I can see some chat on internet about the operational amplifier gain bandwidth product. People are interested in having a better understanding of this parameter, as it appears in any op amp datasheet and it is used in many articles and books. In this article I will describe this parameter and show you an example with Analog Devices’ ADA4004, which is a precision amplifier.
The Gain Bandwidth Product describes the op amp gain behavior with frequency. Manufacturers insert a dominant pole in the op amp frequency response, so that the output voltage versus frequency is predictable. Why do they do that? Because the operational amplifier, which is grown on a silicon die, has many active components, each one with its own cutoff frequency and frequency response. Because of that, the operational amplifier frequency response would be random, with poles and zeros which would differ from op amp to op amp even in the same family. As a consequence, manufacturers thought of introducing a dominant pole in the schematic, so that the op amp response becomes more predictable. It is a way of “standardizing” the op amp frequency response. At the same time, it makes the op amp more user friendly, because its stability in a schematic becomes more predictable.
The dominant pole will make the op amp behave like a single-pole system, which has a drop of 20 dB for every decade of frequency, starting with the cutoff frequency. Such a pole is made with a reactive element, usually a capacitor. The choice of a capacitor on the op amp die is because of the manufacturing process. A capacitor is easier to be grown on silicon, as opposed to an inductor. However, the capacitor value depends on its area, and since the die area is at a premium, capacitors can only be very small, in the picofarads range. Since the pole is made with an RC time constant, we need a large resistor to bring the cutoff frequency at low values, hertz, or tens of hertz. Without going into details, op amp manufacturers achieve these high resistors with active components, like the input resistance seen in a transistor base.
Having said that, the gain bandwidth product shows that the product between the op amp gain and frequency, in any point of the frequency response, is a constant. We can always calculate the bandwidth with the following formula.
In the case of ADA4004, the gain bandwidth product is 12 MHz. This means that, at a gain of one, the bandwidth is 12 MHz, and at the maximum open-loop gain of 500000, the bandwidth is 12 MHz divided by 500000, which is 24 Hz. This is the op amp open-loop cutoff frequency.
The mathematical model of the gain bandwidth product is described by the following single-pole function:
where ω is the variable, or the function argument, which is 2π times frequency, fc is the op amp cutoff frequency, Aol is the open-loop gain at DC, and j is the imaginary unit.
Figure 1 shows a Mathcad plot of this function. The gain starts at low frequencies at 20 log(500000) = 114 dB and then rolls off down to 0 dB. The markers show the cutoff frequency at 24 Hz and 111 dB gain, and the unity-gain frequency at 12 MHz and 0 dB gain. Why is the gain 111 dB at 24 Hz? Because the cutoff frequency is at -3 dB, which means that the corresponding gain is 114 dB – 3dB = 111 dB.
For simplicity and clarity, the gain is always shown in dB. One can see that, at low frequencies, the gain is 20 log(500000) = 114 dB. Starting with the cutoff frequency of 24 Hz, the gain rolls off at a rate of 20dB/decade until 12 MHz, where the gain is 1, or 0 dB. This theoretical graph is almost identical with the ADA4004 open-loop gain characteristic as shown in its datasheet (Rev. F) Figure 14.
The advantage of using the gain bandwidth product parameter lays in the fact that we can always predict the op amp bandwidth for a certain gain.
As an example, Figure 2 shows an inverting amplifier with ADA4004.
The gain is set by the ratio between R2 and R1 (go to this article How to Derive the Inverting Amplifier Transfer Function to see why). The resistor ratio is 10, so the bandwidth is 12 MHz divided by 10 which is 1.2 MHz. This result matches the ADA4004 closed-loop gain characteristic, which is shown in the same datasheet, at Figure 19.
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Categories: Analog Design, Operational Amplifier Formulas