Why is the Op Amp Gain-Bandwidth Product Constant?

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A recurring conversation I have usually starts with two questions: Why is the op amp gain-bandwidth product constant? And, how can we prove that?

The questions refer to the gain-bandwidth product behavior of an op amp after the cutoff frequency. As I showed in this article, Mastering Electronics Design.com: An Op Amp Gain Bandwidth Product, the gain bandwidth product describes the op amp gain dependency on frequency. The open loop graph is shown in Figure 1.

Op Amp open loop graph

Figure 1

Also, in that article, the mathematical expression of the graph is shown as

Op Amp Open Loop Gain Transfer Function


where ω is the variable, or the function argument, which is 2π times frequency, ωc is the op amp cutoff frequency, or 2π times fc, Aol is the open-loop gain at DC, and j is the imaginary unit.

This is a one pole system, therefore it drops constantly at 20 dB/decade after the cutoff frequency, as the graph shows. This means that the amplitude of a signal that goes through this op amp decreases 10 times for each frequency decade. The amplitude decreases as the frequency increases showing a constant gain-bandwidth product.

We can prove that the gain bandwidth product is constant by first finding the absolute value of the gain and then multiply it by ω. Equation 1 is a complex number, so we need to apply some complex numbers transformation to find its absolute value. Let’s multiply the function with the conjugate of 1+jω/ωc.

Op Amp Open Loop Gain Transfer Function


The graph shows the absolute value of Ao(ω). To determine the absolute value, we need to write the square-root of the sum of the real part squared and the imaginary part squared.

Op Amp Open Loop Gain Absolute Value


As ω increases, the ratio ω/ωc becomes quickly large, so 1 can be neglected. The absolute value of Ao(ω) becomes linear after ω > 3.3 ωc. As such, |Ao(ω)| can be written as

Op Amp Gain Absolute Value


where with Gain(ω) I noted the absolute value of the gain which depends on frequency.

The gain-bandwidth product (GBW) is calculated by multiplying the absolute value of the gain with ω.

Op Amp Gain-Bandwidth Product


and so,

Op Amp Gain-Bandwidth Product


which shows that the gain-bandwidth product is a constant, because it is a product between two constants: the op amp open-loop gain and the corner frequency.

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