Jon Slaughter said:
He's simply taking into account that a real op amp does not have infinite
gain... a needless assumption in this case.... don't worry about it. If
you can't do it for the ideal case you can't do it for the non-ideal case.
Well, yes, I was trying to get into the non-ideal case, because I
think it's important that people eventually see where all of these
magic formulas actually come from. If we take the usual
inverting-amp configuration with the non-inverting input grounded
(in other words, I'm going to ignore what was originally called
"V2" here as a needless complication at this point - all it will really
do is offset the output, anyway), the analysis really isn't all
that difficult.
We'll call the input (at the "free" end of R1) Vin, the output Vout,
and the voltage across the op-amp inputs Va. The "ideal"
analysis would have you assume that Va is zero, or call the
inverting input a "virtual ground," or some such, without really
explaining why you do that. But again, it's really not all that hard
to go through the full analysis, and what you'll wind up with is
the general formula for any amplier used in this configuration, no
matter how big the gain is.
We will, however, make the assumption that no current enters the
inverting input (i.e., the input impedance is way bigger than anything
else involved, such that any current into the input is negligible), and
that makes the summing of currents at this node very simple:
(Vin - Va)/R1 = (Va - Vout)/R2
But by definition, the output voltage must be the amp gain times
the voltage at the amplifier's input, which is Va, so:
If Vout = A(Va), then Va = Vout/A, and
(Vin - Vout/A)/R1 = (Vout/A - Vout)/R2
Solving for the overall gain (Vout/Vin), we get
Vout/Vin = R2 / [(1/A)*(R1/R2 + 1) - R1]
Note where the open-loop gain term "A" (i.e., the gain of the amplifier
itself, without the R2 and R1 external components) winds up. If A
gets very large, the 1/A term will approach zero, and take the whole
first part of the denominator with it. If that's the case (and it's a
reasonable
assumption for an op-amp, where the open-loop gain is very commonly
in the tens if not hundreds of thousands), then that first term drops
out completely, and the whole thing winds up with the familiar
Vout/Vin = - (R2/R1)
Simple, no?
Bob M.
