Singularity
Functions
Singularity functions are also called switching
functions. These are very useful in circuit analysis. They are very help use in
neat and compact description of some
circuit phenomena especially step
response of RC and RL circuits…………………
“Singularity
functions are function that either are discontinues or have discontinues
derivatives.”
Three mostly used singularity functions are:
1. Unit
step Functions u(t)
2. Unit
impulse functions d(t)
3. Unit
ramp function r(t)
Unit step functions
u(t)
The Unit step
function u(t) is 0 for negative value of t and u(t) is 1 for positive value of
t
Mathematical form is:
0, t<0
u(t)
=
1, t>0
As at t=0 there is an
abrupt change in the function
So unit step function
is un defined at t=0 t
Unit
step function is dimensionless
Delay of Unit step Function:
If the
Abrupt change occurs at t=to
Instead
of t=0 then we can say that unit step
function is delay by to
seconds.
Then we
replace every t by (t - to) and unit step
function become:
0,
t < t0
u(t – t0)
=
1, t > t0
Advance of unit step function:
If the
Abrupt change in input occurs at t=-to
Instead of t=0 then
we can say that unit step function is Advance by to
seconds
Then we
replace every t by (t + to) and unit step
function become:
0, t <- t0
u(t + t0)
=
1, t >- t0
Applications of Step Function:
We use the step function to represent an Abrupt change in voltage or
current and this phenomena is used in Computers and Control Systems
For
example: the voltage
0, t< t0
V(t)
=
V0, t>t0
Where
t0 may be any value for which
we want to on the switch
we may
express V(t) in this term……….
V(t)=V0 u(t –t0)
0, t< t0
V(t)= V0
1, t>t0
When t0=0
then V(t)=V0 u(t)
Unit impulse Function d(t)
“The derivate of
unit step function is known as delta Function”
It is also known as delta
function d(t)
0, t<0
d(t) = Undefined t=0
0, t>0
1.
Unit
impulse function is zero whenever except at t=0
where it is undefined
2.
Unit
impulse function is like an ideal source ( it is not physically realizable)
3.
It
is very useful tool in mathematics
4. It
may be visualized as a very short duration pulse of unit area
Mathematically: when we take integration
from-0 to +0
Where t=-0 is time just before t=0 and t=+0 is time just after
the t=0
In
this 1 is not magnitude it denoting unit area
The
unit is known as the strength of the impulse function
We
an impulse function has area other then unity, then area of
Impulse
equal to is strength.
For example:
10d(t)
For impulse function
10 d(t) it has area 10
Strength
of impulse function
Unit impulse function
can not apply practically because it is an impulse for undefined time
may
be for 109 second
A impulse may be in
positive cycle and may be in negative cycle
(like for impulse -4d(t-3)
)
Unit Ramp Functions r(t)
Integration
of unit step function is called Unit ramp function
0, t≤0
r(t)=
t, t≥0
“The unit ramp function is zero for
negative values of t and has a unit slope for positive values of t”
In
general, a ramp is a function which is changing at a constant rate.
for more information you download this pdf file
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Thank
you
