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Previously we saw that in order to have power
amplification or power gain, we need to have
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a nonlinear circuit and we also need an additional
source, because every element is passive.
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If you just have a single source which delivers
the signal power then what comes out, the
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power that comes out is going to be smaller
than the input signal power. So, we need to
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have a nonlinear device as well as two sources;
one is the source which provides the desired
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signals and also another is sort of a power
supply usually it is a dc source. And somehow
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some of the power that is taken from the dc
source will be converted to useful form in
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the form of the signal. So, that is how we
get power gain.
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Now, before we can discuss amplifier devices,
we have to learn how to analyze nonlinear
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circuits, because this is usually not what
is taught in basic circuit classes. So, we
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will go step by step and the analysis of nonlinear
circuits is not as easy as linear circuits,
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but we can make some approximations and make
some progress with analyzing nonlinear circuits.
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Now, eventually we want to have two port elements
that is an element where you can provide an
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input from one side and take the output from
the other side. But because we are yet unfamiliar
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with nonlinear circuits, we will first consider
nonlinear one port and then go to nonlinear
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two ports. But the general method of analysis
shown with the nonlinear one port will also
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be applicable to nonlinear two ports. Now,
I will represent a general nonlinear one port
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like this. If it has voltage V across it,
it has certain current I through it; and I
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will be some function of V where this f is
nonlinear. Of course, in case of resistor,
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which is linear we have a similar relationship
where I is G times V where G is the conductance.
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So, the single number conductance is describes
the resistor; and in case when nonlinear element
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the function f describes the nonlinear element.
Now of course, we can talk in abstract terms
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about a general nonlinear element but it is
also useful to learn about a very frequently
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used one port nonlinear element which is the
diode. So if the voltage across this is V
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D, the current I D through it will be given
by some nonlinear function. I am going to
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show that in a moment, but one thing I have
to show is the diode is represented by a symbol
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like this and it is asymmetrical. So this
side is known as the anode or the plus terminal
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and that side is the cathode. And while describing
the characteristic, this V D is defined with
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the anode being positive and I D is defined
with the current going into the anode, so
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that’s the usual passive sign convention.
So with this polarity, this I D will be I
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s exponential V D divided by V t minus 1.
Now this I S is called the saturation current,
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it is some property of the diode; it is somewhat
similar to the conductance of the resistor.
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So you can have a resistor with different
conductance similarly you can have diodes
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with different saturation currents. And the
saturation current can vary widely just like
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you can have one ohm resistor or one mega
ohm resistor, but I just give you some value
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which you can find this I s itself will be
a very small number ten to the minus fifteen
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amperes.
And there is another constant here which is
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V t, and this is not really a property of
the diode, but it is the fundamental constant
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and it is given by k T by q, where k is Boltzmann’s
constant you would be familiar with this and
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T is absolute temperature and q of course
is the electron charge. And this number k
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T by q, this is known as the thermal voltage,
obviously it has dimensions of volts and you
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have see that it must be like that to balance
the dimensions here, because V D is volt,
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V t also has to be in volts so that the argument
of this exponential is dimensionless. Now
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of course this is something that is directly
proportional to the absolute temperature and
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it turns out that at room temperature, this
V t is k T by q and which equal to 25.9 milli
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volts. And we will frequently approximated
with just 25 milli volts. So this is the number
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worthwhile remembering, the thermal voltage
V t is 25 milli volts at room temperature.
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So sometimes it is approximated to twenty-six;
or sometimes, we use 25.9, but this is good
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enough 25 milli volts.
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So now let see what this characteristic looks
like. As usual as for any element, we can
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also graphically depict the characteristic,
I D versus V D; and we said I D is I s exponential
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V D by V t minus 1. It is some nonlinear curve,
but we can see how it behaves. So first of
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all, when V D is very negative that is V D
by V t is large negative number. This exponential
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of large negative number is close to zero;
of course, it will never be exactly zero,
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but it will be very small. So in this parenthesis,
you will just have minus 1, so I D will be
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approximately minus I s when minus V D that
is the negative value of V D is much more
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than V t. So what I am saying here is V D
is large and negative. So I D approximately
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minus I s, it means that I D will be something
like this, and V D is large and negative,
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and this number is minus I s. So what it means
is in this region when V D is negative by
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the way that region is called reverse bias,
the current will be almost constant and it
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will be equal to minus I s. And in fact this
is the reason why I s is called the saturation
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current or the reverse saturation current.
Now when V D equal zero, this number is one,
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and I D will be equal to zero, so that means
that this I D will be zero when V D is zero.
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And as V D becomes positive, this exponential
grows rapidly with V D, and the current increases
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rather steeply, something like this. So this
region where V D is greater than zero, significant
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currents can flow and this is known as forward
bias. And V D less than zero, the current
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is very small; we saw that I gave you typical
value of I s which is ten to the minus fifteen
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amperes, so it is very small and more than
the absolute value of that the significant
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thing is that when V D is more than zero the
current is much, much more than when V D is
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less than 0. So when we say small and large
currents, we have to relate it to something,
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what we mean is that the reverse bias currents
are much smaller than forward bias currents.
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This is reverse bias.
And for V D equal to 0, I D will be equal
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to 0. In fact, this is the condition that
is necessary for passivity. We of course know
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that resistor characteristic would look like
this right, it will be a straight line passing
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through the origin and it will have the slope
of conductance or the reciprocal of resistance,
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and it is passive. Now you must know this
from your basic electrical circuit is courses.
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If the I D characteristic of a one port or
a two terminal element is in the first or
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third quadrant, it is dissipating power, because
v times i is positive. Here in the first quadrant
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both v and i are positive; in the third quadrant,
both v and i are negative. So in both of these,
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it will be dissipating power. In these quadrants
– second and fourth quadrants v times i
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will be negative and the element will be generating
power. So if you have a passive element, it
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cannot have any part of its characteristic
in either the second or fourth quadrants.
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So passive element means the
characteristic only in first and third quadrants;
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and this of course implies that it has to
pass through the origin. So now this is for
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a diode and some array diode has a very small
current in reverse bias that is when the diode
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voltage is smaller than zero and note that
the diode voltage is defined with the specific
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polarity and the symbol of diode is asymmetrical.
So please keep that in mind while figuring
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out forward and reverse bias. And when it
is in forward bias, it can have significant
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currents. Now of course, in general nonlinear
element can have arbitrary characteristics,
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but if it is a passive nonlinear element,
it is going to have characteristic which are
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going to pass through the origin and be only
in the first and third quadrants, it can be
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anything, so in general f of v. So this is
just an introduction to a two terminal nonlinear
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element, and we have seen the characteristic,
we can describe it with an expression or we
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can draw a graph.