VE and
ii. compute the integral
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∫∫∫ (VE)dV=∫∫∫ (VE)dxdydzV V
Divergence theorem of Gauss: The divergence theorem states
that
∫∫ F•dS=∫∫ F•n dS=∫∫∫ ∇•FdVS S V
Green’s theorem: The Green’s theorem states that
∫ CMdx+Ndy=∫∫ [(∂ N/∂x)−(∂M/∂y)]dxdyR
Area bounded by a closed curve C: The area, A bounded
by a closed curve C on thez=0 plane is given by
A=(1/2)∫ C[xdy−ydx]
Stoke’s theorem: The Stoke’s theorem states that
∫ CF•dr=∫∫ (∇×F)•n dS=∫∫ (∇×F)•dS S S
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COMMUNICATION SYSTEMSModulation: The process of
up-shifting the message frequencies to a range more useful
for transmission is called the modulation.
Need for modulation: Ø
Several programs occupying the same base frequency band
and being conducted at
the same time are separated by shifting each program to a
different frequency band.Ø As the antenna size is inversely
proportional to the frequency to be radiated and its
gain is directly proportional to square of the frequency to be
radiated, shifting themessage frequencies to a higher band is
advantageous.
Ø Higher frequency signals will travel a longer
distance.
(I) Amplitude modulation: In this process of modulation the
amplitude of the carrier ismade proportional to the instantaneous
value of the message signal.
Single tone AM: Mathematics of AM
wave: Let the carrier and modulating signals be
ec(t)=Eccosωctem(t)=Emcosωmt,
respectively. Then the modulated wave would be
eAM(t)=(Ec+Emcosωmt)cosωcteAM(t)=Ec(1+mcosωmt)cosωcteAM(t)=Eccosωct+(mEc/2)cos(ωc-ωm)t
+(mEc/2)cos(ωc+ωm)t
carrier LSB USB
where
m=(Em/Ec)
is the modulation index. Power calculation: The total
power in the AM wave is given by
Pt=Pc+PLSB+PUSB
where
Pc=(Ec,rms2/R)=(Ec/√2)2/R=Ec2/2R
PLSB=PUSB=PSB=(ESB,rms2/R)=(mEc/2√2)2/R=(m2/4)(Ec2/2R)=(m2/4)Pc
⇒Pt=Pc[1+(m2/2)]
The total voltage in the AM wave is given by
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Et2=2PtR=2PcR[1+(m
2/2)]=Ec
2[1+(m
2/2)] since Pc=(Ec
2/2R)
⇒Et=Ec[1+(m2/2)]½
The total current in the AM wave is given by
It2=(2Pt/R)=(2Pc/R)[1+(m
2/2)]=Ic2[1+(m2/2)] since Pc=(Ic
2R/2)
⇒It=Ic[1+(m2/2)]½
Multi-tone AM : Mathematics of AM
wave: The AM wave resulting from modulating a complex
wave
em(t)=E1cosω1t+E2cosω2t+...+Encosωnt
by a carrier signal
ec(t)=Eccosωct
is given by
eAM(t)=Ec(1+E1cosω1t+E2cosω2t+...+Encosωnt)cosωcteAM(t)=Eccosωct+[(m1Ec/2)
cos(ωc-ω1)t +(m1Ec/2) cos(ωc+ω1)t+[(m2Ec/2) cos(ωc-ω2)t
+(m2Ec/2) cos(ωc+ω2)t]+...+ [(mnEc/2) cos(ωc-ωn)t +(mnEc/2)
cos(ωc+ωn)t] eAM(t)=Eccosωct+[(m1Ec/2) cos(ωc-ω1)t +(m2Ec/2)
cos(ωc-ω2)t+...+(mnEc/2) cos(ωc-ωn)t]
+ [(m1Ec/2) cos(ωc+ω1)t+(m2Ec/2) cos(ωc+ω2)t]+...+(mnEc/2)
cos(ωc+ωn)t]
where
m1=(E1/Ec)m2=(E2/Ec)...
mn=(En/Ec)
are the individual modulation indices. Power
calculations: The total power in the modulated wave is given
by
Pt=Pc+[PLSB1+PLSB2+...+PLSBn]+[PUSB1+ PUSB2+...+ PUSBn]
Since
Pc=(Ec2/2R)
PLSB1=PUSB1=(m12Ec
2/8R)=(m1
2Pc/4)
PLSB2=PUSB2=(m22Ec
2/8R)=(m22Pc/4)...
