Poles and Zeros Video Lecture - Electrical Engineering (EE)

so by now you should be a pro at
modeling we've gone over how to model
systems using the transfer function and
also in the state space domain so now
we're going to try to use those models
what can they tell us about the system
and to start with there's a few
different ways to break down a system
we're going to start with transfer
functions so in the time frequency
domain and we're going to look at the
poles and zeros of system so we're going
to define that right now so in a
transfer function we usually have
so we'll say have a plant some transfer
function we have usually a numerator and
a denominator and usually it's something
let's say s a s squared plus B plus C
and the bottom you have something that
alpha I squared plus beta s plus I'm
just going to use gamma because I don't
know what the C is Greek all right so we
have some system here sorry equals and
we are going to use some expressions
here we're going to look at the poles
and zeros of the system so what are
those things well okay what is a pole
alright so a pole is essentially the
value of s that makes this system the
transfer function go to infinity so
valdés that makes a transfer function
it's going to use TF here transfer
function go to infinity okay
so that would pretty much mean the
values of s that are the roots of the
denominator okay
and so we denote a pole as an X and I'll
show you how we plot that in a minute
and then the zeroes are essentially the
opposite it is the value of s that makes
the transfer function go to zero so
essentially it's going to be the
roots of the numerator here so the value
of s that makes this becomes zero
or the value so that's the zero and we
denote that as a circle and then the
poles are the values of F s that make
this expression the denominator zero so
let's just do a quick example so these
are the definitions we're going to take
an example we're going to use G of s and
we're going to say we did our model now
we figured out a system and we end up
with this transfer function s plus 2 in
the numerator and we put in our standard
form so we had s squared plus 5x so if
this is our starting function we can
find the poles and 0 a bit so first
let's find actually the spread the zeros
first so the zeros is going to be
whatever makes it 0 so s plus 2 equals 0
relatively easy if we can see that s
equals negative 2 is a 0 so this would
be our answer for the 0 and then what
about our poles so we take this
expression and we want to make it 0 so x
squared plus 5 s equals 0 we see you can
take an S out of here so we have s times
s plus 5 is equal to 0 so here we
actually have two poles so if s is equal
to 0 then that is one of the poles or if
s is equal to negative 5 we have that
expression is also equal to 0 so our
poles here would be 0 and negative 5 and
our zeros here would be negative 2 and
what we can do is actually map these on
a what we call a pole
zero plot so this is our I'm going to
write up here pull zero plot and
essentially all we're doing is mapping s
on the real and imaginary axis imaginary
that looks weird right the real and
imaginary axis right so all we have to
do is use these appropriate symbols to
map them correctly so we have a zero at
negative two so we go to we'll say this
is do this right here is negative five
five here and stop saying here okay so
we have negative two is one of our zeros
there's no imaginary component it's all
real so we're just going to keep on the
real axis and we go to negative two and
we put a circle that's our zero at
negative two and then for the poles
remember we use an X we have s equals
zero so again only real and full no
imaginary or real components though just
an X right at the origin and then s
equals negative five again only a real
component no imaginary so we would put
an X out here so this is how we would
plot our poles and zeros on the pole
zero plane and we're going to talk about
this a lot more later but we're going to
use these values to determine how the
system acts so these values can tell us
quite a bit about how we react to the
system reacts to a step function or to
various changes so we'll look at that
later but right here we just use our
system find these zeros and poles and
then plot them on this type of plot