# 1.3 Telegrapher's equation in real lines

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Explains the behavior of "real lines" in terms of the telegrapher's equation.

So far, the transmission lines we have looked at have been "ideal". That is they have been lossless anddispersionless. Lest you leave the course with a false idea of how things really work, we should go back to our model and try to get things adjusted just a bit.

As you can probably imagine, a real transmission line is going to have some series resistance, associated with the real lossesin the copper wire. There may also be some shunt conductance, if the insulating material holding the two conductors has someleakage current. We will need to include these effects along with the distributed inductance and capacitance which we havealready talked about. Fixing up the model accordingly, we now draw a section of line $(x)$ long as shown in . Taking the same voltage loop and current sum that we did back in thediscussion of transmission lines , we come up with the following version of the telegrapher's equations .

$\frac{\partial^{1}V(x, t)}{\partial x}=-RI(x, t)-L\frac{\partial^{1}I(x, t)}{\partial t}$
and
$\frac{\partial^{1}I(x, t)}{\partial x}=-(GV(x, t)+C\frac{\partial^{1}V(x, t)}{\partial t})$

Clearly, we would like to simplify things if we can. Let's again make a sinusoidal time excitation assumption, and let $I(x, t)$ and $V(x, t)$ become phasors. Since the time variation is now represented by a simple $e^{iL}$ the time derivatives become just $i$ . We have

$\frac{\partial^{1}V(x)}{\partial x}=-((R+iL)I(x))$
and
$\frac{\partial^{1}I(x)}{\partial x}=-((G+iC)V(x))$

The way to get a solution is, of course, just like we have always done. Take the derivative with respect to $x$ of

$\frac{\partial^{2}V(x)}{\partial x^{2}}=-((R+iL)\frac{\partial^{1}I(x)}{\partial x})$
and then plug in
$\frac{\partial^{2}V(x)}{\partial x^{2}}=(R+iL)(G+iC)V(x)$

The obvious solution to this (See how easy this gets after you've done it once or twice) is

$V(x)={V}_{0}e^{(x)}$
with
$=\sqrt{(R+iL)(G+iC)}$

This number,  is called the complex propagation constant . Obviously, in general, it will have both a real and an imaginary part:

$=+i$
and we have
$V(x)={V}_{0}e^{((+i)x)}$
Let's choose the minus sign in the exponent, and write the twoterms as a product.
$V(x)={V}_{0}e^{-(x)}e^{-(ix)}$
We see we have something similar to what we had before, but with just a minor difference. The $e^{-(ix)}$ term is the propagating term which tells us how the phase angle of the phasor changes as we move along the line, andacts just like the  term we had before. Thus
$=\frac{2\pi }{}$
and
${}_{p}=\frac{}{}$

The  is called the attenuation coefficient , and obviously, the $e^{-(x)}$ term in causes the amplitude of the wave to decrease as it moves down the line. is a sketch of what a wave would look like if it is both propagating down the transmissionline and also being attenuated. In a distance $\frac{1}{}$ the amplitude of the propagating wave has fallen to $e^{-1}$ of the value it had when it started.

Let's take the minus sign solution in and substitute back into

$\frac{\partial^{1}V(x)}{\partial x}=-({V}_{0}e^{-(x)})=-((R+iL)I(x))$
From which we get
$I(x)=\frac{}{R+iL}{V}_{0}e^{-(x)}=\frac{\sqrt{(R+iL)(G+iC)}}{R+iL}V(x)=\sqrt{\frac{G+iC}{R+iL}}V(x)$
Thus we can say
$V(x)={Z}_{0}I(x)$
where
${Z}_{0}=\sqrt{\frac{R+iL}{G+iC}}={R}_{0}+i{X}_{0}$

In general, in order to find  ,  , ${R}_{0}$ , and ${X}_{0}$ , we would have to find the square root given in and for specific values of $R$ , $L$ , $G$ , and $C$ . On the other hand, we could maybe come up with some reasonable approximations which mightsuffice for cases of real interest. Obviously, if a line is very lossy, we would not be very interested in using it, and soexcept in some very special cases where an extremely lossy line is unavoidable (usually having to do with signals at very highfrequencies) we might see if we can find a low loss approximation.

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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