PLSBn=PUSBn=(mn2Ec
2/8R)=(mn
2Pc/4)
Pt=Pc[1+½(m12+m2
2+...+mn2)]=Pc[1+(meff
2/2)]
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where
meff =[(m12+m2
2+...+mn
2)]
½
is the effective modulation index.
Similarly, the total voltage and the total current in the AM
wave are given by
Et=Ec[1+(meff 2/2)]
½
The total current in an AM wave is given by
It=Ic[1+(meff 2/2)]
½
Bandwidth of an AM system:
BW=2f m
(II) Angle modulation: An angle-modulated wave is given
by
eAN(t)=Eccos[ωct+φ(t)]
where φ(t) is the instantaneous phase and the quantity
ω={d[ωct+φ(t)]/dt}
⇒ f={(1/2π)d[ωct+φ(t)]/dt}=f c+(1/2π)[dφ(t)/dt]
is the instantaneous frequency.
(i ) Phase modulation: Phase modulation is a form of
angle modulation, in which
the instantaneous phase φ(t) is made proportional to the
modulating signal m(t) i.e.,
φ(t)=k pm(t)
where k p is a constant. Thus, the phase-modulated
wave is given by
ePM(t)=Eccos[ωct+k pm(t)]
(ii ) Frequency modulati on: Frequency modulation is
a form of angle modulation,in which the instantaneous frequency f
is varied linearly with the modulating signal m(t)i.e.,
f=f c+k f m(t)
where k f is a constant. Thus, the
frequency-modulated wave is given by
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t
eFM(t)=Eccos[ωct+2πk f ∫ m(t)dt]0
Relatinship between PM and FM : (i) An FM wave can be
generated from a signal by first integrating it and then
phase-modulating the resultant.
m(t)→integrator →mi(t)→ phase
modulator →eFM(t)
(ii) A PM wave can be generated from a signal by first
differentiating it and thenfrequency-modulating the resultant.
m(t)→differentiator →mi(t)→frequency
modulator →ePM(t)
Single-tone FM : Sinusoidal modulation: The FM wave
resulting from modulatinga sinusoidal signal
em(t)=Emcosωmt
by a carrier signal
ec(t)=Eccosωct
is given byt
eFM(t)=Eccos[ωct+mf sinωmt] since
φ(t)=2πk f ∫ m(t)dt0
where mf is the modulation index and is given by
mf =(max. frequency deviation)/(modulating
frequency)=[∆f/f m]
mf is also the maximum phase deviation.
Narrowband FM (m f
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A narrowband FM wave resembles an AM wave except that the LSB is
reversed in phaseand hence the wave is out of phase with the
carrier.
To limit the harmonic distortion, mf must be less
than or equal to 0.3 rad in MBFM i.e.,
mf
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Multi-tone FM wave:
eFM(t)=Eccos[ωct+mf1sinω1t+mf2sinω2t]∞ ∞
⇒ eFM(t)=EcΣ
Σ Jn(mf1)Jm(mf2)cos[2π(f c+nf 1+mf 2)t]n=−∞ m=−∞
The components of eFM(t) are (i) A carrier component of
amplitude J0(mf1)J0(mf2) andfrequency f c. (ii) A set of
sideband components with amplitudes Jn(mf1)J0(mf2) and
frequencies (f c±nf 1), where n=1,2,3,.... (iii) A set
of sideband components withamplitudes J0(mf1)Jm(mf2) and
frequencies (f c±mf 2). where m=1,2,3,... (iv) A set
ofsideband components with amplitudes Jn(mf1)Jm(mf2) and
frequencies (f c±nf 1±mf 2),
where n, m=1,2,3,....
Frequency Multi plication applied to FM signal: If an
FM signal passes through amultiplier with a multiplication factor
n, both the carrier frequency f c and the frequency
deviation ∆f are multiplied by n. Thus, for a fixed modulation
frequency f m, themodulation index mf is also
multiplied by the same factor n. That is
(i) New carrier frequency f c/=nf c
(ii) New frequency deviation ∆f /=n∆f(iii) New modulation
index mf
/=nmf
Deviation sensitivity: The deviation sensitivity is
defined asmeasured deviation in f c
Deviation sensitivity= peak of the modulating voltage
Deviation ratio: The deviation ratio is defined
asmaximum allowable deviation in f c
Deviation ratio=highest allowable message frequency
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Superheterodyne receiver: The superheterodyne principle is
such that when two sinusoidal signals of different
frequencies are mixed together so that they multiply, then the
output signal will containsignal components at the sum, the
difference and each of the two original frequencies.
