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The operational amplifier has a very high input impedance which makes it very good for amplifying low voltage signals.
Basically the OpAmp is a function like this:
\displaystyle Y = A_v (X_1 - X_2)
Where:
\begin{flalign}
&Y = \text{Output Voltage}&\\\
&A_v = \text{Open Loop Gain}\\
&X_1 = \text{Input V1 (Non Inverting Input)}\\
&X_2 = \text{Input V2 (Inverting Input)}\\
\end{flalign}
Rules
1. No Current flows in or out of the outputs 2. The op-amp tries to keep the input voltages the same The second rule only applies when the op-amp is in closed loop configuration
Regions
Op Amps functions in different regions, just like diodes, and transistors.
Regions
Linear Region This is how the Op-Amp normally functions.
Saturation Region
When the output of the op-amp would be higher than +V_{CC} or lower than -V_{CC} the output value is clamped to those values.
In real life OpAmps have A_V values as high as 10^8 or 10^9 due to this even very small input voltages would quickly leave the linear region. That is why we need
Negative Feedback
To use negative feedback we connect the output of the OpAmp to one of its inputs. This connection is modified by a feedback factor (\beta) which can be in the range 0 \le \beta \le 1.
Due to this feedback the new formula for the output V_O is now:
\begin{flalign}
&V_o = A_V * V_\Delta&\\\
\\
&V_- = \beta * V_o\\
&\text{Now we can say that }V_\Delta \text{is equal to:}\\
&V_\Delta = V_+ - \beta *V_o &| \textit{ Solve for }V_o \\
&V_o = \frac{V_+ - V_\Delta}{\beta}
\end{flalign}
Configurations
Open Loop When the output of the Op Amp is not connected to any of its inputs, it is in the so called "open loop configurations"
Closed Loop
When we connect the output of the OpAmp to either V_+ or V_- the OpAmp is in the "closed loop configuration".
Bandwidth Limitations
Real op-amps behave differently depending on the input signals frequency. Usually the internal open-loop gain gets lower as the input frequency gets higher like this.
The op-amps bandwidth is the frequency range in which the voltage gain is above 70.7% (3dB) of its maximum output. The point at which it is below that gain, is called the breakpoint.
This is also one of the reason we use op-amps in closed loop configuration. Because it allows is to trade maximum gain for a larger bandwidth.
!op-amp-bandwidth-closed-loop.png
If we want to find out the bandwidth of an op-amp, we can check the datasheet. The LM741 OpAmp for example:
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The thing is that this frequency only applies when the op-amp has a gain of 1, this frequency point is also called unity gain. It is called the Gain Bandwidth Product, which is calculated as follows:
GBP = A_V * f_c
Where A_V is the voltage gain, and f_c is the cutoff frequency.
With this equation we can also solve for f_c like so:
\displaystyle f_c = \frac{GBP}{A_V}