I have been looking at AC frequency responses for op-amps and I was wondering how an op-amp's pole and the frequency at the -3dB point are the same? They both mark the location where the gain starts to decrease by -20dB/decade, but I am not sure why.
How are the -3dB point and the pole of an op-amp related?
Because a typical op-amp is dominant pole compensated, it has a single pole in the open-loop transfer function at low frequency, and a -20 dB/decade roll off.
This leads to the concept of "gain-bandwidth". The open loop gain times the open loop pole frequency will be the "gain-bandwidth product". From that you can calculate the new pole position (bandwidth) of the op-amp by just dividing the GBW product by the noise gain (gain as seen from the non-inverting input).
As an example, the old 741 op-amps had a GBW product of 1 MHz. So if I set up my op-amp with a non-inverting gain of 2, my amplifier would have a bandwidth of 500 kHz, and the open loop pole at 10 Hz would shift to a closed loop pole at 500 kHz.
The math works out such that the magnitude of the transfer function at the pole is 0.707 of the low frequency magnitude, which is -3 dB, so a single pole at a given frequency marks the 3 dB point.
They are the same because that's was a convenient place to mark the location of the 'start' of a filter pole and it works out numerically. The pole is convenient because we can use a single time constant to describe where the filter starts
The -3dB point for a low pass RC filter is at the frequency \$1/(RC)\$ which is easy to remember and calculate in the frequency space
The real response of an RC low pass filter is \$V_{\text{out}}(t) = V_{\text{in}}(1-e^{-RCt})\$ which is not as easy to work with
Take an op-amp, with open-loop-gain of \${A_{OL}}\$, and gain-bandwidth product \$f_{GBW}\$ (often written just GBW, but it will be clear in a moment)
The closed loop gain of the non-inverting case, with closed-loop-gain G = \$\frac{R_1+R_2}{R_1}\$ is
\$\frac{A_{OL}}{1+A_{OL}/G} = \frac{1}{1/G+1/A_{OL}} \approx G{\quad\quad\quad}(1)\$
In the above approximation, we are simply saying "\$A_{OL}\$ is large".
Let's fill in more detail.
The next closest approximation for the op-amp open-loop gain -- still ignoring the finite low-frequency gain, but now introducing the high-frequency -20dB/decade behavior -- is the following
\$A_{OL} = {\omega_{GBW}}/{j\omega} {\quad\quad\quad}(2)\$
where \$\omega_{GBW} = 2{\pi}f_{GBW}\$
Substituting (2) into (1), we get
\$\frac{1}{1/G+j\omega/{\omega}_{GBW}} = \frac{G}{1+(G/\omega_{GBW})j\omega}{\quad\quad\quad}(3)\$
And in this expression we should clearly see the single pole at \$j\omega=-\omega_{GBW}/G{\quad\quad\quad}\$
and so the -3dB point is at the frequency \$f=f_{GBW}/G\$
There are 3 concepts here that we need to distinguish: the opamp's open-loop -3dB point, it's GBW, and the closed-loop -3dB bandwidth.
The opamp's open-loop -3dB point is defined as the point where the open-loop gain has dropped by 3dB from its DC value, and by which phase at the output lags by 45 degrees. This drop in gain is caused by the (intentional) placement of a pole inside the opamp. It is not uncommon for this value to be very low (few kHz or less). However, it usually is of little importance for most circuits.
Instead, the number you care about is the gain-bandwidth product. Past the dominant pole, the opamp's gain drops by -20 db/decade, i.e. inversely proportional to the frequency. In turn, this means that the product of the frequency and the gain is constant, hence the name gain-bandwidth product. Naturally, the GBW is also the frequency at which the amplifier's gain drops to unity.
Lastly, there is the closed-loop gain, that is the gain you see after closing the feed-back loop with e.g. a resistor network. Naturally, the amplifier will fail to deliver the desired closed-loop gain is higher than the available open-loop gain, so for a closed-loop gain e.g. x10, the amplifier will be useful only up to approximately GBW/10. Depending on the phase margin, there can be some peaking, and the actual -3dB bandwidth of the closed loop can be somewhat higher.
How to relate them?
As you can tell, the closed-loop bandwidth is usually just \$GBW/G\$, if \$G\$ is the closed-loop gain.
The location of the dominant pole is given by \$GBW/A_{OL}\$ with \$A_{OL}\$ being the DC open-loop gain. Note that the GBW usually is spec'ced to much tighter tolerances than the open loop gain. Hence also the dominant pole is similarly poorly defined.
As an example, here is a graph from the MCP6001 CMOS opamp's datasheet.
The GBW product is 1 MHz. You can also read from the graph the open loop gain at DC: 110 dB. This puts the opamp's dominant pole (and 3dB bandwidth) at just a few hertz!