There will also be mixtures of the harmonics of these signals,
but if the two originalfrequencies are carefully chosen, these
harmonics will not interfere.
Main parts of a super heterodyne receiver are: (1) the RF
amplifier to provide gainat the point of lowest noise in the
system, (2) mixer and oscillator to downshift the carrier
to an intermediate frequency (for a low-carrier frequency AM
receiver the IF is anindustry standard of 455 kHz), (3) the IF
amplifier(s) to provide high gain, (4) the
detector, (5) the AGC circuit to automatically control the gain
of the receiver by sendinga dc bias voltage, derived from the
detector, to a selected number of RF, IF and mixerstages to modify
their gain ( a large negative dc voltage is sent to the selected
number ofRF, IF and mixer stages to reduce their gain when the
strength of the signal is too high
and a small negative dc voltage is sent to the selected number
of RF, IF and mixer stagesto increase their gain when the strength
of the signal is too low) and (6) the final poweramplifier(s) to
improve the power level of the detected signal to drive the
low-impedanceoutput reproducers such as loud speakers, CRTs
etc.
Detectors: Either an envelop detector or a product
detector (balanced demodulator)can be used. The envelope detector
uses diodes. The product detector (balanceddemodulator) uses the
Class B push-pull differential amplifier.
I ntermediate fr equency:
IF=f s−f o if f s >
f o IF=f o−f s if f s <
f o
I mage fr equency: The image frequency is defined
as
Image frequency=RF+2IF if f lo>RF or
Image frequency=RF−2IF if f lo
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Noise in AM: (i ) SSB-SC:
(1) (So/No)=(Si/ηf m)=(Si/Ni) i.e., output signal-to-noise
ratio is equal to inputsignal-to-noise ratio.
(2) Figure of merit, which is defined as the ratio of the output
signal-to-noiseratio to the input signal-to-noise ratio is unity
(1).
(ii ) DSB-SC:
(1) (So/No)=(Si/ηf m)=(Si/Ni) i.e., output signal-to-noise
ratio is equal to inputsignal-to-noise ratio.
(2) Figure of merit, which is defined as the ratio of the output
signal-to-noise
ratio to the input signal-to-noise ratio is unity (1).
(ii i) DSB-FC (any arbi trary modulating signal
):
(1) (So/No)=[mav2/(1+mav
2)](Si/ηf m), where mav is the time average of
themodulating signal.
(2) Figure of merit, which is defined as the ratio of the output
signal-to-noiseratio to the input signal-to-noise ratio is [mav
2/(1+mav2)].
(ii i) DSB-FC (sinusoidal modulating signal):
(1) (So/No)=[m2/(2+m
2)](Si/ηf m), where m is the modulation index.
(2) Figure of merit, which is defined as the ratio of the output
signal-to-noise
ratio to the input signal-to-noise ratio is [m2/(2+m
2)].
Noise in FM:
(1) (So/No)=(3/2)∆f 2(Si/ηf m), where ∆f is the
maximum frequency deviation.(2) Figure of merit, which is defined
as the ratio of the output signal-to-noise
ratio to the input signal-to-noise ratio is (3/2)∆f 2.
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CONTROL SYSTEMS
(I) Linear systems and differential equations: L inear
independence and fundamental sets: A sufficient
condition for a set of n functions f 1, f 2,...,
f n to be linearly independent
is that
f 1 f 2 . . . f n
(df 1/dt) (df 2/dt) . . . (df n/dt)
.
.
.
(dn-1f 1/dtn-1) (dn-1f 2/dt
n-1) . . . (dn-1f n/dtn-1) ≠0
But this is not a necessary condition since there are linearly
independent functions forwhich the above determinant is zero.
Second order dif ferential equations: A linear
constant coefficient second order differential equation, in the
study of
control systems, of the form
(d2y/dt
2)+αω0(dy/dt)+ω02y=ω02x
is very important where
α=damping coefficient andω0=undamped natural frequency.
(II) Stability: A system is said to be stable if (i) its
impulse response approaches zero (i.e.,
converges) as time approaches infinity or (ii) if every bounded
input produces a boundedoutput or (iii) if the transfer function of
the system has poles only with negative real
parts.
