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."
The line s'
The line
multiplied by
Inverse of a rotated line
Questions & Answers
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
