Laplace Transform | Short Notes for Electrical Engineering - Electrical Engineering (EE) PDF Download

 Page 1


Poles and zeros:
Poles are the roots of the denominator of the fraction in the Laplace transform
and zeroes are the roots of numerator of the fraction. For example as follows: -

 




















where a
1
, a
2
, …, an are known as zeroes and b
1
, b
2
, …. , bn are known as poles.
They are represented in a pole zero diagram as
Properties of Laplace transform:
If x(t), x
1
(t), and x
2
(t) are three signals and X(s), X
1
(s), X
2
(s) are their Laplace
transform respectively and a, b are some constant then,
1. Linearity: ax
1
(t) + bx
2
(t) = aX
1
(s) + bX
2
(s)
2. Time shifting: x(t-t
0
) = e
-sto
.X(s)
3. Shifting in s domain: e
sto
x(t) = X(s-s
0
)
4. Time scaling: x(at) = 1/|a| X(s/a)
5. Conjugation: x*(t) = X*(s*)
6. Convolution: x
1
(t)*x
2
(t) = X
1
(s).X
2
(s)
7. Differential in time domain: dx(t)/dt = sX(s)
8. Differential in frequency domain: -tx(t) = d/ds X(s)
9. Integration in time domain: Integration in time domain is division in frequency
domain.
t
i.e. ? x(t)d(t) = 1/s . X(s)
-8
Applications of Laplace transform:
• For a system with a rational system function causality of the system is equivalent
to the ROC being to the right half plane to the right of the right most pole.
• An LTI system is stable if and only if the ROC of its system function H(s) include
the entire j? axis. (i.e. Re(s) = 0)
Page 2


Poles and zeros:
Poles are the roots of the denominator of the fraction in the Laplace transform
and zeroes are the roots of numerator of the fraction. For example as follows: -

 




















where a
1
, a
2
, …, an are known as zeroes and b
1
, b
2
, …. , bn are known as poles.
They are represented in a pole zero diagram as
Properties of Laplace transform:
If x(t), x
1
(t), and x
2
(t) are three signals and X(s), X
1
(s), X
2
(s) are their Laplace
transform respectively and a, b are some constant then,
1. Linearity: ax
1
(t) + bx
2
(t) = aX
1
(s) + bX
2
(s)
2. Time shifting: x(t-t
0
) = e
-sto
.X(s)
3. Shifting in s domain: e
sto
x(t) = X(s-s
0
)
4. Time scaling: x(at) = 1/|a| X(s/a)
5. Conjugation: x*(t) = X*(s*)
6. Convolution: x
1
(t)*x
2
(t) = X
1
(s).X
2
(s)
7. Differential in time domain: dx(t)/dt = sX(s)
8. Differential in frequency domain: -tx(t) = d/ds X(s)
9. Integration in time domain: Integration in time domain is division in frequency
domain.
t
i.e. ? x(t)d(t) = 1/s . X(s)
-8
Applications of Laplace transform:
• For a system with a rational system function causality of the system is equivalent
to the ROC being to the right half plane to the right of the right most pole.
• An LTI system is stable if and only if the ROC of its system function H(s) include
the entire j? axis. (i.e. Re(s) = 0)
Laplace transform pairs:
x(t) X(s) ROC
x(t)
Z
x(t)e
-st
dt (def.)
d(t) 1 all s
u(t)
1
s
Re(s) > 0
e
-at
u(t)
1
s+a
Re(s) >-a
cos(?ot)u(t)
s
s
2
+?
o
2
Re(s) > 0
sin(?ot)u(t)
?o
s
2
+?
o
2
Re(s) > 0
e
-at
cos(?ot)u(t)
s+a
(s+a)
2
+?
o
2
Re(s) >-a
e
-at
sin(?ot)u(t)
?o
(s+a)
2
+?
o
2
Re(s) >-a
; a is real.
Complex arithmetic operations:
Page 3


Poles and zeros:
Poles are the roots of the denominator of the fraction in the Laplace transform
and zeroes are the roots of numerator of the fraction. For example as follows: -

 




