(i) Routh stabil ity cr iteri on: The Routh stability
criterion is a method for determining whether the roots of the
given nth order characteristic equation of the form
a0sn+a1s
n-1+...+an=0
have only negative real parts thereby determining the system
stability. The criterion isapplied through the Routh table defined
as follows:
sn a0 a2 a4 ...
sn-1
a1 a3 a5 ...s
n-2 b1 b2 b3 ...
sn-3
c1 c2 c3 ....
........................
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where a0, a1, a3, a4, a5 ... are the coefficients of the
given characteristic equation and
b1=[(a2a1-a0a3)/a1], b2=[(a4a1-a0a5)/a1], etc.
c1=[(a3 b1-a1 b2)/b1],c2=[(a5 b1-a1 b3)/b1],
etc.
The table is continued horizontally and vertically until only
zeros are obtained. Any row
can be multiplied by a constant before the next row is computed
without disturbing the properties of the table. All the roots
of the given characteristic equation have negative
real parts if and only if the elements of the first column of
the Routh table have the samesign. Otherwise, the number of roots
with positive real parts is equal to the number ofsign changes in
the first column of the Routh table.
General input/output gain formula: The general input/output
gain formula for any signal flow graph is given by
T=(C/R)=[Σ pi∆i]/∆ [T is also known as the transfer
function of the system]i
where
pi =ith forward path gain
∆=1−{sum of all loop gains}+{sum of all gain-products of 2
non-touching loops}− −{sum of all gain-products of 3
non-touching loops}+...
∆i=∆ evaluated with all loops touching
pi eliminated.
T=(C/R)=G/(1±GH)
G=direct/forward transfer function of the system
H=feedback transfer function of the systemGH=open-loop or loop
transfer function of the system
C/R=closed-loop transfer function or control ratio
E/R=error ratio=1/(1±GH)
B/R=primary feedback ratio=GH/(1±GH)
System classification: If the open-loop transfer function
GH of the system has k poles at origin, then the
system is called the type k system.
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Error constants of stable type k, Unity feedback system:
(i) Position error constant:
K p=lim G(s)
s→0
(ii ) Velocity error constant:
K v=lim sG(s)
s→0
(ii i) Acceleration err or constant:
K a=lim s2G(s)
s→0
Steady state error of a stable type k, unity-feedback
system:
(i) when the input is unit step function is given
e(α)=[1/(1+K p)]
(ii) when the input is unit ramp function is given by
e(α)=[1/K v]
(iii) when the input is unit parabola is given by
e(α)=[1/K a]
Error constants of general system:
(i ) Step er ror constant:
1K s=
lim [TD-C/R]
s→0
e(α)=1/K s
TD=desired transfer function that the system
approximatesC/R=actual transfer function of the system.
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(ii ) Ramp err or constant:
1K r =
lim (1/s)[TD-C/R]
s→0
e(α)=1/K r
(ii i) Parabolic error constant:
1K pa=
lim (1/s2)[TD-C/R]
s→0
e(α)=1/K pa
Second order control system:
The second order control systems are generally desired by the
following linear,constant coefficient differential equation
[d2y/dt2]+[2αω0dy/dt]+ω02y=ω02x
α=damping coefficientω=undamped natural frequency.
Root-locus analysis:
Var iation of the closed-loop system poles: The open
loop transfer function GH can
generally be expressed as
GH=KN(s)/D(s)
whereK=open-loop gain factor N(s)=numerator polynomial in
s.D(s)=denominator polynomial in s.
Therefore, the closed-loop transfer function C/R can be
expressed as
C/R=G/(1±GH)=GD(s)/[D(s)±KN(s)]
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The roots of the equation
D(s)±KN(s)=0(called the characteristic equation) are the poles
of the closed-loop system. The locus ofthese poles plotted in the
s-plane as a function of K is called the root-locus.
Number of root-loci: The number of root-loci is equal
to the number of poles of the
open-loop transfer function GH.
Angle and magnitude cr iterion: Given the open-loop
transfer function
GH=KN(s)/D(s)
(i) for a particular point s1 to be on the root-locus of
G
arg[GH(s1)]=(2n+1)π, n=0, ±1, ±2, ...{for K>0}
(or) arg[N(s1)/D(s1)]=(2n+1)π, n=0, ±1, ±2, ...
arg[GH(s1)]=2nπ, n=0, ±1, ±2, ...{for K0.
(ii) The points on the real axis to the left of an even number
of finite poles and zeros
will be on the root locus for K
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Σ Zi=sum of zeros of GHi
n=no. of poles
m=no. of zeros.
