# Gain-bandwidth product proof for open loop/closed loop opamp

On the forum, post #12, Ghar claimed that the proof gain-bandwidth product from wikipedia is awful and he gave another proof but with feedback.

The wikipedia proves gain-bandwidth product constant for open-loop opamp while Ghar proved for closed loop opamp.

Could you please tell me that if the wikipedia proof is wrong? I don't see any thing wrong with it. However, reading Ghar's post makes me confused.

I attach the Ghar's post in the image below just in case some member can't access it.

Note: Let's consider single pole opamp only. This site also gave the same proof as wikipedia.

http://masteringelectronicsdesign.com/why-is-the-op-amp-gain-bandwidth-product-constant/

• true flat response is only possible with negative feedback using R ratios, regardless of configuration Dec 2, 2016 at 0:30

Could you please tell me that if the wikipedia proof is wrong? I don't see any thing wrong with it.

The so called "proof" itself is wrong simply because they are using the Open Loop Gain to calculate it, and that is not the definition of GBWP, in addition to several unexplained simplifications (see appendix for details).

What the Wikipedia article is trying to convey can be seen graphically in the following figure, the 'Open Loop Gain' plot of a system with a single dominant pole:

The dotted lines show you that for different "Closed Loop Gains" and the GBPW product will be the same, as long as there is a single dominant pole AND a constant -20 dB/decade slope.

"The wikipedia proves gain-bandwidth product constant for open-loop opamp while Ghar proved for closed loop opamp."

"However, reading Ghar's post makes me confused."

Let's look at Ghar's response!

GBP = A_0 w_o doesn't equal unity frequency unless A0^2 >> 1

I think his point is that this is not a correct expression, which is what is explored in the Wikipedia article, again, the GBWP is defined by the Closed Loop Gain no the 'Open Loop Gain'.

During the ret of his calculations Ghar expands on the frequency dependency of the "Closed Loop Gain" and its relationship with the "Open Loop Gain" and "Loop Gain"

Below a graphical summary of of said relationship:

## Appendix

Gain Bandwidth Product, GBWP (a.k.a. GBW), can be defined as:

$$GBWP = A_{CL} \, \cdot \, BW_{CL}$$ where:

• A_CL represents the "Closed Loop Voltage Gain".
• BW_CL represents "Closed Loop Bandwidth".

Another way to look at this is that the "Closed Loop Bandwith" of your Op Amp will be the GBWP divided by your "Closed Loop Gain", that is:

Note: the gain-bandwidth product is only valid if the Op Amp's "Open Loop Gain" has a single dominant pole.

Another relevant concept is the Unity Gain Bandwidth, UGBW, can be defined as: $$UBWP = BW{({f_{unity}})}$$

where, BW(f_unity) represents "Closed Loop Bandwidth" at unity gain cross over frequency (i.e. when the closed loop gain crosses 0 dB or 1 V/V).

The UGBW should not be confused with the GBWP, although they can be the same, as in the case of the figure below.

Note that it is possible during certain circumstances to have GBWP = UGBP

## References

• What an excellent answer! Just wanted to up-vote more than once. Dec 2, 2016 at 5:09
• @Victor, nice explanation - however, one must add that the information contained in the last figure apply to a non-inverting opamp only (Acl~1/beta for large Aol ). For inverting gains the closed loop gain Acl is below the red line (which is 1/beta).
– LvW
Dec 2, 2016 at 9:05
• Thanks a lot for the thorough answer. I am trying hand plot for the Bode plot above and prove the close loop gain curve and open loop gain curve are exactly the same at high frequency. Dec 2, 2016 at 12:45
• @LvW, that is correct, in an ideal scenario (i.e. Aol_dc -->∞) for a non-inverting amplifier the Acl(ideal) = 1/β, while an inverting aplifier it would be Acl(ideal) = α/β. The "Operational Amplifier Gain Stability" series found in my references does a great job explaining this matter. Dec 2, 2016 at 18:45
• @anhnha, the third reference that I listed, “Operational Amplifier Gain Stability, Part 3: AC gain-error analysis”, gives you an equation for Open-Loop Gain as a function of frequency Aol(f) (Equation 10), and also gives you an equation for the Closed-Loop Gain (Equation 2, for a non-intervening configuration). You can plot them separately or just identify Aol within your Acl equation. Dec 2, 2016 at 20:20