Editing Negative-feedback amplifier (section)

===Bandwidth extension===
[[Image:Bandwidth comparison.JPG|thumb|380px|Figure 2: Gain vs. frequency for a single-pole amplifier with and without feedback; corner frequencies are labeled]]
Feedback can be used to extend the bandwidth of an amplifier at the cost of lowering the amplifier gain.<ref>R. W. Brodersen. [http://bwrc.eecs.berkeley.edu/classes/ee140/Lectures/10_stability.pdf ''Analog circuit design: lectures on stability''].</ref> Figure 2 shows such a comparison. The figure is understood as follows. Without feedback the so-called '''open-loop''' gain in this example has a single-time-constant frequency response given by

:<math> A_\text{OL}(f) = \frac{A_0}{1 + j f / f_\text{C}},</math>

where ''f''<sub>C</sub> is the [[cutoff frequency|cutoff]] or [[corner frequency]] of the amplifier: in this example ''f''<sub>C</sub> = 10<sup>4</sup> Hz, and the gain at zero frequency ''A''<sub>0</sub> = 10<sup>5</sup> V/V. The figure shows that the gain is flat out to the corner frequency and then drops. When feedback is present, the so-called '''closed-loop''' gain, as shown in the formula of the previous section, becomes

:<math>\begin{align}
 A_\text{FB}(f) &= \frac{A_\text{OL}}{1 + \beta A_\text{OL}} \\
  &= \frac{A_0 / (1 + jf/f_\text{C})}{1 + \beta A_0 / (1 + jf/f_\text{C})} \\
  &= \frac{A_0}{1 + jf/f_\text{C} + \beta A_0} \\
  &= \frac{A_0}{(1 + \beta A_0) \left(1 + j \frac{f}{(1 + \beta A_0) f_\text{C}}\right)}.
\end{align}</math>

The last expression shows that the feedback amplifier still has a single-time-constant behavior, but the corner frequency is now increased by the improvement factor (1 + β ''A''<sub>0</sub>), and the gain at zero frequency has dropped by exactly the same factor. This behavior is called the '''[[Gain–bandwidth product|gain–bandwidth tradeoff]]'''. In Figure 2, (1 + β ''A''<sub>0</sub>) = 10<sup>3</sup>, so ''A''<sub>FB</sub>(0) = 10<sup>5</sup> / 10<sup>3</sup> = 100 V/V, and ''f''<sub>C</sub> increases to 10<sup>4</sup> × 10<sup>3</sup> = 10<sup>7</sup> Hz.