(2k+1)π/(n-m), K >0(2) Angle between the asymptotes,
β=
2k π/(n-m), K
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ELECTRICAL MACHINES
(I) DC generators: Types of dc generators:
(i) Separately excited generator: The field coil is
energized by an external dcsource.
(ii) Self-excited generator: The field coil is energized by
the current induced bythe generator itself. It is further
classified into
(iia) Shunt generator: The field coil is in parallel with the
armature coil.
Ia=IL+If & Eg=VL+IaR a+BCD
(iib) Series generator: The field coil is in series with the
armature coil.
Ia=IL=If &
Eg=VL+Ia(R a+R f )+BCD
where (1) Ia=armature current, (2) IL=load current, (3)
If =field current, (4) Eg=generatedemf, (5) VL=load voltage,
(6) R a=armature resistance, (7) R f =field
resistance, (8) R L=load
resistance and (9) BCD=brush contact drop.(iic) Compound
generator: Both the shunt field coil and series field coil
are present.(1) The compound generators are further classified
into two depending on
the position of the series field coil: (1a) the long shunt
generator , in which the ser iesfi eld coil is in ser ies with
the armature coil and (1b) the shor t shun t
generator , in whichthe ser ies fi eld coi l i s in ser ies
with the load .
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(2) The compound generators are further classified into
two depending onthe behaviour of the shunt and series field fluxes:
(2a) the cumulative compound
generator , in which the shunt f ield fl ux aids the series
field flux and (2b) thedif ferential compound
generator , in which the shun t f ield f lux opposes the ser
ies field
flux .
Ia=Ifp+IL & Eg=VL+Ia(R a+R fs)+BCD
Ia=Ifp+IL & Eg=VL+IaR a+ILR fs+BCD
where (1) Ia=armature current, (2) Ifp=shunt field current, (3)
IL=load current, (4)R a=armature resistance, (5)
R fs=series field coil resistance, (6) R fp=shunt field
coilresistance, (7) Eg=generated emf, (8) VL=load voltage and (9)
BCD=brush contact drop.
Generator EMF equation: Let
φ=flux per pole in webers,Z=total number of armature
conductors,P=number of poles,
N=armature speed in rpmA=number of parallel paths and
Eg=emf induced in any parallel path.
By definition, the average emf induced in any conductor=(Φ/T)
voltswhereΦ is the total flux linking any conductor and T is
the time taken by any conductorfor one revolution. Hence
Φ=Pφ webers
Since the number of revolutions per second is (N/60), the time
taken for onerevolution is found to be
T=(60/N) seconds
Hence,
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Average emf generated per conductor=(NPφ/60) volts
Since the number of conductors in any parallel path is (Z/A),
the average emfgenerated in any parallel path is found to be
Eg=(NPφ/60)(Z/A) volts
Note: (1) For a wave winding, A=2 (i.e., number of
parallel paths=2). (2) For a lapwinding, A=P (i.e., number of
parallel paths=number of poles).
Total losses in a dc generator : (i) Constant
losses: The shunt field copper loss and the stray losses are
termed as
the constant losses.The stray losses are (1) the magnetic losses
and (2) the mechanical losses.
(1) The magnetic losses are (1a) the hysteresis loss and (1b)
the eddy currentloss.
(2) The mechanical losses are (2a) the friction loss at bearings
and commutatorand (2b) the air-friction loss of rotating
armature.
(ii) Variable loss: The armature copper loss is the
variable loss.Condition for maximum efficiency: Constant
losses=Variable lossI mpor tant terms:
(i) Pole pitch: The number of armature conductors per pole
is termed as the pole pitch.
(ii) Coil span or coil pitch: The number of armature slots
(or conductors) between two sides of a coil is termed as the
coil span or coil pitch. (1) If coil pitch=pole pitch, then
the winding is said to be full-pitched. (2) If coil pitch < pole
pitch, then thewinding is said to be fractionally pitched. (3) For
a full-pitched winding, the induced emfis maximum and the coil span
is 180 electrical degrees.
I mportant character istics of dc generators: There
are three types of characteristics: (1) Open circuit
characteristics (between
open circuit voltage VLO and field current If ), (2)
Internal or total characteristics (betweengenerated emf Eg and
armature current Ia) and (3) External characteristics or
voltageregulation curve (between load voltage VL and load
current IL).
