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This document examines how signals behave as they travel along transmission lines. It also introduces distributed parameters.

Having learned something about how we generate signals with bipolar and field effect transistors, wenow turn our attention to the problem of getting those signals from one place to the next. Ever since Samuel Morse (and thefounder of my alma mater , Ezra Cornell) demonstrated the first working telegraph, engineers andscientists have been working on the problem of describing and predicting how electrical signals behave as they travel downspecific structures called transmission lines .

Any electrical structure which carries a signal from one point to another can be considered a transmissionline. Be it a long-haul coaxial cable used in the Internet, a twisted pair in a building as part of a local-area network, acable connecting a PC to a printer, a bus layout on a motherboard, or a metallization layer on a integrated circuit,the fundamental behavior of all of these structures are described by the same basic equations. As computer switchingspeeds run into the 100s of MHz, into the GHz range, considerations of transmission line behavior are ever morecritical, and become a more dominant force in the performance limitations of any system.

For our initial purposes, we will introduce a "generic" transmission line , which will incorporate most (but not all) features of real transmissionlines. We will then make some rather broad simplifications, which, while rendering our results less applicable to real-lifesituations, nevertheless greatly simplify the solutions, and lead us to insights that we can indeed applyto a broad range of situations.

"generic" transmission line

The generic line consists of two conductors. We will suppose a potential difference V x exists between the two conductors, and that a current I x flows down one conductor, and returns via the other. For the time being, we will let the transmission line be"semi-infinite", which means we have access to the line at some point x , but the line then extends out in the x direction to infinity. (Such lines are a bit difficult to handle in the lab!)

In order to be able to describe how V x and I x behave on this line, we have to make some kind of model of the electrical characteristics of the line itself. We can not just make up any model we want however;we have to base the model on physical realities.

Let's start out by just considering one of the conductors and the physical effects of current flowing though thatconductor. We know from freshman physics that a current flowing in a wire gives rise to a magnetic field, H ( ). Multiply H by and you get B , the magnetic flux density, and then integrate B over a plane parallel to the wires and you get , the magnetic flux "linking" the circuit. This is shown in for at least part of the surface. The definition of L , the inductance of a circuit element, is just

where is the flux linking the circuit element, and I is the current flowing through it. Our only problem in finding is that the longer a section of wire we take, the more we have for the same I . Thus, we will introduce the concept of a distributed parameter.
distributed parameter
A distributed parameter is a parameter which is spread throughout astructure and is not confined to a lumped element such as a coil of wire.

Build up of magnetic field

Likewise, if we have two conductors separated by some distance, and if there is a potential difference V between the conductors, thenthere must be some charge Q on the two conductors which gives rise to that potential difference. We can imagine a linear charge distribution on thetransmission line, (C/m), where we have Coulombs/m on one conductor, and Coulombs/m on the other conductor. For a line of length x 0 , we would have Q x 0 on each section of wire. Whenever you have two charged conductors with a voltage difference between them, you candescribe the ratio of the charge to the voltage as a capacitance. The two conductors would have a capacitance
C Q V x 0 V
and a distributed capacitance C (F/m) which is just V . A length of line x 0 long would have a capacitance C C x 0 Farads associated with it .

Find the flux linkage

Line capacitance

Thus, we see that the transmission line has both a distributed inductance L and a distributed capacitance C which are tied up with each other. There is really no way in which we can separate one from the other. In other words, we cannot have only the capacitance, or only the inductance, there will always be some of each associated with each section of linenow matter how small or how big we make it.

We are now ready to build our model. What we want to do is to come up with some arrangement of inductors andcapacitors which will represent electrically, the properties of the distributed capacitance and inductance we discussedabove. As a length of line gets longer, its capacitance increases, so we had better put the distributed capacitances inparallel with one another, since that is the way capacitors add up. Also, as the line gets longer, its total inductanceincreases, so we had better put the distributed inductances in series with one another, for that is the way inductances addup. is a representation of the distributed inductance and capacitance of the generic transmission line.

Distributed parameter model

We break the line up into sections x long, each one with an inductance L x and a capacitance C x . If we halve x , we would halve the inductance and capacitance of each section, but we'd have twice as many of them per unitlength. Duh! The point is no matter how fine we make C x , we still have Ls and Cs arranged like we see in , with the two kinds of components intermixed.

We could make a more realistic model and realize that all real wires have seriesresistance associated with them and that whatever we use to keep the two conductors separated will have some leakage conductanceassociated it. To account for this we would introduce a series resistance R (ohms/unit length) and a series conductance G (ohms/unit length). One section of our line model then looks like .

Complete distributed model

Although this is a more realistic model, it leads to much more complicated math. We will start out anyway,ignoring the series resistance R and the shunt conductance G . This "approximation" turns out to be pretty good as long as eitherthe line is not too long, or the frequencies of the signals we are sending down the line do not get too high. Without theseries resistance or parallel conductance we have what is called an ideal lossless transmission line .

Questions & Answers

how can chip be made from sand
Eke Reply
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are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
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where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
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Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Source:  OpenStax, Communications b : filters and transmission lines. OpenStax CNX. Nov 30, 2012 Download for free at http://cnx.org/content/col11169/1.2
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