Electronic – How to use Gain Bandwidth Product to estimate bandwidth at different gains

GBP is the to do with the open loop gain of the op-amp. If you have a closed-loop circuit then GBP can help you find where the flatness of the frequency response starts to be eroded.

GBP - if it has an open loop DC gain of 1 million and unity gain at 1MHz then the GBP is said to be 1 million.

This is useful to know because if you have an op-amp that has a GBP of 1 million and you have a manufactured a closed loop gain of 100 then I would expect the gain to remain flat up to about 10kHz then roll-off gently at 6 dB / octave: -

Here is an extract from the data sheet for the AD8606 op-amp and I've drawn four red lines on it at 10kHz, 100kHz, 1MHz and 10MHz. The line at 10MHz is important because this is the unity gain point of the op-amp i.e. it has a GBP of 10,000,000. If this was all we knew we could predict the open loop gain at 10kHz by dividing 10,000,000 by 10,000 to get 1,000 (this would be the open loop gain at 10kHz and of course a gain of 1,000 is 60 dB - exactly as seen in the graph.

This is called constant gain-bandwidth product but it isn't true for every op amp. It is only true for voltage feedback op amps which use dominant pole compensation for stability. Such op amps can be approximated as a first order system since one pole dominates all others and the others can be ignored. (However, this is not true of current feedback op amps since current feedback op amps do not have a constant gain-bandwidth product.)

A first order system has a transfer function of the form

$$H(j\omega) = \frac{H_0}{j\omega\tau + 1} = \frac{H_0}{j\omega/\omega_c + 1}$$

where \$H_0\$ is the DC and passband gain, \$\tau\$ is the time constant of the dominant pole and \$\omega_c\$ is the cutoff frequency (bandwidth). The gain of this system is

$$|H(j\omega)| = \frac{H_0}{\sqrt{(\omega/\omega_c)^2+1}}$$

For \$\omega << \omega_c\$ the gain is approximately \$H_0\$ and the bandwidth does not come into play. If \$\omega >> \omega_c\$ the gain-bandwidth product can be approximated as

$$|H(j\omega)|\omega = \frac{H_0}{\sqrt{(\omega/\omega_c)^2+1}}\omega \approx \frac{H_0}{\sqrt{(\omega/\omega_c)^2}}\omega = H_0\omega_c$$

which is a constant. Since it is a constant, increasing the gain requires a decrease in the bandwidth while decreasing the gain allows an increase in the bandwidth.