(i) Shunt generator: Except the open circuit
characteristics, the other twocharacteristics are dropping in
nature i.e., the voltage drops as the current raises. Note:For
voltage buildup, (1) there must be some residual magnetism in the
field winding, (2)
the shunt field resistance must be less the critical resistance
when excited on open circuiti.e., on no load and (3) the shunt
field coil must be properly connected to the armatureterminals, for
the given direction of rotation.
(ii) Series generator: All the three characteristics are
raising in nature i.e., thevoltage raises as the current raises.
The series generator cannot build up on open circuit.
(iii) Cumulative compound generator: (1) In case of
over-compound generator,the full load voltage is somewhat more than
the no load voltage. (2) In case of flat
compound generator, the full load voltage is equal to the no
load voltage. (3) In case ofunder- compound generator, the full
load voltage is somewhat less than the no load
voltage.
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Vol tage regulation: The voltage regulation is
defined as
% voltage regulation=[(VOL-VFL)/VOL]×100
For an ideal generator, the voltage regulation is zero.
Appl ications of dc generators: (i) Shunt
generator: used (1) for charging batteries, (2) for lighting
and (3) in
power supplies in conjunction with voltage regulators.(ii)
Series generator: used (1) as boosters to increase the voltage
across the
feeders in the railway service and (2) to supply the field
current for regenerative brakingof dc locomotive.
(iii) Compound generators: (iia) Cumulative: used (1) as
drives that require a constant voltage, (2) for
lamp loads and (3) for heavy power services such as
railways.(iib) Differential: used in arc-welding.
(II) DC motors: Types of dc motors: (same as in
the case of dc generators)
Equation for back EMF: Let
φ=flux per pole in webers,Z=total number of armature
conductors,P=number of poles,
N=armature speed in rpmA=number of parallel paths
andE b=back emf.
Then
E b=(NPφ/60)(Z/A) volts
Equation for armature torque: Since
power developed=work done per second
⇒E bIa=Ta×(2π N/60)⇒Ta=E bIa×(60/2π N)⇒Ta=E bIa×(30/π N)⇒ Ta=(PZ/2πA)×φIa
Note: (1) For a shunt motor, φ=constant. Hence Ta∝Ia. (2)
For a series motor, φ∝Ia.Hence Ta∝Ia2.
Equation for shaft torque: Since
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power output=work done per second
⇒output power=Ta×(2π N/60)⇒Ta=output
power × (60/2π N)⇒Ta=output power ×(30/π N)
I mportant character istics of dc motors: There are
three types of characteristics: (1) (Ta/Ia) characteristics, (2)
(N/Ia)
characteristics and (3) (N/Ta) characteristics.
(i) Shunt motor: (1) Since Ta∝φIa and, for a shunt
motor, φ=constant,Ta∝Ia⇒Ta=kIa. (2) Since N∝(E b/φ) and, for a
shunt motor, φ=constant, N∝E b⇒ N=kE b.Since
E b is also almost constant for any motor, N is almost
constant for a shunt motor. (3)
N decreases slightly with increase in Ta.
(ii) Series motor: (1) Since Ta∝φIa and, for a series
motor, φ∝Ia,Ta∝Ia2⇒Ta=kIa2. (2) Since N∝(E b/φ) and, for a
series motor, φ∝Ia,
N∝(E b/Ia)⇒ N=k(E b/Ia). Since
E b is also almost constant for any motor, N=k(1/Ia). (3)
N
decreases with increase in Ta.
(iii) Cumulative compound motor: (1) Ta increases more
rapidly than the shunt
motor but less rapidly than the series motor with increase in
Ia. (2) N drops more rapidly
than the shunt motor but less rapidly than the series motor with
increase in Ia. (3) N dropsmore rapidly than the shunt motor but
less rapidly than the series motor with increase inTa.
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(iv) Differential compound motor: The characteristic
curves are just opposite tothose of the cumulative compound
motor.
Appl ications of dc motors: (i) Shunt
motor: (constant speed). used for lathes, centrifugal
pumps,
reciprocating pumps, machine tools, blowers, fans, line-shaft
drivers, wood workingmachines, grinders and shapers.
(ii) Series motor: (very high starting torque). used for
electric traction, tram cars,railway cars, cranes, hoists,
elevators, winches and conveyors.
(iii) Cumulative compound motor: used for shears, punches,
rolling mills,elevators, conveyors, printing presses, heavy planers
and heavy machine tools.
