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We have seen how cumbersome it is to analyze
circuits with nonlinear elements.
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What we will do now is to look at approximate
ways of solving nonlinear circuits.
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We have previously considered an example with
a two terminal nonlinear element.
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And we can solve this
numerically or graphically, but the real difficult
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is that for every change in the input, we
have to redo this.
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This by itself is cumbersome and then for
every value of input we have to redo this.
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And of course, when you have time varying
signals in our circuit, the input will be
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changing; so for every point of the time varying
signal, you have to redo the numerical or
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graphical calculation, and this is very difficult.
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At least to get some insight into the circuit
and for hand analysis, we would like to avoid
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these.
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So let us see what we can do.
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I will repeat the graphical solution which
we had got.
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Essentially, we are solving for this equation,
the current in the resistor being equal to
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the current in the diode.
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The current in the resistor is V s minus V
d by R and that is equal to the current in
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the diode, which is the diode saturation current
times exponential V d by V t minus 1.
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So, what we do is we draw a graph for the
right side and the left side and see where
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they meet.
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So, this is the plot of the right hand side,
this is the plot of the left hand side, and
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they meet here, which is the solution.
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Now, as V s changes, we have to repeat this.
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So, for a larger value of V s, in this particular
case the right hand side does not change,
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but the left hand side does, and we have a
new solution, new point of intersection with
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the nonlinear curve.
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The simplification that we will make is the
following.
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You see that the original solution is here,
and the new one is there.
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We will assume that as V s changes, it will
move to a new point on the nonlinear curve,
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but it will not move too far away, that is
this point is close to this point.
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Now, we will see later what close means, the
point is that if you do not move too far away
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on the nonlinear curve, then around this point,
if you do not move too far away on nonlinear
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curve, we can approximate the nonlinear curve
by straight line that is you take any curve,
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you zoom into very small part of it, it will
look like a straight line, so that is the
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principle we use.
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So the idea here is that if we do not move
too far away, let me call this as the original
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solution, this is really just the first solution
that you calculate.
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And this has to be got from nonlinear numerical
or graphical solutions, there is no shortcut
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to this.
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What we are trying to do is at least to save
some trouble while calculating the solution
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for subsequent points when V s changes.
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So, for one value of V s, we calculated exactly
and the other values we approximate this nonlinearity
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by a straight line, assuming that we do not
go too far on the nonlinear curve.
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Now, what is the straight line that we can
use, so it is the tangent of this curve at
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the original point, and the original solution
is known as the operating point of the circuit.
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So, I first show this graphically, because
it is very easy to understand, it can also
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be proved formally from the equations.
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So, we are approximating this curve, which
is the blue curve - the nonlinear one by its
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tangent which is of course this straight line
at the operating point.
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And operating point for now you can consider
it as the very first value that you calculate.
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For some value of V s, you calculated the
exact solution that is the operating point.
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So, now, how does this help, because you know
this is the straight line.
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For all other points, we are computing intersection
of two straight lines and this is very easy.
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In fact, we do not even have to calculate
the intersection explicitly, I will show the
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method by which we will do that, because when
V s changes to this corresponding to this
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red curve the actual solution is that one,
but if you consider the intersection with
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this approximate straight line the approximation
solution is there, and they are sort of close
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to each other.
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And if you consider a point that is closer
to the original operating point, they will
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be even closer to each other.
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So for now we will state the criteria vaguely;
if
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the change
is small then approximating the nonlinear
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curve by its tangent yields good enough solutions.
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So this is what we will do.
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Now this is approximate, but it turns out
this approximation is quite good and lot of
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practical context, so we will use this widely.
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But of course, you must understand the limitations
of this approximation; you must not go too
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far from the operating point.
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And how far is too far very much depends on
the context that you are working in.
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And it also depends on how much accuracy you
want in the first place.
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If you want the solution to be very accurate
then you cannot go too far from the operating
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point, but if crude solutions are ok in some
cases; many cases you just want to estimates,
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so in those cases, you could go further from
the operating point.