There
is a way that we can
make things a good bit easier for ourselves however. The onlydrawback is that we have to do some complex analysis first, and
look at a
bilinear transform ! Let's do one more
substitution, and define another complex vector, which we cancall
:
The vector
is just the rotating part of the crank diagram which
we have been looking at
. It has a
magnitude equal to that of the reflection coefficient, and itrotates around at a rate
as we move down the line. For every
there is a corresponding
which is given by:
The vector r(s)
Now, it turns out to be easier if we talk about a
normalized
impedance , which we get by dividing
by
.
which we can solve for
This relationship is called a
bilinear
transform . For every
that we can imagine, there is one and only one
and for every
there is one and only one
. What we would like to be able to do, is find
, given an
. The reason for this should be readily
apparent. Whereas, as we move along in
,
behaves in a most difficult manner (dividing one
phasor by another),
simply rotates around on the complex plane. Given one
it is
easy to find another
. We just rotate around!
We shall find the required relationship in a
graphical manner. Suppose I have a complex plane, representing
. And then suppose I have some point "A" on that plane
and I want to know what impedance it represents. I just readalong the two axes, and find that, for the example in
, "A" represents an impedance of
. What I would like to do would be to get a grid
similar to that on the
plane, but on the
plane instead. That way, if I knew one impedence (say
then I could find any other impedance, at any other
, by simply rotating
around by
, and then reading off the new
from the grid I had developed. This is what we shall
attempt to do.
The complex impedance plane
Let's start with
and re-write it as:
In order to use
, we are going to have to
interpret it in a way which might seem a little odd to you. Theway we will read the equation is to say: "Take
and add 1 to it. Invert what you get, and multiply by
-2. Then add 1 to the result." Simple isn't it? The only hardpart we have in doing this is inverting
. This, it turns out, is pretty easy once we learn one
very important fact.
The
one fact about algebra
on the complex plane that we need is as follows. Consider avertical line,
, on the complex
plane, located a distance
away
from the imaginary axis
. There are a lot
of ways we could express the line
, but we will choose one which
will turn out to be convenient for us. Let's let:
A vertical line, s, a distance, d, away from the
imaginary axis
Now we ask ourselves the question: what is the inverse of s?
We can substitute for
:
And then, since
A plot of 1/s
A careful look at
should allow you to
convince yourself that
is an equation for
a circle on the complex plane, with a diameter
. If
is not parallel to
the imaginary axis, but rather has its perpendicular to theorigin at some angle
, to make a line
. Since
, taking
simply will give us a circle with a diameter of
, which has been rotated by an angle
from the real axis
. And so we come to the
one fact we have to keep in mind:
"The inverse of a
straight line on the complex plane is a circle, whose diameteris the inverse of the distance between the line and the
origin."
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