Construction detail s of dc generator/motor : The
main parts of a dc generator/motor are:
(i) Yoke to provide support to other parts(ii) Pole
core and pole shoe to provide support to pole coil, to spread the
magnetic flux
in the air gap and to reduce the reluctance of the magnetic path
because of its largecross-sectional area.
(iii) Pole coil or field coil or exciting coil to produce
the magnetic field when carryinga current.
(iv) Armature core to provide support to armature
winding.(v) Armature winding: there are two types, namely,
(1) lap winding, in which the
finishing end of one coil is connected to the starting end of
the next coil and,which is suitable for high current and low
voltage applications and requiresequalizer rings, (2) wave winding,
in which the finishing end of one coil isconnected to the starting
end of another coil far away from it and, which is suitablefor low
current and high voltage applications and does not require
equalizer rings.
(vi)
Commutator to convert the alternating emf induced in the
armature winding intounidirectional voltage across the external
circuit.
(vii) Brushes
(viii) Bearings
(III) AC generator: Synchronous generator:
Alternator: Construction details: The main parts of
the synchronous generator are:
(i) Stator: Types: (1) wide open type, (2) Semi-closed
type and (3) closed type.
(ii) Rotor: Types: (1) sailent pole rotor, which has large
diameter and small axiallength, and is used for low-speed
applications and (2) smooth cylindrical type
rotor, which has small diameter and large axial length, and is
used for high-speedapplications.
EM F equation: Let
φ=flux per pole in weberf=frequency of the emf generatedT=number
of coils or turns per phase
Z=number of coil sides or conductors per phase (∴Z=2T since any
coil or turn has twosides.)
P=number of poles and
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N=speed of rotor in rpm.
Then,
average emf induced/conductor=[Pφ N/60] volts
But N=[120f/P]
∴average emf induced/conductor=[Pφ/60]×[120f/P] volts⇒average
emf induced/conductor=2f φ volts⇒average emf
induced/phase=2f φZ=4f φT volts⇒RMS value of the induced
emf /phase=2.22f φZ=4.44f φT volts
(IV) AC motor: 3-φ Induction motor: Principle of
operation:
When a 3-φ supply is given to the stator of a
3-φ induction motor, a rotatingmagnetic field of constant
magnitude is created. The speed of this rotating magnetic fieldis
called the synchronous speed given by
Ns=[120f/P]
where
f=supply frequency in Hz and
P=number of stator poles.
The rotating magnetic field produces an emf in the rotor
conductors. The magnitude of
this induced emf depends on the relative speed between the
rotating magnetic field andthe rotor.
Construction details: The main parts of the induction
motor are:
(i) Stator and(ii) Rotor
(i) Stator: (ii) Rotor: There are two types: (1)
squirrel cage rotor and (2) slip ring rotor. The
squirrel cage induction motor has a low starting torque whereas
the slip ring inductionmotor has a high starting torque.
Slip: The slip (S) is defined as
S=[(Ns− N)/Ns]%S=[(Ns− N)/Ns]×100
where
Ns=synchronous speed (i.e., speed of the rotating magnetic
field) and
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N=speed of the rotor.
Frequency of r otor curr ent (i.e., frequency of the emf induced
in the rotor): The frequency of the rotor current
induced is given by
f r =Sf
where
S=slip andf=supply frequency in Hz.
(V) Single phase induction motor: There are two types: (i)
split-phase induction motorand (ii) shaded pole induction
motor.
(i) Spli t-phase induction motor:
There are four types: (1) resistance start induction motor, (2)
capacitance startinduction motor, (3) capacitance start and run
induction motor and (4) two-valuecapacitor-run induction motor.
The two-value capacitor-run motor has (i) the highest starting
pf, (ii) the highestefficiency, (iii) the most noiseless operating
conditions and (iv) the most improvedoverload capacity.
(VI) Transformer: Some importan t terms:
(i) Form factor: The form factor is defined as
Form factor=[RMS value/Average value]=1.11
(ii) Peak or Crest or Amplitude factor: The peak factor is
defined as
Peak factor=[Peak value/RMS value]=1.414
EM F equation of a transformer: Let
φm=maximum value of the flux in the core in webersf=input
frequency in Hz
T=input time period in seconds (i.e., T=[1/f])A=area of the
coreBm=maximum value of the flux density in the core in
webers/sq.mE1=self-induced primary voltage in voltsE2=induced
secondary voltage in volts
N1=number of turns in the primary winding N2=number
of turns in the secondaryI1=full load primary current in amps
andI2=full load secondary current in amps.
