# Gas station without pumps

## 2019 August 21

### Gain-bandwidth product

Filed under: Circuits course — gasstationwithoutpumps @ 17:15
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I spent yesterday afternoon collecting data for two figures for my book illustrating the limitation on gain caused by the gain-bandwidth product:

This graph shows the measured gain of non-inverting amplifiers built using the MCP6004 op amp.
It also includes a line for f_{GB}/f fit to the data for the highest-gain amplifier.

This graph compares the measured data (thin lines) to the model used in the book for amplifiers with gain-bandwidth limitations (thick yellow lines) for three of the amplifier configurations.

I did all the data collection with my Analog Discovery 2, using the network analyzer.

The data looks easy to collect, and I expected it to take me just an hour to gather all the data, but it was a bit trickier than I thought. I used a symmetric ±2.5V power supply from the Analog Discovery 2, so all the signals could be centered at 0V.

One problem was that I initially had not set the offset and gain of the oscilloscope channels to “auto”, and the output was not centered precisely at 0V. The network analyzer does not do as good a job of compensating for DC offsets as I think it should. I set the channels to automatic gain and offset and I tweaked the offset of the waveform generator by 1mV to make the output be centered correctly. I think that the 1mV offset is compensating for the offset of the op-amp chip, but it may be correcting an offset in the waveform generator.

For the high-gain amplifiers, I needed to reduce the signal from waveform generator, because the smallest signal the network analyzer allows is ±10mV, and with a gain of 393, that would cause clipping. My first voltage divider reduced the signal sufficiently, but I got very noisy results. It took me quite a while to realize that the problem was not just loose wires (though those were also a problem), but that I had used resistors that were too large, so that the input of the amplifier was amplifying capacitively coupled 60Hz noise. Reducing the input voltage divider to 1kΩ resistors solved that problem.

I was having another problem in which the shape of the curve for the low-gain amplifiers changed at the high-frequency end as I changed the amplitude of the signal. It took me an embarrassingly long time to realize that the problem was that I was hitting slew-rate limits before hitting the gain-bandwidth product limits. The high-gain amplifiers all had much lower gain at 1MHz than at low frequencies, so an input signal small enough to avoid clipping at low frequencies produced such a small output at high frequencies that slew rate was not limiting. But for low-gain amplifiers, I had been increasing the amplitude for better measurements, and the gain at 1MHz was only a little less than the gain in the passband, so I was asking for over 8V/µs, and the amplifier’s slew rate is only 0.6V/µs. I realized the problem when I used the oscilloscope to look at what the amplifier was producing, and saw that it was not a sine wave, but a small weirdly distorted signal.

After I finally got everything set up and working with small enough inputs to avoid clipping in high-gain amplifiers and slew-rate limitations in low-gain amplifiers, I finally was able to collect a consistent set of data.

The model for the op amp is that the open gain is $A = \frac{1}{1/A_0+ jf/f_{GB}} \approx -j f_{GB}/f$, and the gain of the non-inverting amplifier is $\frac{A}{1+AB} \approx \frac{1}{B}\; \frac{1}{1+jf/(B f_{GB})}$, where $B$ is just the gain of the voltage divider used as a feedback network. The model breaks down at high frequencies, because the op amp has further filtering above 1MHz, and for very high gain, where the DC amplification limit $A_0$ matters. We don’t design op-amp circuits to use such a high gain that the DC open-loop gain matters, but pushing the frequency limit is common.

But the simple gain-bandwidth model does a very good job of fitting the data, as long as you avoid signal levels that cause clipping or slew-rate limitations.