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.

**Figure 1**

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

(1) |

where ω is the variable, or the function argument, which is 2π times frequency, ω_{c} is the op amp cutoff frequency, or 2π times f_{c}, 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/ω_{c}.

(2) |

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.

(3) |

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

(4) |

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.

(5) |

and so,

(6) |

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.

nice blog

Nice proof. Thank you.