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Since the applied voltage is sinusoidal, the flux
produced is also sinusoidal. As theflux attains its maximum value
in T/4 seconds, the rate of change of the flux is found to
be
(dφ/dt)=[φm/(T/4)]=[4φm/T]=4f φm Wb/s∴Average emf
induced/turn=4f φm volts∴RMS value of the induced
emf=4.44f φm volts∴Total emf induced in the primary
E1=4.44f φm N1=4.44fABm N1 volts∴Total emf
induced in the secondary
E2=4.44f φm N2=4.44fABm N1 volts
Transformation ratio: The transformation ratio is
defined as
k=(E1/E2)=(I2/I1)=(R 1/R 2)
½
Transformer on no-load: The primary current
Iop in a transformer on no-load has two components: (i)
the
active or iron loss or wattful component Iw given by
Iw=Iopcosφ
and (ii) the reactive or magnetizing or wattless component
Iµ given by
Iµ=Iopsinφ
where cosφ is the no-load pf and
Iop=Iw+jIµ⇒Iop=(Iw+Iµ)½
Losses in a tr ansformer: The losses in a transformer
are (i) the copper loss and (ii) the iron loss.(i) Copper
loss: The copper loss in a transformer is given by
Total copper loss=I12R 1+I2
2R 2=I1
2R 01=I2
2R 02
where
R 01=R 1+k 2R 2, k
2=(I2/I1)
2
R 02=(R 1/k 2)+R 2, k
2=(I2/I1)2
(ii) Iron loss: The iron loss is also termed as the
magnetic loss.
Ef ficiency of a transformer: The efficiency of a
transformer is defined as
Efficiency=[output/input]
Efficiency=[(input−total losses)/input]Efficiency=1−[total
losses/input]
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where
output=rated KVA×1000× pf ×(x)x=fraction of load given
(e.g., x=1 for full load, x=0.5 for half load etc.)
Condition for maximum efficiency: Copper loss=iron
loss.
Vol tage Regulation of a transformer:
The voltage regulation of a transformer is defined as
Voltage regulation=[(no-load sec. voltage−full load sec.
voltage)/ no-load sec. voltage]
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Miscellaneous topics: 1. Commutator brushes are made
of carbon.2.
Carbon brush causes high voltage drop because of high
resistance.
3. Copper brushes are used in the dc machines where low
voltage and high currentare involved.
4. In dc machines, the mechanical losses are primarily the
function of the speed.5. In dc machines, the magnetic neutral
axis coincides with the geometrical neutral
axis on no load.6. In dc machines, the dummy coils are
used for the mechanical balance.
7. In a dc generator, the ripples in the direct emf
generated are reduced by usingcommutator with large number of
segments.
8. The armature core is usually made of silicon
steel.9. Creating the residual magnetism in the field winding
by an external dc source is
termed as flashing the field.
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Miscellaneous points:
(i) 3 induction motor: 1.
On no load, the slip is generally less than 1%.
2. Slip rings are usually made of phosphor bronze.3.
In case the air gap in an induction motor is increased, the power
factor is
increased.4. If the number of the rotor slots is equal to
the number of stator slots, the motor
refuses to start due to the fact that the speeds of all
harmonics produced by thestator slots coincide with the speeds of
the corresponding rotor harmonics andthese harmonics tend to exert
synchronous torques at their own synchronousspeeds. This condition
is known as the cogging (or magnetic locking). This
isovercome in the squirrel cage induction motor by making the
number of rotorslots prime to the number of stator slots.
5.
The relation between the full load torque and the maximum torque
is given by(Tf /Tmax)=[2aS/(a
2+S
2)]
where S=slip and a=ratio of rotor resistance to rotor standstill
reactance.6. The relation between the starting torque and the
maximum torque is given by
(Tst/Tmax)=[2a/(a2+1)]
where S=slip and a=ratio of rotor resistance to rotor standstill
reactance.7. Maximum torque is obtained when
S=(R r /Xr )where R r =rotor resistance
and Xr =rotor standstill reactance.
8. The synchronous speed of the nth harmonic is (1/n)th of
the synchronous
speed of the fundamental.
9.
In an induction motor, the number of rotor poles is always equal
to that ofstator poles.
10. The tendency of an induction motor to run stably at a
speed ≤ (1/7)th of thenormal speed.
11. If the load on an induction motor goes on increasing,
then the power factorgoes on increasing up to full load and starts
decreasing.