where a
1
, a
2
, …, an are known as zeroes and b
1
, b
2
, …. , bn are known as poles.
They are represented in a pole zero diagram as
Properties of Laplace transform:
If x(t), x
1
(t), and x
2
(t) are three signals and X(s), X
1
(s), X
2
(s) are their Laplace
transform respectively and a, b are some constant then,
1. Linearity: ax
1
(t) + bx
2
(t) = aX
1
(s) + bX
2
(s)
2. Time shifting: x(t-t
0
) = e
-sto
.X(s)
3. Shifting in s domain: e
sto
x(t) = X(s-s
0
)
4. Time scaling: x(at) = 1/|a| X(s/a)
5. Conjugation: x*(t) = X*(s*)
6. Convolution: x
1
(t)*x
2
(t) = X
1
(s).X
2
(s)
7. Differential in time domain: dx(t)/dt = sX(s)
8. Differential in frequency domain: -tx(t) = d/ds X(s)
9. Integration in time domain: Integration in time domain is division in frequency
domain.
t
i.e. ? x(t)d(t) = 1/s . X(s)
-8
Applications of Laplace transform:
• For a system with a rational system function causality of the system is equivalent
to the ROC being to the right half plane to the right of the right most pole.
• An LTI system is stable if and only if the ROC of its system function H(s) include
the entire j? axis. (i.e. Re(s) = 0)
Laplace transform pairs:
x(t) X(s) ROC
x(t)
Z
x(t)e
-st
dt (def.)
d(t) 1 all s
u(t)
1
s
Re(s) > 0
e
-at
u(t)
1
s+a
Re(s) >-a
cos(?ot)u(t)
s
s
2
+?
o
2
Re(s) > 0
sin(?ot)u(t)
?o
s
2
+?
o
2
Re(s) > 0
e
-at
cos(?ot)u(t)
s+a
(s+a)
2
+?
o
2
Re(s) >-a
e
-at
sin(?ot)u(t)
?o
(s+a)
2
+?
o
2
Re(s) >-a
; a is real.
Complex arithmetic operations:
Z-transform and discrete Fourier transform
Z-transform definition:
If x(t) is the signal then its z transform X(z) is defined as follows: -
8
X(z) = S x[n]. z
-n
n=-8
Note: Every transform is of the form ?x(t).k(s,t)dt.
For example of discrete Fourier transforms
8 8
X(re
j?
) = S x[n].(r.e
j?
)
-n
= S x[n].e
-j?n
.r
-n
n=-8
8
n=-8
If r = 1 then, X(re
j?
) = S x[n].e
-j?n
n=-8
ROC of Z transform:
ROC (region of convergence) of Z transform is the region in the x-y plane where Z
transform is valid. It dose not have any pole. It is a ring in z- plane centered about
origin. If x[n] is of finite duration then ROC in the entire Z plane except possibly z =
0 and/or z = 8.
For e.g. X(z) = 1/z+a is Valid if z > -a and we draw a circle with center origin and
radius |a| in the z plane and if the transform is valid for values greater than a then the
ROC is exterior of the circle and if less than a then interior of the circle.. If there are
more than one root the overlapping area is the ROC.
For e.g. X(z) = 1/((z+a)(z+b)) where |a| < |b| say then,
Initial value theorem:
If x[n] = 0 for n < 0 then x[0] = Lim
z->8
X(z)
Page 4


Poles and zeros:
Poles are the roots of the denominator of the fraction in the Laplace transform
and zeroes are the roots of numerator of the fraction. For example as follows: -

 




















where a
1
, a
2
, …, an are known as zeroes and b
1
, b
2
, …. , bn are known as poles.
They are represented in a pole zero diagram as
Properties of Laplace transform:
If x(t), x
1
(t), and x
2
(t) are three signals and X(s), X
1
(s), X
2
(s) are their Laplace
transform respectively and a, b are some constant then,
1. Linearity: ax
1
(t) + bx
2
(t) = aX
1
(s) + bX
2
(s)
2. Time shifting: x(t-t
0
) = e
-sto
.X(s)
3. Shifting in s domain: e
sto
x(t) = X(s-s
0
)
4. Time scaling: x(at) = 1/|a| X(s/a)
5. Conjugation: x*(t) = X*(s*)
6. Convolution: x
1
(t)*x
2
(t) = X
1
(s).X
2
(s)
7. Differential in time domain: dx(t)/dt = sX(s)
8. Differential in frequency domain: -tx(t) = d/ds X(s)
9. Integration in time domain: Integration in time domain is division in frequency
domain.
t
i.e. ? x(t)d(t) = 1/s . X(s)
-8
Applications of Laplace transform:
• For a system with a rational system function causality of the system is equivalent
to the ROC being to the right half plane to the right of the right most pole.
• An LTI system is stable if and only if the ROC of its system function H(s) include
the entire j? axis. (i.e. Re(s) = 0)
Laplace transform pairs:
x(t) X(s) ROC
x(t)
Z
x(t)e
-st
dt (def.)
d(t) 1 all s
u(t)
1
s
Re(s) > 0
e
-at
u(t)
1
s+a
Re(s) >-a
cos(?ot)u(t)
s
s
2
+?
o
2
Re(s) > 0
sin(?ot)u(t)
?o
s
2
+?
o
2
Re(s) > 0
e
-at
cos(?ot)u(t)
s+a
(s+a)
2
+?
o
2
Re(s) >-a
e
-at
sin(?ot)u(t)
?o
(s+a)
2
+?
o
2
Re(s) >-a
; a is real.
Complex arithmetic operations:
Z-transform and discrete Fourier transform
Z-transform definition:
If x(t) is the signal then its z transform X(z) is defined as follows: -
8
X(z) = S x[n]. z
-n
n=-8
Note: Every transform is of the form ?x(t).k(s,t)dt.
For example of discrete Fourier transforms
8 8
X(re
j?
) = S x[n].(r.e
j?
)
-n
= S x[n].e
-j?n
.r
-n
n=-8
8
n=-8
If r = 1 then, X(re
j?
) = S x[n].e
-j?n
n=-8
ROC of Z transform:
ROC (region of convergence) of Z transform is the region in the x-y plane where Z
transform is valid. It dose not have any pole. It is a ring in z- plane centered about
origin. If x[n] is of finite duration then ROC in the entire Z plane except possibly z =
0 and/or z = 8.
For e.g. X(z) = 1/z+a is Valid if z > -a and we draw a circle with center origin and
radius |a| in the z plane and if the transform is valid for values greater than a then the
ROC is exterior of the circle and if less than a then interior of the circle.. If there are
more than one root the overlapping area is the ROC.
For e.g. X(z) = 1/((z+a)(z+b)) where |a| < |b| say then,
Initial value theorem:
If x[n] = 0 for n < 0 then x[0] = Lim
z->8
X(z)