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This lecture is part of lecture series on Electrical Circuit Analysis course. It was delivered by Prof. Mursleen Sayed at Bengal Engineering and Science University. It includes: Impulse, Function, Discontinuous, Discontinuity, Abrupt, Derivative, Parameter, Variable, Duration, Amplitude
Typology: Slides
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^
Derivative of u(t) at t=0 doesn’texist ^
Discontinuous function
^
Assume function varies linearlyacross the discontinuity ^
As
0 abrupt discontinuity
occurs at origin ^
Derivative between
& -
^
is
0.5/
^
Derivative for t >
is –ae
-a(t-
)
t
f(t) 1
(^5). 0 5 t
. 0
) t( a
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^
(t) =du(t)/dt
t
f(t) 1
(^5). 0
5 t
. 0
) t( a e
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^
Amplitude approaches infinity ^
Duration of function approaches zero ^
Area under the variable-parameter function is constant asthe parameter changes
f(t) = 0.5t/
^
(linear
function) ^
as
approaches 0, function becomes
^
time decays to 0 ^
Area under the function isindependent of
(^5). 0 (^5). t 0
) t( a e^
Area = ^
Therefore as
0, f(t)
(t)
^
Mathematically Impulse Function is defined as
(t) = 0, t
dt 0 e K 2
dt e K 2
/t
0
/t
0
^
e K 2
e / 1 K 2
0 /t
0 /t
^
Function f(t) iscontinuous at t = a atlocation of pulse
)t( δK^
)a t(δ K^
)a (f
dt) a t( δ) t( f^
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Sifting property of the impulse function isused to find its Laplace transform
ℒ
{
(t)} =
This is an important Laplace transformused in circuit analysis.
0
st
0
Derivatives of the ImpulseFunction ^
The impulse functiongenerates an impulsefunction as
0
^
Defined as derivative of animpulse ^
[’(t)] as
0
^
Derivative of an impulsefunction is referred to as amoment function or unitdoublet
f(t)
t
0
/^1 (^2) /t
^
/ 1 (^2) /t
f(t)
t
0
2 / 1
2 / 1
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^
All functions zero for t < 0
^
ℒ
{u(t)} =
^
ℒ
{e
-at
} =
s/ 1
s e dt e 1
dt e) t( f
0 st
0
st
0
st^
)s a/( 1
)s a( e
dt
e
dt e e
0 t)s a(
0
t)s a(
0
st at^
Laplace transform of sin
t
^
dt e
e j 2
e
dt e) t ω (sin
0
st tωj
tωj
0
st
2
2
0
t) ωj s(
t) ωj s(
ω s
ω
ωj s
ωj s
1 j 2
dt
e j 2
e
^
