# 2.1 Transistor equations

 Page 1 / 1
Transistor Equations

There are several "figures of merit" for the operation of the transistor. The first of these is called the emitter injection efficiency , $\gamma$ . The emitter injection efficiency is just the ratio of the electron current flowing in the emitter to the totalcurrent across the emitter base junction:

$\gamma =\frac{{I}_{e}}{{I}_{\mathrm{Ee}}+{I}_{\mathrm{Eh}}}$

If you go back and look at the diode equation you will note that the electron forward current across a junction is proportional to ${N}_{d}$ the doping on the n-side of the junction. Clearly the hole current will be proportional to ${N}_{a}$ , the acceptor doping on the p-side of the junction. Thus, atleast to first order

$\gamma =\frac{{N}_{{d}_{E}}}{{N}_{{d}_{E}}+{N}_{{a}_{B}}}$

(There are some other considerations which we are ignoring in obtaining this expression, but to first order, and for most"real" transistors, is a very good approximation.)

The second "figure of merit" is the base transport factor, ${\alpha }_{T}$ . The base transport factor tells us what fraction of the electroncurrent which is injected into the base actually makes it to collector junction. This turns out to be given, to a very goodapproximation, by the expression

${\alpha }_{T}=1-\frac{1}{2}\left(\frac{{W}_{B}}{{L}_{e}}\right)^{2}$

Where ${W}_{B}$ is the physical width of the base region, and ${L}_{e}$ is the electron diffusion length, defined in the electron diffusion length equation .

${L}_{e}=\sqrt{{D}_{e}{\tau }_{r}}$

Clearly, if the base is very narrow compared to the diffusion length, and since the electron concentration is falling off like $e^{\frac{-x}{{L}_{e}}}$ the shorter the base is compared to ${L}_{e}$ the greater the fraction of electrons who will actually make it across. We saw before that a typical value for ${L}_{e}$ might be on the order of 0.005 cm or 50 μm. In a typical bipolar transistor, the base width, ${W}_{B}$ is usually only a few μm and so $\alpha$ can be quite close to unity as well.

Looking back at this figure , it should be clear that, so long as the collector-base junction remains reverse-biased, the collectorcurrent ${I}_{\mathrm{Ce}}$ , will only depend on how much of the total emitter current actually gets collected by thereverse-biased base-collector junction. That is, the collector current IC is just some fraction of the total emitter current ${I}_{E}$ . We introduce yet one more constant which reflects the ratio between these two currents, and call it simply" $\alpha$ ." Thus we say

${I}_{C}=\alpha {I}_{E}$

Since the electron current into the base is just $\gamma {I}_{E}$ and ${\alpha }_{T}$ of that current reaches the collector, we can write:

${I}_{C}=\alpha {I}_{E}={\alpha }_{T}\gamma {I}_{E}$

Looking back at the structure of an npn bipolar transistor , we can use Kirchoff's current law for the transistor and say:

${I}_{C}+{I}_{B}={I}_{E}$
or
${I}_{B}={I}_{E}-{I}_{C}=\frac{{I}_{C}}{\alpha }-{I}_{C}$

This can be re-written to express ${I}_{C}$ in terms of ${I}_{B}$ as:

${I}_{C}=\frac{\alpha }{1-\alpha }{I}_{B}\equiv \beta {I}_{B}$

This is the fundamental operational equation for the bipolar equation. It says that the collector current is dependent onlyon the base current. Note that if $\alpha$ is a number close to (but still slightly less than) unity, then $\beta$ which is just given by

$\beta =\frac{\alpha }{1-\alpha }$
will be a fairly large number. Typical values for a will be on the order of 0.99 or greater, which puts $\beta$ , the current gain, at around 100 or more! This means that we can control, or amplifythe current going into the collector of the transistor with a current 100 times smaller going into the base. This all occursbecause the ratio of the collector current to the base current is fixed by the conditions across the emitter-base junction, andthe ratio of the two, ${I}_{C}$ to ${I}_{B}$ is always the same.

#### Questions & Answers

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
characteristics of micro business
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
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!