Editing Bode plot (section)

===Examples using Bode plots===
Figures 6 and 7 illustrate the gain behavior and terminology. For a three-pole amplifier, Figure 6 compares the Bode plot for the gain without feedback (the ''open-loop'' gain) ''A''<sub>OL</sub> with the gain with feedback ''A''<sub>FB</sub> (the ''closed-loop'' gain). See [[negative feedback amplifier]] for more detail.

In this example, ''A''<sub>OL</sub> = 100&nbsp;dB at low frequencies, and 1 / β = 58&nbsp;dB. At low frequencies, ''A''<sub>FB</sub> ≈ 58&nbsp;dB as well.

Because the open-loop gain ''A''<sub>OL</sub> is plotted and not the product β ''A''<sub>OL</sub>, the condition ''A''<sub>OL</sub> = 1 / β decides ''f''<sub>0&nbsp;dB</sub>. The feedback gain at low frequencies and for large ''A''<sub>OL</sub> is ''A''<sub>FB</sub> ≈ 1 / β (look at the formula for the feedback gain at the beginning of this section for the case of large gain ''A''<sub>OL</sub>), so an equivalent way to find ''f''<sub>0&nbsp;dB</sub> is to look where the feedback gain intersects the open-loop gain. (Frequency ''f''<sub>0&nbsp;dB</sub> is needed later to find the phase margin.)

Near this crossover of the two gains at ''f''<sub>0&nbsp;dB</sub>, the Barkhausen criteria are almost satisfied in this example, and the feedback amplifier exhibits a massive peak in gain (it would be infinity if β ''A''<sub>OL</sub> = −1). Beyond the unity gain frequency ''f''<sub>0&nbsp;dB</sub>, the open-loop gain is sufficiently small that ''A''<sub>FB</sub> ≈ ''A''<sub>OL</sub> (examine the formula at the beginning of this section for the case of small ''A''<sub>OL</sub>).

Figure 7 shows the corresponding phase comparison: the phase of the feedback amplifier is nearly zero out to the frequency ''f''<sub>180</sub> where the open-loop gain has a phase of −180°. In this vicinity, the phase of the feedback amplifier plunges abruptly downward to become almost the same as the phase of the open-loop amplifier. (Recall, ''A''<sub>FB</sub> ≈ ''A''<sub>OL</sub> for small ''A''<sub>OL</sub>.)

Comparing the labeled points in Figure 6 and Figure 7, it is seen that the unity gain frequency ''f''<sub>0&nbsp;dB</sub> and the phase-flip frequency ''f''<sub>180</sub> are very nearly equal in this amplifier, ''f''<sub>180</sub> ≈ ''f''<sub>0&nbsp;dB</sub> ≈ 3.332&nbsp;kHz, which means the gain margin and phase margin are nearly zero. The amplifier is borderline stable.

Figures 8 and 9 illustrate the gain margin and phase margin for a different amount of feedback β. The feedback factor is chosen smaller than in Figure 6 or 7, moving the condition | β ''A''<sub>OL</sub> | = 1 to lower frequency. In this example, 1 / β = 77&nbsp;dB, and at low frequencies ''A''<sub>FB</sub> ≈ 77&nbsp;dB as well.

Figure 8 shows the gain plot. From Figure 8, the intersection of 1 / β and ''A''<sub>OL</sub> occurs at  ''f''<sub>0&nbsp;dB</sub> = 1&nbsp;kHz. Notice that the peak in the gain ''A''<sub>FB</sub> near ''f''<sub>0&nbsp;dB</sub> is almost gone.<ref group="note">The critical amount of feedback where the peak in the gain ''just'' disappears altogether is the ''maximally flat'' or [[Butterworth filter#Maximal flatness|Butterworth]] design.</ref><ref name=Sansen>
{{cite book 
|author=Willy M C Sansen
|title=Analog design essentials
|pages=157–163
|year= 2006
|publisher=Springer
|location=Dordrecht, The Netherlands
|isbn=0-387-25746-2
|url=http://worldcat.org/isbn/0-387-25746-2}}
</ref>

Figure 9 is the phase plot. Using the value of ''f''<sub>0&nbsp;dB</sub> = 1&nbsp;kHz found above from the magnitude plot of Figure 8, the open-loop phase at ''f''<sub>0&nbsp;dB</sub> is −135°, which is a phase margin of 45° above −180°.

Using Figure 9, for a phase of −180° the value of ''f''<sub>180</sub> = 3.332&nbsp;kHz (the same result as found earlier, of course<ref group="note">The frequency where the open-loop gain flips sign ''f''<sub>180</sub> does not change with a change in feedback factor; it is a property of the open-loop gain. The value of the gain at ''f''<sub>180</sub> also does not change with a change in β. Therefore, we could use the previous values from Figures 6 and 7. However, for clarity the procedure is described using only Figures 8 and 9.</ref>). The open-loop gain from Figure 8 at ''f''<sub>180</sub> is 58&nbsp;dB, and 1 / β = 77&nbsp;dB, so the gain margin is 19&nbsp;dB.

Stability is not the sole criterion for amplifier response, and in many applications a more stringent demand than stability is good [[Step response#Step response of feedback amplifiers|step response]]. As a [[rule of thumb]], good step response requires a phase margin of at least 45°, and often a margin of over 70° is advocated, particularly where component variation due to manufacturing tolerances is an issue.<ref name=Sansen/> See also the discussion of phase margin in the [[Step response#Phase margin|step response]] article.

<gallery caption="Examples" widths="300px" perrow="2" class="skin-invert-image">
Image:Magnitude of feedback amplifier.PNG|Figure 6: Gain of feedback amplifier ''A''<sub>FB</sub> in dB and corresponding open-loop amplifier ''A''<sub>OL</sub>. Parameter 1/β = 58&nbsp;dB, and at low frequencies ''A''<sub>FB</sub> ≈ 58&nbsp;dB as well. The gain margin in this amplifier is nearly zero because &#124; β''A''<sub>OL</sub>&#124; = 1 occurs at almost ''f'' = ''f''<sub>180°</sub>.
Image:Phase of feedback amplifier.PNG|Figure 7: Phase of feedback amplifier ''°A''<sub>FB</sub> in degrees and corresponding open-loop amplifier ''°A''<sub>OL</sub>. The phase margin in this amplifier is nearly zero because the phase-flip occurs at almost the unity gain frequency ''f'' = ''f''<sub>0&nbsp;dB</sub> where &#124; β''A''<sub>OL</sub>&#124; = 1.
Image:Gain Margin.PNG|Figure 8: Gain of feedback amplifier ''A''<sub>FB</sub> in dB and corresponding open-loop amplifier ''A''<sub>OL</sub>. In this example, 1 / β = 77&nbsp;dB. The gain margin in this amplifier is 19&nbsp;dB.
Image:Phase Margin.PNG|Figure 9: Phase of feedback amplifier ''A''<sub>FB</sub> in degrees and corresponding open-loop amplifier ''A''<sub>OL</sub>. The phase margin in this amplifier is 45°.
</gallery>