Gain, Gain?  What's this "Gain" Anyway?

        When you use an operational amplifier in a circuit you often need to remember that the operational amplifier has a very high gain.  What is gain, anyway?  There's a very simple answer.

• Gain is normally defined as the ratio of the output of a device/system/amplifier to the input of said device/system/amplifier.

In an operational amplifier, we have a relationship between the output voltage and the input voltages at the input terminals of the operational amplifier.

• Op-amp output voltage = A (V₊ - V_-) where

• A is the gain - usually a pretty large number, often greater than 100,000 or 200,000.

• V₊  is the voltage at the non-inverting input (measured to ground).

• V_- is the voltage at the inverting input (measured to ground).

• The difference between the two inputs (V₊ and V_-) is the real input to the operational amplifier.  In other words, that difference is the thing that actually gets amplified.  And in an op-amp it gets amplified a lot since A is a "really big" number.  It's so big that we often treat the circuit as though A was infinite.

        In the case of the inverting amplifier (Which is an operational amplifier circuit.  In other words, it is a circuit which uses an op-amp and some other circuitry.) we might use the word "gain" to describe something entirely different.  Here's an inverting amplifier with lots of variables defined.  We want to focus on the input voltage, V₁ and the output voltage, V_out.

If you analyze this circuit, you can find that the output voltage and the input voltage are related by this equation.

In this expression, we can see that the input voltage is multiplied by a factor of  -R₀/R₁.  For the circuit, the gain is -R₀/R₁.  That gain might have a value of -10 or something like that.

• The gain of the circuit is always going to be smaller than the gain of the operational amplifier itself.

• The gain of the circuit is negative.  If you have a philosophical problem with the gain being negative, remember that all that means is that this circuit (and some others, but not all others) changes the sign of the signal that enters the circuit.

• So, if you put in +0.7 volts and the gain is -10, you will get out -7.0 volts.

        You need to distinguish between the gain of the operational amplifier (which is large and postive) and the gain of the circuit which is much smaller and can be negative (like it is for this circuit).

        Now, if you followed all of that, you might be tempted to assume that gain has no units.  That is not always true.  In the circuits above, you put in a voltage and you get out a voltage.  When you divide the output by the input, the units go away.  If you have a motor, you put in a voltage you get out so many rpm.  The gain has units of rpm/volt.

        And, that's the end of this story.